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740-fall10-lecture5-afterlecture-preciseexceptions	15-740/18-740 Computer Architecture Lecture 5: Precise Exceptions Prof. Onur Mutlu Carnegie Mellon University Last Time … 2 Performance Metrics Amdahl’s Law Single-cycle, multi-cycle machines Pipelining Stalls Dependencies Control dependency stall: what to fetch next Solution: predict which instruction comes next What if prediction is wrong? Another solution: hardware-based fine-grained multithreading Can tolerate both data and control dependencies Read: James Thornton, “Parallel operation in the Control Data 6600,” AFIPS 1964. Read: Burton Smith, “A pipelined, shared resource MIMD computer,” ICPP 1978. Issues in Pipelining: Increased CPI 3 BEQ R1, R2, TARGET F D E W F F F D E W Issues in Pipelining: Increased CPI Resource Contention Stall What if two concurrent operations need the same resource? Examples: Instruction fetch and data fetch both need memory. Solution? Register read and register write both need the register file A store instruction and a load instruction both need to access memory. Solution? 4 F D E W F D E W F F D E W LD R1 Å R2(4) ADD R2 Å R1, R5 ADD R6 Å R3, R4 Issues in Pipelining: Multi-Cycle Execute Instructions can take different number of cycles in EXECUTE stage Integer ADD versus FP MULtiply What is wrong with this picture? What if FMUL incurs an exception? Sequential semantics of the ISA NOT preserved! 5 F D E W F D E WE E E E E E EFMUL R4 Å R1, R2 ADD R3 Å R1, R2 F D E W F D E W F D E W F D E W FMUL R4 Å R5, R6 ADD R3 Å R5, R6 F D E WE E E E E E E Handling Exceptions in Pipelining Exceptions versus interrupts Cause Exceptions: internal to the running thread Interrupts: external to the running thread When to Handle Exceptions: when detected (and known to be non-speculative) Interrupts: when convenient Except for very high priority ones Power failure Machine check Priority: process (exception), depends (interrupt) Handling Context: process (exception), system (interrupt) 6 Precise Exceptions/Interrupts The architectural state should be consistent when the exception/interrupt is ready to be handled 1. All previous instructions should be completely retired. 2. No later instruction should be retired. Retire = commit = finish execution and update arch. state 7 Ensuring Precise Exceptions in Pipelining Idea: Make each operation take the same amount of time Downside What about memory operations? Each functional unit takes 500 cycles? 8 F D E W F D E WE E E E E E E F D E W F D E W F D E W F D E W F D E W E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E FMUL R3 Å R1, R2 ADD R4 Å R1, R2 Solutions Reorder buffer History buffer Future register file Checkpointing Reading Smith and Plezskun, “Implementing Precise Interrupts in Pipelined Processors” IEEE Trans on Computers 1988 and ISCA 1985. Hwu and Patt, “Checkpoint Repair for Out-of-order Execution Machines,” ISCA 1987. 9 Solution I: Reorder Buffer (ROB) Idea: Complete instructions out-of-order, but reorder them before making results visible to architectural state When instruction is decoded it reserves an entry in the ROB When instruction completes, it writes result into ROB entry When instruction oldest in ROB, its result moved to reg. file or memory 10 Register File Func Unit Func Unit Func Unit Reorder Buffer Instruction Cache Reorder Buffer: Independent Operations Results first written to ROB, then to register file at commit time What if a later operation needs a value in the reorder buffer? Read reorder buffer in parallel with the register file. How? 11 F D E W F D E RE E E E E E E F D E W F D E R F D E R F D E R F D E RE E E E E E E W R R W W W W Reorder Buffer: How to Access? A register value can be in the register file, reorder buffer, (or bypass paths) 12 Register File Func Unit Func Unit Func UnitReorder Buffer Instruction Cache bypass path Content Addressable Memory (searched with register ID) Simplifying Reorder Buffer Access Idea: Use indirection Access register file first If register not valid, register file stores the ID of the reorder buffer entry that contains (or will contain) the value of the register Mapping of the register to a ROB entry Access reorder buffer next What is in a reorder buffer entry? Can it be simplified further? 13 V DestRegID DestRegVal StoreAddr StoreData BranchTarget PC/IP Control/valid bits What is Wrong with This Picture? What is R4’s value at the end? The first FMUL’s result Output dependency not respected 14 F D E W F D E WE E E E E E EFMUL R4 Å R1, R2 ADD R3 Å R1, R2 F D E W F D E W F D E W F D E W FMUL R2 Å R5, R6 ADD R4 Å R5, R6 F D E WE E E E E E E Register Renaming with a Reorder Buffer Output and anti dependencies are not true dependencies WHY? The same register refers to values that have nothing to do with each other They exist due to lack of register ID’s (i.e. names) in the ISA The register ID is renamed to the reorder buffer entry that will hold the register’s value Register ID Æ ROB entry ID Architectural register ID Æ Physical register ID After renaming, ROB entry ID used to refer to the register This eliminates anti- and output- dependencies Gives the illusion that there are a large number of registers 15 Solution II: History Buffer (HB) Idea: Update architectural state when instruction completes, but UNDO UPDATES when an exception occurs When instruction is decoded, it reserves an HB entry When the instruction completes, it stores the old value of its destination in the HB When instruction is oldest and no exceptions/interrupts, the HB entry discarded When instruction is oldest and an exception needs to be handled, old values in the HB are written back into the architectural state from tail to head 16 History Buffer Advantage: Register file contains up-to-date values. History buffer access not on critical path Disadvantage: Need to read the old value of the destination What about stores? 17 Register File Func Unit Func Unit Func Unit History Buffer Instruction Cache Used only on exceptions Solution III: Future File (FF) Idea: Keep two register files: Arch reg file: Updated in program order for precise exceptions Future reg file: Updated as soon as an instruction completes (if the instruction is the youngest one to write to a register) Future file is used for fast access to latest register values Architectural file is used for recovery on exceptions 18 Future File Advantage No sequential scanning of history buffer: Upon exception, simply copy arch file to future file No need for extra read of destination value Disadvantage Multiple register files + reorder buffer 19 Future File Func Unit Func Unit Func Unit Arch. FileInstruction Cache Used only on exceptions ROB VData or Tag Checkpointing Idea: Periodically checkpoint the register file state. When exception/interrupt occurs, go back to the most recent checkpoint and re-execute instructions one by one to re- generate exception. State guaranteed to be precise only at checkpoints. Advantage: Allows for aggressive execution between checkpoints Per-instruction reorder buffer is not needed Disadvantage: Interrupt latency depends on distance from checkpoint Hwu and Patt, “Checkpoint Repair for Out-of-order Execution Machines,” ISCA 1987. 20 Summary: Precise Exceptions in Pipelining When the oldest instruction ready-to-be-retired is detected to have caused an exception, the control logic Recovers architectural state (register file, IP, and memory) Flushes all younger instructions in the pipeline Saves IP and registers (as specified by the ISA) Redirects the fetch engine to the exception handling routine Vectored exceptions 21 Pipelining Issues: Branch Mispredictions A branch misprediction resembles an “exception” Except it is not visible to software What about branch misprediction recovery? Similar to exception handling except can be initiated before the branch is the oldest instruction All three state recovery methods can be used Difference between exceptions and branch mispredictions? Branch mispredictions more common: need fast recovery 22 Pipelining Issues: Stores Handling out-of-order completion of memory operations UNDOing a memory write more difficult than UNDOing a register write. Why? One idea: Keep store address/data in reorder buffer How does a load instruction find its data? Store/write buffer: Similar to reorder buffer, but used only for store instructions Program-order list of un-committed store operations When store is decoded: Allocate a store buffer entry When store address and data become available: Record in store buffer entry When the store is the oldest instruction in the pipeline: Update the memory address (i.e. cache) with store data 23
Crypto101	Crypto101 lvh Copyright 2013-2017, Laurens Van Houtven (lvh) This work is available under the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) license. You can find the full text of the license athttps: //creativecommons.org/licenses/by-nc/4.0/. The following is a human-readable summary of (and not a substitute for) the license. You can: • Share: copy and redistribute the material in any medium or format • Adapt: remix, transform, and build upon the material The licensor cannot revoke these freedoms as long as you follow the license terms: • Attribution: you must give appropriate credit, provide a link to the license, and indicate if changes were made. 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For example, other rights such as publicity, privacy, or moral rights may limit how you use the material. 2 Pomidorkowi 3 Contents Contents 4 I Foreword 9 1 Aboutthisbook 10 2 Advancedsections 12 3 Development 13 4 Acknowledgments 14 II Buildingblocks 16 5 Exclusiveor 17 5.1 Description . . . . . . . . . . . . . . . . . . . 17 5.2 A few properties of XOR . . . . . . . . . . . . 18 5.3 Bitwise XOR . . . . . . . . . . . . . . . . . . . 19 5.4 One-time pads . . . . . . . . . . . . . . . . . . 19 5.5 Attacks on “one-time pads”. . . . . . . . . . . 21 5.6 Remaining problems . . . . . . . . . . . . . . 26 6 Blockciphers 28 6.1 Description . . . . . . . . . . . . . . . . . . . 28 6.2 AES . . . . . . . . . . . . . . . . . . . . . . . . 33 4 6.3 DES and 3DES . . . . . . . . . . . . . . . . . . 37 6.4 Remaining problems . . . . . . . . . . . . . . 40 7 Streamciphers 41 7.1 Description . . . . . . . . . . . . . . . . . . . 41 7.2 A naive attempt with block ciphers. . . . . . 41 7.3 Block cipher modes of operation. . . . . . . . 48 7.4 CBC mode . . . . . . . . . . . . . . . . . . . . 48 7.5 Attacks on CBC mode with predictable IVs. . 50 7.6 Attacks on CBC mode with the key as the IV. . 52 7.7 CBC bit flipping attacks. . . . . . . . . . . . . 53 7.8 Padding . . . . . . . . . . . . . . . . . . . . . 56 7.9 CBC padding attacks . . . . . . . . . . . . . . 57 7.10 Native stream ciphers. . . . . . . . . . . . . . 65 7.11 RC4 . . . . . . . . . . . . . . . . . . . . . . . . 66 7.12 Salsa20 . . . . . . . . . . . . . . . . . . . . . . 75 7.13 Native stream ciphers versus modes of opera- tion . . . . . . . . . . . . . . . . . . . . . . . . 77 7.14 CTR mode . . . . . . . . . . . . . . . . . . . . 78 7.15 Stream cipher bit flipping attacks. . . . . . . 79 7.16 Authenticating modes of operation. . . . . . 80 7.17 Remaining problems . . . . . . . . . . . . . . 80 8 Keyexchange 81 8.1 Description . . . . . . . . . . . . . . . . . . . 81 8.2 Abstract Diffie-Hellman . . . . . . . . . . . . . 82 8.3 Diffie-Hellman with discrete logarithms. . . . 86 8.4 Diffie-Hellman with elliptic curves. . . . . . . 87 8.5 Remaining problems . . . . . . . . . . . . . . 88 9 Public-keyencryption 90 9.1 Description . . . . . . . . . . . . . . . . . . . 90 9.2 Why not use public-key encryption for every- thing? . . . . . . . . . . . . . . . . . . . . . . 91 9.3 RSA . . . . . . . . . . . . . . . . . . . . . . . 92 9.4 Elliptic curve cryptography . . . . . . . . . . . 96 5 9.5 Remaining problem: unauthenticated en- cryption . . . . . . . . . . . . . . . . . . . . . 96 10 Hashfunctions 98 10.1 Description . . . . . . . . . . . . . . . . . . . 98 10.2 MD5 . . . . . . . . . . . . . . . . . . . . . . . 100 10.3 SHA-1 . . . . . . . . . . . . . . . . . . . . . . 101 10.4 SHA-2 . . . . . . . . . . . . . . . . . . . . . . 102 10.5 Keccak and SHA-3. . . . . . . . . . . . . . . . 103 10.6 Password storage . . . . . . . . . . . . . . . . 104 10.7 Length extension attacks. . . . . . . . . . . . 108 10.8 Hash trees . . . . . . . . . . . . . . . . . . . . 110 10.9 Remaining issues . . . . . . . . . . . . . . . . 110 11 Messageauthenticationcodes 111 11.1 Description . . . . . . . . . . . . . . . . . . . 111 11.2 Combining MAC and message. . . . . . . . . 113 11.3 A naive attempt with hash functions. . . . . . 115 11.4 HMAC . . . . . . . . . . . . . . . . . . . . . . 119 11.5 One-time MACs . . . . . . . . . . . . . . . . . 120 11.6 Carter-Wegman MAC . . . . . . . . . . . . . . 123 11.7 Authenticated encryption modes . . . . . . . 124 11.8 OCB mode . . . . . . . . . . . . . . . . . . . . 126 11.9 GCM mode. . . . . . . . . . . . . . . . . . . . 128 12 Signaturealgorithms 130 12.1 Description . . . . . . . . . . . . . . . . . . . 130 12.2 RSA-based signatures . . . . . . . . . . . . . . 131 12.3 DSA . . . . . . . . . . . . . . . . . . . . . . . 131 12.4 ECDSA . . . . . . . . . . . . . . . . . . . . . . 136 12.5 Repudiable authenticators . . . . . . . . . . . 136 13 Keyderivationfunctions 137 13.1 Description . . . . . . . . . . . . . . . . . . . 137 13.2 Password strength . . . . . . . . . . . . . . . 138 13.3 PBKDF2 . . . . . . . . . . . . . . . . . . . . . 139 13.4 bcrypt . . . . . . . . . . . . . . . . . . . . . . 139 13.5 scrypt . . . . . . . . . . . . . . . . . . . . . . 139 6 13.6 HKDF . . . . . . . . . . . . . . . . . . . . . . 139 14 Randomnumbergenerators 143 14.1 Introduction . . . . . . . . . . . . . . . . . . . 143 14.2 True random number generators. . . . . . . 144 14.3 Cryptographically secure pseudorandom gen- erators . . . . . . . . . . . . . . . . . . . . . . 146 14.4 Yarrow . . . . . . . . . . . . . . . . . . . . . . 147 14.5 Blum Blum Shub . . . . . . . . . . . . . . . . 148 14.6 Dual_EC_DRBG . . . . . . . . . . . . . . . . . 148 14.7 Mersenne Twister. . . . . . . . . . . . . . . . 155 IIICompletecryptosystems 162 15 SSLandTLS 163 15.1 Description . . . . . . . . . . . . . . . . . . . 163 15.2 Handshakes . . . . . . . . . . . . . . . . . . . 164 15.3 Certificate authorities. . . . . . . . . . . . . . 165 15.4 Self-signed certificates . . . . . . . . . . . . . 166 15.5 Client certificates . . . . . . . . . . . . . . . . 166 15.6 Perfect forward secrecy. . . . . . . . . . . . . 166 15.7 Attacks . . . . . . . . . . . . . . . . . . . . . . 168 15.8 HSTS . . . . . . . . . . . . . . . . . . . . . . . 171 15.9 Certificate pinning . . . . . . . . . . . . . . . 172 15.10Secure configurations. . . . . . . . . . . . . . 173 16 OpenPGPandGPG 175 16.1 Description . . . . . . . . . . . . . . . . . . . 175 16.2 The web of trust. . . . . . . . . . . . . . . . . 176 17 Off-The-RecordMessaging(OTR) 179 17.1 Description . . . . . . . . . . . . . . . . . . . 179 17.2 Key exchange . . . . . . . . . . . . . . . . . . 180 17.3 Data exchange. . . . . . . . . . . . . . . . . . 184 7 8 IV Appendices 185 A Modulararithmetic 186 A.1 Addition and subtraction . . . . . . . . . . . . 186 A.2 Prime numbers . . . . . . . . . . . . . . . . . 189 A.3 Multiplication . . . . . . . . . . . . . . . . . . 190 A.4 Division and modular inverses. . . . . . . . . 191 A.5 Exponentiation . . . . . . . . . . . . . . . . . 192 A.6 Exponentiation by squaring . . . . . . . . . . 193 A.7 Montgomery ladder exponentiation . . . . . . 195 A.8 Discrete logarithm . . . . . . . . . . . . . . . 200 A.9 Multiplicative order . . . . . . . . . . . . . . . 201 B Ellipticcurves 202 B.1 The elliptic curve discrete log problem. . . . 204 C Side-channelattacks 205 C.1 Timing attacks . . . . . . . . . . . . . . . . . . 205 C.2 Power measurement attacks . . . . . . . . . . 205 V Glossary 206 Index 212 VI References 215 Bibliography 216 PartI Foreword 9 1 Aboutthisbook Lots of people working in cryptography have no deep concern with real application issues. They are trying to discover things clever enough to write papers about. Whitfield Diffie This book is intended as an introduction to cryptography for programmers of any skill level. Itʼs a continuation of a talk of the same name, which was given by the author at Py- Con 2013. The structure of this book is very similar: it starts with very simple primitives, and gradually introduces new ones, demonstrating why theyʼre necessary. Eventually, all of this is put together into complete, practical cryptosystems, such as TLS, GPG andOTR. The goal of this book is not to make anyone a cryptog- rapher or a security researcher. The goal of this book is to understand how complete cryptosystems work from a birdʼs eye view, and how to apply them in real software. The exercises accompanying this book focus on teaching cryptography by breaking inferior systems. That way, you 10 CHAPTER 1. ABOUT THIS BOOK 11 wonʼt just “know” that some particular thing is broken; youʼll know exactlyhow itʼs broken, and that you, yourself, armed with little more than some spare time and your favorite pro- gramming language, can break them. By seeing how these ostensibly secure systems are actually completely broken, you will understandwhy all these primitives and construc- tions are necessary for complete cryptosystems. Hopefully, these exercises will also leave you with healthy distrust of DIY cryptography in all its forms. For a long time, cryptography has been deemed the ex- clusive realm of experts. From the many internal leaks weʼve seen over the years of the internals of both large and small corporations alike, it has become obvious that that ap- proach is doing more harm than good. We can no longer afford to keep the two worlds strictly separate. We must join them into one world where all programmers are educated in the basic underpinnings of information security, so that they can work together with information security profes- sionals to produce more secure software systems for every- one. That does not make people such as penetration testers and security researchers obsolete or less valuable; quite the opposite, in fact. By sensitizing all programmers to security concerns, the need for professional security audits will be- come more apparent, not less. This book hopes to be a bridge: to teach everyday pro- grammers from any field or specialization to understand just enough cryptography to do their jobs, or maybe just satisfy their appetite. 2 Advancedsections This book is intended as a practical guide to cryptography for programmers. Some sections go into more depth than they need to in order to achieve that goal. Theyʼre in the book any- way, just in case youʼre curious; but I generally recommend skipping these sections. Theyʼll be marked like this: This is an optional, in-depth section. It almost certainly wonʼt help you write bet- ter software, so feel free to skip it. It is only here to satisfy your inner geekʼs curiosity. 12 3 Development The entire Crypto 101 project is publicly developed on GitHub under thecrypto101 organization, includingthis book. This is an early pre-release of this book. All of your questions, comments and bug reports are highly appreci- ated. If you donʼt understand something after reading it, or a sentence is particularly clumsily worded,that’s a bugand I would very much like to fix it! Of course, if I never hear about your issue, itʼs very hard for me to address… The copy of this book that you are reading right now is based on the git commit with hash 64e8ccf, also known as 0.6.0-95-g64e8ccf. 13 4 Acknowledgments This book would not have been possible without the support and contributions of many people, even before the first pub- lic release. Some people reviewed the text, some people pro- vided technical review, and some people helped with the original talk. In no particular order: • My wife, Ewa • Brian Warner • Oskar Żabik • Ian Cordasco • Zooko Wilcox-OʼHearn • Nathan Nguyen (@nathanhere) Following the public release, many more people con- tributed changes. Iʼd like to thank the following people in particular (again, in no particular order): • coh2, for work on illustrations 14 CHAPTER 4. ACKNOWLEDGMENTS 15 • TinnedTuna, for review work on the XOR section (and others) • dfc, for work on typography and alternative formats • jvasile, for work on typefaces and automated builds • hmmueller, for many, many notes and suggestions • postboy (Ivan Zuboff), for many reported issues • EdOverflow, for many contributions • gliptak (Gábor Lipták) for work on automating builds, as well as the huge number of people that contributed spelling, grammar and content improvements. Thank you! PartII Buildingblocks 16 5 Exclusiveor 5.1 Description Exclusive or, often called “XOR”, is a Boolean1 binary2 op- erator that is true when either the first input or the second input, but not both, are true. Another way to think of XOR is as something called a “programmable inverter”: one input bit decides whether to invert the other input bit, or to just pass it through un- changed. “Inverting” bits is colloquially called “flipping” bits, a term weʼll use often throughout the book. In mathematics and cryptography papers, exclusive or is generally represented by a cross in a circle:⊕. Weʼll use the same notation in this book: 1 Uses only “true” and “false” as input and output values. 2 Takes two parameters. 17 CHAPTER 5. EXCLUSIVE OR 18 The inputs and output here are named as if weʼre using XOR as an encryption operation. On the left, we have the plaintext bit Pi. The i is just an index, since weʼll usually deal with more than one such bit. On top, we have the key bit ki, that decides whether or not to invertPi. On the right, we have the ciphertext bit,Ci, which is the result of the XOR operation. 5.2 AfewpropertiesofXOR Since weʼll be dealing with XOR extensively during this book, weʼll take a closer look at some of its properties. If youʼre already familiar with how XOR works, feel free to skip this section. We saw that the output of XOR is 1 when one input or the other (but not both) is 1: 0 ⊕0 = 0 1 ⊕0 = 1 0 ⊕1 = 1 1 ⊕1 = 0 There are a few useful arithmetic tricks we can derive from that. 1. You can apply XOR in any order:a⊕(b⊕c) = ( a⊕b)⊕c 2. You can flip the operands around:a ⊕b = b ⊕a 3. Any bit XOR itself is 0:a ⊕a = 0 . If a is 0, then itʼs 0 ⊕0 = 0 ; ifa is 1, then itʼs1 ⊕1 = 0 . 4. Any bit XOR 0 is that bit again:a ⊕0 = a. Ifa is 0, then itʼs 0 ⊕0 = 0 ; ifa is 1, then itʼs1 ⊕0 = 1 . CHAPTER 5. EXCLUSIVE OR 19 These rules also implya ⊕b ⊕a = b: a ⊕b ⊕a = a ⊕a ⊕b (second rule) = 0 ⊕b (third rule) = b (fourth rule) Weʼll use this property often when using XOR for encryption; you can think of that first XOR witha as encrypting, and the second one as decrypting. 5.3 BitwiseXOR XOR, as weʼve just defined it, operates only on single bits or Boolean values. Since we usually deal with values com- prised of many bits, most programming languages provide a “bitwise XOR” operator: an operator that performs XOR on the respective bits in a value. Python, for example, provides the^ (caret) operator that performs bitwise XOR on integers. It does this by first ex- pressing those two integers in binary3, and then performing XOR on their respective bits. Hence the name,bitwise XOR. 73 ⊕87 = 0 b1001001 ⊕0b1010111 = 1 0 0 1 0 0 1 (left) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 1 0 1 0 1 1 1 (right) = 0 0 1 1 1 1 0 = 0 b0011110 = 30 5.4 One-timepads XOR may seem like an awfully simple, even trivial operator. Even so, thereʼs an encryption scheme, called a one-time 3 Usually, numbers are already stored in binary internally, so this doesnʼt actually take any work. When you see a number prefixed with “0b”, the remaining digits are a binary representation. CHAPTER 5. EXCLUSIVE OR 20 pad, which consists of just that single operator. Itʼs called a one-time pad because it involves a sequence (the “pad”) of random bits, and the security of the scheme depends on only using that pad once. The sequence is called a pad because it was originally recorded on a physical, paper pad. This scheme is unique not only in its simplicity, but also because it has the strongest possible security guarantee. If the bits are truly random (and therefore unpredictable by an attacker), and the pad is only used once, the attacker learns nothing about the plaintext when they see a ciphertext.4 Suppose we can translate our plaintext into a sequence of bits. We also have the pad of random bits, shared between the sender and the (one or more) recipients. We can com- pute the ciphertext by taking the bitwise XOR of the two se- quences of bits. If an attacker sees the ciphertext, we can prove that they will learn zero information about the plaintext without the key. This property is calledperfect security. The proof can be understood intuitively by thinking of XOR as a pro- grammable inverter, and then looking at a particular bit in- tercepted by Eve, the eavesdropper. Letʼs say Eve sees that a particular ciphertext bitci is 1. She has no idea if the matching plaintext bitpi was 0 or 1, because she has no idea if the key bitki was 0 or 1. Since all of the key bits are truly random, both options are exactly equally probable. 4 The attacker does learn that the message exists, and, in this simple scheme, the length of the message. While this typically isnʼt too impor- tant, there are situations where this might matter, and there are secure cryptosystems to both hide the existence and the length of a message. CHAPTER 5. EXCLUSIVE OR 21 5.5 Attackson“one-timepads” The one-time pad security guarantee only holds if it is used correctly. First of all, the one-time pad has to consist of truly random data. Secondly, the one-time pad can only be used once (hence the name). Unfortunately, most commercial products that claim to be “one-time pads” are snake oil5, and donʼt satisfy at least one of those two properties. Notusingtrulyrandomdata The first issue is that they use various deterministic con- structs to produce the one-time pad, instead of using truly random data. That isnʼt necessarily insecure: in fact, the most obvious example, a synchronous stream cipher, is something weʼll see later in the book. However, it does inval- idate the “unbreakable” security property of one-time pads. The end user would be better served by a more honest cryp- tosystem, instead of one that lies about its security proper- ties. Reusingthe“one-time”pad The other issue is with key reuse, which is much more seri- ous. Suppose an attacker gets two ciphertexts with the same “one-time” pad. The attacker can then XOR the two cipher- 5 “Snake oil” is a term for all sorts of dubious products that claim ex- traordinary benefits and features, but donʼt really realize any of them. CHAPTER 5. EXCLUSIVE OR 22 texts, which is also the XOR of the plaintexts: c1 ⊕c2 = ( p1 ⊕k) ⊕(p2 ⊕k) ( definition) = p1 ⊕k ⊕p2 ⊕k (reorder terms) = p1 ⊕p2 ⊕k ⊕k (a ⊕b = b ⊕a) = p1 ⊕p2 ⊕0 (x ⊕x = 0) = p1 ⊕p2 (x ⊕0 = x) At first sight, that may not seem like an issue. To extract ei- ther p1 or p2, youʼd need to cancel out the XOR operation, which means you need to know the other plaintext. The problem is that even the result of the XOR operation on two plaintexts contains quite a bit information about the plain- texts themselves. Weʼll illustrate this visually with some im- ages from a broken “one-time” pad process, starting with Figure 5.1. Crib-dragging A classical approach to breaking multi-time pad systems in- volves “crib-dragging”, a process that uses small sequences that are expected to occur with high probability. Those se- quences are called “cribs”. The name crib-dragging origi- nated from the fact that these small “cribs” are dragged from left to right across each ciphertext, and from top to bottom across the ciphertexts, in the hope of finding a match some- where. Those matches form the sites of the start, or “crib”, if you will, of further decryption. The idea is fairly simple. Suppose we have several en- crypted messages Ci encrypted with the same “one-time” pad K6. If we could correctly guess the plaintext for one of 6 We use capital letters when referring to an entire message, as op- posed to just bits of a message. CHAPTER 5. EXCLUSIVE OR 23 (a) First plaintext. (b) Second plaintext. (c) First ciphertext. (d) Second ciphertext. (e) Reused key. (f) XOR of ciphertexts. Figure 5.1: Two plaintexts, the re-used key, their respective ciphertexts, and the XOR of the ciphertexts. Information about the plaintexts clearly leaks through when we XOR the ciphertexts. CHAPTER 5. EXCLUSIVE OR 24 the messages, letʼs sayCj, weʼd knowK: Cj ⊕Pj = ( Pj ⊕K) ⊕Pj = K ⊕Pj ⊕Pj = K ⊕0 = K Since K is the shared secret, we can now use it to decrypt all of the other messages, just as if we were the recipient: Pi = Ci ⊕K for alli Since we usually canʼt guess an entire message, this doesnʼt actually work. However, we might be able to guess parts of a message. If we guess a few plaintext bitspi correctly forany of the messages, that would reveal the key bits at that position for all of the messages, sincek = ci ⊕pi. Hence, all of the plain- text bits at that position are revealed: using that value fork, we can compute the plaintext bitspi = ci ⊕k for all the other messages. Guessing parts of the plaintext is a lot easier than guess- ing the entire plaintext. Suppose we know that the plaintext is in English. There are some sequences that we know will occur very commonly, for example (the␣ symbol denotes a space): • ␣the␣ and variants such as.␣The␣ • ␣of␣ and variants • ␣to␣ and variants • ␣and␣(no variants; only occurs in the middle of a sen- tence) • ␣a␣ and variants CHAPTER 5. EXCLUSIVE OR 25 If we know more about the plaintext, we can make even better guesses. For example, if itʼs HTTP serving HTML, we would expect to see things likeContent-Type, <a>, and so on. That only tells us which plaintext sequences are likely, giving us likely guesses. How do we tell if any of those guesses are correct? If our guess is correct, we know all the other plaintexts at that position as well, using the technique described earlier. We could simply look at those plaintexts and decide if they look correct. In practice, this process needs to be automated because there are so many possible guesses. Fortunately thatʼs quite easy to do. For example, a very simple but effective method is to count how often different symbols occur in the guessed plaintexts: if the messages contain English text, weʼd expect to see a lot of letters e, t, a, o, i, n. If weʼre seeing binary nonsense instead, we know that the guess was probably in- correct, or perhaps that message is actually binary data. These small, highly probable sequences are called “cribs” because theyʼre the start of a larger decryption pro- cess. Suppose your crib,the, was successful and found the five-letter sequence t thr in another message. You can then use a dictionary to find common words starting with thr, such asthrough. If that guess were correct, it would reveal four more bytes in all of the ciphertexts, which can be used to reveal even more. Similarly, you can use the dic- tionary to find words ending int. This becomes even more effective for some plaintexts that we know more about. If some HTTP data has the plaintext ent-Len in it, then we can expand that to Content-Length:, revealing many more bytes. While this technique works as soon as two messages are encrypted with the same key, itʼs clear that this becomes even easier with more ciphertexts using the same key, since all of the steps become more effective: • We get more cribbing positions. CHAPTER 5. EXCLUSIVE OR 26 • More plaintext bytes are revealed with each successful crib and guess, leading to more guessing options else- where. • More ciphertexts are available for any given position, making guess validation easier and sometimes more accurate. These are just simple ideas for breaking multi-time pads. While theyʼre already quite effective, people have invented even more effective methods by applying advanced, statis- tical models based on natural language analysis. This only demonstrates further just how broken multi-time pads are. [MWES06] 5.6 Remainingproblems Real one-time pads, implemented properly, have an ex- tremely strong security guarantee. It would appear, then, that cryptography is over: encryption is a solved problem, and we can all go home. Obviously, thatʼs not the case. One-time pads are rarely used, because they are horri- bly impractical: the key is at least as large as all informa- tion youʼd like to transmit,put together. Plus, youʼd have to exchange those keys securely, ahead of time, with all peo- ple youʼd like to communicate with. Weʼd like to communi- cate securely with everyone on the Internet, and thatʼs a very large number of people. Furthermore, since the keys have to consist of truly random data for its security property to hold, key generation is fairly difficult and time-consuming without specialized hardware. One-time pads pose a trade-off. Itʼs an algorithm with a solid information-theoretic security guarantee, which you can not get from any other system. On the other hand, it also has extremely impractical key exchange requirements. However, as weʼll see throughout this book, secure sym- metric encryption algorithms arenʼt the pain point of mod- ern cryptosystems. Cryptographers have designed plenty of CHAPTER 5. EXCLUSIVE OR 27 those, while practical key management remains one of the toughest challenges facing modern cryptography. One-time pads may solve a problem, but itʼs the wrong problem. While they may have their uses, theyʼre obviously not a panacea. We need something with manageable key sizes while maintaining secrecy. We need ways to negotiate keys over the Internet with people weʼve never met before. 6 Blockciphers Few false ideas have more firmly gripped the minds of so many intelligent men than the one that, if they just tried, they could invent a cipher that no one could break. David Kahn 6.1 Description A block cipheris an algorithm that encrypts blocks of a fixed length. The encryption function E transforms plaintext blocks P into ciphertext blocksC by using a secret keyk: C = E(k, P ) Plaintext and ciphertext blocks are sequences of bits and al- ways match in size. The block cipherʼsblock size is a fixed size. Keyspaceis the set of all possible keys. Once we encrypt plaintext blocks into ciphertext blocks, they are later decrypted to recover original plaintext block. 28 CHAPTER 6. BLOCK CIPHERS 29 The original plaintext blockP is produced using a decryp- tion functionD. It takes the ciphertext blockC and the key k (the same one used to encrypt the block) as inputs. P = D(k, C) Or, visually represented in blocks: A block cipher is an example of asymmetric-key encryp- tion scheme, also known as asecret-key encryption scheme. The same secret key is used for both encryption and decryp- tion. Later in the book, we contrast this withpublic-key en- cryptionalgorithms, which have a distinct key for encryption and decryption. A block cipher is akeyed permutation. It is apermutation because the block cipher maps each possible block to an- other block. It is also akeyed permutation because the key determines exactly which blocks map to which. It is impor- tant for the block cipher to be a permutation because the recipient must map blocks back to the original blocks. We illustrate this by looking at a block cipher with an impractical, tiny 4-bit block size.24 = 16 possible blocks. Since each of the blocks map to a hexadecimal digit, we rep- resent the blocks by that digit.Figure 6.1illustrates blocks that the cipher operates on. Once we select a secret key, the block cipher uses it to determine the encryption of any given block. We illustrate that relationship with an arrow. The tail of the arrow has the block encrypted withE under keyk and the arrowhead is mapped to the block. In Figure 6.2, note that the permutation is not just one big cycle. It contains a large cycle of 7 elements, and several CHAPTER 6. BLOCK CIPHERS 30 Figure 6.1: All 16 nodes operated on by the block cipher. Each node is designated by a hexadecimal digit. smaller cycles of 4, 3 and 2 elements each. It is also perfectly possible that an element encrypts to itself. This is to be ex- pected when selecting random permutations, which is ap- proximately what a block cipher is doing; it doesnʼt demon- strate a bug in the block cipher. When you decrypt instead of encrypt, the block cipher computes the inverse permutation. InFigure 6.3, we get the same illustration. The difference between the illustrations is that all arrowheads point in the opposite direction. The key defines which blocks map to which blocks. A different key would lead to a different set of arrows, as you CHAPTER 6. BLOCK CIPHERS 31 Figure 6.2: An encryption permutation made by a block ci- pher under a particular keyk. can see inFigure 6.4. In this illustration, youʼll even notice that there are two permutations of length 1: an element that maps to itself. This is again something to be expected when selecting ran- dom permutations. Knowing a bunch of (input, output) pairs for a given key shouldnʼt give you any information about any other (input, output) pairs under that key1. As long as weʼre talking about a hypothetical perfect block cipher, thereʼs no easier way to decrypt a block other than to “brute-force” the key: i.e. just try every single one of them until you find the right one. 1 The attentive reader may have noticed that this breaks in the ex- tremes: if you know all but one of the pairs, then you know the last one by exclusion. CHAPTER 6. BLOCK CIPHERS 32 Figure 6.3: The decryption permutation produced by the block cipher under the same keyk. It is the inverse of the encryption permutation in that all arrowheads reverse. Our toy illustration block cipher only has 4 bit blocks, or 24 = 16 possibilities. Real, modern block ciphers have much larger block sizes, such as 128 bits, or2128 (slightly more than 1038.5) possible blocks. Mathematics tells us that there are n! (pronounced “n factorial”) different permutations of ann element set. Itʼs defined as the product of all of the numbers from 1 up to and includingn: n! = 1 ·2 ·3 ·. . . ·(n −1) ·n Factorials grow incredibly quickly. For example,5! = 120 , 10! = 3628800 , and the rate continues to increase. The num- ber of permutations of the set of blocks of a cipher with a 128 bit block size is(2128)!. Just 2128 is large already (it takes 39 digits to write it down), so(2128)! is a mind-bogglingly huge number, impossible to comprehend. Common key sizes are only in the range of 128 to 256 bits, so there are only between 2128 and 2256 permutations a cipher can perform. Thatʼs just a tiny fraction of all possible permutations of the blocks, CHAPTER 6. BLOCK CIPHERS 33 Figure 6.4: An encryption permutation produced by the block cipher under a different key. but thatʼs okay: that tiny fraction is still nowhere near small enough for an attacker to just try them all. Of course, a block cipher should be as easy to compute as possible, as long as it doesnʼt sacrifice any of the above properties. 6.2 AES The most common block cipher in current use is AES. Contrary to its predecessor DES (which weʼll look at in more detail in the next chapter), AES was selected through CHAPTER 6. BLOCK CIPHERS 34 a public, peer-reviewed competition following an open call for proposals. This competition involved several rounds where all of the contestants were presented, subject to ex- tensive cryptanalysis, and voted upon. The AES process was well-received among cryptographers, and similar processes are generally considered to be the preferred way to select cryptographic standards. Prior to being chosen as the Advanced Encryption Stan- dard, the algorithm was known as Rijndael, a name derived from the two last names of the Belgian cryptographers that designed it: Vincent Rijmen and Joan Daemen. The Rijn- dael algorithm defined a family of block ciphers, with block sizes and key sizes that could be any multiple of 32 bits be- tween 128 bits and 256 bits. [DR02] When Rijndael became AES through the FIPS standardization process, the parame- ters were restricted to a block size of 128 bits and keys sizes of 128, 192 and 256 bits. [fip01] There are no practical attacks known against AES. While there have been some developments in the last few years, most of them involve related-key attacks [BK09], some of them only on reduced-round versions of AES [BDK+09].2 2 Symmetric algorithms usually rely on a round function to be re- peated a number of times. Typically each invocation involves a “round key” derived from the main key. A reduced-round version is intention- ally easier to attack. These attacks can give insight as to how resistant the full cipher is. A related key attack involves making some predictions about how AES will behave under several different keys with some specific mathemati- cal relation. These relations are fairly simple, such as XORing with an attacker-chosen constant. If an attacker is allowed to encrypt and decrypt a large number of blocks with these related keys, they can attempt to re- cover the original key with significantly less computation than would or- dinarily be necessary to crack it. While a theoretically ideal block cipher wouldnʼt be vulnerable to a related key attack, these attacks arenʼt considered practical concerns. In practice cryptographic keys are generated via a cryptographically se- cure pseudorandom number generator, or a similarly securekey agree- ment scheme or key derivation scheme (weʼll see more about those later). Therefore, the odds of selecting two such related keys by accident is nonexistent. These attacks are interesting from an academic perspec- CHAPTER 6. BLOCK CIPHERS 35 AcloserlookatRijndael This is an optional, in-depth section. It almost certainly wonʼt help you write bet- ter software, so feel free to skip it. It is only here to satisfy your inner geekʼs curiosity. AES consists of several independent steps. At a high level, AES is asubstitution-permutation network. Keyschedule AES requires separate keys for each round in the next steps. The key schedule is the process which AES uses to derive 128- bit keys for each round from one master key. First, the key is separated into 4 byte columns. The key is rotated and then each byte is run through an S-box (sub- stitution box) that maps it to something else. Each column is then XORed with a round constant. The last step is to XOR the result with the previous round key. The other columns are then XORed with the previous round key to produce the remaining columns. SubBytes SubBytes is the step that applies the S-box (substitution box) in AES. The S-box itself substitutes a byte with another byte, and this S-box is applied to each byte in the AES state. It works by taking the multiplicative inverse over the Ga- lois field, and then applying an affine transformation so that there are no valuesx so thatx⊕S(x) = 0 or x⊕S(x) = 0xff. To rephrase: there are no values ofx that the substitution box maps tox itself, orx with all bits flipped. This makes tive: they can help provide insight in the workings of the cipher, guiding cryptographers in designing future ciphers and attacks against current ciphers. CHAPTER 6. BLOCK CIPHERS 36 the cipher resistant to linear cryptanalysis, unlike the ear- lier DES algorithm, whose fifth S-box caused serious security problems.3 ShiftRows After having applied the SubBytes step to the 16 bytes of the block, AES shifts the rows in the4 ×4 array: 3 In its defense, linear attacks were not publicly known back when DES was designed. CHAPTER 6. BLOCK CIPHERS 37 MixColumns MixColumns multiplies each column of the state with a fixed polynomial. ShiftRows and MixColumns represent the diffusion prop- erties of AES. AddRoundKey As the name implies, the AddRoundKey step adds the bytes from the round key produced by the key schedule to the state of the cipher. 6.3 DESand3DES The DES is one of the oldest block ciphers that saw widespread use. It was published as an official FIPS standard in 1977. It is no longer considered secure, mainly due to its tiny key size of 56 bits. (The DES algorithm actually takes a 64 bit key input, but the remaining 8 bits are only used for par- ity checking, and are discarded immediately.) It shouldnʼt be used in new systems. On modern hardware, DES can be brute forced in less than a day. [Gmb08] CHAPTER 6. BLOCK CIPHERS 38 In an effort to extend the life of the DES algorithm, in a way that allowed much of the spent hardware development effort to be reused, people came up with 3DES: a scheme where input is first encrypted, then decrypted, then en- crypted again: C = EDES (k1, DDES (k2, EDES (k3, p))) This scheme provides two improvements: • By applying the algorithm three times, the cipher be- comes harder to attack directly through cryptanalysis. • By having the option of using many more total key bits, spread over the three keys, the set of all possible keys CHAPTER 6. BLOCK CIPHERS 39 becomes much larger, making brute-forcing impracti- cal. The three keys could all be chosen independently (yield- ing 168 key bits), ork3 = k1 (yielding 112 key bits), or k1 = k2 = k3, which, of course, is just plain old DES (with 56 key bits). In the last keying option, the middle decryption reverses the first encryption, so you really only get the ef- fect of the last encryption. This is intended as a backwards compatibility mode for existing DES systems. If 3DES had been defined asE(k1, E(k2, E(k3, p))), it would have been impossible to use 3DES implementations for systems that required compatibility with DES. This is particularly impor- tant for hardware implementations, where it is not always possible to provide a secondary, regular “single DES” inter- face next to the primary 3DES interface. Some attacks on 3DES are known, reducing their effec- tive security. While breaking 3DES with the first keying op- tion is currently impractical, 3DES is a poor choice for any modern cryptosystem. The security margin is already small, and continues to shrink as cryptographic attacks improve and processing power grows. Far better alternatives, such as AES, are available. Not only are they more secure than 3DES, they are also gener- ally much, much faster. On the same hardware and in the same mode of operation(weʼll explain what that means in the next chapter), AES-128 only takes 12.6 cycles per byte, while 3DES takes up to 134.5 cycles per byte. [Dai] Despite being worse from a security point of view, it is literally an order of magnitude slower. While more iterations of DES might increase the security margin, they arenʼt used in practice. First of all, the process has never been standardized beyond three iterations. Also, the performance only becomes worse as you add more itera- tions. Finally, increasing the key bits has diminishing secu- rity returns, only increasing the security level of the result- ing algorithm by a smaller amount as the number of key bits CHAPTER 6. BLOCK CIPHERS 40 increases. While 3DES with keying option 1 has a key length of 168 bits, the effective security level is estimated at only 112 bits. Even though 3DES is significantly worse in terms of per- formance and slightly worse in terms of security, 3DES is still the workhorse of the financial industry. With a plethora of standards already in existence and new ones continuing to be created, in such an extremely technologically conser- vative industry where Fortran and Cobol still reign supreme on massive mainframes, it will probably continue to be used for many years to come, unless there are some large crypt- analytic breakthroughs that threaten the security of 3DES. 6.4 Remainingproblems Even with block ciphers, there are still some unsolved prob- lems. For example, we can only send messages of a very lim- ited length: the block length of the block cipher. Obviously, weʼd like to be able to send much larger messages, or, ideally, streams of indeterminate size. Weʼll address this problem with astream cipher(page 41). Although we have reduced the key size drastically (from the total size of all data ever sent under a one-time pad scheme versus a few bytes for most block ciphers), we still need to address the issue of agreeing on those few key bytes, potentially over an insecure channel. Weʼll address this problem in a later chapter with akey exchange protocol (page 81). 7 Streamciphers 7.1 Description A stream cipher is asymmetric-key encryption algorithm that encrypts a stream of bits. Ideally, that stream could be as long as weʼd like; real-worldstream ciphers have limits, but they are normally sufficiently large that they donʼt pose a practical problem. 7.2 Anaiveattemptwithblockciphers Letʼs try to build astream cipher using the tools we already have. Since we already have block ciphers, we could simply divide an incoming stream into different blocks, and encrypt each block: abcdefgh\| {z } ijklmno\| {z } pqrstuvw\| {z } ... ↓ ↓ ↓z }\| { APOHGMMW z }\| { PVMEHQOM z }\| { MEEZSNFM ... This scheme is called ECB mode (Electronic Code Book Mode), and it is one of the many ways that block ciphers can 41 CHAPTER 7. STREAM CIPHERS 42 be used to construct stream ciphers. Unfortunately, while being very common in home-grown cryptosystems, it poses very serious security flaws. For example, in ECB mode, iden- tical input blocks will always map to identical output blocks: abcdefgh\| {z } abcdefgh\| {z } abcdefgh\| {z } ... ↓ ↓ ↓z }\| { APOHGMMW z }\| { APOHGMMW z }\| { APOHGMMW ... At first, this might not seem like a particularly serious prob- lem. Assuming the block cipher is secure, it doesnʼt look like an attacker would be able to decrypt anything. By dividing the ciphertext stream up into blocks, an attacker would only be able to see that a ciphertext block, and therefore a plain- text block, was repeated. Weʼll now illustrate the many flaws ofECB modewith two attacks. First, weʼll exploit the fact that repeating plaintext blocks result in repeating ciphertext blocks, by visually in- specting an encrypted image. Then, weʼll demonstrate that attackers can often decrypt messages encrypted inECBmode by communicating with the person performing the encryp- tion. Visualinspectionofanencryptedstream To demonstrate that this is, in fact, a serious problem, weʼll use a simulated block cipher of various block sizes and ap- ply it to an image1. Weʼll then visually inspect the different outputs. Because identical blocks of pixels in the plaintext will map to identical blocks of pixels in the ciphertext, the global structure of the image is largely preserved. As you can see, the situation appears to get slightly better with larger block sizes, but the fundamental problem still re- mains: the macrostructure of the image remains visible in 1 This particular demonstration only works on uncompressed bitmaps. For other media, the effect isnʼt significantly less damning: itʼs just less visual. CHAPTER 7. STREAM CIPHERS 43 (a) Plaintext image, 2000 by 1400 pixels, 24 bit color depth. (b) ECB mode ciphertext, 5 pixel (120 bit) block size. (c) ECB mode ciphertext, 30 pixel (720 bit) block size. (d) ECB mode ciphertext, 100 pixel (2400 bit) block size. (e) ECB mode ciphertext, 400 pixel (9600 bit) block size. (f) Ciphertext under idealized encryption. Figure 7.1: Plaintext image with ciphertext images under idealized encryption andECB modeencryption with various block sizes. Information about the macro-structure of the image clearly leaks. This becomes less apparent as block sizes increase, but only at block sizes far larger than typical block ciphers. Only the first block size (Figure 7.1f, a block size of 5 pixels or 120 bits) is realistic. CHAPTER 7. STREAM CIPHERS 44 all but the most extreme block sizes. Furthermore, all but the smallest of these block sizes are unrealistically large. For an uncompressed bitmap with three color channels of 8 bit depth, each pixel takes 24 bits to store. Since the block size of AES is only 128 bits, that would equate to128 24 or just over 5 pixels per block. Thatʼs significantly fewer pixels per block than the larger block sizes in the example. But AES is the workhorse of modern block ciphers—it canʼt be at fault, cer- tainly not because of an insufficient block size. When we look at a picture of what would happen with an idealized encryption scheme, we notice that it looks like ran- dom noise. Keep in mind that “looking like random noise” doesnʼt mean something is properly encrypted: it just means that we canʼt inspect it using methods this trivial. Encryptionoracleattack In the previous section, weʼve focused on how an attacker can inspect a ciphertext encrypted usingECB mode. Thatʼs a passive, ciphertext-only attack. Itʼs passive because the attacker doesnʼt really interfere in any communication; theyʼre simply examining a ciphertext. In this section, weʼll study a different,active attack, where the attacker actively communicates with their target. Weʼll see how the active at- tack can enable an attacker to decrypt ciphertexts encrypted using ECB mode. To do this, weʼll introduce a new concept called anoracle. Formally definedoracles are used in the study of computer science, but for our purposes itʼs sufficient to just say that an oracle is something that will compute some particular func- tion for you. In our case, theoraclewill perform a specific encryption for the attacker, which is why itʼs called anencryption oracle. Given some dataA chosen by the attacker, theoraclewill en- crypt that data, followed by a secret suffixS, in ECB mode. Or, in symbols: C = ECB (Ek, A∥S) CHAPTER 7. STREAM CIPHERS 45 The secret suffixS is specific to this system. The attackerʼs goal is to decrypt it. Weʼll see that being able to encrypt other messages surprisingly allows the attacker to decrypt the suf- fix. This oracle might seem artificial, but is quite common in practice. A simple example would be a cookie encrypted with ECB, where the prefixA is a name or an e-mail address field, controlled by the attacker. You can see why the concept of anoracle is important here: the attacker would not be able to computeC them- selves, since they do not have access to the encryption key k or the secret suffixS. The goal of theoracle is for those values to remain secret, but weʼll see how an attacker will be able to recover the secret suffixS (but not the keyk) any- way. The attacker does this by inspecting the ciphertextC for many carefully chosen values of the attacker-chosen pre- fix A. Assuming that an attacker would have access to such an oraclemight seem like a very artificial scenario. It turns out that in practice, a lot of software can be tricked into behaving like one. Even if an attacker canʼt control the real software as precisely as they can query anoracle, the attacker gen- erally isnʼt thwarted. Time is on their side: they only have to convince the software to give the answer they wantonce. Systems where part of the message is secret and part of the message can be influenced by the attacker are actually very common, and, unfortunately, so isECB mode. Decryptingablockusingtheoracle The attacker starts by sending in a plaintextA thatʼs just one byte shorter than the block size. That means the block thatʼs being encrypted will consist of those bytes, plus the first byte ofS, which weʼll calls0. The attacker remembers the encrypted block. They donʼt know the value ofs0 yet, but now they do know the value of the first encrypted block: Ek(A∥s0). In the illustration, this is blockCR1: Then, the attacker tries a full-size block, trying all pos- CHAPTER 7. STREAM CIPHERS 46 sible values for the final byte. Eventually, theyʼll find the value ofs0; they know the guess is correct because the re- sulting ciphertext block will match the ciphertext blockCR1 they remembered earlier. The attacker can repeat this for the penultimate byte. They submit a plaintextA thatʼs two bytes shorter than the block size. Theoraclewill encrypt a first block consisting of that A followed by the first two bytes of the secret suffix,s0s1. The attacker remembers that block. Since the attacker already knowss0, they tryA∥s0 fol- lowed by all possible values ofs1. Eventually theyʼll guess correctly, which, again, theyʼll know because the ciphertext blocks match: The attacker can then rinse and repeat, eventually de- crypting an entire block. This allows them to brute-force a CHAPTER 7. STREAM CIPHERS 47 block inp ·b attempts, wherep is the number of possible val- ues for each byte (so, for 8-bit bytes, thatʼs28 = 256 ) andb is the block size. This is much better than a regular brute- force attack, where an attacker has to try all of the possible blocks, which would be: p ·p . . . ·p\| {z } b positions = pb For a typical block size of 16 bytes (or 128 bits), brute forc- ing would mean trying25616 combinations. Thatʼs a huge, 39-digit number. Itʼs so large that trying all of those combi- nations is considered impossible. An ECBencryption oracle allows an attacker to do it in at most256 ·16 = 4096 tries, a far more manageable number. CHAPTER 7. STREAM CIPHERS 48 Conclusion In the real world, block ciphers are used in systems that en- crypt large amounts of data all the time. Weʼve seen that when usingECB mode, an attacker can both analyze cipher- texts to recognize repeating patterns, and even decrypt mes- sages when given access to anencryption oracle. Even when we use idealized block ciphers with unrealis- tic properties, such as block sizes of more than a thousand bits, an attacker ends up being able to decrypt the cipher- texts. Real world block ciphers only have more limitations than our idealized examples, such as much smaller block sizes. We arenʼt even taking into account any potential weak- nesses in the block cipher. Itʼs not AES (or our test block ciphers) that cause this problem, itʼs our ECB construction. Clearly, we need something better. 7.3 Blockciphermodesofoperation One of the more common ways of producing astream cipher is to use a block cipher in a particular configuration. The compound system behaves like astream cipher. These con- figurations are commonly calledmode of operations. They arenʼt specific to a particular block cipher. ECB mode, which weʼve just seen, is the simplest such mode of operation. The lettersECB stand for electronic code book2. For reasons weʼve already gone into,ECBmode is very ineffective. Fortunately, there are plenty of other choices. 7.4 CBCmode CBC mode, which stands for cipher block chaining, is a very common modeofoperation where plaintext blocks are XORed 2 Traditionally, modes of operation seem to be referred to by a three- letter acronym. CHAPTER 7. STREAM CIPHERS 49 with the previous ciphertext block before being encrypted by the block cipher. Of course, this leaves us with a problem for the first plain- text block: there is no previous ciphertext block to XOR it with. Instead, we pick an IV: a random number that takes the place of the “first” ciphertext in this construction.initial- ization vectors also appear in many other algorithms. Anini- tialization vectorshould be unpredictable; ideally, they will be cryptographically random. They do not have to be secret: IVs are typically just added to ciphertext messages in plain- text. It may sound contradictory that something has to be unpredictable, but doesnʼt have to be secret; itʼs important to remember that an attacker must not be able to predictahead of time what a given IV will be. We will illustrate this later with an attack on predictable CBC IVs. The following diagram demonstrates encryption inCBC mode: CHAPTER 7. STREAM CIPHERS 50 Decryption is the inverse construction, with block ci- phers in decryption mode instead of encryption mode: While CBC modeitself is not inherently insecure (unlike ECB mode), its particular use in TLS 1.0 was. This eventu- ally led to the BEAST attack, which weʼll cover in more detail in the section on SSL/TLS. The short version is that instead of using unpredictableinitialization vectors, for example by choosing random IVs, the standard used the previous cipher- text block as the IV for the next message. Unfortunately, it turns out that attackers figured out how to exploit that prop- erty. 7.5 Attacks on CBC mode with predictable IVs Suppose thereʼs a database that stores secret user informa- tion, like medical, payroll or even criminal records. In or- der to protect that information, the server that handles it encrypts it using a strong block cipher inCBC modewith a fixed key. For now, weʼll assume that that server is secure, and thereʼs no way to get it to leak the key. CHAPTER 7. STREAM CIPHERS 51 Mallory gets a hold of all of the rows in the database. Perhaps she did it through a SQL injection attack, or maybe with a little social engineering.3 Everything is supposed to remain secure: Mallory only has the ciphertexts, but she doesnʼt have the secret key. Mallory wants to figure out what Aliceʼs record says. For simplicityʼs sake, letʼs say thereʼs only one ciphertext block. That means Aliceʼs ciphertext consists of an IV and one ci- phertext block. Mallory can still try to use the application as a normal user, meaning that the application will encrypt some data of Malloryʼs choosing and write it to the database. Suppose that through a bug in the server, Mallory can predict the IV that will be used for her ciphertext. Perhaps the server always uses the same IV for the same person, or always uses an all- zero IV , or… Mallory can construct her plaintext using Aliceʼs IVIVA (which Mallory can see) and her own predicted IVIVM . She makes a guessG as to what Aliceʼs data could be. She asks the server to encrypt: PM = IVM ⊕IVA ⊕G The server dutifully encrypts that message using the pre- dicted IVIVM . It computes: CM = E(k, IVM ⊕PM ) = E(k, IVM ⊕(IVM ⊕IVA ⊕G)) = E(k, IVA ⊕G) That ciphertext, CM, is exactly the ciphertext block Alice would have had if her plaintext block was G. So, depending on what the data is, Mallory has figured out if Alice has a 3 Social engineering means tricking people into things they shouldnʼt be doing, like giving out secret keys, or performing certain operations. Itʼs usually the most effective way to break otherwise secure cryptosys- tems. CHAPTER 7. STREAM CIPHERS 52 criminal record or not, or perhaps some kind of embarrass- ing disease, or some other issue that Alice really expected the server to keep secret. Lessons learned: donʼt let IVs be predictable. Also, donʼt roll your own cryptosystems. In a secure system, Alice and Malloryʼs records probably wouldnʼt be encrypted using the same key. 7.6 AttacksonCBCmodewiththekeyasthe IV Many CBC systems set the key as theinitialization vector. This seems like a good idea: you always need a shared se- cret key already anyway. It yields a nice performance ben- efit, because the sender and the receiver donʼt have to com- municate the IV explicitly, they already know the key (and therefore the IV) ahead of time. Plus, the key is definitely unpredictable because itʼs secret: if it were predictable, the attacker could just predict the key directly and already have won. Conveniently, many block ciphers have block sizes that are the same length or less than the key size, so the key is big enough. This setup is completely insecure. If Alice sends a mes- sage to Bob, Mallory, an active adversary who can intercept and modify the message, can perform a chosen ciphertext attack to recover the key. Alice turns her plaintext messageP into three blocks P1P2P3 and encrypts it inCBC mode with the secret keyk and also usesk as the IV . She gets a three block ciphertext C = C1C2C3, which she sends to Bob. Before the message reaches Bob, Mallory intercepts it. She modifies the message to beC′ = C1ZC1, whereZ is a block filled with null bytes (value zero). Bob decrypts C′, and gets the three plaintext blocks CHAPTER 7. STREAM CIPHERS 53 P′ 1, P ′ 2, P ′ 3: P′ 1 = D(k, C1) ⊕IV = D(k, C1) ⊕k = P1 P′ 2 = D(k, Z) ⊕C1 = R P′ 3 = D(k, C1) ⊕Z = D(k, C1) = P1 ⊕IV R is some random block. Its value doesnʼt matter. Under the chosen-ciphertext attack assumption, Mallory recovers that decryption. She is only interested in the first block (P′ 1 = P1) and the third block (P′ 3 = P1 ⊕IV ). By XORing those two together, she finds(P1 ⊕IV ) ⊕P1 = IV . But, the IV is the key, so Mallory successfully recovered the key by modifying a single message. Lesson learned: donʼt use the key as an IV . Part of the fal- lacy in the introduction is that it assumed secret data could be used for the IV , because it only had to be unpredictable. Thatʼs not true: “secret” is just a different requirement from “not secret”, not necessarily astronger one. It is not gener- ally okay to use secret information where it isnʼt required, precisely because if itʼs not supposed to be secret, the algo- rithm may very well treat it as non-secret, as is the case here. There are plenty of systems where it is okay to use a secret where it isnʼt required. In some cases you might even get a stronger system as a result, but the point is that it is not generally true, and depends on what youʼre doing. 7.7 CBCbitflippingattacks An interesting attack onCBC modeis called a bit flipping at- tack. Using a CBC bit flipping attack, attackers can modify CHAPTER 7. STREAM CIPHERS 54 ciphertexts encrypted inCBC modeso that it will have a pre- dictable effect on the plaintext. This may seem like a very strange definition of “attack” at first. The attacker will not even attempt to decrypt any messages, but they will just be flipping some bits in a plain- text. We will demonstrate that the attacker can turn the abil- ity to flip some bits in the plaintext into the ability to have the plaintext saywhatever they want it to say, and, of course, that can lead to very serious problems in real systems. Suppose we have a CBC encrypted ciphertext. This could be, for example, a cookie. We take a particular ciphertext block, and we flip some bits in it. What happens to the plain- text? When we “flip some bits”, we do that by XORing with a sequence of bits, which weʼll callX. If the corresponding bit inX is 1, the bit will be flipped; otherwise, the bit will remain the same. When we try to decrypt the ciphertext block with the flipped bits, we will get indecipherable4 nonsense. Remem- 4 Excuse the pun. CHAPTER 7. STREAM CIPHERS 55 ber how CBC decryption works: the output of the block ci- pher is XORed with the previous ciphertext block to produce the plaintext block. Now that the input ciphertext blockCi has been modified, the output of the block cipher will be some random unrelated block, and, statistically speaking, nonsense. After being XORed with that previous ciphertext block, it will still be nonsense. As a result, the produced plaintext block is still just nonsense. In the illustration, this unintelligible plaintext block isP′ i . However, in the blockafter that, the bits we flipped in the ciphertext will be flipped in the plaintext as well! This is be- cause, in CBC decryption, ciphertext blocks are decrypted by the block cipher, and the result is XORed with the previ- ous ciphertext block. But since we modified the previous ci- phertext block by XORing it withX, the plaintext blockPi+1 will also be XORed withX. As a result, the attacker com- pletely controls that plaintext blockPi+1, since they can just flip the bits that arenʼt the value they want them to be. TODO: add previous illustration, but mark the path X takes to influence P prime {i + 1} in red or something This may not sound like a huge deal at first. If you donʼt know the plaintext bytes of that next block, you have no idea which bits to flip in order to get the plaintext you want. To illustrate how attackers can turn this into a practical attack, letʼs consider a website using cookies. When you reg- ister, your chosen user name is put into a cookie. The web- site encrypts the cookie and sends it to your browser. The next time your browser visits the website, it will provide the encrypted cookie; the website decrypts it and knows who you are. An attacker can often control at least part of the plaintext being encrypted. In this example, the user name is part of the plaintext of the cookie. Of course, the website just lets you provide whatever value for the user name you want at registration, so the attacker can just add a very long string of Z bytes to their user name. The server will happily en- crypt such a cookie, giving the attacker an encrypted cipher- CHAPTER 7. STREAM CIPHERS 56 text that matches a plaintext with many suchZ bytes in them. The plaintext getting modified will then probably be part of that sequence ofZ bytes. An attacker may have some target bytes that theyʼd like to see in the decrypted plaintext, for example,;admin=1;. In order to figure out which bytes they should flip (so, the value ofX in the illustration), they just XOR the filler bytes (~ZZZ~…) with that target. Because two XOR operations with the same value cancel each other out, the two filler values (~ZZZ~…) will cancel out, and the attacker can expect to see ;admin=1; pop up in the next plaintext block: P′ i+1 = Pi+1 ⊕X = Pi+1 ⊕ZZZZZZZZZ ⊕; admin = 1; = ZZZZZZZZZ ⊕ZZZZZZZZZ ⊕; admin = 1; = ; admin = 1; This attack is another demonstration of an important crypto- graphic principle: encryption is not authentication! Itʼs vir- tually never sufficient to simply encrypt a message. Itmay prevent an attacker from reading it, but thatʼs often not even necessary for the attacker to be able to modify it to say what- ever they want it to. This particular problem would be solved by also securely authenticating the message. Weʼll see how you can do that later in the book; for now, just remember that weʼre going to need authentication in order to produce secure cryptosystems. 7.8 Padding So far, weʼve conveniently assumed that all messages just happened to fit exactly in our system of block ciphers, be it CBC or ECB. That means that all messages happen to be a multiple of the block size, which, in a typical block cipher such as AES, is 16 bytes. Of course, real messages can be of arbitrary length. We need some scheme to make them fit. That process is called padding. CHAPTER 7. STREAM CIPHERS 57 Paddingwithzeroes(orsomeotherpadbyte) One way to pad would be to simply append a particular byte value until the plaintext is of the appropriate length. To undo the padding, you just remove those bytes. This scheme has an obvious flaw: you canʼt send messages that end in that particular byte value, or you will be unable to distinguish between padding and the actual message. PKCS#5/PKCS#7padding A better, and much more popular scheme, is PKCS#5/PKCS#7 padding. PKCS#5, PKCS#7 and later CMS padding are all more or less the same idea5. Take the number of bytes you have to pad, and pad them with that many times the byte with that value. For example, if the block size is 8 bytes, and the last block has the three bytes12 34 45, the block becomes12 34 45 05 05 05 05 05 after padding. If the plaintext happened to be exactly a multiple of the block size, an entire block of padding is used. Otherwise, the recipient would look at the last byte of the plaintext, treat it as a padding length, and almost certainly conclude the mes- sage was improperly padded. This scheme is described in [Hou]. 7.9 CBCpaddingattacks We can refine CBC bit flipping attacks to trick a recipient into decrypting arbitrary messages! As weʼve just discussed,CBC moderequires padding the message to a multiple of the block size. If the padding is in- correct, the recipient typically rejects the message, saying 5 Technically, PKCS#5 padding is only defined for 8 byte block sizes, but the idea clearly generalizes easily, and itʼs also the most commonly used term. CHAPTER 7. STREAM CIPHERS 58 that the padding was invalid. We can use that tiny bit of in- formation about the padding of the plaintext to iteratively decrypt the entire message. The attacker will do this, one ciphertext block at a time, by trying to get an entire plaintext block worth of valid padding. Weʼll see that this tells them the decryption of their target ciphertext block, under the block cipher. Weʼll also see that you can do this efficiently and iteratively, just from that little leak of information about the padding being valid or not. It may be helpful to keep in mind that a CBC padding attack does not actually attack the padding for a given mes- sage; instead the attacker will beconstructing paddings to de- crypt a message. To mount this attack, an attacker only needs two things: 1. A target ciphertext to decrypt 2. A padding oracle: a function that takes ciphertexts and tells the attacker if the padding was correct As with the ECBencryption oracle, the availability of a padding oracle may sound like a very unrealistic assump- tion. The massive impact of this attack proves otherwise. For a long time, most systems did not even attempt to hide if the padding was valid or not. This attack remained dan- gerous for a long time after it was originally discovered, be- cause it turns out that in many systems it is extremely diffi- cult to actually hide if padding is valid or not. We will go into this problem in more detail both in this chapter and in later chapters. In this chapter, weʼll assume that PKCS#5/PKCS#7 padding is being used, since thatʼs the most popular op- tion. The attack is general enough to work on other kinds of padding, with minor modifications. CHAPTER 7. STREAM CIPHERS 59 Decryptingthefirstbyte The attacker fills a block with arbitrary bytesR = r1, r2 . . . rb. They also pick a target blockCi from the ciphertext that theyʼd like to decrypt. The attacker asks the padding ora- cle if the plaintext ofR∥Ci has valid padding. Statistically speaking, such a random plaintext probably wonʼt have valid padding: the odds are in the half-a-percent ballpark. If by pure chance the message happens to already have valid padding, the attacker can simply skip the next step. Next, the attacker tries to modify the message so that it does have valid padding. They can do that by indirectly mod- ifying the last byte of the plaintext: eventually that byte will be 01, which is always valid padding. In order to modify the last byte of a plaintext block, the attacker modifies the last byte of thepreviousciphertext block. This works exactly like it did with CBC bit flipping attacks. That previous ciphertext block is the blockR, so the byte being modified is the last byte ofR, rb. The attacker tries all possible values for that last byte. There are several ways of doing that: modular addition, XOR- CHAPTER 7. STREAM CIPHERS 60 ing it with all values up to 256, or even picking randomly; the only thing that matters is that the attacker tries all of them. Eventually, the padding oracle will report that for some ci- phertext blockR, the decrypted plaintext ofR∥Ci has valid padding. Discoveringthepaddinglength The oracle has just told the attacker that for our chosen value of R, the plaintext ofR∥Ci has valid padding. Since weʼre working with PKCS#5 padding, that means that the plaintext block Pi ends in one of the following byte sequences: • 01 • 02 02 • 03 03 03 • … The first option (01) is much more likely than the others, since it only requires one byte to have a particular value. The attacker is modifying that byte to takeevery possible value, so it is quite likely that they happened to stumble upon01. All of the other valid padding options not only require that byte to have some particular value, but also one or more other bytes. For an attacker to be guaranteed a message with a valid01 padding, they just have to try every possible byte. For an attacker to end up with a message with a valid02 02 padding, they have to try every possible byteand happen to have picked a combination ofC and R that causes the plain- text to have a02 in that second-to-last position. (To rephrase: the second-to-last byte of the decryption of the ciphertext block, XORed with the second-to-last byte ofR, is02.) In order to successfully decrypt the message, we still need to figure out which one of those options is the actual value of the padding. To do that, we try to discover the length of the padding by modifying bytes starting at the left-hand CHAPTER 7. STREAM CIPHERS 61 side ofPi until the padding becomes invalid again. As with everything else in this attack, we modify those bytes inPi by modifying the equivalent bytes in our chosen blockR. As soon as padding breaks, you know that the last byte you mod- ified was part of the valid padding, which tells you how many padding bytes there are. Since weʼre using PKCS#5 padding, that also tells you what their value is. Letʼs illustrate this with an example. Suppose weʼve suc- cessfully found some blockR so that the plaintext ofR∥Ci has valid padding. Letʼs say that padding is03 03 03. Nor- mally, the attacker wouldnʼt know this; the point of this pro- cedure is to discover what that padding is. Suppose the block size is 8 bytes. So, we (but not the attacker) know thatPi is currently: p0p1p2p3p4030303 In that equation,p0 . . . are some bytes of the plaintext. Their actual value doesnʼt matter: the only thing that matters is that theyʼre not part of the padding. When we modify the first byte ofR, weʼll cause a change in the first byte ofPi, so that p0 becomes some other bytep′ 0: p′ 0p1p2p3p4030303 As you can see, this doesnʼt affect the validity of the padding. It also does not affectp1, p2, p3 or p4. However, when we continue modifying subsequent bytes, we will eventually hit a byte thatis part of the padding. For example, letʼs say we turn that first03 into 02 by modifyingR. Pi now looks like this: p′ 0p′ 1p′ 2p′ 3p′ 4020303 Since 02 03 03 isnʼt valid PKCS#5 padding, the server will reject the message. At that point, we know that once we mod- ify six bytes, the padding breaks. That means the sixth byte is the first byte of the padding. Since the block is 8 bytes long, CHAPTER 7. STREAM CIPHERS 62 we know that the padding consists of the sixth, seventh and eighth bytes. So, the padding is three bytes long, and, in PKCS#5, equal to03 03 03. A clever attacker whoʼs trying to minimize the number of oracle queries can leverage the fact that longer valid padding becomes progressively more rare. They can do this by starting from the penultimate byte instead of the beginning of the block. The advantage to this method is that short paddings (which are more common) are detected more quickly. For example, if the padding is0x01 and an at- tacker starts modifying the penultimate byte, they only need one query to learn what the padding was. If the penultimate byte is changed to any other value and the padding is still valid, the padding must be0x01. If the padding is not valid, the padding must be at least0x02 0x02. So, they go back to the original block and start modifying the third byte from the back. If that passes, the padding was indeed0x02 0x02, otherwise the padding must be at least0x03 0x03 0x03. The process repeats until theyʼve found the correct length. This is a little trickier to implement; you canʼt just keep mod- ifying the same block (if itʼs mutable), and youʼre waiting for the oracle to fail instead of pass, which can be confusing. But other than being faster at the cost of being slightly more complex, this technique is equivalent to the one described above. For the next section, weʼll assume that it was just01, since that is the most common case. The attack doesnʼt re- ally change depending on the length of the padding. If you guess more bytes of padding correctly, that just means that there are fewer remaining bytes you will have to guess man- ually. (This will become clear once you understand the rest of the attack.) Decryptingonebyte At this point, the attacker has already successfully decrypted the last byte of the target block of ciphertext! Actually, weʼve CHAPTER 7. STREAM CIPHERS 63 decrypted as many bytes as we have valid padding; weʼre just assuming the worst case scenario where there is only a sin- gle byte. How? The attacker knows that the last byte of the decrypted ciphertext blockCi (weʼll call that byteD(Ci)[b]), XORed with the iteratively found valuerb, is01: D(Ci)[b] ⊕rb = 01 By moving the XOR operation to the other side, the attacker gets: D(Ci)[b] = 01 ⊕rb The attacker has now tricked the receiver into revealing the value of the last byte of the block cipher decryption ofCi. Decryptingsubsequentbytes Next, the attacker tricks the receiver into decrypting the next byte. Remember the previous equation, where we reasoned that the last byte of the plaintext was01: D(Ci)[b] ⊕rb = 01 Now, weʼd like to get that byte to say02, to produce anal- most valid padding: the last byte would be correct for a 2- byte PKCS#5 padding (02 02), but that second-to-last byte probably isnʼt02 yet. To do that, we XOR with01 to cancel the 01 thatʼs already there (since two XORs with the same value cancel each other out), and then we XOR with02 to get 02: D(Ci)[b] ⊕rb ⊕01 ⊕02 = 01 ⊕01 ⊕02 = 02 So, to produce a value of02 in the final position of the de- crypted plaintext, the attacker replacesrb with: r′ b = rb ⊕01 ⊕02 CHAPTER 7. STREAM CIPHERS 64 This accomplishes the goal of almost valid padding. Then, they try all possible values for the second-to-last byte (index b −1). Eventually, one of them will cause the message to have valid padding. Since we modified the random block so that the final byte of the plaintext will be02, the only byte in the second-to-last position that can cause valid padding is 02 as well. Using the same math as above, the attacker has recovered the second-to-last byte. Then, itʼs just rinse and repeat. The last two bytes are modified to create an almost-valid padding of03 03, then the third byte from the right is modified until the padding is valid, and so on. Repeating this for all the bytes in the block means the attacker can decrypt the entire block; repeating it for different blocks means the attacker can read the entire message. This attack has proven to be very subtle and hard to fix. First of all, messages should be authenticated, as well as en- crypted. That would cause modified messages to be rejected. However, many systems decrypt (and remove padding) be- fore authenticating the message; so the information about the padding being valid or not has already leaked. We will discuss secure ways of authenticating messages later in the book. You might consider just getting rid of the “invalid padding” message; declaring the message invalid without specifying why it was invalid. That turns out to only be a partial solution for systems that decrypt before authenticat- ing. Those systems would typically reject messages with an invalid paddingslightly fasterthan messages with a valid padding. After all, they didnʼt have to do the authentication step: if the padding is invalid, the message canʼt possibly be valid. An attack that leaks secret information through timing differences is called atiming attack, which is a special case of aside-channel attack: attacks on the practical implementa- tion of a cryptosystem rather than its “perfect” abstract rep- resentation. We will talk about these kinds of attacks more later in the book. CHAPTER 7. STREAM CIPHERS 65 That discrepancy was commonly exploited as well. By measuring how long it takes the recipient to reject the mes- sage, the attacker can tell if the recipient performed the au- thentication step. That tells them if the padding was correct or not, providing the padding oracle to complete the attack. The principal lesson learned here is, again, not to design your own cryptosystems. The main way to avoid this partic- ular problem is by performing constant time authentication, and authenticating the ciphertext before decrypting it. We will talk more about this in a later chapter on message au- thentication. 7.10 Nativestreamciphers In addition to block ciphers being used in a particularmode ofoperation, there are also “native”streamcipher s algorithms that are designed from the ground up to be astream cipher. The most common type ofstream cipher is called asyn- chronous stream cipher. These algorithms produce a long stream of pseudorandom bits from a secret symmetric key. This stream, called the keystream, is then XORed with the plaintext to produce the ciphertext. Decryption is the iden- tical operation as encryption, just repeated: the keystream is produced from the key, and is XORed with the ciphertext to produce the plaintext. You can see how this construction looks quite similar to a one-time pad, except that the truly random one-time pad has been replaced by a pseudorandomstream cipher. CHAPTER 7. STREAM CIPHERS 66 There are alsoasynchronous or self-synchronizing stream ciphers, where the previously produced ciphertext bits are used to produce the current keystream bit. This has the in- teresting consequence that a receiver can eventually recover if some ciphertext bits are dropped. This is generally not considered to be a desirable property anymore in modern cryptosystems, which instead prefer to send complete, au- thenticated messages. As a result, thesestream ciphers are very rare, and we donʼt talk about them explicitly in this book. Whenever someone says “stream cipher”, itʼs safe to assume they mean the synchronous kind. Historically, nativestream ciphers have had their issues. NESSIE, an international competition for new cryptographic primitives, for example, did not result in any newstream ci- phers, because all of the participants were broken before the competition ended. RC4, one of the most popular native stream ciphers, has had serious known issues for years. By comparison, some of the constructions using block ciphers seem bulletproof. Fortunately, more recently, several new cipher algo- rithms provide new hope that we can get practical, secure and performantstream ciphers. 7.11 RC4 By far the most common nativestreamcipher in common use on desktop and mobile devices is RC4. RC4 is sometimes also called ARCFOUR or ARC4, which stands forallegedRC4. While its source code has been leaked and its implementation is now well-known, RSA Security (the company that authored RC4 and still holds the RC4 trademark) has never acknowledged that it is the real algo- rithm. It quickly became popular because itʼs very simple and very fast. Itʼs not just extremely simple to implement, itʼs also extremely simple to apply. Being a synchronousstream cipher, thereʼs little that can go wrong; with a block cipher, CHAPTER 7. STREAM CIPHERS 67 youʼd have to worry about things like modes of operation and padding. Clocking in at around 13.9 cycles per byte, itʼs comparable to AES-128 in CTR (12.6 cycles per byte) or CBC (16.0 cycles per byte) modes. AES came out a few years after RC4; when RC4 was designed, the state of the art was 3DES, which was excruciatingly slow by comparison (134.5 cycles per byte inCTR mode). [Dai] Anin-depthlookatRC4 This is an optional, in-depth section. It almost certainly wonʼt help you write bet- ter software, so feel free to skip it. It is only here to satisfy your inner geekʼs curiosity. On the other hand, RC4 is incredibly simple, and it may be worth skimming this section. RC4 is, unfortunately, quite broken. To better under- stand just how broken, weʼll take a look at how RC4 works. The description requires understanding modular addition; if you arenʼt familiar with it, you may want to reviewthe ap- pendix on modular addition(page 186). Everything in RC4 revolves around a state array and two indexes into that array. The array consists of 256 bytes form- ing apermutation: that is, all possible index values occur ex- actly once as a value in the array. That means it maps ev- ery possible byte value to every possible byte value: usually different, but sometimes the same one. We know that itʼs a permutation becauseS starts as one, and all operations that modify S always swap values, which obviously keeps it a per- mutation. RC4 consists of two major components that work on two indexes i, j and the state arrayS: 1. The key scheduling algorithm, which produces an ini- tial state arrayS for a given key. CHAPTER 7. STREAM CIPHERS 68 2. The pseudorandom generator, which produces the ac- tual keystream bytes from the state arrayS which was produced by the key scheduling algorithm. The pseu- dorandom generator itself modifies the state array as it produces keystream bytes. Thekeyschedulingalgorithm The key scheduling algorithm starts with theidentity permu- tation. That means that each byte is mapped to itself. Then, the key is mixed into the state. This is done by let- ting index i iterate over every element of the state. Thej index is found by adding the current value ofj (starting at 0) with the next byte of the key, and the current state element: Once j has been found,S[i] and S[j] are swapped: This process is repeated for all the elements ofS. If you run out of key bytes, you just wrap around on the key. This explains why RC4 accepts keys from anywhere between 1 CHAPTER 7. STREAM CIPHERS 69 and 256 bytes long. Usually, 128 bit (16 byte) keys are used, which means that each byte in the key is used 16 times. Or, in Python: from itertools import cycle def key_schedule(key): s = range(256) key_bytes = cycle(ord(x) for x in key) j = 0 for i in range(256): j = (j + s[i] + next(key_bytes)) % 256 s[i], s[j] = s[j], s[i] return s Thepseudorandomgenerator The pseudorandom generator is responsible for producing pseudorandom bytes from the stateS. These bytes form the keystream, and are XORed with the plaintext to produce the ciphertext. For each indexi, it computesj = j+S[i] (j starts at 0). Then,S[i] and S[j] are swapped: To produce the output byte,S[i] and S[j] are added to- gether. Their sum is used as an index intoS; the value at CHAPTER 7. STREAM CIPHERS 70 S[S[i] + S[j]] is the keystream byteKi: We can express this in Python: def pseudorandom_generator(s): j = 0 for i in cycle(range(256)): j = (j + s[i]) % 256 s[i], s[j] = s[j], s[i] k = (s[i] + s[j]) % 256 yield s[k] Attacks This is an optional, in-depth section. It almost certainly wonʼt help you write bet- ter software, so feel free to skip it. It is only here to satisfy your inner geekʼs curiosity. The section on the attacks on RC4 is a good deal more com- plicated than RC4 itself, so you may want to skip this even if youʼve read this far. There are many attacks on RC4-using cryptosystems where RC4 isnʼt really the issue, but are caused by things like key reuse or failing to authenticate the message. We wonʼt discuss these in this section. Right now, weʼre only talking about issues specific to the RC4 algorithm itself. CHAPTER 7. STREAM CIPHERS 71 Intuitively, we can understand how an idealstreamcipher would produce a stream of random bits. After all, if thatʼs what it did, weʼd end up in a situation quite similar to that of a one-time pad. Figure 7.2: A one-time pad scheme. Figure 7.3: A synchronous stream cipher scheme. Note similarity to the one-time pad scheme. The critical differ- ence is that while the one-time padki is truly random, the keystream Ki is only pseudorandom. The streamcipher is ideal if the best way we have to attack it is to try all of the keys, a process called brute-forcing the key. If thereʼs an easier way, such as through a bias in the output bytes, thatʼs a flaw of thestream cipher. Throughout the history of RC4, people have found many such biases. In the mid-nineties, Andrew Roos noticed two such flaws: • The first three bytes of the key are correlated with the first byte of the keystream. • The first few bytes of the state are related to the key with a simple (linear) relation. CHAPTER 7. STREAM CIPHERS 72 For an idealstream cipher, the first byte of the keystream should tell me nothing about the key. In RC4, it gives me some information about the first three bytes of the key. The latter seems less serious: after all, the attacker isnʼt sup- posed to know the state of the cipher. As always, attacks never get worse. They only get better. Adi Shamir and Itsik Mantin showed that the second byte produced by the cipher istwice as likely to be zero as it should be. Other researchers showed similar biases in the first few bytes of the keystream. This sparked further re- search by Mantin, Shamir and Fluhrer, showing large bi- ases in the first bytes of the keystream. [FMS01] They also showed that knowing even small parts of the key would al- low attackers to make strong predictions about the state and outputs of the cipher. Unlike RC4, most modern stream ciphers provide a way to combine a long-term key with a nonce (a number used once), to produce multiple different keystreams from the same long-term key. RC4, by itself, doesnʼt do that. The most common approach was also the simplest: concatenate6 the long-term keyk with thenoncen: k∥n, taking advantage of RC4ʼs flexible key length require- ments. In this context, concatenation means the bits ofn are appended to the bits ofk. This scheme meant attackers could recover parts of the combined key, eventually allow- ing them to slowly recover the long-term key from a large amount of messages (around224 to 226, or tens of millions of messages). WEP, a standard for protecting wireless networks that was popular at the time, was heavily affected by this attack, because it used this simplisticnoncecombination scheme. A scheme where the long-term key and thenonce had been se- curely combined (for example using a key derivation func- tion or a cryptographic hash function) wouldnʼt have had 6 Here we use∥ as the operator for concatenation. Other common symbols for concatenation include+ (for some programming languages, such as Python) and (for formal languages). CHAPTER 7. STREAM CIPHERS 73 this weakness. Many other standards including TLS were therefore not affected. Again, attacks only get better. Andreas Klein showed more extensive correlation between the key and the keystream. [Kle08] Instead of tens of millions of messages with the Fluhrer, Mantin, Shamir attacks, attackers now only needed several tens of thousands of messages to make the attack practical. This was applied against WEP with great effect. In 2013, a team of researchers at Royal Holloway in Lon- don produced a combination of two independent practical attacks [ABP+]. These attacks proved to be very damning for RC4: while RC4ʼs weaknesses had been known for a long time, they finally drove the point home for everyone that it really shouldnʼt be used anymore. The first attack is based on single-byte biases in the first 256 bytes of the keystream. By performing statistical analy- sis on the keystreams produced by a large number of keys, they were able to analyze the already well-known biases in the early keystream bytes of RC4 in much greater detail. TODO: illustrate: http://www.isg.rhul.ac.uk/ tls/RC4_keystream_dist_2_45.txt The second attack is based on double byte biases any- where in the keystream. It turns out that adjacent bytes of the keystream have an exploitable relation, whereas in an ideal stream cipheryou would expect them to be completely independent. CHAPTER 7. STREAM CIPHERS 74 Byte pair Byte position (mod 256) i Probability (0, 0) i = 1 2−16(1 + 2 −9) (0, 0) i ̸∈{1, 255} 2−16(1 + 2 −8) (0, 1) i ̸∈{0, 1} 2−16(1 + 2 −8) (0, i + 1) i ̸∈{0, 255} 2−16(1 + 2 −8) (i + 1, 255) i ̸= 254 2−16(1 + 2 −8) (255, i + 1) i ̸∈{1, 254} 2−16(1 + 2 −8) (255, i + 2) i ̸∈{0, 253, 254, 255} 2−16(1 + 2 −8) (255, 0) i = 254 2−16(1 + 2 −8) (255, 1) i = 255 2−16(1 + 2 −8) (255, 2) i ∈{0, 1} 2−16(1 + 2 −8) (255, 255) i ̸= 254 2−16(1 + 2 −8) (129, 129) i = 2 2−16(1 + 2 −8) This table may seem a bit daunting at first. The probabil- ity expression in the rightmost column may look a bit com- plex, but thereʼs a reason itʼs expressed that way. Suppose that RC4 was a goodstream cipher, and all values occurred with equal probability. Then youʼd expect the probability for any given byte value to be2−8 since there are28 different byte values. If RC4 was a goodstream cipher, two adjacent bytes would each have probability2−8, so any given pair of two bytes would have probability2−8 ·2−8 = 2 −16. However, RC4 isnʼt an idealstream cipher, so these properties arenʼt true. By writing the probability in the2−16(1 + 2 −k) form, itʼs easier to see how much RC4 deviates from what youʼd ex- pect from an idealstream cipher. So, letʼs try to read the first line of the table. It says that when the first bytei = 1 of any 256-byte chunk from the cipher is0, then the byte following it is slightly more likely (1 + 2 −9 times as likely, to be exact) to be 0 than for it to be any other number. We can also see that when one of the keystream bytes is255, you can make many predictions about the next byte, depending on where it occurs in the keystream. Itʼs more likely to be0, 1, 2, 255, or the position in the keystream plus one or two. CHAPTER 7. STREAM CIPHERS 75 TODO: demonstrate attack success Again, attacks only get better. These attacks have pri- marily focused on the cipher itself, and havenʼt been fully optimized for practical attacks on, say, web services. The at- tacks can be greatly improved with some extra information about the plaintext youʼre attempting to recover. For exam- ple, HTTP cookies are often base-64 or hex encoded. Thereʼs no way around it: we need to stop using RC4. Fortunately, weʼve also developed many secure alternatives. The continuing advances in cryptanalysis of RC4 helped con- tribute to a sense of urgency regarding the improvement of commonly available cryptographic primitives. Throughout 2013 in particular, this led to large improvements in, for ex- ample, browser cryptography (we will discuss browser cryp- tography, notably SSL/TLS, in a later chapter). 7.12 Salsa20 Salsa20 is a newerstream cipherdesigned by Dan Bernstein. Bernstein is well-known for writing a lot of open source (public domain) software, most of which is either directly se- curity related or built with information security very much in mind. There are two minor variants of Salsa20, called Salsa20/12 and Salsa20/8, which are simply the same algorithm except with 12 and 8 rounds7 respectively, down from the original 20. ChaCha is another, orthogonal tweak of the Salsa20 cipher, which tries to increase the amount of diffusion per round while maintaining or improving performance. ChaCha doesnʼt have a “20” after it; spe- cific algorithms do have a number after them (ChaCha8, ChaCha12, ChaCha20), which refers to the number of rounds. 7 Rounds are repetitions of an internal function. Typically a number of rounds are required to make an algorithm work effectively; attacks of- ten start on reduced-round versions of an algorithm. CHAPTER 7. STREAM CIPHERS 76 Salsa20 and ChaCha are among the state of the art of modern stream ciphers. There are currently no publicly known attacks against Salsa20, ChaCha, nor against any of their recommended reduced-round variants, that break their practical security. Both cipher families are also pretty fast. For long streams, Salsa20 takes about 4 cycles per byte for the full- round version, about 3 cycles per byte for the 12-round ver- sion and about 2 cycles per byte for the 8-round version, on modern Intel processors [Ber] and modern AMD processors [Dai]. ChaCha is (on most platforms) slightly faster still. To put that into comparison, thatʼs more than three times faster than RC48, approximately three times faster than AES-CTR with a 128 bit key at 12.6 cycles per byte, and roughly in the ballpark of AESGCM mode9 with specialized hardware in- structions. Salsa20 has two particularly interesting properties. Firstly, it is possible to “jump” to a particular point in the keystream without computing all previous bits. This can be useful, for example, if a large file is encrypted, and youʼd like to be able to do random reads in the middle of the file. While many encryption schemes require the entire file to be decrypted, with Salsa20, you can just select the portion you need. Another construction that has this property is amode of operationcalled CTR mode, which weʼll talk about later. This ability to “jump” also means that blocks from Salsa20 can be computed independently of one another, allowing for encryption or decryption to work in parallel, which can increase performance on multi-core CPUs. Secondly, it is resistant to many side-channel attacks. This is done by ensuring that no key material is ever used 8 The quoted benchmarks donʼt mention RC4 but MARC4, which stands for “modified alleged RC4”. The RC4 section explains why itʼs “al- leged”, and “modified” means it throws away the first 256 bytes because of a weakness in RC4. 9 GCM modeis an authenticated encryption mode, which we will see in more detail in a later chapter. CHAPTER 7. STREAM CIPHERS 77 to choose between different code paths in the cipher, and that every round is made up of a fixed-number of constant- time operations. The result is that every block is produced with exactly the same number of operations, regardless of what the key is. Both stream ciphers are based on an ARX design. One benefit of ARX ciphers is that they are intrinsically con- stant time. There are no secret memory access patterns that might leak information, as with AES. These ciphers also perform well on modern CPU architectures without need- ing cipher-specific optimizations. They take advantage of generic vector instructions, where the CPU performs related operations on multiple pieces of data in a single instruction. As a result, ChaCha20 performance is competitive with AES on modern Intel CPUs, even though the latter has specialized hardware. Here is an example ARX operation: x ←x ⊕(y ⊞z) ≪ n To find the new value ofx, first we perform a modular addi- tion (⊞) ofy and z, then we XOR (⊕) the result with x and finally we rotate left (≪) byn bits. This is the core round primitive of Salsa20. 7.13 Nativestreamciphersversusmodesof operation Some texts only consider nativestream ciphers to bestream ciphers. This book emphasizes what the functionality of the algorithm is. Since both block ciphers in amode of operation and a nativestreamcipher take a secret key and can be used to encrypt a stream, and the two can usually replace each other in a cryptosystem, we just call both of themstream ciphers and are done with it. We will further emphasize the tight link between the two with CTR mode, a mode of operation which produces a syn- CHAPTER 7. STREAM CIPHERS 78 chronous streamcipher. While there are also modes of opera- tion (like OFB and CFB) that can produce self-synchronizing streamcipher s, these are far less common, and not discussed here. 7.14 CTRmode CTRmode, short for counter mode, is amodeofoperation that works by concatenating anonce with a counter. The counter is incremented with each block, and padded with zeroes so that the whole is as long as the block size. The resulting con- catenated string is run through a block cipher. The outputs of the block cipher are then used as the keystream. Figure 7.4: CTR mode: a singlenonce N with a zero-padded counter i is encrypted by the block cipher to produce a keystream block; this block is XORed with the plaintext block Pi to produce the ciphertext blockCi. This illustration shows a single input blockN∥00 . . . ∥i, consisting ofnonce N, current counter valuei and padding, being encrypted by the block cipherE using key k to pro- duce keystream blockSi, which is then XORed with the plain- text blockPi to produce ciphertext blockCi. Obviously, to decrypt, you do the exact same thing again, since XORing a bit with the same value twice always pro- duces the original bit:pi ⊕si ⊕si = pi. As a consequence, CTR encryption and decryption is the same thing: in both cases you produce the keystream, and you XOR either the CHAPTER 7. STREAM CIPHERS 79 plaintext or the ciphertext with it in order to get the other one. For CTR modeto be secure, it is critical thatnonces arenʼt reused. If they are, the entire keystream will be repeated, allowing an attacker to mount multi-time pad attacks. This is different from aninitialization vectorsuch as the one used by CBC. An IV has to be unpredictable. An attacker being able to predict a CTRnoncedoesnʼt really matter: with- out the secret key, they have no idea what the output of the block cipher (the sequence in the keystream) would be. Like Salsa20,CTR modehas the interesting property that you can jump to any point in the keystream easily: just incre- ment the counter to that point.The Salsa20 paragraph on this topic (page 76) explains why that might be useful. Another interesting property is that since any keystream block can be computed completely separately from any other keystream block, both encryption and decryption are very easy to compute in parallel. 7.15 Streamcipherbitflippingattacks Synchronous stream ciphers, such as nativestream ciphers or a block cipher inCTR mode, are also vulnerable to a bit flip- ping attack. Itʼs similar to CBC bit flipping attacks in the sense that an attacker flips several bits in the ciphertext, and that causes some bits to be flipped in the plaintext. This attack is actually much simpler to perform onstream ciphers than it is onCBC mode. First of all, a flipped bit in the ciphertext results in the same bit being flipped in the plain- text, not the corresponding bit in the following block. Addi- tionally, it only affects that bit; in CBC bit flipping attacks, the plaintext of the modified block is scrambled. Finally, since the attacker is modifying a sequence of bytes and not a sequence of blocks, the attacks are not limited by the spe- cific block size. In CBC bit flipping attacks, for example, an attacker can adjust a single block, but canʼt adjust the adja- cent block. CHAPTER 7. STREAM CIPHERS 80 TODO illustrate This is yet another example of why authentication has to go hand in hand with encryption. If the message is properly authenticated, the recipient can simply reject the modified messages, and the attack is foiled. 7.16 Authenticatingmodesofoperation There are other modes of operation that provide authentica- tion as well as encryption at the same time. Since we havenʼt discussed authentication at all yet, weʼll handle these later. 7.17 Remainingproblems We now have tools that will encrypt large streams of data using a small key. However, we havenʼt actually discussed how weʼre going to agree on that key. As noted in a previous chapter, to communicate betweenn people, we needn(n−1) 2 key exchanges. The number of key exchanges grows about as fast as the number of peoplesquared. While the key to be exchanged is a lot smaller now than it was with one-time pads, the fundamental problem of the impossibly large num- ber of key exchanges hasnʼt been solved yet. We will tackle that problem in the next section, where weʼll look at key ex- change protocols: protocols that allow us to agree on a secret key over an insecure medium. Additionally, weʼve seen that encryption isnʼt enough to provide security: without authentication, itʼs easy for attack- ers to modify the message, and in many flawed systems even decrypt messages. In a future chapter, weʼll discuss how to authenticate messages, to prevent attackers from modifying them. 8 Keyexchange 8.1 Description Key exchange protocols attempt to solve a problem that, at first glance, seems impossible. Alice and Bob, whoʼve never met before, have to agree on a secret value. The chan- nel they use to communicate is insecure: weʼre assuming that everything they send across the channel is being eaves- dropped on. Weʼll demonstrate such a protocol here. Alice and Bob will end up having a shared secret, only communicating over the insecure channel. Despite Eve having literally all of the information Alice and Bob send to each other, she canʼt use any of that information to figure out their shared secret. That protocol is called Diffie-Hellman, named after Whit- field Diffie and Martin Hellman, the two cryptographic pio- neers who discovered it. They suggested calling the proto- col Diffie-Hellman-Merklekey exchange, to honor the contri- butions of Ralph Merkle. While his contributions certainly deserve honoring, that term hasnʼt really caught on. For the benefit of the reader weʼll use the more common term. 81 CHAPTER 8. KEY EXCHANGE 82 Practical implementations of Diffie-Hellman rely on mathematical problems that are believed to be very complex to solve in the “wrong” direction, but easy to compute in the “right” direction. Understanding the mathematical im- plementation isnʼt necessary to understand the principle be- hind the protocol. Most people also find it a lot easier to un- derstand without the mathematical complexity. So, weʼll ex- plain Diffie-Hellman in the abstract first, without any math- ematical constructs. Afterwards, weʼll look at two practical implementations. 8.2 AbstractDiffie-Hellman In order to describe Diffie-Hellman, weʼll use an analogy based on mixing colors. We can mix colors according to the following rules: • Itʼs very easy to mix two colors into a third color. • Mixing two or more colors in different order results in the same color. • Mixing colors isone-way. Itʼs impossible to determine if, let alone which, multiple colors were used to pro- duce a given color. Even if you know it was mixed, and even if you know some of the colors used to produce it, you have no idea what the remaining color(s) were. Weʼll demonstrate that with a mixing function like this one, we can produce a secret color only known by Alice and Bob. Later, weʼll simply have to describe the concrete imple- mentation of those functions to get a concretekey exchange scheme. To illustrate why this remains secure in the face of eaves- droppers, weʼll walk through an entire exchange with Eve, the eavesdropper, in the middle. Eve is listening to all of the messages sent across the network. Weʼll keep track of ev- erything she knows and what she can compute, and end up seeing why Eve canʼt compute Alice and Bobʼs shared secret. CHAPTER 8. KEY EXCHANGE 83 To start the protocol, Alice and Bob have to agree on a base color. They can communicate that across the network: itʼs okay if Eve intercepts the message and finds out what the color is. Typically, this base color is a fixed part of the proto- col; Alice and Bob donʼt need to communicate it. After this step, Alice, Bob and Eve all have the same information: the base color. Alice and Bob both pick a random color, and they mix it with the base color. At the end of this step, Alice and Bob know their respec- tive secret color, the mix of the secret color and the base color, and the base color itself. Everyone, including Eve, knows the base color. Then, Alice and Bob both send their mixed colors over the network. Eve sees both mixed colors, but she canʼt fig- CHAPTER 8. KEY EXCHANGE 84 ure out what either of Alice and Bobʼssecret colors are. Even though she knows the base, she canʼt “un-mix” the colors sent over the network.1 At the end of this step, Alice and Bob know the base, their respective secrets, their respective mixed colors, and each otherʼs mixed colors. Eve knows the base color and both mixed colors. Once Alice and Bob receive each otherʼs mixed color, they add their own secret color to it. Since the order of the mixing doesnʼt matter, theyʼll both end up with the same se- cret. 1 While this might seem like an easy operation with black-and-white approximations of color mixing, keep in mind that this is just a failure of the illustration: our assumption was that this was hard. CHAPTER 8. KEY EXCHANGE 85 Eve canʼt perform that computation. She could finish the computation with either Alice or Bobʼs secret color, since she has both mixed colors, but she has neither of those se- cret colors. She can also try to mix the two mixed colors, which would have both Alice and Bobʼs secret colors mixed into them. However, that would have the base color in it twice, resulting in a different color than the shared secret color that Alice and Bob computed, which only has the base color in it once. CHAPTER 8. KEY EXCHANGE 86 8.3 Diffie-Hellmanwithdiscretelogarithms This section describes a practical implementation of the Diffie-Hellman algorithm, based on the discrete logarithm problem. It is intended to provide some mathematical back- ground, and requires modular arithmetic to understand. If you are unfamiliar with modular arithmetic, you can either skip this chapter, or first read themathematical background appendix (page 186). Discrete log Diffie-Hellman is based on the idea that com- puting y in the following equation is easy (at least for a com- puter): y ≡gx (mod p) However, computingx given y, g and p is believed to be very hard. This is called the discrete logarithm problem, because a similar operation without the modular arithmetic is called a logarithm. This is just a concrete implementation of the abstract Diffie-Hellman process we discussed earlier. The common base color is a large primep and the baseg. The “color mix- ing” operation is the equation given above, wherex is the input value andy is the resulting mixed value. When Alice or Bob select their random numbersrA and rB, they mix them with the base to produce the mixed num- bers mA and mB: mA ≡grA (mod p) mB ≡grB (mod p) These numbers are sent across the network where Eve can see them. The premise of the discrete logarithm problem is that it is okay to do so, because figuring outr in m ≡gr (mod p) is supposedly very hard. Once Alice and Bob have each otherʼs mixed numbers, they add their own secret number to it. For example, Bob CHAPTER 8. KEY EXCHANGE 87 would compute: s ≡(grA)rB (mod p) While Aliceʼs computation looks different, they get the same result, because (grA)rB ≡ (grB )rA (mod p). This is the shared secret. Because Eve doesnʼt haverA or rB, she can not perform the equivalent computation: she only has the base number g and mixed numbersmA ≡ grA (mod p) and mB ≡ grB (mod p) , which are useless to her. She needs eitherrA or rB (or both) to make the computation Alice and Bob do. TODO: Say something about active MITM attacks where the attacker picks smooth values to produce weak secrets? 8.4 Diffie-Hellmanwithellipticcurves This section describes a practical implementation of the Diffie-Hellman algorithm, based on the elliptic curve dis- crete logarithm problem. It is intended to provide some mathematical background, and requires a (very basic) un- derstanding of the mathematics behind elliptic curve cryp- tography. If you are unfamiliar with elliptic curves, you can either skip this chapter, or first read themathematical back- ground appendix(page 202). One of the benefits of the elliptic curve Diffie-Hellman variant is that the required key size is much, much smaller than the variant based on the discrete log problem. This is because the fastest algorithms for breaking the discrete log problem have a larger asymptotic complexity than their el- liptic curve variants. For example, the number field sieve for discrete logarithms, a state of the art algorithm for attacking discrete logarithm-based Diffie-Hellman, has time complex- ity: L h 1/3, 3 p 64/9 i CHAPTER 8. KEY EXCHANGE 88 Which is more than polynomial (but less than exponential) in the number of digits. On the other hand, the fastest algo- rithms that could be used to break the elliptic curve discrete log problem all have complexity: L [1, 1/2] = O(√n) Relatively speaking, that means that itʼs much harder to solve the elliptic curve problem than it is to solve the regular dis- crete log problem, using state of the art algorithms for both. The flip side of that is that for equivalent security levels, the elliptic curve algorithm needs much smaller key sizes [Lab] [InstitutefStandardsTechnology]2: Security level in bits Discrete log key bits Elliptic curve key bits 56 512 112 80 1024 160 112 2048 224 128 3072 256 256 15360 512 8.5 Remainingproblems Using Diffie-Hellman, we can agree on shared secrets across an insecure Internet, safe from eavesdroppers. However, while an attacker may not be able to simply get the secret from eavesdropping, an active attacker can still break the system. If such an attacker, usually called Mallory, is in between Alice and Bob, she can still perform the Diffie- Hellman protocol twice: once with Alice, where Mallory pre- tends to be Bob, and once with Bob, where Mallory pretends to be Alice. 2 These figures are actually for the RSA problem versus the equivalent elliptic curve problem, but their security levels are sufficiently close to give you an idea. CHAPTER 8. KEY EXCHANGE 89 There are two shared secrets here: one between Alice and Mallory, and one between Mallory and Bob. The at- tacker (Mallory) can then simply take all the messages they get from one person and send them to the other, they can look at the plaintext messages, remove messages, and they can also modify them in any way they choose. To make matters worse, even if one of the two partici- pants was somehow aware that this was going on, they would have no way to get the other party to believe them. After all: Mallory performed the successful Diffie-Hellman exchange with the unwitting victim, she has all the correct shared se- crets. Bob has no shared secrets with Alice, just with Mal- lory; thereʼs no way for him to prove that heʼs the legitimate participant. As far as Alice can tell, Bob just chose a few ran- dom numbers. Thereʼs no way to link any key that Bob has with any key that Alice has. Attacks like these are called MITM attacks, because the attacker (Mallory) is in between the two peers (Alice and Bob). Given that the network infrastructure that we typically use to send messages is run by many different operators, this kind of attack scenario is very realistic, and a secure cryptosystem will have to address them somehow. While the Diffie-Hellman protocol successfully pro- duced a shared secret between two peers, there are clearly some pieces of the puzzle still missing to build those cryp- tosystems. We need tools that help us authenticate Alice to Bob and vice versa, and we need tools that help guarantee message integrity, allowing the receiver to verify that the received messages are in fact the messages the sender in- tended to send. 9 Public-keyencryption 9.1 Description So far, we have only donesecret-key encryption. Suppose, that you could have a cryptosystem that didnʼt involve a single secret key, but instead had a key pair: one public key, which you freely distribute, and a private one, which you keep to yourself. People can encrypt information intended for you by us- ing your public key. The information is then impossible to decipher without your private key. This is calledpublic-key encryption. For a long time, people thought this was impossible. However, starting in the 1970s, such algorithms started ap- pearing. The first publicly available encryption scheme was produced by three cryptographers from MIT: Ron Rivest, Adi Shamir and Leonard Adleman. The algorithm they pub- lished is still the most common one today, and carries the first letters of their last names: RSA. public-key algorithms arenʼt limited to encryption. In fact, youʼve already seen apublic-key algorithmin this book that 90 CHAPTER 9. PUBLIC-KEY ENCRYPTION 91 isnʼt directly used for encryption. There are actually three related classes ofpublic-key algorithms: 1. Key exchange algorithms, such as Diffie-Hellman, which allow you to agree on a shared secret across an insecure medium. 2. Encryption algorithms, such as the ones weʼll discuss in this chapter, which allow people to encrypt without having to agree on a shared secret. 3. Signature algorithms, which weʼll discuss in a later chapter, which allow you to sign any piece of informa- tion using your private key in a way that allows anyone else to easily verify it using your public key. 9.2 Why not use public-key encryption for everything? At face value, it seems thatpublic-key encryptionalgorithms obsolete all our previous secret-key encryption algorithms. We could just use public key encryption for everything, avoiding all the added complexity of having to dokey agree- ment for our symmetric algorithms. However, when we look at practical cryptosystems, we see that theyʼre almost always hybridcryptosystems: whilepublic-keyalgorithms play a very important role, the bulk of the encryption and authentica- tion work is done by secret-key algorithms. By far the most important reason for this is performance. Compared to our speedystreamcipher s (native or otherwise), public-key encryption mechanisms are extremely slow. RSA is limited to at most its key size, which for 2048-bit means 256 bytes. Under these circumstances encryption takes 0.29 megacycles, and decryption takes a whopping 11.12 megacy- cles. [Dai] To put this into perspective, symmetric key algo- rithms work within an order of magnitude of 10 or so cycles per byte in either direction. This means it will take a sym- metric key algorithm approximately 3 kilocycles in order to CHAPTER 9. PUBLIC-KEY ENCRYPTION 92 decrypt 256 bytes, which is about 4000 times faster than the asymmetric version. The state of the art in secure symmet- ric ciphers is even faster: AES-GCM with hardware acceler- ation or Salsa20/ChaCha20 only need about 2 to 4 cycles per byte, further widening the performance gap. There are a few other problems with most practical cryp- tosystems. For example, RSA canʼt encrypt anything larger than its modulus, which is generally less than or equal 4096 bits, far smaller than the largest messages weʼd like to send. Still, the most important reason is the speed argument given above. 9.3 RSA As we already mentioned, RSA is one of the first practical public-key encryptionschemes. It remains the most common one to this day. Encryptionanddecryption RSA encryption and decryption relies on modular arith- metic. You may want to review themodulararithmeticprimer (page 186) before continuing. This section describes the simplified math problem be- hind RSA, commonly referred to as “textbook RSA”. By itself, this doesnʼt produce a secure encryption scheme. Weʼll see a secure construction called OAEP that builds on top of it in a later section. In order to generate a key, you pick two large prime num- bers p and q. These numbers have to be picked at random, and in secret. You multiply them together to produce the modulus N, which is public. Then, you pick anencryption exponent e, which is also public. Usually, this value is either 3 or 65537. Because those numbers have a small number of 1ʼs in their binary expansion, you can compute the exponen- tiation more efficiently. Put together,(N, e) is the public key. Anyone can use the public key to encrypt a messageM into CHAPTER 9. PUBLIC-KEY ENCRYPTION 93 a ciphertextC: C ≡Me (mod N) The next problem is decryption. It turns out that there is a value d, thedecryption exponent, that can turnC back intoM. That value is fairly easy to compute assuming that you know p and q, which we do. Usingd, you can decrypt the message like so: M ≡Cd (mod N) The security of RSA relies on that decryption operation be- ing impossible without knowing the secret exponentd, and that the secret exponentd is very hard (practically impos- sible) to compute from the public key(N, e). Weʼll see ap- proaches for breaking RSA in the next section. BreakingRSA Like many cryptosystems, RSA relies on the presumed dif- ficulty of a particular mathematical problem. For RSA, this is the RSA problem, specifically: to find the plaintext mes- sage M, given a ciphertextC, and public key(N, e) in the equation: C ≡Me (mod N) The easiest way we know how to do that is to factorN back into p ·q. Givenp and q, the attacker can just repeat the pro- cess that the legitimate owner of the key does during key gen- eration in order to compute the private exponentd. Fortunately, we donʼt have an algorithm that can factor such large numbers in reasonable time. Unfortunately, we also havenʼt proven it doesnʼt exist. Even more unfortunate is that there is a theoretical algorithm, called Shorʼs algo- rithm, that would be able to factor such a number in rea- sonable time on a quantum computer. Right now, quantum CHAPTER 9. PUBLIC-KEY ENCRYPTION 94 computers are far from practical, but it does appear that if someone in the future manages to build one thatʼs suffi- ciently large, RSA becomes ineffective. In this section, we have only considered a private key re- covery attack that attacks the purely abstract mathematical RSA problem by factoring the modulus. In the next section, we will see all sorts of realistic attacks on RSA that rely on flaws in theimplementation, rather than the mathematical problem stated above. Implementationpitfalls Right now, there are no known practical complete breaks against RSA. Thatʼs not to say that systems employing RSA arenʼt routinely broken. Like with most broken cryptosys- tems, there are plenty of cases where sound components, improperly applied, result in a useless system. For a more complete overview of the things that can go wrong with RSA implementations, please refer to [Bon99] and [A V96]. In this book, weʼll just highlight a few interesting ones. PKCSv1.5padding Salt Salt1 is a provisioning system written in Python. It has one major flaw: it has a module namedcrypt. Instead of reusing existing complete cryptosystems, it implements its own, using RSA and AES provided by a third party package. For a long time, Salt used a public exponent (e) of 1, which meant the encryption phase didnʼt actually do any- thing: Pe ≡P1 ≡P (mod N). This meant that the result- ing ciphertext was in fact just the plaintext. While this issue has now been fixed, this only goes to show that you probably 1 So, thereʼs Salt the provisioning system,salts the things used in bro- ken password stores, NaCl pronounced “salt” the cryptography library, and NaCl which runs native code in some browsers, and probably a bunch Iʼm forgetting. Can we stop naming things after it? CHAPTER 9. PUBLIC-KEY ENCRYPTION 95 shouldnʼt implement your own cryptography. Salt currently also supports SSH as a transport, but the aforementioned DIY RSA/AES system remains, and is at time of writing still the recommended and the default transport. OAEP OAEP, short for optimal asymmetric encryption padding, is the state of the art in RSA padding. It was introduced by Mi- hir Bellare and Phillip Rogaway in 1995. [BR95]. Its structure looks like this: The thing that eventually gets encrypted isX∥Y , which is n bits long, wheren is the number of bits ofN, the RSA modulus. It takes a random blockR thatʼs k bits long, where k is a constant specified by the standard. The message is first padded with zeroes to ben −k bits long. If you look at the above “ladder”, everything on the left half isn −k bits long, and everything on the right half isk bits long. The random block R and zero-padded messageM∥000 . . . are combined using two “trapdoor” functions,G and H. A trapdoor func- tion is a function thatʼs very easy to compute in one direc- tion and very hard to reverse. In practice, these are crypto- graphic hash functions; weʼll see more about those later. As you can tell from the diagram,G takes k bits and turns them inton −k bits, andH is the other way around, taking n −k bits and turning them intok bits. The resulting blocksX and Y are concatenated, and the result is encrypted using the standard RSA encryption prim- itive, to produce the ciphertext. CHAPTER 9. PUBLIC-KEY ENCRYPTION 96 To see how decryption works, we reverse all the steps. The recipient getsX∥Y when decrypting the message. They know k, since it is a fixed parameter of the protocol, so they can split upX∥Y into X (the firstn −k bits) andY (the final k bits). In the previous diagram, the directions are for padding being applied. Reverse the arrows on the side of the ladder, and you can see how to revert the padding: TODO: reverse arrows We want to get toM, which is inM∥000 . . .. Thereʼs only one way to compute that, which is: M∥000 . . . = X ⊕G(R) Computing G(R) is a little harder: G(R) = G(H(X) ⊕Y ) As you can see, at least for some definitions of the functions H and G, we need all ofX and all ofY (and hence the en- tire encrypted message) in order to learn anything aboutM. There are many functions that would be a good choice forH and G; based on cryptographic hash functions, which weʼll discuss in more detail later in the book. 9.4 Ellipticcurvecryptography TODO: This 9.5 Remaining problem: unauthenticated encryption Most public-key encryption schemes can only encrypt small chunks of data at a time, much smaller than the messages we want to be able to send. They are also generally quite slow, much slower than their symmetric counterparts. Therefore CHAPTER 9. PUBLIC-KEY ENCRYPTION 97 public-key cryptosystems are almost always used in conjunc- tion with secret-key cryptosystems. When we discussedstream ciphers, one of the remaining issues that we were facing was that we still had to exchange secret keys with a large number of people. With public-key cryptosystems such as public encryption andkey exchange protocols, weʼve now seen two ways that we can solve that problem. That means that we can now communicate with anyone, using only public information, completely secure from eavesdroppers. So far weʼve only been talking about encryption without any form of authentication. That means that while we can encrypt and decrypt messages, we cannot verify that the message is what the sender actually sent. While unauthenticated encryption may provide secrecy, we have already seen that without authentication an active attacker can generally modify valid encrypted messages suc- cessfully, despite the fact that they donʼt necessarily know the corresponding plaintext. Accepting these messages can often lead to secret information being leaked, meaning we donʼt even get secrecy. The CBC padding attacks weʼve al- ready discussed illustrate this. As a result it has become evident that we need ways to authenticate as well as encrypt our secret communications. This is done by adding extra information to the message that only the sender could have computed. Just like encryp- tion, authentication comes in both private-key (symmetric) and public-key (asymmetric) forms. Symmetric authenti- cation schemes are typically calledmessage authentication codes, while the public-key equivalent is typically called a sig- nature. First, we will introduce a new cryptographic primitive: hash functions. These can be used to produce both signa- ture schemes as well as message authentication schemes. Unfortunately, they are also very often abused to produce entirely insecure systems. 10 Hashfunctions 10.1 Description Hash functions are functions that take an input of indeter- minate length and produce a fixed-length value, also known as a “digest”. Simple hash functions have many applications. Hash ta- bles, a common data structure, rely on them. These sim- ple hash functions really only guarantee one thing: for two identical inputs, theyʼll produce an identical output. Impor- tantly, thereʼs no guarantee that two identical outputs imply that the inputs were the same. That would be impossible: thereʼs only a finite amount of digests, since theyʼre fixed size, but thereʼs an infinite amount of inputs. A good hash function is also quick to compute. Since this is a book on cryptography, weʼre particularly interested in cryptographic hash functions. Cryptographic hash functions can be used to build secure (symmetric) mes- sage authentication algorithms, (asymmetric) signature al- gorithms, and various other tools such as random number generators. Weʼll see some of these systems in detail in fu- ture chapters. 98 CHAPTER 10. HASH FUNCTIONS 99 Cryptographic hash functions have much stronger prop- erties than regular hash functions, such as one that you might find in a hash table. For a cryptographic hash func- tion, we want it to be impossibly hard to: 1. modify a message without changing the hash. 2. generate a message that has a given hash. 3. find two different messages with the same hash. The first property implies that cryptographic hash func- tions will exhibit something known as the “avalanche ef- fect”. Changing even a single bit in the input will produce an avalanche of changes through the entire digest: each bit of the digest will have approximately 50% chance of flip- ping. That doesnʼt mean that every changewillcause approx- imately half of the bits to flip, but the cryptographic hash function does guarantee that the odds of that happening are extremely large. More importantly it is impossibly hard to find such collisions or near-collisions. The second property, which states that it should be dif- ficult to find a messagem that has a given hash valueh, is called pre-imageresistance. This makes a hash function a one- way function: itʼs very easy to compute a hash for a given message, but itʼs very hard to compute a message for a given hash. The third property talks about finding messages with the same hash value, comes in two flavors. In the first one, thereʼs a given messagem, and it should be difficult to find another messagem′ with the same hash value: thatʼs called second pre-image resistance. The second one is stronger, stat- ing that it should be hard to find any two messagesm, m′ that have the same hash value. This is calledcollision resistance. Because collision resistance is a stronger form of second pre- image resistance, theyʼre sometimes also called weak and strong collision resistance. These concepts are often named from the point of view of an attack, rather than the resistance to an attack. For ex- CHAPTER 10. HASH FUNCTIONS 100 ample, youʼll often hear about a collision attack, which is an attack that attempts to generate a hash collision, or a second pre-image attack, which attempts to find a second pre-image that hashes to the same value as a given pre-image, et cetera. TODO: Maybe link tohttp://www.cs.ucdavis.edu/ ~rogaway/papers/relates.pdf for further reading 10.2 MD5 MD5 is a hash function designed by Ronald Rivest in 1991 as an extension of MD4. This hash function outputs 128- bit digests. Over the course of the years, the cryptographic community has repeatedly uncovered MD5ʼs weaknesses. In 1993, Bert den Boer and Antoon Bosselaers published a pa- per demonstrating “pseudo-collisions” for the compression function of MD5. [dBB93] Dobbertin expanded upon this re- search and was able to produce collisions for the compres- sion function. In 2004, based on Dobbertinʼs work, Xiaoyun Wang, Dengguo Feng, Xuejia Lai and Hongbo Yu showed that MD5 is vulnerable to real collision attacks. [LWdW05] The last straw came when Xiaoyun Wang et al. managed to gen- erate colliding X.509 certificates and then presented a distin- guishing attack on HMAC-MD5. [LWdW05] [WYW+09] Nowadays, it is not recommended to use MD5 for gen- erating digital signatures, but it is important to note that HMAC-MD5 is still a secure form of message authentication; however, it probably shouldnʼt be implemented in new cryp- tosystems. Five steps are required to compute an MD5 message di- gest: 1. Add padding. First, 1 bit is appended to the message and then 0 bits are added to the end until the length is 448 ( mod 512). 2. Fill up the remaining 64 bits with the the length of the original message modulo264, so that the entire mes- sage is a multiple of 512 bits. CHAPTER 10. HASH FUNCTIONS 101 3. Initialize the state as four 32-bit words, A, B, C and D. These are initialized with constants defined in the spec. 4. Process the input in 512 bit blocks; for each block, run four “rounds” consisting of 16 similar operations each. The operations all consist of shifts, modular addition, and a specific nonlinear function, different for each round. Once done, A∥B∥C∥D is the output of the hash. This padding style combined with the concatenation at the end is what makes MD5 vulnerable to length extension attacks; more on that later. In Python one can use the hashlib module to create an MD5 digest as follows: import hashlib hashlib.md5(b”crypto101”).hexdigest() 10.3 SHA-1 SHA-1 is another hash function from the MD4 family de- signed by the NSA, which produces a 160-bit digest. Just like MD5, SHA-1 is no longer considered secure for digital signa- tures. Many software companies and browsers, including Google Chrome, have started to retire support of the signa- ture algorithm of SHA-1. On February 23, 2017 researchers from CWI Amsterdam and Google managed to produce a col- lision on the full SHA-1 function. [SBK+] In the past methods to cause collisions on reduced versions of SHA-1 have been published, including one by Xiaoyun Wang. “The SHAppen- ing” demonstrated freestart collisions for SHA-1. A freestart collision allows one to pick the initial value known as the initialization vectorat the start of the compression function. [SKP15] Once again the hashlib Python module can be used to generate a SHA-1 hash: CHAPTER 10. HASH FUNCTIONS 102 import hashlib hashlib.sha1(b”crypto101”).hexdigest() 10.4 SHA-2 SHA-2 is a family of hash functions including SHA-224, SHA- 256, SHA-384, SHA-512, SHA-512/224 and SHA-512/256 and their digest sizes 224, 256, 384, 512, 224 and 256 respectively. These hash functions are based on the Merkle–Damgård construction and can be used for digital signatures, message authentication and random number generators. SHA-2 not only performs better than SHA-1, it also provides better se- curity, because of its increase in collision resistance. SHA-224 and SHA-256 were designed for 32-bit proces- sor registers, while SHA-384 and SHA-512 for 64-bit registers. The 32-bit register variants will therefore run faster on a 32- bit CPU and the 64-bit variants will perform better on a 64-bit CPU. SHA-512/224 and SHA-512/256 are truncated versions of SHA-512 allowing use of 64-bit words with an output size equivalent to the 32-bit register variants (i.e., 224 and 256 di- gest sizes and better performance on a 64-bit CPU). The following is a table that gives a good overview of the SHA-2 family: Hash func- tion Message size Block size Word size Digest size SHA-224 < 264 512 32 224 SHA-256 < 264 512 32 256 SHA-384 < 2128 1024 64 384 SHA-512 < 2128 1024 64 512 SHA- 512/224 < 2128 1024 64 224 SHA- 512/256 < 2128 1024 64 256 You can hash an empty string with the hashlib module CHAPTER 10. HASH FUNCTIONS 103 and compare digest sizes as follows: >>> import hashlib >>> len(hashlib.sha224(b””).hexdigest()) 56 >>> len(hashlib.sha256(b””).hexdigest()) 64 >>> len(hashlib.sha384(b””).hexdigest()) 96 >>> len(hashlib.sha512(b””).hexdigest()) 128 AttacksonSHA-2 Several (pseudo-)collision and preimage attacks have been demonstrated using SHA-256 and SHA-512 with less rounds. It is important to note that by removing a certain amount of rounds one canʼt attack the entire algorithm. For instance, Somitra Kumar Sanadhya and Palash Sarkar were able to cause collisions with SHA-256 using 24 of 64 rounds (remov- ing the last 40 rounds). [SS08] 10.5 KeccakandSHA-3 Keccak is a family of sponge functions designed by Guido Bertoni, Joan Daemen, Gilles Van Assche and Michaël Peeters, which won NISTʼs Secure Hash Algorithm Competi- tion in 2012. Keccak has since been standardized in form of the SHA3-224, SHA3-256, SHA3-384 and SHA3-512 hash func- tions. Although SHA-3 sounds like it might come from the same family as SHA-2, the two are designed very differently. SHA- 3 is very efficient in hardware [Hua], but is relatively slow in software in comparison to SHA-2. [ECR] Later in the book, you will find the security aspects of SHA-3, such as prevent- ing length extension attacks. The SHA-3 hash functions were introduced in Python ver- sion 3.6 and can be used as follows: CHAPTER 10. HASH FUNCTIONS 104 import hashlib hashlib.sha3_224(b”crypto101”).hexdigest() hashlib.sha3_256(b”crypto101”).hexdigest() hashlib.sha3_384(b”crypto101”).hexdigest() hashlib.sha3_512(b”crypto101”).hexdigest() 10.6 Passwordstorage One of the most common use cases for cryptographic hash functions, and unfortunately one which is also completely and utterly broken, is password storage. Suppose you have a service where people log in using a username and a password. Youʼd have to store the password somewhere, so that next time the user logs in, you can verify the password they supplied. Storing the password directly has several issues. Besides an obvious timing attack in the string comparison, if the password database were to be compromised, an attacker would be able to just go ahead and read all of the passwords. Since many users re-use passwords, thatʼs a catastrophic fail- ure. Most user databases also contain their e-mail addresses, so it would be very easy to hi-jack a bunch of your userʼs ac- counts that are unrelated to this service. Hashfunctionstotherescue An obvious approach would be to hash the password using a cryptographically secure hash function. Since the hash function is easy to compute, whenever the user provides their password, you can just compute the hash value of that, and compare that to what you stored in the database. If an attacker were to steal the user database, they could only see the hash values, and not the actual passwords. Since the hash function is impossible for an attacker to in- verse, they wouldnʼt be able to turn those back into the orig- inal passwords. Or so people thought. CHAPTER 10. HASH FUNCTIONS 105 Rainbowtables It turns out that this reasoning is flawed. The amount of passwords that people actually use is very limited. Even with very good password practices, theyʼre strings some- where between 10 and 20 characters, consisting mostly of things that you can type on common keyboards. In practice though, people use even worse passwords: things based on real words (password, swordfish), consisting of few sym- bols and few symbol types (1234), or with predictable mod- ifications of the above (passw0rd). To make matters worse, hash functions are the same ev- erywhere. If a user re-uses the same password on two sites, and both of them hash the password using MD5, the values in the password database will be the same. It doesnʼt even have to be per-user: many passwords are extremely com- mon (password), so many users will use the same one. Keep in mind that a hash function is easy to evaluate. What if we simply try many of those passwords, creating huge tables mapping passwords to their hash values? Thatʼs exactly what some people did, and the tables were just as effective as youʼd expect them to be, completely break- ing any vulnerable password store. Such tables are called rainbowtables. This is because theyʼre essentially sorted lists of hash function outputs. Those outputs will be more or less randomly distributed. When written down in hexadecimal formats, this reminded some people of color specifications like the ones used in HTML, e.g.#52f211, which is lime green. Salts The reason rainbow tables were so incredibly effective was because everyone was using one of a handful of hash func- tions. The same password would result in the same hash ev- erywhere. This problem was generally solved by usingsalts. By mix- CHAPTER 10. HASH FUNCTIONS 106 ing (appending or prepending1) the password with some ran- dom value before hashing it, you could produce completely different hash values out of the same hash function. It ef- fectively turns a hash function into a whole family of related hash functions, with virtually identical security and perfor- mance properties, except with completely different output values. The salt value is stored next to the password hash in the database. When the user authenticates using the password, you just combine thesalt with the password, hash it, and compare it against the stored hash. If you pick a sufficiently large (say, 160 bits/32 bytes), cryptographically randomsalt, youʼve completely defeated ahead-of-time attacks like rainbow tables. In order to suc- cessfully mount a rainbow table attack, an attacker would have to have a separate table for each of thosesalt values. Since even a single table was usually quite large, storing a large amount of them would be impossible. Even if an at- tacker would be able to store all that data, theyʼd still have to compute it first. Computing a single table takes a decent amount of time; computing2160 different tables is impossi- ble. Many systems used a singlesalt for all users. While that prevented an ahead-of-time rainbow table attack, it still allowed attackers to attack all passwords simultaneously, once they knew the value of thesalt. An attacker would simply compute a single rainbow table for thatsalt, and compare the results with the hashed passwords from the database. While this would have been prevented by using a different salt for each user, systems that use a cryptographic hash with a per-usersalt are still considered fundamentally broken today; they are justharder to crack, but not at all se- cure. 1 While you could also do this with XOR, itʼs needlessly more error- prone, and doesnʼt provide better results. Unless you zero-pad both the password and thesalt, you might be truncating either one. CHAPTER 10. HASH FUNCTIONS 107 Perhaps the biggest problem with salts is that many programmers were suddenly convinced they were doing the right thing. Theyʼd heard of broken password storage schemes, and they knew what to do instead, so they ignored all talk about how a password database could be compro- mised. They werenʼt the ones storing passwords in plain- text, or forgetting tosalt their hashes, or re-usingsalts for different users. It was all of those other people that didnʼt know what they were doing that had those problems. Un- fortunately, thatʼs not true. Perhaps thatʼs why broken pass- word storage schemes are still the norm. Modernattacksonweakpasswordsystems To a modern attack,salts quite simply donʼt help. Modern at- tacks take advantage of the fact that the hash function being used is easy to compute. Using faster hardware, in particular video cards, we can simply enumerate all of the passwords, regardless ofsalt. TODO: more concrete performance numbers about GPUs Salts may make precomputed attacks impossible, but they do very little against an attacker that actually knows the salt. One approach you might be inclined to take is to at- tempt to hide thesalt from the attacker. This typically isnʼt very useful: if an attacker can manage to access the database, attempts to hide thesalt are unlikely to be successful. Like many ineffective home-grown crypto schemes, this only pro- tects against an incredibly improbable event. It would be much more useful to just use a good password store to begin with, than trying to fix a broken one. Sowheredowegofromhere? In order to protect passwords, you need a (low-entropy)key derivation function (page 137). Weʼll discuss them in more detail in a future chapter. CHAPTER 10. HASH FUNCTIONS 108 While key derivation functions can be built using cryp- tographic hash functions, they have very different perfor- mance properties. This is a common pattern: while crypto- graphic hash functions are incredibly important primitives for building secure tools (such as key derivation functions or message authentication algorithms), they are routinely abused as those tools themselves. In the rest of this chap- ter, we will see other examples of how cryptographic hash functions can be used and abused. 10.7 Lengthextensionattacks In many hash functions, particularly the previous genera- tions, the internal state kept by the hash function is used as the digest value. In some poorly engineered systems, that causes a critical flaw: if an attacker knowsH(M1), itʼs very simple to compute H(M1∥M2), without actually knowing the value ofM1. Since you knowH(M1), you know the state of the hash function after itʼs hashedM1. You can use that to reconstruct the hash function, and ask it to hash more bytes. Setting the hash functionʼs internal state to a known state you got from somewhere else (such asH(M1)) is calledfixa- tion. For most real-world hash functions, itʼs a little bit more complicated than that. They commonly have a padding step that an attacker needs to recreate. MD5 and SHA-1 have the same padding step. Itʼs fairly simple, so weʼll go through it: 1. Add a 1 bit to the message. 2. Add zero bits until the length is448 ( mod 512). 3. Take the total length of the message, before padding, and add it as a 64-bit integer. For the attacker to be able to computeH(M1∥M2) given H(M1), the attacker needs to fake that padding, as well. The attacker will actually computeH(M1∥G∥M2), whereG is the CHAPTER 10. HASH FUNCTIONS 109 glue padding, called that way because itglues the two mes- sages together. The hard part is knowing the length of the message M1. In many systems, the attacker can actually make fairly educated guesses about the length ofM1, though. As an ex- ample, consider the common (broken) example of a secret- prefix authentication code. People send messagesMi, au- thenticated usingAi = H(S∥Mi), whereS is a shared secret. Weʼll see (and break) this MAC algorithm in a future section. Itʼs very easy for the recipient to compute the same func- tion, and verify the code is correct. Any change to the mes- sage Mi will change the value ofAi drastically, thanks to the avalanche effect. Unfortunately, itʼs quite easy for attackers to forge messages. Since the MAC is usually sent together with the original message, the attacker knows the length of the original message. Then, the attacker only has to guess at the length of the secret, which is often fixed as part of the protocol, and, even if it isnʼt, the attacker will probably get in a hundred tries or less. Contrast this with guessing the secret itself, which is impossible for any reasonably chosen secret. There are secure authentication codes that can be de- signed using cryptographic hash functions: this one just isnʼt it. Weʼll see better ones in a later chapter. Some hash functions, particularly newer ones such as SHA-3 competition finalists, do not exhibit this property. The digest is computed from the internal state, instead of using the internal state directly. This makes the SHA-3-era hash functions not only a bit more fool-proof, but also enables them to produce simpler schemes for message authentication. (Weʼll elaborate on those in a later chapter.) While length extension attacks only affected systems where cryptographic hash functions were being abused in the first place, thereʼs something to be said for preventing them anyway. People will end up making mis- takes, we might as well mitigate where we can. TODO: say why this prevents meet in the middle attacks? CHAPTER 10. HASH FUNCTIONS 110 10.8 Hashtrees Hash trees are trees2 where each node is identified by a hash value, consisting of its contents and the hash value of its ancestor. The root node, not having an ancestor, simply hashes its own contents. This definition is very wide: practical hash trees are often more restricted. They might be binary trees 3, or perhaps only leaf nodes carry data of their own, and par- ent nodes only carry derivative data. Particularly these re- stricted kinds are often called Merkle trees. Systems like these or their variants are used by many sys- tems, particularly distributed systems. Examples include distributed version control systems such as Git, digital cur- rencies such as Bitcoin, distributed peer-to-peer networks like Bittorrent, and distributed databases such as Cassandra. 10.9 Remainingissues Weʼve already illustrated that hash functions, by themselves, canʼt authenticate messages, because anyone can compute them. Also, weʼve illustrated that hash functions canʼt be used to secure passwords. Weʼll tackle both of these prob- lems in the following chapters. While this chapter has focused heavily on what hash functions can’t do, it canʼt be stressed enough that they are still incredibly important cryptographic primitives. They just happen to be commonlyabused cryptographic primi- tives. 2 Directed graphs, where each node except the root has exactly one ancestor. 3 Each non-leaf node has no more than two children 11 Messageauthentication codes 11.1 Description A MAC is a small bit of information that can be used to check the authenticity and the integrity of a message. These codes are often called “tags”. A MAC algorithm takes a message of arbitrary length and a secret key of fixed length, and pro- duces the tag. The MAC algorithm also comes with a verifi- cation algorithm that takes a message, the key and a tag, and tells you if the tag was valid or not. (It is not always sufficient to just recompute a tag and check if they are the same; many secure MAC algorithms are randomized, and will produce different tags every time you apply them.) Note that we say “message” here instead of “plaintext” or “ciphertext”. This ambiguity is intentional. In this book weʼre mostly interested in MACs as a way to achieve authenti- cated encryption, so the message will always be a ciphertext. That said, thereʼs nothing wrong with a MAC being applied to a plaintext message. In fact, we will be seeing examples 111 CHAPTER 11. MESSAGE AUTHENTICATION CODES 112 of secure authenticated encryption schemes that explicitly allow for authenticated (but not encrypted) information to be sent along with the authenticated ciphertext. Often, when you just want to talk about the authenticity and integrity of a particular message, it may be more prac- tical to use asignature algorithm, which weʼll talk about in a later chapter. For now, all you need to know is that the term “signature” is normally reserved for asymmetric algorithms, whereas this chapter deals with symmetric algorithms. SecureMACs We havenʼt quite defined yet exactly which properties we want from a secure MAC. We will be defending against an active attacker. The attacker will be performing achosen message attack. That means that an attacker will ask us the tag for any number of messagesmi, and weʼll answer truthfully with the appro- priate tagti. An attacker will then attempt to produce anexistential forgery, a fancy way of saying that they will produce some new valid combination of(m, t). The obvious target for the attacker is the ability to produce valid tagst′ for new mes- sages m′ of their choosing. We will also consider the MAC insecure if an attacker can compute a new, different valid tag t′ for a messagemi that we previously gave them a valid tag for. WhydoesaMACtakeasecretkey? If youʼve had to deal with verifying the integrity of a mes- sage before, you may have used checksums (like CRC32 or Adler32) or even cryptographic hashes (like the SHA family) in order to compute a checksum for the message (depending on the algorithm and who youʼre talking to, they may have called it “hash” or “digest”, too). Letʼs say that youʼre distributing a software package. You have some tarballs with source code in them, and maybe CHAPTER 11. MESSAGE AUTHENTICATION CODES 113 some binary packages for popular operating systems. Then you put some (cryptographically secure!) hashes right next to them, so that anyone who downloads them can verify the hashes and be confident that they downloaded what they think they downloaded. Of course, this scheme is actually totally broken. Com- puting those hashes is something everyone can do. Youʼre even relying on that fact for your user to be able to verify their download. That also means that an attacker that mod- ified any of the downloads can just compute the hash again for the modified download and save that value. A user down- loading the modified file will compute its hash and com- pare it against the modified hash, and conclude that the download worked. The scheme provided no help whatso- ever against an attacker modifying the download, either as stored, or in transit. In order to do this securely, you would either apply a sig- nature algorithm to the binaries directly, or by signing the digests, as long as the hash function used to produce the di- gest is secure against second-preimage attacks. The impor- tant difference is that producing a signature (using either a pre-shared key with your users, or, preferably, a public-key signature algorithm) isnot something that an attacker can do. Only someone who has the secret keys can do that. 11.2 CombiningMACandmessage As weʼve mentioned before, unauthenticated encryption is bad. Thatʼs why we introduced MACs. Of course, for a MAC to be useful, it has to make it to the recipient. Since weʼre ex- plicitly talking about authenticating encryption, now, weʼll stop using the word “message” and instead use the less am- biguous “plaintext” and “ciphertext”. There are three common ways to combine a ciphertext with a MAC. 1. Authenticate and encrypt. You authenticate and en- CHAPTER 11. MESSAGE AUTHENTICATION CODES 114 crypt the plaintext separately. This is how SSH does it. In symbols:C = E(KC, P ), t = MAC (KM , P ), and you send both ciphertextC and tagt. 2. Authenticate, then encrypt. You authenticate the plaintext and then encrypt the combination of the plaintext and the authentication tag. This is how TLS usually does it. In symbols:t = MAC (KM , P ), C = E(KC, P ∥t), and you only sendC. (You donʼt need to send t, because itʼs already an encrypted part ofC.) 3. Encrypt, then authenticate. You encrypt the plaintext, compute the MAC of that ciphertext. This is how IPSec does it. In symbols:C = E(KC, P ), t = MAC (KM , C), and you send bothC and t. All of these options were studied and compared exten- sively. [Kra01] [BN07] We now know that out of all of these, encrypt-then-authenticate is unequivocally the best option. Itʼs so emphatically the best option that Moxie Marlinspike, a well-respected information security researcher, has a prin- ciple called “The Cryptographic Doom Principle” for any system that does not follow this pattern [Mar11]. Moxie claims that any system that does anything before checking the MAC is doomed. Both authenticate-and-encrypt and authenticate-then-encrypt require you to decrypt something before you can verify the authentication. Authenticate-then-encrypt Authenticate-then-encrypt is a poor choice, but itʼs a subtle poor choice. It can still be provably secure, but only under certain conditions. [Kra01] At first sight, this scheme appears to work. Sure, you have to decrypt before you can do anything, but to many cryptographers, including the designers of TLS, this did not appear to pose a problem. In fact, prior to rigorous comparative study of differ- ent composition mechanisms, many preferred this setup. CHAPTER 11. MESSAGE AUTHENTICATION CODES 115 In a critique of IPSec, Schneier and Ferguson, two vet- eran cryptographers, considered IPSecʼs use of encrypt- then-authenticate was a flaw, preferring TLSʼs authenticate- then-encrypt. [FS99] While they may have had a plausible (albeit mostly heuristic) argument for the time, this criti- cism is completely superseded by theprovable security of encrypt-then-authenticate schemes. [Kra01] [BN07] TODO: Explain Vaudenay CBC attack [Vau] Authenticate-and-encrypt Authenticate-and-encrypt has some serious problems. Since the tag authenticates the plaintext and that tag is part of the transmitted message, an attacker will be able to recognize two plaintext messages are the same because their tags will also be the same. This essentially leads to the same problem we saw with ECB mode, where an attacker can identify identical blocks. Thatʼs a serious problem, even if they canʼt decrypt those blocks. TODO: Explain how this works in SSH (see Moxieʼs Doom article) 11.3 Anaiveattemptwithhashfunctions Many ways of constructing MACs involve hash functions. Perhaps one of the simplest ways you could imagine doing that is to just prefix the message with the secret key and hash the whole thing: t = H(k∥m) This scheme is most commonly called “Prefix-MAC”, be- cause it is a MAC algorithm that works by using the secret key as a prefix. The cryptographically secure hash functionH guaran- tees a few things that are important to us here: CHAPTER 11. MESSAGE AUTHENTICATION CODES 116 • The tag t will be easy to compute; the hash function H itself is typically very fast. In many cases we can compute the common key part ahead of time, so we only have to hash the message itself. • Given any number of tags, there is no way for an at- tacker to “invert” the hash function to recoverk, which would allow them to forge arbitrary messages. • Given any number of tags, there is no way for an at- tacker to “rewind” the hash function to recoverH(k), which may allow them to forgealmost arbitrary mes- sages. One small caveat: weʼre assuming that the secret keyk has enough entropy. Otherwise, we have the same issue that we had for password storage using hash functions: an at- tacker could just try every singlek until one of them matches. Once theyʼve done that, theyʼve almost certainly found the correct k. Thatʼs not really a failure of the MAC though: if your secret key contains so little entropy that itʼs feasible for an attacker to try all of them, youʼve already lost, no matter which MAC algorithm you pick. Breakingprefix-MAC Despite being quite common, this MAC is actually com- pletely insecure for most (cryptographically secure!) hash functions H, including SHA-2. As we saw in the chapter on hash functions, many hash functions, such as MD5, SHA-0, SHA-1 and SHA-2, pad the message with a predictable padding before producing the output digest. The output digest is the same thing as the internal state of the hash function. Thatʼs a problem: the attacker can use those properties to forge messages. First, they use the digest as the internal state of the hash function. That state matches the state you get when you hash k∥m∥p, wherek is the secret key,m is the message, andp is CHAPTER 11. MESSAGE AUTHENTICATION CODES 117 that predictable padding. Now, the attacker gets the hash function to consume some new bytes: the attackerʼs chosen message m′. The internal state of the hash function is now what you get when you feed itk∥m∥p∥m′. Then, the attacker tells the hash function to produce a digest. Again, the hash function appends a padding, so weʼre now atk∥m∥p∥m′∥p′. The attacker outputs that digest as the tag. That isexactly the same thing as what happens when you try to compute the tag for the messagem∥p∥m′ under the secret keyk. So, the attacker has successfully forged a tag for a new message, and, by our definition, the MAC is insecure. This attack is called a length extension attack, because you are extending a valid message. The padding in the mid- dle p, which started out as the padding for the original mes- sage but has become just some data in the middle, is called gluepadding, because it glues the original messagem and the attackerʼs messagem′ together. This attack might sound a little academic, and far from a practical problem. We may have proven that the MAC is insecure by our definition, but the only tags the attacker can successfully forge are for very limited modifications of real messages. Specifically, the attacker can only forge tags for a message that consists of a message we sent, followed by some binary junk, followed by something the attacker chooses. However, it turns out that for many systems, this is plenty to result in real breaks. Consider the following Python code that parses a sequence of key-value pairs that look likek1=v1&k2=v2&...:1 def parse(s): pairs = s.split(”&”) parsed = {} for pair in pairs: key, value = pair.split(”=”) (continues on next page) 1 I realize there are briefer ways to write that function. I am trying to make it comprehensible to most programmers; not pleasing to advanced Pythonistas. CHAPTER 11. MESSAGE AUTHENTICATION CODES 118 (continued from previous page) parsed[key] = value return parsed The parsing function only remembers the last value for a given key: previous values in the dictionary are overwritten. As a result, an attacker mounting a length extension attack can effectively control the parsed dictionary entirely. If youʼre thinking that this code has many issues; sure, it does. For example, it doesnʼt handle escaping correctly. But even if it did, that wouldnʼt really fix the length extension at- tack problem. Most parsing functions will perfectly happily live with that binary junk in the middle. Hopefully it con- vinces you that there is in fact a pretty good chance that an attacker can produce messages with valid tags that say some- thing entirely different from what you intended. The prefix-MAC construction is actually secure with many current (SHA-3-era) hash functions, such as Keccak and BLAKE(2). The specifications for these hash functions even recommend it as a secure and fast MAC. They use various techniques to foil length extension attacks: for ex- ample, BLAKE keeps track of the number of bits that have been hashed so far, while BLAKE2 has a finalization flag that marks a specific block as the last. Variants Issues with prefix-MAC has tempted people to come up with all sorts of clever variations. For example, why not add the key to the end instead of the beginning (t = H(m∥k), or “suffix-MAC”, if you will)? Or maybe we should append the key to both ends for good measure (t = H(k∥m∥k), “sandwich-MAC” perhaps?)? For what itʼs worth, both of these are at least better than prefix-MAC, but both of these have serious issues. For exam- ple, a suffix-MAC system is more vulnerable to weaknesses in the underlying hash function; a successful collision attack CHAPTER 11. MESSAGE AUTHENTICATION CODES 119 breaks the MAC. Sandwich-MAC has other, more complex is- sues. Cryptography has produced much stronger MACs, which weʼll see in the next few sections. There are no good reasons not to use them. 11.4 HMAC HMAC is a standard to produce a MAC with a cryptographic hash function as a parameter. It was introduced in 1996 in a paper by Bellare, Canetti and Krawczyk. Many protocols at the time implemented their own attempt at message authen- tication using hash functions. Most of these attempts failed. The goal of that paper specifically was to produce a provably secure MAC that didnʼt require anything beyond a secret key and a hash function. One of the nice features of HMAC is that it has a fairly strong security proof. As long as the underlying hash func- tion is a pseudorandom function, HMAC itself is also a pseu- dorandom function. The underlying hash function doesnʼt even have to be collision resistant for HMAC to be a secure MAC. [Bel06] This proof was introduced after HMAC itself, and matched real-world observations: even though MD5 and to a lesser extent SHA-0 had serious collision attacks, HMAC constructions built from those hash functions still ap- peared to be entirely secure. The biggest difference between HMAC and prefix-MAC or its variants is that the message passes through a hash func- tion twice, and is combined with the key before each pass. Visually, HMAC looks like this: The only surprising thing here perhaps are the two con- stants pinner (the inner padding, one hash functionʼs block length worth of0x36 bytes) andpouter (the outer padding, one block length worth of0x5c bytes). These are necessary for the security proof of HMAC to work; their particular val- ues arenʼt very important, as long as the two constants are different. CHAPTER 11. MESSAGE AUTHENTICATION CODES 120 The two pads are XORed with the key before use. The re- sult is either prepended to the original message (for the in- ner paddingpinner) or to the intermediate hash output (for the outer paddingpouter). Because theyʼre prepended, the internal state of the hash function after processing the pre- fixes can be computed ahead of time, shaving a few cycles off the MAC computation time. 11.5 One-timeMACs So far, weʼve always assumed that MAC functions can be used with a single key to produce secure MACs for a very large number of messages. By contrast,one-time MACs are CHAPTER 11. MESSAGE AUTHENTICATION CODES 121 MAC functions that can only securely be used once with a single key. That might sound like a silly idea, since weʼve already talked about regular secure MACs. An algorithm that only works once just seems objectively worse. However, they have several big advantages: • They can be incredibly fast to evaluate, even for very large messages. • They have a compelling security proof based on the in- formation content of the tag. • A construction exists to turn aone-time MACinto a se- cure multiple-use MAC, removing the principal prob- lem. A typical simple example of suchone-time MACs consists of a simple multiplication and addition modulo some large prime p. In this case, the secret key consists of two truly random numbersa and b, both between 1 andp. t ≡m ·a + b (mod p) This simple example only works for one-block messagesm, and some primep slightly bigger than the biggestm. It can be extended to support bigger messagesM consisting of blocks mi by using a message-specific polynomialP: t ≡(mn ·an + ··· + m1 ·a)\| {z } P(M,a) +b (mod p) This might look like a lot of computation, but this polyno- mial can be efficiently evaluated by iteratively factoring out the common factora (also known as Hornerʼs rule): P(M, a) ≡a ·(a ·(a ·(···) + m2) + m1) + b (mod p) By computing each multiplication modulop, the numbers will remain conveniently small. CHAPTER 11. MESSAGE AUTHENTICATION CODES 122 In many ways, aone-time MACis to authentication what a one-time pad is to encryption. The security argument is similar: as long as the key is only used once, an attacker learns no information about the key or the message, because they are being irreversibly mixed. This demonstrates that the MAC is secure against attackers trying to produce exis- tential forgeries, even when that attacker has infinite com- putational power. Also like a one-time pad, the security argument relies on two very important properties about the keysa, b: • They have to be truly random. • They have to be used at most once. Re-usinga andb Weʼll illustrate that our example MAC is insecure if it is used to authenticate two messagesm1, m2 with the same key (a, b): t1 ≡m1 ·a + b (mod p) t2 ≡m2 ·a + b (mod p) An attacker can reconstructa, b with some simple modular arithmetic:2 t1 −t2 ≡(m1 ·a + b) −(m2 ·a + b) ( mod p) ⇓(remove parentheses) t1 −t2 ≡m1 ·a + b −m2 ·a −b (mod p) ⇓(b and −b cancel out) t1 −t2 ≡m1 ·a −m2 ·a (mod p) ⇓(factor outa) t1 −t2 ≡a ·(m1 −m2) ( mod p) ⇓(flip sides, multiply by inverse of(m1 −m2)) a ≡(t1 −t2)(m1 −m2)−1 (mod p) 2 For a refresher on modular arithmetic, including an explanation of the modular inverse, please refer tothe appendix(page 186). CHAPTER 11. MESSAGE AUTHENTICATION CODES 123 Plugging a into either the equation fort1 or t2 gets b: t1 ≡m1 ·a + b (mod p) ⇓(reorder terms) b ≡t1 −m1 ·a (mod p) As you can see, as with one-time pads, re-using the key even once leads to a complete failure of the cryptosystem to preserve privacy or integrity, as the case may be. As a re- sult, one-time MACs are a bit dangerous to use directly. For- tunately, this weakness can be solved with a construction called aCarter-Wegman MAC, which weʼll see in the next sec- tion. 11.6 Carter-WegmanMAC As weʼve already stated, the obvious problem withone-time MACs is their limited practicality. Fortunately, it turns out that there is a construction, called aCarter-Wegman MAC, that turns any secure one-time MAC into a secure many-time MAC while preserving most of the performance benefit. The idea behind aCarter-WegmanMAC is that you can use a one-time MACO to produce a tag for the bulk of the data, and then encrypt anonce n with a pseudorandom function F, such as a block cipher, to protect that one-time tag: CW ((k1, k2), n, M) = F(k1, n) ⊕O(k2, M) As long asF is a secure pseudorandom function, thenonceʼs encryption is totally unpredictable. In the eyes of an at- tacker, that means the XOR operation will randomly flip the bits of theone-time MACtag O(k2, M). Because this masks the real value of theone-time MACtag, the attacker can not perform the algebraic tricks we saw forone-timeMACs recov- ering the key when it is used more than once. Keep in mind that whileCarter-Wegman MACs take two distinct keysk1 and k2, and thatCarter-Wegman MACs are re- lated toone-time MACs, some of which also take two distinct CHAPTER 11. MESSAGE AUTHENTICATION CODES 124 keys a and b, they are not the same two keys. The Carter- Wegman MACʼsk2 is the only key passed to the fastone-time MACO. If that fastone-time MACis our earlier example that takes two keysa and b, thatk2 would have to get split up into those two keys. TheCarter-Wegman MACkey would then be (k1, k2) = ( k1, (a, b)). You can tell how aCarter-Wegman MACexploits the ben- efits of both kinds of MACs by considering the two terms of the equation separately. InF(k1, n), F is just a regular pseu- dorandom function, such as a block cipher. It is quite slow by comparison to the one-time MAC. However, its input, the nonce, is very small. The unpredictable output of the block cipher masks the output of theone-time MAC. In the second term, O(k2, M), the large input messageM is only handled by the very fastone-time MACO. These constructions, in particular Poly1305-AES, cur- rently represent some of the state of the art in MAC func- tions. The paper ([BHK+99]) and RFC ([BHK+]) for an older, related MAC function called UMAC may also be good sources of extra background information, since they go into ex- tensive details of the hows and whys of a practicalCarter- Wegman MAC. 11.7 Authenticatedencryptionmodes So far, weʼve always clearly distinguished encryption from authentication, and explained the need for both. The major- ity of secure connections that are set up every day have that distinction as well: they treat encryption and authentication as fundamentally different steps. Alternatively, we could make authentication a funda- mental part of themode of operation. After all, weʼve already seen that unauthenticated encryption is virtually never what you want; it is, at best, something you occasionally have to live with. It makes sense to use constructions that not only guarantee the privacy of an arbitrary stream, but also its in- tegrity. CHAPTER 11. MESSAGE AUTHENTICATION CODES 125 As weʼve already seen, many of the methods of compos- ing authentication and encryption are inherently insecure. By doing that in a fixed, secure way such as a properly de- signed authenticated encryption mode, an application de- veloper no longer has to make that choice, which means they also canʼt inadvertently make thewrong choice. AEAD AEAD is a feature of certain modes of authenticated encryp- tion. Such modes of operation are calledAEAD modes. It starts with the premise that many messages actually consist of two parts: • The actual content itself • Metadata: dataabout the content In many cases the metadata should be plaintext, but the content itself should be encrypted. The entire message should be authenticated: it should not be possible for an at- tacker to mess with the metadata and have the resulting mes- sage still be considered valid. Consider an e-mail alternative as an example cryptosys- tem. The metadata about the content might contain the in- tended recipient. We definitely want to encrypt and authen- ticate the content itself, so that only the recipient can read it. The metadata, however, has to be in plaintext: the e- mail servers performing the message delivery have to know which recipient to send the message to. Many systems would leave this metadata unauthenti- cated, allowing attackers to modify it. In our case, that looks like it may just lead to messages being delivered to the wrong inbox. That also means that an attacker can force e-mail to be delivered to the wrong person, or not delivered at all. AEAD modes address this issue by providing a specified way to add metadata to encrypted content, so that the whole of the encrypted content and the metadata is authenticated, and not the two pieces separately: CHAPTER 11. MESSAGE AUTHENTICATION CODES 126 11.8 OCBmode This is an optional, in-depth section. It almost certainly wonʼt help you write bet- ter software, so feel free to skip it. It is only here to satisfy your inner geekʼs curiosity. Usually, you will want to use a much more high level cryp- tosystem, such as OpenPGP, NaCl or TLS. OCB modeis anAEAD modeof operation. It is one of the earliest developedAEAD modes. As you can see, most of this scheme looks quite similar to ECB mode. The name OCB is quite similar to electronic code- book, as well. OCB does not share the security issues ECB mode has, however, as there are several important differ- ences, such as the offsets∆i introduced in each individual block encryption. Being anAEAD mode, OCB modeprovides a cryptograph- ically secure authentication tagt, which is built fromX, a very simple (not cryptographically secure by itself) check- sum of the plaintext. There is also another, separate tagta, which authenticates the AEAD associated data. That associ- ated data tagta is computed as follows: This design has a number of interesting properties. For example, it is very fast: only requiring roughly one block cipher operation per encrypted or associate data block, as well as one additional block cipher operation for the final tag. The offsets (∆i) are also extremely easy to compute. The CHAPTER 11. MESSAGE AUTHENTICATION CODES 127 checksum blockX is just all of the plaintext blocksPi XORed together. Finally, OCB mode is easy to compute in parallel; only the final authentication tag is dependent on all the pre- ceding information. OCB modealso comes with a built-in padding scheme: it behaves slightly differently when the plaintexts or authen- tication text is not exactly a multiple of the block size. This means that, unlike with PKCS#5/PKCS#7 padding, there isnʼt an entire block of “wasted” padding if the plaintext happens to be a multiple of the block size. Despite having several interesting properties going for it, OCB mode has not received as much attention as some of the alternatives; one of the main reasons being that it is patent encumbered. Even though a number of patent li- censes are available, including a free-of-charge one for open source software, this does not appear to have significantly impacted how muchOCB modeis used in the field. [Rog] CHAPTER 11. MESSAGE AUTHENTICATION CODES 128 11.9 GCMmode This is an optional, in-depth section. It almost certainly wonʼt help you write bet- ter software, so feel free to skip it. It is only here to satisfy your inner geekʼs curiosity. Usually, you will want to use a much more high level cryp- tosystem, such as OpenPGP, NaCl or TLS. GCM mode is an AEAD mode with an unfortunate case of RAS (redundant acronym syndrome) syndrome: GCM it- self stands for “Galois Counter Mode”. It is formalized in a NIST Special Publication [gcm07] and roughly boils down to a combination of classical CTR mode with aCarter-Wegman CHAPTER 11. MESSAGE AUTHENTICATION CODES 129 MAC. That MAC can be used by itself as well, which is called GMAC. Authentication GCM mode(and by extensionGMAC) 12 Signaturealgorithms 12.1 Description A signature algorithm is the public-key equivalent of a mes- sage authentication code. It consists of three parts: 1. a key generation algorithm, which can be shared with other public-key algorithms 2. a signature generation algorithm 3. a signature verification algorithm Signature algorithms can be built using encryption algo- rithms. Using the private key, we produce a value based on the message, usually using a cryptographic hash function. Anyone can then use the public key to retrieve that value, compute what the value should be from the message, and compare the two to verify. The obvious difference between this andpublic-key encryption is that in signing, the private key is used to produce the message (in this case the signa- ture) and the public key is used to interpret it, which is the opposite of how encryption and decryption work. 130 CHAPTER 12. SIGNATURE ALGORITHMS 131 The above explanation glosses over many important de- tails. Weʼll discuss real schemes in more detail below. 12.2 RSA-basedsignatures PKCS#1v1.5 TODO (see #48) PSS TODO (see #49) 12.3 DSA The Digital Signature Algorithm (DSA) is a US Federal Gov- ernment standard for digital signatures. It was first pro- posed by the National Institute of Standards and Technology (NIST) in 1991, to be used in the Digital Signature Standard (DSS). The algorithm is attributed to David W . Kravitz, a for- mer technical advisor at the NSA. DSA key generation happens in two steps. The first step is a choice of parameters, which can be shared between users. The second step is the generation of public and pri- vate keys for a single user. Parametergeneration We start by picking an approved cryptographic hash func- tion H. We also pick a key lengthL and a prime lengthN. While the original DSS specified thatL be between 512 and 1024, NIST now recommends a length of 3072 for keys with a security lifetime beyond 2030. AsL increases, so shouldN. Next we choose a primeq of length N bits; N must be less than or equal to the length of the hash output. We also pick anL-bit primep such thatp −1 is a multiple ofq. CHAPTER 12. SIGNATURE ALGORITHMS 132 The last part is the most confusing. We have to find a number g whose multiplicative order(page 201) (mod p) is q. The easy way to do this is to setg ≡2(p−1)/q (mod p). We can try another number greater than 2, and less thanp −1, if g comes out to equal 1. Once we have parameters(p, q, g), they can be shared be- tween users. Keygeneration Armed with parameters, itʼs time to compute public and pri- vate keys for an individual user. First, select a randomx with 0 < x < q . Next, calculatey where y ≡gx (mod p). This de- livers a public key(p, q, g, y ), and private keyx. Signingamessage In order to sign a message, the signer picks a randomk be- tween 0 andq. Picking thatk turns out to be a fairly sensitive and involved process; but weʼll go into more detail on that later. Withk chosen, they then compute the two parts of the signature r, s of the messagem: r ≡(gk (mod p)) ( mod q) s ≡k−1(H(m) + xr) ( mod q) If either of these happen to be 0 (a rare event, with 1 inq odds, andq being a pretty large number), pick a differentk. TODO: Talk about k-1, the modular inverse (see #52) Verifyingasignature Verifying the signature is a lot more complex. Given the mes- sage m and signature(r, s): w ≡s−1 (mod q) u1 ≡wH(m) ( mod q) CHAPTER 12. SIGNATURE ALGORITHMS 133 u2 ≡wr (mod q) v ≡(gu1yu2 (mod p)) ( mod q) If the signature is valid that final resultv will be equal tor, the second part of the signature. Thetroublewith k While there is nothing wrong with DSA done right, itʼs very easy to get it wrong. Furthermore, DSA is quite sensitive: even a small implementation mistake results in a broken scheme. In particular, the choice of the signature parameterk is critical. The requirements for this number are among the strictest of all random numbers in cryptographic algorithms. For example, many algorithms require anonce. Anonce just has to be unique: you can use it once, and then you can never use it again. It doesnʼt have to be secret. It doesnʼt even have to be unpredictable. Anonce can be implemented by a simple counter, or a monotonic clock. Many other al- gorithms, such asCBC mode, use an initialization vector. It doesnʼt have to be unique: it only has to be unpredictable. It also doesnʼt have to be secret: initialization vectors are typi- cally tacked on to the ciphertext. DSAʼs requirements for the k value are a combination of all of these: • It has to be unique. • It has to be unpredictable. • It has to be secret. Muddle with any of these properties, and an attacker can probably retrieve your secret key, even with a modest amount of signatures. For example, an attacker can recover the secret key knowing only a few bits ofk, plus a large amount of valid signatures. [NS00] CHAPTER 12. SIGNATURE ALGORITHMS 134 It turns out that many implementations of DSA donʼt even get the uniqueness part right, happily reusingk val- ues. That allows a direct recovery of the secret key using basic arithmetic. Since this attack is much simpler to under- stand, very commonly applicable, and equally devastating, weʼll discuss it in detail. Suppose that an attacker sees multiple signatures(ri, si), for different messagesmi, all with the samek. The attacker picks any two signatures(r1, s1) and (r2, s2) of messagesm1 and m2 respectively. Writing down the equations fors1 and s2: s1 ≡k−1(H(m1) + xr1) ( mod q) s2 ≡k−1(H(m2) + xr2) ( mod q) The attacker can simplify this further:r1 and r2 must be equal, following the definition: ri ≡gk (mod q) Since the signer is reusingk, and the value ofr only depends on k, allri will be equal. Since the signer is using the same key,x is equal in the two equations as well. Subtract the twosi equations from each other, followed by some other arithmetic manipulations: s1 −s2 ≡ k−1(H(m1) + xr) −k−1(H(m2) + xr) ( mod q) ≡ k−1 ((H(m1) + xr) −(H(m2) + xr)) ( mod q) ≡ k−1(H(m1) + xr −H(m2) −xr) ( mod q) ≡ k−1(H(m1) −H(m2)) ( mod q) This gives us the simple, direct solution fork: k ≡(H(m1) −H(m2)) (s1 −s2)−1 (mod q) The hash values H(m1) and H(m2) are easy to compute. Theyʼre not secret: the messages being signed are public. CHAPTER 12. SIGNATURE ALGORITHMS 135 The two valuess1 and s2 are part of the signatures the at- tacker saw. So, the attacker can computek. That doesnʼt give him the private keyx yet, though, or the ability to forge signatures. Letʼs write the equation fors down again, but this time thinking ofk as something we know, andx as the variable weʼre trying to solve for: s ≡k−1(H(m) + xr) ( mod q) All (r, s) that are valid signatures satisfy this equation, so we can just take any signature we saw. Solve forx with some algebra: sk ≡H(m) + xr (mod q) sk −H(m) ≡xr (mod q) r−1(sk −H(m)) ≡x (mod q) Again, H(m) is public, plus the attacker needed it to com- pute k, anyway. Theyʼve already computed k, and s is plucked straight from the signature. That just leaves us with r−1 (mod q) (read as: “the modular inverse ofr modulo q”), but that can be computed efficiently as well. (For more in- formation, see the appendix on modular arithmetic; keep in mind that q is prime, so the modular inverse can be com- puted directly.) That means that the attacker, once theyʼve discovered the k of any signature, can recover the private key directly. So far, weʼve assumed that the broken signer would al- ways use the samek. To make matters worse, a signer only has to re-usek once in any two signatures that the attacker can see for the attack to work. As weʼve seen, ifk is repeated, the ri values repeat as well. Sinceri is a part of the signature, itʼs very easy to see when the signer has made this mistake. So, even if reusingk is something the signer only does rarely CHAPTER 12. SIGNATURE ALGORITHMS 136 (because their random number generator is broken, for ex- ample), doing it once is enough for the attacker to break the DSA scheme. In short, reusing thek parameter of a DSA signing oper- ation means an attacker recovers the private key. TODO: Debian http://rdist.root.org/2009/05/ 17/the-debian-pgp-disaster-that-almost-was/ 12.4 ECDSA TODO: explain (see #53) As with regular DSA, the choice ofk is extremely critical. There are attacks that manage to recover the signing key us- ing a few thousand signatures when only a few bits of the nonce leak. [MHMP13] 12.5 Repudiableauthenticators Signatures like the ones we described above provide a prop- erty callednon-repudiation. In short, it means that you canʼt later deny being the sender of the signed message. Anyone can verify that the signature was made using your private key, something only you could do. That may not always be a useful feature; it may be more prudent to have a scheme where only the intended recipient can verify the signature. An obvious way to design such a scheme would be to make sure that the recipient (or, in fact, anyone else) could have computed an identical value. Such messages can be repudiated; such a scheme is often called “deniable authentication”. While it authenticates the sender to the intended recipient, the sender can later deny (to third parties) having sent the message. Equivalently, the recipient canʼt convince anyone else that the sender sent that particular message. 13 Keyderivationfunctions 13.1 Description A key derivation function is a function that derives one or more secret values (thekeys) from one secret value. Many key derivation functions can also take a (usually op- tional) salt parameter. This parameter causes the key deriva- tion function to not always return the same output keys for the same input secret. As with other cryptosystems,salts are fundamentally different from the secret input:salts gen- erally do not have to be secret, and can be re-used. Key derivation functions can be useful, for example, when a cryptographic protocol starts with a single secret value, such as a shared password or a secret derived using Diffie-Hellman key exchange, but requires multiple secret values to operate, such as encryption and MAC keys. An- other use case of key derivation functions is in cryptograph- ically secure random number generators, which weʼll see in more detail in a following chapter, where they are used to extract randomness with high entropy density from many sources that each have low entropy density. 137 CHAPTER 13. KEY DERIVATION FUNCTIONS 138 There are two main categories of key derivation func- tions, depending on the entropy content of the secret value, which determines how many different possible values the secret value can take. If the secret value is a user-supplied password, for exam- ple, it typically contains very little entropy. There are very few values the password will take. As weʼve already estab- lished inaprevioussectiononpasswordstorage (page 104), that means it is necessary that the key derivation function is hard to compute. That means it requires a non-trivial amount of computing resources, such as CPU cycles or memory. If the key derivation function were easy to compute, an attacker could simply enumerate all possible values of the shared se- cret, since there are few possibilities, and then compute the key derivation function for all of them. As weʼve seen in that previous section on password storage, this is how most mod- ern attacks on password stores work. Using an appropriate key derivation function would prevent these attacks. In this chapter, weʼll see scrypt, as well as other key derivation func- tions in this category. On the other hand, the secret value could also have a high entropy content. For example, it could be a shared se- cret derived from a Diffie-Hellmankey agreement protocol, or an API key consisting of cryptographically random bytes (weʼll discuss cryptographically secure random number gen- eration in the next chapter). In that case, it isnʼt necessary to have a key derivation function thatʼs hard to compute: even if the key derivation function is trivial to compute, there are too many possible values the secret can take, so an attacker would not be able to enumerate them all. Weʼll see the best- of-breed of this kind of key derivation function, HKDF, in this chapter. 13.2 Passwordstrength TODO: NIST Special Publication 800-63 CHAPTER 13. KEY DERIVATION FUNCTIONS 139 13.3 PBKDF2 13.4 bcrypt 13.5 scrypt 13.6 HKDF The HKDF, defined in RFC 5869 [KE] and explained in detail in a related paper [Kra10], is a key derivation function de- signed for high entropy inputs, such as shared secrets from a Diffie-Hellman key exchange. It is specificallynot designed to be secure for low-entropy inputs such as passwords. HKDF exists to give people an appropriate, off-the-shelf key derivation function. Previously, key derivation was of- ten something that was done ad hoc for a particular standard. Usually these ad hoc solutions did not have the extra provi- sions HKDF does, such assalts or the optional info parame- ter (which weʼll discuss later in this section); and thatʼs only in the best case scenario where the KDF wasnʼt fundamen- tally broken to begin with. HKDF is based on HMAC. Like HMAC, it is a generic con- struction that uses hash functions, and can be built using any cryptographically secure hash function you want. AcloserlookatHKDF This is an optional, in-depth section. It almost certainly wonʼt help you write bet- ter software, so feel free to skip it. It is only here to satisfy your inner geekʼs curiosity. HKDF consists of two phases. In the first phase, called the extraction phase, a fixed-length key is extracted from the in- CHAPTER 13. KEY DERIVATION FUNCTIONS 140 put entropy. In the second phase, called theexpansion phase, that key is used to produce a number of pseudorandom keys. Theextractionphase The extraction phase is responsible for extracting a small amount of data with a high entropy content from a poten- tially large amount of data with a smaller entropy density. The extraction phase just uses HMAC with asalt: def extract(salt, data): return hmac(salt, data) The salt value is optional. If thesalt is not specified, a string of zeroes equal to the length of the hash functionʼs out- put is used. While thesalt is technically optional, the design- ers stress its importance, because it makes the independent uses of the key derivation function (for example, in different applications, or with different users) produce independent results. Even a fairly low-entropysalt can already contribute significantly to the security of the key derivation function. [KE] [Kra10] The extraction phase explains why HKDF is not suitable for deriving keys from passwords. While the extraction phase is very good atconcentrating entropy, it is not capable of amplifying entropy. It is designed for compacting a small amount of entropy spread out over a large amount of data into the same amount of entropy in a small amount of data, but is not designed for creating a set of keys that are diffi- cult to compute in the face of a small amount of available entropy. There are also no provisions for making this phase computationally intensive. [KE] In some cases, it is possible to skip the extraction phase, if the shared secret already has all the right properties, for example, if it is a pseudorandom string of sufficient length, and with sufficient entropy. However, sometimes this should not be done at all, for example when dealing with a Diffie-Hellman shared secret. The RFC goes into slightly CHAPTER 13. KEY DERIVATION FUNCTIONS 141 more detail on the topic of whether or not to skip this step; but it is generally inadvisable. [KE] Theexpansionphase In the expansion phase, the random data extracted from the inputs in the extraction phase is expanded into as much data as is required. The expansion step is also quite simple: chunks of data are produced using HMAC, this time with the extracted se- cret, not with the publicsalt, until enough bytes are pro- duced. The data being HMACed is the previous output (start- ing with an empty string), an “info” parameter (by default also the empty string), and a counter byte that counts which block is currently being produced. def expand(key, info=””): ”””Expands the key, with optional info.””” output = ”” for byte in map(chr, range(256)): output = hmac(key, output + info + byte) yield output def get_output(desired_length, key, info=””): ”””Collects output from the expansion step ,→until enough has been collected; then returns that output. ,→””” outputs, current_length = [], 0 for output in expand(key, info): outputs.append(output) current_length += len(output) if current_length >= desired_length: break else: # This block is executed when the for ,→loop isn't # terminated by the ``break`` statement, ,→ which (continues on next page) CHAPTER 13. KEY DERIVATION FUNCTIONS 142 (continued from previous page) # happens when we run out of ``expand`` ,→outputs # before reaching the desired length. raise RuntimeError(”Desired length too ,→long”) return ””.join(outputs)[:desired_length] Like thesalt in the extraction phase, the “info” parame- ter is entirely optional, but can actually greatly increase the security of the application. The “info” parameter is intended to contain some application-specific context in which the key derivation function is being used. Like thesalt, it will cause the key derivation function to produce different val- ues in different contexts, further increasing its security. For example, the info parameter may contain information about the user being dealt with, the part of the protocol the key derivation function is being executed for or the like. [KE] 14 Randomnumber generators The generation of random numbers is too impor- tant to be left to chance. Robert R. Coveyou 14.1 Introduction Many cryptographic systems require random numbers. So far, weʼve just assumed that theyʼre available. In this chapter, weʼll go more in depth about the importance and mechanics of random numbers in cryptographic systems. Producing random numbers is a fairly intricate process. Like with so many other things in cryptography, itʼs quite easy to get it completely wrong but have everythinglookcom- pletely fine to the untrained eye. There are three categories of random number genera- tion that weʼll consider separately: • True random number generators 143 CHAPTER 14. RANDOM NUMBER GENERATORS 144 • Cryptographically secure pseudorandom number gen- erators • Pseudorandom number generators 14.2 Truerandomnumbergenerators Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. John von Neumann John von Neumann, father of the modern model of computing, made an obvious point. We canʼt expect to produce random numbers using predictable, deterministic arithmetic. We need a source of randomness that isnʼt a con- sequence of deterministic rules. True random number generators get their randomness from physical processes. Historically, many systems have been used for producing such numbers. Systems like dice are still in common use today. However, for the amount of randomness we need for practical cryptographic algorithms, these are typically far too slow, and often quite unreliable. Weʼve since come up with more speedy and reliable sources of randomness. There are several categories of physical processes that are used for hardware random num- ber generation: • Quantum processes • Thermal processes • Oscillator drift • Timing events Keep in mind that not all of these options necessarily gen- erate high-quality, truly random numbers. Weʼll elaborate further on how they can be applied successfully anyway. CHAPTER 14. RANDOM NUMBER GENERATORS 145 Radioactivedecay One example of a quantum physical process used to pro- duce random numbers is radioactive decay. We know that radioactive substances will slowly decay over time. Itʼs im- possible to know when the next atom will decay; that pro- cess is entirely random. Detecting when such a decay has occurred, however, is fairly easy. By measuring the time be- tween individual decays, we can produce random numbers. Shotnoise Shot noise is another quantum physical process used to pro- duce random numbers. Shot noise is based on the fact that light and electricity are caused by the movement of indivisi- ble little packets: photons in the case of light, and electrons in the case of electricity. Nyquistnoise An example of a thermal process used to produce random numbers is Nyquist noise. Nyquist noise is the noise that occurs from charge carriers (typically electrons) traveling through a medium with a certain resistance. That causes a tiny current to flow through the resistor (or, alternatively put, causes a tiny voltage difference across the resistor). i = r 4kBT∆f R v = p 4kBTR∆f These formulas may seem a little scary to those who havenʼt seen the physics behind them before, but donʼt worry too much: understanding them isnʼt really necessary to go along with the reasoning. These formulas are for theroot mean square. If youʼve never heard that term before, you can roughly pretend that means “average”.∆f is the bandwidth, CHAPTER 14. RANDOM NUMBER GENERATORS 146 T is the temperature of the system in Kelvins,kB is Boltz- mannʼs constant. As you can see from the formula, Nyquist noise isther- mal, or temperature-dependent. Fortunately, an attacker generally canʼt use that property to break the generator: the temperature at which it would become ineffective is so low that the system using it has probably already failed at that point. By evaluating the formula, we can see that Nyquist noise is quite small. At room temperature with reasonable assumptions (10 kHz bandwidth and a 1kΩ resistor), the Nyquist voltage is in the order of several hundred nanovolts. Even if you round up liberally to a microvolt (a thousand nanovolts), thatʼs still a thousandth of a thousandth of a volt, and even a tiny AA battery produces 1.5V . While the formulas describe the root mean square, the value you can measure will be randomly distributed. By re- peatedly measuring it, we can produce high-quality random numbers. For most practical applications, thermal noise numbers are quite high quality and relatively unbiased. TODO: weʼve never actually explained the word entropy; “resistance an attacker perceives” is necessary in a good def- inition TODO: explain synchronous stream ciphers as CSPRNGs 14.3 Cryptographically secure pseudoran- domgenerators While weʼll see several examples of cryptographically secure pseudorandom generators in the next few sections, keep in mind that they are all just algorithms thatcould be used. As an application developer, you shouldnever be making a choice between one of them. Instead, in the few cases you really want to pick a random number manually, you shouldalways use the cryptographi- cally secure random number generator provided by your op- CHAPTER 14. RANDOM NUMBER GENERATORS 147 erating system:/dev/urandom on NIX (Linux, BSDs, and OS X), orCryptGenRandom on Windows. Python provides handy interfaces to these in the form ofos.urandom and random.SystemRandom. While they can be implemented securely, try to avoid using userspace cryptographically secure random number generators such as the one in OpenSSL. There are far more things that can go wrong with them, usually involving their internal state: either they remain uninitialized, poorly ini- tialized, or end up re-using the same state in different lo- cations. In all of these cases, the resulting cryptosystem is completely and utterly broken. TODO: talk about the FUD in the Linux man page for urandom This is an optional, in-depth section. It almost certainly wonʼt help you write bet- ter software, so feel free to skip it. It is only here to satisfy your inner geekʼs curiosity. Since this is a specific cryptograph- ically secure pseudorandom number generator algo- rithm, you donʼt actually need to know how it works to write good software. Just use ~urandom~. 14.4 Yarrow This is an optional, in-depth section. It almost certainly wonʼt help you write bet- ter software, so feel free to skip it. It is only here to satisfy your inner geekʼs curiosity. The Yarrow algorithm is a cryptographically secure pseudo- random number generator. TODO: actually explain Yarrow CHAPTER 14. RANDOM NUMBER GENERATORS 148 This algorithm is used as the CSPRNG for FreeBSD, and was inherited by Mac OS X. On both of these operating systems, itʼs used to implement/dev/random. Unlike on Linux, /dev/urandom is just an alias for/dev/random. 14.5 BlumBlumShub TODO: explain this, and why itʼs good (provable), but why we donʼt use it (slow) 14.6 Dual_EC_DRBG This is an optional, in-depth section. It almost certainly wonʼt help you write bet- ter software, so feel free to skip it. It is only here to satisfy your inner geekʼs curiosity. Dual_EC_DRBG is a NIST standard for a cryptographically secure pseudorandom bit generator. It sparked a large amount of controversy: despite being put forth as an offi- cial, federal cryptographic standard, it quickly became evi- dent that it wasnʼt very good. Cryptanalysis eventually demonstrated that the standard could contain a back door hidden in the constants specified by the standard, potentially allowing an unspecified attacker to completely break the random number generator. Several years afterwards, leaked documents suggested a backdoor in an unnamed NIST standard released in the same year asDual_EC_DRBG, fueling the suspicions further. This led to an official recommendation from the standards body to stop using the standard, which was previously unheard of under such circumstances. CHAPTER 14. RANDOM NUMBER GENERATORS 149 Background For a long time, the official standards produced by NIST lacked good, modern cryptographically secure pseudoran- dom number generators. It had a meager choice, and the ones that had been standardized had several serious flaws. NIST hoped to address this issue with a new publication called SP 800-90, that contained several new cryptographi- cally secure pseudorandom number generators. This docu- ment specified a number of algorithms, based on different cryptographic primitives: 1. Cryptographic hash functions 2. HMAC 3. Block ciphers 4. Elliptic curves Right off the bat, that last one jumps out. Using elliptic curves for random number generation was unusual. Stan- dards like these are expected to be state-of-the-art, while still staying conservative. Elliptic curves had been considered before in an academic context, but that was a far cry from being suggested as a standard for common use. There is a second reason elliptic curves seem strange. HMAC and block ciphers are obviously symmetric algo- rithms. Hash functions have their applications in asymmet- ric algorithms such as digital signatures, but arenʼt them- selves asymmetric. Elliptic curves, on the other hand, are exclusively used for asymmetric algorithms: signatures, key exchange, encryption. That said, the choice didnʼt come entirely out of the blue. A choice for a cryptographically secure pseudorandom num- ber generator with a strong number-theoretical basis isnʼt unheard of: Blum Blum Shub is a perfect example. Those generators are typically much slower than the alternatives. Dual_EC_DRBG, for example, is three orders of magnitude CHAPTER 14. RANDOM NUMBER GENERATORS 150 slower than its peers presented in the same standard. The idea is that the extra confidence inspired by the stronger mathematical guarantees is worth the performance penalty. For example, weʼre fairly confident that factoring numbers is hard, but weʼre a lot less sure about our hash functions and ciphers. RSA came out in 1977 and has stood the test of time quite well since then. DES came out two years later, and is now considered completely broken. MD4 and MD5 came out over a decade later, and are completely broken as well. The problem is, though, that the standard didnʼt actually provide the security proof. The standard specifies the gen- erator but then merely suggests that it would be at least as hard as solving the elliptic curve discrete log problem. Blum Blum Shub, by contrast, has a proof that shows that break- ing it is at least as hard as solving the quadratic residuosity problem. The best algorithm we have for that is factoring numbers, which weʼre fairly sure is pretty hard. The omission of the proof is a bit silly, because thereʼs no reason youʼd use a pseudorandom number generator as slow asDual_EC_DRBG unless you had proof that you were getting something in return for the performance hit. Cryptographers later did the homework that NIST should have provided in the specification [ SS06] [BGjosteen07]. Those analyses quickly highlighted a few issues. Aquickoverviewofthealgorithm The algorithm consists of two parts: 1. Generating pseudorandom points on the elliptic curve, which are turned into the internal state of the genera- tor; 2. Turning those points into pseudorandom bits. Weʼll illustrate this graphically, with an illustration based on the work by Shumow and Ferguson, two cryptog- CHAPTER 14. RANDOM NUMBER GENERATORS 151 raphers who highlighted some of the major issues with this algorithm: Throughout the algorithm,ϕ is a function that takes a curve point and turns it into an integer. The algorithm needs two given points on the curve:P and Q. These are fixed, and defined in the specification. The algorithm has an internal state s. When producing a new block of bits, the algorithm turns s into a different valuer using theϕ function and ellip- tic curve scalar multiplication withP: r = ϕ(sP) That value,r, is used both for producing the output bits and updating the internal state of the generator. In order to pro- duce the output bits, a different elliptic curve point,Q, is used. The output bits are produced by multiplyingr with Q, and running the result through a transformationθ: o = θ(ϕ(rQ)) In order to perform the state update,r is multiplied withP again, and the result is converted to an integer. That integer is used as the new states. s = ϕ(rP ) Issuesandquestionmarks First of all,ϕ is extremely simple: it just takes thex coordi- nate of the curve point, and discards they coordinate. That CHAPTER 14. RANDOM NUMBER GENERATORS 152 means that itʼs quite easy for an attacker who sees the output value ofϕ to find points that could have produced that value. In itself, thatʼs not necessarily a big deal; but, as weʼll see, itʼs one factor that contributes to the possibility of a backdoor. Another flaw was shown where points were turned into pseudorandom bits. Theθ function simply discards the 16 most significant bits. Previous designs discarded signifi- cantly more: for 256-bit curves such as these, they discarded somewhere in the range of 120 and 175 bits. Failing to discard sufficient bits gave the generator a small bias. The next-bit property was violated, giving attack- ers a better than 50% chance of guessing the next bit cor- rectly. Granted, that chance was only about one in a thou- sand better than 50%; but thatʼs still unacceptable for whatʼs supposed to be the state-of-the-art in cryptographically se- cure pseudorandom number generators. Discarding only those 16 bits has another consequence. Because only 16 bits were discarded, we only have to guess 216 possibilities to find possible values ofϕ(rQ) that pro- duced the output. That is a very small number: we can sim- ply enumerate all of them. Those values are the outputs of ϕ, which as we saw just returns thex coordinate of a point. Since we know it came from a point on the curve, we just have to check if our guess is a solution for the curve equa- tion: y2 ≡x3 + ax + b (mod p) The constants a, b, p are specified by the curve. Weʼve just guessed a value forx, leaving only one unknown,y. We can solve that quite efficiently. We compute the right hand side and see if itʼs a perfect square:y2 ≡q ≡ √ x3 + ax + b (mod p). If it is,A = ( x, √q) = ( x, y) is a point on the curve. This gives us a number of possible pointsA, one of which is rQ used to produce the output. This isnʼt a big deal at face value. To find the state of the algorithm, an attacker needs to findr, so they can compute CHAPTER 14. RANDOM NUMBER GENERATORS 153 s. They still need to solve the elliptic curve discrete log prob- lem to findr from rQ, givenQ. Weʼre assuming that problem is hard. Keep in mind that elliptic curves are primitives used for asymmetric encryption. That problem is expected to be hard to solve in general, but what if we have some extra in- formation? What if thereʼs a secret valuee so thateQ = P? Letʼs put ourselves in the shoes of an attacker knowinge. We repeat our math from earlier. One of those pointsA we just found is therQ weʼre looking for. We can compute: ϕ(eA) ≡ϕ(erQ) ≡ϕ(rP ) ( mod p) That last step is a consequence of the special relationship between e, P, Q. Thatʼs pretty interesting, becauseϕ(rP ) is exactly the computation the algorithm does to computes, the new state of the algorithm! That means that an attacker that knowse can, quite efficiently, compute the new states from any outputo, allowing them to predict all future values of the generator! This assumes that the attacker knows whichA is theright A. Because only 16 bits were discarded there are only 16 bits left for us to guess. That gives us216 candidate x coordinates. Experimentally, we find that roughly half of the possiblex coordinates correspond to points on the curve, leaving us with 215 possible curve pointsA, one of which isrQ. Thatʼs a pretty small number for a bit of computer-aided arithmetic: plenty small for us to try all options. We can therefore say that an attacker that does know the secret valuee most defi- nitely can break the generator. So, weʼve now shown that if there is a magicale for which eQ = P, and you can pickP and Q (and you donʼt have to explain where you got them from), that you could break the generator. How do you pick such values? To demonstrate just how possible it is, the researchers started from the NIST curveʼsP and p values, but came up with their ownQ′. They did this by starting withP, picking a random d (keeping it secret), and settingQ′ = dP. The trick CHAPTER 14. RANDOM NUMBER GENERATORS 154 is that thereʼs an efficient algorithm for computinge in eQ′ = P if you know thed in Q′ = dP. This is thee we need for our earlier attack. When they tried this out, they discovered that in all cases (that is, for many randomd), seeing 32 bytes of output was enough to determine the states. All of this, of course, only demonstrates that it is possi- ble for the specified values ofP and Q to be special values with a secret back door. It doesnʼt provide any evidence that the actual values have a backdoor in them. However, given that the standard never actually explainshow they got the magical value forQ, it doesnʼt really inspire a lot of con- fidence. Typically, cryptographic standards use “nothing- up-my-sleeve” numbers, such as the value of some constant such asπ or the natural logarithm base,e. If someone does know the backdoor, the consequences are obviously devastating. Weʼve already argued for the ne- cessity of cryptographically secure pseudorandom number generators: having a broken one essentially means that all cryptosystems that use this generator are completely and ut- terly defeated. There are two ways one might try to fix this particular algorithm: • Make the θ function more complex to invert, rather than just discarding 16 bits. This makes it harder to find candidate points, and hence, harder to perform the attack. One obvious way would be to discard more bits. Another option would be to use a cryptographi- cally secure hash, or a combination of both. • Generate randomQ every time you start the algorithm, possibly by picking a randomd and settingQ = dP. Of course, d has to be sufficiently large and truly random: if θ is unchanged, and there are only a few valuesd can have, the attacker can just perform the above attack for all values ofd. CHAPTER 14. RANDOM NUMBER GENERATORS 155 Both of these are really just band-aid solutions; it would be a much better idea to just use a different algorithm al- together. These suggestions donʼt resolve the issue that itʼs slow, exotic, and now a retracted standard. Aftermath TODO: Talk about RSA guyʼs comments + snowden leaks 14.7 MersenneTwister Mersenne Twister is a very common pseudorandom number generator. It has many nice properties, such as high perfor- mance, a huge period1 of 219937 −1 ≈4 ·106001, and it passes all but the most demanding randomness tests. Despite all of these wonderful properties, it isnot cryptographically se- cure. Anin-depthlookattheMersenneTwister This is an optional, in-depth section. It almost certainly wonʼt help you write bet- ter software, so feel free to skip it. It is only here to satisfy your inner geekʼs curiosity. To demonstrate why Mersenne Twister isnʼt cryptographi- cally secure, weʼll take a look at how the algorithm works. Fortunately, itʼs not very complex. The standard Mersenne Twister algorithm operates on an internal state arrayS consisting of 624 unsigned 32-bit integers, and an indexi pointing to the current integer. It consists of three steps: 1 The period of a pseudorandom number generator is how many ran- dom numbers it produces before the entire sequence repeats. CHAPTER 14. RANDOM NUMBER GENERATORS 156 1. An optional initialization function, which produces an initial state from a small random value called aseed. 2. A state generation function, which produces a new state from the old state. 3. An extraction function, also called thetempering func- tion, that produces a random number from the current element of the state (the element pointed at by the in- dex i). Whenever the extraction function is called, the index to the current integer is incremented. When all of the current elements of the state have been used to produce a number, the state initialization function is called again. The state ini- tialization function is also called right before the first num- ber is extracted. So, to recap: the state is regenerated, then the extraction function goes over each of the elements in the state, until it runs out. This process repeats indefinitely. TODO: illustrate Weʼll look at each of the parts briefly. The exact work- ings of them is outside the scope of this book, but weʼll look at them just long enough to get some insight into why Mersenne Twister is unsuitable as a cryptographically se- cure random number generator. Theinitializationfunction The initialization function creates an instance of Mersenne Twisterʼs state array, from a small initial random number called aseed. The array starts with the seed itself. Then, each next el- ement is produced from a constant, the previous element, and the index of the new element. Elements are produced until there are 624 of them. Hereʼs the Python source code: CHAPTER 14. RANDOM NUMBER GENERATORS 157 def uint32(n): return 0xFFFFFFFF & n def initialize_state(seed): state = [seed] for i in range(1, 624): prev = state[-1] elem = 0x6c078965 (prev ^ (prev >> ,→30)) + i state.append(uint32(elem)) return state For those of you who havenʼt worked with Python or its bitwise operators: • >> and << are right-shift and left-shift • & is binary AND:0&0 = 0 &1 = 1 &0 = 0 , and1&1 = 1 . • ^ is binary XOR,^= XORs and assigns the result to the name on the left-hand side, sox ^= k is the same thing asx = x ^ k. REVIEW: Bitwise arithmetic appendix? Thestateregenerationfunction The state regeneration function takes the current state and produces a new state. It is called right before the first num- ber is extracted, and every time all 624 elements of the state have been used up. The Python source code for this function is fairly simple. Note that it modifies the state array in place, instead of re- turning a new one. def regenerate(s): for i in range(624): y = s[i] & 0x80000000 (continues on next page) CHAPTER 14. RANDOM NUMBER GENERATORS 158 (continued from previous page) y += s[(i + 1) % 624] & 0x7fffffff z = s[(i + 397) % 624] s[i] = z ^ (y >> 1) if y % 2: s[i] ^= 0x9908b0df The % in an expression likes[(i + n) % 624] means that a next element of the state is looked at, wrapping around to the start of the state array if there is no next element. The values 0x80000000 and 0x7fffffff have a spe- cific meaning when interpreted as sequences of 32 bits. 0x80000000 has only the first bit set;0x7fffffff has ev- ery bit except the first bit set. Because these are bitwise ANDʼed together (&), this effectively means that after the first two lines in the loop,y consists of the first bit of the cur- rent state element and all the subsequent bits of the next el- ement. Thetemperingfunction The tempering function is applied to the current element of the state before returning it as the produced random num- ber. Itʼs easier to just show the code instead of explaining how it works: _TEMPER_MASK_1 = 0x9d2c5680 _TEMPER_MASK_2 = 0xefc60000 def temper(y): y ^= uint32(y >> 11) y ^= uint32((y << 7) & _TEMPER_MASK_1) y ^= uint32((y << 15) & _TEMPER_MASK_2) y ^= uint32(y >> 18) return y It may not be obvious, especially if youʼre not used to bi- nary arithmetic, but this function isbijective or one-to-one: CHAPTER 14. RANDOM NUMBER GENERATORS 159 each 32 bit integer input maps to exactly one output, and vice versa: for each 32 bit integer we get as an output there was exactly one 32 bit integer it could have come from. Because it uses right and left shifts, it might look like it throws away data at first glance, and hence canʼt possibly be reversible. Itʼs true that those shifts throw some bits away, however, the critical operation here is the inline XOR (^=): those shifts are just used to compute masks that the value to be tempered is XORʼd with. The XOR operations themselves are reversible, and because each independent operation is reversible, their composition is too. Because the tempering function is one-to-one, there is an inverse function: a function that gives you the untem- pered equivalent of a number. It may not be obvious to you how to construct that function unless youʼre a bitwise arith- metic wizard, but thatʼs okay; in the worst case scenario we could still brute-force it. Suppose we just try every single 32 bit integer, and remember the result in a table. Then, when we get a result, we look it up in the table, and find the origi- nal. That table would have to be at least232 ·32 bits in length, or a good 17 gigabytes; big, but not impossibly so. Fortunately, thereʼs a much simpler method to compute the inverse of the temper function. Weʼll see why thatʼs in- teresting when we evaluate the cryptographic security of the Mersenne Twister in the next section. For those interested in the result, the untempering function looks like this: def untemper(y): y ^= y >> 18 y ^= ((y << 15) & _TEMPER_MASK_2) y = _undo_shift_2(y) y = _undo_shift_1(y) return y def _undo_shift_2(y): t = y (continues on next page) CHAPTER 14. RANDOM NUMBER GENERATORS 160 (continued from previous page) for _ in range(5): t <<= 7 t = y ^ (t & _TEMPER_MASK_1) return t def _undo_shift_1(y): t = y for _ in range(2): t >>= 11 t ^= y return t Cryptographicsecurity Remember that for cryptographic security, it has to be im- possible to predict future outputs or recover past outputs given present outputs. The Mersenne Twister doesnʼt have that property. Itʼs clear that pseudorandom number generators, both those cryptographically secure and those that arenʼt, are en- tirely defined by their internal state. After all, they are deter- ministic algorithms: theyʼre just trying very hard to pretend not to be. Therefore, you could say that the principal differ- ence between cryptographically secure and ordinary pseu- dorandom number generators is that the cryptographically secure ones shouldnʼt leak information about their internal state, whereas it doesnʼt matter for regular ones. Remember that in Mersenne Twister, a random number is produced by taking the current element of the state, apply- ing the tempering function, and returning the result. Weʼve also seen that the tempering function has an inverse func- tion. So, if I can see the output of the algorithm and apply the inverse of the tempering function, Iʼve recovered one el- CHAPTER 14. RANDOM NUMBER GENERATORS 161 ement out of the 624 in the state. Suppose that I happen to be the only person seeing the outputs of the algorithm, and you begin at the start of the state, such as with a fresh instance of the algorithm, that means that I can clone the state by just having it produce 624 random numbers. Even if an attacker doesnʼt see all 624 numbers, they can often still recreate future states, thanks to the simple rela- tions between past states and future states produced by the state regeneration function. Again, this is not a weakness of Mersenne Twister. Itʼs designed to be fast and have strong randomness properties. It is not designed to be unpredictable, which is the defining property of a cryptographically secure pseudorandom num- ber generator. PartIII Completecryptosystems 162 15 SSLandTLS 15.1 Description SSL, short for Secure Socket Layer, is a cryptographic proto- col originally introduced by Netscape Communications1 for securing traffic on the Web. The standard is now superseded by TLS (Transport Layer Security), a standard publicized in RFCs by the IETF. The term SSL is still commonly used, even when the speaker actually means a TLS connection. From now on, this book will only use the term TLS, unless we re- ally mean the old SSL standard. Its first and foremost goal is to transport bytes securely, over the Internet or any other insecure medium. [DR] Itʼs a hybrid cryptosystem: it uses both symmetric and asym- metric algorithms in unison. For example, asymmetric algo- rithms such as signature algorithms can be used to authenti- cate peers, while public key encryption algorithms or Diffie- Hellman exchanges can be used to negotiate shared secrets and authenticate certificates. On the symmetric side,stream 1 For those too young to remember, Netscape is a company that used to make browsers. 163 CHAPTER 15. SSL AND TLS 164 ciphers (both native ones and block ciphers in amode of oper- ation) are used to encrypt the actual data being transmitted, and MAC algorithms are used to authenticate that data. TLS is the worldʼs most common cryptosystem, and hence probably also the most studied. Over the years, many flaws have been discovered in SSL and TLS, despite many of the worldʼs top cryptographers contributing to and examin- ing the standard2. As far as we know, the current versions of TLS are secure, or at least can be configured to be secure. 15.2 Handshakes TODO: explain a modern TLS handshake Downgradeattacks SSL 2.0 made the mistake of not authenticating handshakes. This made it easy to mount downgrade attacks. A down- grade attack is a man-in-the-middle attack where an attacker modifies the handshake messages that negotiate which ci- phersuite is being used. That way, he can force the clients to set up the connection using an insecure block cipher, for example. Due to cryptographic export restrictions at the time, many ciphers were only 40 or 56 bit. Even if the attacker couldnʼt break the best encryption both client and server supported, he could probably break the weakest, which is all that is necessary for a downgrade attack to succeed. This is one of the many reasons that there is an explicit RFC [TP] prohibiting new TLS implementations from having SSL v2.0 support. 2 In case I havenʼt driven this point home yet: it only goes to show that designing cryptosystems is hard, and you probably shouldnʼt do it yourself. CHAPTER 15. SSL AND TLS 165 15.3 Certificateauthorities TLS certificates can be used to authenticate peers, but how do we authenticate the certificate? My bank may very well have a certificate claiming to be that particular bank, but how do I know itʼs actually my bank, and not just someone pretending to be my bank? Why should I trust this partic- ular certificate? As weʼve seen when we discussed these al- gorithms, anyone can generate as many key pairs as theyʼd like. Thereʼs nothing stopping someone from generating a key pair pretending to be your bank. When someone actually tries to use a certificate to im- personate a bank, real browsers donʼt believe them. They notify the user that the certificate is untrusted. They do this using the standard TLS trust model of certificate author- ities. TLS clients come with a list of trusted certificate au- thorities, commonly shipped with your operating system or your browser. These are special, trusted certificates, that are carefully guarded by their owners. For a fee, these owners will use their certificate author- ity to sign other certificates. The idea is that the certificate authority wouldnʼt sign a certificate for Facebook or a bank or anyone else, unless you could prove youʼre actually them. When a TLS client connects to a server, that server pro- vides a certificate chain. Typically, their own certificate is signed by an intermediary CA certificate, which is signed by another, and another, and one that is signed by a trusted root certificate authority. Since the client already has a copy of that root certificate, they can verify the signature chain start- ing with the root. Your fake certificate doesnʼt have a chain leading up to a trusted root certificate, so the browser rejects it. TODO: Explain why this is a total racket CHAPTER 15. SSL AND TLS 166 15.4 Self-signedcertificates 15.5 Clientcertificates In TLS, certificates are usually only used to identify the server. This satisfies a typical use case: users want to com- municate securely with their banks and e-mail providers, and the certificate authenticates the service theyʼre talking to. The service usually authenticates the user using pass- words, and, occasionally, two-factor authentication. In public-key schemes weʼve seen so far, all peers typi- cally had one or more key pairs of their own. Thereʼs no rea- son users canʼt have their own certificates, and use them to authenticate to the server. The TLS specification explicitly supports client certificates. This feature is only rarely used, even though it clearly has very interesting security benefits. The main reason for that is probably rooted in the poor user experience. There are no systems that rely on client cer- tificates that are easy to use for non-technical people. Since there are few such systems, even tech-savvy people donʼt know about them, which means new systems arenʼt created. Client certificates are a great solution for when you con- trol both ends of the wire and want to securely authenticate both peers in a TLS connection. By producing your own cer- tificate authority, you can even sign these client certificates to authenticate them. 15.6 Perfectforwardsecrecy Historically, the most common way to agree on the pre- master secret is for the client to select a random number and encrypt it, typically using RSA. This has a few nice prop- erties. For example, it means the server can make do with less entropy: since the random bits are handed to the server by the client, the server doesnʼt need to produce any cryp- tographically random bits. It also makes the handshake CHAPTER 15. SSL AND TLS 167 slightly faster, since thereʼs no need for back-and-forth com- munication to agree on a shared secret. However, it has one major flaw. Suppose an attacker gets access to the serverʼs private key. Perhaps they managed to factor the modulus of the RSA key, or perhaps they broke in and stole it, or perhaps they used legal force to get the owner to hand over the key. Regardless of how they acquired it, getting access to the key allows the attacker to decrypt all past communication. The key allows them to decrypt the encrypted pre-master secrets, which allows them to derive all of the symmetric encryption keys, and therefore decrypt everything. There are obvious alternatives to this scheme. Weʼve al- ready seen Diffie-Hellman key exchange, allowing two peers to agree on secret keys over an insecure medium. TLS allows for peers to agree on the pre-master secret using a Diffie- Hellman exchange, either based on discrete logs or elliptic curves. Assuming both peers discard the keys after use like theyʼre supposed to, getting access to the secret keys wouldnʼt allow an attacker to decrypt previous communi- cation. That property is calledperfect forward secrecy. The term “perfect” is a little contested, but the term “forward” means that communications canʼt be decrypted later if the long-term keys (such as the serverʼs private key) fall into the wrong hands. Of course, this is only true if Diffie-Hellman exchanges are secure. If an attacker has a significant mathematical and computational advantage over everyone else, such as an algorithm for solving the discrete log problem more ef- ficiently than thought possible, combined with many data centers filled with number-crunching computers, itʼs possi- ble that theyʼll break the key exchange itself. CHAPTER 15. SSL AND TLS 168 15.7 Attacks As with most attacks, attacks on TLS can usually be grouped into two distinct categories: 1. Attacks on the protocol itself, such as subverting the CA mechanism; 2. Attacks on a particular implementation or cipher, such as cryptanalytic attacks exploiting weaknesses in RC4, or timing attacks in a particular AES implementation. Unfortunately, SSL/TLS has had many successful attacks in both categories. This section is particularly about the lat- ter. CRIMEandBREACH CRIME3 is an attack by the authors of BEAST. Itʼs an innova- tive side channel attack that relies on TLS compression leak- ing information about secrets in the plaintext. In a related attack called BREACH4, the attackers accomplish the same effect using HTTP compression. That was predicted by the authors of the original paper, but the BREACH authors were the first to demonstrate it as a practical attack. The BREACH attack was more practically applicable, though: HTTP com- pression is significantly more common than TLS compres- sion. Both of these rely on encryption of a compressed plain- text, and their mechanisms are virtually identical: only the specific details related to HTTP compression or TLS com- pression are relevant. The largest difference is that with TLS compression, the entire stream can be attacked; with HTTP compression, only the body is compressed, so HTTP head- ers are safe. Since the attacks are otherwise extremely simi- lar, weʼll just talk about how the attack works in the abstract, 3 Compression Ratio Info-leak Made Easy 4 Browser Reconnaissance and Exfiltration via Adaptive Compression of Hypertext CHAPTER 15. SSL AND TLS 169 by explaining how attackers can learn information about the plaintext if it is compressed before encryption. The most common algorithm used to compress both HTTP and TLS [Hol] is called DEFLATE. The exact mechan- ics of DEFLATE arenʼt too important, but the important fea- ture is that byte sequences that occur more than once can be efficiently stored. When a byte sequence recurs5, instead of recording the same sequence, a reference is provided to the previous sequence: instead of repeating the sequence, it says “go back and look at the thing I wrote N bytes ago”. Suppose an attacker can control the plaintext. For ex- ample, the attacker injects an invisible iframe6 or some JavaScript code that fires off many requests. The attacker needs some way to inject their guess of the secret so that their guess occurs in the plaintext, such as the query param- eters7. Usually, they can prefix their guess with something known. Suppose theyʼre trying to intercept an authentica- tion token being supplied in the body of the web page: <input type=”hidden” name=”csrf-token” value=”TOKEN_VALUE_HERE”> … they can prefix the guess with the known part of that. In this case, itʼs aCSRF token; a random token selected by the server and given to the client. This token is intended to prevent malicious third party websites from using the ambi- ent authority present in the browser (such as session cook- ies) to make authenticated requests. Without a CSRF token, a third party website might just make a request to the vulner- able website; the web browser will provide the stored cookie, and the vulnerable website will mistake that for an authen- ticated request. 5 Within limits; specifically within a sliding window, usually 32kB big. Otherwise, the pointers would grow bigger than the sequences theyʼre meant to compress. 6 An iframe is a web page embedded within a page. 7 The key-value pairs in a URL after the question mark, e.g. the x=1&y=2 in http://example.test/path?x=1&y=2. CHAPTER 15. SSL AND TLS 170 The attacker makes guesses at the value of the token, starting with the first byte, and moving on one byte at a time.8 When they guess a byte correctly, the ciphertext will be just a little shorter: the compression algorithm will no- tice that itʼs seen this pattern before, and be able to compress the plaintext before encrypting. The plaintext, and hence the compressed ciphertext, will therefore be smaller. They can do this directly when the connection is using astream ci- pher or a similar construction such asCTR mode, since they produce ciphertexts that are exactly as long as the plaintexts. If the connection is using a block-oriented mode such asCBC mode, the difference might get lost in the block padding. The attacker can solve that by simply controlling the prefix so that the difference in ciphertext size will be an entire block. Once theyʼve guessed one byte correctly, they can move on to the next byte, until they recover the entire token. This attack is particularly interesting for a number of reasons. Not only is it a completely new class of attack, widely applicable to many cryptosystems, but compressing the plaintext prior to encryption was actively recommended by existing cryptographic literature. It doesnʼt require any particularly advanced tools: you only need to convince the user to make requests to a vulnerable website, and you only need to be able to measure the size of the responses. Itʼs also extremely effective: the researchers that published BREACH report being able to extract secrets, such asCSRF tokens, within one minute. In order to defend against CRIME, disable TLS compres- sion. This is generally done in most systems by default. In order to defend against BREACH, there are a number of pos- sible options: • Donʼt allow the user to inject arbitrary data into the re- quest. • Donʼt put secrets in the response bodies. 8 They may be able to move more quickly than just one byte at a time, but this is the simplest way to reason about. CHAPTER 15. SSL AND TLS 171 • Regenerate secrets such as CSRF tokens liberally, for example, each request. Itʼs a bad idea to simply unconditionally turn off HTTP compression. While it does successfully stop the attack, HTTP compression is a critical tool for making the Web faster. Web apps that consist of a static front-end (say, using HTML5, JS, CSS) and that only operate using an API, say, JSON over REST, are particularly easy to immunize against this attack. Just disable compression on the channel that ac- tually contains secrets. It makes things slower, of course, but at least the majority of data can still be served over a CDN. 15.8 HSTS HSTS is a way for web servers to communicate that what theyʼre saying should only ever be transferred over a secure transport. In practice, the only secure transport that is ever used for HTTP is TLS. Using HSTS is quite simple; the web server just adds an extra Strict-Transport-Security header to the response. The header value contains a maximum age (max-age), which determines how long into the future the browser can trust that this website will be HSTS-enabled. This is typically a large value, such as a year. Browsers successfully remembering that a particular host is HSTS- enabled is very important to the effectiveness of the scheme, as weʼll see in a bit. Optionally, the HSTS header can include the includeSubDomains directive, which details the scope of the HSTS policy. [HJB] There are several things that a conforming web browser will do when communicating with an HSTS-enabled web- site: • Whenever there is any attempt to make any connec- tion to this website, it will always be done over HTTPS. CHAPTER 15. SSL AND TLS 172 The browser does this completely by itself,before mak- ing the request to the website. • If there is an issue setting up a TLS connection, the website will not be accessible, instead of simply dis- playing a warning. Essentially, HSTS is a way for websites to communicate that they only support secure transports. This helps protect the users against all sorts of attacks including both passive eavesdroppers (that were hoping to see some credentials ac- cidentally sent in plaintext), and active man-in-the-middle attacks such as SSL stripping. HSTS also defends against mistakes on the part of the web server. For example, a web server might accidentally pull in some executable code, such as some JavaScript, over an insecure connection. An active attacker that can inter- cept and modify that JavaScript would then have complete control over the (supposedly secure) web site. As with many TLS improvements, HSTS is not a panacea: it is just one tool in a very big toolbox of stuff that we have to try and make TLS more secure. HSTS only helps to en- sure that TLS is actually used; it does absolutely nothing to prevent attacks against TLS itself. HSTS can suffer from a chicken-or-egg problem. If a browser has never visited a particular HSTS-enabled website before, itʼs possible that the browser doesnʼt know that the website is HSTS-enabled yet. Therefore, the browser may still attempt a regular HTTP connection, vulnerable to an SSL stripping attack. Some browsers have attempted to mit- igate this issue by having browsers come pre-loaded with a list of HSTS websites. 15.9 Certificatepinning Certificate pinning is an idea thatʼs very similar to HSTS, taken a little further: instead of just remembering that a CHAPTER 15. SSL AND TLS 173 particular server promises to support HTTPS, weʼll remem- ber information about their certificates (in practice, weʼll re- member a hash of the public key). When we connect to a server that we have some stored information about, weʼll ver- ify their certificates, making it much harder for an impostor to pretend to be the website weʼre connecting to using a dif- ferent certificate. Browsers originally implemented certificate pinning by coming shipped with a list of certificates from large, high- profile websites. For example, Google included whitelisted certificates for all of their services in their Chrome browser. 15.10 Secureconfigurations In this section, we are only talking about configuration op- tions such as which ciphers to use, TLS/SSL versions, etc. Weʼre specificallynot talking about TLS configurations in the sense of trust models, key management, etc. There are several issues with configuring TLS securely: 1. Often, the defaults are unsafe, and people are unaware that they should be changed. 2. The things that constitute a secure TLS configuration can change rapidly, because cryptanalysis and practi- cal attacks are continuously improving. 3. Old clients that still need to be supported sometimes mean that you have to hang on to broken configuration options. A practical example of some of these points coming to- gether is the BEAST attack. That attack exploited weak- nesses in CBC ciphersuites in TLSv1.0, which were parts of the default ciphersuite specifications everywhere. Many people recommended defending against it by switching to RC4. RC4 was already considered cryptographically weak, later cryptanalysis showed that RC4 was even more broken CHAPTER 15. SSL AND TLS 174 than previously suspected. The attack had been known for years before being practically exploited; it was already fixed in TLSv1.1 in 2006, years before the BEAST paper being pub- lished. However, TLSv1.1 had not seen wide adoption. Good advice necessarily changes over time, and itʼs im- possible to do so in a persistent medium such as a book. In- stead, you should look at continuously updated third party sources such asQualys SSL Labs. They provide tests for both SSL clients and servers, and extensive advice on how to im- prove configurations. That said, there are certainly some general things we want from a TLS configuration. TODO: say stuff we generally want from TLS configura- tions TODO: http://tools.ietf.org/html/ draft-agl-tls-chacha20poly1305-01 16 OpenPGPandGPG 16.1 Description OpenPGP is an open standard that describes a method for en- crypting and signing messages. GPG is the most popular im- plementation of that standard1, available under a free soft- ware license. Unlike TLS, which focuses on data in motion, OpenPGP focuses on data at rest. A TLS session is active: bytes fly back and forth as the peers set up the secure channel. An OpenPGP interaction is, by comparison, static: the sender computes the entire message up front using information shared ahead of time. In fact, OpenPGP doesnʼt insist that anything is sent at all: for example, it can be used to sign software releases. Like TLS, OpenPGP is a hybrid cryptosystem. Users have key pairs consisting of a public key and a private key. Pub- lic key algorithms are used both for signing and encryption. Symmetric key algorithms are used to encrypt the message 1 GPG 2 also implements S/MIME, which is unrelated to the OpenPGP standard. This chapter only discusses OpenPGP. 175 CHAPTER 16. OPENPGP AND GPG 176 body; the symmetric key itself is protected usingpublic-key encryption. This also makes it easy to encrypt a message for multiple recipients: only the secret key has to be encrypted multiple times. 16.2 Theweboftrust Earlier, we saw that TLS typically uses trusted root certifi- cates to establish that a particular peer is who they claim to be. OpenPGP does not operate using such trusted roots. In- stead, it relies on a system called the Web of Trust: a friend- of-a-friend honor system that relies on physical meetings where people verify identities. The simplest case is a directly trusted key. If we meet up in person, we can verify each otherʼs identities. Perhaps we know each other, or perhaps weʼd check some form of identification. Then, we sign each otherʼs keys. Because I know the key is yours, I know that you can read the messages encrypted by it, and the other way around. Pro- vided you donʼt share your key, I know thatonly you can read those messages. No-one can replace my copy of your key, because they wouldnʼt be able to forge my signature on it. Thereʼs a direct trust link between the two of us, and we can communicate securely. A slightly more complicated case is when a friend of yours would like to send me a message. Weʼve never met: heʼs never signed my key, nor have I signed theirs. However, I have signed your key, and vice versa. Youʼve signed your friendʼs key, and vice versa. Your friend can choose to lever- age your assertion that Iʼm indeed the person in possession of that key you signed, and use that to communicate with me securely. You might wonder how your friend would ever see sig- natures that you placed on my key. This is because keys and signatures are typically uploaded to a network of key servers, making them freely available to the world. CHAPTER 16. OPENPGP AND GPG 177 The above system can be extended to multiple layers of friends. It relies in no small part in communities being linked by signatures, which is why many community events include key signing parties, where people sign each otherʼs keys. For large events, such as international programming conferences, this system is very effective. The main weak- ness in this system are “islands” of trust: individuals or small groups with no connections to the rest of the web. Of course, this is only the default way to use OpenPGP. Thereʼs nothing stopping you from shipping a particular pub- lic key as a part of a software package, and using that to sign messages or verify messages. This is analogous to how you might want to ship a key with a client certificate, or a custom CHAPTER 16. OPENPGP AND GPG 178 root CA certificate, with TLS. 17 Off-The-Record Messaging(OTR) 17.1 Description OTR messagingis a protocol for securing instant messaging communication between people [BGB04]. It intends to be the online equivalent of a private, real-life conversation. It encrypts messages, preventing eavesdroppers from reading them. It also authenticates peers to each other, so they know who theyʼre talking to. Despite authenticating peers, it is de- signed to be deniable: participants can later deny to third parties anything they said to each other. It is also designed to have perfect forward secrecy: even a compromise of a long-term public key pair doesnʼt compromise any previous conversations. The deniability and perfect forward secrecy prop- erties are very different from those of other systems such as OpenPGP. OpenPGP intentionally guarantees non- repudiability. Itʼs a great property if youʼre signing software packages, talking on mailing lists or signing business in- 179 CHAPTER 17. OFF-THE-RECORD MESSAGING (OTR) 180 voices, but the authors ofOTR argue that those arenʼt desir- able properties for the online equivalent of one-on-one con- versations. Furthermore, OpenPGPʼs static model of com- munication makes the constant key renegotiation to facili- tate OTRʼs perfect forward secrecy impossible. OTR is typically configured opportunistically, which means that it will attempt to secure any communication be- tween two peers, if both understand the protocol, without interfering with communication where the other peer does not. The protocol is supported in many different instant messaging clients either directly, or with a plugin. Because it works over instant messages, it can be used across many different instant messaging protocols. A peer can signal that they would like to speakOTR with an explicit message, called theOTR Query message. If the peer is just willing to speakOTR but doesnʼt require it, they can optionally invisibly add that information to a plaintext message. That happens with a clever system of whitespace tags: a bunch of whitespace such as spaces and tab charac- ters are used to encode that information. AnOTR-capable client can interpret that tag and start anOTR conversation; an client that isnʼt OTR-capable just displays some extra whitespace. OTR uses many of the primitives weʼve seen so far: • Symmetric key encryption (AES in CTR mode) • Message authentication codes(HMAC with SHA-1) • Diffie-Hellman key exchange OTRalso utilizes another mechanism, called the SMP, to check if peers arrived at the same shared secret. 17.2 Keyexchange In OTR, AKE relies heavily on Diffie-Hellman key exchange, extended with a significant number of extra, interlocking CHAPTER 17. OFF-THE-RECORD MESSAGING (OTR) 181 checks. The Diffie-Hellman exchange itself uses a fixed 1536- bit prime with a fixed generatorg. We suppose that two participants, named Alice and Bob want to communicate and are willing to exchange sensitive data with each other. Alice and Bob have a long-term DSA authentication key pair each, which weʼll call (pA, sA) and (pB, sB) respectively. The protocol also relies on a number of other primitives: • A 128-bit block cipher. InOTR, this is always AES. In this section, weʼll call block cipher encryption and de- cryption E and D, respectively. • A hash function,H. InOTR, this is SHA1. • A messageauthenticationcode , M. InOTR, this is HMAC- SHA1. • A signing function,S. Commitmessage Initially Alice and Bob are in a protocol state where they wait for the peer to initiate anOTR connection, and adver- tise their own capability of speakingOTR. Letʼs suppose that Bob chooses to initiate anOTRconver- sation with Alice. His client sends anOTR Commit Message, and then transitions to a state where he waits for a reply from from Aliceʼs client. To send a commit message, a client picks a random 128- bit valuer and a random 320-bit (or larger) Diffie-Hellman secret x. It then sendsE(r, gx) and H(gx) to the peer. Keymessage Aliceʼs client has received Bobʼs clientʼs advertisement to start anOTR session. Her client replies with a key message, which involves creating a new Diffie-Hellman key pair. She picks a 320-bit (or larger) Diffie-Hellman secrety and sends gy to Bob. CHAPTER 17. OFF-THE-RECORD MESSAGING (OTR) 182 RevealSignatureMessage Now that Alice has sent her public Diffie-Hellman key, Bob can complete his part of the Diffie-Hellman protocol. Alice canʼt continue yet, because she hasnʼt seen Bobʼs public key. When we discussed Diffie-Hellman, we noted that it does not authenticate the peer. Bob can compute a secret, but doesnʼt know heʼs talking to Alice. As with TLS and other systems using Diffie-Hellman, this problem is solved by au- thenticating the key exchange. After verifying that Aliceʼs public key is a valid value, Bob computes the shared secrets = ( gy)x. Using a key derivation function, he derives several keys froms: two AES keysc, c′, and four MAC keysm1, m′ 1, m2, m′ 2. He chooses an identification numberiB for his current Diffie-Hellman key pair(x, gx). This will be important once Alice and Bob generate new key pairs, which they will do later on in theOTR protocol. Bob computes: MB = Mm1(gx, gy, pB, iB) XB = ( pB, iB, S(pB, MB)) He sends Alicer, Ec(XB), Mm2(Ec(XB)). SignatureMessage Alice can now confirm sheʼs talking to Bob directly, because Bob signed the authenticator for the exchangeMB with his long-term DSA key. Alice can now also compute the shared secret: Bob has sent herr, which was previously used to encrypt Bobʼs Diffie- Hellman public key. She then computesH(gx) herself, to compare it against what Bob sent. By completing her side of the Diffie-Hellman exchange (s = ( gx)y), she derives the same keys: c, c′, m1, m′ 1, m2, m′ 2. Using m2, she can verify Mm2(Ec(XB)). Once that message is verified, she can safely decrypt it using her computedc. CHAPTER 17. OFF-THE-RECORD MESSAGING (OTR) 183 She can then also computeMB = Mm1(gx, gy, pB, iB), and verifies that it is the same as Bob sent. By verifying the signed portionS(pB, MB) against Bobʼs public key, she has now unambiguously tied the current interaction to Bobʼs long-term authentication key. She then computes the same values Bob computed to tie his long-term key to the short-term handshake, so that Bob can also authenticate her. She chooses an identifica- tion numberiA for her current DH keypair(y, gy), computes MA = Mm′ 1 (gy, gx, pA, iA) and XA = pA, iA, S(pA, MA). Fi- nally, she sends BobEc′ (XA), Mm′ 2 (Ec(XB)). AuthenticatingAlice Now Bob can also authenticate Alice, again by mirroring steps. First, he verifiesMm′ 2 (Ec(XB)). This allows him to check that Alice saw the sameXB he sent. Once he decryptsEc′ (XA), he has access toXA, which is Aliceʼs long-term public key information. He can then com- pute MA = Mm′ 1 (gy, gx, pA, iA) to compare it with the ver- sion Alice sent. Finally, he verifiesS(pA, MA) with Aliceʼs public key. Whathaveweaccomplished? If all checks succeed then Alice and Bob have completed an authenticated Diffie-Hellman exchange and have a shared secret that only the two of them know. Now that youʼve seen both sides of the authenticated handshake, you can see why so many different keys are de- rived from the Diffie-Hellman secret. Keys marked with a prime (′) are for messages originating from the second peer (the one responding to the advertisement, in our case, Al- ice); keys without a prime are for the initiating peer (in our case, Bob). CHAPTER 17. OFF-THE-RECORD MESSAGING (OTR) 184 17.3 Dataexchange TODO: Explain ( https://otr.cypherpunks.ca/ Protocol-v3-4.0.0.html), #33 PartIV Appendices 185 A Modulararithmetic Modular arithmetic is used for many public key cryptosys- tems, including public-key encryption algorithms like RSA and key exchange protocols like Diffie-Hellman. Modular arithmetic is something most people actually al- ready understand, they just donʼt know itʼs called that. We can illustrate the principles of modular arithmetic using a clock. For simplicityʼs sake, our demonstration 12-hour clock only shows hours, not minutes or seconds. Also unlike real clocks, the hour hand is never halfway in between two hours: it always shows an exact hour, such as 2 or 9. A.1 Additionandsubtraction It obviously makes sense to add hours on our clock: if itʼs 2 oʼclock now, and youʼd like to know what time it is five hours from now, you can add 5, and end up with 7, as you can see in Figure 1.2. 186 APPENDIX A. MODULAR ARITHMETIC 187 Figure 1.1: A clock, pointing to 2. Figure 1.2:2 + 5 = 7 , on the clock. APPENDIX A. MODULAR ARITHMETIC 188 Similarly, we can subtract times. If itʼs 10 oʼclock now, and youʼd like to know what time it was two hours ago, you subtract 2 and end up with 8. Figure 1.3:10 −2 = 8 , on the clock. The “weird” part is when you cross the boundary at 12. As far as the clock is concerned, thereʼs no real difference between 12 and 0. If itʼs 10 oʼclock now, itʼll be 2 oʼclock in four hours. If itʼs 2 oʼclock now, it was 9 oʼclock five hours ago. This is an example of whatʼs called “modular arithmetic”. The modulus, in this case, is 12. We can write the above equations as: (10 + 4) mod 12 = 2 (2 −5) mod 12 = 9 In these equations, the mod is an operator, giving the re- mainder after division. When we are dealing with modular APPENDIX A. MODULAR ARITHMETIC 189 arithmetic, where all operations are affected by the modu- lus instead of a simple single operation, weʼll instead write (mod 12) at the end of the equation and use an≡sign instead of an equals sign (=): 10 + 4 ≡2 ( mod 12) 2 −5 ≡9 ( mod 12) This is read as “ten plus four is equivalent to two, modulo twelve” and “two minus five is equivalent to nine, modulo twelve”. That might seem like a trivial notational hack now, but the difference will become apparent once we start apply- ing tricks for doing more complex modular computations, like multiplication and exponentiation. In general, we call two numbersequivalent modulo some modulusif dividing them by the modulus leaves the same re- mainder. We can illustrate this with our previous examples: 10 + 4 = 14 leaves a remainder of 2 when divided by 12, so it is equivalent to 2 modulo 12. For negative numbers, weʼll always use positive remainders. For example,2 −5 ≡ 9 (mod 12). This is exactly the way a clock works as well: if itʼs 2 oʼclock now, then five hours ago was “nine oʼclock”, not “minus three oʼclock”. A.2 Primenumbers Prime numbers are wonderful kinds of numbers that come back in many branches of mathematics. Anything I say about them probably wonʼt do them justice; but weʼre in a practical book about applied cryptography, so weʼll only see a few properties. A prime number is a number that is divisible only by two numbers: 1 and itself. For example, 3 is a prime number, but 4 is not, because it can be divided by 2. Any number can be written as a product of prime factors: a bunch of prime numbers multiplied together. That prod- uct is called a prime factorization. For example, 30 can be APPENDIX A. MODULAR ARITHMETIC 190 factorized into 2, 3 and 5: 30 = 2 ·3 ·5 Sometimes, a prime number will occur more than once in a factorization. For example, the factorization of 360 has 2 in it three times, and three in it twice: 360 = 2 3 ·32 ·5 The factorization of any prime number is just that prime number itself. Modern mathematics no longer considers 1 to be a prime number, even though it is only divisible by 1 and itself (1 again). Under this convention, every number not onlyhas a factorization, but that factorization isunique. Otherwise, 4 could be factored not only as2·2, but also as2·2·1, 2·2·1·1, and so on. The uniqueness of factorization helps in some important proofs in number theory. Also, 0 isnot a prime number, as it is divisible by many numbers: all numbers except 0 itself. Two numbers are called coprime when their greatest common divisor is 1, or, to put it in another way, they donʼt share any prime factors. Since the only prime factor a prime has is itself, that means that all prime numbers are also co- prime. More generally, a prime is coprime to any number that isnʼt a multiple of that prime. A.3 Multiplication You might remember you were first taught multiplication as repeated addition: n ·x = x + x + . . . + x\| {z } n times Modular multiplication is no different. You can compute modular multiplication by adding the numbers together, APPENDIX A. MODULAR ARITHMETIC 191 and taking the modulus whenever the sum gets larger than the modulus. You can also just do regular multiplication, and then take the modulus at the end. A.4 Divisionandmodularinverses Division is defined as the inverse of multiplication. So,a·b ≡ c (mod m), thenc b ≡a (mod m). For example,5 ·6 ≡2 ( mod 7); so: 2 6 ≡5 ( mod 7). This is because5 ·6 = 30 , which leaves a remainder of 2 when divided by 7. Usually, instead of using division directly, weʼll multiply using something called a modular inverse. The modular in- verse ofa is a number, that when you multiply it witha, you get 1. This is just like the inverse of a number in regular arith- metic: x ·1 x = 1 . Like in regular arithmetic, not all numbers have modular inverses. This is the equivalent of dividing by zero in regular arithmetic. There are two algorithms that are used to compute mod- ular inverses: the extended Euclidean algorithm, and with the help of Eulerʼs theorem. TheextendedEuclideantheorem TODO: explain, and how you can get modular inverses with it UsingEulerʼstheorem Eulerʼs theorem states that if two numbersa and n are co- prime, then: aϕ(n) ≡1 ( mod n) In that equation,ϕ is Eulerʼs totient function, which counts the amount of numbers that are coprime to (and less than or APPENDIX A. MODULAR ARITHMETIC 192 equal to) its argument. As an example, the totient of 10 is 4, as 1, 3, 7, and 9 do not have common prime factors with 10. We can use Eulerʼs theorem to find the multiplicative in- verse ofa. If we just multiply both sides of the equation by a−1, we get: aϕ(n)−1 ≡a−1 (mod n) That gives us a direct formula for computinga−1. Unfortu- nately, this is still generally less interesting than using the extended Euclidean algorithm, for two reasons: 1. It requires computing the totient function, which is harder than running the extended Euclidean algo- rithm in the first place, unless you happen to know the prime factors ofn. 2. Modular exponentiation is computationally expen- sive. One exception to that rule is for prime moduli. Since a prime is coprime to every other number, and since there are p −1 numbers smaller thanp, ϕ(p) = p −1. So, for a prime modulus, the modular inverse ofa is simply: a−1 ≡aϕ(p)−1 ≡ap−2 (mod p) This still requires us to be able to efficiently raisea to a power using modular arithmetic. Weʼll discuss how you can do that efficiently in the next section. A.5 Exponentiation Like multiplication is taught as repeated addition, exponen- tiation can be thought of as repeated multiplication: an = a ·a ·. . . ·a\| {z } n times APPENDIX A. MODULAR ARITHMETIC 193 As with multiplication, itʼs possible to compute modular exponentiation by performing regular exponentiation, and then taking the modulus at the end. However, this is very inefficient, particularly for largen: the product quickly be- comes far too large. Fortunately, it is possible to compute modular exponen- tiation much more efficiently. This is done by splitting the problem up into smaller sub-problems. For example, in- stead of computing220 directly you could split it up: 220 = (2 10)2 210 is something you can compute on your hands: start at 2, which is21, and then keep multiplying by two. Every time you multiply by two, the exponent goes up by 1, so by the time youʼve counted all your fingers (assuming you have ten of them), youʼre done. The result is 1024. So: 220 ≡(210 mod 15)2 (mod 15) ≡(1024 mod 15)2 (mod 15) ≡42 (mod 15) ≡16 ( mod 15) ≡1 ( mod 15) A.6 Exponentiationbysquaring A particularly efficient way to do it on computers is split- ting the exponent up into a sum of powers of two. This is called exponentiation by squaring, or sometimes also bi- nary exponentiation. Suppose we want to compute 3209 (mod 19). First, we split up 209 into a sum of powers of two. This process is essentially just writing 209 down in binary: 0b11010001. Thatʼs very practical if the computation is be- ing performed by a computer, because thatʼs typically how APPENDIX A. MODULAR ARITHMETIC 194 the computer had the number stored in the first place. 209 = 1 ·27 +1 ·26 +0 ·25 +1 ·24 +0 ·23 +0 ·22 +0 ·21 +1 ·20 = 1 ·128 +1 ·64 +0 ·32 +1 ·16 +0 ·8 +0 ·4 +0 ·2 +1 ·1 = 128 +64 +16 +1 We use that expansion into a sum of powers of two to rewrite the equation: 3209 = 3 128+64+16+1 = 3 128 ·364 ·316 ·31 Now, we need to compute those individual powers of 3: 1, 16, 64 and 128. A nice property of this algorithm is that we donʼt actually have to compute the big powers separately from scratch. We can use previously computed smaller powers to compute the larger ones. For example, we need both3128 (mod 19) and 364 (mod 19), but you can write the former in terms of the latter: 3128 mod 19 = (3 64 mod 19)2 (mod 19) Letʼs compute all the powers of 3 we need. For sake of brevity, we wonʼt write these out entirely, but remember that all tricks weʼve already seen to compute these still apply: 316 ≡17 ( mod 19) 364 ≡(316)4 ≡174 ≡16 ( mod 19) 3128 ≡(364)2 ≡162 ≡9 ( mod 19) Filling these back in to our old equation: 3209 = 3 128 ·364 ·316 ·31 (mod 19) ≡9 ·16 ·17 ·3 ( mod 19) This trick is particularly interesting when the exponent is a very large number. That is the case in many cryptographic applications. For example, in RSA decryption, the exponent APPENDIX A. MODULAR ARITHMETIC 195 is the private keyd, which is usually more than a thousand bits long. Keep in mind that this method will still leak tim- ing information, so itʼs only suitable for offline computation. Modular exponentiation can also be computed using a tech- nique called a Montgomery ladder, which weʼll see in the next section. Many programming languages provide access to specific modular exponentiation functions. For example, in Python, pow(e, x, m) performs efficient modular exponentiation. However, the expression(e x) % m will still use the in- efficient method. A.7 Montgomeryladderexponentiation As we mentioned before, the exponentiation by squaring algorithm is simple and fast, but the time it takes to com- plete depends on the value of the exponent. Thatʼs bad, be- cause the exponent is usually a secret value, such as a Diffie- Hellman secret or the private exponentd in RSA. The Montgomery ladder is an algorithm that resolves this by guaranteeing the same number of operations irre- spective of the particular value of the exponent. It was origi- nally applied for efficient scalar multiplications over elliptic curves, but the mathematics works for many other systems: specifically, for any abelian group. [JY02] Derivingtheladder This is an optional, in-depth section. It almost certainly wonʼt help you write bet- ter software, so feel free to skip it. It is only here to satisfy your inner geekʼs curiosity. This section involves a good deal of arithmetic tricks. You might want to get out some paper and pencil to follow along. APPENDIX A. MODULAR ARITHMETIC 196 Like with exponentiation by squaring, we start by looking at the binary expansion of the exponentk. Generally, anyk can be written as a sum (P) of some powers of two (2i). If 2j appears in the binary expansion, weʼll say thatkj = 1 ; if it doesnʼt, weʼll say thatkj = 0 . That gives us: k = t−1X i=0 2iki That definition might look scary, but all youʼre really doing here is definingki as bit ofk at positioni. The sum goes over all the bits: ifk is t bits long, and we start indexing at 0, the index of the highest bit ist −1, and the index of the lowest bit is 0. For example, the binary expansion of the number 6 is 0b110. That number is three bits long, sot = 3 . So: 6 = t−1X i=0 2iki = 2X i=0 2iki = k2 ·22 + k1 ·21 + k0 ·20 = 1 ·22 + 1 ·21 + 0 ·20 So, (k2, k1, k0) = (1 , 1, 0). The next few steps donʼt make a lot of sense until you see them come together at the end, so bear with me and check that the math works out. Weʼll define a related sum,Lj: Lj = t−1X i=j 2i−jki APPENDIX A. MODULAR ARITHMETIC 197 For example,L1 (still withk = 6 ) becomes: L1 = 2X i=1 2i−1ki = 2 1 ·k2\| {z } i=2 + 20 ·k1\| {z } i=1 = 2 ·1 + 1 ·1 = 3 Essentially, Lj is justk shifted to the right byj bits. Shifting to the right by one bit is the same thing as flooring division by two, just like right-shifting by a decimal digit is the same thing as flooring division by 10. For example: 73, shifted one decimal digit to the right is 7; 0b101 (5) shifted one binary digit (bit) to the right is 0b10 (2). Analogously, shifting left is the inverse operation, and is equivalent tomultiplying by two. Next, weʼll perform a little arithmetical hocus pocus. First of all: Lj = 2 ·Lj+1 + kj While you can verify this arithmetically, the easiest way to check this is to think of it in terms of right and left shifts. If you shiftk to the right byj positions, that k = 0b110010111 Lj = L2 = 0b1100101 Lj+1 = L3 = 0b110010 2 ·Lj+1 = 2 ·L3 = 0b1100100 You can visually verify thatL2 is indeed L3, shifted one to the left (which is the same thing as multiplying by two), plus that one bitkj that “fell off” when shifting right.kj is the last bit ofLj; in this case it happens to be 1, but it could equally well have been 0. APPENDIX A. MODULAR ARITHMETIC 198 We define another very simple functionHj: Hj = Lj + 1 ⇐⇒Lj = Hj −1 Starting from our previous result: Lj = 2 ·Lj+1 + kj ⇓(Lj+1 = Hj+1 −1) Lj = Lj+1 + kj + Hj+1 −1 ⇓(Lj+1 = Hj+1 −1) Lj = 2 ·Hj+1 + kj −2 We can combine these to produce an inductive way to com- pute Lj and Hj: Lj = ( 2Lj+1 if kj = 0 , Lj+1 + Hj+1 if kj = 1 . Hj = ( Lj+1 + Hj+1 if kj = 0 , 2Hj+1 if kj = 1 . Remember that weʼre doing this to computegk. Letʼs write the exponentiation out: gLj = ( g2Lj+1 = gLj+1 2 if kj = 0 , gLj+1+Hj+1 = gLj+1 ·gHj+1 if kj = 1 . gHj = ( gLj+1+Hj+1 = gLj+1 ·gHj+1 if kj = 0 , g2Hj+1 = gHj+1 2 if kj = 1 . Remember that Lj is k right-shifted by j bits, so L0 is k shifted right by 0 bits, or justk itself. That means gk, the number weʼre trying to compute, is the same thing asgL0. By starting atgLt−1 (g raised to the power of the leftmost bit of k) and iteratively making our way down togL0 = gk, we have an elegant inductive method for computinggk based on two simple recursive rules. APPENDIX A. MODULAR ARITHMETIC 199 The important part about this algorithm is the constant number of operations. Ifkj = 0 , computinggLj involves one squaring andgHj involves one multiplication; ifkj = 1 , itʼs the other way around. No matter what any of the bits ofk are, you need one squaring operation and one multiplication per bit. ImplementingtheMontgomeryladderinPython The Python implementation of this algorithm, applied to modular exponentiation, is surprisingly terse: def montgomery(x, exponent, modulus): x1, x2 = x, x 2 high_bit, remaining_bits = bits(exponent) for bit in remaining_bits: if bit == 0: x2 = x1 x2 x1 = x1 ** 2 else: x1 = x1 * x2 x2 = x2 ** 2 x1, x2 = x1 % modulus, x2 % modulus return x1 This code block doesnʼt show the definition ofbits: it produces the binary expansion of its argument. Python doesnʼt provide that by default;bin is close, but that pro- duces a string:bin(100) evaluates to0b1100100. The a, *b = bits(...) construct assigns the first item inbits(. ..) to a, and all remaining bits tob, effectively just skipping the first bit. The important thing to note here is that no matter what the particular value of the exponent is, there is one squaring, one multiplication, and one modulo operation per bit. Keep in mind that this doesnʼt necessarily make the entire algo- rithm take constant time, because the individual squaring and multiplication operations are not necessarily constant time. APPENDIX A. MODULAR ARITHMETIC 200 A.8 Discretelogarithm Just like subtraction is the inverse of addition, and division is the inverse of multiplication, logarithms are the inverse of exponentiation. In regular arithmetic,bx = y, ifx = logb y. This is pronounced “b raised to the powerx is y”, and “the logarithm ofy with respect tob is x”. The equivalent of this in modular arithmetic is called a “discrete logarithm”. As with division, if you start from the definition as the inverse of a different operator, itʼs easy to come up with ex- amples. For example, since36 ≡ 9 ( mod 15), we can de- fine 6 ≡log3 9 ( mod 15). Unlike modular inverses, comput- ing discrete logarithms is generally hard. There is no for- mal proof that computing discrete logarithms isintrinsically complex; we just havenʼt found any efficient algorithms to do it. Because this field has gotten extensive research and we still donʼt have very fast general algorithms, we consider it safe to base the security of protocols on the assumption that computing discrete logs is hard. There is one theoretical algorithm for computing dis- crete logarithms efficiently. However, it requires a quantum computer, which is a fundamentally different kind of com- puter from the classical computers we use today. While we can build such computers, we can only build very small ones. The limited size of our quantum computers strongly limits which problems we can solve. So far, theyʼre much more in the realm of the kind of arithmetic a child can do in their head, than ousting the top of the line classical computers from the performance throne. The complexity of computing discrete logarithms, to- gether with the relative simplicity of computing its inverse, modular exponentiation, is the basis for many public key cryptosystems. Common examples include the RSA encryp- tion primitive, and the Diffie-Hellman key exchange proto- col. While cryptosystems based on the discrete logarithm problem are currently considered secure with appropri- APPENDIX A. MODULAR ARITHMETIC 201 ate parameter choices, there are certainly ways that could change in the future. For example: • Theoretical breakthroughs in number theory could make discrete logarithms significantly easier to com- pute than we currently think. • Technological breakthroughs in quantum computing could lead to large enough quantum computers. • Technological breakthroughs in classical computing as well as the continuous gradual increases in perfor- mance and decreases in cost could increase the size of some problems that can be tackled using classical computers. Discrete logarithm computation is tightly linked to the problem of number factorization. They are still areas of ac- tive mathematical research; the links between the two prob- lems are still not thoroughly understood. That said, there are many similarities between the two: • Both are believed to be hard to compute on classical computers, but neither has a proof of that fact. • They can both be efficiently computed on quantum computers using Shorʼs algorithm. • Mathematical advances in one are typically quickly turned into mathematical advances in the other. A.9 Multiplicativeorder Given integera and positive integerb with gcd(a, b) = 1 , the multiplicative orderof a (mod b) is the smallest positive inte- ger k such thatak = 1 ( mod b). B Ellipticcurves Like modular arithmetic, elliptic curve arithmetic is used for many public key cryptosystems. Many cryptosystems that traditionally work with modular arithmetic, such as Diffie- Hellman and DSA, have an elliptic curve counterpart. Elliptic curves are curves with the following form: y2 = x3 + ax + b This is called the “short Weierstrass form”, and is the most common form when talking about elliptic curves in general. There are several other forms which mostly have applica- tions in cryptography, notably the Edwards form: x2 + y2 = 1 + dx2y2 We can define addition of points on the curve. TODO: Move the Abelian group thing somewhere else, since it applies to our fields thing as well All of this put together form something called an Abelian group. Thatʼs a scary-sounding mathematical term that al- most everyone already understands the basics of. Specifi- cally, if you know how to add integers (. . . −2, −1, 0, 1, 2, . . .) 202 APPENDIX B. ELLIPTIC CURVES 203 together, you already know an Abelian group. An Abelian group satisfies five properties: 1. If a and b are members of the Abelian group and⋆ is the operator, thena⋆b is also a member of that Abelian group. Indeed, any two integers added together always get you another integer. This property is calledclosure, or, we say that the group isclosed under addition (or whatever the name is of the operation weʼve defined). 2. If a, b and c are members of the Abelian group, the or- der of operations doesnʼt matter; to put it differently: we can move the brackets around. In equation form: (a ⋆ b ) ⋆ c = a ⋆ (b ⋆ c ). Indeed, the order in which you add integers together doesnʼt matter; they will always sum up to the same value. This property is calledasso- ciativity, and the group is said to beassociative. 3. Thereʼs exactly one identity elementi, for whicha⋆i = i ⋆ a = a. For integer addition, thatʼs zero:a + 0 = 0 + a = a for all a. 4. For each element a, thereʼs exactly one inverse ele- ment b, for whicha ⋆ b = b ⋆ a = i, wherei is the iden- tity element. Indeed, for integer addition,a + (−a) = (−a) + a = 0 for all a. 5. The order of elements doesnʼt matter for the result of the operation. For all elementsa, b, a ⋆ b = b ⋆ a . This is known ascommutativity, and the group is said to be commutative. The first four properties are called group properties and make something a group; the last property is what makes a group Abelian. We can see that our elliptic curve, with the point at infin- ity and the addition operator, forms an Abelian group: 1. If P and Q are two points on the elliptic curve, then P + Q is also always a point on the curve. APPENDIX B. ELLIPTIC CURVES 204 2. If P, Q, andR are all points on the curve, thenP +(Q+ R) = ( P + Q) + R, so the elliptic curve is associative. 3. Thereʼs an identity element, our point at infinityO. For all points on the curveP, P + O = O + P = P. 4. Each element has an inverse element. This is easiest explained visually TODO: Explain visually 5. The order of operations doesnʼt matter,P +Q = Q+P for allP, Q on the curve. B.1 Theellipticcurvediscretelogproblem TODO: explain fully As with the regular discrete log problem, the elliptic curve discrete log problem doesnʼt actually have a formal proof that the operation is “hard” to perform: we just know that there is no publicly available algorithm to do it effi- ciently. Itʼs possible, however unlikely, that someone has a magical algorithm that makes the problem easy, and that would break elliptic curve cryptography completely. Itʼs far more likely that we will see a stream of continuous improvements, which coupled with increased computing power eventually eat away at the security of the algorithm. C Side-channelattacks C.1 Timingattacks AEScachetiming http://tau.ac.il/~tromer/papers/cache.pdf Ellipticcurvetimingattacks TODO: Explain why the edwards form is great? C.2 Powermeasurementattacks TODO: Say something here. 205 PartV Glossary 206 APPENDIX C. SIDE-CHANNEL ATTACKS 207 AEAD Authenticated Encryption with Associated Data AEADmode Class ofblock ciphermode of operationthat pro- vides authenticated encryption, as well as authenticat- ing some unencrypted associated data AES Advanced Encryption Standard AKE authenticated key exchange ARX add, rotate, XOR asymmetric-keyalgorithm See public-key algorithm asymmetric-keyencryption See public-key encryption BEAST Browser Exploit Against SSL/TLS blockcipher Symmetric encryption algorithm that en- crypts and decrypts blocks of fixed size Carter-WegmanMAC Reusable message authentication code scheme built from aone-time MAC. Combines benefits of performance and ease of use CBC cipher block chaining CBCmode Cipher block chaining mode; common mode of operation where the previous ciphertext block is XORed with the plaintext block during encryption. Takes an initialization vector, which assumes the role of the “block before the first block” CDN content distribution network cross-siterequestforgery Kind of attack where a mali- cious website tricks the browser into making requests to another website. Can be prevented by properly au- thenticating requests instead of relying on ambient au- thority such as session cookies APPENDIX C. SIDE-CHANNEL ATTACKS 208 CSPRNG cryptographically secure pseudorandom number generator CSRF cross-site request forgery CTRmode Counter mode; anoncecombined with a counter produces a sequence of inputs to the block cipher; the resulting ciphertext blocks are the keystream DES Data Encryption Standard ECBmode Electronic code book mode; mode of operation where plaintext is separated into blocks that are en- crypted separately under the same key. The default mode in many cryptographic libraries, despite many security issues encryptionoracle An oraclethat will encrypt some data FIPS Federal Information Processing Standards GCM Galois Counter Mode GCMmode Galois counter mode; AEAD mode combining CTR modewith aCarter-Wegman MAC GMAC message authentication code part of GCM mode used separately HKDF HMAC-based (Extract-and-Expand) Key Derivation Function HMAC Hash-based Message Authentication Code HSTS HTTP Strict Transport Security initializationvector Data used to initialize some algo- rithms such asCBC mode. Generally not required to be secret, but required to be unpredictable. Compare nonce, salt IV initialization vector APPENDIX C. SIDE-CHANNEL ATTACKS 209 KDF key derivation function keyagreement See key exchange keyexchange The process of exchanging keys across an in- secure medium using a particular cryptographic proto- col. Typically designed to be secure against eavesdrop- pers. Also known as key agreement keyspace The set of all possible keys MAC message authentication code messageauthenticationcode Small piece of information used to verify authenticity and integrity of a message. Often called a tag MITM man-in-the-middle modeofoperation modesofoperation Generic construction that encrypts and decrypts streams, built from a block cipher nonce Number used once. Used in many cryptographic protocols. Generally does not have to be secret or un- predictable, but does have to be unique. Compareini- tialization vector, salt OCB offset codebook OCBmode Offset codebook mode; high-performance AEAD mode, unfortunately encumbered by patents one-timeMAC message authentication codethat can only be used securely for a single message. Main benefit is in- creased performance over re-usableMAC oracle A “black box” that will perform some computation for you OTR off-the-record APPENDIX C. SIDE-CHANNEL ATTACKS 210 OTRmessaging Off-the-record messaging, messaging pro- tocol that intends to mimic the properties of a real- life private conversation. Piggy-backs onto existing in- stant messaging protocols PRF pseudorandom function PRNG pseudorandom number generator PRP pseudorandom permutation public-keyalgorithm Algorithm that uses a pair of two re- lated but distinct keys. Also known asasymmetric-key algorithm. Examples includepublic-key encryptionand most key exchangeprotocols public-keyencryption Encryption using a pair of distinct keys for encryption and decryption. Also known as asymmetric-key encryption. Contrast with secret-key encryption RSA Rivest Shamir Adleman salt Random data that is added to a cryptographic primitive (usually a one-way function such as a cryptographic hash function or a key derivation function) Customizes such functions to produce different outputs (provided the salt is different). Can be used to prevent e.g. dictio- nary attacks. Typically does not have to be secret, but secrecy may improve security properties of the system. Compare nonce, initialization vector secret-keyencryption Encryption that uses the same key for both encryption and decryption. Also known as symmetric-key encryption. Contrast withpublic-keyen- cryption SMP socialist millionaire protocol streamcipher Symmetric encryption algorithm that en- crypts streams of arbitrary size APPENDIX C. SIDE-CHANNEL ATTACKS 211 substitution-permutationnetwork Generic design for block ciphers where the block is enciphered by repeated substitutions and permutations symmetric-keyencryption See secret-key encryption Index A AEAD, 207 AEAD mode, 207 AES, 207 AKE, 207 ARX, 207 asymmetric-key algorithm, 207 asymmetric-key encryption, 207 B BEAST, 207 block cipher, 207 C Carter-Wegman MAC, 207 CBC, 207 CBC mode, 207 CDN, 207 cross-site request forgery, 207 CSPRNG, 208 CSRF, 208 CTR mode, 208 D DES, 208 212 APPENDIX C. SIDE-CHANNEL ATTACKS 213 E ECB mode, 208 encryption oracle, 208 F FIPS, 208 G GCM, 208 GCM mode, 208 GMAC, 208 H HKDF, 208 HMAC, 208 HSTS, 208 I initialization vector, 208 IV, 208 K KDF, 209 key agreement, 209 key exchange, 209 keyspace, 209 M MAC, 209 message authentication code, 209 MITM, 209 mode of operation, 209 modes of operation, 209 N nonce, 209 APPENDIX C. SIDE-CHANNEL ATTACKS 214 O OCB, 209 OCB mode, 209 one-time MAC, 209 oracle, 209 OTR, 209 OTR messaging, 210 P PRF, 210 PRNG, 210 PRP, 210 public-key algorithm, 210 public-key encryption, 210 R RSA, 210 S salt, 210 secret-key encryption, 210 SMP, 210 stream cipher, 210 substitution-permutation network, 211 symmetric-key encryption, 211 PartVI References 215 Bibliography [fip01] Specification for the Advanced Encryption Standard (AES). Federal Information Process- ing Standards Publication 197, 2001. URL: http://csrc.nist.gov/publications/ fips/fips197/fips-197.pdf. 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EECS-2011-62	The RISC-V Instruction Set Manual, Volume I: Base User-Level ISA Andrew Waterman Yunsup Lee David A. Patterson Krste Asanovi c Electrical Engineering and Computer Sciences University of California at Berkeley Technical Report No. UCB/EECS-2011-62 http://www.eecs.berkeley.edu/Pubs/TechRpts/2011/EECS-2011-62.html May 13, 2011 Copyright © 2011, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. The RISC-V Instruction Set Manual Volume I: Base User-Level ISA Version 1.0 Andrew Waterman, Yunsup Lee, David Patterson, Krste Asanovi´ c CS Division, EECS Department, University of California, Berkeley {waterman\|yunsup\|pattrsn\|krste}@eecs.berkeley.edu May 13, 2011 1 Introduction RISC-V is a new instruction set architecture (ISA) designed to support computer architecture research and education. Our goals in deﬁning RISC-V include: • Provide a realistic but open ISA that captures important details of commercial general- purpose ISA designs and that is suitable for direct hardware implementation. • Provide a small but complete base ISA that avoids “over-architecting” for a particular mi- croarchitecture style (e.g., microcoded, in-order, decoupled, out-of-order) or implementation technology (e.g., full-custom, ASIC, FPGA), but which allows eﬃcient implementation in any of these. • Support both 32-bit and 64-bit address space variants for applications, operating system kernels, and hardware implementations. • Support highly-parallel multicore or manycore implementations, including heterogeneous mul- tiprocessors. • Support an eﬃcient dense instruction encoding with variable-length instructions, improving performance and reducing energy and code size. • Support the revised 2008 IEEE 754 ﬂoating-point standard. • Be fully virtualizable. • Be simple to subset for educational purposes and to reduce complexity of bringing up new implementations. • Support experimentation with user-level ISA extensions and specialized variants. • Support independent experimentation with new supervisor-level ISA designs. This manual is structured into two volumes. This volume covers the base user-level ISA design and provides examples of possible ISA extensions. The second volume provides examples of supervisor- level ISA design. This manual represents only a snapshot of the RISC-V ISA, which is still under active development; some aspects of the instruction set may change in future revisions. Commentary on our design decisions is formatted as in this paragraph, and can be skipped if the reader is only interested in the speciﬁcation itself. The name RISC-V was chosen to represent the ﬁfth major RISC ISA design from UC Berkeley (RISC-I, RISC-II, SOAR, and SPUR were the ﬁrst four). We also pun on the use of the Roman numeral “V” to signify “variations” and “vectors”, as support for a range of architecture research, including various data-parallel accelerators, is an explicit goal of the ISA design. Our intent is to provide a long-lived open ISA with signiﬁcant infrastructure support, includ- ing documentation, compiler tool chains, operating system ports, reference SAME simulators, cycle-accurate FAME-7 FPGA simulators, high-performance FPGA computers, eﬃcient ASIC 2 RISC-V Speciﬁcation implementations of various target platform designs, conﬁgurable processor generators, architec- ture test suites, and teaching materials. Initial versions of all of these have been developed or are under active development. This material is to be made available under open licenses (either modiﬁed BSD or GPL/LGPL). 2 Base User-Level ISA This section deﬁnes the standard base user-level ISA, which has two variants, RV32 and RV64, providing 32-bit or 64-bit user-level address spaces respectively. Hardware implementations and operating systems might provide only one or both of RV32 and RV64 for user programs. The ISA may be subset by a hardware implementation, but opcode traps and software emulation must then be used to implement functionality not provided by hardware. The base ISA may be extended with new instructions, but the base instructions cannot be redeﬁned. Several standard extensions have been deﬁned and are described in subsequent sections. Although 64-bit address spaces are a requirement for larger systems, we believe 32-bit address spaces will remain adequate for many embedded and client devices for decades to come and will be desirable to lower memory traﬃc and energy consumption. In addition, 32-bit address spaces are suﬃcient for educational purposes. 2.1 Base Programmers’ Model Figure 1 shows the base user-visible state in a RISC-V CPU. There are 31 general-purpose registers x1–x31, which hold ﬁxed-point values. Register x0 is hardwired to the constant 0. For RV64, the x registers are 64 bits wide, and for RV32, they are 32 bits wide. This document uses the term XPRLEN to refer to the current width of an x register in bits (either 32 or 64). Additionally, there are 32 64-bit registers f0–f31, which hold single- or double-precision ﬂoating-point values. There are also two special user-visible registers deﬁned in the architecture. The program counter pc holds the address of the current instruction. The ﬂoating-point status register fsr contains the operating mode and exception status of the ﬂoating-point unit. We considered a uniﬁed register ﬁle for both ﬁxed-point and ﬂoating-point values as this simpliﬁes software register allocation and calling conventions, and reduces total user state. However, a split organization increases the total number of registers accessible with a given instruction width, simplﬁes provision of enough regﬁle ports for wide superscalar issue, supports decoupled ﬂoating-point unit architectures, and simpliﬁes use of internal ﬂoating-point encoding techniques. Compiler support and calling conventions for split register ﬁle architectures are well understood, and using dirty bits on ﬂoating-point register ﬁle state can reduce context-switch overhead. The number of available architectural registers can have large impacts on performance and energy consumption. For the base ISA, we chose a conventional size of 32 integer plus 32 ﬂoating-point registers based on the behavior of standard compilers on existing code. Register usage tends to be dominated by a few frequently accessed registers, and regﬁle implementations can be optimized to reduce access energy for the frequently accessed registers. The optional compressed 16-bit instruction format mostly only accesses 8 registers, while instruction-set ex- tensions could support a much larger register space (either ﬂat or hierarchical) if desired. Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 3 XPRLEN-1 0 63 0 x0 / zero f0 x1 / ra f1 x2 f2 x3 f3 x4 f4 x5 f5 x6 f6 x7 f7 x8 f8 x9 f9 x10 f10 x11 f11 x12 f12 x13 f13 x14 f14 x15 f15 x16 f16 x17 f17 x18 f18 x19 f19 x20 f20 x21 f21 x22 f22 x23 f23 x24 f24 x25 f25 x26 f26 x27 f27 x28 f28 x29 f29 x30 f30 x31 f31 XPRLEN 64 XPRLEN-1 0 31 0 pc fsr XPRLEN 32 Figure 1: RISC-V base user-level programmer state. 4 RISC-V Speciﬁcation 2.2 Instruction Length Encoding The base RISC-V ISA has ﬁxed-length 32-bit instructions that must be naturally aligned on 32- bit boundaries. However, the RISC-V encoding scheme is designed to support ISA extensions with variable-length instructions, where each instruction can be any number of 16-bit instruction parcels in length and parcels are naturally aligned on 16-bit boundaries. A standard compressed ISA extension described in the following section reduces code size by providing compressed 16-bit instructions and relaxes the alignment constraints to allow all instructions (16 bit and 32 bit) to be aligned on any 16-bit boundary to improve code density. Figure 2 illustrates the RISC-V instruction length encoding convention. All the 32-bit instructions in the base ISA have their lowest two bits set to11. The compressed 16-bit instruction-set extensions have their lowest two bits equal to 00, 01, or 10. Instruction-set extensions encoded with more than 32 bits have additional low-order bits set to 1. xxxxxxxxxxxxxxaa 16-bit (aa ̸= 11) xxxxxxxxxxxxxxxx xxxxxxxxxxxbbb11 32-bit (bbb ̸= 111) ···xxxx xxxxxxxxxxxxxxxx xxxxxxxxxxx11111 >32-bit Byte Address: base+4 base+2 base Figure 2: RISC-V instruction length encoding. RISC-V can be implemented with either big-endian or little-endian memory systems. Instructions are stored in memory with each 16-bit parcel stored in a memory halfword according to the imple- mentation’s natural endianess. Parcels comprising one instruction are stored at increasing halfword addresses, with the lowest addressed parcel holding the lowest numbered bits in the instruction spec- iﬁcation, i.e., instructions are always stored in a little-endian sequence of parcels regardless of the memory system endianess. The code sequence in Figure 3 will store a 32-bit instruction to memory correctly regardless of memory system endianess. // Store 32-bit instruction in x2 register to location pointed to by x3. sh x2, 0(x3) // Store low bits of instruction in first parcel. srli x2, x2, 16 // Move high bits down to low bits, overwriting x2. sh x2, 2(x3) // Store high bits in second parcel. Figure 3: Recommended code sequence to store 32-bit instruction from register to memory. Oper- ates correctly on both big- and little-endian memory systems and avoids misaligned accesses when used with variable-length instruction-set extensions. Given the code size and energy savings of a compressed format, we wanted to build in support for a compressed format to the base ISA rather than adding this as an afterthought, but to allow simpler implementations we didn’t want to make the compressed format mandatory. We also wanted to optionally allow longer instructions to support experimentation and instruction-set extensions. Although our encoding convention reduces opcode space for the base 32-bit ISA, 32-bit RISC ISAs are generally very loosely encoded, and our scheme simpliﬁes hardware for variable-length instructions, which support a much larger potential instruction encoding space. Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 5 A base implementation need only hold the most-signiﬁcant 30 bits in instruction caches (a 6.25% saving). On instruction cache reﬁlls, any instructions encountered with either low bit clear should be recoded into trap instructions before storing in the cache to preserve illegal instruction trap behavior. We have to ﬁx the order in which parcels are stored in memory, independent of memory system endianess, to ensure that the length-encoding bits always appear ﬁrst in halfword address order. This allows the length of a variable-length instruction to be quickly determined by an instruction fetch unit by examining only the ﬁrst few bits of the ﬁrst 16-bit instruction parcel. The parcel ordering could have been ﬁxed to be either big-endian (most-signiﬁcant parcel ﬁrst) or little-endian (least-signiﬁcant parcel ﬁrst). We chose to ﬁx the parcel order to be little-endian, as little-endian systems are currently dominant commercially (all x86 systems; iOS, Android, and Windows for ARM). Once we had decided to ﬁx on a little-endian instruction parcel ordering, this naturally led to placing the length-encoding bits in the LSB positions of the instruction format to avoid breaking up opcode ﬁelds. 2.3 Base Instruction Formats In the base ISA, there are six basic instruction formats as shown in Table 1. These are a ﬁxed 32 bits in length, and must be aligned on a four-byte boundary in memory. An instruction address misaligned exception is generated if the PC is not four-byte aligned on an instruction fetch. 31 27 26 22 21 17 16 12 11 10 9 7 6 0 rd rs1 rs2 funct10 opcode R-type rd rs1 rs2 rs3 funct5 opcode R4-type rd rs1 imm[11:7] imm[6:0] funct3 opcode I-type imm[11:7] rs1 rs2 imm[6:0] funct3 opcode B-type rd LUI immediate[19:0] opcode L-type jump oﬀset [24:0] opcode J-type Table 1: RISC-V base instruction formats. R-Type 31 27 26 22 21 17 16 7 6 0 rd rs1 rs2 funct10 opcode 5 5 5 10 7 R-type instructions specify two source registers ( rs1 and rs2) and a destination register ( rd). The funct10 ﬁeld is an additional opcode ﬁeld. R4-Type 31 27 26 22 21 17 16 12 11 7 6 0 rd rs1 rs2 rs3 funct5 opcode 5 5 5 5 5 7 R4-type instructions specify three source registers ( rs1, rs2, and rs3) and a destination register (rd). The funct5 ﬁeld is a second opcode ﬁeld. This format is only used by the ﬂoating-point fused multiply-add instructions. 6 RISC-V Speciﬁcation I-Type 31 27 26 22 21 17 16 10 9 7 6 0 rd rs1 imm[11:7] imm[6:0] funct3 opcode 5 5 5 7 3 7 I-type instructions specify one source register ( rs1) and a destination register ( rd). The second source operand is a sign-extended 12-bit immediate, encoded contiguously in bits 21–10. The funct3 ﬁeld is a second opcode ﬁeld. B-Type 31 27 26 22 21 17 16 10 9 7 6 0 imm[11:7] rs1 rs2 imm[6:0] funct3 opcode 5 5 5 7 3 7 B-type instructions specify two source registers ( rs1 and rs2) and a third source operand encoded as a sign-extended 12-bit immediate. The immediate is encoded as the concatenation of the upper 5 bits in bits 31–27, and a lower 7 bits in bits 16–10. The funct3 ﬁeld is a second opcode ﬁeld. L-Type 31 27 26 7 6 0 rd LUI immediate[19:0] opcode 5 20 7 L-type instructions specify a destination register (rd) and a 20-bit immediate value. lui is the only instruction of this format. J-Type 31 7 6 0 Jump oﬀset[24:0] opcode 25 7 J-type instructions encode a 25-bit jump target address as a PC-relative oﬀset. The 25-bit imme- diate value is shifted left one bit and added to the current PC to form the target address. Decoding register speciﬁers is usually on the critical paths in implementations, and so the in- struction format was chosen to keep all register speciﬁers at the same position in all formats at the expense of having to move low immediate bits across some formats (a property shared with SPUR aka. RISC-IV). We also took the opportunity to pack all opcode-related ﬁelds (opcode + functX) together at the low end of the word. In practice, most immediates are small or require all 32 bits (or all 64 bits). We chose an asymmetric immediate split (12 bits in regular instructions plus a special load upper immediate instruction with 20 bits) to increase the opcode space available for regular instructions. In addition, the ISA only has sign-extended immediates. We did not observe a beneﬁt to using zero-extension for some immediates and wanted to keep the ISA as simple as possible. Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 7 Major Opcode Map Table 2 shows a map of the major opcodes for the base ISA. inst[4:2] 000 001 010 011 100 101 110 111 inst[6:5] (> 32) 00 LOAD LOAD-FP OP-IMM OP-IMM-32 01 STORE STORE-FP AMO MISC-MEM OP LUI OP-32 10 MADD MSUB NMSUB NMADD OP-FP 11 BRANCH J JALR JAL SYSTEM Table 2: RISC-V base opcode map, inst[1:0]=11 8 RISC-V Speciﬁcation 2.4 Load and Store Instructions RISC-V provides a byte-addressed user memory address space and is a load-store architecture, where only load and store instructions access memory and arithmetic instructions only operate on CPU registers. The memory system can be either big-endian or little-endian depending on the implementation. Byte addresses are 64 bits wide for RV64, and 32 bits wide for RV32. 31 27 26 22 21 17 16 10 9 7 6 0 rd rs1 imm[11:7] imm[6:0] funct3 opcode 5 5 5 7 3 7 dest base oﬀset[11:0] width LOAD dest base oﬀset[11:0] width LOAD-FP 31 27 26 22 21 17 16 10 9 7 6 0 imm[11:7] rs1 rs2 imm[6:0] funct3 opcode 5 5 5 7 3 7 oﬀset[11:7] base src oﬀset[6:0] width STORE oﬀset[11:7] base src oﬀset[6:0] width STORE-FP Load and store instructions transfer a value between the registers and memory. Loads are encoded in the I-type format, and stores are B-type. The eﬀective byte address is obtained by adding register rs1 to the sign-extended immediate. Loads write to register rd a value in memory. Stores write to memory the value in register rs2. The LD instruction loads a 64-bit value from memory into register rd for RV64. LD is illegal for RV32. The LW instruction loads a 32-bit value from memory for RV32, and sign-extends this to 64 bits before storing it in register rd for RV64. The LWU instruction, on the other hand, zero-extends the 32-bit value from memory for RV64, but is illegal for RV32. LH and LHU are deﬁned analogously for 16-bit values, as are LB and LBU for 8-bit values. The SD, SW, SH, and SB instructions store 64-bit, 32-bit, 16-bit, and 8-bit values in register rd to memory, with SD only being valid for RV64. The FLD instruction loads a 64-bit double-precision ﬂoating-point value from memory into ﬂoating- point register rd, and the FLW instruction loads a 32-bit single-precision ﬂoating-point value. FSD and FSW store double- and single-precision values, respectively, from ﬂoating-point registers to memory. For best performance, the eﬀective address for all loads and stores should be naturally aligned for each data type (i.e., on an eight-byte boundary for 64-bit accesses, a four-byte boundary for 32-bit accesses, and a two-byte boundary for 16-bit accesses). The base ISA supports misaligned accesses, but these might run extremely slowly depending on the implementation. Furthermore, naturally aligned loads and stores are guaranteed to execute atomically, whereas misaligned loads and stores might not, and hence require additional synchronization to ensure atomicity. Misaligned accesses are occasionally required when porting legacy code, and are essential for good performance on many applications when using any form of packed SIMD extension. Our ratio- nale for supporting misaligned accesses via the regular load and store instructions is to simplify Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 9 the addition of misaligned hardware support. One option would have been to disallow misaligned accesses in the base ISA and then provide some separate ISA support for misaligned accesses, either special instructions to help software handle misaligned accesses or a new hardware ad- dressing mode for misaligned accesses. Special instructions are diﬃcult to use, complicate the ISA, and often add new processor state (e.g., SPARC VIS align address oﬀset register) or com- plicate access to existing processor state (e.g., MIPS LWL/LWR partial register writes). In addition, for loop-oriented packed SIMD code, the extra overhead when operands are misaligned motivates software to provide multiple forms of loop depending on operand alignment, which complicates code generation and adds to startup overhead. New misaligned hardware addressing modes take considerable space in the instruction encoding or require very simpliﬁed addressing modes (e.g., register indirect only). We do not mandate atomicity for misaligned accesses so simple implementations can just use a machine trap and software handler to handle misaligned accesses. If hardware misaligned support is provided, software can exploit this by simply using regular load and store instruc- tions. Hardware can automatically optimize accesses depending on whether runtime addresses are aligned. Atomic Memory Operation Instructions 31 27 26 22 21 17 16 10 9 7 6 0 rd rs1 rs2 funct7 funct3 opcode 5 5 5 7 3 7 dest addr src operation width AMO The atomic memory operation (AMO) instructions perform read-modify-write operations for mul- tiprocessor synchronization and are encoded with an R-type instruction format. These AMO in- structions atomically load a data value from the address in rs1, place the value into register rd, apply a binary operator to the loaded value and the value in rs2, then store the result back to the address in rs1. AMOs can either operate on 32-bit or 64-bit words in memory. For RV64, 32-bit AMOs always sign-extend the value placed in rd. The address held in rs1 must be naturally aligned to the size of the operand (i.e., eight-byte aligned for 64-bit words and four-byte aligned for 32-bit words). If the address is not naturally aligned, a misaligned address trap will be generated. The operations supported are integer add, logical AND, logical OR, swap, and signed and unsigned integer maximum and minimum. Even uniprocessor systems need atomic instructions to support operating systems. We selected fetch-and-op style synchronization primitives for the base ISA as they guarantee forward progress unlike compare-and-swap (CAS) or load-linked/store-conditional (LLSC) constructs, and scale better to highly parallel systems. CAS or LLSC could help in the implementation of lock-free data structures, but CAS suﬀers from the ABA problem and would require a new integer in- struction format to support three source operands (address, compare value, swap value) as well as a diﬀerent memory system message format. LLSC can avoid the ABA problem but is more susceptible to livelock, and implementations usually impose strict constraints or prohibit access to other memory locations while a reservation is held. In general, a multi-word atomic primitive is desirable but there is still considerable debate about what form this should take. Our current thoughts are to include a small limited-capacity transactional memory buﬀer along the lines of the original transactional memory proposals. A simple microarchitecture can implement AMOs by locking a private cache line for the duration. More complex implementations might also implement AMOs at memory controllers, and can optimize away fetching the original value when the destination is x0. 10 RISC-V Speciﬁcation 2.5 Integer Computational Instructions Integer computational instructions are either encoded as register-immediate operations using the I-type format or as register-register operations using the R-type format. The destination is reg- ister rd for both register-immediate and register-register instructions. No integer computational instructions cause arithmetic traps. Most integer instructions operate on XPRLEN bits of values held in the ﬁxed-point register ﬁle. Additional instruction variants are provided to manipulate 32-bit values in RV64. These are in- dicated with a ‘W’ suﬃx to the opcode; they ignore the upper 32 bits of their inputs and always produce 32-bit signed values, i.e. bits XPRLEN-1 through 31 are equal. These instructions cause an illegal instruction trap in RV32. Integer Register-Immediate Instructions 31 27 26 22 21 17 16 10 9 7 6 0 rd rs1 imm[11:7] imm[6:0] funct3 opcode 5 5 5 7 3 7 dest src immediate[11:0] ADDI/SLTI[U] OP-IMM dest src immediate[11:0] ANDI/ORI/XORI OP-IMM dest src immediate[11:0] ADDIW OP-IMM-32 ADDI and ADDIW add the sign-extended 12-bit immediate to register rs1. ADDIW is an RV64- only instruction that produces the proper sign-extension of a 32-bit result. Note, ADDIW rd, rs1, 0 writes the sign-extension of the lower 32 bits of register rs1 into register rd. SLTI (set less than immediate) places the value 1 in register rd if register rs1 is less than the sign-extended immediate when both are treated as signed numbers, else 0 is written to rd. SLTIU is similar but compares the values as unsigned numbers. ANDI, ORI, XORI are logical operations that perform bit-wise AND, OR, and XOR on register rs1 and the sign-extended 12-bit immediate and place the result in rd. Note, XORI rd, rs1, -1 performs a logical inversion (NOT) of register rs1. 31 27 26 22 21 16 15 14 10 9 7 6 0 rd rs1 imm[11:6] imm[5] imm[4:0] funct3 opcode 5 5 6 1 5 3 7 dest src SRA/SRL shamt[5] shamt[4:0] SRxI OP-IMM dest src SRA/SRL 0 shamt[4:0] SRxIW OP-IMM-32 dest src 0 shamt[5] shamt[4:0] SLLI OP-IMM dest src 0 0 shamt[4:0] SLLIW OP-IMM-32 Shifts by a constant are also encoded as a specialization of the I-type format. The operand to be shifted is in rs1, and the shift amount is encoded in the lower 6 bits of the immediate ﬁeld for RV64, and in the lower 5 bits for RV32. The shift type is encoded in the upper bits of the immediate ﬁeld. Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 11 SLLI is a logical left shift (zeros are shifted into the lower bits); SRLI is a logical right shift (zeros are shifted into the upper bits); and SRAI is an arithmetic right shift (the original sign bit is copied into the vacated upper bits). In RV32, SLLI, SRLI, and SRAI generate an illegal instruction trap if imm[5] ̸= 0. SLLIW, SRLIW, and SRAIW are RV64-only instructions that are analogously deﬁned but operate on 32-bit values and produce signed 32-bit results. SLLIW, SRLIW, and SRAIW generate an illegal instruction trap if imm[5] ̸= 0. 31 27 26 7 6 0 rd immediate[19:0] opcode 5 20 7 dest 20-bit upper immediate LUI LUI (load upper immediate) is used to build 32-bit constants. LUI shifts the 20-bit immediate left 12 bits, ﬁlling in the vacated bits with zeros, then places the result in register rd. For RV64, the 32-bit result is sign-extended to 64 bits. Integer Register-Register Operations RISC-V deﬁnes several arithmetic R-type operations. All operations read the rs1 and rs2 registers as source operands and write the result into registerrd. The funct ﬁeld selects the type of operation. 31 27 26 22 21 17 16 7 6 0 rd rs1 rs2 funct10 opcode 5 5 5 10 7 dest src1 src2 ADD/SUB/SLT/SLTU OP dest src1 src2 AND/OR/XOR OP dest src1 src2 SLL/SRL/SRA OP dest src1 src2 ADDW/SUBW OP-32 dest src1 src2 SLLW/SRLW/SRAW OP-32 ADD and SUB perform addition and subtraction respectively. SLT and SLTU perform signed and unsigned compares respectively, writing 1 to rd if rs1 < rs2, 0 otherwise. AND, OR, and XOR perform bitwise logical operations. ADDW and SUBW are RV64-only instructions that are deﬁned analogously to ADD and SUB but operate on 32-bit values and produce signed 32-bit results. SLL, SRL, and SRA perform logical left, logical right, and arithmetic right shifts on the value in register rs1 by the shift amount held in register rs2. In RV64, only the low 6 bits of rs2 are considered for the shift amount. Similarly for RV32, only the low 5 bits of rs2 are considered. SLLW, SRLW, and SRAW are RV64-only instructions that are analogously deﬁned but operate on 32-bit values and produce signed 32-bit results. The shift amount is given by rs2[4:0]. 12 RISC-V Speciﬁcation 31 27 26 22 21 17 16 7 6 0 rd rs1 rs2 funct10 opcode 5 5 5 10 7 dest src1 src2 MUL/MULH[[S]U] OP dest dividend divisor DIV[U]/REM[U] OP dest src1 src2 MUL[U]W OP-32 dest dividend divisor DIV[U]W/REM[U]W OP-32 MUL performs an XPRLEN-bit ×XPRLEN-bit multiplication and places the lower XPRLEN bits in the destination register. MULH, MULHU, and MULHSU perform the same multiplication but re- turn the upper XPRLEN bits of the full 2×XPRLEN-bit product, for signed×signed, unsigned×unsigned, and signed×unsigned multiplication respectively. If both the high and low bits of the same product are required, then the recommended code sequence is: MULH[[S]U] rdh, rs1, rs2; MUL rdl, rs1, rs2 (source register speciﬁers must be in same order and rdh cannot be the same as rs1 or rs2). Microarchitectures can then fuse these into a single multiply operation instead of performing two separate multiplies. MULW is an RV64-only instruction that multiplies the lower 32 bits of the source registers, placing the sign-extension of the lower 32 bits of the result into the destination register. MUL can be used to obtain the upper 32 bits of the 64-bit product, but signed arguments must be proper 32-bit signed values, whereas unsigned arguments must have their upper 32 bits clear. DIV and DIVU perform signed and unsigned integer division of XPRLEN bits by XPRLEN bits. REM and REMU provide the remainder of the corresponding division operation. If both the quo- tient and remainder are required from the same division, the recommended code sequence is: DIV[U] rdq, rs1, rs2; REM[U] rdr, rs1, rs2 (rdq cannot be the same as rs1 or rs2). Microarchitectures can then fuse these into a single divide operation instead of performing two separate divides. DIVW and DIVUW are RV64-only instructions that divide the lower 32 bits rs1 by the lower 32 bits of rs2, treating them as signed and unsigned integers respectively, placing the 32-bit quotient in rd. REMW and REMUW are RV64-only instructions that provide the corresponding signed and unsigned remainder operations respectively. The quotient of division by 0 has all bits set, i.e. 2 XPRLEN −1 for unsigned division or −1 for signed division. The remainder of division by 0 equals the dividend. Signed division overﬂow occurs only when the most-negative integer, −(2XPRLEN −1), is divided by −1. The quotient of signed division overﬂow is equal to the dividend, and the remainder is 0. Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 13 2.6 Control Transfer Instructions RISC-V provides two types of control transfer instructions: unconditional jumps and conditional branches. Control transfer instructions in RISC-V do not have architecturally visible delay slots. Unconditional Jumps Absolute jumps (J) and jump and link (JAL) instructions use the J-type format. The 25-bit jump target oﬀset is sign-extended and shifted left one bit to form a byte oﬀset, then added to the pc to form the jump target address. Jumps can therefore target a ±32 MB range. JAL stores the address of the instruction following the jump ( pc+4) into register x1. 31 7 6 0 Jump oﬀset[24:0] opcode 25 7 target oﬀset J/JAL The indirect jump instruction JALR (jump and link register) uses the I-type encoding. It has three variants that are functionally identical but provide hints to the implementation: JALR.C is used to call subroutines; JALR.R is used to return from subroutines; and JALR.J is used for indirect jumps. The target address is obtained by sign-extending the 12-bit immediate then adding it to the address contained in register rs1. The address of the instruction following the jump ( pc+4) is written to register rd. Register x0 can be used as the destination if the result is not required. The JALR major opcode is also used to encode the RDNPC instruction, which writes the address of the following instruction ( pc+4) to register rd without changing control ﬂow. 31 27 26 22 21 17 16 10 9 7 6 0 rd rs1 imm[11:7] imm[6:0] funct3 opcode 5 5 5 7 3 7 dest base oﬀset[11:7] oﬀset[6:0] C/R/J JALR dest 0 0 0 RDNPC JALR The unconditional jump instructions all use PC-relative addressing to help support position- independent code. The JALR instruction was deﬁned to enable a two-instruction sequence to jump anywhere in a 32-bit address range. A LUI instruction can ﬁrst load rs1 with the upper 20 bits of a target address, then JALR can add in the lower bits. Note that the JALR instruction does not shift the 12-bit immediate by one bit, unlike the conditional branch instructions. This is to allow the same linker relocation format to be used for JALR as for global loads and stores. For implementations with dedicated branch target address adders, this is only a minor inconvenience, as some of the immediate ﬁeld is already in a diﬀerent position than for conditional branches. For implementations that use the execute- stage adders to perform jump target arithmetic, this reuses the same datapath required for load address calculations. The JALR hints are used to guide an implementation’s instruction-fetch predictors, indicat- ing whether JALR instructions should push (C), pop (R), or not touch (J/RDNPC) a return- address stack. 14 RISC-V Speciﬁcation Conditional Branches All branch instructions use the B-type encoding. The 12-bit immediate is sign-extended, shifted left one bit, then added to the current pc to give the target address. 31 27 26 22 21 17 16 10 9 7 6 0 imm[11:7] rs1 rs2 imm[6:0] funct3 opcode 5 5 5 7 3 7 oﬀset[11:7] src1 src2 oﬀset[6:0] BEQ/BNE BRANCH oﬀset[11:7] src1 src2 oﬀset[6:0] BLT[U] BRANCH oﬀset[11:7] src1 src2 oﬀset[6:0] BGE[U] BRANCH Branch instructions compare two registers. BEQ and BNE take the branch if registers rs1 and rs2 are equal or unequal respectively. BLT and BLTU take the branch if rs1 is less than rs2, using signed and unsigned comparison respectively. BGE and BGEU take the branch if rs1 is greater than or equal to rs2, using signed and unsigned comparison respectively. Note, BGT, BGTU, BLE, and BLEU can be synthesized by reversing the operands to BLT, BLTU, BGE, and BGEU, respectively. Software should be optimized such that the sequential code path is the most common path, with less-frequently-taken code paths placed out of line. Software should also assume that backward branches will be predicted taken and forward branches as not-taken, at least the ﬁrst time they are encountered. Dynamic predictors should quickly learn any predictable branch behavior. The conditional branches were designed to include arithmetic comparison operations between two registers, rather than use condition codes (x86, ARM, SPARC, PowerPC), or to only com- pare one register against zero (Alpha, MIPS), or two registers only for equality (MIPS). This design was motivated by the observation that a combined compare-and-branch instruction ﬁts into a regular pipeline, avoids additional condition code state or use of a temporary register, and reduces static code size and dynamic instruction fetch traﬃc. Another point is that com- parisons against zero require non-trivial circuit delay (especially after the move to static logic in advanced processes) and so are almost as expensive as arithmetic magnitude compares. Another advantage of a fused compare-and-branch instruction is that branches are observed earlier in the front-end instruction stream, and so can be predicted earlier. There is perhaps an advantage to a design with condition codes in the case where multiple branches can be taken based on the same condition codes, but we believe this case to be relatively rare. We considered but did not include static branch hints in the instruction encoding. These can reduce the pressure on dynamic predictors, but require more instruction encoding space and software proﬁling for best results. We considered but did not include conditional moves or predicated instructions, which can eﬀectively replace unpredictable short forward branches. Conditional move and predicated in- structions cause complications in out-of-order microarchitectures, due to the need to copy the original value of the destination architectural register into the renamed destination physical register if the predicate is false, adding an implicit third source operand. Predicates also add additional user state and require additional instruction encoding space. Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 15 2.7 Floating-Point Instructions The base RISC-V ISA provides both single- and double-precision ﬂoating-point computational instructions compliant with the IEEE 754-2008 ﬂoating-point arithmetic standard. Most ﬂoating- point instructions operate on values in the 32-entry ﬂoating-point register ﬁle. Floating-point load and store instructions transfer ﬂoating-point values between registers and memory, as described in Section 2.4. Instructions to transfer values to and from the ﬁxed-point register ﬁle are also provided. Floating-Point Status Register 32 8 7 5 4 3 2 1 0 0 Rounding Mode Accrued Exceptions NV DZ OF UF NX 24 3 1 1 1 1 1 Figure 4: Floating-point status register. The fsr register is a 32-bit read/write register that selects the dynamic rounding mode for ﬂoating- point arithmetic operations and holds the accrued exception ﬂags. The fsr is read and written with the MFFSR and MTFSR ﬂoating-point instructions, described below. Floating-point operations use either a static rounding mode encoded in the instruction, or a dynamic rounding mode held in the Rounding Mode ﬁeld and encoded as shown in Table 3. If the Rounding Mode ﬁeld is set to an invalid value (101–111), any subsequent attempt to execute a ﬂoating-point operation with a dynamic rounding mode will cause an illegal instruction trap. Some instructions are never aﬀected by rounding mode, and should have their rm ﬁeld set to RNE (000). Rounding Mode Mnemonic Meaning 000 RNE Round to Nearest, ties to Even 001 RTZ Round toward Zero 010 RDN Round Down (towards −∞) 011 RUP Round Up (towards +∞) 100 RMM Round to Nearest, ties to Max Magnitude 101–111 Invalid. Table 3: Rounding Mode ﬁeld encoding. The accrued exception ﬂags indicate the exception conditions that have arisen on any ﬂoating-point arithmetic instruction since the ﬁeld was last reset by software, as shown in Table 4. NaN Generation and Propagation If a ﬂoating-point operation on non-NaN inputs is invalid, e.g. √ −1.0, the result is the canonical NaN: the sign bit is 0, and the fraction and exponent have all bits set. As the MSB of the signiﬁcand (aka. the quiet bit) is set, the canonical NaN is quiet. 16 RISC-V Speciﬁcation Flag Mnemonic Flag Meaning NV Invalid Operation DZ Divide by Zero OF Overﬂow UF Underﬂow NX Inexact Table 4: Current and accrued exception ﬂag encoding. With the exception of the FMIN and FMAX operations, if a ﬂoating-point operation has at least one signaling NaN input, the ﬁrst such input ( rs1, rs2, or rs3, in that order) is returned with its quiet bit set. Otherwise, if a ﬂoating-point operation has at least one quiet NaN input, the ﬁrst such input is returned. For FMIN and FMAX, if at least one input is a signaling NaN, the ﬁrst such input is returned with its quiet bit set. If both inputs are quiet NaNs, the ﬁrst input is returned. If just one input is a quiet NaN, the non-NaN input is returned. If a NaN value is converted to a larger ﬂoating-point type, the signiﬁcand of the input becomes the MSBs of the signiﬁcand of the output; the LSBs are cleared. If a NaN value is converted to a smaller ﬂoating-point type, the LSBs of the signiﬁcand are discarded. In both cases, the quiet bit of the output is set, even for signaling NaN inputs. Floating-Point Computational Instructions Floating-point arithmetic instructions with one or two source operands use the R-type format with the OP-FP major opcode. FADD.fmt, FSUB.fmt, FMUL.fmt, and FDIV.fmt perform ﬂoating-point addition, subtraction, multiplication, and division, respectively, between rs1 and rs2, writing the result to rd. FMIN. fmt and FMAX.fmt write, respectively, the smaller or larger of rs1 and rs2 to rd. FSQRT.fmt computes the square root of rs1 and writes the result to rd. The fmt ﬁeld encodes the datatype of the operands and destination: S for single-precision or D for double-precision. All ﬂoating-point operations that perform rounding can select the rounding mode statically using the rm ﬁeld with the same encoding as shown in Table 3. A value of 111 in the instruction’s rm ﬁeld selects the dynamic rounding mode held in the fsr. Any attempt to execute a ﬂoating-point operation that performs rounding with an invalid value for rm, or with dynamic rounding and an invalid value for rm in the fsr, will cause an illegal instruction trap. 31 27 26 22 21 17 16 12 11 9 8 7 6 0 rd rs1 rs2 funct rm fmt opcode 5 5 5 5 3 2 7 dest src1 src2 FADD/FSUB RM S/D OP-FP dest src1 src2 FMUL/FDIV RM S/D OP-FP dest src1 src2 FMIN/FMAX 000 S/D OP-FP dest src 0 FSQRT RM S/D OP-FP Floating-point fused multiply-add instructions are encoded as R4-type instructions and multiply Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 17 the values in rs1 and rs2, optionally negate the result, then add or subtract the value in rs3 to or from that result. FMADD.fmt computes rs1×rs2+rs3; FMSUB.fmt computes rs1×rs2-rs3; FNMSUB.fmt computes -(rs1×rs2-rs3); and FNMADD.fmt computes -(rs1×rs2+rs3). 31 27 26 22 21 17 16 12 11 9 8 7 6 0 rd rs1 rs2 rs3 rm fmt opcode 5 5 5 5 3 2 7 dest src1 src2 src3 RM S/D F[N]MADD/F[N]MSUB The 2-bit ﬂoating-point format ﬁeld fmt is encoded as shown in Table 5. fmt ﬁeld Mnemonic Meaning 00 S 32-bit single-precision 01 D 64-bit double-precision 10 - reserved 11 - reserved Table 5: Format ﬁeld encoding. 18 RISC-V Speciﬁcation Floating-Point Conversion and Move Instructions Floating-point-to-integer and integer-to-ﬂoating-point conversion instructions are encoded in the OP-FP major opcode space. The fmt ﬁeld encodes the datatype of the lone ﬂoating-point operand. FCVT.W.fmt or FCVT.L.fmt converts a ﬂoating-point number in ﬂoating-point register rs1 to a signed 32-bit or 64-bit integer, respectively, in ﬁxed-point register rd. FCVT.fmt.W or FCVT.fmt.L converts a 32-bit or 64-bit signed integer, respectively, in ﬁxed-point register rs1 into a ﬂoating- point number in ﬂoating-point register rd. FCVT.WU.fmt, FCVT.LU. fmt, FCVT.fmt.WU, and FCVT.fmt.LU variants convert to or from unsigned integer values. FCVT.L[U].fmt and FCVT.fmt.L[U] are illegal in RV32. All ﬂoating-point to integer and integer to ﬂoating-point conversion instructions round according to the rm ﬁeld. Note FCVT.D.W[U] always produces an exact result and is unaﬀected by rounding mode. A ﬂoating-point register can be initialized to ﬂoating-point positive zero using FCVT.fmt.W rd, x0, which will never raise any exceptions. 31 27 26 22 21 17 16 12 11 9 8 7 6 0 rd rs1 rs2 funct rm fmt opcode 5 5 5 5 3 2 7 dest src 0 FCVT.W[U].fmt RM S/D OP-FP dest src 0 FCVT.fmt.W[U] RM S/D OP-FP dest src 0 FCVT.L[U]. fmt RM S/D OP-FP dest src 0 FCVT.fmt.L[U] RM S/D OP-FP The double-precision to single-precision and single-precision to double-precision conversion instruc- tions, FCVT.S.D and FCVT.D.S, are encoded in the OP-FP major opcode space and both the source and destination are ﬂoating-point registers. The fmt ﬁeld encodes the datatype of the result. FCVT.S.D rounds according to the RM ﬁeld; FCVT.D.S will never round. 31 27 26 22 21 17 16 12 11 9 8 7 6 0 rd rs1 rs2 funct rm fmt opcode 5 5 5 5 3 2 7 dest src 0 FCVT.S.D RM S OP-FP dest src 0 FCVT.D.S RM D OP-FP Floating-point to ﬂoating-point sign-injection instructions, FSGNJ.fmt, FSGNJN. fmt, and FS- GNJX.fmt, produce a result that takes all bits except the sign bit from rs1. For FSGNJ, the result’s sign bit is rs2’s sign bit; for FSGNJN, the result’s sign bit is the opposite of rs2’s sign bit; and for FSGNJX, the sign bit is the XOR of the sign bits of rs1 and rs2. Sign-injection instructions do not set ﬂoating-point exception ﬂags. Note, FSGNJ rx, ry, ry moves ry to rx; FSGNJN rx, ry, ry moves the the negation of ry to rx; and FSGNJX rx, ry, ry moves the absolute value of ry to rx. 31 27 26 22 21 17 16 12 11 9 8 7 6 0 rd rs1 rs2 funct rm fmt opcode 5 5 5 5 3 2 7 dest src1 src2 FSGNJ[N] 000 S/D OP-FP dest src1 src2 FSGNJX 000 S/D OP-FP Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 19 Instructions are provided to move bit patterns between the ﬂoating-point and ﬁxed-point registers. MFTX.S moves the single-precision value in ﬂoating-point register rs2 represented in IEEE 754- 2008 encoding to the lower 32 bits of ﬁxed-point register rd. For RV64, the higher 32 bits of the destination register are ﬁlled with copies of the ﬂoating-point number’s sign bit. MXTF.S moves the single-precision value encoded in IEEE 754-2008 standard encoding from the lower 32 bits of ﬁxed-point register rs1 to the ﬂoating-point register rd. MFTX.D and MXTF.D are deﬁned analogously for double-precision values in RV64, but are illegal in RV32. RV32 can use stores and loads to transfer double-precision values between ﬁxed-point and ﬂoating-point registers. 31 27 26 22 21 17 16 12 11 9 8 7 6 0 rd rs1 rs2 funct rm fmt opcode 5 5 5 5 3 2 7 dest 0 src MFTX. fmt 000 S/D OP-FP dest src 0 MXTF. fmt 000 S/D OP-FP The Floating-point Status Register fsr can be read and written with the MFFSR and MTFSR instructions. MFFSR copies fsr into ﬁxed-point register rd. MTFSR writes fsr with the value in ﬁxed-point register rs1, and also copies the original value of fsr into ﬁxed-point register rd. 31 27 26 22 21 17 16 12 11 9 8 7 6 0 rd rs1 rs2 funct rm fmt opcode 5 5 5 5 3 2 7 dest 0 0 MFFSR 000 S OP-FP dest src 0 MTFSR 000 S OP-FP Floating-Point Compare Instructions Floating-point compare instructions perform the speciﬁed comparison (equal, less than, or less than or equal) between ﬂoating-point registers rs1 and rs2 and record the boolean result in ﬁxed-point register rd. 31 27 26 22 21 17 16 12 11 9 8 7 6 0 rd rs1 rs2 funct rm fmt opcode 5 5 5 5 3 2 7 dest src1 src2 FEQ/FLT/FLE. fmt 000 S/D OP-FP The base ﬂoating-point ISA was deﬁned so as to allow implementations to employ an internal recoding of the ﬂoating-point format in registers to simplify handling of subnormal values and possibly to reduce functional unit latency. To this end, the base ISA avoids representing integer values in the ﬂoating-point registers by deﬁning conversion and comparison operations that read and write the ﬁxed-point register ﬁle directly. This also removes many of the common cases where explicit moves between integer and ﬂoating-point registers are required, reducing instruction count and critical paths for common mixed-format code sequences. We require implementations to return the standard-mandated default values in the case of exceptional conditions, without any further intervention on the part of user-level software (unlike 20 RISC-V Speciﬁcation the Alpha ISA ﬂoating-point trap barriers). We believe full hardware handling of exceptional cases will become more common, and so wish to avoid complicating the user-level ISA to optimize other approaches. As allowed by the standard, we do not support traps on ﬂoating-point exceptions in the base ISA, but instead require explicit checks of the ﬂags in software. We are contemplating addition of a branch controlled directly by the contents of the ﬂoating-point accrued exception ﬂags to support fast user-level exception handling. The desire to support IEEE 754-2008 requires the addition of the three-source-operands fused multiply-add instructions, and the ﬁfth rounding mode. The C99 language standard mandates the provision of a dynamic rounding mode register. The MTFSR instruction was deﬁned to both read and write the ﬂoating-point status register to allow rapid save and restore of ﬂoating-point context. The operation MTFSR x0, rd will save the accrued exception ﬂags and rounding mode in ﬁxed-point register rd, then clear the ﬂags. Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 21 2.8 Memory Model In the base RISC-V ISA, each hardware thread observes its own memory operations as if they executed sequentially in program order. RISC-V has a relaxed memory model between diﬀerent hardware threads, requiring an explicit FENCE instruction to guarantee any speciﬁc ordering between memory operations from diﬀerent threads. 31 27 26 22 21 10 9 7 6 0 rd rs1 imm[11:0] funct3 opcode 5 5 12 3 7 - - - FENCE MISC-MEM The FENCE instruction is used to order loads and stores as viewed by other hardware threads. Informally, no other hardware thread can observe any memory operations (LOAD/STORE/AMO) following a FENCE before any memory operations preceding the FENCE. We chose a relaxed memory model to allow high performance from simple machine implemen- tations. The base ISA provides only a global FENCE operation, but suﬃcient encoding space is reserved to allow ﬁner-grain FENCE instructions in optional extensions. A base implementation should ignore the higher-order bits in a FENCE instruction and simply execute a conservative global fence to provide forwards compatibility with ﬁner-grain fences. 31 27 26 22 21 10 9 7 6 0 rd rs1 imm[11:0] funct3 opcode 5 5 12 3 7 - - - FENCE.I MISC-MEM The FENCE.I instruction is used to synchronize the instruction and data streams. RISC-V does not guarantee that stores to instruction memory will be made visible to instruction fetches until a FENCE.I instruction is executed. A FENCE.I instruction only ensures that a subsequent in- struction fetch on the same hardware thread will see any previous data stores. FENCE.I does not ensure that other hardware threads’ instruction fetches will observe the local thread’s stores in a multiprocessor system. The FENCE.I instruction was designed to support a wide variety of implementations. A sim- ple implementation can ﬂush the local instruction cache and the instruction pipeline when the FENCE.I is decoded. A more complex implementation might snoop the instruction (data) cache on every data (instruction) cache miss, or use an inclusive uniﬁed private L2 cache to invalidate lines from the primary instruction cache when they are being written by a local store instruction. If instruction and data caches are kept coherent in this way, then only the pipeline needs to be ﬂushed at a FENCE.I. Extensions might deﬁne ﬁner-grain FENCE.I instructions targeting speciﬁc instruction ad- dresses, so a base implementation should ignore the higher-order bits in a FENCE.I instruction and simply execute a conservative local FENCE.I to provide forwards compatibility. To make a store to instruction memory visible to all hardware threads, the writing thread has to issue a global FENCE before requesting that all remote hardware threads execute a FENCE.I. We considered but did not include a “store instruction” instruction (as in MAJC). JIT compilers may generate a large trace of instructions before a single FENCE.I, and amortize any instruction cache snooping/invalidation overhead. 22 RISC-V Speciﬁcation 2.9 System Instructions SYSTEM instructions are used to access system functionality that might require privileged access and are encoded as an R-type instruction. The SYSTEM instructions are deﬁned to allow simpler implementations to always trap to a single software exception handler. More sophisticated implementations might execute more of each system instruction in hardware. SYSCALL and BREAK 31 27 26 22 21 17 16 7 6 0 rd rs1 rs2 funct10 opcode 5 5 5 10 7 0 0 0 SYSCALL SYSTEM 0 0 0 BREAK SYSTEM The SYSCALL instruction is used to make a request to an operating system environment. The ABI for the operating system will deﬁne how parameters for the OS request are passed, but usually these will be in deﬁned locations in the ﬁxed-point register ﬁle. The BREAK instruction is used by debuggers to cause control to be transferred back to the de- bugging environment. Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 23 Timers and Counters 31 27 26 22 21 17 16 7 6 0 rd rs1 rs2 funct10 opcode 5 5 5 10 7 dest 0 0 RDCYCLE SYSTEM dest 0 0 RDTIME SYSTEM dest 0 0 RDINSTRET SYSTEM The RDCYCLE instruction writes ﬁxed-point register dest with a count of the number of clock cycles executed by the processor on which the hardware thread is running from an arbitrary start time in the past. In RV32, this returns a 32-bit unsigned integer value that will wrap around when the count value overﬂows (modulo arithmetic). In RV64, this will return a 64-bit unsigned integer value, which will never overﬂow. The rate at which the cycle counter advances will depend on the implementation and operating environment. The software environment should provide a means to determine the current rate (cycles/second) at which the cycle counter is incrementing. The RDTIME instruction writes ﬁxed-point register dest with an integer value corresponding to the wall-clock real time that has passed from an arbitrary start time in the past. In RV32, this returns a 32-bit unsigned integer value that will wrap around when the time value overﬂows (modulo arithmetic). In RV64, this will return a 64-bit unsigned integer value, which should never overﬂow. The software environment should provide a means of determining the period of the real-time counter (seconds/tick). The period must be constant and should be no greater than 100 ns (at least 10 MHz rate). For RV32, the real-time clock period should be no shorter than 10 ns to allow periods of up to 4 seconds to be measured simply. The real-time clocks of all hardware threads in a single user application should be synchronized to within one tick of the real-time clock. The environment should provide a means to determine the accuracy of the clock. The RDINSTRET instruction writes ﬁxed-point registerdest with the number of instructions retired by this hardware thread from some arbitrary start point in the past. In RV32, this returns an unsigned 32-bit integer value that will wrap around when the count overﬂows. In RV64, this returns an unsigned 64-bit integer value that will never overﬂow. We mandate these basic counters be provided in all implementations as they are essential for basic performance analysis, adaptive and dynamic optimization, and to allow an application to work with real-time streams. Additional counters should be provided to help diagnose performance problems and these should be made accessible from user-level application code with low overhead. In some applications, it is important to be able to read multiple counters at the same instant in time. When run under a multitasking environment, a user thread can suﬀer a context switch while attempting to read the counters. One solution is for the user thread to read the real-time counter before and after reading the other counters to determine if a context switch occured in the middle of the sequence, in which case the reads can be retried. We considered adding output latches to allow a user thread to snapshot the counter values atomically, but this would increase the size of the user context especially for implementations with a richer set of counters. 24 RISC-V Speciﬁcation 3 Compressed Instruction Set Extension The RISC-V compressed instruction set extension reduces static and dynamic code size by adding short 16-bit instruction encodings for common integer operations. The compressed instruction encodings can be added to both RV64 and RV32, forming RVC64 and RVC32 respectively. The RVC64 and RVC32 ISAs allow 16-bit instructions to be freely intermixed with the 32-bit base instructions, with the latter now able to start on any 16-bit boundary. All of the 16-bit instructions can be expanded into one or more of the base RISC-V instructions. The RVC ISAs are still under development, but we expect a 25–30% reduction in static and dynamic code size. Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 25 31 27 26 22 21 17 16 15 14 12 11 10 9 8 7 6 0 jump target opcode J-type rd LUI-immediate opcode LUI-type rd rs1 imm[11:7] imm[6:0] funct3 opcode I-type imm[11:7] rs1 rs2 imm[6:0] funct3 opcode B-type rd rs1 rs2 funct10 opcode R-type rd rs1 rs2 rs3 funct5 opcode R4-type Unimplemented Instruction Control Transfer Instructions imm25 1100111 J imm25 imm25 1101111 JAL imm25 imm12hi rs1 rs2 imm12lo 000 1100011 BEQ rs1,rs2,imm12 imm12hi rs1 rs2 imm12lo 001 1100011 BNE rs1,rs2,imm12 imm12hi rs1 rs2 imm12lo 100 1100011 BLT rs1,rs2,imm12 imm12hi rs1 rs2 imm12lo 101 1100011 BGE rs1,rs2,imm12 imm12hi rs1 rs2 imm12lo 110 1100011 BLTU rs1,rs2,imm12 imm12hi rs1 rs2 imm12lo 111 1100011 BGEU rs1,rs2,imm12 rd rs1 imm12 000 1101011 JALR.C rd,rs1,imm12 rd rs1 imm12 001 1101011 JALR.R rd,rs1,imm12 rd rs1 imm12 010 1101011 JALR.J rd,rs1,imm12 rd 00000 000000000000 100 1101011 RDNPC rd Memory Instructions rd rs1 imm12 000 0000011 LB rd,rs1,imm12 rd rs1 imm12 001 0000011 LH rd,rs1,imm12 rd rs1 imm12 010 0000011 LW rd,rs1,imm12 rd rs1 imm12 011 0000011 LD rd,rs1,imm12 rd rs1 imm12 100 0000011 LBU rd,rs1,imm12 rd rs1 imm12 101 0000011 LHU rd,rs1,imm12 rd rs1 imm12 110 0000011 LWU rd,rs1,imm12 imm12hi rs1 rs2 imm12lo 000 0100011 SB rs1,rs2,imm12 imm12hi rs1 rs2 imm12lo 001 0100011 SH rs1,rs2,imm12 imm12hi rs1 rs2 imm12lo 010 0100011 SW rs1,rs2,imm12 imm12hi rs1 rs2 imm12lo 011 0100011 SD rs1,rs2,imm12 Atomic Memory Instructions rd rs1 rs2 0000000 010 0101011 AMOADD.W rd,rs1,rs2 rd rs1 rs2 0000001 010 0101011 AMOSWAP.W rd,rs1,rs2 rd rs1 rs2 0000010 010 0101011 AMOAND.W rd,rs1,rs2 rd rs1 rs2 0000011 010 0101011 AMOOR.W rd,rs1,rs2 rd rs1 rs2 0000100 010 0101011 AMOMIN.W rd,rs1,rs2 rd rs1 rs2 0000101 010 0101011 AMOMAX.W rd,rs1,rs2 rd rs1 rs2 0000110 010 0101011 AMOMINU.W rd,rs1,rs2 rd rs1 rs2 0000111 010 0101011 AMOMAXU.W rd,rs1,rs2 rd rs1 rs2 0000000 011 0101011 AMOADD.D rd,rs1,rs2 rd rs1 rs2 0000001 011 0101011 AMOSWAP.D rd,rs1,rs2 rd rs1 rs2 0000010 011 0101011 AMOAND.D rd,rs1,rs2 rd rs1 rs2 0000011 011 0101011 AMOOR.D rd,rs1,rs2 rd rs1 rs2 0000100 011 0101011 AMOMIN.D rd,rs1,rs2 rd rs1 rs2 0000101 011 0101011 AMOMAX.D rd,rs1,rs2 rd rs1 rs2 0000110 011 0101011 AMOMINU.D rd,rs1,rs2 rd rs1 rs2 0000111 011 0101011 AMOMAXU.D rd,rs1,rs2 26 RISC-V Speciﬁcation 31 27 26 22 21 17 16 15 14 12 11 10 9 8 7 6 0 jump target opcode J-type rd LUI-immediate opcode LUI-type rd rs1 imm[11:7] imm[6:0] funct3 opcode I-type imm[11:7] rs1 rs2 imm[6:0] funct3 opcode B-type rd rs1 rs2 funct10 opcode R-type rd rs1 rs2 rs3 funct5 opcode R4-type Integer Compute Instructions rd rs1 imm12 000 0010011 ADDI rd,rs1,imm12 rd rs1 000000 shamt 001 0010011 SLLI rd,rs1,shamt rd rs1 imm12 010 0010011 SLTI rd,rs1,imm12 rd rs1 imm12 011 0010011 SLTIU rd,rs1,imm12 rd rs1 imm12 100 0010011 XORI rd,rs1,imm12 rd rs1 000000 shamt 101 0010011 SRLI rd,rs1,shamt rd rs1 000001 shamt 101 0010011 SRAI rd,rs1,shamt rd rs1 imm12 110 0010011 ORI rd,rs1,imm12 rd rs1 imm12 111 0010011 ANDI rd,rs1,imm12 rd rs1 rs2 0000000 000 0110011 ADD rd,rs1,rs2 rd rs1 rs2 1000000 000 0110011 SUB rd,rs1,rs2 rd rs1 rs2 0000000 001 0110011 SLL rd,rs1,rs2 rd rs1 rs2 0000000 010 0110011 SLT rd,rs1,rs2 rd rs1 rs2 0000000 011 0110011 SLTU rd,rs1,rs2 rd rs1 rs2 0000000 100 0110011 XOR rd,rs1,rs2 rd rs1 rs2 0000000 101 0110011 SRL rd,rs1,rs2 rd rs1 rs2 1000000 101 0110011 SRA rd,rs1,rs2 rd rs1 rs2 0000000 110 0110011 OR rd,rs1,rs2 rd rs1 rs2 0000000 111 0110011 AND rd,rs1,rs2 rd rs1 rs2 0000001 000 0110011 MUL rd,rs1,rs2 rd rs1 rs2 0000001 001 0110011 MULH rd,rs1,rs2 rd rs1 rs2 0000001 010 0110011 MULHSU rd,rs1,rs2 rd rs1 rs2 0000001 011 0110011 MULHU rd,rs1,rs2 rd rs1 rs2 0000001 100 0110011 DIV rd,rs1,rs2 rd rs1 rs2 0000001 101 0110011 DIVU rd,rs1,rs2 rd rs1 rs2 0000001 110 0110011 REM rd,rs1,rs2 rd rs1 rs2 0000001 111 0110011 REMU rd,rs1,rs2 rd imm20 0110111 LUI rd,imm20 32-bit Integer Compute Instructions rd rs1 imm12 000 0011011 ADDIW rd,rs1,imm12 rd rs1 0000000 shamtw 001 0011011 SLLIW rd,rs1,shamtw rd rs1 0000000 shamtw 101 0011011 SRLIW rd,rs1,shamtw rd rs1 0000010 shamtw 101 0011011 SRAIW rd,rs1,shamtw rd rs1 rs2 0000000 000 0111011 ADDW rd,rs1,rs2 rd rs1 rs2 1000000 000 0111011 SUBW rd,rs1,rs2 rd rs1 rs2 0000000 001 0111011 SLLW rd,rs1,rs2 rd rs1 rs2 0000000 101 0111011 SRLW rd,rs1,rs2 rd rs1 rs2 1000000 101 0111011 SRAW rd,rs1,rs2 rd rs1 rs2 0000001 000 0111011 MULW rd,rs1,rs2 rd rs1 rs2 0000001 100 0111011 DIVW rd,rs1,rs2 rd rs1 rs2 0000001 101 0111011 DIVUW rd,rs1,rs2 rd rs1 rs2 0000001 110 0111011 REMW rd,rs1,rs2 rd rs1 rs2 0000001 111 0111011 REMUW rd,rs1,rs2 Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 27 31 27 26 22 21 17 16 15 14 12 11 10 9 8 7 6 0 jump target opcode J-type rd LUI-immediate opcode LUI-type rd rs1 imm[11:7] imm[6:0] funct3 opcode I-type imm[11:7] rs1 rs2 imm[6:0] funct3 opcode B-type rd rs1 rs2 funct10 opcode R-type rd rs1 rs2 rs3 funct5 opcode R4-type Floating-Point Memory Instructions rd rs1 imm12 010 0000111 FLW rd,rs1,imm12 rd rs1 imm12 011 0000111 FLD rd,rs1,imm12 imm12hi rs1 rs2 imm12lo 010 0100111 FSW rs1,rs2,imm12 imm12hi rs1 rs2 imm12lo 011 0100111 FSD rs1,rs2,imm12 Floating-Point Compute Instructions rd rs1 rs2 00000 rm 00 1010011 FADD.S rd,rs1,rs2[,rm] rd rs1 rs2 00001 rm 00 1010011 FSUB.S rd,rs1,rs2[,rm] rd rs1 rs2 00010 rm 00 1010011 FMUL.S rd,rs1,rs2[,rm] rd rs1 rs2 00011 rm 00 1010011 FDIV.S rd,rs1,rs2[,rm] rd rs1 00000 00100 rm 00 1010011 FSQRT.S rd,rs1[,rm] rd rs1 rs2 11000 000 00 1010011 FMIN.S rd,rs1,rs2 rd rs1 rs2 11001 000 00 1010011 FMAX.S rd,rs1,rs2 rd rs1 rs2 00000 rm 01 1010011 FADD.D rd,rs1,rs2[,rm] rd rs1 rs2 00001 rm 01 1010011 FSUB.D rd,rs1,rs2[,rm] rd rs1 rs2 00010 rm 01 1010011 FMUL.D rd,rs1,rs2[,rm] rd rs1 rs2 00011 rm 01 1010011 FDIV.D rd,rs1,rs2[,rm] rd rs1 00000 00100 rm 01 1010011 FSQRT.D rd,rs1[,rm] rd rs1 rs2 11000 000 01 1010011 FMIN.D rd,rs1,rs2 rd rs1 rs2 11001 000 01 1010011 FMAX.D rd,rs1,rs2 rd rs1 rs2 rs3 rm 00 1000011 FMADD.S rd,rs1,rs2,rs3[,rm] rd rs1 rs2 rs3 rm 00 1000111 FMSUB.S rd,rs1,rs2,rs3[,rm] rd rs1 rs2 rs3 rm 00 1001011 FNMSUB.S rd,rs1,rs2,rs3[,rm] rd rs1 rs2 rs3 rm 00 1001111 FNMADD.S rd,rs1,rs2,rs3[,rm] rd rs1 rs2 rs3 rm 01 1000011 FMADD.D rd,rs1,rs2,rs3[,rm] rd rs1 rs2 rs3 rm 01 1000111 FMSUB.D rd,rs1,rs2,rs3[,rm] rd rs1 rs2 rs3 rm 01 1001011 FNMSUB.D rd,rs1,rs2,rs3[,rm] rd rs1 rs2 rs3 rm 01 1001111 FNMADD.D rd,rs1,rs2,rs3[,rm] 28 RISC-V Speciﬁcation 31 27 26 22 21 17 16 15 14 12 11 10 9 8 7 6 0 jump target opcode J-type rd LUI-immediate opcode LUI-type rd rs1 imm[11:7] imm[6:0] funct3 opcode I-type imm[11:7] rs1 rs2 imm[6:0] funct3 opcode B-type rd rs1 rs2 funct10 opcode R-type rd rs1 rs2 rs3 funct5 opcode R4-type Floating-Point Move & Conversion Instructions rd rs1 rs2 00101 000 00 1010011 FSGNJ.S rd,rs1,rs2 rd rs1 rs2 00110 000 00 1010011 FSGNJN.S rd,rs1,rs2 rd rs1 rs2 00111 000 00 1010011 FSGNJX.S rd,rs1,rs2 rd rs1 rs2 00101 000 01 1010011 FSGNJ.D rd,rs1,rs2 rd rs1 rs2 00110 000 01 1010011 FSGNJN.D rd,rs1,rs2 rd rs1 rs2 00111 000 01 1010011 FSGNJX.D rd,rs1,rs2 rd rs1 00000 10001 rm 00 1010011 FCVT.S.D rd,rs1[,rm] rd rs1 00000 10000 rm 01 1010011 FCVT.D.S rd,rs1[,rm] Integer to Floating-Point Move & Conversion Instructions rd rs1 00000 01100 rm 00 1010011 FCVT.S.L rd,rs1[,rm] rd rs1 00000 01101 rm 00 1010011 FCVT.S.LU rd,rs1[,rm] rd rs1 00000 01110 rm 00 1010011 FCVT.S.W rd,rs1[,rm] rd rs1 00000 01111 rm 00 1010011 FCVT.S.WU rd,rs1[,rm] rd rs1 00000 01100 rm 01 1010011 FCVT.D.L rd,rs1[,rm] rd rs1 00000 01101 rm 01 1010011 FCVT.D.LU rd,rs1[,rm] rd rs1 00000 01110 rm 01 1010011 FCVT.D.W rd,rs1[,rm] rd rs1 00000 01111 rm 01 1010011 FCVT.D.WU rd,rs1[,rm] rd rs1 00000 11110 000 00 1010011 MXTF.S rd,rs1 rd rs1 00000 11110 000 01 1010011 MXTF.D rd,rs1 rd rs1 00000 11111 000 00 1010011 MTFSR rd,rs1 Floating-Point to Integer Move & Conversion Instructions rd rs1 00000 01000 rm 00 1010011 FCVT.L.S rd,rs1[,rm] rd rs1 00000 01001 rm 00 1010011 FCVT.LU.S rd,rs1[,rm] rd rs1 00000 01010 rm 00 1010011 FCVT.W.S rd,rs1[,rm] rd rs1 00000 01011 rm 00 1010011 FCVT.WU.S rd,rs1[,rm] rd rs1 00000 01000 rm 01 1010011 FCVT.L.D rd,rs1[,rm] rd rs1 00000 01001 rm 01 1010011 FCVT.LU.D rd,rs1[,rm] rd rs1 00000 01010 rm 01 1010011 FCVT.W.D rd,rs1[,rm] rd rs1 00000 01011 rm 01 1010011 FCVT.WU.D rd,rs1[,rm] rd 00000 rs2 11100 000 00 1010011 MFTX.S rd,rs2 rd 00000 rs2 11100 000 01 1010011 MFTX.D rd,rs2 rd 00000 00000 11101 000 00 1010011 MFFSR rd Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 29 31 27 26 22 21 17 16 15 14 12 11 10 9 8 7 6 0 jump target opcode J-type rd LUI-immediate opcode LUI-type rd rs1 imm[11:7] imm[6:0] funct3 opcode I-type imm[11:7] rs1 rs2 imm[6:0] funct3 opcode B-type rd rs1 rs2 funct10 opcode R-type rd rs1 rs2 rs3 funct5 opcode R4-type Floating-Point Compare Instructions rd rs1 rs2 10101 000 00 1010011 FEQ.S rd,rs1,rs2 rd rs1 rs2 10110 000 00 1010011 FLT.S rd,rs1,rs2 rd rs1 rs2 10111 000 00 1010011 FLE.S rd,rs1,rs2 rd rs1 rs2 10101 000 01 1010011 FEQ.D rd,rs1,rs2 rd rs1 rs2 10110 000 01 1010011 FLT.D rd,rs1,rs2 rd rs1 rs2 10111 000 01 1010011 FLE.D rd,rs1,rs2 Miscellaneous Memory Instructions rd rs1 imm12 001 0101111 FENCE.I rd,rs1,imm12 rd rs1 imm12 010 0101111 FENCE rd,rs1,imm12 System Instructions 00000 00000 00000 0000000 000 1110111 SYSCALL 00000 00000 00000 0000000 001 1110111 BREAK rd 00000 00000 0000000 100 1110111 RDCYCLE rd rd 00000 00000 0000001 100 1110111 RDTIME rd rd 00000 00000 0000010 100 1110111 RDINSTRET rd Table 6: Instruction listing for RISC-V 30 RISC-V Speciﬁcation 4 Floating-Point Extensions This section describes the optional 128-bit binary ﬂoating-point instructions, and how we would intend to support the decimal ﬂoating-point arithmetic deﬁned in the IEEE 754-2008 standard. 4.1 Quad-Precision Binary Floating-Point Extension The 128-bit or quad-precision binary ﬂoating-point extensions are built upon the base ﬂoating-point instructions, and are only available as an extension to RV[C]64. The ﬂoating-point registers are now extended to hold either a single, double, or quad-precision ﬂoating-point value. A new supported format is added to the format ﬁeld of most instructions, as shown in Table 7. fmt ﬁeld Mnemonic Meaning 00 S 32-bit single-precision 01 D 64-bit double-precision 10 - reserved 11 Q 128-bit quad-precisionn Table 7: Format ﬁeld encoding. The following instructions support the quad-precision format: FADD/FSUB, FMUL/FDIV, FSQRT, F[N]MADD/F[N]MSUB, FCVT.fmt.W[U], FCVT.W[U].fmt, FCVT.fmt.L[U], FCVT.L[U].fmt, FS- GNJ[N], FSGNX, and FEQ/FLT/FLE. New ﬂoating-point to ﬂoating-point conversion instructions FCVT.S.Q, FCVT.Q.S, FCVT.D.Q, FCVT.Q.D are added. MFTX.Q and MXTF.Q instructions are not provided, so quad-precision bit patterns must be moved to the integer registers via memory. New 128-bit variants of LOAD-FP and STORE-FP instructions are added, encoded with a new value for the width ﬁeld. 4.2 Decimal Floating-Point Extension The existing ﬂoating-point registers can be used to hold 64-bit and 128-bit decimal ﬂoating-point values, with the existing ﬂoating-point load and store instructions used to move values to and from memory. Due to the large opcode space required by the fused multiply-add instructions, the decimal ﬂoating- point instruction extension requires a 48-bit instruction encoding. Copyright (c) 2010, 2011, The Regents of the University of California. All rights reserved. 31 5 Packed-SIMD Extensions In this section, we outline how a packed-SIMD extension could be added to RISC-V (any of RV[C]64 or RV[C]32). A packed-SIMD extension will have some ﬁxed register width of 64 bits, 128 bits, or larger. These registers will be overlaid on the RISC-V ﬂoating-point registers. Each register can be treated as N×64-bit, 2N ×32-bit, 4N ×16-bit, or 8N ×8-bit packed variables, where N is 1 for the base 64-bit ﬂoating-point ISA but can be extended up to N = 64 (4096-bit registers). Packed-SIMD instructions operate on these packed values in FP registers. The existing ﬂoating-point load and store instructions can be used to load and store various-sized words from memory. The base ISA supports 32-bit and 64-bit loads and stores, but the LOAD-FP and STORE-FP instruction encodings allow up to 8 diﬀerent widths to be encoded (32, 64, 128, 256, 512, 1024, 2048, 4096). When used with packed-SIMD operations, it is desirable to support non-naturally aligned loads and stores in hardware. Simple packed-SIMD extensions might ﬁt in unused 32-bit instruction opcodes, but more extensive packed-SIMD extensions will likely require a 48-bit instruction encoding. It is natural to use the ﬂoating-point registers for packed-SIMD values rather than the integer registers (PA-RISC and Alpha packed-SIMD extensions) as this frees the integer registers for control and address values and leads naturally to a decoupled integer/ﬂoating-point unit hardware design. The ﬂoating-point load and store instruction encodings also have space to handle wider packed-SIMD registers. Reusing the ﬂoating-point registers for packed-SIMD values does make it more diﬃcult to use a recoded internal format for ﬂoating-point values. 32 RISC-V Speciﬁcation 6 History and Acknowledgements The RISC-V ISA and instruction set manual builds up several earlier projects. Several aspects of the supervisor-level machine and the overall format of the manual date back to the T0 (Torrent-0) vector microprocessor project at UC Berkeley and ICSI, begun in 1992. T0 was a vector processor based on the MIPS-II ISA, with Krste Asanovi´ c as main architect and RTL designer, and Brian Kingsbury and Bertrand Irrisou as principal VLSI implementers. David Johnson at ICSI was a major contributor to the T0 ISA design, particularly supervisor mode, and to the manual text. John Hauser also provided considerable feedback on the T0 ISA design. The Scale (Software-Controlled Architecture for Low Energy) project at MIT, begun in 2000, built upon the T0 project infrastructure, reﬁned the supervisor-level interface, and moved away from the MIPS scalar ISA by dropping the branch delay slot. Ronny Krashinsky and Christopher Batten were the principal architects of the Scale Vector-Thread processor at MIT, while Mark Hampton ported the GCC-based compiler infrastructure and tools for Scale. A lightly edited version of the T0 MIPS scalar processor speciﬁcation (MIPS-6371) was used in teaching a new version of the MIT 6.371 Introduction to VLSI Systems class in the Fall 2002 semester, with Chris Terman and Krste Asanovi´ c as lecturers. Chris Terman contributed most of the lab material for the class (there was no TA!). The 6.371 class evolved into the trial 6.884 Complex Digital Design class at MIT, taught by Arvind and Krste Asanovi´ c in Spring 2005, which became a regular Spring class 6.375. A reduced version of the Scale MIPS-based scalar ISA, named SMIPS, was used in 6.884/6.375. Christopher Batten was the TA for the early oﬀerings of these classes and developed a considerable amount of documentation and lab material based around the SMIPS ISA. This same SMIPS lab material was adapted and enhanced by TA Yunsup Lee for the UC Berkeley Fall 2009 CS250 VLSI Systems Design class taught by John Wawrzynek, Krste Asanovi´ c, and John Lazzaro. The Maven (Malleable Array of Vector-thread ENgines) project was a second-generation vector- thread architecture, whose design was led by Christopher Batten while an Exchange Scholar at UC Berkeley starting in summer 2007. Hidetaka Aoki, a visiting industrial fellow from Hitachi gave considerable feedback on the early Maven ISA and microarchitecture design. The Maven infrastructure was based on the Scale infrastructure but the Maven ISA moved further away from the MIPS ISA variant deﬁned in Scale, with a uniﬁed ﬂoating-point and integer register ﬁle. Maven was designed to support experimentation with alternative data-parallel accelerators. Yunsup Lee was the main implementor of the various Maven vector units, while Rimas Avizienis was the main implementor of the various Maven scalar units. Christopher Batten and Yunsup Lee ported GCC to work with the new Maven ISA. Christopher Celio provided the initial deﬁnition of a traditional vector instruction set (“Flood”) variant of Maven. Based on experience with all these previous projects, the RISC-V ISA deﬁnition was begun in Summer 2010. An initial version of the RISC-V 32-bit instruction subset was used in the UC Berkeley Fall 2010 CS250 VLSI Systems Design class. RISC-V is a clean break from the earlier MIPS-inspired designs. John Hauser contributed to the ﬂoating-point ISA deﬁnition.
Handbook_of_Applied_Cryptography	HANDBOOK of APPLIED CRYPTOGRAPHY Alfred J. Menezes P a u l C . v a n O o r s c h o t S c o t t A . V a n s t o n e Foreword by R.L. Rivest As we draw near to closing out the twentieth century, we see quite clearly that the information-processing and telecommunications revolutions now underway will continue vigorously into the twenty-first. We interact and transact by directing flocks of digital packets towards each other through cyberspace, carrying love notes, digital cash, and secret corporate documents. Our personal and economic lives rely more and more on our ability to let such ethereal carrier pigeons mediate at a distance what we used to do with face-to-face meetings, paper documents, and a firm handshake. Unfortunately, the technical wizardry enabling remote collaborations is founded on broadcasting everything as sequences of zeros and ones that one's own dog wouldn't recognize. What is to distinguish a digital dollar when it is as easily reproducible as the spoken word? How do we converse privately when every syllable is bounced off a satellite and smeared over an entire continent? How should a bank know that it really is Bill Gates requesting from his laptop in Fiji a transfer of $10,000,000,000 to another bank? Fortunately, the magical mathematics of cryptography can help. Cryptography provides techniques for keeping information secret, for determining that information has not been tampered with, and for determining who authored pieces of information. Cryptography is fascinating because of the close ties it forges between theory and practice, and because today's practical applications of cryptography are pervasive and critical components of our information-based society. Information-protection protocols designed on theoretical foundations one year appear in products and standards documents the next. Conversely, new theoretical developments sometimes mean that last year's proposal has a previously unsuspected weakness. While the theory is advancing vigorously, there are as yet few true guarantees; the security of many proposals depends on unproven (if plausible) assumptions. The theoretical work refines and improves the practice, while the practice challenges and inspires the theoretical work. When a system is "broken," our knowledge improves, and next year's system is improved to repair the defect. (One is reminded of the long and intriguing battle between the designers of bank vaults and their opponents.) Cryptography is also fascinating because of its game-like adversarial nature. A good cryptographer rapidly changes sides back and forth in his or her thinking, from attacker to defender and back. Just as in a game of chess, sequences of moves and counter- moves must be considered until the current situation is understood. Unlike chess players, cryptographers must also consider all the ways an adversary might try to gain by breaking the rules or violating expectations. (Does it matter if she measures how long I am computing? Does it matter if her "random" number isn't one?) The current volume is a major contribution to the field of cryptography. It is a rigorous encyclopedia of known techniques, with an emphasis on those that are both (believed to be) secure and practically useful. It presents in a coherent manner most of the important cryptographic tools one needs to implement secure cryptographic systems, and explains many of the cryptographic principles and protocols of existing systems. The topics covered range from low-level considerations such as random-number generation and efficient modular exponentiation algorithms and medium-level items such as public- key signature techniques, to higher-level topics such as zero-knowledge protocols. This book's excellent organization and style allow it to serve well as both a self-contained tutorial and an indispensable desk reference. In documenting the state of a fast-moving field, the authors have done incredibly well at providing error-free comprehensive content that is up-to-date. Indeed, many of the chapters, such as those on hash functions or key-establishment protocols, break new ground in both their content and their unified presentations. In the trade-off between comprehensive coverage and exhaustive treatment of individual items, the authors have chosen to write simply and directly, and thus efficiently, allowing each element to be explained together with their important details, caveats, and comparisons. While motivated by practical applications, the authors have clearly written a book that will be of as much interest to researchers and students as it is to practitioners, by including ample discussion of the underlying mathematics and associated theoretical considerations. The essential mathematical techniques and requisite notions are presented crisply and clearly, with illustrative examples. The insightful historical notes and extensive bibliography make this book a superb stepping-stone to the literature. (I was very pleasantly surprised to find an appendix with complete programs for the CRYPTO and EUROCRYPT conferences!) It is a pleasure to have been asked to provide the foreword for this book. I am happy to congratulate the authors on their accomplishment, and to inform the reader that he/she is looking at a landmark in the development of the field. Ronald L. Rivest Webster Professor of Electrical Engineering and Computer Science Massachusetts Institute of Technology June 1996 Preface This book is intended as a reference for professional cryptographers, presenting the techniques and algorithms of greatest interest to the current practitioner, along with the sup- porting motivation and background material. It also provides a comprehensive source from which to learn cryptography, serving both students and instructors. In addition, the rigor- ous treatment, breadth, and extensive bibliographic material should make it an important reference for research professionals. Our goal was to assimilate the existing cryptographic knowledge of industrial interest into one consistent, self-contained volume accessible to engineers in practice, to computer scientists and mathematicians in academia, and to motivated non-specialists with a strong desire to learn cryptography. Such a task is beyond the scope of each of the following: re- search papers, which by nature focus on narrow topics using very specialized (and often non-standard) terminology; survey papers, which typically address, at most, a small num- ber of major topics at a high level; and (regretably also) most books, due to the fact that many book authors lack either practical experience or familiarity with the research litera- ture or both. Our intent was to provide a detailed presentation of those areas of cryptogra- phy which we have found to be of greatest practical utility in our own industrial experience, while maintaining a sufﬁciently formal approach to be suitable both as a trustworthy refer- ence for those whose primary interest is further research, and to provide a solid foundation for students and others ﬁrst learning the subject. Throughout each chapter, we emphasize the relationship between various aspects of cryptography. Background sections commence most chapters, providing a framework and perspective for the techniques which follow. Computer source code (e.g. C code) for algo- rithms has been intentionally omitted, in favor of algorithms speciﬁed in sufﬁcient detail to allow direct implementation without consulting secondary references. We believe this style of presentation allows a better understanding of how algorithms actually work, while at the same time avoiding low-level implementation-speciﬁc constructs (which some readers will invariably be unfamiliar with) of various currently-popular programming languages. The presentation also strongly delineates what has been established as fact (by math- ematical arguments) from what is simply current conjecture. To avoid obscuring the very applied nature of the subject, rigorous proofs of correctness are in most cases omitted; how- ever, references given in the Notes section at the end of each chapter indicate the original or recommended sources for these results. The trailing Notes sections also provide infor- mation (quite detailed in places) on various additional techniques not addressed in the main text, and provide a survey of research activities and theoretical results; references again in- dicate where readers may pursue particular aspects in greater depth. Needless to say, many results, and indeed some entire research areas, have been given far less attention than they warrant, or have been omitted entirely due to lack of space; we apologize in advance for such major omissions, and hope that the most signiﬁcant of these are brought to our atten- tion. To provide an integrated treatment of cryptography spanning foundational motivation through concrete implementation, it is useful to consider a hierarchy of thought ranging from conceptual ideas and end-user services, down to the tools necessary to complete ac- tual implementations. Table 1 depicts the hierarchical structure around which this book is organized. Corresponding to this, Figure 1 illustrates how these hierarchical levels map xxiii xxiv Preface Information Security Objectives Conﬁdentiality Data integrity Authentication (entity and data origin) Non-repudiation Cryptographic functions Encryption Chapters 6, 7, 8 Message authentication and data integrity techniques Chapter 9 Identiﬁcation/entity authentication techniques Chapter 10 Digital signatures Chapter 11 Cryptographic building blocks Stream ciphers Chapter 6 Block ciphers (symmetric-key) Chapter 7 Public-key encryption Chapter 8 One-way hash functions (unkeyed) Chapter 9 Message authentication codes Chapter 9 Signature schemes (public-key, symmetric-key) Chapter 11 Utilities Public-key parameter generation Chapter 4 Pseudorandom bit generation Chapter 5 Efﬁcient algorithms for discrete arithmetic Chapter 14 Foundations Introduction to cryptography Chapter 1 Mathematical background Chapter 2 Complexity and analysis of underlying problems Chapter 3 Infrastructure techniques and commercial aspects Key establishment protocols Chapter 12 Key installation and key management Chapter 13 Cryptographic patents Chapter 15 Cryptographic standards Chapter 15 Table 1:Hierarchical levels of applied cryptography. onto the various chapters, and their inter-dependence. Table 2 lists the chapters of the book, along with the primary author(s) of each who should be contacted by readers with comments on speciﬁc chapters. Each chapter was writ- ten to provide a self-contained treatment of one major topic. Collectively, however, the chapters have been designed and carefully integrated to be entirely complementary with respect to deﬁnitions, terminology, and notation. Furthermore, there is essentially no du- plication of material across chapters; instead, appropriate cross-chapter references are pro- vided where relevant. While it is not intended that this book be read linearly from front to back, the material has been arranged so that doing so has some merit. Two primary goals motivated by the “handbook” nature of this project were to allow easy access to stand-alone results, and to al- low results and algorithms to be easily referenced (e.g., for discussion or subsequent cross- reference). To facilitate the ease of accessing and referencing results, items have been cate- gorized and numbered to a large extent, with the followingclasses of items jointlynumbered consecutively in each chapter:Deﬁnitions,Examples,Facts,Notes,Remarks,Algorithms, Protocols, andMechanisms. In more traditional treatments,Factsare usually identiﬁed as propositions, lemmas, or theorems. We use numberedNotesfor additional technical points, Preface xxv authenticationdata integrityconfidentiality data integrity techniques message authentication identification Chapter 9 Chapter 9Chapters 6,7,8 encryption Chapter 9 hash functions Chapter 9 signatures Chapter 11 (symmetric-key) number random Chapter 5 generation Chapter 4 non-repudiation Chapter 10 Chapter 11 signatures digital hash functions Chapter 13 key management (keyed)(unkeyed) stream ciphers Chapter 8 (public-key) Chapter 7 block ciphers (symmetric-key) signatures Chapter 11 (public-key) Chapter 3 public-key parameters public-key security foundationsestablishment of secret keys Chapter 12 Chapter 6 encryption Chapter 14 implementation efficient patents and standards Chapter 15 Chapter 2 background math Chapter 1 introduction Figure 1:Roadmap of the book. xxvi Preface Chapter Primary Author AJM PVO SA V 1. Overview of Cryptography * * * 2. Mathematical Background * 3. Number-Theoretic Reference Problems * 4. Public-Key Parameters * * 5. Pseudorandom Bits and Sequences * 6. Stream Ciphers * 7. Block Ciphers * 8. Public-Key Encryption * 9. Hash Functions and Data Integrity * 10. Identiﬁcation and Entity Authentication * 11. Digital Signatures * 12. Key Establishment Protocols * 13. Key Management Techniques * 14. Efﬁcient Implementation * 15. Patents and Standards * — Overall organization * * Table 2:Primary authors of each chapter. while numberedRemarks identify non-technical (often non-rigorous) comments, observa- tions, and opinions.Algorithms,Protocolsand Mechanisms refer to techniques involving a series of steps.Examples,Notes, andRemarks generally begin with parenthetical sum- mary titles to allow faster access, by indicating the nature of the content so that the entire item itself need not be read in order to determine this. The use of a large number of small subsections is also intended to enhance the handbook nature and accessibility to results. Regarding the partitioning of subject areas into chapters, we have used what we call a functional organization(based on functions of interest to end-users). For example, all items related to entity authentication are addressed in one chapter. An alternative would have been what may be called anacademic organization, under which perhaps, all protocols based on zero-knowledge concepts (including both a subset of entity authentication protocols and signature schemes) might be covered in one chapter. We believe that a functional organi- zation is more convenient to the practitioner, who is more likely to be interested in options available for an entity authentication protocol (Chapter 10) or a signature scheme (Chapter 11), than to be seeking a zero-knowledge protocol with unspeciﬁed end-purpose. In the front matter, a top-level Table of Contents (giving chapter numbers and titles only) is provided, as well as a detailed Table of Contents (down to the level of subsections, e.g., x 5.1.1). This is followed by a List of Figures, and a List of Tables. At the start of each chapter, a brief Table of Contents (specifying section number and titles only, e.g.,x 5.1,x 5.2) is also given for convenience. At the end of the book, we have included a list of papers presented at each of the Crypto, Eurocrypt, Asiacrypt/Auscrypt and Fast Software Encryption conferences to date, as well as a list of all papers published in theJournal of Cryptologyup to V olume 9. These are in addition to theReferencessection, each entry of which is cited at least once in the body of the handbook. Almost all of these references have been veriﬁed for correctness in their exact titles, volume and page numbers, etc. Finally, an extensive Index prepared by the authors is included. The Index begins with a List of Symbols. Our intention was not to introduce a collection of new techniques and protocols, but Preface xxvii rather to selectively present techniques from those currently available in the public domain. Such a consolidation of the literature is necessary from time to time. The fact that many good books in this ﬁeld include essentially no more than what is covered here in Chapters 7, 8 and 11 (indeed, these might serve as an introductory course along with Chapter 1) illus- trates that the ﬁeld has grown tremendously in the past 15 years. The mathematical foun- dation presented in Chapters 2 and 3 is hard to ﬁnd in one volume, and missing from most cryptography texts. The material in Chapter 4 on generation of public-key parameters, and in Chapter 14 on efﬁcient implementations, while well-known to a small body of specialists and available in the scattered literature, has previously not been available in general texts. The material in Chapters 5 and 6 on pseudorandom number generation and stream ciphers is also often absent (many texts focus entirely on block ciphers), or approached only from a theoretical viewpoint. Hash functions (Chapter 9) and identiﬁcation protocols (Chapter 10) have only recently been studied in depth as specialized topics on their own, and along with Chapter 12 on key establishment protocols, it is hard to ﬁnd consolidated treatments of these now-mainstream topics. Key management techniques as presented in Chapter 13 have traditionally not been given much attention by cryptographers, but are of great impor- tance in practice. A focused treatment of cryptographic patents and a concise summary of cryptographic standards, as presented in Chapter 15, are also long overdue. In most cases (with some historical exceptions), where algorithms are known to be in- secure, we have chosen to leave out speciﬁcation of their details, because most such tech- niques are of little practical interest. Essentially all of the algorithms included have been veriﬁed for correctness by independent implementation, conﬁrming the test vectors speci- ﬁed. Acknowledgements This project would not have been possible without the tremendous efforts put forth by our peers who have taken the time to read endless drafts and provide us with technical correc- tions, constructive feedback, and countless suggestions. In particular, the advice of our Ad- visory Editors has been invaluable, and it is impossible to attribute individualcredit for their many suggestions throughout this book. Among our Advisory Editors, we would particu- larly like to thank: Mihir Bellare Don Coppersmith Dorothy Denning Walter Fumy Burt Kaliski Peter Landrock Arjen Lenstra Ueli Maurer Chris Mitchell Tatsuaki Okamoto Bart Preneel Ron Rivest Gus Simmons Miles Smid Jacques Stern Mike Wiener Yacov Yacobi In addition, we gratefully acknowledge the exceptionally large number of additional indi- viduals who have helped improve the quality of this volume, by providing highly appreci- ated feedback and guidance on various matters. These individuals include: Carlisle Adams Rich Ankney Tom Berson Simon Blackburn Ian Blake Antoon Bosselaers Colin Boyd J¨ orgen Brandt Mike Burmester Ed Dawson Peter de Rooij Yvo Desmedt Whit Difﬁe Hans Dobbertin Carl Ellison Luis Encinas Warwick Ford Amparo Fuster Shuhong Gao Will Gilbert Marc Girault Jovan Goli´c Dieter Gollmann Li Gong xxviii Preface Carrie Grant Blake Greenlee Helen Gustafson Darrel Hankerson Anwar Hasan Don Johnson Mike Just Andy Klapper Lars Knudsen Neal Koblitz C ¸ etin Koc¸ Judy Koeller Evangelos Kranakis David Kravitz Hugo Krawczyk Xuejia Lai Charles Lam Alan Ling S. Mike Matyas Willi Meier Peter Montgomery Mike Mosca Tim Moses Serge Mister V olker M¨ueller David Naccache James Nechvatal Kaisa Nyberg Andrew Odlyzko Richard Outerbridge Walter Penzhorn Birgit Pﬁtzmann Kevin Phelps Leon Pintsov Fred Piper Carl Pomerance Matt Robshaw Peter Rodney Phil Rogaway Rainer Rueppel Mahmoud Salmasizadeh Roger Schlaﬂy Jeff Shallit Jon Sorenson Doug Stinson Andrea V anstone Serge V audenay Klaus V edder Jerry V eeh Fausto Vitini Lisa Yin Robert Zuccherato We apologize to those whose names have inadvertently escaped this list. Special thanks are due to Carrie Grant, Darrel Hankerson, Judy Koeller, Charles Lam, and Andrea V anstone. Their hard work contributed greatly to the quality of this book, and it was truly a pleasure working with them. Thanks also to the folks at CRC Press, including Tia Atchison, Gary Bennett, Susie Carlisle, Nora Konopka, Mary Kugler, Amy Morrell, Tim Pletscher, Bob Stern, and Wayne Y uhasz. The second author would like to thank his colleagues past and present at Nortel Secure Networks (Bell-Northern Research), many of whom are mentioned above, for their contributions on this project, and in particular Brian O’Higgins for his en- couragement and support; all views expressed, however, are entirely that of the author. The third author would also like to acknowledge the support of the Natural Sciences and Engi- neering Research Council. Any errors that remain are, of course, entirely our own. We would be grateful if readers who spot errors, missing references or credits, or incorrectly attributed results would contact us with details. It is our hope that this volume facilitates further advancement of the ﬁeld, and that we have helped play a small part in this. Alfred J. Menezes Paul C. van Oorschot Scott A. V anstone August, 1996 Table of Contents List of Tables xv List of Figures xix Foreword by R.L. Rivest xxi Preface xxiii 1 Overview of Cryptography 1 1.1 Introduction /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 1 1.2 Information security and cryptography/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 2 1.3 Background on functions/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 6 1.3.1 Functions (1-1, one-way, trapdoor one-way)/: /:/:/: /:/:/:/: /:/:/: /: 6 1.3.2 Permutations/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 10 1.3.3 Involutions/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 10 1.4 Basic terminology and concepts/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 11 1.5 Symmetric-key encryption /:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 15 1.5.1 Overview of block ciphers and stream ciphers/:/:/: /:/:/:/: /:/:/: /: 15 1.5.2 Substitution ciphers and transposition ciphers/:/:/: /:/:/:/: /:/:/: /: 17 1.5.3 Composition of ciphers/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 19 1.5.4 Stream ciphers/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 20 1.5.5 The key space/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 21 1.6 Digital signatures/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 22 1.7 Authentication and identiﬁcation/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 24 1.7.1 Identiﬁcation/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 24 1.7.2 Data origin authentication/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 25 1.8 Public-key cryptography/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 25 1.8.1 Public-key encryption/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 25 1.8.2 The necessity of authentication in public-key systems/:/:/: /:/:/: /: 27 1.8.3 Digital signatures from reversible public-key encryption/:/: /:/:/: /: 28 1.8.4 Symmetric-key vs. public-key cryptography/: /:/:/: /:/:/:/: /:/:/: /: 31 1.9 Hash functions /: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 33 1.10 Protocols and mechanisms/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 33 1.11 Key establishment, management, and certiﬁcation/:/: /:/:/: /:/:/:/: /:/:/: /: 35 1.11.1 Key management through symmetric-key techniques/:/:/: /:/:/: /: 36 1.11.2 Key management through public-key techniques/:/: /:/:/:/: /:/:/: /: 37 1.11.3 Trusted third parties and public-key certiﬁcates/:/: /:/:/:/: /:/:/: /: 39 1.12 Pseudorandom numbers and sequences/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 39 1.13 Classes of attacks and security models/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 41 1.13.1 Attacks on encryption schemes/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 41 1.13.2 Attacks on protocols/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 42 1.13.3 Models for evaluating security/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 42 1.13.4 Perspective for computational security/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 44 1.14 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 45 v vi Table of Contents 2 Mathematical Background 49 2.1 Probability theory/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 50 2.1.1 Basic deﬁnitions/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 50 2.1.2 Conditional probability/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 51 2.1.3 Random variables/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 51 2.1.4 Binomial distribution/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 52 2.1.5 Birthday attacks/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 53 2.1.6 Random mappings /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 54 2.2 Information theory /:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 56 2.2.1 Entropy /: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 56 2.2.2 Mutual information/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 57 2.3 Complexity theory/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 57 2.3.1 Basic deﬁnitions/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 57 2.3.2 Asymptotic notation/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 58 2.3.3 Complexity classes/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 59 2.3.4 Randomized algorithms/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 62 2.4 Number theory /: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 63 2.4.1 The integers/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 63 2.4.2 Algorithms inZ /: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 66 2.4.3 The integers modulon /: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 67 2.4.4 Algorithms inZ n /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 71 2.4.5 The Legendre and Jacobi symbols/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 72 2.4.6 Blum integers/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 74 2.5 Abstract algebra/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 75 2.5.1 Groups /:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 75 2.5.2 Rings /:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 76 2.5.3 Fields /:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 77 2.5.4 Polynomial rings/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 78 2.5.5 V ector spaces/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 79 2.6 Finite ﬁelds/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 80 2.6.1 Basic properties/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 80 2.6.2 The Euclidean algorithm for polynomials/:/: /:/:/: /:/:/:/: /:/:/: /: 81 2.6.3 Arithmetic of polynomials/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 83 2.7 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 85 3 Number-Theoretic Reference Problems 87 3.1 Introduction and overview/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 87 3.2 The integer factorization problem/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 89 3.2.1 Trial division/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 90 3.2.2 Pollard’s rho factoring algorithm/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 91 3.2.3 Pollard’sp /; /1 factoring algorithm/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 92 3.2.4 Elliptic curve factoring/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 94 3.2.5 Random square factoring methods/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 94 3.2.6 Quadratic sieve factoring/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 95 3.2.7 Number ﬁeld sieve factoring/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 98 3.3 The RSA problem /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 98 3.4 The quadratic residuosity problem/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 99 3.5 Computing square roots inZ n /: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 99 3.5.1 Case (i):n prime /: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 100 3.5.2 Case (ii):n composite /: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 101 Table of Contents vii 3.6 The discrete logarithm problem/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 103 3.6.1 Exhaustive search/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 104 3.6.2 Baby-step giant-step algorithm/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 104 3.6.3 Pollard’s rho algorithm for logarithms/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 106 3.6.4 Pohlig-Hellman algorithm/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 107 3.6.5 Index-calculus algorithm/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 109 3.6.6 Discrete logarithm problem in subgroups ofZ / p /:/: /:/:/:/: /:/:/: /: 113 3.7 The Difﬁe-Hellman problem /:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: 113 3.8 Composite moduli /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 114 3.9 Computing individual bits/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 114 3.9.1 The discrete logarithm problem inZ / p — individual bits/:/: /:/:/: /: 116 3.9.2 The RSA problem — individual bits/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 116 3.9.3 The Rabin problem — individual bits/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 117 3.10 The subset sum problem/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 117 3.10.1 TheL /3 -lattice basis reduction algorithm/:/:/: /:/:/: /:/:/:/: /:/:/: /: 118 3.10.2 Solving subset sum problems of low density/: /:/:/: /:/:/:/: /:/:/: /: 120 3.10.3 Simultaneous diophantine approximation/:/: /:/:/: /:/:/:/: /:/:/: /: 121 3.11 Factoring polynomials over ﬁnite ﬁelds/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 122 3.11.1 Square-free factorization/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 123 3.11.2 Berlekamp’sQ -matrix algorithm/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 124 3.12 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 125 4 Public-Key Parameters 133 4.1 Introduction/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 133 4.1.1 Generating large prime numbers naively/:/:/: /:/:/: /:/:/:/: /:/:/: /: 134 4.1.2 Distribution of prime numbers/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 134 4.2 Probabilistic primality tests/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 135 4.2.1 Fermat’s test/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 136 4.2.2 Solovay-Strassen test/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 137 4.2.3 Miller-Rabin test/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 138 4.2.4 Comparison: Fermat, Solovay-Strassen, and Miller-Rabin/:/: /: /:/: 140 4.3 (True) Primality tests/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 142 4.3.1 Testing Mersenne numbers/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 142 4.3.2 Primality testing using the factorization ofn /; /1 /: /:/:/:/: /:/:/: /: 143 4.3.3 Jacobi sum test/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 144 4.3.4 Tests using elliptic curves/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 145 4.4 Prime number generation/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 145 4.4.1 Random search for probable primes/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 145 4.4.2 Strong primes/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 149 4.4.3 NIST method for generating DSA primes/:/: /:/:/: /:/:/:/: /:/:/: /: 150 4.4.4 Constructive techniques for provable primes/: /:/:/: /:/:/:/: /:/:/: /: 152 4.5 Irreducible polynomials overZ p /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 154 4.5.1 Irreducible polynomials/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 154 4.5.2 Irreducible trinomials/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 157 4.5.3 Primitive polynomials/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 157 4.6 Generators and elements of high order/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 160 4.6.1 Selecting a primep and generator ofZ / p /:/:/: /:/:/: /:/:/:/: /:/:/: /: 164 4.7 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 165 viii Table of Contents 5 Pseudorandom Bits and Sequences 169 5.1 Introduction/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 169 5.1.1 Background and Classiﬁcation/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 170 5.2 Random bit generation /: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 171 5.3 Pseudorandom bit generation/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 173 5.3.1 ANSI X9.17 generator/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 173 5.3.2 FIPS 186 generator/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 174 5.4 Statistical tests/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 175 5.4.1 The normal and chi-square distributions/:/:/: /:/:/: /:/:/:/: /:/:/: /: 176 5.4.2 Hypothesis testing/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 179 5.4.3 Golomb’s randomness postulates/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 180 5.4.4 Five basic tests/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 181 5.4.5 Maurer’s universal statistical test/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 183 5.5 Cryptographically secure pseudorandom bit generation/:/: /:/:/:/: /:/:/: /: 185 5.5.1 RSA pseudorandom bit generator/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 185 5.5.2 Blum-Blum-Shub pseudorandom bit generator/:/:/: /:/:/:/: /:/:/: /: 186 5.6 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 187 6 Stream Ciphers 191 6.1 Introduction/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 191 6.1.1 Classiﬁcation/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 192 6.2 Feedback shift registers/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 195 6.2.1 Linear feedback shift registers/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 195 6.2.2 Linear complexity/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 198 6.2.3 Berlekamp-Massey algorithm/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 200 6.2.4 Nonlinear feedback shift registers/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 202 6.3 Stream ciphers based on LFSRs/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 203 6.3.1 Nonlinear combination generators/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 205 6.3.2 Nonlinear ﬁlter generators/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 208 6.3.3 Clock-controlled generators/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 209 6.4 Other stream ciphers/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 212 6.4.1 SEAL /:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 213 6.5 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 216 7 Block Ciphers 223 7.1 Introduction and overview/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 223 7.2 Background and general concepts/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 224 7.2.1 Introduction to block ciphers/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 224 7.2.2 Modes of operation/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 228 7.2.3 Exhaustive key search and multiple encryption/:/: /:/:/:/: /:/:/: /: 233 7.3 Classical ciphers and historical development/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 237 7.3.1 Transposition ciphers (background)/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 238 7.3.2 Substitution ciphers (background)/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 238 7.3.3 Polyalphabetic substitutions and Vigen`ere ciphers (historical)/:/: /: 241 7.3.4 Polyalphabetic cipher machines and rotors (historical)/:/:/: /:/:/: /: 242 7.3.5 Cryptanalysis of classical ciphers (historical)/:/:/: /:/:/:/: /:/:/: /: 245 7.4 DES /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 250 7.4.1 Product ciphers and Feistel ciphers/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 250 7.4.2 DES algorithm/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 252 7.4.3 DES properties and strength/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 256 Table of Contents ix 7.5 FEAL /:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 259 7.6 IDEA /:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 263 7.7 SAFER, RC5, and other block ciphers/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 266 7.7.1 SAFER /: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 266 7.7.2 RC5 /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 269 7.7.3 Other block ciphers/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 270 7.8 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 271 8 Public-Key Encryption 283 8.1 Introduction/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 283 8.1.1 Basic principles/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 284 8.2 RSA public-key encryption/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 285 8.2.1 Description/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 286 8.2.2 Security of RSA/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 287 8.2.3 RSA encryption in practice/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 290 8.3 Rabin public-key encryption/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 292 8.4 ElGamal public-key encryption/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 294 8.4.1 Basic ElGamal encryption/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 294 8.4.2 Generalized ElGamal encryption/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 297 8.5 McEliece public-key encryption/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 298 8.6 Knapsack public-key encryption/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 300 8.6.1 Merkle-Hellman knapsack encryption/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 300 8.6.2 Chor-Rivest knapsack encryption/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 302 8.7 Probabilistic public-key encryption/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 306 8.7.1 Goldwasser-Micali probabilistic encryption/: /:/:/: /:/:/:/: /:/:/: /: 307 8.7.2 Blum-Goldwasser probabilistic encryption/:/: /:/:/: /:/:/:/: /:/:/: /: 308 8.7.3 Plaintext-aware encryption/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 311 8.8 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 312 9 Hash Functions and Data Integrity 321 9.1 Introduction/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 321 9.2 Classiﬁcation and framework/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 322 9.2.1 General classiﬁcation/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 322 9.2.2 Basic properties and deﬁnitions/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 323 9.2.3 Hash properties required for speciﬁc applications/:/: /: /: /:/: /: /:/: 327 9.2.4 One-way functions and compression functions/:/:/: /:/:/:/: /:/:/: /: 327 9.2.5 Relationships between properties/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 329 9.2.6 Other hash function properties and applications/:/: /:/:/:/: /:/:/: /: 330 9.3 Basic constructions and general results/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 332 9.3.1 General model for iterated hash functions/:/: /:/:/: /:/:/:/: /:/:/: /: 332 9.3.2 General constructions and extensions/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 333 9.3.3 Formatting and initialization details/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 334 9.3.4 Security objectives and basic attacks/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 335 9.3.5 Bitsizes required for practical security/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 337 9.4 Unkeyed hash functions (MDCs)/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 338 9.4.1 Hash functions based on block ciphers/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 338 9.4.2 Customized hash functions based on MD4/:/: /:/:/: /:/:/:/: /:/:/: /: 343 9.4.3 Hash functions based on modular arithmetic/: /:/:/: /:/:/:/: /:/:/: /: 351 9.5 Keyed hash functions (MACs) /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 352 9.5.1 MACs based on block ciphers/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 353 x Table of Contents 9.5.2 Constructing MACs from MDCs/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 354 9.5.3 Customized MACs /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 356 9.5.4 MACs for stream ciphers/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 358 9.6 Data integrity and message authentication/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 359 9.6.1 Background and deﬁnitions/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 359 9.6.2 Non-malicious vs. malicious threats to data integrity/:/:/:/: /:/:/: /: 362 9.6.3 Data integrity using a MAC alone/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 364 9.6.4 Data integrity using an MDC and an authentic channel/:/: /:/:/: /: 364 9.6.5 Data integrity combined with encryption/:/:/: /:/:/: /:/:/:/: /:/:/: /: 364 9.7 Advanced attacks on hash functions/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 368 9.7.1 Birthday attacks/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 369 9.7.2 Pseudo-collisions and compression function attacks/:/:/:/: /:/:/: /: 371 9.7.3 Chaining attacks/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 373 9.7.4 Attacks based on properties of underlying cipher/: /:/:/:/: /:/:/: /: 375 9.8 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 376 10 Identiﬁcation and Entity Authentication 385 10.1 Introduction/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 385 10.1.1 Identiﬁcation objectives and applications/:/: /:/:/: /:/:/:/: /:/:/: /: 386 10.1.2 Properties of identiﬁcation protocols/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 387 10.2 Passwords (weak authentication)/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 388 10.2.1 Fixed password schemes: techniques/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 389 10.2.2 Fixed password schemes: attacks/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 391 10.2.3 Case study – UNIX passwords/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 393 10.2.4 PINs and passkeys/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 394 10.2.5 One-time passwords (towards strong authentication)/:/:/:/: /:/:/: /: 395 10.3 Challenge-response identiﬁcation (strong authentication)/: /:/:/:/: /:/:/: /: 397 10.3.1 Background on time-variant parameters/:/:/: /:/:/: /:/:/:/: /:/:/: /: 397 10.3.2 Challenge-response by symmetric-key techniques/:/: /: /: /:/: /: /:/: 400 10.3.3 Challenge-response by public-key techniques/:/:/: /:/:/:/: /:/:/: /: 403 10.4 Customized and zero-knowledge identiﬁcation protocols/: /:/:/:/: /:/:/: /: 405 10.4.1 Overview of zero-knowledge concepts/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 405 10.4.2 Feige-Fiat-Shamir identiﬁcation protocol/:/: /:/:/: /:/:/:/: /:/:/: /: 410 10.4.3 GQ identiﬁcation protocol/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 412 10.4.4 Schnorr identiﬁcation protocol/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 414 10.4.5 Comparison: Fiat-Shamir, GQ, and Schnorr/: /:/:/: /:/:/:/: /:/:/: /: 416 10.5 Attacks on identiﬁcation protocols/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 417 10.6 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 420 11 Digital Signatures 425 11.1 Introduction/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 425 11.2 A framework for digital signature mechanisms/:/:/: /:/:/: /:/:/:/: /:/:/: /: 426 11.2.1 Basic deﬁnitions/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 426 11.2.2 Digital signature schemes with appendix/:/:/: /:/:/: /:/:/:/: /:/:/: /: 428 11.2.3 Digital signature schemes with message recovery/: /:/:/:/: /:/:/: /: 430 11.2.4 Types of attacks on signature schemes/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 432 11.3 RSA and related signature schemes/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 433 11.3.1 The RSA signature scheme/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 433 11.3.2 Possible attacks on RSA signatures/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 434 11.3.3 RSA signatures in practice/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 435 Table of Contents xi 11.3.4 The Rabin public-key signature scheme/:/:/: /:/:/: /:/:/:/: /:/:/: /: 438 11.3.5 ISO/IEC 9796 formatting/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 442 11.3.6 PKCS #1 formatting/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 445 11.4 Fiat-Shamir signature schemes/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 447 11.4.1 Feige-Fiat-Shamir signature scheme/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 447 11.4.2 GQ signature scheme/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 450 11.5 The DSA and related signature schemes/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 451 11.5.1 The Digital Signature Algorithm (DSA)/:/:/: /:/:/: /:/:/:/: /:/:/: /: 452 11.5.2 The ElGamal signature scheme/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 454 11.5.3 The Schnorr signature scheme/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 459 11.5.4 The ElGamal signature scheme with message recovery/:/: /:/:/: /: 460 11.6 One-time digital signatures/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 462 11.6.1 The Rabin one-time signature scheme/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 462 11.6.2 The Merkle one-time signature scheme/:/:/: /:/:/: /:/:/:/: /:/:/: /: 464 11.6.3 Authentication trees and one-time signatures/: /:/:/: /:/:/:/: /:/:/: /: 466 11.6.4 The GMR one-time signature scheme/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 468 11.7 Other signature schemes/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 471 11.7.1 Arbitrated digital signatures/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 472 11.7.2 ESIGN /:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 473 11.8 Signatures with additional functionality/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 474 11.8.1 Blind signature schemes/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 475 11.8.2 Undeniable signature schemes/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 476 11.8.3 Fail-stop signature schemes/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 478 11.9 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 481 12 Key Establishment Protocols 489 12.1 Introduction/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 489 12.2 Classiﬁcation and framework/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 490 12.2.1 General classiﬁcation and fundamental concepts/:/: /:/:/:/: /:/:/: /: 490 12.2.2 Objectives and properties/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 493 12.2.3 Assumptions and adversaries in key establishment protocols/:/:/: /: 495 12.3 Key transport based on symmetric encryption/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 497 12.3.1 Symmetric key transport and derivation without a server/:/: /: /:/: 497 12.3.2 Kerberos and related server-based protocols/: /:/:/: /:/:/:/: /:/:/: /: 500 12.4 Key agreement based on symmetric techniques/:/:/: /:/:/: /:/:/:/: /:/:/: /: 505 12.5 Key transport based on public-key encryption/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 506 12.5.1 Key transport using PK encryption without signatures/:/:/: /:/:/: /: 507 12.5.2 Protocols combining PK encryption and signatures/:/:/:/: /:/:/: /: 509 12.5.3 Hybrid key transport protocols using PK encryption/:/:/:/: /:/:/: /: 512 12.6 Key agreement based on asymmetric techniques/:/: /:/:/: /:/:/:/: /:/:/: /: 515 12.6.1 Difﬁe-Hellman and related key agreement protocols/:/: /: /: /:/: /: /: 515 12.6.2 Implicitly-certiﬁed public keys/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 520 12.6.3 Difﬁe-Hellman protocols using implicitly-certiﬁed keys/:/:/: /:/:/: 522 12.7 Secret sharing/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 524 12.7.1 Simple shared control schemes/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 524 12.7.2 Threshold schemes/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 525 12.7.3 Generalized secret sharing/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 526 12.8 Conference keying /:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 528 12.9 Analysis of key establishment protocols/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 530 12.9.1 Attack strategies and classic protocol ﬂaws/: /:/:/: /:/:/:/: /:/:/: /: 530 xii Table of Contents 12.9.2 Analysis objectives and methods/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 532 12.10 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 534 13 Key Management Techniques 543 13.1 Introduction/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 543 13.2 Background and basic concepts/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 544 13.2.1 Classifying keys by algorithm type and intended use/:/:/:/: /:/:/: /: 544 13.2.2 Key management objectives, threats, and policy/:/: /:/:/:/: /:/:/: /: 545 13.2.3 Simple key establishment models/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 546 13.2.4 Roles of third parties/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 547 13.2.5 Tradeoffs among key establishment protocols/: /:/:/: /:/:/:/: /:/:/: 550 13.3 Techniques for distributing conﬁdential keys/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 551 13.3.1 Key layering and cryptoperiods/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 551 13.3.2 Key translation centers and symmetric-key certiﬁcates/:/:/: /:/:/: /: 553 13.4 Techniques for distributing public keys/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 555 13.4.1 Authentication trees/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 556 13.4.2 Public-key certiﬁcates/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 559 13.4.3 Identity-based systems/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 561 13.4.4 Implicitly-certiﬁed public keys/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 562 13.4.5 Comparison of techniques for distributing public keys/:/:/: /:/:/: /: 563 13.5 Techniques for controlling key usage/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 567 13.5.1 Key separation and constraints on key usage/: /:/:/: /:/:/:/: /:/:/: /: 567 13.5.2 Techniques for controlling use of symmetric keys/: /:/:/:/: /:/:/: /: 568 13.6 Key management involving multiple domains/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 570 13.6.1 Trust between two domains/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 570 13.6.2 Trust models involving multiple certiﬁcation authorities/:/: /:/:/: /: 572 13.6.3 Certiﬁcate distribution and revocation/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 576 13.7 Key life cycle issues/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 577 13.7.1 Lifetime protection requirements/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 578 13.7.2 Key management life cycle/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 578 13.8 Advanced trusted third party services/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 581 13.8.1 Trusted timestamping service/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 581 13.8.2 Non-repudiation and notarization of digital signatures/:/:/: /:/:/: /: 582 13.8.3 Key escrow/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 584 13.9 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 586 14 Efﬁcient Implementation 591 14.1 Introduction/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 591 14.2 Multiple-precision integer arithmetic/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 592 14.2.1 Radix representation/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 592 14.2.2 Addition and subtraction/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 594 14.2.3 Multiplication/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 595 14.2.4 Squaring/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 596 14.2.5 Division/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 598 14.3 Multiple-precision modular arithmetic/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 599 14.3.1 Classical modular multiplication/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 600 14.3.2 Montgomery reduction/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 600 14.3.3 Barrett reduction/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 603 14.3.4 Reduction methods for moduli of special form/: /: /:/: /: /: /:/: /: /:/: 605 14.4 Greatest common divisor algorithms/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 606 Table of Contents xiii 14.4.1 Binary gcd algorithm/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 606 14.4.2 Lehmer’s gcd algorithm/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 607 14.4.3 Binary extended gcd algorithm/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 608 14.5 Chinese remainder theorem for integers/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 610 14.5.1 Residue number systems/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 611 14.5.2 Garner’s algorithm/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 612 14.6 Exponentiation/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 613 14.6.1 Techniques for general exponentiation/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 614 14.6.2 Fixed-exponent exponentiation algorithms/:/: /:/:/: /:/:/:/: /:/:/: /: 620 14.6.3 Fixed-base exponentiation algorithms/:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 623 14.7 Exponent recoding /:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 627 14.7.1 Signed-digit representation/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 627 14.7.2 String-replacement representation/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 628 14.8 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 630 15 Patents and Standards 635 15.1 Introduction/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 635 15.2 Patents on cryptographic techniques/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 635 15.2.1 Five fundamental patents/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 636 15.2.2 Ten prominent patents/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 638 15.2.3 Ten selected patents/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 641 15.2.4 Ordering and acquiring patents/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 645 15.3 Cryptographic standards/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 645 15.3.1 International standards – cryptographic techniques/: /:/:/:/: /:/:/: /: 645 15.3.2 Banking security standards (ANSI, ISO)/:/:/: /:/:/: /:/:/:/: /:/:/: /: 648 15.3.3 International security architectures and frameworks/:/:/:/: /:/:/: /: 653 15.3.4 U.S. government standards (FIPS)/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 654 15.3.5 Internet standards and RFCs/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 655 15.3.6 De facto standards/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 656 15.3.7 Ordering and acquiring standards/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 656 15.4 Notes and further references/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 657 A Bibliography of Papers from Selected Cryptographic Forums 663 A.1 Asiacrypt/Auscrypt Proceedings/:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 663 A.2 Crypto Proceedings /:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 667 A.3 Eurocrypt Proceedings /: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 684 A.4 Fast Software Encryption Proceedings/:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 698 A.5 Journal of Cryptology papers/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /:/:/:/: /:/:/: /: 700 References 703 Index 755 Chapter /1 Overview of Cryptography Contents in Brief 1.1 Introduction ............................. 1 1.2 Information security and cryptography.............. 2 1.3 Background on functions ...................... 6 1.4 Basic terminology and concepts................... 11 1.5 Symmetric-key encryption ..................... 15 1.6 Digital signatures.......................... 22 1.7 Authentication and identiﬁcation.................. 24 1.8 Public-key cryptography ...................... 25 1.9 Hash functions ........................... 33 1.10 Protocols and mechanisms ..................... 33 1.11 Key establishment, management, and certiﬁcation......... 35 1.12 Pseudorandom numbers and sequences .............. 39 1.13 Classes of attacks and security models............... 41 1.14 Notes and further references.................... 45 1.1 Introduction Cryptography has a long and fascinating history. The most complete non-technical account of the subject is Kahn’sThe Codebreakers. This book traces cryptography from its initial and limited use by the Egyptians some 4000 years ago, to the twentieth century where it played a crucial role in the outcome of both world wars. Completed in 1963, Kahn’s book covers those aspects of the history which were most signiﬁcant (up to that time) to the devel- opment of the subject. The predominant practitioners of the art were those associated with the military, the diplomatic service and government in general. Cryptography was used as a tool to protect national secrets and strategies. The proliferation of computers and communications systems in the 1960s brought with it a demand from the private sector for means to protect information in digital form and to provide security services. Beginning with the work of Feistel at IBM in the early 1970s and culminating in 1977 with the adoption as a U.S. Federal Information Processing Standard for encrypting unclassiﬁed information, DES, the Data Encryption Standard, is the most well-known cryptographic mechanism in history. It remains the standard means for secur- ing electronic commerce for many ﬁnancial institutions around the world. The most striking development in the history of cryptography came in 1976 when Difﬁe and Hellman publishedNew Directions in Cryptography. This paper introduced the revolu- tionary concept of public-key cryptography and also provided a new and ingenious method 1 2 Ch. 1 Overview of Cryptography for key exchange, the security of which is based on the intractability of the discrete loga- rithm problem. Although the authors had no practical realization of a public-key encryp- tion scheme at the time, the idea was clear and it generated extensive interest and activity in the cryptographic community. In 1978 Rivest, Shamir, and Adleman discovered the ﬁrst practical public-key encryption and signature scheme, now referred to as RSA. The RSA scheme is based on another hard mathematical problem, the intractability of factoring large integers. This application of a hard mathematical problem to cryptography revitalized ef- forts to ﬁnd more efﬁcient methods to factor. The 1980s saw major advances in this area but none which rendered the RSA system insecure. Another class of powerful and practical public-key schemes was found by ElGamal in 1985. These are also based on the discrete logarithm problem. One of the most signiﬁcant contributions provided by public-key cryptography is the digital signature. In 1991 the ﬁrst international standard for digital signatures (ISO/IEC 9796) was adopted. It is based on the RSA public-key scheme. In 1994 the U.S. Govern- ment adopted the Digital Signature Standard, a mechanism based on the ElGamal public- key scheme. The search for new public-key schemes, improvements to existing cryptographic mec- hanisms, and proofs of security continues at a rapid pace. Various standards and infrastruc- tures involving cryptography are being put in place. Security products are being developed to address the security needs of an information intensive society. The purpose of this book is to give an up-to-date treatise of the principles, techniques, and algorithms of interest in cryptographic practice. Emphasis has been placed on those aspects which are most practical and applied. The reader will be made aware of the basic issues and pointed to speciﬁc related research in the literature where more indepth discus- sions can be found. Due to the volume of material which is covered, most results will be stated without proofs. This also serves the purpose of not obscuring the very applied nature of the subject. This book is intended for both implementers and researchers. It describes algorithms, systems, and their interactions. Chapter 1 is a tutorial on the many and various aspects of cryptography. It does not attempt to convey all of the details and subtleties inherent to the subject. Its purpose is to introduce the basic issues and principles and to point the reader to appropriate chapters in the book for more comprehensive treatments. Speciﬁc techniques are avoided in this chapter. 1.2 Information security and cryptography The concept ofinformationwill be taken to be an understood quantity. To introduce cryp- tography, an understanding of issues related to information security in general is necessary. Information security manifests itself in many ways according to the situation and require- ment. Regardless of who is involved, to one degree or another, all parties to a transaction must have conﬁdence that certain objectives associated with information security have been met. Some of these objectives are listed in Table 1.1. Over the centuries, an elaborate set of protocols and mechanisms has been created to deal with information security issues when the information is conveyed by physical doc- uments. Often the objectives of information security cannot solely be achieved through mathematical algorithms and protocols alone, but require procedural techniques and abid- ance of laws to achieve the desired result. For example, privacy of letters is provided by sealed envelopes delivered by an accepted mail service. The physical security of the en- velope is, for practical necessity, limited and so laws are enacted which make it a criminal §1.2 Information security and cryptography 3 privacy or conﬁdentiality keeping information secret from all but those who are autho- rized to see it. data integrity ensuring information has not been altered by unauthorized or unknown means. entity authentication or identiﬁcation corroboration of the identity of an entity (e.g., a person, a computer terminal, a credit card, etc.). message authentication corroborating the source of information; also known as data origin authentication. signature a means to bind information to an entity. authorization conveyance, to another entity, of ofﬁcial sanction to do or be something. validation a means to provide timeliness of authorization to use or ma- nipulate information or resources. access control restricting access to resources to privileged entities. certiﬁcation endorsement of information by a trusted entity. timestamping recording the time of creation or existence of information. witnessing verifying the creation or existence of information by an entity other than the creator. receipt acknowledgement that information has been received. conﬁrmation acknowledgement that services have been provided. ownership a means to provide an entity with the legal right to use or transfer a resource to others. anonymity concealing the identity of an entity involved in some process. non-repudiation preventing the denial of previous commitments or actions. revocation retraction of certiﬁcation or authorization. Table 1.1:Some information security objectives. offense to open mail for which one is not authorized. It is sometimes the case that security is achieved not through the information itself but through the physical document recording it. For example, paper currency requires special inks and material to prevent counterfeiting. Conceptually, the way information is recorded has not changed dramatically over time. Whereas information was typically stored and transmitted on paper, much of it now re- sides on magnetic media and is transmitted via telecommunications systems, some wire- less. What has changed dramatically is the ability to copy and alter information. One can make thousands of identical copies of a piece of information stored electronically and each is indistinguishable from the original. With information on paper, this is much more difﬁ- cult. What is needed then for a society where information is mostly stored and transmitted in electronic form is a means to ensure information security which is independent of the physical medium recording or conveying it and such that the objectives of information se- curity rely solely on digital information itself. One of the fundamental tools used in information security is the signature. It is a build- ing block for many other services such as non-repudiation, data origin authentication, iden- tiﬁcation, and witnessing, to mention a few. Having learned the basics in writing, an indi- vidual is taught how to produce a handwritten signature for the purpose of identiﬁcation. At contract age the signature evolves to take on a very integral part of the person’s identity. This signature is intended to be unique to the individual and serve as a means to identify, authorize, and validate. With electronic information the concept of a signature needs to be 4 Ch. 1 Overview of Cryptography redressed; it cannot simply be something unique to the signer and independent of the in- formation signed. Electronic replication of it is so simple that appending a signature to a document not signed by the originator of the signature is almost a triviality. Analogues of the “paper protocols” currently in use are required. Hopefully these new electronic based protocols are at least as good as those they replace. There is a unique op- portunity for society to introduce new and more efﬁcient ways of ensuring information se- curity. Much can be learned from the evolution of the paper based system, mimicking those aspects which have served us well and removing the inefﬁciencies. Achieving information security in an electronic society requires a vast array of techni- cal and legal skills. There is, however, no guarantee that all of the information security ob- jectives deemed necessary can be adequately met. The technical means is provided through cryptography. 1.1 DeﬁnitionCryptographyis the study of mathematical techniques related to aspects of in- formation security such as conﬁdentiality, data integrity, entity authentication, and data ori- gin authentication. Cryptography is not the only means of providing information security, but rather one set of techniques. Cryptographic goals Of all the information security objectives listed in Table 1.1, the following four form a framework upon which the others will be derived: (1) privacy or conﬁdentiality (§1.5,§1.8); (2) data integrity (§1.9); (3) authentication (§1.7); and (4) non-repudiation (§1.6). 1. Conﬁdentialityis a service used to keep the content of information from all but those authorized to have it.Secrecyis a term synonymous with conﬁdentiality and privacy. There are numerous approaches to providing conﬁdentiality, ranging from physical protection to mathematical algorithms which render data unintelligible. 2. Data integrityis a service which addresses the unauthorized alteration of data. To assure data integrity, one must have the ability to detect data manipulation by unau- thorized parties. Data manipulation includes such things as insertion, deletion, and substitution. 3. Authenticationis a service related to identiﬁcation. This function applies to both enti- ties and information itself. Two parties entering into a communication should identify each other. Information delivered over a channel should be authenticated as to origin, date of origin, data content, time sent, etc. For these reasons this aspect of cryptog- raphy is usually subdivided into two major classes:entity authenticationand data origin authentication. Data origin authentication implicitly provides data integrity (for if a message is modiﬁed, the source has changed). 4. Non-repudiationis a service which prevents an entity from denying previous commit- ments or actions. When disputes arise due to an entity denying that certain actions were taken, a means to resolve the situation is necessary. For example, one entity may authorize the purchase of property by another entity and later deny such autho- rization was granted. A procedure involving a trusted third party is needed to resolve the dispute. A fundamental goal of cryptography is to adequately address these four areas in both theory and practice. Cryptography is about the prevention and detection of cheating and other malicious activities. This book describes a number of basiccryptographic tools(primitives) used to provide information security. Examples of primitives include encryption schemes (§1.5 and§1.8), §1.2 Information security and cryptography 5 hash functions (§1.9), and digital signature schemes (§1.6). Figure 1.1 provides a schematic listing of the primitives considered and how they relate. Many of these will be brieﬂy intro- duced in this chapter, with detailed discussion left to later chapters. These primitives should Symmetric-key ciphers Primitives Unkeyed Arbitrary length hash functions hash functions (MACs) Arbitrary length ciphers Block Stream ciphers Pseudorandom sequences Random sequences Public-key Primitives Public-key ciphers Identiﬁcation primitives Signatures Identiﬁcation primitives Primitives Security Symmetric-key Primitives One-way permutations Signatures Figure 1.1:A taxonomy of cryptographic primitives. be evaluated with respect to various criteria such as: 1. level of security.This is usually difﬁcult to quantify. Often it is given in terms of the number of operations required (using the best methods currently known) to defeat the intended objective. Typically the level of security is deﬁned by an upper bound on the amount of work necessary to defeat the objective. This is sometimes called the work factor (see§1.13.4). 2. functionality.Primitives will need to be combined to meet various information se- curity objectives. Which primitives are most effective for a given objective will be determined by the basic properties of the primitives. 3. methods of operation.Primitives, when applied in various ways and with various in- puts, will typically exhibit different characteristics; thus, one primitive could provide 6 Ch. 1 Overview of Cryptography very different functionality depending on its mode of operation or usage. 4. performance.This refers to the efﬁciency of a primitive in a particular mode of op- eration. (For example, an encryption algorithm may be rated by the number of bits per second which it can encrypt.) 5. ease of implementation.This refers to the difﬁculty of realizing the primitive in a practical instantiation. This might include the complexity of implementing the prim- itive in either a software or hardware environment. The relative importance of various criteria is very much dependent on the application and resources available. For example, in an environment where computing power is limited one may have to trade off a very high level of security for better performance of the system as a whole. Cryptography, over the ages, has been an art practised by many who have devised ad hoc techniques to meet some of the information security requirements. The last twenty years have been a period of transition as the discipline moved from an art to a science. There are now several international scientiﬁc conferences devoted exclusively to cryptography and also an international scientiﬁc organization, the International Association for Crypto- logic Research (IACR), aimed at fostering research in the area. This book is about cryptography: the theory, the practice, and the standards. 1.3 Background on functions While this book is not a treatise on abstract mathematics, a familiarity with basic mathe- matical concepts will prove to be useful. One concept which is absolutely fundamental to cryptography is that of afunctionin the mathematical sense. A function is alternately re- ferred to as amapping or atransformation. 1.3.1 Functions (1-1, one-way, trapdoor one-way) A setconsists of distinct objects which are calledelementsof the set. For example, a setX might consist of the elementsa,b,c, and this is denotedX = {a,b,c }. 1.2 DeﬁnitionA functionis deﬁned by two setsX and Y and arulef which assigns to each element inX precisely one element inY . The setX is called thedomain of the function and Y thecodomain.I fx is an element ofX (usually writtenx ∈X) theimage ofx is the element inY which the rulef associates withx; the imagey ofx is denoted byy = f(x). Standard notation for a functionf from setX to setY isf : X −→Y .I fy ∈Y , then a preimageofy is an elementx ∈X for whichf(x)= y. The set of all elements inY which have at least one preimage is called theimage off, denotedIm(f). 1.3 Example (function) Consider the setsX = {a,b,c },Y = {1,2,3,4}, and the rulef from X toY deﬁned asf(a)=2 ,f(b)=4 ,f(c)=1 . Figure 1.2 shows a schematic of the setsX,Y and the functionf. The preimage of the element2 isa. The image off is {1,2,4}. □ Thinking of a function in terms of the schematic (sometimes called afunctional dia- gram) given in Figure 1.2, each element in the domainX has precisely one arrowed line originating from it. Each element in the codomainY can have any number of arrowed lines incident to it (including zero lines). §1.3 Background on functions 7 1 3 4 c b a 2 f YX Figure 1.2:A functionf from a setX of three elements to a setY of four elements. Often only the domainX and the rulef are given and the codomain is assumed to be the image off. This point is illustrated with two examples. 1.4 Example (function) TakeX = {1,2,3,..., 10}and letf be the rule that for eachx ∈X, f(x)= rx, whererx is the remainder whenx2 is divided by11. Explicitly then f(1) = 1 f(2) = 4 f(3) = 9 f(4) = 5 f(5) = 3 f(6) = 3 f(7) = 5 f(8) = 9 f(9) = 4 f(10) = 1. The image off is the setY = {1,3,4,5,9}. □ 1.5 Example (function) TakeX = {1,2,3,..., 1050}and letf be the rulef(x)= rx, where rx is the remainder whenx2 is divided by1050 +1 for allx ∈X. Here it is not feasible to write downf explicitly as in Example 1.4, but nonetheless the function is completely speciﬁed by the domain and the mathematical description of the rulef. □ (i) 1-1 functions 1.6 DeﬁnitionA function (or transformation) is1 −1 (one-to-one)if each element in the codomain Y is the image of at most one element in the domainX. 1.7 DeﬁnitionA function (or transformation) isonto if each element in the codomainY is the image of at least one element in the domain. Equivalently, a functionf : X −→Y is onto ifIm(f)= Y . 1.8 DeﬁnitionIf a functionf : X −→Y is1−1 and Im(f)= Y , thenf is called abijection. 1.9 FactIff : X −→Y is1 −1 thenf : X −→Im(f) is a bijection. In particular, if f : X −→Y is1 −1, andX and Y are ﬁnite sets of the same size, thenf is a bijection. In terms of the schematic representation, iff is a bijection, then each element inY has exactly one arrowed line incident with it. The functions described in Examples 1.3 and 1.4 are not bijections. In Example 1.3 the element3 is not the image of any element in the domain. In Example 1.4 each element in the codomain has two preimages. 1.10 DeﬁnitionIff is a bijection fromX toY then it is a simple matter to deﬁne a bijectiong fromY toX as follows: for eachy ∈Y deﬁne g(y)= xwhere x ∈X and f(x)= y. This functiong obtained fromf is called theinverse functionoff and is denoted byg = f−1. 8 Ch. 1 Overview of Cryptography b c d e 2 3 4 5 1 2 3 4 5 b c d e 1aa f XY g XY Figure 1.3:A bijectionf and its inverseg = f−1. 1.11 Example (inverse function) LetX = {a,b,c,d,e }, andY = {1,2,3,4,5}, and consider the rulef given by the arrowed edges in Figure 1.3.f is a bijection and its inverseg is formed simply by reversing the arrows on the edges. The domain ofg isY and the codomain isX. □ Note that iff is a bijection, then so isf−1. In cryptography bijections are used as the tool for encrypting messages and the inverse transformations are used to decrypt. This will be made clearer in§1.4 when some basic terminology is introduced. Notice that if the transformations were not bijections then it would not be possible to always decrypt to a unique message. (ii) One-way functions There are certain types of functions which play signiﬁcant roles in cryptography. At the expense of rigor, an intuitive deﬁnition of a one-way function is given. 1.12 DeﬁnitionA functionf from a setX to a setY is called aone-way functioniff(x) is “easy” to compute for allx ∈X but for “essentially all” elementsy ∈Im(f) it is “com- putationally infeasible” to ﬁnd anyx ∈X such thatf(x)= y. 1.13 Note (clariﬁcation of terms in Deﬁnition 1.12) (i) A rigorous deﬁnition of the terms “easy” and “computationally infeasible” is neces- sary but would detract from the simple idea that is being conveyed. For the purpose of this chapter, the intuitive meaning will sufﬁce. (ii) The phrase “for essentially all elements inY ” refers to the fact that there are a few valuesy ∈Y for which it is easy to ﬁnd anx ∈X such thaty = f(x). For example, one may computey = f(x) for a small number ofx values and then for these, the inverse is known by table look-up. An alternate way to describe this property of a one-way function is the following: for a randomy ∈Im(f) it is computationally infeasible to ﬁnd anyx ∈X such thatf(x)= y. The concept of a one-way function is illustrated through the following examples. 1.14 Example (one-way function) TakeX = {1,2,3,..., 16}and deﬁnef(x)= r x for all x ∈X where rx is the remainder when3x is divided by17. Explicitly, x 1 234 56789 1 0 1 1 1 2 1 3 1 4 1 5 1 6 f(x) 3 9 1 0 1 3 5 1 5 1 1 1 6 1 4874 1 2261 Given a number between1 and 16, it is relatively easy to ﬁnd the image of it underf. How- ever, given a number such as7, without having the table in front of you, it is harder to ﬁnd §1.3 Background on functions 9 xgiven thatf(x)=7 . Of course, if the number you are given is3 then it is clear thatx =1 is what you need; but for most of the elements in the codomain it is not that easy.□ One must keep in mind that this is an example which uses very small numbers; the important point here is that there is a difference in the amount of work to computef(x) and the amount of work to ﬁndx givenf(x). Even for very large numbers,f(x) can be computed efﬁciently using the repeated square-and-multiply algorithm (Algorithm 2.143), whereas the process of ﬁndingx from f(x) is much harder. 1.15 Example (one-way function)A prime numberis a positive integer greater than 1 whose only positive integer divisors are 1 and itself. Select primesp = 48611,q = 53993, form n = pq = 2624653723, and letX = {1,2,3,...,n −1}. Deﬁne a functionf on X by f(x)= rx for eachx ∈X, whererx is the remainder whenx3 is divided byn. For instance,f(2489991) = 1981394214since24899913 = 5881949859·n + 1981394214. Computing f(x) is a relatively simple thing to do, but to reverse the procedure is much more difﬁcult; that is, given a remainder to ﬁnd the valuex which was originally cubed (raised to the third power). This procedure is referred to as the computation of a modular cube root with modulusn. If the factors ofn are unknown and large, this is a difﬁcult problem; how- ever, if the factorsp and q ofn are known then there is an efﬁcient algorithm for computing modular cube roots. (See§8.2.2(i) for details.) □ Example 1.15 leads one to consider another type of function which will prove to be fundamental in later developments. (iii) Trapdoor one-way functions 1.16 DeﬁnitionA trapdoor one-way functionis a one-way functionf : X −→Y with the additional property that given some extra information (called thetrapdoor information)i t becomes feasible to ﬁnd for any giveny ∈Im(f),a nx ∈X such thatf(x)= y. Example 1.15 illustrates the concept of a trapdoor one-way function. With the addi- tional information of the factors ofn = 2624653723(namely,p = 48611and q = 53993, each of which is ﬁve decimal digits long) it becomes much easier to invert the function. The factors of2624653723 are large enough that ﬁnding them by hand computation would be difﬁcult. Of course, any reasonable computer program could ﬁnd the factors relatively quickly. If, on the other hand, one selectsp and q to be very large distinct prime numbers (each having about 100 decimal digits) then, by today’s standards, it is a difﬁcult problem, even with the most powerful computers, to deducep and q simply fromn. This is the well- known integer factorization problem(see§3.2) and a source of many trapdoor one-way functions. It remains to be rigorously established whether there actually are any (true) one-way functions. That is to say, no one has yet deﬁnitively proved the existence of such func- tions under reasonable (and rigorous) deﬁnitions of “easy” and “computationally infeasi- ble”. Since the existence of one-way functions is still unknown, the existence of trapdoor one-way functions is also unknown. However, there are a number of good candidates for one-way and trapdoor one-way functions. Many of these are discussed in this book, with emphasis given to those which are practical. One-way and trapdoor one-way functions are the basis for public-key cryptography (discussed in§1.8). The importance of these concepts will become clearer when their appli- cation to cryptographic techniques is considered. It will be worthwhile to keep the abstract concepts of this section in mind as concrete methods are presented. 10 Ch. 1 Overview of Cryptography 1.3.2 Permutations Permutations are functions which are often used in various cryptographic constructs. 1.17 DeﬁnitionLetSbe a ﬁnite set of elements. Apermutationp on Sis a bijection (Deﬁni- tion 1.8) fromSto itself (i.e.,p: S− →S). 1.18 Example (permutation) LetS= {1,2,3,4,5}. A permutationp: S− →Sis deﬁned as follows: p(1) = 3,p (2) = 5,p (3) = 4,p (4) = 2,p (5) = 1. A permutation can be described in various ways. It can be displayed as above or as an array: p = ( 12345 35421 ) , (1.1) where the top row in the array is the domain and the bottom row is the image under the mapping p. Of course, other representations are possible. □ Since permutations are bijections, they have inverses. If a permutation is written as an array (see 1.1), its inverse is easily found by interchanging the rows in the array and reorder- ing the elements in the new top row if desired (the bottom row would have to be reordered correspondingly). The inverse ofp in Example 1.18 isp −1 = ( 12345 54132 ) . 1.19 Example (permutation) LetX be the set of integers{0,1,2,...,pq −1}where p and q are distinctlargeprimes (for example,p and q are each about 100 decimal digits long), and suppose that neitherp−1 norq−1 is divisible by 3. Then the functionp(x)= rx, whererx is the remainder whenx3 is divided bypq, can be shown to be a permutation. Determining the inverse permutation is computationally infeasible by today’s standards unlessp and q are known (cf. Example 1.15). □ 1.3.3 Involutions Another type of function which will be referred to in§1.5.3 is an involution. Involutions have the property that they are their own inverses. 1.20 DeﬁnitionLetSbe a ﬁnite set and letf be a bijection fromStoS(i.e.,f : S− →S). The functionf is called aninvolutioniff = f−1. An equivalent way of stating this is f(f(x)) =x for allx ∈S. 1.21 Example (involution) Figure 1.4 is an example of an involution. In the diagram of an involution, note that ifj is the image ofi theni is the image ofj. □ §1.4 Basic terminology and concepts 11 1 2 3 4 5 2 3 4 5 1 SS Figure 1.4:An involution on a setS of 5 elements. 1.4 Basic terminology and concepts The scientiﬁc study of any discipline must be built upon rigorous deﬁnitions arising from fundamental concepts. What follows is a list of terms and basic concepts used throughout this book. Where appropriate, rigor has been sacriﬁced (here in Chapter 1) for the sake of clarity. Encryption domains and codomains •A denotes a ﬁnite set called thealphabet of deﬁnition. For example,A= {0,1}, the binary alphabet, is a frequently used alphabet of deﬁnition. Note that any alphabet can be encoded in terms of the binary alphabet. For example, since there are32 binary strings of length ﬁve, each letter of the English alphabet can be assigned a unique binary string of length ﬁve. •M denotes a set called themessage space. Mconsists of strings of symbols from an alphabet of deﬁnition. An element ofMis called aplaintext messageor simply a plaintext. For example,Mmay consist of binary strings, English text, computer code, etc. •C denotes a set called theciphertext space.Cconsists of strings of symbols from an alphabet of deﬁnition, which may differ from the alphabet of deﬁnition forM.A n element ofCis called aciphertext. Encryption and decryption transformations •K denotes a set called thekey space. An element ofKis called akey. •Each elemente ∈K uniquely determines a bijection fromMtoC, denoted byE e. Ee is called anencryption functionor anencryption transformation. Note thatEe must be a bijection if the process is to be reversed and a unique plaintext message recovered for each distinct ciphertext.1 •For eachd ∈K,Dd denotes a bijection fromCtoM(i.e.,Dd : C− →M).Dd is called adecryption functionordecryption transformation. •The process of applying the transformationEe to a messagem ∈M is usually re- ferred to asencryptingm or theencryptionofm. •The process of applying the transformationDd to a ciphertextc is usually referred to asdecryptingc or thedecryptionofc. 1More generality is obtained ifEe is simply deﬁned as a1 −1 transformation fromMtoC. That is to say, Ee is a bijection fromMtoIm(Ee) where Im(Ee) is a subset ofC. 12 Ch. 1 Overview of Cryptography •An encryption schemeconsists of a set{Ee : e ∈K} of encryption transformations and a corresponding set{Dd : d ∈K} of decryption transformations with the prop- erty that for eache ∈K there is a unique keyd ∈K such thatDd = E−1 e ; that is, Dd(Ee(m)) =m for allm ∈M. An encryption scheme is sometimes referred to as acipher. •The keyse and d in the preceding deﬁnition are referred to as akey pairand some- times denoted by(e,d). Note thate and d could be the same. •To constructan encryption scheme requires one to select a message spaceM, a ci- phertext spaceC, a key spaceK, a set of encryption transformations{Ee : e ∈K}, and a corresponding set of decryption transformations{Dd : d ∈K}. Achieving conﬁdentiality An encryption scheme may be used as follows for the purpose of achieving conﬁdentiality. Two parties Alice and Bob ﬁrst secretly choose or secretly exchange a key pair(e,d).A ta subsequent point in time, if Alice wishes to send a messagem ∈M to Bob, she computes c = Ee(m) and transmits this to Bob. Upon receivingc, Bob computesDd(c)= m and hence recovers the original messagem. The question arises as to why keys are necessary. (Why not just choose one encryption function and its corresponding decryption function?) Having transformations which are very similar but characterized by keys means that if some particular encryption/decryption transformation is revealed then one does not have to redesign the entire scheme but simply change the key. It is sound cryptographic practice to change the key (encryption/decryption transformation) frequently. As a physical analogue, consider an ordinary resettable combi- nation lock. The structure of the lock is available to anyone who wishes to purchase one but the combination is chosen and set by the owner. If the owner suspects that the combination has been revealed he can easily reset it without replacing the physical mechanism. 1.22 Example (encryption scheme) LetM = {m 1,m2,m3}and C = {c1,c2,c3}. There are precisely3! = 6bijections fromMtoC. The key spaceK = {1,2,3,4,5,6}has six elements in it, each specifying one of the transformations. Figure 1.5 illustrates the six encryption functions which are denoted byE i,1 ≤i ≤6. Alice and Bob agree on a trans- E1 m1 m2 m3 c1 c2 E2 m1 m2 m3 m1 m2 m3 E3 E4 m1 m2 m3 m1 m2 m3 E5 m1 m2 m3 E6 c1 c2 c1 c2c2 c1 c1 c2 c1 c2 c3 c3 c3 c3 c3 c3 Figure 1.5:Schematic of a simple encryption scheme. formation, sayE1. To encrypt the messagem1, Alice computesE1(m1)= c3 and sends c3 to Bob. Bob decryptsc3 by reversing the arrows on the diagram forE1 and observing thatc3 points tom1. §1.4 Basic terminology and concepts 13 When Mis a small set, the functional diagram is a simple visual means to describe the mapping. In cryptography, the setMis typically of astronomical proportions and, as such, the visual description is infeasible. What is required, in these cases, is some other simple means to describe the encryption and decryption transformations, such as mathematical al- gorithms. □ Figure 1.6 provides a simple model of a two-party communication using encryption. m c m Dd(c)= mEe(m)= c plaintext source Alice Bob UNSECURED CHANNEL Adversary decryptionencryption destination Figure 1.6:Schematic of a two-party communication using encryption. Communication participants Referring to Figure 1.6, the following terminology is deﬁned. •An entityorpartyis someone or something which sends, receives, or manipulates information. Alice and Bob are entities in Example 1.22. An entity may be a person, a computer terminal, etc. •A senderis an entity in a two-party communication which is the legitimate transmitter of information. In Figure 1.6, the sender is Alice. •A receiveris an entity in a two-party communication which is the intended recipient of information. In Figure 1.6, the receiver is Bob. •An adversaryis an entity in a two-party communication which is neither the sender nor receiver, and which tries to defeat the information security service being provided between the sender and receiver. Various other names are synonymous with adver- sary such as enemy, attacker, opponent, tapper, eavesdropper, intruder, and interloper. An adversary will often attempt to play the role of either the legitimate sender or the legitimate receiver. Channels •A channelis a means of conveying information from one entity to another. •A physically secure channelorsecure channelis one which is not physically acces- sible to the adversary. •An unsecured channelis one from which parties other than those for which the in- formation is intended can reorder, delete, insert, or read. •A secured channelis one from which an adversary does not have the ability to reorder, delete, insert, or read. 14 Ch. 1 Overview of Cryptography One should note the subtle difference between a physically secure channel and a se- cured channel – a secured channel may be secured by physical or cryptographic techniques, the latter being the topic of this book. Certain channels are assumed to be physically secure. These include trusted couriers, personal contact between communicating parties, and a ded- icated communication link, to name a few. Security A fundamental premise in cryptography is that the setsM,C,K,{Ee : e ∈K},{Dd : d ∈ K}are public knowledge. When two parties wish to communicate securely using an en- cryption scheme, the only thing that they keep secret is the particular key pair(e,d) which they are using, and which they must select. One can gain additional security by keeping the class of encryption and decryption transformations secret but one should not base the secu- rity of the entire scheme on this approach. History has shown that maintaining the secrecy of the transformations is very difﬁcult indeed. 1.23 DeﬁnitionAn encryption scheme is said to bebreakableif a third party, without prior knowledge of the key pair(e,d), can systematically recover plaintext from corresponding ciphertext within some appropriate time frame. An appropriate time frame will be a function of the useful lifespan of the data being protected. For example, an instruction to buy a certain stock may only need to be kept secret for a few minutes whereas state secrets may need to remain conﬁdential indeﬁnitely. An encryption scheme can be broken by trying all possible keys to see which one the communicating parties are using (assuming that the class of encryption functions is public knowledge). This is called anexhaustive searchof the key space. It follows then that the number of keys (i.e., the size of the key space) should be large enough to make this approach computationally infeasible. It is the objective of a designer of an encryption scheme that this be the best approach to break the system. Frequently cited in the literature areKerckhoffs’ desiderata, a set of requirements for cipher systems. They are given here essentially as Kerckhoffs originally stated them: 1. the system should be, if not theoretically unbreakable, unbreakable in practice; 2. compromise of the system details should not inconvenience the correspondents; 3. the key should be rememberable without notes and easily changed; 4. the cryptogram should be transmissible by telegraph; 5. the encryption apparatus should be portable and operable by a single person; and 6. the system should be easy, requiring neither the knowledge of a long list of rules nor mental strain. This list of requirements was articulated in 1883 and, for the most part, remains useful today. Point 2 allows that the class of encryption transformations being used be publicly known and that the security of the system should reside only in the key chosen. Information security in general So far the terminology has been restricted to encryption and decryption with the goal of pri- vacy in mind. Information security is much broader, encompassing such things as authen- tication and data integrity. A few more general deﬁnitions, pertinent to discussions later in the book, are given next. •An information security serviceis a method to provide some speciﬁc aspect of secu- rity. For example, integrity of transmitted data is a security objective, and a method to ensure this aspect is an information security service. §1.5 Symmetric-key encryption 15 •Breakingan information security service (which often involves more than simply en- cryption) implies defeating the objective of the intended service. •A passive adversaryis an adversary who is capable only of reading information from an unsecured channel. •An active adversaryis an adversary who may also transmit, alter, or delete informa- tion on an unsecured channel. Cryptology •Cryptanalysisis the study of mathematical techniques for attempting to defeat cryp- tographic techniques, and, more generally, information security services. •A cryptanalystis someone who engages in cryptanalysis. •Cryptologyis the study of cryptography (Deﬁnition 1.1) and cryptanalysis. •A cryptosystemis a general term referring to a set of cryptographic primitives used to provide information security services. Most often the term is used in conjunction with primitives providing conﬁdentiality, i.e., encryption. Cryptographic techniques are typically divided into two generic types:symmetric-key and public-key. Encryption methods of these types will be discussed separately in§1.5 and §1.8. Other deﬁnitions and terminology will be introduced as required. 1.5 Symmetric-key encryption §1.5 considers symmetric-key encryption. Public-key encryption is the topic of§1.8. 1.5.1 Overview of block ciphers and stream ciphers 1.24 DeﬁnitionConsider an encryption scheme consisting of the sets of encryption and de- cryption transformations{Ee : e ∈K} and {Dd : d ∈K}, respectively, whereKis the key space. The encryption scheme is said to besymmetric-keyif for each associated encryp- tion/decryption key pair(e,d), it is computationally “easy” to determined knowing onlye, and to determinee from d. Sincee = din most practical symmetric-key encryption schemes, the term symmetric- key becomes appropriate. Other terms used in the literature aresingle-key,one-key,private- key,2 and conventionalencryption. Example 1.25 illustrates the idea of symmetric-key en- cryption. 1.25 Example (symmetric-key encryption) LetA = {A,B,C,..., X,Y,Z}be the English alphabet. LetMand Cbe the set of all strings of length ﬁve overA. The keye is chosen to be a permutation onA. To encrypt, an English message is broken up into groups each having ﬁve letters (with appropriate padding if the length of the message is not a multiple of ﬁve) and a permutatione is applied to each letter one at a time. To decrypt, the inverse permutationd = e−1 is applied to each letter of the ciphertext. For instance, suppose that the keye is chosen to be the permutation which maps each letter to the one which is three positions to its right, as shown below e = ( ABCDEFGH I J KLMNOPQR S T UVWXYZ DEFGHI J KLMNO P QRSTUVWXY Z ABC ) 2Private key is a term also used in quite a different context (see§1.8). The term will be reserved for the latter usage in this book. 16 Ch. 1 Overview of Cryptography A message m = THISC IPHER ISCER TAINL YNOTS ECURE is encrypted to c = Ee(m) = WKLVF LSKHU LVFHU WDLQO BQRWV HFXUH. □ A two-party communication using symmetric-key encryption can be described by the block diagram of Figure 1.7, which is Figure 1.6 with the addition of the secure (both con- m e c SECURE CHANNEL Dd(c)= mEe(m)= c e m UNSECURED CHANNEL encryption plaintext source Alice Adversary source key decryption destination Bob Figure 1.7:Two-party communication using encryption, with a secure channel for key exchange. The decryption keyd can be efﬁciently computed from the encryption keye. ﬁdential and authentic) channel. One of the major issues with symmetric-key systems is to ﬁnd an efﬁcient method to agree upon and exchange keys securely. This problem is referred to as thekey distribution problem(see Chapters 12 and 13). It is assumed that all parties know the set of encryption/decryptiontransformations (i.e., they all know the encryption scheme). As has been emphasized several times the only infor- mation which should be required to be kept secret is the keyd. However, in symmetric-key encryption, this means that the keye must also be kept secret, asd can be deduced from e. In Figure 1.7 the encryption keye is transported from one entity to the other with the understanding that both can construct the decryption keyd. There are two classes of symmetric-key encryption schemes which are commonly dis- tinguished:block ciphersand stream ciphers. 1.26 DeﬁnitionA block cipheris an encryption scheme which breaks up the plaintext mes- sages to be transmitted into strings (calledblocks) of a ﬁxed lengtht over an alphabetA, and encrypts one block at a time. Most well-known symmetric-key encryption techniques are block ciphers. A number of examples of these are given in Chapter 7. Two important classes of block ciphers are substitution ciphersand transposition ciphers(§1.5.2). Product ciphers (§1.5.3) combine §1.5 Symmetric-key encryption 17 these. Stream ciphers are considered in§1.5.4, while comments on the key space follow in §1.5.5. 1.5.2 Substitution ciphers and transposition ciphers Substitution ciphers are block ciphers which replace symbols (or groups of symbols) by other symbols or groups of symbols. Simple substitution ciphers 1.27 DeﬁnitionLet Abe an alphabet ofq symbols andMbe the set of all strings of length t overA. LetKbe the set of all permutations on the setA. Deﬁne for eache ∈K an encryption transformationEe as: Ee(m)=( e(m1)e(m2) ···e(mt)) = (c1c2 ···ct)= c, where m =( m1m2 ···mt) ∈M. In other words, for each symbol in at-tuple, replace (substitute) it by another symbol fromAaccording to some ﬁxed permutatione. To decrypt c =( c1c2 ···ct) compute the inverse permutationd = e−1 and Dd(c)=( d(c1)d(c2) ···d(ct)) = (m1m2 ···mt)= m. Ee is called asimple substitution cipheror amono-alphabetic substitution cipher. The number of distinct substitution ciphers isq! and is independent of the block size in the cipher. Example 1.25 is an example of a simple substitution cipher of block length ﬁve. Simple substitution ciphers over small block sizes provide inadequate security even when the key space is extremely large. If the alphabet is the English alphabet as in Exam- ple 1.25, then the size of the key space is26! ≈4 ×1026, yet the key being used can be determined quite easily by examining a modest amount of ciphertext. This follows from the simple observation that the distribution of letter frequencies is preserved in the ciphertext. For example, the letterE occurs more frequently than the other letters in ordinary English text. Hence the letter occurring most frequently in a sequence of ciphertext blocks is most likely to correspond to the letterE in the plaintext. By observing a modest quantity of ci- phertext blocks, a cryptanalyst can determine the key. Homophonic substitution ciphers 1.28 DeﬁnitionTo each symbola ∈A , associate a setH(a) of strings oft symbols, with the restriction that the setsH(a),a ∈A, be pairwise disjoint. Ahomophonic substitution cipherreplaces each symbola in a plaintext message block with a randomly chosen string from H(a). To decrypt a stringc oft symbols, one must determine ana ∈A such that c ∈H(a). The key for the cipher consists of the setsH(a). 1.29 Example (homophonic substitution cipher) ConsiderA= {a,b},H(a)= {00,10}, and H(b)= {01,11}. The plaintext message blockab encrypts to one of the following:0001, 0011,1001,1011. Observe that the codomain of the encryption function (for messages of length two) consists of the following pairwise disjoint sets of four-element bitstrings: aa −→ {0000,0010,1000,1010} ab −→ {0001,0011,1001,1011} ba −→ {0100,0110,1100,1110} bb −→ {0101,0111,1101,1111} Any 4-bitstring uniquely identiﬁes a codomain element, and hence a plaintext message.□ 18 Ch. 1 Overview of Cryptography Often the symbols do not occur with equal frequency in plaintext messages. With a simple substitution cipher this non-uniform frequency property is reﬂected in the ciphertext as illustrated in Example 1.25. A homophonic cipher can be used to make the frequency of occurrence of ciphertext symbols more uniform, at the expense of data expansion. Decryp- tion is not as easily performed as it is for simple substitution ciphers. Polyalphabetic substitution ciphers 1.30 DeﬁnitionA polyalphabetic substitution cipheris a block cipher with block lengthtover an alphabetAhaving the following properties: (i) the key spaceKconsists of all ordered sets oft permutations(p1,p2,...,p t), where each permutationpi is deﬁned on the setA; (ii) encryption of the messagem =( m1m2 ···mt) under the keye =( p1,p2,...,p t) is given byEe(m)=( p1(m1)p2(m2) ···pt(mt)); and (iii) the decryption key associated withe =( p1,p2,...,p t) isd =( p−1 1 ,p−1 2 ,...,p −1 t ). 1.31 Example (Vigen`ere cipher) LetA= {A,B,C,..., X,Y,Z}and t =3 . Choosee = (p1,p2,p3), wherep1 maps each letter to the letter three positions to its right in the alphabet, p2 to the one seven positions to its right, andp3 ten positions to its right. If m = THI SCI PHE RIS CER TAI NLY NOT SEC URE then c = Ee(m) = WOS VJS SOO UPC FLB WHS QSI QVD VLM XYO. □ Polyalphabetic ciphers have the advantage over simple substitution ciphers that symbol frequencies are not preserved. In the example above, the letter E is encrypted to both O and L. However, polyalphabetic ciphers are not signiﬁcantly more difﬁcult to cryptanalyze, the approach being similar to the simple substitution cipher. In fact, once the block lengtht is determined, the ciphertext letters can be divided intot groups (where groupi,1 ≤i ≤t, consists of those ciphertext letters derived using permutationpi), and a frequency analysis can be done on each group. Transposition ciphers Another class of symmetric-key ciphers is the simple transposition cipher, which simply permutes the symbols in a block. 1.32 DeﬁnitionConsider a symmetric-key block encryption scheme with block lengtht. LetK be the set of all permutations on the set{1,2,...,t }. For eache ∈Kdeﬁne the encryption function Ee(m)=( me(1)me(2) ···me(t)) where m =( m1m2 ···mt) ∈M, the message space. The set of all such transformations is called asimple transposition cipher.The decryption key corresponding toeis the inverse permutationd = e−1. To decryptc =( c1c2 ···ct), computeDd(c)=( cd(1)cd(2) ···cd(t)). A simple transposition cipher preserves the number of symbols of a given type within a block, and thus is easily cryptanalyzed. §1.5 Symmetric-key encryption 19 1.5.3 Composition of ciphers In order to describe product ciphers, the concept of composition of functions is introduced. Compositions are a convenient way of constructing more complicated functions from sim- pler ones. Composition of functions 1.33 DeﬁnitionLetS,T, andUbe ﬁnite sets and letf : S− →Tand g: T− →Ube func- tions. Thecompositionofg withf, denotedg ◦f (or simplygf), is a function fromSto Uas illustrated in Figure 1.8 and deﬁned by(g ◦f)(x)= g(f(x)) for allx ∈S. s t u v 1 2 3 4 s t u v a b c a b c S TU SU g ◦f f g Figure 1.8:The compositiong ◦f of functionsg and f. Composition can be easily extended to more than two functions. For functionsf1,f2, ...,f t, one can deﬁneft ◦···◦ f2 ◦f1, provided that the domain offt equals the codomain offt−1 and so on. Compositions and involutions Involutions were introduced in§1.3.3 as a simple class of functions with an interesting prop- erty:Ek(Ek(x)) =x for allx in the domain ofEk; that is,Ek ◦Ek is the identity function. 1.34 Remark (composition of involutions) The composition of two involutions is not necessar- ily an involution, as illustrated in Figure 1.9. However, involutions may be composed to get somewhat more complicated functions whose inverses are easy to ﬁnd. This is an important feature for decryption. For example ifEk1 ,Ek2 ,...,E kt are involutions then the inverse ofEk = Ek1 Ek2 ···Ekt isE−1 k = Ekt Ekt−1 ···Ek1 , the composition of the involutions in the reverse order. 1 2 3 44 3 2 1 4 3 2 11 2 3 44 2 1 3 4 3 2 1 fg g ◦f Figure 1.9:The compositiong ◦f of involutionsg and f is not an involution. 20 Ch. 1 Overview of Cryptography Product ciphers Simple substitution and transposition ciphers individually do not provide a very high level of security. However, by combining these transformations it is possible to obtain strong ci- phers. As will be seen in Chapter 7 some of the most practical and effective symmetric-key systems are product ciphers. One example of aproduct cipheris a composition oft ≥2 transformationsEk1 Ek2 ···Ekt where eachEki ,1 ≤i ≤t, is either a substitution or a transposition cipher. For the purpose of this introduction, let the composition of a substitu- tion and a transposition be called around. 1.35 Example (product cipher) LetM= C= Kbe the set of all binary strings of length six. The number of elements inMis2 6 =6 4. Letm =( m1m2 ···m6) and deﬁne E(1) k (m)= m ⊕k, where k ∈K, E(2)(m)=( m4m5m6m1m2m3). Here,⊕is theexclusive-OR(XOR) operation deﬁned as follows:0 ⊕0=0 ,0 ⊕1=1 , 1 ⊕0=1 ,1 ⊕1=0 . E(1) k is a polyalphabetic substitution cipher andE(2) is a trans- position cipher (not involving the key). The productE(1) k E(2) is a round. While here the transposition cipher is very simple and is not determined by the key, this need not be the case. □ 1.36 Remark (confusion and diffusion) A substitution in a round is said to addconfusionto the encryption process whereas a transposition is said to adddiffusion. Confusion is intended to make the relationship between the key and ciphertext as complex as possible. Diffusion refers to rearranging or spreading out the bits in the message so that any redundancy in the plaintext is spread out over the ciphertext. A round then can be said to add both confu- sion and diffusion to the encryption. Most modern block cipher systems apply a number of rounds in succession to encrypt plaintext. 1.5.4 Stream ciphers Stream ciphers form an important class of symmetric-key encryption schemes. They are, in one sense, very simple block ciphers having block length equal to one. What makes them useful is the fact that the encryption transformation can change for each symbol of plain- text being encrypted. In situations where transmission errors are highly probable, stream ciphers are advantageous because they have no error propagation. They can also be used when the data must be processed one symbol at a time (e.g., if the equipment has no memory or buffering of data is limited). 1.37 DeﬁnitionLetKbe the key space for a set of encryption transformations. A sequence of symbolse 1e2e3 ···ei ∈K, is called akeystream. 1.38 DeﬁnitionLetAbe an alphabet ofq symbols and letEe be a simple substitution cipher with block length1 where e ∈K. Letm1m2m3 ··· be a plaintext string and lete1e2e3 ··· be a keystream fromK.A stream ciphertakes the plaintext string and produces a ciphertext stringc1c2c3 ··· where ci = Eei (mi).I fdi denotes the inverse ofei, thenDdi(ci)= mi decrypts the ciphertext string. §1.5 Symmetric-key encryption 21 A stream cipher applies simple encryption transformations according to the keystream being used. The keystream could be generated at random, or by an algorithm which gen- erates the keystream from an initial small keystream (called aseed), or from a seed and previous ciphertext symbols. Such an algorithm is called akeystream generator. The Vernam cipher A motivating factor for the Vernam cipher was its simplicity and ease of implementation. 1.39 DeﬁnitionThe Vernam Cipheris a stream cipher deﬁned on the alphabetA= {0,1}.A binary messagem1m2 ···mt is operated on by a binary key stringk1k2 ···kt of the same length to produce a ciphertext stringc1c2 ···ct where ci = mi ⊕ki, 1 ≤i ≤t. If the key string is randomly chosen and never used again, the Vernam cipher is called a one-time systemor aone-time pad. To see how the Vernam cipher corresponds to Deﬁnition 1.38, observe that there are precisely two substitution ciphers on the setA. One is simply the identity mapE0 which sends0 to0 and 1 to1; the otherE1 sends0 to1 and 1 to0. When the keystream contains a 0, applyE0 to the corresponding plaintext symbol; otherwise, applyE1. If the key string is reused there are ways to attack the system. For example, ifc1c2 ···ct and c′ 1c′ 2 ···c′ t are two ciphertext strings produced by the same keystreamk1k2 ···kt then ci = mi ⊕ki,c ′ i = m′ i ⊕ki and ci ⊕c′ i = mi ⊕m′ i . The redundancy in the latter may permit cryptanalysis. The one-time pad can be shown to be theoretically unbreakable. That is, if a cryptana- lyst has a ciphertext stringc1c2 ···ct encrypted using a random key string which has been used only once, the cryptanalyst can do no better than guess at the plaintext being any bi- nary string of lengtht (i.e.,t-bit binary strings are equally likely as plaintext). It has been proven that to realize an unbreakable system requires a random key of the same length as the message. This reduces the practicality of the system in all but a few specialized situations. Reportedly until very recently the communication line between Moscow and Washington was secured by a one-time pad. Transport of the key was done by trusted courier. 1.5.5 The key space The size of the key space is the number of encryption/decryption key pairs that are available in the cipher system. A key is typically a compact way to specify the encryption transfor- mation (from the set of all encryption transformations) to be used. For example, a transpo- sition cipher of block lengtht hast! encryption functions from which to select. Each can be simply described by a permutation which is called the key. It is a great temptation to relate the security of the encryption scheme to the size of the key space. The following statement is important to remember. 1.40 FactA necessary, but usually not sufﬁcient, condition for an encryption scheme to be se- cure is that the key space be large enough to preclude exhaustive search. For instance, the simple substitution cipher in Example 1.25 has a key space of size 26! ≈4 ×1026. The polyalphabetic substitution cipher of Example 1.31 has a key space of size(26!)3 ≈7 ×1079. Exhaustive search of either key space is completely infeasible, yet both ciphers are relatively weak and provide little security. 22 Ch. 1 Overview of Cryptography 1.6 Digital signatures A cryptographic primitive which is fundamental in authentication, authorization, and non- repudiation is thedigital signature. The purpose of a digital signature is to provide a means for an entity to bind its identity to a piece of information. The process ofsigningentails transforming the message and some secret information held by the entity into a tag called a signature. A generic description follows. Nomenclature and set-up •M is the set of messages which can be signed. •S is a set of elements calledsignatures, possibly binary strings of a ﬁxed length. •SA is a transformation from the message setMto the signature setS, and is called a signing transformationfor entityA.3 The transformationSA is kept secret byA, and will be used to create signatures for messages fromM. •VA is a transformation from the setM×S to the set{true,false}.4 VA is called a veriﬁcation transformationforA’s signatures, is publicly known, and is used by other entities to verify signatures created byA. 1.41 DeﬁnitionThe transformationsSA and VA provide adigital signature schemeforA. Oc- casionally the termdigital signature mechanismis used. 1.42 Example (digital signature scheme)M= {m1,m2,m3}and S= {s1,s2,s3}. The left side of Figure 1.10 displays a signing functionSA from the setMand, the right side, the corresponding veriﬁcation functionVA. □ SA VA False True m1 m2 m3 s2 s1 s3 (m1,s 1) (m1,s 2) (m1,s 3) (m2,s 1) (m2,s 2) (m2,s 3) (m3,s 1) (m3,s 2) (m3,s 3) Figure 1.10:A signing and veriﬁcation function for a digital signature scheme. 3The names of Alice and Bob are usually abbreviated toA and B, respectively. 4M×S consists of all pairs(m, s) where m ∈M,s ∈S, called theCartesian productofMand S. §1.6 Digital signatures 23 Signing procedure EntityA (thesigner) creates a signature for a messagem ∈M by doing the following: 1. Compute s = SA(m). 2. Transmit the pair(m,s).s is called thesignaturefor messagem. Veriﬁcation procedure To verify that a signatures on a messagem was created byA, an entityB (theveriﬁer) performs the following steps: 1. Obtain the veriﬁcation functionV A ofA. 2. Compute u = VA(m,s). 3. Accept the signature as having been created byAifu = true, and reject the signature ifu = false. 1.43 Remark (concise representation) The transformationsSA and VA are typically character- ized more compactly by a key; that is, there is a class of signing and veriﬁcation algorithms publicly known, and each algorithm is identiﬁed by a key. Thus the signing algorithmSA ofA is determined by a keykA and A is only required to keepkA secret. Similarly, the veriﬁcation algorithmVA ofA is determined by a keylA which is made public. 1.44 Remark (handwritten signatures) Handwritten signatures could be interpreted as a spe- cial class of digital signatures. To see this, take the set of signaturesSto contain only one element which is the handwritten signature ofA, denoted bysA. The veriﬁcation function simply checks if the signature on a message purportedly signed byA issA. An undesirable feature in Remark 1.44 is that the signature is not message-dependent. Hence, further constraints are imposed on digital signature mechanisms as next discussed. Properties required for signing and veriﬁcation functions There are several properties which the signing and veriﬁcation transformations must satisfy. (a)s is a valid signature ofA on messagem if and only ifVA(m,s)= true. (b) It is computationally infeasible for any entity other thanA to ﬁnd, for anym ∈M, an s ∈S such thatVA(m,s)= true. Figure 1.10 graphically displays property (a). There is an arrowed line in the diagram forVA from (mi,sj) totrueprovided there is an arrowed line frommi tosj in the diagram forSA. Property (b) provides the security for the method – the signature uniquely bindsA to the message which is signed. No one has yet formally proved that digital signature schemes satisfying (b) exist (al- though existence is widely believed to be true); however, there are some very good can- didates.§1.8.3 introduces a particular class of digital signatures which arise from public- key encryption techniques. Chapter 11 describes a number of digital signature mechanisms which are believed to satisfy the two properties cited above. Although the description of a digital signature given in this section is quite general, it can be broadened further, as pre- sented in§11.2. 24 Ch. 1 Overview of Cryptography 1.7 Authentication and identiﬁcation Authentication is a term which is used (and often abused) in a very broad sense. By itself it has little meaning other than to convey the idea that some means has been provided to guarantee that entities are who they claim to be, or that information has not been manip- ulated by unauthorized parties. Authentication is speciﬁc to the security objective which one is trying to achieve. Examples of speciﬁc objectives include access control, entity au- thentication, message authentication, data integrity, non-repudiation, and key authentica- tion. These instances of authentication are dealt with at length in Chapters 9 through 13. For the purposes of this chapter, it sufﬁces to give a brief introduction to authentication by describing several of the most obvious applications. Authentication is one of the most important of all information security objectives. Un- til the mid 1970s it was generally believed that secrecy and authentication were intrinsically connected. With the discovery of hash functions (§1.9) and digital signatures (§1.6), it was realized that secrecy and authentication were truly separate and independent information security objectives. It may at ﬁrst not seem important to separate the two but there are situ- ations where it is not only useful but essential. For example, if a two-party communication between Alice and Bob is to take place where Alice is in one country and Bob in another, the host countries might not permit secrecy on the channel; one or both countries might want the ability to monitor all communications. Alice and Bob, however, would like to be assured of the identity of each other, and of the integrity and origin of the information they send and receive. The preceding scenario illustrates several independent aspects of authentication. If Al- ice and Bob desire assurance of each other’s identity, there are two possibilities to consider. 1. Alice and Bob could be communicating with no appreciable time delay. That is, they are both active in the communication in “real time”. 2. Alice or Bob could be exchanging messages with some delay. That is, messages might be routed through various networks, stored, and forwarded at some later time. In the ﬁrst instance Alice and Bob would want to verify identities in real time. This might be accomplished by Alice sending Bob some challenge, to which Bob is the only entity which can respond correctly. Bob could perform a similar action to identify Alice. This type of authentication is commonly referred to asentity authenticationor more simply identiﬁcation. For the second possibility, it is not convenient to challenge and await response, and moreover the communication path may be only in one direction. Different techniques are now required to authenticate the originator of the message. This form of authentication is calleddata origin authentication. 1.7.1 Identiﬁcation 1.45 DeﬁnitionAn identiﬁcationorentity authenticationtechnique assures one party (through acquisition of corroborative evidence) of both the identity of a second party involved, and that the second was active at the time the evidence was created or acquired. Typically the only data transmitted is that necessary to identify the communicating par- ties. The entities are both active in the communication, giving a timeliness guarantee. §1.8 Public-key cryptography 25 1.46 Example (identiﬁcation)A callsB on the telephone. IfA and B know each other then entity authentication is provided through voice recognition. Although not foolproof, this works effectively in practice. □ 1.47 Example (identiﬁcation) PersonA provides to a banking machine a personal identiﬁca- tion number (PIN) along with a magnetic stripe card containing information aboutA. The banking machine uses the information on the card and the PIN to verify the identity of the card holder. If veriﬁcation succeeds,A is given access to various services offered by the machine. □ Example 1.46 is an instance ofmutual authenticationwhereas Example 1.47 only pro- videsunilateral authentication. Numerous mechanisms and protocols devised to provide mutual or unilateral authentication are discussed in Chapter 10. 1.7.2 Data origin authentication 1.48 DeﬁnitionData origin authenticationor message authenticationtechniques provide to one party which receives a message assurance (through corroborative evidence) of the iden- tity of the party which originated the message. Often a message is provided toB along with additional information so thatB can de- termine the identity of the entity who originated the message. This form of authentication typically provides no guarantee of timeliness, but is useful in situations where one of the parties is not active in the communication. 1.49 Example (need for data origin authentication)A sends toB an electronic mail message (e-mail). The message may travel through various network communications systems and be stored forB to retrieve at some later time.AandB are usually not in direct communication. B would like some means to verify that the message received and purportedly created by A did indeed originate fromA. □ Data origin authentication implicitly provides data integrity since, if the message was modiﬁed during transmission,A would no longer be the originator. 1.8 Public-key cryptography The concept of public-key encryption is simple and elegant, but has far-reaching conse- quences. 1.8.1 Public-key encryption Let{Ee : e ∈K}be a set of encryption transformations, and let{Dd : d ∈K} be the set of corresponding decryption transformations, whereKis the key space. Consider any pair of associated encryption/decryption transformations(Ee,Dd) and suppose that each pair has the property that knowingEe it is computationally infeasible, given a random ciphertext c ∈C, to ﬁnd the messagem ∈M such thatEe(m)= c. This property implies that given e it is infeasible to determine the corresponding decryption keyd. (Of coursee and d are 26 Ch. 1 Overview of Cryptography simply means to describe the encryption and decryption functions, respectively.)Ee is be- ing viewed here as a trapdoor one-way function (Deﬁnition 1.16) withd being the trapdoor information necessary to compute the inverse function and hence allow decryption. This is unlike symmetric-key ciphers wheree and d are essentially the same. Under these assumptions, consider the two-party communication between Alice and Bob illustrated in Figure 1.11. Bob selects the key pair(e,d). Bob sends the encryption key e(called thepublic key) to Alice over any channel but keeps the decryption keyd(called the private key) secure and secret. Alice may subsequently send a messagemto Bob by apply- ing the encryption transformation determined by Bob’s public key to getc = Ee(m). Bob decrypts the ciphertextc by applying the inverse transformationDd uniquely determined by d. e m c Dd(c)= m d m UNSECURED CHANNEL Ee(m)= c UNSECURED CHANNEL Alice Bob encryption destinationplaintext source key source decryption Passive Adversary Figure 1.11:Encryption using public-key techniques. Notice how Figure 1.11 differs from Figure 1.7 for a symmetric-key cipher. Here the encryption key is transmitted to Alice over an unsecured channel. This unsecured channel may be the same channel on which the ciphertext is being transmitted (but see§1.8.2). Since the encryption keye need not be kept secret, it may be made public. Any entity can subsequently send encrypted messages to Bob which only Bob can decrypt. Figure 1.12 illustrates this idea, whereA1,A2, andA3 are distinct entities. Note that ifA1 destroys message m1 after encrypting it toc1, then evenA1 cannot recoverm1 from c1. As a physical analogue, consider a metal box with the lid secured by a combination lock. The combination is known only to Bob. If the lock is left open and made publicly available then anyone can place a message inside and lock the lid. Only Bob can retrieve the message. Even the entity which placed the message into the box is unable to retrieve it. Public-key encryption, as described here, assumes that knowledge of the public keye does not allow computation of the private keyd. In other words, this assumes the existence of trapdoor one-way functions (§1.3.1(iii)). 1.50 DeﬁnitionConsider an encryption scheme consisting of the sets of encryption and decryp- §1.8 Public-key cryptography 27 c2 Dd(c1)= m1 e Dd(c2)= m2 Dd(c3)= m3 Ee(m2)= c2 e c1 c3 Ee(m1)= c1 Ee(m3)= c3 e A1 A2 A3 Bob Figure 1.12:Schematic use of public-key encryption. tion transformations{Ee : e ∈K}and{Dd : d ∈K}, respectively. The encryption method is said to be apublic-key encryption schemeif for each associated encryption/decryption pair(e,d), one keye (thepublic key) is made publicly available, while the otherd (thepri- vate key) is kept secret. For the scheme to besecure, it must be computationally infeasible to computed from e. 1.51 Remark (private key vs. secret key) To avoid ambiguity, a common convention is to use the termprivate keyin association with public-key cryptosystems, andsecret keyin associ- ation with symmetric-key cryptosystems. This may be motivated by the following line of thought: it takes two or more parties tosharea secret, but a key is trulyprivateonly when one party alone knows it. There are many schemes known which are widely believed to be secure public-key encryption methods, but none have been mathematically proven to be secure independent of qualifying assumptions. This is not unlike the symmetric-key case where the only system which has been proven secure is the one-time pad (§1.5.4). 1.8.2 The necessity of authentication in public-key systems It would appear that public-key cryptographyis an ideal system, not requiring a secure chan- nel to pass the encryption key. This would imply that two entities could communicate over an unsecured channel without ever having met to exchange keys. Unfortunately, this is not the case. Figure 1.13 illustrates how an active adversary can defeat the system (decrypt messages intended for a second entity) without breaking the encryption system. This is a type ofimpersonationand is an example ofprotocol failure(see§1.10). In this scenario the adversary impersonates entityB by sending entityA a public keye ′which A assumes (incorrectly) to be the public key ofB. The adversary intercepts encrypted messages from A toB, decrypts with its own private keyd′, re-encrypts the message underB’s public key e, and sends it on toB. This highlights the necessity toauthenticatepublic keys to achieve data origin authentication of the public keys themselves.A must be convinced that she is 28 Ch. 1 Overview of Cryptography encrypting under the legitimate public key ofB. Fortunately, public-key techniques also allow an elegant solution to this problem (see§1.11). e′ m c′ e c m A B Ee′(m)= c′ m Dd(c)= m d d′ Ee(m)= c Dd′(c′)= m destination key source plaintext source encryption decryption Adversary key source encryption decryption Figure 1.13:An impersonation attack on a two-party communication. 1.8.3 Digital signatures from reversible public-key encryption This section considers a class of digital signature schemes which is based on public-key encryption systems of a particular type. SupposeEe is a public-key encryption transformation with message spaceMand ci- phertext spaceC. Suppose further thatM= C.I fDd is the decryption transformation corresponding toEe then sinceEe and Dd are both permutations, one has Dd(Ee(m)) =Ee(Dd(m)) =m, for allm ∈M. A public-key encryption scheme of this type is calledreversible.5 Note that it is essential thatM = Cfor this to be a valid equality for allm ∈M ; otherwise,Dd(m) will be meaningless form ̸∈C. 5There is a broader class of digital signatures which can be informally described as arising fromirreversible cryptographic algorithms. These are described in§11.2. §1.8 Public-key cryptography 29 Construction for a digital signature scheme 1. LetMbe the message space for the signature scheme. 2. LetC= Mbe the signature spaceS. 3. Let(e,d) be a key pair for the public-key encryption scheme. 4. Deﬁne the signing functionSA to beDd. That is, the signature for a messagem ∈M iss = Dd(m). 5. Deﬁne the veriﬁcation functionVA by VA(m,s)= { true, ifEe(s)= m, false, otherwise. The signature scheme can be simpliﬁed further ifA only signs messages having a spe- cial structure, and this structure is publicly known. LetM′be a subset ofMwhere ele- ments ofM′have a well-deﬁned special structure, such thatM′contains only a negligi- ble fraction of messages from the set. For example, suppose thatMconsists of all binary strings of length2tfor some positive integert. LetM′be the subset ofMconsisting of all strings where the ﬁrstt bits are replicated in the lastt positions (e.g.,101101 would be in M′fort =3 ). IfA only signs messages within the subsetM′, these are easily recognized by a veriﬁer. Redeﬁne the veriﬁcation functionVA as VA(s)= { true, ifEe(s) ∈M′, false, otherwise. Under this new scenarioA only needs to transmit the signatures since the messagem = Ee(s) can be recovered by applying the veriﬁcation function. Such a scheme is called a digital signature scheme with message recovery. Figure 1.14 illustrates how this signature function is used. The feature of selecting messages of special structure is referred to as selecting messages withredundancy. Dd(m)= s e Ee(s) ifm′∈M′ key message M′ m′ d m Accept source source s VeriﬁerB SignerA Figure 1.14:A digital signature scheme with message recovery. The modiﬁcation presented above is more than a simpliﬁcation; it is absolutely crucial if one hopes to meet the requirement of property (b) of signing and veriﬁcation functions (see page 23). To see why this is the case, note that any entityB can select a random ele- ment s ∈S as a signature and applyEe to getu = Ee(s), sinceS= Mand Ee is public 30 Ch. 1 Overview of Cryptography knowledge.B may then take the messagem = u and the signature onm to bes and trans- mits(m,s). It is easy to check thats will verify as a signature created byA form but in which A has had no part. In this caseB hasforgeda signature ofA. This is an example of what is calledexistential forgery.(B has producedA’s signature on some message likely not ofB’s choosing.) IfM′contains only a negligible fraction of messages fromM, then the probability of some entity forging a signature ofA in this manner is negligibly small. 1.52 Remark (digital signatures vs. conﬁdentiality) Although digital signature schemes based on reversible public-key encryption are attractive, they require an encryption method as a primitive. There are situations where a digital signature mechanism is required but encryp- tion is forbidden. In such cases these digital signature schemes are inappropriate. Digital signatures in practice For digital signatures to be useful in practice, concrete realizations of the preceding con- cepts should have certain additional properties. A digital signature must 1. be easy to compute by the signer (the signing function should be easy to apply); 2. be easy to verify by anyone (the veriﬁcation function should be easy to apply); and 3. have an appropriate lifespan, i.e., be computationally secure from forgery until the signature is no longer necessary for its original purpose. Resolution of disputes The purpose of a digital signature (or any signature method) is to permit the resolution of disputes. For example, an entityA could at some point deny having signed a message or some other entityB could falsely claim that a signature on a message was produced byA. In order to overcome such problems atrusted third party(TTP) orjudgeis required. The TTP must be some entity which all parties involved agree upon in advance. IfA denies that a messagem held byB was signed byA, thenB should be able to present the signaturesA form to the TTP along withm. The TTP rules in favor ofB if VA(m,sA)= trueand in favor ofA otherwise.B will accept the decision ifB is conﬁdent that the TTP has the same verifying transformationVA asAdoes.Awill accept the decision ifA is conﬁdent that the TTP usedVA and thatSA has not been compromised. Therefore, fair resolution of disputes requires that the following criteria are met. Requirements for resolution of disputed signatures 1. SA and VA have properties (a) and (b) of page 23. 2. The TTP has an authentic copy ofVA. 3. The signing transformationSA has been kept secret and remains secure. These properties are necessary but in practice it might not be possible to guarantee them. For example, the assumption thatSA and VA have the desired characteristics given in property 1 might turn out to be false for a particular signature scheme. Another possi- bility is thatA claims falsely thatSA was compromised. To overcome these problems re- quires an agreed method to validate the time period for whichA will accept responsibility for the veriﬁcation transformation. An analogue of this situation can be made with credit card revocation. The holder of a card is responsible until the holder notiﬁes the card issuing company that the card has been lost or stolen.§13.8.2 gives a more indepth discussion of these problems and possible solutions. §1.8 Public-key cryptography 31 1.8.4 Symmetric-key vs. public-key cryptography Symmetric-key and public-key encryption schemes have various advantages and disadvan- tages, some of which are common to both. This section highlights a number of these and summarizes features pointed out in previous sections. (i) Advantages of symmetric-key cryptography 1. Symmetric-key ciphers can be designed to have high rates of data throughput. Some hardware implementations achieve encrypt rates of hundreds of megabytes per sec- ond, while software implementations may attain throughput rates in the megabytes per second range. 2. Keys for symmetric-key ciphers are relatively short. 3. Symmetric-key ciphers can be employed as primitives to construct various crypto- graphic mechanisms including pseudorandom number generators (see Chapter 5), hash functions (see Chapter 9), and computationally efﬁcient digital signature sch- emes (see Chapter 11), to name just a few. 4. Symmetric-key ciphers can be composed to produce stronger ciphers. Simple trans- formations which are easy to analyze, but on their own weak, can be used to construct strong product ciphers. 5. Symmetric-key encryption is perceived to have an extensive history, although it must be acknowledged that, notwithstanding the invention of rotor machines earlier, much of the knowledge in this area has been acquired subsequent to the invention of the digital computer, and, in particular, the design of the Data Encryption Standard (see Chapter 7) in the early 1970s. (ii) Disadvantages of symmetric-key cryptography 1. In a two-party communication, the key must remain secret at both ends. 2. In a large network, there are many key pairs to be managed. Consequently, effective key management requires the use of an unconditionally trusted TTP (Deﬁnition 1.65). 3. In a two-party communication between entitiesA and B, sound cryptographic prac- tice dictates that the key be changed frequently, and perhaps for each communication session. 4. Digital signature mechanisms arising from symmetric-key encryption typically re- quire either large keys for the public veriﬁcation function or the use of a TTP (see Chapter 11). (iii) Advantages of public-key cryptography 1. Only the private key must be kept secret (authenticity of public keys must, however, be guaranteed). 2. The administration of keys on a network requires the presence of only a functionally trusted TTP (Deﬁnition 1.66) as opposed to an unconditionally trusted TTP. Depend- ing on the mode of usage, the TTP might only be required in an “off-line” manner, as opposed to in real time. 3. Depending on the mode of usage, a private key/public key pair may remain unchang- ed for considerable periods of time, e.g., many sessions (even several years). 4. Many public-key schemes yield relatively efﬁcient digital signature mechanisms. The key used to describe the public veriﬁcation function is typically much smaller than for the symmetric-key counterpart. 32 Ch. 1 Overview of Cryptography 5. In a large network, the number of keys necessary may be considerably smaller than in the symmetric-key scenario. (iv) Disadvantages of public-key encryption 1. Throughput rates for the most popular public-key encryption methods are several or- ders of magnitude slower than the best known symmetric-key schemes. 2. Key sizes are typically much larger than those required for symmetric-key encryption (see Remark 1.53), and the size of public-key signatures is larger than that of tags providing data origin authentication from symmetric-key techniques. 3. No public-key scheme has been proven to be secure (the same can be said for block ciphers). The most effective public-key encryption schemes found to date have their security based on the presumed difﬁculty of a small set of number-theoretic problems. 4. Public-key cryptography does not have as extensive a history as symmetric-key en- cryption, being discovered only in the mid 1970s.6 Summary of comparison Symmetric-key and public-key encryption have a number of complementary advantages. Current cryptographic systems exploit the strengths of each. An example will serve to il- lustrate. Public-key encryption techniques may be used to establish a key for a symmetric-key system being used by communicating entitiesA and B. In this scenarioA and B can take advantage of the long term nature of the public/private keys of the public-key scheme and the performance efﬁciencies of the symmetric-key scheme. Since data encryption is fre- quently the most time consuming part of the encryption process, the public-key scheme for key establishment is a small fraction of the total encryption process betweenA and B. To date, the computational performance of public-key encryption is inferior to that of symmetric-key encryption. There is, however, no proof that this must be the case. The important points in practice are: 1. public-key cryptographyfacilitates efﬁcient signatures (particularly non-repudiation) and key mangement; and 2. symmetric-key cryptography is efﬁcient for encryption and some data integrity ap- plications. 1.53 Remark (key sizes: symmetric key vs. private key) Private keys in public-key systems must be larger (e.g., 1024 bits for RSA) than secret keys in symmetric-key systems (e.g., 64 or 128 bits) because whereas (for secure algorithms) the most efﬁcient attack on symmetric- key systems is an exhaustive key search, all known public-key systems are subject to “short- cut” attacks (e.g., factoring) more efﬁcient than exhaustive search. Consequently, for equiv- alent security, symmetric keys have bitlengths considerably smaller than that of private keys in public-key systems, e.g., by a factor of 10 or more. 6It is, of course, arguable that some public-key schemes which are based on hard mathematical problems have a long history since these problems have been studied for many years. Although this may be true, one must be wary that the mathematics was not studied with this application in mind. §1.9 Hash functions 33 1.9 Hash functions One of the fundamental primitives in modern cryptography is the cryptographic hash func- tion, often informally called a one-way hash function. A simpliﬁed deﬁnition for the present discussion follows. 1.54 DeﬁnitionA hash functionis a computationally efﬁcient function mapping binary strings of arbitrary length to binary strings of some ﬁxed length, calledhash-values. For a hash function which outputsn-bit hash-values (e.g.,n = 128or160) and has de- sirable properties, the probability that a randomly chosen string gets mapped to a particular n-bit hash-value (image) is2−n. The basic idea is that a hash-value serves as a compact representative of an input string. To be of cryptographic use, a hash functionh is typically chosen such that it is computationally infeasible to ﬁnd two distinct inputs which hash to a common value (i.e., twocollidinginputsx and y such thath(x)= h(y)), and that given a speciﬁc hash-valuey, it is computationally infeasible to ﬁnd an input (pre-image)x such thath(x)= y. The most common cryptographic uses of hash functions are with digital signatures and for data integrity. With digital signatures, a long message is usually hashed (using a pub- licly available hash function) and only the hash-value is signed. The party receiving the message then hashes the received message, and veriﬁes that the received signature is cor- rect for this hash-value. This saves both time and space compared to signing the message directly, which would typically involve splitting the message into appropriate-sized blocks and signing each block individually. Note here that the inability to ﬁnd two messages with the same hash-value is a security requirement, since otherwise, the signature on one mes- sage hash-value would be the same as that on another, allowing a signer to sign one message and at a later point in time claim to have signed another. Hash functions may be used for data integrity as follows. The hash-value correspond- ing to a particular input is computed at some point in time. The integrity of this hash-value is protected in some manner. At a subsequent point in time, to verify that the input data has not been altered, the hash-value is recomputed using the input at hand, and compared for equality with the original hash-value. Speciﬁc applications include virus protection and software distribution. A third application of hash functions is their use in protocols involving a priori com- mitments, including some digital signature schemes and identiﬁcation protocols (e.g., see Chapter 10). Hash functions as discussed above are typically publicly known and involve no secret keys. When used to detect whether the message input has been altered, they are calledmodi- ﬁcation detection codes(MDCs). Related to these are hash functions which involve a secret key, and provide data origin authentication (§9.76) as well as data integrity; these are called message authentication codes(MACs). 1.10 Protocols and mechanisms 1.55 DeﬁnitionA cryptographic protocol(protocol) is a distributed algorithm deﬁned by a se- quence of steps precisely specifying the actions required of two or more entities to achieve a speciﬁc security objective. 34 Ch. 1 Overview of Cryptography 1.56 Remark (protocol vs. mechanism) As opposed to a protocol, amechanism is a more gen- eral term encompassing protocols, algorithms (specifying the steps followed by a single en- tity), and non-cryptographic techniques (e.g., hardware protection and procedural controls) to achieve speciﬁc security objectives. Protocols play a major role in cryptography and are essential in meeting cryptographic goals as discussed in§1.2. Encryption schemes, digital signatures, hash functions, and ran- dom number generation are among the primitives which may be utilized to build a protocol. 1.57 Example (a simple key agreement protocol) Alice and Bob have chosen a symmetric-key encryption scheme to use in communicating over an unsecured channel. To encrypt infor- mation they require a key. The communication protocol is the following: 1. Bob constructs a public-key encryption scheme and sends his public key to Alice over the channel. 2. Alice generates a key for the symmetric-key encryption scheme. 3. Alice encrypts the key using Bob’s public key and sends the encrypted key to Bob. 4. Bob decrypts using his private key and recovers the symmetric (secret) key. 5. Alice and Bob begin communicating with privacy by using the symmetric-key sys- tem and the common secret key. This protocol uses basic functions to attempt to realize private communications on an unse- cured channel. The basic primitives are the symmetric-key and the public-key encryption schemes. The protocol has shortcomings including the impersonation attack of§1.8.2, but it does convey the idea of a protocol. □ Often the role of public-key encryption in privacy communications is exactly the one suggested by this protocol – public-key encryption is used as a means to exchange keys for subsequent use in symmetric-key encryption, motivated by performance differences be- tween symmetric-key and public-key encryption. Protocol and mechanism failure 1.58 DeﬁnitionA protocol failureormechanism failureoccurs when a mechanism fails to meet the goals for which it was intended, in a manner whereby an adversary gains advantage not by breaking an underlying primitive such as an encryption algorithm directly, but by manipulating the protocol or mechanism itself. 1.59 Example (mechanism failure) Alice and Bob are communicating using a stream cipher. Messages which they encrypt are known to have a special form: the ﬁrst twenty bits carry information which represents a monetary amount. An active adversary can simply XOR an appropriate bitstring into the ﬁrst twenty bits of ciphertext and change the amount. While the adversary has not been able to read the underlying message, she has been able to alter the transmission. The encryption has not been compromised but the protocol has failed to perform adequately; the inherent assumption that encryption provides data integrity is in- correct. □ 1.60 Example (forward search attack) Suppose that in an electronic bank transaction the32- bit ﬁeld which records the value of the transaction is to be encrypted using a public-key scheme. This simple protocol is intended to provide privacy of the value ﬁeld – but does it? An adversary could easily take all2 32 possible entries that could be plaintext in this ﬁeld and encrypt them using the public encryption function. (Remember that by the very nature of public-key encryption this function must be available to the adversary.) By comparing §1.11 Key establishment, management, and certiﬁcation 35 each of the232 ciphertexts with the one which is actually encrypted in the transaction, the adversary can determine the plaintext. Here the public-key encryption function is not com- promised, but rather the way it is used. A closely related attack which applies directly to authentication for access control purposes is the dictionary attack (see§10.2.2). □ 1.61 Remark (causes of protocol failure) Protocols and mechanisms may fail for a number of reasons, including: 1. weaknesses in a particular cryptographic primitive which may be ampliﬁed by the protocol or mechanism; 2. claimed or assumed security guarantees which are overstated or not clearly under- stood; and 3. the oversight of some principle applicable to a broad class of primitives such as en- cryption. Example 1.59 illustrates item 2 if the stream cipher is the one-time pad, and also item 1. Example 1.60 illustrates item 3. See also§1.8.2. 1.62 Remark (protocol design) When designing cryptographic protocols and mechanisms, the following two steps are essential: 1. identifyallassumptions in the protocol or mechanism design; and 2. for each assumption, determine the effect on the security objective if that assumption is violated. 1.11 Key establishment, management, and certiﬁcation This section gives a brief introduction to methodology for ensuring the secure distribution of keys for cryptographic purposes. 1.63 DeﬁnitionKey establishmentis any process whereby a shared secret key becomes avail- able to two or more parties, for subsequent cryptographic use. 1.64 DeﬁnitionKey management is the set of processes and mechanisms which support key establishment and the maintenance of ongoing keying relationships between parties, includ- ing replacing older keys with new keys as necessary. Key establishment can be broadly subdivided intokey agreementand key transport. Many and various protocols have been proposed to provide key establishment. Chapter 12 describes a number of these in detail. For the purpose of this chapter only a brief overview of issues related to key management will be given. Simple architectures based on symmetric- key and public-key cryptography along with the concept of certiﬁcation will be addressed. As noted in§1.5, a major issue when using symmetric-key techniques is the establish- ment of pairwise secret keys. This becomes more evident when considering a network of entities, any two of which may wish to communicate. Figure 1.15 illustrates a network con- sisting of 6 entities. The arrowed edges indicate the15 possible two-party communications which could take place. Since each pair of entities wish to communicate, this small net- work requires the secure exchange of ( 6 2 ) =1 5key pairs. In a network withn entities, the number of secure key exchanges required is (n 2 ) = n(n−1) 2 . 36 Ch. 1 Overview of Cryptography A1 A2 A5 A3A6 A4 Figure 1.15:Keying relationships in a simple 6-party network. The network diagram depicted in Figure 1.15 is simply the amalgamation of15 two- party communications as depicted in Figure 1.7. In practice, networks are very large and the key management problem is a crucial issue. There are a number of ways to handle this problem. Two simplistic methods are discussed; one based on symmetric-key and the other on public-key techniques. 1.11.1 Key management through symmetric-key techniques One solution which employs symmetric-key techniques involves an entity in the network which is trusted by all other entities. As in§1.8.3, this entity is referred to as atrusted third party(TTP). Each entityAi shares a distinct symmetric keyki with the TTP. These keys are assumed to have been distributed over a secured channel. If two entities subsequently wish to communicate, the TTP generates a keyk (sometimes called asession key) and sends it encrypted under each of the ﬁxed keys as depicted in Figure 1.16 for entitiesA 1 and A5. A1 Ek(m) Ek5 (k) A5 TTP A6 k5 k Ek1 (k) k1 k6 A2 A3 A4 k2 k3 k4 source key Figure 1.16:Key management using a trusted third party (TTP). Advantages of this approach include: 1. It is easy to add and remove entities from the network. 2. Each entity needs to store only one long-term secret key. Disadvantages include: 1. All communications require initial interaction with the TTP. 2. The TTP must storen long-term secret keys. §1.11 Key establishment, management, and certiﬁcation 37 3. The TTP has the ability to read all messages. 4. If the TTP is compromised, all communications are insecure. 1.11.2 Key management through public-key techniques There are a number of ways to address the key management problem through public-key techniques. Chapter 13 describes many of these in detail. For the purpose of this chapter a very simple model is considered. Each entity in the network has a public/private encryption key pair. The public key along with the identity of the entity is stored in a central repository called apublic ﬁle.I f an entityA1 wishes to send encrypted messages to entityA6,A1 retrieves the public key e6 ofA6 from the public ﬁle, encrypts the message using this key, and sends the ciphertext toA6. Figure 1.17 depicts such a network. private keyd5 c private keyd6 private keyd1 c = Ee6 (m) Public ﬁle e6 A1 : e1 A2 : e2 A3 : e3 A4 : e4 A5 : e5 A6 : e6 private keyd2 private keyd3 A5 A4 private keyd4 A3A6 A1 A2 m = Dd6 (c) Figure 1.17:Key management using public-key techniques. Advantages of this approach include: 1. No trusted third party is required. 2. The public ﬁle could reside with each entity. 3. Onlyn public keys need to be stored to allow secure communications between any pair of entities, assuming the only attack is that by a passive adversary. The key management problem becomes more difﬁcult when one must take into account an adversary who isactive(i.e. an adversary who can alter the public ﬁle containing public keys). Figure 1.18 illustrates how an active adversary could compromise the key manage- ment scheme given above. (This is directly analogous to the attack in§1.8.2.) In the ﬁgure, the adversary alters the public ﬁle by replacing the public keye 6 of entityA6 by the adver- sary’s public keye∗. Any message encrypted forA6 using the public key from the public ﬁle can be decrypted by only the adversary. Having decrypted and read the message, the 38 Ch. 1 Overview of Cryptography adversary can now encrypt it using the public key ofA6 and forward the ciphertext toA6. A1 however believes that onlyA6 can decrypt the ciphertextc. c A1 e∗ A1 : e1 A2 : e2 A3 : e3 A4 : e4 A5 : e5 A6 : e6 e∗ Dd6 (c′)= mEe6 (m)= c′ c′ Adversary private key d6 Public ﬁle Ee∗(m)= c private key A6 d∗ Dd∗(c)= m Figure 1.18:An impersonation ofA6 by an active adversary with public keye∗. To prevent this type of attack, the entities may use a TTP tocertifythe public key of each entity. The TTP has a private signing algorithmST and a veriﬁcation algorithmVT (see§1.6) assumed to be known by all entities. The TTP carefully veriﬁes the identity of each entity, and signs a message consisting of an identiﬁer and the entity’s authentic public key. This is a simple example of acertiﬁcate, binding the identity of an entity to its public key (see§1.11.3). Figure 1.19 illustrates the network under these conditions.A1 uses the public key ofA6 only if the certiﬁcate signature veriﬁes successfully. e6,s6 A1,e1,ST (A1∥e1)= s1 A2,e2,ST (A2∥e2)= s2 A3,e3,ST (A3∥e3)= s3 A4,e4,ST (A4∥e4)= s4 A5,e5,ST (A5∥e5)= s5 A6,e6,ST (A6∥e6)= s6 A1 Ee6 (m) VT (A6∥e6,s 6) Public ﬁle Dd6 (c)= m d6 c = A6 veriﬁcation private key Figure 1.19:Authentication of public keys by a TTP .∥denotes concatenation. Advantages of using a TTP to maintain the integrity of the public ﬁle include: 1. It prevents an active adversary from impersonation on the network. 2. The TTP cannot monitor communications. Entities need trust the TTP only to bind identities to public keys properly. 3. Per-communication interaction with the public ﬁle can be eliminated if entities store certiﬁcates locally. Even with a TTP, some concerns still remain: 1. If the signing key of the TTP is compromised, all communications become insecure. 2. All trust is placed with one entity. §1.12 Pseudorandom numbers and sequences 39 1.11.3 Trusted third parties and public-key certiﬁcates A trusted third party has been used in§1.8.3 and again here in§1.11. The trust placed on this entity varies with the way it is used, and hence motivates the following classiﬁcation. 1.65 DeﬁnitionA TTP is said to beunconditionally trustedif it is trusted on all matters. For example, it may have access to the secret and private keys of users, as well as be charged with the association of public keys to identiﬁers. 1.66 DeﬁnitionA TTP is said to befunctionally trustedif the entity is assumed to be honest and fair but it does not have access to the secret or private keys of users. §1.11.1 provides a scenario which employs an unconditionally trusted TTP.§1.11.2 uses a functionally trusted TTP to maintain the integrity of the public ﬁle. A functionally trusted TTP could be used to register or certify users and contents of documents or, as in §1.8.3, as a judge. Public-key certiﬁcates The distribution of public keys is generally easier than that of symmetric keys, since secrecy is not required. However, the integrity (authenticity) of public keys is critical (recall§1.8.2). A public-key certiﬁcateconsists of adata partand asignature part. The data part con- sists of the name of an entity, the public key corresponding to that entity, possibly additional relevant information (e.g., the entity’s street or network address, a validity period for the public key, and various other attributes). The signature part consists of the signature of a TTP over the data part. In order for an entityB to verify the authenticity of the public key of an entityA,B must have an authentic copy of the public signature veriﬁcation function of the TTP. For simplicity, assume that the authenticity of this veriﬁcation function is provided toB by non- cryptographic means, for example byB obtaining it from the TTP in person.B can then carry out the following steps: 1. Acquire the public-key certiﬁcate ofA over some unsecured channel, either from a central database of certiﬁcates, fromA directly, or otherwise. 2. Use the TTP’s veriﬁcation function to verify the TTP’s signature onA’s certiﬁcate. 3. If this signature veriﬁes correctly, accept the public key in the certiﬁcate asA’s au- thentic public key; otherwise, assume the public key is invalid. Before creating a public-key certiﬁcate forA, the TTP must take appropriate measures to verify the identity ofA and the fact that the public key to be certiﬁcated actually belongs toA. One method is to require thatA appear before the TTP with a conventional passport as proof of identity, and obtainA’s public key fromA in person along with evidence that A knows the corresponding private key. Once the TTP creates a certiﬁcate for a party, the trust that all other entities have in the authenticity of the TTP’s public key can be used tran- sitively to gain trust in the authenticity of that party’s public key, through acquisition and veriﬁcation of the certiﬁcate. 1.12 Pseudorandom numbers and sequences Random number generation is an important primitive in many cryptographic mechanisms. For example, keys for encryption transformations need to be generated in a manner which is 40 Ch. 1 Overview of Cryptography unpredictable to an adversary. Generating a random key typically involves the selection of random numbers or bit sequences. Random number generation presents challenging issues. A brief introduction is given here with details left to Chapter 5. Often in cryptographic applications, one of the following steps must be performed: (i) From a ﬁnite set ofn elements (e.g.,{1,2,...,n }), select an element at random. (ii) From the set of all sequences (strings) of lengthm over some ﬁnite alphabetAofn symbols, select a sequence at random. (iii) Generate a random sequence (string) of symbols of lengthmover a set ofn symbols. It is not clear what exactly it means toselect at randomorgenerate at random. Calling a number random without a context makes little sense. Is the number23 a random number? No, but if49 identical balls labeled with a number from1 to49 are in a container, and this container mixes the balls uniformly, drops one ball out, and this ball happens to be labeled with the number23, then one would say that23 was generated randomly from a uniform distribution. Theprobabilitythat23 drops out is1 in49 or 1 49 . If the number on the ball which was dropped from the container is recorded and the ball is placed back in the container and the process repeated6 times, then a random sequence of length6 deﬁned on the alphabetA= {1,2,..., 49}will have been generated. What is the chance that the sequence17,45,1,7,23,35occurs? Since each element in the sequence has probability1 49 of occuring, the probability of the sequence17,45,1,7,23,35occurring is 1 49 × 1 49 × 1 49 × 1 49 × 1 49 × 1 49 = 1 13841287201. There are precisely13841287201 sequences of length6 over the alphabetA. If each of these sequences is written on one of13841287201 balls and they are placed in the container (ﬁrst removing the original49 balls) then the chance that the sequence given above drops out is the same as if it were generated one ball at a time. Hence, (ii) and (iii) above are essentially the same statements. Finding good methods to generate random sequences is difﬁcult. 1.67 Example (random sequence generator) To generate a random sequence of0’s and1’s, a coin could be tossed with a head landing up recorded as a1 and a tail as a0. It is assumed that the coin isunbiased, which means that the probability of a1 on a given toss is exactly1 2 . This will depend on how well the coin is made and how the toss is performed. This method would be of little value in a system where random sequences must be generated quickly and often. It has no practical value other than to serve as an example of the idea of random number generation. □ 1.68 Example (random sequence generator)A noise diodemay be used to produce random binary sequences. This is reasonable if one has some way to be convinced that the proba- bility that a1 will be produced on any given trial is1 2 . Should this assumption be false, the sequence generated would not have been selected from a uniform distribution and so not all sequences of a given length would be equally likely. The only way to get some feeling for the reliability of this type of random source is to carry out statistical tests on its output. These are considered in Chapter 5. If the diode is a source of a uniform distribution on the set of all binary sequences of a given length, it provides an effective way to generate ran- dom sequences. □ Since mosttrue sourcesof random sequences (if there is such a thing) come fromphys- ical means, they tend to be either costly or slow in their generation. To overcome these §1.13 Classes of attacks and security models 41 problems, methods have been devised to constructpseudorandom sequencesin a determin- istic manner from a shorter random sequence called aseed. The pseudorandom sequences appear to be generated by a truly random source to anyone not knowing the method of gen- eration. Often the generation algorithm is known to all, but the seed is unknown except by the entity generating the sequence. A plethora of algorithms has been developed to generate pseudorandom bit sequences of various types. Many of these are completely unsuitable for cryptographic purposes and one must be cautious of claims by creators of such algorithms as to the random nature of the output. 1.13 Classes of attacks and security models Over the years, many different types of attacks on cryptographic primitives and protocols have been identiﬁed. The discussion here limits consideration to attacks on encryption and protocols. Attacks on other cryptographic primitives will be given in appropriate chapters. In§1.11 the roles of an active and a passive adversary were discussed. The attacks these adversaries can mount may be classiﬁed as follows:. 1. A passive attackis one where the adversary only monitors the communication chan- nel. A passive attacker only threatens conﬁdentiality of data. 2. An active attackis one where the adversary attempts to delete, add, or in some other way alter the transmission on the channel. An active attacker threatens data integrity and authentication as well as conﬁdentiality. A passive attack can be further subdivided into more specialized attacks for deducing plaintext from ciphertext, as outlined in§1.13.1. 1.13.1 Attacks on encryption schemes The objective of the following attacks is to systematically recover plaintext from ciphertext, or even more drastically, to deduce the decryption key. 1. A ciphertext-only attackis one where the adversary (or cryptanalyst) tries to deduce the decryption key or plaintext by only observing ciphertext. Any encryption scheme vulnerable to this type of attack is considered to be completely insecure. 2. A known-plaintext attackis one where the adversary has a quantity of plaintext and corresponding ciphertext. This type of attack is typically only marginally more dif- ﬁcult to mount. 3. A chosen-plaintext attackis one where the adversary chooses plaintext and is then given corresponding ciphertext. Subsequently, the adversary uses any information deduced in order to recover plaintext corresponding to previously unseen ciphertext. 4. An adaptive chosen-plaintext attackis a chosen-plaintext attack wherein the choice of plaintext may depend on the ciphertext received from previous requests. 5. A chosen-ciphertext attackis one where the adversary selects the ciphertext and is then given the corresponding plaintext. One way to mount such an attack is for the adversary to gain access to the equipment used for decryption (but not the decryption key, which may be securely embedded in the equipment). The objective is then to be able, without access to such equipment, to deduce the plaintext from (different) ciphertext. 42 Ch. 1 Overview of Cryptography 6. An adaptive chosen-ciphertext attackis a chosen-ciphertext attack where the choice of ciphertext may depend on the plaintext received from previous requests. Most of these attacks also apply to digital signature schemes and message authentication codes. In this case, the objective of the attacker is to forge messages or MACs, as discussed in Chapters 11 and 9, respectively. 1.13.2 Attacks on protocols The following is a partial list of attacks which might be mounted on various protocols. Until a protocol is proven to provide the service intended, the list of possible attacks can never be said to be complete. 1. known-key attack. In this attack an adversary obtains some keys used previously and then uses this information to determine new keys. 2. replay. In this attack an adversary records a communication session and replays the entire session, or a portion thereof, at some later point in time. 3. impersonation. Here an adversary assumes the identity of one of the legitimate par- ties in a network. 4. dictionary. This is usually an attack against passwords. Typically, a password is stored in a computer ﬁle as the image of an unkeyed hash function. When a user logs on and enters a password, it is hashed and the image is compared to the stored value. An adversary can take a list of probable passwords, hash all entries in this list, and then compare this to the list of true encrypted passwords with the hope of ﬁnding matches. 5. forward search. This attack is similar in spirit to the dictionary attack and is used to decrypt messages. An example of this method was cited in Example 1.60. 6. interleaving attack. This type of attack usually involves some form of impersonation in an authentication protocol (see§12.9.1). 1.13.3 Models for evaluating security The security of cryptographic primitives and protocols can be evaluated under several dif- ferent models. The most practical security metrics are computational, provable, and ad hoc methodology, although the latter is often dangerous. The conﬁdence level in the amount of security provided by a primitive or protocol based on computational or ad hoc security increases with time and investigation of the scheme. However, time is not enough if few people have given the method careful analysis. (i) Unconditional security The most stringent measure is an information-theoretic measure – whether or not a sys- tem hasunconditional security. An adversary is assumed to have unlimited computational resources, and the question is whether or not there is enough information available to de- feat the system. Unconditional security for encryption systems is calledperfect secrecy. For perfect secrecy, the uncertainty in the plaintext, after observing the ciphertext, must be equal to the a priori uncertainty about the plaintext – observation of the ciphertext provides no information whatsoever to an adversary. A necessary condition for a symmetric-key encryption scheme to be unconditionally secure is that the key be at least as long as the message. The one-time pad (§1.5.4) is an ex- ample of an unconditionally secure encryption algorithm. In general, encryption schemes §1.13 Classes of attacks and security models 43 do not offer perfect secrecy, and each ciphertext character observed decreases the theoreti- cal uncertainty in the plaintext and the encryption key. Public-key encryption schemes can- not be unconditionally secure since, given a ciphertextc, the plaintext can in principle be recovered by encrypting all possible plaintexts untilc is obtained. (ii) Complexity-theoretic security An appropriate model of computation is deﬁned and adversaries are modeled as having polynomial computational power. (They mount attacks involving time and space polyno- mial in the size of appropriate security parameters.) A proof of security relative to the model is then constructed. An objective is to design a cryptographic method based on the weakest assumptions possible anticipating a powerful adversary. Asymptotic analysis and usually also worst-case analysis is used and so care must be exercised to determine when proofs have practical signiﬁcance. In contrast, polynomial attacks which are feasible under the model might, in practice, still be computationally infeasible. Security analysis of this type, although not of practical value in all cases, may nonethe- less pave the way to a better overall understanding of security. Complexity-theoretic anal- ysis is invaluable for formulating fundamental principles and conﬁrming intuition. This is like many other sciences, whose practical techniques are discovered early in the develop- ment, well before a theoretical basis and understanding is attained. (iii) Provablesecurity A cryptographic method is said to beprovably secureif the difﬁculty of defeating it can be shown to be essentially as difﬁcult as solving a well-known andsupposedlydifﬁcult (typ- ically number-theoretic) problem, such as integer factorization or the computation of dis- crete logarithms. Thus, “provable” here means provable subject to assumptions. This approach is considered by some to be as good a practical analysis technique as exists. Provable security may be considered part of a special sub-class of the larger class of computational security considered next. (iv) Computational security This measures the amount of computational effort required, by the best currently-known methods, to defeat a system; it must be assumed here that the system has been well-studied to determine which attacks are relevant. A proposed technique is said to becomputation- ally secureif the perceived level of computation required to defeat it (using the best attack known) exceeds, by a comfortable margin, the computational resources of the hypothesized adversary. Often methods in this class are related to hard problems but, unlike for provable secu- rity, no proof of equivalence is known. Most of the best known public-key and symmetric- key schemes in current use are in this class. This class is sometimes also calledpractical security. (v) Ad hoc security This approach consists of any variety of convincing arguments that every successful attack requires a resource level (e.g., time and space) greater than the ﬁxed resources of a perceived adversary. Cryptographic primitives and protocols which survive such analysis are said to haveheuristic security, with security here typically in the computational sense. Primitives and protocols are usually designed to counter standard attacks such as those given in§1.13. While perhaps the most commonly used approach (especially for protocols), it is, in some ways, the least satisfying. Claims of security generally remain questionable and unforeseen attacks remain a threat. 44 Ch. 1 Overview of Cryptography 1.13.4 Perspective for computational security To evaluate the security of cryptographic schemes, certain quantities are often considered. 1.69 DeﬁnitionThe work factorWd is the minimum amount of work (measured in appropriate units such as elementary operations or clock cycles) required to compute the private keyd given the public keye, or, in the case of symmetric-key schemes, to determine the secret keyk. More speciﬁcally, one may consider the work required under a ciphertext-only attack givenn ciphertexts, denotedWd(n). IfWd istyears, then for sufﬁciently largetthe cryptographic scheme is, for all practical purposes, a secure system. To date no public-key system has been found where one can prove a sufﬁciently large lower bound on the work factorWd. The best that is possible to date is to rely on the following as a basis for security. 1.70 DeﬁnitionThe historical work factor Wd is the minimum amount of work required to compute the private keyd from the public keye using the best known algorithms at a given point in time. The historical work factor Wd varies with time as algorithms and technology improve. It corresponds to computational security, whereasWd corresponds to the true security level, although this typically cannot be determined. How large is large? §1.4 described how the designer of an encryption system tries to create a scheme for which the best approach to breaking it is through exhaustive search of the key space. The key space must then be large enough to make an exhaustive search completely infeasible. An important question then is “How large is large?”. In order to gain some perspective on the magnitude of numbers, Table 1.2 lists various items along with an associated magnitude. Reference Magnitude Seconds in a year ≈3 ×107 Age of our solar system (years) ≈6 ×109 Seconds since creation of solar system ≈2 ×1017 Clock cycles per year, 50 MHz computer ≈1.6 ×1015 Binary strings of length 64 264 ≈1.8 ×1019 Binary strings of length 128 2128 ≈3.4 ×1038 Binary strings of length 256 2256 ≈1.2 ×1077 Number of 75-digit prime numbers ≈5.2 ×1072 Electrons in the universe ≈8.37 ×1077 Table 1.2:Reference numbers comparing relative magnitudes. Some powers of10 are referred to by preﬁxes. For example, high-speed modern com- puters are now being rated in terms ofteraﬂopswhere a teraﬂop is1012 ﬂoating point op- erations per second. Table 1.3 provides a list of commonly used preﬁxes. §1.14 Notes and further references 45 Preﬁx Symbol Magnitude exa E 1018 peta P 1015 tera T 1012 giga G 109 mega M 106 kilo k 103 hecto h 102 deca da 10 Preﬁx Symbol Magnitude deci d 10−1 centi c 10−2 milli m 10−3 micro µ 10−6 nano n 10−9 pico p 10−12 femto f 10−15 atto a 10−18 Table 1.3:Preﬁxes used for various powers of 10. 1.14 Notes and further references §1.1 Kahn [648] gives a thorough, comprehensive, and non-technical history of cryptography, published in 1967. Feistel [387] provides an early exposition of block cipher ideas. The original speciﬁcation of DES is the 1977 U.S. Federal Information Processing Standards Publication 46 [396]. Public-key cryptography was introduced by Difﬁe and Hellman [345]. The ﬁrst concrete realization of a public-key encryption scheme was the knapsack scheme by Merkle and Hellman [857]. The RSA public-key encryption and signature sch- eme is due to Rivest, Shamir, and Adleman [1060], while the ElGamal public-key encryp- tion and signature schemes are due to ElGamal [368]. The two digital signature standards, ISO/IEC 9796 [596] and the Digital Signature Standard [406], are discussed extensively in Chapter 11. Cryptography has used specialized areas of mathematics such as number theory to realize very practical mechanisms such as public-key encryption and digital signatures. Such usage was not conceived as possible a mere twenty years ago. The famous mathematician, Hardy [539], went as far as to boast about its lack of utility: “ ... both Gauss and lesser mathematicians may be justiﬁed in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.” §1.2 This section was inspired by the foreword to the bookContemporary Cryptology, The Sci- ence of Information Integrity, edited by Simmons [1143]. The handwritten signature came into the British legal system in the seventeenth century as a means to provide various func- tions associated with information security. See Chapter 9 of Meyer and Matyas [859] for details. This book only considers cryptography as it applies to information in digital form. Chapter 9 of Beker and Piper [84] provides an introduction to the encryption of analogue signals, in particular, speech. Although in many cases physical means are employed to facilitate privacy, cryptography plays the major role. Physical means of providing privacy include ﬁber optic communication links, spread spectrum technology, TEMPEST techniques, and 46 Ch. 1 Overview of Cryptography tamper-resistant hardware.Steganographyis that branch of information privacy which at- tempts to obscure the existence of data through such devices as invisible inks, secret com- partments, the use of subliminal channels, and the like. Kahn [648] provides an historical account of various steganographic techniques. Excellent introductions to cryptography can be found in the articles by Difﬁe and Hellman [347], Massey [786], and Rivest [1054]. A concise and elegant way to describe cryptogra- phy was given by Rivest [1054]:Cryptography is about communication in the presence of adversaries. The taxonomy of cryptographic primitives (Figure 1.1) was derived from the classiﬁcation given by Bosselaers, Govaerts, and Vandewalle [175]. §1.3 The theory of functions is fundamental in modern mathematics. The termrangeis often used in place of image of a function. The latter, being more descriptive, is preferred. An alternate term for one-to-one isinjective; an alternate term for onto issurjective. One-way functions were introduced by Difﬁe and Hellman [345]. A more extensive history is given on page 377. Trapdoor one-way functions were ﬁrst postulated by Difﬁe and Hell- man [345] and independently by Merkle [850] as a means to obtain public-key encryption schemes; several candidates are given in Chapter 8. §1.4 The basic concepts of cryptography are treated quite differently by various authors, some being more technical than others. Brassard [192] provides a concise, lucid, and technically accurate account. Schneier [1094] gives a less technical but very accessible introduction. Salomaa [1089], Stinson [1178], and Rivest [1054] present more mathematical approaches. Davies and Price [308] provide a very readable presentation suitable for the practitioner. The comparison of an encryption scheme to a resettable combination lock is from Difﬁe and Hellman [347]. Kerckhoffs’ desiderata [668] were originally stated in French. The translation stated here is given in Kahn [648]. Shannon [1121] also gives desiderata for encryption schemes. §1.5 Symmetric-key encryption has a very long history, as recorded by Kahn [648]. Most sys- tems invented prior to the 1970s are now of historical interest only. Chapter 2 of Denning [326] is also a good source for many of the more well known schemes such as the Caesar cipher, Vigen`ere and Beaufort ciphers, rotor machines (Enigma and Hagelin), running key ciphers, and so on; see also Davies and Price [308] and Konheim [705]. Beker and Piper [84] give an indepth treatment, including cryptanalysis of several of the classical systems used in World War II. Shannon’s paper [1121] is considered the seminal work on secure communications. It is also an excellent source for descriptions of various well-known his- torical symmetric-key ciphers. Simple substitution and transposition ciphers are the focus of§1.5. Hill ciphers [557], a class of substitution ciphers which substitute blocks using matrix methods, are covered in Example 7.52. The idea of confusion and diffusion (Remark 1.36) was introduced by Shan- non [1121]. Kahn [648] gives 1917 as the date when Vernam discovered the cipher which bears Ver- nam’s name, however, Vernam did not publish the result until 1926 [1222]; see page 274 for further discussion. Massey [786] states that reliable sources have suggested that the Moscow-Washington hot-line (channel for very high level communications) is no longer secured with a one-time pad, which has been replaced by a symmetric-key cipher requiring a much shorter key. This change would indicate that conﬁdence and understanding in the §1.14 Notes and further references 47 ability to construct very strong symmetric-key encryption schemes exists. The one-time pad seems to have been used extensively by Russian agents operating in foreign countries. The highest ranking Russian agent ever captured in the United States was Rudolph Abel. When apprehended in 1957 he had in his possession a booklet the size of a postage stamp (17 8 ×7 8 ×7 8 inches) containing a one-time key; see Kahn [648, p.664]. §1.6 The concept of a digital signature was introduced by Difﬁe and Hellman [345] and indepen- dently by Merkle [850]. The ﬁrst practical realization of a digital signature scheme appeared in the paper by Rivest, Shamir, and Adleman [1060]. Rabin [1022] (see also [1023]) also claims to have independently discovered RSA but did not publish the result. Most introductory sources for digital signatures stress digital signatures with message re- covery coming from a public-key encryption system. Mitchell, Piper, and Wild [882] give a good general treatment of the subject. Stinson [1178] provides a similar elementary but general introduction. Chapter 11 generalizes the deﬁnition of a digital signature by allowing randomization. The scheme described in§1.8 is referred to asdeterministic. Many other types of digital signatures with speciﬁc properties have been created, such as blind signa- tures, undeniable signatures, and failstop signatures (see Chapter 11). §1.7 Much effort has been devoted to developing a theory of authentication. At the forefront of this is Simmons [1144], whose contributions are nicely summarized by Massey [786]. For a more concrete example of the necessity for authentication without secrecy, see the article by Simmons [1146]. §1.8 1976 marked a major turning point in the history of cryptography. In several papers that year, Difﬁe and Hellman introduced the idea of public-key cryptography and gave concrete examples of how such a scheme might be realized. The ﬁrst paper on public-key cryptog- raphy was “Multiuser cryptographic techniques” by Difﬁe and Hellman [344], presented at the National Computer Conference in June of 1976. Although the authors were not sat- isﬁed with the examples they cited, the concept was made clear. In their landmark paper, Difﬁe and Hellman [345] provided a more comprehensive account of public-key cryptog- raphy and described the ﬁrst viable method to realize this elegant concept. Another good source for the early history and development of the subject is Difﬁe [343]. Nechvatal [922] also provides a broad survey of public-key cryptography. Merkle [849, 850] independently discovered public-key cryptography, illustrating how this concept could be realized by giving an elegant and ingenious example now commonly re- ferred to as theMerkle puzzle scheme. Simmons [1144, p.412] notes the ﬁrst reported ap- plication of public-key cryptography was ﬁelded by Sandia National Laboratories (U.S.) in 1978. §1.9 Much of the early work on cryptographic hash functions was done by Merkle [850]. The most comprehensive current treatment of the subject is by Preneel [1004]. §1.10 A large number of successful cryptanalytic attacks on systems claiming security are due to protocol failure. An overview of this area is given by Moore [899], including classiﬁcations of protocol failures and design principles. 48 Ch. 1 Overview of Cryptography §1.11 One approach to distributing public-keys is the so-calledMerkle channel(see Simmons [1144, p.387]). Merkle proposed that public keys be distributed over so many independent public channels (newspaper, radio, television, etc.) that it would be improbable for an ad- versary to compromise all of them. In 1979 Kohnfelder [702] suggested the idea of usingpublic-key certiﬁcatesto facilitate the distribution of public keys over unsecured channels, such that their authenticity can be veriﬁed. Essentially the same idea, but by on-line requests, was proposed by Needham and Schroeder (ses Wilkes [1244]). A provably secure key agreement protocol has been proposed whose security is based on the Heisenberg uncertainty principle of quantum physics. The security of so-calledquantum cryptographydoes not rely upon any complexity-theoretic assumptions. For further details on quantum cryptography, consult Chapter 6 of Brassard [192], and Bennett, Brassard, and Ekert [115]. §1.12 For an introduction and detailed treatment of many pseudorandom sequence generators, see Knuth [692]. Knuth cites an example of a complex scheme to generate random numbers which on closer analysis is shown to produce numbers which are far from random, and con- cludes:...random numbers should not be generated with a method chosen at random. §1.13 The seminal work of Shannon [1121] on secure communications, published in 1949, re- mains as one of the best introductions to both practice and theory, clearly presenting many of the fundamental ideas including redundancy, entropy, and unicity distance. Various mod- els under which security may be examined are considered by Rueppel [1081], Simmons [1144], and Preneel [1003], among others; see also Goldwasser [476]. Chapter /2 Mathematical Background Contents in Brief 2.1 Probability theory.......................... 50 2.2 Information theory ......................... 56 2.3 Complexity theory ......................... 57 2.4 Number theory ........................... 63 2.5 Abstract algebra .......................... 75 2.6 Finite ﬁelds ............................. 80 2.7 Notes and further references.................... 85 This chapter is a collection of basic material on probability theory, information the- ory, complexity theory, number theory, abstract algebra, and ﬁnite ﬁelds that will be used throughout this book. Further background and proofs of the facts presented here can be found in the references given in§2.7. The following standard notation will be used through- out: 1. Z denotes the set ofintegers; that is, the set{..., −2,−1,0,1,2,... }. 2. Q denotes the set ofrational numbers; that is, the set{a b \|a,b∈Z,b̸=0}. 3. R denotes the set ofreal numbers. 4. πis the mathematical constant;π≈3.14159. 5. eis the base of the natural logarithm;e≈2.71828. 6. [a,b]denotes the integersxsatisfyinga≤x≤b. 7. ⌊x⌋is the largest integer less than or equal tox. For example,⌊5.2⌋=5 and ⌊−5.2⌋=−6. 8. ⌈x⌉is the smallest integer greater than or equal tox. For example,⌈5.2⌉=6 and ⌈−5.2⌉=−5. 9. IfAis a ﬁnite set, then\|A\|denotes the number of elements inA, called thecardinality ofA. 10. a∈Ameans that elementais a member of the setA. 11. A⊆Bmeans thatAis a subset ofB. 12. A⊂Bmeans thatAis a proper subset ofB; that isA⊆Band A̸=B. 13. Theintersectionof setsAand Bis the setA∩B={x\|x∈Aand x∈B}. 14. Theunionof setsAand Bis the setA∪B={x\|x∈Aorx∈B}. 15. Thedifferenceof setsAand Bis the setA−B={x\|x∈Aand x̸∈B}. 16. TheCartesian productof setsAand Bis the setA×B={(a,b)\|a∈Aand b∈ B}. For example,{a1,a2}×{ b1,b2,b3} = {(a1,b1),(a1,b2),(a1,b3),(a2,b1), (a2,b2),(a2,b3)}. 49 50 Ch. 2 Mathematical Background 17. A functionormapping f:A−→Bis a rule which assigns to each elementainA precisely one elementbinB.I fa∈Ais mapped tob∈Bthenbis called theimage ofa,ais called apreimageofb, and this is writtenf(a)= b. The setAis called the domain off, and the setBis called thecodomain off. 18. A functionf:A−→Bis1−1(one-to-one)o rinjectiveif each element inBis the image of at most one element inA. Hencef(a1)= f(a2)impliesa1 =a2. 19. A functionf:A−→Bisontoorsurjectiveif eachb∈Bis the image of at least one a∈A. 20. A functionf :A−→Bis abijectionif it is both one-to-one and onto. Iffis a bijection between ﬁnite setsAand B, then\|A\|=\|B\|.I ffis a bijection between a setAand itself, thenfis called apermutationon A. 21. lnxis the natural logarithm ofx; that is, the logarithm ofxto the basee. 22. lgxis the logarithm ofxto the base2. 23. exp(x)is the exponential functionex. 24. ∑ n i=1 aidenotes the suma1 +a2 +··· +an. 25. ∏ n i=1 aidenotes the producta1 ·a2 ····· an. 26. For a positive integern, the factorial function isn!= n(n−1)(n−2)···1.B y convention,0!=1 . 2.1 Probability theory 2.1.1 Basic deﬁnitions 2.1 DeﬁnitionAn experimentis a procedure that yields one of a given set of outcomes. The individual possible outcomes are calledsimple events. The set of all possible outcomes is called thesample space. This chapter only considersdiscretesample spaces; that is, sample spaces with only ﬁnitely many possible outcomes. Let the simple events of a sample spaceSbe labeled s1,s2,...,s n. 2.2 DeﬁnitionA probability distributionPon Sis a sequence of numbersp1,p2,...,p nthat are all non-negative and sum to 1. The numberpiis interpreted as theprobabilityofsibeing the outcome of the experiment. 2.3 DeﬁnitionAn eventEis a subset of the sample spaceS. Theprobabilitythat eventE occurs, denotedP(E), is the sum of the probabilitiespiof all simple eventssiwhich belong toE.I fsi∈S,P({si})is simply denoted byP(si). 2.4 DeﬁnitionIfEis an event, thecomplementary eventis the set of simple events not be- longing toE, denoted E. 2.5 FactLetE⊆Sbe an event. (i)0≤P(E)≤1. Furthermore,P(S)=1 and P(∅)=0 .(∅is the empty set.) (ii)P( E)=1 −P(E). §2.1 Probability theory 51 (iii) If the outcomes inSare equally likely, thenP(E)= \|E\| \|S\|. 2.6 DeﬁnitionTwo eventsE1 and E2 are calledmutually exclusiveifP(E1 ∩E2)=0 . That is, the occurrence of one of the two events excludes the possibility that the other occurs. 2.7 FactLetE1 and E2 be two events. (i) IfE1 ⊆E2, thenP(E1)≤P(E2). (ii)P(E1 ∪E2)+P(E1 ∩E2)= P(E1)+P(E2). Hence, ifE1 and E2 are mutually exclusive, thenP(E1 ∪E2)= P(E1)+P(E2). 2.1.2 Conditional probability 2.8 DeﬁnitionLetE1 and E2 be two events withP(E2)>0. Theconditional probability of E1 givenE2, denotedP(E1\|E2),i s P(E1\|E2)= P(E1 ∩E2) P(E2) . P(E1\|E2)measures the probability of eventE1 occurring, given thatE2 has occurred. 2.9 DeﬁnitionEventsE1 and E2 are said to beindependentifP(E1 ∩E2)= P(E1)P(E2). Observe that ifE1 andE2 are independent, thenP(E1\|E2)= P(E1)andP(E2\|E1)= P(E2). That is, the occurrence of one event does not inﬂuence the likelihood of occurrence of the other. 2.10 Fact(Bayes’ theorem)I fE1 and E2 are events withP(E2)>0, then P(E1\|E2)= P(E1)P(E2\|E1) P(E2) . 2.1.3 Random variables LetSbe a sample space with probability distributionP. 2.11 DeﬁnitionA random variableXis a function from the sample spaceSto the set of real numbers; to each simple eventsi∈S,Xassigns a real numberX(si). SinceSis assumed to be ﬁnite,Xcan only take on a ﬁnite number of values. 2.12 DeﬁnitionLetXbe a random variable onS. Theexpected valueormean ofXisE(X)=∑ si∈SX(si)P(si). 2.13 FactLetXbe a random variable onS. ThenE(X)= ∑ x∈R x·P(X=x). 2.14 FactIfX1,X2,...,X mare random variables onS, anda1,a2,...,a mare real numbers, thenE(∑ m i=1 aiXi)= ∑ m i=1 aiE(Xi). 2.15 DeﬁnitionThe varianceof a random variableXof meanµis a non-negative number de- ﬁned by Var(X)= E((X−µ)2). The standard deviationofXis the non-negative square root ofVar(X). 52 Ch. 2 Mathematical Background If a random variable has small variance then large deviations from the mean are un- likely to be observed. This statement is made more precise below. 2.16 Fact(Chebyshev’ s inequality) LetXbe a random variable with meanµ = E(X)and varianceσ2 =Var(X). Then for anyt> 0, P(\|X−µ\|≥ t)≤σ2 t2 . 2.1.4 Binomial distribution 2.17 DeﬁnitionLetnand kbe non-negative integers. Thebinomial coefﬁcient (n k ) is the num- ber of different ways of choosingkdistinct objects from a set ofndistinct objects, where the order of choice is not important. 2.18 Fact(properties of binomial coefﬁcients) Letnand kbe non-negative integers. (i) (n k ) = n! k!(n−k)! . (ii) (n k ) = ( n n−k ) . (iii) (n+1 k+1 ) = (n k ) + ( n k+1 ) . 2.19 Fact(binomial theorem) For any real numbersa,b, and non-negative integern,(a+b)n=∑ n k=0 (n k ) akbn−k. 2.20 DeﬁnitionA Bernoulli trialis an experiment with exactly two possible outcomes, called successand failure. 2.21 FactSuppose that the probability of success on a particular Bernoulli trial isp. Then the probability of exactlyksuccesses in a sequence ofnsuch independent trials is (n k ) pk(1−p)n−k,for each0≤k≤n. (2.1) 2.22 DeﬁnitionThe probability distribution (2.1) is called thebinomial distribution. 2.23 FactThe expected number of successes in a sequence ofnindependent Bernoulli trials, with probabilitypof success in each trial, isnp. The variance of the number of successes isnp(1−p). 2.24 Fact(law of large numbers) LetXbe the random variable denoting the fraction of suc- cesses innindependent Bernoulli trials, with probabilitypof success in each trial. Then for anyϵ>0, P(\|X−p\|>ϵ)−→0,asn−→∞. In other words, asngets larger, the proportion of successes should be close top, the probability of success in each trial. §2.1 Probability theory 53 2.1.5 Birthday problems 2.25 Deﬁnition (i) For positive integersm,nwithm≥n, the numberm(n) is deﬁned as follows: m(n) =m(m−1)(m−2)···(m−n+1). (ii) Letm,nbe non-negative integers withm≥n. TheStirling number of the second kind, denoted {m n } ,i s {m n } = 1 n! n∑ k=0 (−1)n−k (n k ) km, with the exception that {0 0 } =1. The symbol {m n } counts the number of ways of partitioning a set ofmobjects inton non-empty subsets. 2.26 Fact(classical occupancy problem) An urn hasmballs numbered1tom. Suppose thatn balls are drawn from the urn one at a time, with replacement, and their numbers are listed. The probability that exactlytdifferent balls have been drawn is P1(m,n,t)= {n t }m(t) mn , 1≤t≤n. The birthday problem is a special case of the classical occupancy problem. 2.27 Fact(birthday problem) An urn hasmballs numbered1tom. Suppose thatnballs are drawn from the urn one at a time, with replacement, and their numbers are listed. (i) The probability of at least one coincidence (i.e., a ball drawn at least twice) is P2(m,n)=1 −P1(m,n,n)=1 −m(n) mn , 1≤n≤m. (2.2) Ifn=O(√ m)(see Deﬁnition 2.55) andm−→∞, then P2(m,n)−→1−exp ( −n(n−1) 2m +O ( 1 √ m )) ≈1−exp ( −n2 2m ) . (ii) Asm−→∞, the expected number of draws before a coincidence is√ πm 2 . The following explains why probability distribution (2.2) is referred to as thebirthday surpriseorbirthday paradox. The probability that at least2people in a room of23people have the same birthday isP2(365,23)≈0.507, which is surprisingly large. The quantity P2(365,n)also increases rapidly asnincreases; for example,P2(365,30)≈0.706. A different kind of problem is considered in Facts 2.28, 2.29, and 2.30 below. Suppose that there are two urns, one containingmwhite balls numbered1tom, and the other con- tainingmred balls numbered1tom. First,n1 balls are selected from the ﬁrst urn and their numbers listed. Thenn2 balls are selected from the second urn and their numbers listed. Finally, the number of coincidences between the two lists is counted. 2.28 Fact(model A) If the balls from both urns are drawn one at a time, with replacement, then the probability of at least one coincidence is P3(m,n1,n2)=1 − 1 mn1+n2 ∑ t1,t2 m(t1+t2) {n1 t1 }{n2 t2 } , Handbook of Applied Cryptographyby A. Menezes, P. van Oorschot and S. Vanstone. 54 Ch. 2 Mathematical Background where the summation is over all0≤t1 ≤n1,0≤t2 ≤n2.I fn=n1 =n2,n=O(√ m) and m−→∞, then P3(m,n1,n2)−→1−exp ( −n2 m [ 1+O ( 1 √ m )]) ≈1−exp ( −n2 m ) . 2.29 Fact(model B) If the balls from both urns are drawn without replacement, then the prob- ability of at least one coincidence is P4(m,n1,n2)=1 − m(n1+n2) m(n1)m(n2) . Ifn1 =O(√ m),n2 =O(√ m), andm−→∞, then P4(m,n1,n2)−→1−exp ( −n1n2 m [ 1+ n1 +n2 −1 2m +O ( 1 m )]) . 2.30 Fact(model C)I ft h en1 white balls are drawn one at a time, with replacement, and then2 red balls are drawn without replacement, then the probability of at least one coincidence is P5(m,n1,n2)=1 − ( 1−n2 m )n1 . Ifn1 =O(√ m),n2 =O(√ m), andm−→∞, then P5(m,n1,n2)−→1−exp ( −n1n2 m [ 1+O ( 1 √ m )]) ≈1−exp ( −n1n2 m ) . 2.1.6 Random mappings 2.31 DeﬁnitionLetFndenote the collection of all functions (mappings) from a ﬁnite domain of sizento a ﬁnite codomain of sizen. Models where random elements ofFn are considered are calledrandom mappings models. In this section the only random mappings model considered is where every function from Fnis equally likely to be chosen; such models arise frequently in cryptography and algorithmic number theory. Note that\|Fn\|=nn, whence the probability that a particular function fromFnis chosen is1/nn. 2.32 DeﬁnitionLetfbe a function inFnwith domain and codomain equal to{1,2,...,n }. The functional graphoffis a directed graph whosepoints(orvertices) are the elements {1,2,...,n }and whoseedgesare the ordered pairs(x,f(x))for allx∈{1,2,...,n }. 2.33 Example (functional graph) Consider the functionf:{1,2,..., 13}−→{1,2,..., 13} deﬁned byf(1)=4 ,f(2)=11 ,f(3)=1 ,f(4)=6 ,f(5)=3 ,f(6)=9 ,f(7)=3 , f(8)=11 ,f(9)=1 ,f(10)=2 ,f(11)=10 ,f(12)=4 ,f(13)=7 . The functional graph offis shown in Figure 2.1. □ As Figure 2.1 illustrates, a functional graph may have severalcomponents(maximal connected subgraphs), each component consisting of a directedcycleand some directed treesattached to the cycle. 2.34 FactAs ntends to inﬁnity, the following statements regarding the functional digraph of a random functionffrom Fnare true: (i) The expected number of components is1 2 lnn. §2.1 Probability theory 55 13 7 5 3 12 4 1 9 6 8 11 2 10 Figure 2.1:A functional graph (see Example 2.33). (ii) The expected number of points which are on the cycles is √ πn/2. (iii) The expected number ofterminal points(points which have no preimages) isn/e. (iv) The expected number ofk-th iterate image points(xis ak-th iterate image point if x=f(f(···f   ktimes (y)···))for somey)i s(1−τk)n, where theτksatisfy the recurrence τ0 =0,τk+1 =e−1+τk fork≥0. 2.35 DeﬁnitionLetfbe a random function from{1,2,...,n }to{1,2,...,n }and letu∈ {1,2,...,n }. Consider the sequence of pointsu0,u1,u2,... deﬁned byu0 =u,ui = f(ui−1)fori≥1. In terms of the functional graph off, this sequence describes a path that connects to a cycle. (i) The number of edges in the path is called thetail lengthofu, denotedλ(u). (ii) The number of edges in the cycle is called thecycle lengthofu, denotedµ(u). (iii) Therho-lengthofuis the quantityρ(u)= λ(u)+µ(u). (iv) Thetree sizeofuis the number of edges in the maximal tree rooted on a cycle in the component that containsu. (v) Thecomponent sizeofuis the number of edges in the component that containsu. (vi) Thepredecessors sizeofuis the number of iterated preimages ofu. 2.36 Example The functional graph in Figure 2.1 has2components and4terminal points. The pointu=3 has parametersλ(u)=1 ,µ(u)=4 ,ρ(u)=5 . The tree, component, and predecessors sizes ofu=3 are4,9, and3, respectively. □ 2.37 FactAs ntends to inﬁnity, the following are the expectations of some parameters associ- ated with a random point in{1,2,...,n }and a random function fromFn: (i) tail length:√ πn/8(ii) cycle length: √ πn/8(iii) rho-length: √ πn/2(iv) tree size:n/3(v) compo- nent size:2n/3(vi) predecessors size: √ πn/8. 2.38 FactAs ntends to inﬁnity, the expectations of the maximum tail, cycle, and rho lengths in a random function fromFnarec1 √ n,c2 √ n, andc3 √ n, respectively, wherec1 ≈0.78248, c2 ≈1.73746, andc3 ≈2.4149. Facts 2.37 and 2.38 indicate that in the functional graph of a random function, most points are grouped together in one giant component, and there is a small number of large trees. Also, almost unavoidably, a cycle of length about√ narises after following a path of length√ nedges. 56 Ch. 2 Mathematical Background 2.2 Information theory 2.2.1 Entropy LetXbe a random variable which takes on a ﬁnite set of valuesx1,x2,...,x n, with prob- abilityP(X=xi)= pi, where0≤pi≤1for eachi,1≤i≤n, and where∑ n i=1 pi=1. Also, letYand Zbe random variables which take on ﬁnite sets of values. The entropy ofXis a mathematical measure of the amount of information provided by an observation ofX. Equivalently, it is the uncertainity about the outcome before an obser- vation ofX. Entropy is also useful for approximating the average number of bits required to encode the elements ofX. 2.39 DeﬁnitionThe entropyoruncertaintyofXis deﬁned to beH(X)= −∑ n i=1 pilgpi= ∑ n i=1 pilg ( 1 pi ) where, by convention,pi·lgpi=pi·lg ( 1 pi ) =0 ifpi=0. 2.40 Fact(properties of entropy) LetXbe a random variable which takes onnvalues. (i)0≤H(X)≤lgn. (ii)H(X)=0 if and only ifpi=1 for somei, andpj =0 for allj̸=i(that is, there is no uncertainty of the outcome). (iii)H(X)=lg nif and only ifpi=1/nfor eachi,1≤i≤n(that is, all outcomes are equally likely). 2.41 DeﬁnitionThe joint entropyofXand Yis deﬁned to be H(X,Y)= − ∑ x,y P(X=x,Y =y)lg(P(X=x,Y =y)), where the summation indicesxand yrange over all values ofXand Y, respectively. The deﬁnition can be extended to any number of random variables. 2.42 FactIfXand Yare random variables, thenH(X,Y)≤H(X)+H(Y), with equality if and only ifXand Yare independent. 2.43 DeﬁnitionIfX,Yare random variables, theconditional entropy ofXgivenY=yis H(X\|Y=y)= − ∑ x P(X=x\|Y=y)lg(P(X=x\|Y=y)), where the summation indexxranges over all values ofX. Theconditional entropy ofX givenY, also called theequivocation ofYaboutX,i s H(X\|Y)= ∑ y P(Y=y)H(X\|Y=y), where the summation indexyranges over all values ofY. 2.44 Fact(properties of conditional entropy) LetXand Ybe random variables. (i) The quantityH(X\|Y)measures the amount of uncertainty remaining aboutXafter Yhas been observed. §2.3 Complexity theory 57 (ii)H(X\|Y)≥0and H(X\|X)=0 . (iii)H(X,Y)= H(X)+H(Y\|X)= H(Y)+H(X\|Y). (iv)H(X\|Y)≤H(X), with equality if and only ifXand Yare independent. 2.2.2 Mutual information 2.45 DeﬁnitionThe mutual informationortransinformationof random variablesXand Yis I(X;Y)= H(X)−H(X\|Y). Similarly, the transinformation ofXand the pairY,Zis deﬁned to beI(X;Y,Z)= H(X)−H(X\|Y,Z). 2.46 Fact(properties of mutual transinformation) (i) The quantityI(X;Y)can be thought of as the amount of information thatYreveals aboutX. Similarly, the quantityI(X;Y,Z)can be thought of as the amount of in- formation thatYand Ztogether reveal aboutX. (ii)I(X;Y)≥0. (iii)I(X;Y)=0 if and only ifXand Yare independent (that is,Ycontributes no in- formation aboutX). (iv)I(X;Y)= I(Y;X). 2.47 DeﬁnitionThe conditional transinformationof the pairX,YgivenZis deﬁned to be IZ(X;Y)= H(X\|Z)−H(X\|Y,Z). 2.48 Fact(properties of conditional transinformation) (i) The quantityIZ(X;Y)can be interpreted as the amount of information thatYpro- vides aboutX, given thatZhas already been observed. (ii)I(X;Y,Z)= I(X;Y)+IY(X;Z). (iii)IZ(X;Y)= IZ(Y;X). 2.3 Complexity theory 2.3.1 Basic deﬁnitions The main goal of complexity theory is to provide mechanisms for classifying computational problems according to the resources needed to solve them. The classiﬁcation should not depend on a particular computational model, but rather should measure the intrinsic dif- ﬁculty of the problem. The resources measured may include time, storage space, random bits, number of processors, etc., but typically the main focus is time, and sometimes space. 2.49 DeﬁnitionAn algorithmis a well-deﬁned computational procedure that takes a variable input and halts with an output. 58 Ch. 2 Mathematical Background Of course, the term “well-deﬁned computational procedure” is not mathematically pre- cise. It can be made so by using formal computational models such as Turing machines, random-access machines, or boolean circuits. Rather than get involved with the technical intricacies of these models, it is simpler to think of an algorithm as a computer program written in some speciﬁc programming language for a speciﬁc computer that takes a vari- able input and halts with an output. It is usually of interest to ﬁnd the most efﬁcient (i.e., fastest) algorithm for solving a given computational problem. The time that an algorithm takes to halt depends on the “size” of the problem instance. Also, the unit of time used should be made precise, especially when comparing the performance of two algorithms. 2.50 DeﬁnitionThe sizeof the input is the total number of bits needed to represent the input in ordinary binary notation using an appropriate encoding scheme. Occasionally, the size of the input will be the number of items in the input. 2.51 Example (sizes of some objects) (i) The number of bits in the binary representation of a positive integernis1+⌊lgn⌋ bits. For simplicity, the size ofnwill be approximated bylgn. (ii) Iffis a polynomial of degree at mostk, each coefﬁcient being a non-negative integer at mostn, then the size offis(k+1)lg nbits. (iii) IfAis a matrix withrrows,scolumns, and with non-negative integer entries each at mostn, then the size ofAisrslgnbits. □ 2.52 DeﬁnitionThe running timeof an algorithm on a particular input is the number of prim- itive operations or “steps” executed. Often a step is taken to mean a bit operation. For some algorithms it will be more con- venient to take step to mean something else such as a comparison, a machine instruction, a machine clock cycle, a modular multiplication, etc. 2.53 DeﬁnitionThe worst-case running timeof an algorithm is an upper bound on the running time for any input, expressed as a function of the input size. 2.54 DeﬁnitionThe average-case running timeof an algorithm is the average running time over all inputs of a ﬁxed size, expressed as a function of the input size. 2.3.2 Asymptotic notation It is often difﬁcult to derive the exact running time of an algorithm. In such situations one is forced to settle for approximations of the running time, and usually may only derive the asymptoticrunning time. That is, one studies how the running time of the algorithm in- creases as the size of the input increases without bound. In what follows, the only functions considered are those which are deﬁned on the posi- tive integers and take on real values that are always positive from some point onwards. Let fand gbe two such functions. 2.55 Deﬁnition(order notation) (i) (asymptotic upper bound)f(n)= O(g(n))if there exists a positive constantcand a positive integern0 such that0≤f(n)≤cg(n)for alln≥n0. §2.3 Complexity theory 59 (ii) (asymptotic lower bound)f(n)=Ω( g(n))if there exists a positive constantcand a positive integern0 such that0≤cg(n)≤f(n)for alln≥n0. (iii) (asymptotic tight bound)f(n)=Θ( g(n))if there exist positive constantsc1 and c2, and a positive integern0 such thatc1g(n)≤f(n)≤c2g(n)for alln≥n0. (iv) (o-notation)f(n)= o(g(n))if for any positive constantc> 0there exists a constant n0 >0such that0≤f(n)<cg(n)for alln≥n0. Intuitively,f(n)= O(g(n))means thatfgrows no faster asymptotically thang(n)to within a constant multiple, whilef(n)=Ω( g(n))means thatf(n)grows at least as fast asymptotically asg(n)to within a constant multiple.f(n)= o(g(n))means thatg(n)is an upper bound forf(n)that is not asymptotically tight, or in other words, the functionf(n) becomes insigniﬁcant relative tog(n)asngets larger. The expressiono(1)is often used to signify a functionf(n)whose limit asnapproaches∞is0. 2.56 Fact(properties of order notation) For any functionsf(n),g(n),h(n), andl(n), the fol- lowing are true. (i)f(n)= O(g(n))if and only ifg(n)=Ω( f(n)). (ii)f(n)=Θ( g(n))if and only iff(n)= O(g(n))and f(n)=Ω( g(n)). (iii) Iff(n)= O(h(n))and g(n)= O(h(n)), then(f+g)(n)= O(h(n)). (iv) Iff(n)= O(h(n))and g(n)= O(l(n)), then(f·g)(n)= O(h(n)l(n)). (v) (reﬂexivity)f(n)= O(f(n)). (vi) (transitivity)I ff(n)= O(g(n))and g(n)= O(h(n)), thenf(n)= O(h(n)). 2.57 Fact(approximations of some commonly occurring functions) (i) (polynomial function)I ff(n)is a polynomial of degreekwith positive leading term, thenf(n)=Θ( nk). (ii) For any constantc> 0,logcn=Θ(lgn). (iii) (Stirling’ s formula) For all integersn≥1, √ 2πn (n e )n ≤ n! ≤ √ 2πn (n e )n+(1/(12n)) . Thus n!= √ 2πn (n e )n( 1+Θ( 1 n) ) . Also,n!= o(nn)and n!=Ω(2 n). (iv)lg(n!)=Θ( nlgn). 2.58 Example (comparative growth rates of some functions) Letϵand cbe arbitrary constants with0<ϵ<1<c. The following functions are listed in increasing order of their asymp- totic growth rates: 1<lnlnn< lnn< exp( √ lnnlnlnn)<nϵ<nc<nln n<cn<nn<ccn . □ 2.3.3 Complexity classes 2.59 DeﬁnitionA polynomial-time algorithmis an algorithm whose worst-case running time function is of the formO(nk), wherenis the input size andkis a constant. Any algorithm whose running time cannot be so bounded is called anexponential-time algorithm. Roughly speaking, polynomial-time algorithms can be equated withgood orefﬁcient algorithms, while exponential-time algorithms are consideredinefﬁcient. There are, how- ever, some practical situations when this distinction is not appropriate. When considering polynomial-time complexity, the degree of the polynomial is signiﬁcant. For example, even 60 Ch. 2 Mathematical Background though an algorithm with a running time ofO(nln lnn),nbeing the input size, is asymptot- ically slower that an algorithm with a running time ofO(n100), the former algorithm may be faster in practice for smaller values ofn, especially if the constants hidden by the big-O notation are smaller. Furthermore, in cryptography, average-case complexity is more im- portant than worst-case complexity — a necessary condition for an encryption scheme to be considered secure is that the corresponding cryptanalysis problem is difﬁcult on average (or more precisely, almost always difﬁcult), and not just for some isolated cases. 2.60 DeﬁnitionA subexponential-time algorithmis an algorithm whose worst-case running time function is of the forme o(n), wherenis the input size. A subexponential-time algorithm is asymptotically faster than an algorithm whose run- ning time is fully exponential in the input size, while it is asymptotically slower than a polynomial-time algorithm. 2.61 Example (subexponential running time) LetAbe an algorithm whose inputs are either elements of a ﬁnite ﬁeldF q(see§2.6), or an integerq. If the expected running time ofAis of the form Lq[α,c]= O ( exp ( (c+o(1))(lnq)α(lnlnq)1−α)) , (2.3) where cis a positive constant, andαis a constant satisfying0 <α< 1, thenAis a subexponential-time algorithm. Observe that forα=0,Lq[0,c]is a polynomial inlnq, while forα=1,Lq[1,c]is a polynomial inq, and thus fully exponential inlnq. □ For simplicity, the theory of computational complexity restricts its attention todeci- sion problems, i.e., problems which have either YES or NO as an answer. This is not too restrictive in practice, as all the computational problems that will be encountered here can be phrased as decision problems in such a way that an efﬁcient algorithm for the decision problem yields an efﬁcient algorithm for the computational problem, and vice versa. 2.62 DeﬁnitionThe complexity classP is the set of all decision problems that are solvable in polynomial time. 2.63 DeﬁnitionThe complexity classNP is the set of all decision problems for which a YES answer can be veriﬁed in polynomial time given some extra information, called acertiﬁcate. 2.64 DeﬁnitionThe complexity classco-NP is the set of all decision problems for which a NO answer can be veriﬁed in polynomial time using an appropriate certiﬁcate. It must be emphasized that if a decision problem is inNP , it may not be the case that the certiﬁcate of a YES answer can be easily obtained; what is asserted is that such a certiﬁcate does exist, and, if known, can be used to efﬁciently verify the YES answer. The same is true of the NO answers for problems inco-NP. 2.65 Example (problem inNP ) Consider the following decision problem: COMPOSITES INSTANCE: A positive integern. QUESTION: Is ncomposite? That is, are there integersa,b> 1such thatn=ab? COMPOSITES belongs toNP because if an integernis composite, then this fact can be veriﬁed in polynomial time if one is given a divisoraofn, where1<a<n (the certiﬁcate in this case consists of the divisora). It is in fact also the case that COMPOSITES belongs toco-NP. It is still unknown whether or not COMPOSITES belongs toP. □ §2.3 Complexity theory 61 2.66 FactP ⊆NP and P ⊆co-NP. The following are among the outstanding unresolved questions in the subject of com- plexity theory: 1. IsP =NP ? 2. IsNP =co-NP? 3. IsP =NP ∩co-NP? Most experts are of the opinion that the answer to each of the three questions is NO, although nothing along these lines has been proven. The notion of reducibility is useful when comparing the relative difﬁculties of prob- lems. 2.67 DeﬁnitionLetL1 and L2 be two decision problems.L1 is said topolytime reducetoL2, writtenL1 ≤P L2, if there is an algorithm that solvesL1 which uses, as a subroutine, an algorithm for solvingL2, and which runs in polynomial time if the algorithm forL2 does. Informally, ifL1 ≤P L2, thenL2 is at least as difﬁcult asL1, or, equivalently,L1 is no harder thanL2. 2.68 DeﬁnitionLetL1 and L2 be two decision problems. IfL1 ≤P L2 and L2 ≤P L1, then L1 and L2 are said to becomputationally equivalent. 2.69 FactLetL1,L2, andL3 be three decision problems. (i) (transitivity)I fL1 ≤P L2 and L2 ≤P L3, thenL1 ≤P L3. (ii) IfL1 ≤P L2 and L2 ∈P, thenL1 ∈P. 2.70 DeﬁnitionA decision problemLis said to beNP -completeif (i)L∈NP , and (ii)L1 ≤P Lfor everyL1 ∈NP . The class of allNP -complete problems is denoted byNPC . NP -complete problems are the hardest problems inNP in the sense that they are at least as difﬁcult as every other problem inNP . There are thousands of problems drawn from diverse ﬁelds such as combinatorics, number theory, and logic, that are known to beNP - complete. 2.71 Example (subset sum problem) Thesubset sum problemis the following: given a set of positive integers{a1,a2,...,a n}and a positive integers, determine whether or not there is a subset of theaithat sum tos. The subset sum problem isNP -complete. □ 2.72 FactLetL1 and L2 be two decision problems. (i) IfL1 isNP -complete andL1 ∈P, thenP = NP . (ii) IfL1 ∈NP ,L2 isNP -complete, andL2 ≤P L1, thenL1 is alsoNP -complete. (iii) IfL1 isNP -complete andL1 ∈co-NP, thenNP = co-NP. By Fact 2.72(i), if a polynomial-time algorithm is found for any singleNP -complete problem, then it is the case thatP = NP , a result that would be extremely surprising. Hence, a proof that a problem isNP -complete provides strong evidence for its intractability. Fig- ure 2.2 illustrates what is widely believed to be the relationship between the complexity classesP,NP ,co-NP, andNPC . Fact 2.72(ii) suggests the following procedure for proving that a decision problemL1 isNP -complete: 62 Ch. 2 Mathematical Background co-NP NPC NP P NP ∩ co-NP Figure 2.2:Conjectured relationship between the complexity classesP,NP ,co-NP, andNPC . 1. Prove thatL1 ∈NP . 2. Select a problemL2 that is known to beNP -complete. 3. Prove thatL2 ≤P L1. 2.73 DeﬁnitionA problem isNP -hardif there exists someNP -complete problem that polytime reduces to it. Note that theNP -hard classiﬁcation is not restricted to only decision problems. Ob- serve also that anNP -complete problem is alsoNP -hard. 2.74 Example (NP -hard problem) Given positive integersa1,a2,...,a nand a positive inte- gers, the computational version of the subset sum problem would ask to actually ﬁnd a subset of theaiwhich sums tos, provided that such a subset exists. This problem isNP - hard. □ 2.3.4 Randomized algorithms The algorithms studied so far in this section have beendeterministic; such algorithms fol- low the same execution path (sequence of operations) each time they execute with the same input. By contrast, arandomizedalgorithm makes random decisions at certain points in the execution; hence their execution paths may differ each time they are invoked with the same input. The random decisions are based upon the outcome of a random number gen- erator. Remarkably, there are many problems for which randomized algorithms are known that are more efﬁcient, both in terms of time and space, than the best known deterministic algorithms. Randomized algorithms for decision problems can be classiﬁed according to the prob- ability that they return the correct answer. 2.75 DeﬁnitionLetAbe a randomized algorithm for a decision problemL, and letIdenote an arbitrary instance ofL. (i)Ahas0-sided errorifP(Aoutputs YES\|I’s answer is YES)=1 , and P(Aoutputs YES\|I’s answer is NO)=0 . (ii)Ahas1-sided errorifP(Aoutputs YES\|I’s answer is YES)≥1 2 , and P(Aoutputs YES\|I’s answer is NO)=0 . §2.4 Number theory 63 (iii)Ahas2-sided errorifP(Aoutputs YES\|I’s answer is YES)≥2 3 , and P(Aoutputs YES\|I’s answer is NO)≤1 3 . The number 1 2 in the deﬁnition of 1-sided error is somewhat arbitrary and can be re- placed by any positive constant. Similarly, the numbers2 3 and 1 3 in the deﬁnition of 2-sided error, can be replaced by1 2 +ϵand 1 2 −ϵ, respectively, for any constantϵ,0<ϵ< 1 2 . 2.76 DeﬁnitionThe expected running timeof a randomized algorithm is an upper bound on the expected running time for each input (the expectation being over all outputs of the random number generator used by the algorithm), expressed as a function of the input size. The important randomized complexity classes are deﬁned next. 2.77 Deﬁnition(randomized complexity classes) (i) The complexity classZPP (“zero-sided probabilistic polynomial time”) is the set of all decision problems for which there is a randomized algorithm with 0-sided error which runs in expected polynomial time. (ii) The complexity classRP (“randomized polynomial time”) is the set of all decision problems for which there is a randomized algorithm with 1-sided error which runs in (worst-case) polynomial time. (iii) The complexity classBPP (“bounded error probabilistic polynomial time”) is the set of all decision problems for which there is a randomized algorithm with 2-sided error which runs in (worst-case) polynomial time. 2.78 FactP ⊆ZPP ⊆RP ⊆BPP and RP ⊆NP . 2.4 Number theory 2.4.1 The integers The set of integers{..., −3,−2,−1,0,1,2,3,... }is denoted by the symbolZ. 2.79 DeﬁnitionLeta,bbe integers. Thenadividesb(equivalently:ais adivisorofb,o rais a factorofb) if there exists an integercsuch thatb=ac.I fadividesb, then this is denoted by a\|b. 2.80 Example (i)−3\|18, since18=( −3)(−6). (ii)173\|0, since0=(173)(0) . □ The following are some elementary properties of divisibility. 2.81 Fact(properties of divisibility) For alla,b,c∈Z, the following are true: (i)a\|a. (ii) Ifa\|band b\|c, thena\|c. (iii) Ifa\|band a\|c, thena\|(bx+cy)for allx,y∈Z. (iv) Ifa\|band b\|a, thena=±b. 64 Ch. 2 Mathematical Background 2.82 Deﬁnition(division algorithm for integers)I faand bare integers withb≥1, then or- dinary long division ofaby byields integersq(thequotient) andr(theremainder) such that a=qb+r, where 0≤r<b. Moreover,qand rare unique. The remainder of the division is denotedamodb, and the quotient is denotedadivb. 2.83 FactLeta,b∈Z withb̸=0. Thenadivb=⌊a/b⌋and amodb=a−b⌊a/b⌋. 2.84 Example Ifa =7 3,b =1 7, thenq =4 and r =5 . Hence 73mod17 = 5and 73div17=4 . □ 2.85 DeﬁnitionAn integercis acommon divisorofaand bifc\|aand c\|b. 2.86 DeﬁnitionA non-negative integerdis thegreatest common divisorof integersaand b, denotedd=gcd(a,b),i f (i)dis a common divisor ofaand b; and (ii) wheneverc\|aand c\|b, thenc\|d. Equivalently,gcd(a,b)is the largest positive integer that divides bothaand b, with the ex- ception thatgcd(0,0)=0 . 2.87 Example The common divisors of12and18are{±1,±2,±3,±6}, andgcd(12,18)=6 . □ 2.88 DeﬁnitionA non-negative integerdis theleast common multipleof integersaand b, de- notedd=lcm(a,b),i f (i)a\|dand b\|d; and (ii) whenevera\|cand b\|c, thend\|c. Equivalently,lcm(a,b)is the smallest non-negative integer divisible by bothaand b. 2.89 FactIfaand bare positive integers, thenlcm(a,b)= a·b/gcd(a,b). 2.90 Example Sincegcd(12,18)=6 , it follows thatlcm(12,18)=12 ·18/6=36 . □ 2.91 DeﬁnitionTwo integersaandbare said to berelatively primeorcoprimeifgcd(a,b)=1 . 2.92 DeﬁnitionAn integerp≥2is said to beprimeif its only positive divisors are 1 andp. Otherwise,pis calledcomposite. The following are some well known facts about prime numbers. 2.93 FactIfpis prime andp\|ab, then eitherp\|aorp\|b(or both). 2.94 FactThere are an inﬁnite number of prime numbers. 2.95 Fact(prime number theorem) Letπ(x)denote the number of prime numbers≤x. Then lim x→∞ π(x) x/lnx=1. §2.4 Number theory 65 This means that for large values ofx,π(x)is closely approximated by the expres- sionx/lnx. For instance, whenx=1010,π(x)=455 ,052,511, whereas⌊x/lnx⌋= 434,294,481. A more explicit estimate forπ(x)is given below. 2.96 FactLetπ(x)denote the number of primes≤x. Then forx≥17 π(x)> x lnx and forx> 1 π(x)<1.25506 x lnx. 2.97 Fact(fundamental theorem of arithmetic) Every integern ≥ 2has a factorization as a product of prime powers: n=pe1 1 pe2 2 ···pek k , where thepiare distinct primes, and theeiare positive integers. Furthermore, the factor- ization is unique up to rearrangement of factors. 2.98 FactIfa=pe1 1 pe2 2 ···pek k ,b=pf1 1 pf2 2 ···pfk k , where eachei≥0and fi≥0, then gcd(a,b)= pmin(e1,f1) 1 pmin(e2,f2) 2 ···pmin(ek,fk) k and lcm(a,b)= pmax(e1,f1) 1 pmax(e2,f2) 2 ···pmax(ek,fk) k . 2.99 Example Leta=4864=2 8 ·19,b=3458=2 ·7·13·19. Thengcd(4864,3458)= 2·19=38 and lcm(4864,3458)=2 8 ·7·13·19=442624 . □ 2.100 DeﬁnitionFor n≥1, letφ(n)denote the number of integers in the interval[1,n]which are relatively prime ton. The functionφis called theEuler phi function(or theEuler totient function). 2.101 Fact(properties of Euler phi function) (i) Ifpis a prime, thenφ(p)= p−1. (ii) The Euler phi function ismultiplicative. That is, ifgcd(m,n)=1 , thenφ(mn)= φ(m)·φ(n). (iii) Ifn=pe1 1 pe2 2 ···pek k is the prime factorization ofn, then φ(n)= n ( 1− 1 p1 )( 1− 1 p2 ) ··· ( 1− 1 pk ) . Fact 2.102 gives an explicit lower bound forφ(n). 2.102 FactFor all integersn≥5, φ(n)> n 6lnlnn. 66 Ch. 2 Mathematical Background 2.4.2 Algorithms inZ Letaand bbe non-negative integers, each less than or equal ton. Recall (Example 2.51) that the number of bits in the binary representation ofnis⌊lgn⌋+1, and this number is approximated bylgn. The number of bit operations for the four basic integer operations of addition, subtraction, multiplication, and division using the classical algorithms is summa- rized in Table 2.1. These algorithms are studied in more detail in§14.2. More sophisticated techniques for multiplication and division have smaller complexities. Operation Bit complexity Addition a+b O(lga+lgb)= O(lgn) Subtraction a−b O(lga+lgb)= O(lgn) Multiplication a·b O((lga)(lgb))= O((lgn)2) Division a=qb+r O((lgq)(lgb))= O((lgn)2) Table 2.1:Bit complexity of basic operations inZ. The greatest common divisor of two integersaand bcan be computed via Fact 2.98. However, computing a gcd by ﬁrst obtaining prime-power factorizations does not result in an efﬁcient algorithm, as the problem of factoring integers appears to be relatively difﬁ- cult. The Euclidean algorithm (Algorithm 2.104) is an efﬁcient algorithm for computing the greatest common divisor of two integers that does not require the factorization of the integers. It is based on the following simple fact. 2.103 FactIfaand bare positive integers witha>b , thengcd(a,b)=gcd( b,amodb). 2.104 AlgorithmEuclidean algorithm for computing the greatest common divisor of two integers INPUT: two non-negative integersaand bwitha≥b. OUTPUT: the greatest common divisor ofaand b. 1. Whileb̸=0do the following: 1.1 Setr←amodb, a←b, b←r. 2. Return(a). 2.105 FactAlgorithm 2.104 has a running time ofO((lgn)2)bit operations. 2.106 Example (Euclidean algorithm) The following are the division steps of Algorithm 2.104 for computinggcd(4864,3458)=38 : 4864 = 1·3458+1406 3458 = 2·1406+646 1406 = 2·646+114 646 = 5·114+76 114 = 1·76+38 76 = 2·38+0. □ §2.4 Number theory 67 The Euclidean algorithm can be extended so that it not only yields the greatest common divisordof two integersaand b, but also integersxand ysatisfyingax+by=d. 2.107 AlgorithmExtended Euclidean algorithm INPUT: two non-negative integersaand bwitha≥b. OUTPUT: d=gcd(a,b)and integersx,ysatisfyingax+by=d. 1. Ifb=0 then setd←a, x←1, y←0, and return(d,x,y). 2. Setx2←1, x1←0, y2←0, y1←1. 3. Whileb> 0do the following: 3.1 q←⌊a/b⌋, r←a−qb, x←x2 −qx1, y←y2 −qy1. 3.2 a←b, b←r, x2←x1, x1←x, y2←y1, andy1←y. 4. Setd←a, x←x2, y←y2, and return(d,x,y). 2.108 FactAlgorithm 2.107 has a running time ofO((lgn)2)bit operations. 2.109 Example (extended Euclidean algorithm) Table 2.2 shows the steps of Algorithm 2.107 with inputsa=4864 and b=3458. Hencegcd(4864,3458)=38 and (4864)(32)+ (3458)(−45)=38 . □ q r x y a b x2 x1 y2 y1 − − − − 4864 3458 1 0 0 1 1 1406 1 −1 3458 1406 0 1 1 −1 2 646 −2 3 1406 646 1 −2 −1 3 2 114 5 −7 646 114 −2 5 3 −7 5 76 −27 38 114 76 5 −27 −7 38 1 38 32 −45 76 38 −27 32 38 −45 2 0 −91 128 38 0 32 −91 −45 128 Table 2.2:Extended Euclidean algorithm (Algorithm 2.107) with inputsa = 4864,b = 3458. Efﬁcient algorithms for gcd and extended gcd computations are further studied in§14.4. 2.4.3 The integers modulon Letnbe a positive integer. 2.110 DeﬁnitionIfaand bare integers, thenais said to becongruent tobmodulo n, written a≡b(modn),i fndivides(a−b). The integernis called themodulusof the congruence. 2.111 Example (i)24≡9(mod5) since24−9=3 ·5. (ii)−11≡17(mod7) since−11−17= −4·7. □ 2.112 Fact(properties of congruences) For alla,a1,b,b1,c∈Z, the following are true. (i)a≡b(modn)if and only ifaand bleave the same remainder when divided byn. (ii) (reﬂexivity)a≡a(modn). (iii) (symmetry)I fa≡b(modn)thenb≡a(modn). 68 Ch. 2 Mathematical Background (iv) (transitivity)I fa≡b(modn)and b≡c(modn), thena≡c(modn). (v) Ifa ≡ a1 (modn)and b ≡ b1 (modn), thena+b≡ a1 +b1 (modn)and ab≡a1b1 (modn). The equivalence classof an integerais the set of all integers congruent toamodulo n. From properties (ii), (iii), and (iv) above, it can be seen that for a ﬁxednthe relation of congruence modulonpartitionsZ into equivalence classes. Now, ifa=qn+r, where 0≤r<n , thena≡r(modn). Hence each integerais congruent modulonto a unique integer between0and n−1, called theleast residueofamodulo n. Thusaand rare in the same equivalence class, and sormay simply be used to represent this equivalence class. 2.113 DeﬁnitionThe integers modulon, denotedZn, is the set of (equivalence classes of) in- tegers{0,1,2,...,n −1}. Addition, subtraction, and multiplication inZnare performed modulo n. 2.114 Example Z25 = {0,1,2,..., 24}.I nZ25,13+16 = 4, since13+16 = 29≡ 4 (mod25). Similarly,13·16=8 inZ25. □ 2.115 DeﬁnitionLeta∈Zn. Themultiplicative inverseofamodulo nis an integerx∈Zn such thatax≡1(mod n). If such anxexists, then it is unique, andais said to beinvert- ible,o raunit; the inverse ofais denoted bya−1. 2.116 DeﬁnitionLeta,b∈Zn.Divisionofaby bmodulo nis the product ofaandb−1 modulo n, and is only deﬁned ifbis invertible modulon. 2.117 FactLeta∈Zn. Thenais invertible if and only ifgcd(a,n)=1 . 2.118 Example The invertible elements inZ9 are1,2,4,5,7, and8. For example,4−1 =7 because4·7≡1(mod9) . □ The following is a generalization of Fact 2.117. 2.119 FactLetd=gcd(a,n). The congruence equationax≡b(modn)has a solutionxif and only ifddividesb, in which case there are exactlydsolutions between0and n−1; these solutions are all congruent modulon/d. 2.120 Fact(Chinese remainder theorem, CRT) If the integersn1,n2,...,n kare pairwise rela- tively prime, then the system of simultaneous congruences x ≡ a1 (modn1) x ≡ a2 (modn2) ... x ≡ ak (modnk) has a unique solution modulon=n1n2 ···nk. 2.121 Algorithm(Gauss’ s algorithm) The solutionxto the simultaneous congruences in the Chinese remainder theorem (Fact 2.120) may be computed asx=∑ k i=1 aiNiMimodn, where Ni = n/ni and Mi = N−1 i modni. These computations can be performed in O((lgn)2)bit operations. §2.4 Number theory 69 Another efﬁcient practical algorithm for solving simultaneous congruences in the Chinese remainder theorem is presented in§14.5. 2.122 Example The pair of congruencesx≡3(mod7) ,x≡7(mod13) has a unique solu- tionx≡59(mod91) . □ 2.123 FactIfgcd(n1,n2)=1 , then the pair of congruencesx≡a(modn1),x≡a(modn2) has a unique solutionx≡a(modn1n2). 2.124 DeﬁnitionThe multiplicative groupof Zn isZ∗ n = {a ∈ Zn \| gcd(a,n)=1 }.In particular, ifnis a prime, thenZ∗ n ={a\|1≤a≤n−1}. 2.125 DeﬁnitionThe orderofZ∗ n is deﬁned to be the number of elements inZ∗ n , namely\|Z∗ n \|. It follows from the deﬁnition of the Euler phi function (Deﬁnition 2.100) that\|Z∗ n \|= φ(n). Note also that ifa∈Z∗ n and b∈Z∗ n , thena·b∈Z∗ n , and soZ∗ n is closed under multiplication. 2.126 FactLetn≥2be an integer. (i) (Euler’ s theorem)I fa∈Z∗ n , thenaφ(n) ≡1(mod n). (ii) Ifnis a product of distinct primes, and ifr≡s(modφ(n)), thenar≡as (modn) for all integersa. In other words, when working modulo such ann, exponents can be reduced moduloφ(n). A special case of Euler’s theorem is Fermat’s (little) theorem. 2.127 FactLetpbe a prime. (i) (Fermat’ s theorem)I fgcd(a,p)=1 , thenap−1 ≡1(mod p). (ii) Ifr≡s(modp−1), thenar ≡as (modp)for all integersa. In other words, when working modulo a primep, exponents can be reduced modulop−1. (iii) In particular,ap≡a(modp)for all integersa. 2.128 DeﬁnitionLeta∈Z∗ n . Theorderofa, denotedord(a), is the least positive integertsuch thatat≡1(mod n). 2.129 FactIf the order ofa∈Z∗ n ist, andas ≡1(mod n), thentdividess. In particular, t\|φ(n). 2.130 Example Let n=2 1. ThenZ∗ 21 ={1,2,4,5,8,10,11,13,16,17,19,20}. Note that φ(21)= φ(7)φ(3)=12= \|Z∗ 21\|. The orders of elements inZ∗ 21 are listed in Table 2.3.□ a∈Z∗ 21 1 2 4 5 8 10 11 13 16 17 19 20 order ofa 1 6 3 6 2 6 6 2 3 6 6 2 Table 2.3:Orders of elements inZ∗ 21. 2.131 DeﬁnitionLetα∈Z∗ n. If the order ofαisφ(n), thenαis said to be ageneratoror a primitive elementofZ∗ n .I fZ∗ n has a generator, thenZ∗ n is said to becyclic. 70 Ch. 2 Mathematical Background 2.132 Fact(properties of generators ofZ∗ n) (i)Z∗ nhas a generator if and only ifn=2,4,pkor2pk, wherepis an odd prime and k≥1. In particular, ifpis a prime, thenZ∗ p has a generator. (ii) Ifαis a generator ofZ∗ n , thenZ∗ n ={αimodn\|0≤i≤φ(n)−1}. (iii) Suppose thatαis a generator ofZ∗ n . Thenb=αimodnis also a generator ofZ∗ n if and only ifgcd(i,φ(n))=1 . It follows that ifZ∗ n is cyclic, then the number of generators isφ(φ(n)). (iv)α∈Z∗ n is a generator ofZ∗ n if and only ifαφ(n)/p ̸≡1(mod n)for each prime divisorpofφ(n). 2.133 Example Z∗ 21 is not cyclic since it does not contain an element of orderφ(21)=12 (see Table 2.3); note that21does not satisfy the condition of Fact 2.132(i). On the other hand, Z∗ 25 is cyclic, and has a generatorα=2. □ 2.134 DeﬁnitionLeta∈Z∗ n.ais said to be aquadratic residuemodulo n,o rasquaremodulo n, if there exists anx∈Z∗ n such thatx2 ≡a(modn). If no suchxexists, thenais called a quadratic non-residuemodulo n. The set of all quadratic residues modulonis denoted by Qnand the set of all quadratic non-residues is denoted by Qn. Note that by deﬁnition0̸∈Z∗ n , whence0̸∈Qnand 0̸∈ Qn. 2.135 FactLetpbe an odd prime and letαbe a generator ofZ∗ p . Thena∈Z∗ p is a quadratic residue modulopif and only ifa=αimodp, whereiis an even integer. It follows that \|Qp\|=(p−1)/2and \| Qp\|=(p−1)/2; that is, half of the elements inZ∗ p are quadratic residues and the other half are quadratic non-residues. 2.136 Example α=6 is a generator ofZ∗ 13 . The powers ofαare listed in the following table. i 0 1 2 3 4 5 6 7 8 9 10 11 αimod13 1 6 10 8 9 2 12 7 3 5 4 11 Hence Q13 ={1,3,4,9,10,12}and Q13 ={2,5,6,7,8,11}. □ 2.137 FactLetnbe a product of two distinct odd primespand q,n=pq. Thena∈Z∗ n is a quadratic residue modulonif and only ifa∈Qp and a∈Qq. It follows that\|Qn\|= \|Qp\|·\|Qq\|=(p−1)(q−1)/4and \| Qn\|=3(p−1)(q−1)/4. 2.138 Example Letn=21. ThenQ21 ={1,4,16}and Q21 ={2,5,8,10,11,13,17,19,20}. □ 2.139 DeﬁnitionLeta∈Qn.I fx∈Z∗ n satisﬁesx2 ≡a(modn), thenxis called asquare rootofamodulo n. 2.140 Fact(number of square roots) (i) Ifpis an odd prime anda∈Qp, thenahas exactly two square roots modulop. (ii) More generally, letn=pe1 1 pe2 2 ···pek k where thepiare distinct odd primes andei≥ 1.I fa∈Qn, thenahas precisely2kdistinct square roots modulon. 2.141 Example The square roots of12modulo 37are7and 30. The square roots of121modulo 315are11,74,101,151,164,214,241, and304. □ §2.4 Number theory 71 2.4.4 Algorithms inZn Letnbe a positive integer. As before, the elements ofZnwill be represented by the integers {0,1,2,...,n −1}. Observe that ifa,b∈Zn, then (a+b)mod n= { a+b, ifa+b<n, a+b−n, ifa+b≥n. Hence modular addition (and subtraction) can be performed without the need of a long di- vision. Modular multiplication ofaand bmay be accomplished by simply multiplyinga and bas integers, and then taking the remainder of the result after division byn. Inverses inZncan be computed using the extended Euclidean algorithm as next described. 2.142 AlgorithmComputing multiplicative inverses inZn INPUT: a∈Zn. OUTPUT: a−1 modn, provided that it exists. 1. Use the extended Euclidean algorithm (Algorithm 2.107) to ﬁnd integersxandysuch thatax+ny=d, whered=gcd(a,n). 2. Ifd> 1, thena−1 modndoes not exist. Otherwise, return(x). Modular exponentiation can be performed efﬁciently with the repeated square-and- multiply algorithm (Algorithm 2.143), which is crucial for many cryptographic protocols. One version of this algorithm is based on the following observation. Let the binary repre- sentation ofkbe ∑ t i=0 ki2i, where eachki∈{0,1}. Then ak= t∏ i=0 aki2i =(a20 )k0 (a21 )k1 ···(a2t )kt . 2.143 AlgorithmRepeated square-and-multiply algorithm for exponentiation inZn INPUT: a∈Zn, and integer0≤k<n whose binary representation isk=∑ t i=0 ki2i. OUTPUT: akmodn. 1. Setb←1.I fk=0 then return(b). 2. SetA←a. 3. Ifk0 =1 then setb←a. 4. Forifrom 1 totdo the following: 4.1 SetA←A2 modn. 4.2 Ifki=1 then setb←A·bmodn. 5. Return(b). 2.144 Example (modular exponentiation) Table 2.4 shows the steps involved in the computation of5596 mod1234=1013 . □ The number of bit operations for the basic operations inZnis summarized in Table 2.5. Efﬁcient algorithms for performing modular multiplication and exponentiation are further examined in§14.3 and§14.6. 72 Ch. 2 Mathematical Background i 0 1 2 3 4 5 6 7 8 9 ki 0 0 1 0 1 0 1 0 0 1 A 5 25 625 681 1011 369 421 779 947 925 b 1 1 625 625 67 67 1059 1059 1059 1013 Table 2.4:Computation of5596 mod 1234. Operation Bit complexity Modular addition (a+b)mod n O(lgn) Modular subtraction (a−b)mod n O(lgn) Modular multiplication (a·b)mod n O((lgn)2) Modular inversion a−1 modn O((lgn)2) Modular exponentiationakmodn,k<n O((lgn)3) Table 2.5:Bit complexity of basic operations inZn. 2.4.5 The Legendre and Jacobi symbols The Legendre symbol is a useful tool for keeping track of whether or not an integerais a quadratic residue modulo a primep. 2.145 DeﬁnitionLetpbe an odd prime andaan integer. TheLegendre symbol (a p ) is deﬁned to be (a p ) =    0, ifp\|a, 1, ifa∈Qp, −1, ifa∈ Qp. 2.146 Fact(properties of Legendre symbol) Letpbe an odd prime anda,b∈Z. Then the Leg- endre symbol has the following properties: (i) (a p ) ≡a(p−1)/2 (modp). In particular, (1 p ) =1 and (−1 p ) =(−1)(p−1)/2. Hence −1∈Qpifp≡1(mod4) , and−1∈ Qpifp≡3(mod4) . (ii) (ab p ) = (a p )(b p ) . Hence ifa∈Z∗ p, then (a2 p ) =1. (iii) Ifa≡b(modp), then (a p ) = (b p ) . (iv) (2 p ) =(−1)(p2−1)/8. Hence (2 p ) =1 ifp≡1or7(mod8) , and (2 p ) =−1ifp≡3 or5(mod8) . (v) (law of quadratic reciprocity)I fqis an odd prime distinct fromp, then (p q ) = (q p ) (−1)(p−1)(q−1)/4. In other words, (p q ) = (q p ) unless bothpand qare congruent to3modulo 4, in which case (p q ) =− (q p ) . The Jacobi symbol is a generalization of the Legendre symbol to integersnwhich are odd but not necessarily prime. §2.4 Number theory 73 2.147 DeﬁnitionLetn≥3be odd with prime factorizationn=pe1 1 pe2 2 ···pek k . Then theJacobi symbol (a n ) is deﬁned to be (a n ) = ( a p1 )e1 ( a p2 )e2 ··· ( a pk )ek . Observe that ifnis prime, then the Jacobi symbol is just the Legendre symbol. 2.148 Fact(properties of Jacobi symbol) Letm≥3,n≥3be odd integers, anda,b∈Z. Then the Jacobi symbol has the following properties: (i) (a n ) =0,1,or −1. Moreover, (a n ) =0 if and only ifgcd(a,n)̸=1. (ii) (ab n ) = (a n )(b n ) . Hence ifa∈Z∗ n, then (a2 n ) =1. (iii) (a mn ) = (a m )(a n ) . (iv) Ifa≡b(modn), then (a n ) = (b n ) . (v) (1 n ) =1. (vi) (−1 n ) =(−1)(n−1)/2. Hence (−1 n ) =1 ifn≡1(mod4) , and (−1 n ) =−1ifn≡3 (mod4). (vii) (2 n ) =(−1)(n2−1)/8. Hence (2 n ) =1 ifn≡1or7(mod8) , and (2 n ) =−1if n≡3or5(mod8) . (viii) (m n ) = (n m ) (−1)(m−1)(n−1)/4. In other words, (m n ) = (n m ) unless bothmand nare congruent to3modulo 4, in which case (m n ) =− (n m ) . By properties of the Jacobi symbol it follows that ifnis odd anda=2ea1 where a1 is odd, then (a n ) = (2e n )(a1 n ) = (2 n )e(nmoda1 a1 ) (−1)(a1−1)(n−1)/4. This observation yields the following recursive algorithm for computing (a n ) , which does not require the prime factorization ofn. 2.149 AlgorithmJacobi symbol (and Legendre symbol) computation JACOBI( a,n) INPUT: an odd integern≥3, and an integera,0≤a<n . OUTPUT: the Jacobi symbol (a n ) (and hence the Legendre symbol whennis prime). 1. Ifa=0 then return(0). 2. Ifa=1 then return(1). 3. Writea=2ea1, wherea1 is odd. 4. Ifeis even then sets←1. Otherwise sets←1ifn≡1or7(mod8) , or sets←−1 ifn≡3or5(mod8) . 5. Ifn≡3(mod4) and a1 ≡3(mod4) then sets←−s. 6. Setn1←nmoda1. 7. Ifa1 =1 then return(s); otherwise return(s·JACOBI( n1,a1)). 2.150 FactAlgorithm 2.149 has a running time ofO((lgn)2)bit operations. 74 Ch. 2 Mathematical Background 2.151 Remark (ﬁnding quadratic non-residues modulo a primep) Letpdenote an odd prime. Even though it is known that half of the elements inZ∗ pare quadratic non-residues modulo p(see Fact 2.135), there is nodeterministicpolynomial-time algorithm known for ﬁnding one. Arandomizedalgorithm for ﬁnding a quadratic non-residue is to simply select random integersa∈Z∗ p until one is found satisfying (a p ) =−1. The expected number iterations before a non-residue is found is2, and hence the procedure takes expected polynomial-time. 2.152 Example (Jacobi symbol computation) Fora=158and n=235, Algorithm 2.149 com- putes the Jacobi symbol (158 235 ) as follows: (158 235 ) = ( 2 235 )( 79 235 ) =(−1) (235 79 ) (−1)78·234/4 = (77 79 ) = (79 77 ) (−1)76·78/4 = ( 2 77 ) =−1. □ Unlike the Legendre symbol, the Jacobi symbol (a n ) does not reveal whether or nota is a quadratic residue modulon. It is indeed true that ifa∈Qn, then (a n ) =1. However,(a n ) =1 does not imply thata∈Qn. 2.153 Example (quadratic residues and non-residues) Table 2.6 lists the elements inZ∗ 21 and their Jacobi symbols. Recall from Example 2.138 thatQ21 = {1,4,16}. Observe that(5 21 ) =1 but5̸∈Q21. □ a ∈Z∗ 21 1 2 4 5 8 10 11 13 16 17 19 20 a2 mod n 1 4 16 4 1 16 16 1 4 16 4 1 (a 3 ) 1 −1 1 −1 −1 1 −1 1 1 −1 1 −1 (a 7 ) 1 1 1 −1 1 −1 1 −1 1 −1 −1 −1 ( a 21 ) 1 −1 1 1 −1 −1 −1 −1 1 1 −1 1 Table 2.6:Jacobi symbols of elements inZ∗ 21. 2.154 DeﬁnitionLetn≥3be an odd integer, and letJn ={a∈Z∗ n \| (a n ) =1}. The set of pseudosquaresmodulo n, denoted˜Qn, is deﬁned to be the setJn−Qn. 2.155 FactLet n=pqbe a product of two distinct odd primes. Then\|Qn\|=\|˜Qn\|=(p− 1)(q−1)/4; that is, half of the elements inJnare quadratic residues and the other half are pseudosquares. 2.4.6 Blum integers 2.156 DeﬁnitionA Blum integeris a composite integer of the formn=pq, wherepand qare distinct primes each congruent to3modulo 4. 2.157 FactLet n=pqbe a Blum integer, and leta∈Qn. Thenahas precisely four square roots modulon, exactly one of which is also inQn. 2.158 DeﬁnitionLetnbe a Blum integer and leta∈Qn. The unique square root ofainQnis called theprincipal square rootofamodulo n. §2.5 Abstract algebra 75 2.159 Example (Blum integer) For the Blum integern=2 1,Jn ={1,4,5,16,17,20}and ˜Qn={5,17,20}. The four square roots ofa=4 are2,5,16, and19, of which only16is also inQ21. Thus16is the principal square root of4modulo 21. □ 2.160 FactIfn=pqis a Blum integer, then the functionf:Qn −→Qndeﬁned byf(x)= x2 modnis a permutation. The inverse function offis: f−1(x)= x((p−1)(q−1)+4)/8 modn. 2.5 Abstract algebra This section provides an overview of basic algebraic objects and their properties, for refer- ence in the remainder of this handbook. Several of the deﬁnitions in§2.5.1 and§2.5.2 were presented earlier in§2.4.3 in the more concrete setting of the algebraic structureZ∗ n. 2.161 DeﬁnitionA binary operation∗on a setSis a mapping fromS×StoS. That is,∗is a rule which assigns to each ordered pair of elements fromSan element ofS. 2.5.1 Groups 2.162 DeﬁnitionA group(G,∗)consists of a setGwith a binary operation∗on Gsatisfying the following three axioms. (i) The group operation isassociative. That is,a∗(b∗c)=( a∗b)∗cfor alla,b,c ∈G. (ii) There is an element1∈G, called theidentity element, such thata∗1=1 ∗a=a for alla∈G. (iii) For eacha∈Gthere exists an elementa−1 ∈G, called theinverseofa, such that a∗a−1 =a−1 ∗a=1. A groupGisabelian(orcommutative) if, furthermore, (iv)a∗b=b∗afor alla,b∈G. Note that multiplicative group notation has been used for the group operation. If the group operation is addition, then the group is said to be anadditivegroup, the identity ele- ment is denoted by0, and the inverse ofais denoted−a. Henceforth, unless otherwise stated, the symbol∗will be omitted and the group oper- ation will simply be denoted by juxtaposition. 2.163 DeﬁnitionA groupGisﬁniteif\|G\|is ﬁnite. The number of elements in a ﬁnite group is called itsorder. 2.164 Example The set of integersZ with the operation of addition forms a group. The identity element is0and the inverse of an integerais the integer−a. □ 2.165 Example The setZn, with the operation of addition modulon, forms a group of order n. The setZnwith the operation of multiplication modulonis not a group, since not all elements have multiplicative inverses. However, the setZ∗ n (see Deﬁnition 2.124) is a group of orderφ(n)under the operation of multiplication modulon, with identity element1. □ 76 Ch. 2 Mathematical Background 2.166 DeﬁnitionA non-empty subsetHof a groupGis asubgroupofGifHis itself a group with respect to the operation ofG.I fHis a subgroup ofGand H̸=G, thenHis called a propersubgroup ofG. 2.167 DeﬁnitionA groupGiscyclicif there is an elementα∈Gsuch that for eachb∈Gthere is an integeriwithb=αi. Such an elementαis called ageneratorofG. 2.168 FactIfGis a group anda∈G, then the set of all powers ofaforms a cyclic subgroup of G, called the subgroupgenerated bya, and denoted by⟨a⟩. 2.169 DeﬁnitionLetGbe a group anda∈G. Theorderofais deﬁned to be the least positive integertsuch thatat =1, provided that such an integer exists. If such atdoes not exist, then the order ofais deﬁned to be∞. 2.170 FactLetGbe a group, and leta∈Gbe an element of ﬁnite ordert. Then\|⟨a⟩\|, the size of the subgroup generated bya, is equal tot. 2.171 Fact(Lagrange’ s theorem)I fGis a ﬁnite group andHis a subgroup ofG, then\|H\|divides \|G\|. Hence, ifa∈G, the order ofadivides\|G\|. 2.172 FactEvery subgroup of a cyclic groupGis also cyclic. In fact, ifGis a cyclic group of ordern, then for each positive divisordofn,Gcontains exactly one subgroup of orderd. 2.173 FactLetGbe a group. (i) If the order ofa∈Gist, then the order ofakist/gcd(t,k). (ii) IfGis a cyclic group of ordernand d\|n, thenGhas exactlyφ(d)elements of order d. In particular,Ghasφ(n)generators. 2.174 Example Consider the multiplicative groupZ∗ 19 ={1,2,..., 18}of order18. The group is cyclic (Fact 2.132(i)), and a generator isα=2. The subgroups ofZ∗ 19 , and their gener- ators, are listed in Table 2.7. □ Subgroup Generators Order {1} 1 1 {1, 18} 18 2 {1, 7, 11} 7, 11 3 {1, 7, 8, 11, 12, 18} 8, 12 6 {1, 4, 5, 6, 7, 9, 11, 16, 17} 4, 5, 6, 9, 16, 17 9 {1, 2, 3,... , 18} 2, 3, 10, 13, 14, 15 18 Table 2.7:The subgroups ofZ∗ 19. 2.5.2 Rings 2.175 DeﬁnitionA ring(R,+,×)consists of a setRwith two binary operations arbitrarily de- noted+(addition) and×(multiplication) onR, satisfying the following axioms. (i)(R,+)is an abelian group with identity denoted0. §2.5 Abstract algebra 77 (ii) The operation×is associative. That is,a×(b×c)=( a×b)×cfor alla,b,c ∈R. (iii) There is a multiplicative identity denoted1, with1̸=0, such that1×a=a×1= a for alla∈R. (iv) The operation×isdistributiveover+. That is,a×(b+c)=( a×b)+(a×c)and (b+c)×a=(b×a)+(c×a)for alla,b,c ∈R. The ring is acommutative ringifa×b=b×afor alla,b∈R. 2.176 Example The set of integersZ with the usual operations of addition and multiplication is a commutative ring. □ 2.177 Example The setZnwith addition and multiplication performed modulonis a commu- tative ring. □ 2.178 DeﬁnitionAn elementaof a ringRis called aunitor aninvertible elementif there is an elementb∈Rsuch thata×b=1. 2.179 FactThe set of units in a ringRforms a group under multiplication, called thegroup of unitsofR. 2.180 Example The group of units of the ringZnisZ∗ n(see Deﬁnition 2.124). □ 2.5.3 Fields 2.181 DeﬁnitionA ﬁeldis a commutative ring in which all non-zero elements have multiplica- tive inverses. 2.182 DeﬁnitionThe characteristicof a ﬁeld is0if mtimes    1+1+ ··· +1is never equal to0for any m≥1. Otherwise, the characteristic of the ﬁeld is the least positive integermsuch that∑ m i=1 1equals0. 2.183 Example The set of integers under the usual operations of addition and multiplication is not a ﬁeld, since the only non-zero integers with multiplicative inverses are1and−1. How- ever, the rational numbersQ, the real numbersR, and the complex numbersC form ﬁelds of characteristic0under the usual operations. □ 2.184 FactZnis a ﬁeld (under the usual operations of addition and multiplication modulon)i f and only ifnis a prime number. Ifnis prime, thenZnhas characteristicn. 2.185 FactIf the characteristicmof a ﬁeld is not0, thenmis a prime number. 2.186 DeﬁnitionA subsetFof a ﬁeldEis asubﬁeldofEifFis itself a ﬁeld with respect to the operations ofE. If this is the case,Eis said to be anextension ﬁeldofF. 78 Ch. 2 Mathematical Background 2.5.4 Polynomial rings 2.187 DeﬁnitionIfRis a commutative ring, then apolynomialin the indeterminatexover the ringRis an expression of the form f(x)= anxn+··· +a2x2 +a1x+a0 where eachai ∈ Rand n ≥ 0. The elementai is called thecoefﬁcientof xi inf(x). The largest integermfor whicham ̸=0is called thedegreeoff(x), denoteddegf(x); am is called theleading coefﬁcientof f(x).I ff(x)= a0 (aconstant polynomial) and a0 ̸=0, thenf(x)has degree0. If all the coefﬁcients off(x)are0, thenf(x)is called the zero polynomialand its degree, for mathematical convenience, is deﬁned to be−∞. The polynomialf(x)is said to bemonic if its leading coefﬁcient is equal to1. 2.188 DeﬁnitionIfRis a commutative ring, thepolynomial ringR[x]is the ring formed by the set of all polynomials in the indeterminatexhaving coefﬁcients fromR. The two opera- tions are the standard polynomial addition and multiplication, with coefﬁcient arithmetic performed in the ringR. 2.189 Example (polynomial ring) Letf(x)= x3 +x+1 and g(x)= x2 +xbe elements of the polynomial ringZ2[x]. Working inZ2[x], f(x)+g(x)= x3 +x2 +1 and f(x)·g(x)= x5 +x4 +x3 +x. □ For the remainder of this section,Fwill denote an arbitrary ﬁeld. The polynomial ring F[x]has many properties in common with the integers (more precisely,F[x]andZ are both Euclidean domains, however, this generalization will not be pursued here). These similar- ities are investigated further. 2.190 DeﬁnitionLetf(x)∈F[x]be a polynomial of degree at least1. Thenf(x)is said to be irreducible overFif it cannot be written as the product of two polynomials inF[x], each of positive degree. 2.191 Deﬁnition(division algorithm for polynomials)I fg(x),h(x)∈F[x], withh(x)̸=0, then ordinary polynomial long division ofg(x)by h(x)yields polynomialsq(x)andr(x)∈ F[x]such that g(x)= q(x)h(x)+r(x),where degr(x)<degh(x). Moreover,q(x)and r(x)are unique. The polynomialq(x)is called thequotient, while r(x)is called theremainder. The remainder of the division is sometimes denotedg(x)mod h(x), and the quotient is sometimes denotedg(x)div h(x)(cf. Deﬁnition 2.82). 2.192 Example (polynomial division) Consider the polynomialsg(x)= x6+x5+x3+x2+x+1 and h(x)= x4 +x3 +1inZ2[x]. Polynomial long division ofg(x)by h(x)yields g(x)= x2h(x)+(x3 +x+1). Hence g(x)mod h(x)= x3 +x+1and g(x)div h(x)= x2. □ §2.5 Abstract algebra 79 2.193 DeﬁnitionIfg(x),h(x)∈F[x]thenh(x)dividesg(x), writtenh(x)\|g(x),i fg(x)mod h(x)=0 . Letf(x)be a ﬁxed polynomial inF[x]. As with the integers (Deﬁnition 2.110), one can deﬁne congruences of polynomials inF[x]based on division byf(x). 2.194 DeﬁnitionIfg(x),h(x)∈F[x], theng(x)is said to becongruent toh(x)modulo f(x) iff(x)dividesg(x)−h(x). This is denoted byg(x)≡h(x)(mod f(x)). 2.195 Fact(properties of congruences) For allg(x),h(x),g1(x),h1(x),s(x)∈F[x], the fol- lowing are true. (i)g(x)≡h(x)(mod f(x))if and only ifg(x)and h(x)leave the same remainder upon division byf(x). (ii) (reﬂexivity)g(x)≡g(x)(mod f(x)). (iii) (symmetry)I fg(x)≡h(x)(mod f(x)), thenh(x)≡g(x)(mod f(x)). (iv) (transitivity)I fg(x)≡h(x)(mod f(x))and h(x)≡s(x)(mod f(x)), then g(x)≡s(x)(mod f(x)). (v) Ifg(x)≡g1(x)(mod f(x))and h(x)≡h1(x)(mod f(x)), theng(x)+h(x)≡ g1(x)+h1(x)(mod f(x))and g(x)h(x)≡g1(x)h1(x)(mod f(x)). Letf(x)be a ﬁxed polynomial inF[x]. Theequivalence classof a polynomialg(x)∈ F[x]is the set of all polynomials inF[x]congruent tog(x)modulo f(x). From properties (ii), (iii), and (iv) above, it can be seen that the relation of congruence modulof(x)par- titionsF[x]into equivalence classes. Ifg(x)∈F[x], then long division byf(x)yields unique polynomialsq(x),r(x)∈F[x]such thatg(x)= q(x)f(x)+r(x), wheredegr(x) <degf(x). Hence every polynomialg(x)is congruent modulof(x)to a unique polyno- mial of degree less thandegf(x). The polynomialr(x)will be used as representative of the equivalence class of polynomials containingg(x). 2.196 DeﬁnitionF[x]/(f(x))denotes the set of (equivalence classes of) polynomials inF[x] of degree less thann=degf(x). Addition and multiplication are performed modulof(x). 2.197 FactF[x]/(f(x))is a commutative ring. 2.198 FactIff(x)is irreducible overF, thenF[x]/(f(x))is a ﬁeld. 2.5.5 Vector spaces 2.199 DeﬁnitionA vector spaceV over a ﬁeldFis an abelian group(V,+), together with a multiplication operation•:F×V−→V(usually denoted by juxtaposition) such that for alla,b∈Fand v,w ∈V, the following axioms are satisﬁed. (i)a(v+w)= av+aw. (ii)(a+b)v=av+bv. (iii)(ab)v=a(bv). (iv)1v=v. The elements ofVare calledvectors, while the elements ofFare calledscalars. The group operation+is calledvector addition, while the multiplication operation is calledscalar multiplication. 80 Ch. 2 Mathematical Background 2.200 DeﬁnitionLetVbe a vector space over a ﬁeldF.A subspaceofVis an additive subgroup UofVwhich is closed under scalar multiplication, i.e.,av∈Ufor alla∈Fand v∈U. 2.201 FactA subspace of a vector space is also a vector space. 2.202 DeﬁnitionLetS={v1,v2,...,v n}be a ﬁnite subset of a vector spaceVover a ﬁeldF. (i) Alinear combinationofSis an expression of the forma1v1 +a2v2 +··· +anvn, where eachai∈F. (ii) ThespanofS, denoted⟨S⟩, is the set of all linear combinations ofS. The span ofS is a subspace ofV. (iii) IfUis a subspace ofV, thenSis said tospanUif⟨S⟩=U. (iv) The setSislinearly dependentoverFif there exist scalarsa1,a2,...,a n, not all zero, such thata1v1 +a2v2 +··· +anvn =0. If no such scalars exist, thenSis linearly independentoverF. (v) A linearly independent set of vectors that spansVis called abasisforV. 2.203 FactLetVbe a vector space. (i) IfVhas a ﬁnite spanning set, then it has a basis. (ii) IfVhas a basis, then in fact all bases have the same number of elements. 2.204 DeﬁnitionIf a vector spaceVhas a basis, then the number of elements in a basis is called thedimensionofV, denoteddimV. 2.205 Example IfFis any ﬁeld, then then-fold Cartesian productV=F×F×···× Fis a vector space overFof dimensionn. Thestandard basisforVis{e1,e2,...,e n}, where eiis a vector with a1in theith coordinate and0’s elsewhere. □ 2.206 DeﬁnitionLet Ebe an extension ﬁeld ofF. Then Ecan be viewed as a vector space over the subﬁeldF, where vector addition and scalar multiplication are simply the ﬁeld operations of addition and multiplication inE. The dimension of this vector space is called thedegreeofEoverF, and denoted by[E:F]. If this degree is ﬁnite, thenEis called a ﬁnite extensionofF. 2.207 FactLetF,E, andLbe ﬁelds. IfLis a ﬁnite extension ofEand Eis a ﬁnite extension ofF, thenLis also a ﬁnite extension ofFand [L:F]=[ L:E][E:F]. 2.6 Finite ﬁelds 2.6.1 Basic properties 2.208 DeﬁnitionA ﬁnite ﬁeldis a ﬁeldFwhich contains a ﬁnite number of elements. Theorder ofFis the number of elements inF. §2.6 Finite ﬁelds 81 2.209 Fact(existence and uniqueness of ﬁnite ﬁelds) (i) IfFis a ﬁnite ﬁeld, thenFcontainspmelements for some primepand integerm≥1. (ii) For every prime power orderpm, there is a unique (up to isomorphism) ﬁnite ﬁeld of orderpm. This ﬁeld is denoted byFpm , or sometimes byGF(pm). Informally speaking, two ﬁelds areisomorphicif they are structurally the same, al- though the representation of their ﬁeld elements may be different. Note that ifpis a prime thenZpis a ﬁeld, and hence every ﬁeld of orderpis isomorphic toZp. Unless otherwise stated, the ﬁnite ﬁeldFpwill henceforth be identiﬁed withZp. 2.210 FactIfFq is a ﬁnite ﬁeld of orderq=pm,pa prime, then the characteristic ofFq isp. Moreover,Fqcontains a copy ofZpas a subﬁeld. HenceFqcan be viewed as an extension ﬁeld ofZpof degreem. 2.211 Fact(subﬁelds of a ﬁnite ﬁeld) LetFqbe a ﬁnite ﬁeld of orderq=pm. Then every subﬁeld ofFqhas orderpn, for somenthat is a positive divisor ofm. Conversely, ifnis a positive divisor ofm, then there is exactly one subﬁeld ofFqof orderpn; an elementa∈Fqis in the subﬁeldFpn if and only ifapn =a. 2.212 DeﬁnitionThe non-zero elements ofFqform a group under multiplication called themul- tiplicative groupofFq, denoted byF∗ q. 2.213 FactF∗ q is a cyclic group of orderq−1. Henceaq =afor alla∈Fq. 2.214 DeﬁnitionA generator of the cyclic groupF∗ q is called aprimitive elementorgenerator ofFq. 2.215 FactIfa,b∈Fq, a ﬁnite ﬁeld of characteristicp, then (a+b)pt =apt +bpt for allt≥0. 2.6.2 The Euclidean algorithm for polynomials LetZpbe the ﬁnite ﬁeld of orderp. The theory of greatest common divisors and the Eu- clidean algorithm for integers carries over in a straightforward manner to the polynomial ringZp[x](and more generally to the polynomial ringF[x], whereFis any ﬁeld). 2.216 DeﬁnitionLetg(x),h(x)∈Zp[x], where not both are0. Then thegreatest common divi- sorofg(x)and h(x), denotedgcd(g(x),h(x)), is the monic polynomial of greatest degree inZp[x]which divides bothg(x)and h(x). By deﬁnition,gcd(0,0)=0 . 2.217 FactZp[x]is aunique factorization domain. That is, every non-zero polynomialf(x)∈ Zp[x]has a factorization f(x)= af1(x)e1 f2(x)e2 ···fk(x)ek , where thefi(x)are distinct monic irreducible polynomials inZp[x], theeiare positive in- tegers, anda∈Zp. Furthermore, the factorization is unique up to rearrangement of factors. The following is the polynomial version of the Euclidean algorithm (cf. Algorithm 2.104). 82 Ch. 2 Mathematical Background 2.218 AlgorithmEuclidean algorithm forZp[x] INPUT: two polynomialsg(x),h(x)∈Zp[x]. OUTPUT: the greatest common divisor ofg(x)and h(x). 1. Whileh(x)̸=0do the following: 1.1 Setr(x)←g(x)mod h(x), g(x)←h(x), h(x)←r(x). 2. Return(g(x)). 2.219 DeﬁnitionA Zp-operationmeans either an addition, subtraction, multiplication, inver- sion, or division inZp. 2.220 FactSuppose thatdegg(x)≤manddegh(x)≤m. Then Algorithm 2.218 has a running time ofO(m2)Zp-operations, or equivalently,O(m2(lgp)2)bit operations. As with the case of the integers (cf. Algorithm 2.107), the Euclidean algorithm can be extended so that it also yields two polynomialss(x)and t(x)satisfying s(x)g(x)+t(x)h(x)=gcd( g(x),h(x)). 2.221 AlgorithmExtended Euclidean algorithm forZp[x] INPUT: two polynomialsg(x),h(x)∈Zp[x]. OUTPUT: d(x)=g c d (g(x),h(x))and polynomialss(x),t(x) ∈ Zp[x]which satisfy s(x)g(x)+t(x)h(x)= d(x). 1. Ifh(x)=0 then setd(x)←g(x),s(x)←1,t(x)←0, and return(d(x),s(x),t(x)). 2. Sets2(x)←1, s1(x)←0, t2(x)←0, t1(x)←1. 3. Whileh(x)̸=0do the following: 3.1 q(x)←g(x)div h(x), r(x)←g(x)−h(x)q(x). 3.2 s(x)←s2(x)−q(x)s1(x), t(x)←t2(x)−q(x)t1(x). 3.3 g(x)←h(x), h(x)←r(x). 3.4 s2(x)←s1(x), s1(x)←s(x), t2(x)←t1(x), andt1(x)←t(x). 4. Setd(x)←g(x), s(x)←s2(x), t(x)←t2(x). 5. Return(d(x),s(x),t(x)). 2.222 Fact(running time of Algorithm 2.221) (i) The polynomialss(x)and t(x)given by Algorithm 2.221 have small degree; that is, they satisfydegs(x)<degh(x)and degt(x)<degg(x). (ii) Suppose thatdegg(x)≤manddegh(x)≤m. Then Algorithm 2.221 has a running time ofO(m2)Zp-operations, or equivalently,O(m2(lgp)2)bit operations. 2.223 Example (extended Euclidean algorithm for polynomials) The following are the steps of Algorithm 2.221 with inputsg(x)= x10 +x9 +x8 +x6 +x5 +x4 +1 and h(x)= x9 +x6 +x5 +x3 +x2 +1inZ2[x]. Initialization s2(x)←1, s1(x)←0, t2(x)←0, t1(x)←1. §2.6 Finite ﬁelds 83 Iteration 1 q(x)←x+1, r(x)←x8 +x7 +x6 +x2 +x, s(x)←1, t(x)←x+1, g(x)←x9 +x6 +x5 +x3 +x2 +1, h(x)←x8 +x7 +x6 +x2 +1, s2(x)←0, s1(x)←1, t2(x)←1, t1(x)←x+1. Iteration 2 q(x)←x+1, r(x)←x5 +x2 +x+1, s(x)←x+1, t(x)←x2, g(x)←x8 +x7 +x6 +x2 +1, h(x)←x5 +x2 +x+1, s2(x)←1, s1(x)←x+1, t2(x)←x+1, t1(x)←x2. Iteration 3 q(x)←x3 +x2 +x+1, r(x)←x3 +x+1, s(x)←x4, t(x)←x5 +x4 +x3 +x2 +x+1, g(x)←x5 +x2 +x+1, h(x)←x3 +x+1, s2(x)←x+1, s1(x)←x4, t2(x)←x2, t1(x)←x5 +x4 +x3 +x2 +x+1. Iteration 4 q(x)←x2 +1, r(x)←0, s(x)←x6 +x4 +x+1, t(x)←x7 +x6 +x2 +x+1, g(x)←x3 +x+1, h(x)←0, s2(x)←x4, s1(x)←x6 +x4 +x+1, t2(x)←x5 +x4 +x3 +x2 +x+1, t1(x)←x7 +x6 +x2 +x+1. Hence gcd(g(x),h(x))= x3 +x+1and (x4)g(x)+(x5 +x4 +x3 +x2 +x+1)h(x)= x3 +x+1. □ 2.6.3 Arithmetic of polynomials A commonly used representation for the elements of a ﬁnite ﬁeldFq, whereq=pmand p is a prime, is apolynomial basis representation.I fm=1, thenFqis justZpand arithmetic is performed modulop. Since these operations have already been studied in Section 2.4.2, it is henceforth assumed thatm≥2. The representation is based on Fact 2.198. 2.224 FactLetf(x)∈Zp[x]be an irreducible polynomial of degreem. ThenZp[x]/(f(x))is a ﬁnite ﬁeld of orderpm. Addition and multiplication of polynomials is performed modulo f(x). The following fact assures that all ﬁnite ﬁelds can be represented in this manner. 2.225 FactFor eachm≥1, there exists a monic irreducible polynomial of degreemoverZp. Hence, every ﬁnite ﬁeld has a polynomial basis representation. An efﬁcient algorithm for ﬁnding irreducible polynomials over ﬁnite ﬁelds is presented in§4.5.1. Tables 4.6 and 4.7 list some irreducible polynomials over the ﬁnite ﬁeldZ2. Henceforth, the elements of the ﬁnite ﬁeldFpm will be represented by polynomials in Zp[x]of degree<m.I fg(x),h(x)∈Fpm, then addition is the usual addition of polyno- mials inZp[x]. The productg(x)h(x)can be formed by ﬁrst multiplyingg(x)and h(x)as polynomials by the ordinary method, and then taking the remainder after polynomial divi- sion byf(x). Multiplicative inverses inFpm can be computed by using the extended Eu- clidean algorithm for the polynomial ringZp[x]. 84 Ch. 2 Mathematical Background 2.226 AlgorithmComputing multiplicative inverses inFpm INPUT: a non-zero polynomialg(x)∈Fpm. (The elements of the ﬁeldFpm are represented asZp[x]/(f(x)), wheref(x)∈Zp[x]is an irreducible polynomial of degreemoverZp.) OUTPUT: g(x)−1 ∈Fpm. 1. Use the extended Euclidean algorithm for polynomials (Algorithm 2.221) to ﬁnd two polynomialss(x)and t(x)∈Zp[x]such thats(x)g(x)+t(x)f(x)=1 . 2. Return(s(x)). Exponentiation inFpm can be done efﬁciently by the repeated square-and-multiply al- gorithm (cf. Algorithm 2.143). 2.227 AlgorithmRepeated square-and-multiply algorithm for exponentiation inFpm INPUT: g(x) ∈ Fpm and an integer0 ≤ k<p m −1whose binary representation is k=∑ t i=0 ki2i. (The ﬁeldFpm is represented asZp[x]/(f(x)), wheref(x)∈Zp[x]is an irreducible polynomial of degreemoverZp.) OUTPUT: g(x)kmodf(x). 1. Sets(x)←1.I fk=0 then return(s(x)). 2. SetG(x)←g(x). 3. Ifk0 =1 then sets(x)←g(x). 4. Forifrom 1totdo the following: 4.1 SetG(x)←G(x)2 modf(x). 4.2 Ifki=1 then sets(x)←G(x)·s(x)mod f(x). 5. Return(s(x)). The number ofZp-operations for the basic operations inFpm is summarized in Ta- ble 2.8. Operation Number ofZp-operations Addition g(x)+h(x) O(m) Subtraction g(x)−h(x) O(m) Multiplication g(x)·h(x) O(m2) Inversion g(x)−1 O(m2) Exponentiation g(x)k,k<p m O((lgp)m3) Table 2.8:Complexity of basic operations inFpm. In some applications (cf.§4.5.3), it may be preferable to use a primitive polynomial to deﬁne a ﬁnite ﬁeld. 2.228 DeﬁnitionAn irreducible polynomialf(x) ∈ Zp[x]of degreemis called aprimitive polynomialifxis a generator ofF∗ pm , the multiplicative group of all the non-zero elements inFpm =Zp[x]/(f(x)). 2.229 FactThe irreducible polynomialf(x)∈Zp[x]of degreemis a primitive polynomial if and only iff(x)dividesxk−1fork=pm−1and for no smaller positive integerk. §2.7 Notes and further references 85 2.230 FactFor eachm≥1, there exists a monic primitive polynomial of degreemoverZp.I n fact, there are preciselyφ(pm−1)/msuch polynomials. 2.231 Example (the ﬁnite ﬁeldF24 of order16) It can be veriﬁed (Algorithm 4.69) that the poly- nomialf(x)= x4 +x+1is irreducible overZ2. Hence the ﬁnite ﬁeldF24 can be repre- sented as the set of all polynomials overF2 of degree less than4. That is, F24 = {a3x3 +a2x2 +a1x+a0 \|ai∈{0,1}}. For convenience, the polynomiala3x3 +a2x2 +a1x+a0 is represented by the vector (a3a2a1a0)of length4, and F24 = {(a3a2a1a0)\|ai∈{0,1}}. The following are some examples of ﬁeld arithmetic. (i) Field elements are simply added componentwise: for example,(1011)+(1001)= (0010). (ii) To multiply the ﬁeld elements(1101)and (1001), multiply them as polynomials and then take the remainder when this product is divided byf(x): (x3 +x2 +1)·(x3 +1) = x6 +x5 +x2 +1 ≡ x3 +x2 +x+1 (mod f(x)). Hence (1101)·(1001)=(1111) . (iii) The multiplicative identity ofF24 is(0001). (iv) The inverse of(1011)is(0101). To verify this, observe that (x3 +x+1)·(x2 +1) = x5 +x2 +x+1 ≡ 1( m o df(x)), whence (1011)·(0101)=(0001) . f(x)is a primitive polynomial, or, equivalently, the ﬁeld elementx=(0010) is a genera- tor ofF∗ 24 . This may be checked by verifying that all the non-zero elements inF24 can be obtained as a powers ofx. The computations are summarized in Table 2.9. □ A list of some primitive polynomials over ﬁnite ﬁelds of characteristic two is given in Table 4.8. 2.7 Notes and further references §2.1 A classic introduction to probability theory is the ﬁrst volume of the book by Feller [392]. The material on the birthday problem (§2.1.5) is summarized from Nishimura and Sibuya [931]. See also Girault, Cohen, and Campana [460]. The material on random mappings (§2.1.6) is summarized from the excellent article by Flajolet and Odlyzko [413]. §2.2 The concept of entropy was introduced in the seminal paper of Shannon [1120]. These ideas were then applied to develop a mathematical theory of secrecy systems by Shannon [1121]. Hellman [548] extended the Shannon theory approach to cryptography, and this work was further generalized by Beauchemin and Brassard [80]. For an introduction to information theory see the books by Welsh [1235] and Goldie and Pinch [464]. For more complete treat- ments, consult Blahut [144] and McEliece [829]. 86 Ch. 2 Mathematical Background i xi mod x4 + x +1 vector notation 0 1 (0001) 1 x (0010) 2 x2 (0100) 3 x3 (1000) 4 x +1 (0011) 5 x2 + x (0110) 6 x3 + x2 (1100) 7 x3 + x +1 (1011) 8 x2 +1 (0101) 9 x3 + x (1010) 10 x2 + x +1 (0111) 11 x3 + x2 + x (1110) 12 x3 + x2 + x +1 (1111) 13 x3 + x2 +1 (1101) 14 x3 +1 (1001) Table 2.9:The powers ofx modulo f(x)= x4 + x +1 . §2.3 Among the many introductory-level books on algorithms are those of Cormen, Leiserson, and Rivest [282], Rawlins [1030], and Sedgewick [1105]. A recent book on complexity theory is Papadimitriou [963]. Example 2.58 is from Graham, Knuth, and Patashnik [520, p.441]. For an extensive list ofNP -complete problems, see Garey and Johnson [441]. §2.4 Two introductory-level books in number theory are Giblin [449] and Rosen [1069]. Good number theory books at a more advanced level include Koblitz [697], Hardy and Wright [540], Ireland and Rosen [572], and Niven and Zuckerman [932]. The most comprehensive works on the design and analysis of algorithms, including number theoretic algorithms, are the ﬁrst two volumes of Knuth [691, 692]. Two more recent books exclusively devoted to this subject are Bach and Shallit [70] and Cohen [263]. Facts 2.96 and 2.102 are due to Rosser and Schoenfeld [1070]. Shallit [1108] describes and analyzes three algorithms for computing the Jacobi symbol. §2.5 Among standard references in abstract algebra are the books by Herstein [556] and Hunger- ford [565]. §2.6 An excellent introduction to ﬁnite ﬁelds is provided in McEliece [830]. An encyclopedic treatment of the theory and applications of ﬁnite ﬁelds is given by Lidl and Niederreitter [764]. Two books which discuss various methods of representing the elements of a ﬁnite ﬁeld are those of Jungnickel [646] and Menezes et al. [841]. Chapter /3 Number-Theoretic Reference Problems Contents in Brief 3.1 Introduction and overview ..................... 87 3.2 The integer factorization problem................. 89 3.3 The RSA problem .......................... 98 3.4 The quadratic residuosity problem................. 99 3.5 Computing square roots inZn ................... 99 3.6 The discrete logarithm problem .................. 103 3.7 The Difﬁe-Hellman problem .................... 113 3.8 Composite moduli .......................... 114 3.9 Computing individual bits ..................... 114 3.10 The subset sum problem ...................... 117 3.11 Factoring polynomials over ﬁnite ﬁelds............... 122 3.12 Notes and further references.................... 125 3.1 Introduction and overview The security of many public-key cryptosystems relies on the apparent intractability of the computational problems studied in this chapter. In a cryptographic setting, it is prudent to make the assumption that the adversary is very powerful. Thus, informally speaking, a com- putational problem is said to beeasyortractableif it can be solved in (expected)1 polyno- mial time, at least for a non-negligible fraction of all possible inputs. In other words, if there is an algorithm which can solve a non-negligible fraction of all instances of a problem in polynomial time, then any cryptosystem whose security is based on that problem must be considered insecure. The computational problems studied in this chapter are summarized in Table 3.1. The true computational complexities of these problems are not known. That is to say, they are widely believed to be intractable,2 although no proof of this is known. Generally, the only lower bounds known on the resources required to solve these problems are the trivial linear bounds, which do not provide any evidence of their intractability. It is, therefore, of inter- est to study their relative difﬁculties. For this reason, various techniques of reducing one 1For simplicity, the remainder of the chapter shall generally not distinguish between deterministic polynomial- time algorithms and randomized algorithms (see§2.3.4) whoseexpectedrunning time is polynomial. 2More precisely, these problems are intractable if the problem parameters are carefully chosen. 87 88 Ch. 3 Number-Theoretic Reference Problems Problem Description FACTORING Integer factorization problem: given a positive integern, ﬁnd its prime factorization; that is, writen=pe1 1 pe2 2 ...pek k where thepiare pairwise distinct primes and eachei≥1. RSAP RSA problem(also known asRSA inversion): given a positive integernthat is a product of two distinct odd primespand q,a positive integeresuch thatgcd(e,(p−1)(q−1))=1 , and an integerc, ﬁnd an integermsuch thatme≡c(modn). QRP Quadratic residuosity problem: given an odd composite inte- gernand an integerahaving Jacobi symbol (a n ) =1, decide whether or notais a quadratic residue modulon. SQROOT Square roots modulon: given a composite integernanda∈Qn (the set of quadratic residues modulon), ﬁnd a square root ofa modulo n; that is, an integerxsuch thatx2 ≡a(modn). DLP Discrete logarithm problem: given a primep, a generatorαof Z∗ p, and an elementβ∈Z∗ p , ﬁnd the integerx,0≤x≤p−2, such thatαx≡β(modp). GDLP Generalized discrete logarithm problem: given a ﬁnite cyclic groupGof ordern, a generatorαofG, and an elementβ∈G, ﬁnd the integerx,0≤x≤n−1, such thatαx=β. DHP Difﬁe-Hellman problem: given a primep, a generatorαofZ∗ p, and elementsαamodpand αbmodp, ﬁndαabmodp. GDHP Generalized Difﬁe-Hellman problem: given a ﬁnite cyclic group G, a generatorαofG, and group elementsαaand αb, ﬁndαab. SUBSET-SUM Subset sum problem: given a set of positive integers {a1,a2,...,a n}and a positive integers, determine whether or not there is a subset of theajthat sums tos. Table 3.1:Some computational problems of cryptographic relevance. computational problem to another have been devised and studied in the literature. These re- ductions provide a means for converting any algorithm that solves the second problem into an algorithm for solving the ﬁrst problem. The following intuitive notion of reducibility (cf.§2.3.3) is used in this chapter. 3.1 DeﬁnitionLetAand Bbe two computational problems.Ais said topolytime reduceto B, writtenA≤P B, if there is an algorithm that solvesAwhich uses, as a subroutine, a hypothetical algorithm for solvingB, and which runs in polynomial time if the algorithm forBdoes.3 Informally speaking, ifApolytime reduces toB, thenBis at least as difﬁcult asA; equivalently,Ais no harder thanB. Consequently, ifAis a well-studied computational problem that is widely believed to be intractable, then proving thatA≤P Bprovides strong evidence of the intractability of problemB. 3.2 DeﬁnitionLetAand Bbe two computational problems. IfA≤P Band B≤P A, then Aand Bare said to becomputationally equivalent, writtenA≡P B. 3In the literature, the hypothetical polynomial-time subroutine forBis sometimes called anoracleforB. §3.2 The integer factorization problem 89 Informally speaking, ifA≡P BthenAand Bare either both tractable or both in- tractable, as the case may be. Chapter outline The remainder of the chapter is organized as follows. Algorithms for the integer factoriza- tion problem are studied in§3.2. Two problems related to factoring, the RSA problem and the quadratic residuosity problem, are brieﬂy considered in§3.3 and§3.4. Efﬁcient algo- rithms for computing square roots inZp,pa prime, are presented in§3.5, and the equiva- lence of the problems of ﬁnding square roots modulo a composite integernand factoring nis established. Algorithms for the discrete logarithm problem are studied in§3.6, and the related Difﬁe-Hellman problem is brieﬂy considered in§3.7. The relation between the problems of factoring a composite integernand computing discrete logarithms in (cyclic subgroups of) the groupZ∗ nis investigated in§3.8. The tasks of ﬁnding partial solutions to the discrete logarithm problem, the RSA problem, and the problem of computing square roots modulo a composite integernare the topics of§3.9. TheL3-lattice basis reduction algorithm is presented in§3.10, along with algorithms for the subset sum problem and for simultaneous diophantine approximation. Berlekamp’sQ-matrix algorithm for factoring polynomials is presented in§3.11. Finally,§3.12 provides references and further chapter notes. 3.2 The integer factorization problem The security of many cryptographic techniques depends upon the intractability of the in- teger factorization problem. A partial list of such protocols includes the RSA public-key encryption scheme (§8.2), the RSA signature scheme (§11.3.1), and the Rabin public-key encryption scheme (§8.3). This section summarizes the current knowledge on algorithms for the integer factorization problem. 3.3 DeﬁnitionThe integer factorization problem(FACTORING) is the following: given a positive integern, ﬁnd its prime factorization; that is, writen=pe1 1 pe2 2 ···pek k where the piare pairwise distinct primes and eachei≥1. 3.4 Remark (primality testing vs. factoring) The problem ofdecidingwhether an integer is composite or prime seems to be, in general, much easier than the factoring problem. Hence, before attempting to factor an integer, the integer should be tested to make sure that it is indeed composite. Primality tests are a main topic of Chapter 4. 3.5 Remark (splitting vs. factoring)A non-trivial factorizationofnis a factorization of the form n=abwhere 1<a<n and 1<b<n ;aand bare said to benon-trivial factors ofn. Hereaand bare not necessarily prime. To solve the integer factorization problem, it sufﬁces to study algorithms thatsplitn, that is, ﬁnd a non-trivial factorizationn=ab. Once found, the factorsaandbcan be tested for primality. The algorithm for splitting integers can then be recursively applied toaand/orb, if either is found to be composite. In this manner, the prime factorization ofncan be obtained. 3.6 Note (testing for perfect powers)I fn≥2, it can be efﬁciently checked as follows whether or notnis aperfect power, i.e.,n=x kfor some integersx≥2,k≥2. For each prime 90 Ch. 3 Number-Theoretic Reference Problems p≤lgn, an integer approximationxofn1/pis computed. This can be done by performing a binary search forxsatisfyingn=xpin the interval[2,2⌊lg n/p⌋+1]. The entire procedure takesO((lg3 n)lglglg n)bit operations. For the remainder of this section, it will always be assumed thatnis not a perfect power. It follows that ifnis composite, thennhas at least two distinct prime factors. Some factoring algorithms are tailored to perform better when the integernbeing fac- tored is of a special form; these are calledspecial-purposefactoring algorithms. The run- ning times of such algorithms typically depend on certain properties of the factors ofn. Ex- amples of special-purpose factoring algorithms include trial division (§3.2.1), Pollard’s rho algorithm (§3.2.2), Pollard’sp−1algorithm (§3.2.3), the elliptic curve algorithm (§3.2.4), and the special number ﬁeld sieve (§3.2.7). In contrast, the running times of the so-called general-purposefactoring algorithms depend solely on the size ofn. Examples of general- purpose factoring algorithms include the quadratic sieve (§3.2.6) and the general number ﬁeld sieve (§3.2.7). Whenever applicable, special-purpose algorithms should be employed as they will gen- erally be more efﬁcient. A reasonable overall strategy is to attempt to ﬁnd small factors ﬁrst, capitalize on any particular special forms an integer may have, and then, if all else fails, bring out the general-purpose algorithms. As an example of a general strategy, one might consider the following. 1. Apply trial division by small primes less than some boundb 1. 2. Next, apply Pollard’s rho algorithm, hoping to ﬁnd any small prime factors smaller than some boundb2, whereb2 >b1. 3. Apply the elliptic curve factoring algorithm, hoping to ﬁnd any small factors smaller than some boundb3, whereb3 >b2. 4. Finally, apply one of the more powerful general-purpose algorithms (quadratic sieve or general number ﬁeld sieve). 3.2.1 Trial division Once it is established that an integernis composite, before expending vast amounts of time with more powerful techniques, the ﬁrst thing that should be attempted is trial division by all “small” primes. Here, “small” is determined as a function of the size ofn. As an extreme case, trial division can be attempted by all primes up to√ n. If this is done, trial division will completely factornbut the procedure will take roughly√ ndivisions in the worst case when nis a product of two primes of the same size. In general, if the factors found at each stage are tested for primality, then trial division to factorncompletely takesO(p+lgn) divisions, wherepis the second-largest prime factor ofn. Fact 3.7 indicates that if trial division is used to factor a randomly chosen large integer n, then the algorithm can be expected to ﬁnd some small factors ofnrelatively quickly, and expend a large amount of time to ﬁnd the second largest prime factor ofn. 3.7 FactLetnbe chosen uniformly at random from the interval[1,x]. (i) If1 2 ≤ α ≤ 1, then the probability that the largest prime factor ofnis≤ xα is approximately1+ln α. Thus, for example, the probability thatnhas a prime factor >√ xisln2≈0.69. (ii) The probability that the second-largest prime factor ofnis≤x0.2117 is about1 2 . (iii) The expected total number of prime factors ofnislnlnx+O(1). (Ifn=∏ pei i , the totalnumber of prime factors ofnis∑ ei.) §3.2 The integer factorization problem 91 3.2.2 Pollard’s rho factoring algorithm Pollard’s rho algorithm is a special-purpose factoring algorithm for ﬁnding small factors of a composite integer. Letf :S−→Sbe a random function, whereSis a ﬁnite set of cardinalityn. Let x0 be a random element ofS, and consider the sequencex0,x1,x2,... deﬁned byxi+1 = f(xi)fori≥0. SinceSis ﬁnite, the sequence must eventually cycle, and consists of a tailof expected length √ πn/8followed by an endlessly repeatingcycleof expected length√ πn/8(see Fact 2.37). A problem that arises in some cryptanalytic tasks, including integer factorization (Algorithm 3.9) and the discrete logarithm problem (Algorithm 3.60), is of ﬁnding distinct indicesiand jsuch thatx i=xj(acollisionis then said to have occurred). An obvious method for ﬁnding a collision is to compute and storexifori=0,1,2,... and look for duplicates. The expected number of inputs that must be tried before a duplicate is detected is √ πn/2(Fact 2.27). This method requiresO(√ n)memory and O(√ n)time, assuming thexiare stored in a hash table so that new entries can be added in constant time. 3.8 Note (Floyd’ s cycle-ﬁnding algorithm) The large storage requirements in the above tech- nique for ﬁnding a collision can be eliminated by usingFloyd’ s cycle-ﬁnding algorithm. In this method, one starts with the pair(x1,x2), and iteratively computes(xi,x2i)from the previous pair(xi−1,x2i−2), untilxm =x2mfor somem. If the tail of the sequence has lengthλand the cycle has lengthµ, then the ﬁrst time thatxm =x2mis whenm= µ(1+⌊λ/µ⌋). Note thatλ<m ≤λ+µ, and consequently the expected running time of this method isO(√ n). Now, letpbe a prime factor of a composite integern. Pollard’s rho algorithm for fac- toringnattempts to ﬁnd duplicates in the sequence of integersx0,x1,x2,... deﬁned by x0 =2,xi+1 =f(xi)= x2 i +1mod pfori≥0. Floyd’s cycle-ﬁnding algorithm is uti- lized to ﬁndxmand x2msuch thatxm≡x2m (modp). Sincepdividesnbut is unknown, this is done by computing the termsximodulo nand testing ifgcd(xm−x2m,n)>1. If alsogcd(xm−x2m,n)<n, then a non-trivial factor ofnis obtained. (The situation gcd(xm−x2m,n)= noccurs with negligible probability.) 3.9 AlgorithmPollard’s rho algorithm for factoring integers INPUT: a composite integernthat is not a prime power. OUTPUT: a non-trivial factordofn. 1. Seta←2,b←2. 2. Fori=1,2,... do the following: 2.1 Compute a←a2 +1mod n, b←b2 +1mod n, b←b2 +1mod n. 2.2 Compute d=gcd(a−b,n). 2.3 If1<d<n then return(d) and terminate with success. 2.4 Ifd=nthen terminate the algorithm with failure (see Note 3.12). 3.10 Example (Pollard’ s rho algorithm for ﬁnding a non-trivial factor ofn=455459) The following table lists the values of variablesa,b, anddat the end of each iteration of step 2 of Algorithm 3.9. 92 Ch. 3 Number-Theoretic Reference Problems a b d 5 26 1 26 2871 1 677 179685 1 2871 155260 1 44380 416250 1 179685 43670 1 121634 164403 1 155260 247944 1 44567 68343 743 Hence two non-trivial factors of455459are743and 455459/743=613 . □ 3.11 FactAssuming that the functionf(x)= x2 +1mod pbehaves like a random function, the expected time for Pollard’s rho algorithm to ﬁnd a factorpofnisO(√ p)modular mul- tiplications. This implies that the expected time to ﬁnd a non-trivial factor ofnisO(n1/4) modular multiplications. 3.12 Note (options upon termination with failure) If Pollard’s rho algorithm terminates with failure, one option is to try again with a different polynomialfhaving integer coefﬁcients instead off(x)= x2 +1. For example, the polynomialf(x)= x2 +cmay be used as long asc̸=0,−2. 3.2.3 Pollard’sp −1 factoring algorithm Pollard’sp−1factoring algorithm is a special-purpose factoring algorithm that can be used to efﬁciently ﬁnd any prime factorspof a composite integernfor whichp−1is smooth (see Deﬁnition 3.13) with respect to some relatively small boundB. 3.13 DeﬁnitionLetBbe a positive integer. An integernis said to beB-smooth,o rsmooth with respect to a boundB, if all its prime factors are≤B. The idea behind Pollard’sp−1algorithm is the following. LetBbe a smoothness bound. LetQbe the least common multiple of all powers of primes≤Bthat are≤n.I f ql≤n, thenllnq≤lnn, and sol≤⌊ln n ln q⌋. Thus Q = ∏ q≤B q⌊ln n/ln q⌋, where the product is over all distinct primesq≤B.I fpis a prime factor ofnsuch thatp−1 isB-smooth, thenp−1\|Q, and consequently for anyasatisfyinggcd(a,p)=1 , Fermat’s theorem (Fact 2.127) implies thataQ ≡1(mod p). Hence ifd=gcd(aQ−1,n), then p\|d. It is possible thatd=n, in which case the algorithm fails; however, this is unlikely to occur ifnhas at least two large distinct prime factors. §3.2 The integer factorization problem 93 3.14 AlgorithmPollard’sp−1 algorithm for factoring integers INPUT: a composite integernthat is not a prime power. OUTPUT: a non-trivial factordofn. 1. Select a smoothness boundB. 2. Select a random integera,2≤a≤n−1, and computed=gcd(a,n).I fd≥2 then return(d). 3. For each primeq≤Bdo the following: 3.1 Compute l=⌊ln n ln q⌋. 3.2 Compute a←aql modn(using Algorithm 2.143). 4. Compute d=gcd(a−1,n). 5. Ifd=1 ord=n, then terminate the algorithm with failure. Otherwise, return(d). 3.15 Example (Pollard’ sp−1algorithm for ﬁnding a non-trivial factor ofn=19048567) 1. Select the smoothness boundB=19. 2. Select the integera=3 and computegcd(3,n)=1 . 3. The following table lists the intermediate values of the variablesq,l, andaafter each iteration of step 3 in Algorithm 3.14: q l a 2 24 2293244 3 15 13555889 5 10 16937223 7 8 15214586 11 6 9685355 13 6 13271154 17 5 11406961 19 5 554506 4. Compute d=gcd(554506−1,n)=5281 . 5. Two non-trivial factors ofnarep=5281 and q=n/p=3607 (these factors are in fact prime). Notice thatp−1=5280=2 5 ×3×5×11, andq−1=3606=2 ×3×601. That is,p−1is19-smooth, whileq−1is not19-smooth. □ 3.16 FactLetnbe an integer having a prime factorpsuch thatp−1isB-smooth. The run- ning time of Pollard’sp−1algorithm for ﬁnding the factorpisO(Blnn/lnB)modular multiplications. 3.17 Note (improvements) The smoothness boundBin Algorithm 3.14 is selected based on the amount of time one is willing to spend on Pollard’sp−1algorithm before moving on to more general techniques. In practice,Bmay be between105 and 106. If the algorithm terminates withd =1 , then one might try searching over prime numbersq1,q2,...,q l larger thanBby ﬁrst computinga←aqi modnfor1≤i≤l, and then computingd= gcd(a−1,n). Another variant is to start with a large boundB, and repeatedly execute step 3 for a few primesqfollowed by the gcd computation in step 4. There are numerous other practical improvements of the algorithm (see page 125). Handbook of Applied Cryptographyby A. Menezes, P. van Oorschot and S. Vanstone. 94 Ch. 3 Number-Theoretic Reference Problems 3.2.4 Elliptic curve factoring The details of theelliptic curve factoring algorithmare beyond the scope of this book; nev- ertheless, a rough outline follows. The success of Pollard’sp−1algorithm hinges onp−1 being smooth for some prime divisorpofn; if no suchpexists, then the algorithm fails. Observe thatp−1is the order of the groupZ∗ p. The elliptic curve factoring algorithm is a generalization of Pollard’sp−1algorithm in the sense that the groupZ∗ pis replaced by a random elliptic curve group overZp. The order of such a group is roughly uniformly dis- tributed in the interval[p+1−2√ p,p+1+2√ p]. If the order of the group chosen is smooth with respect to some pre-selected bound, the elliptic curve algorithm will, with high prob- ability, ﬁnd a non-trivial factor ofn. If the group order is not smooth, then the algorithm will likely fail, but can be repeated with a different choice of elliptic curve group. The elliptic curve algorithm has an expected running time ofLp[1 2 , √ 2](see Exam- ple 2.61 for deﬁnition ofLp) to ﬁnd a factorpofn. Since this running time depends on the size of the prime factors ofn, the algorithm tends to ﬁnd small such factors ﬁrst. The elliptic curve algorithm is, therefore, classiﬁed as a special-purpose factoring algorithm. It is currently the algorithm of choice for ﬁndingt-decimal digit prime factors, fort≤40,o f very large composite integers. In the hardest case, whennis a product of two primes of roughly the same size, the expected running time of the elliptic curve algorithm isLn[1 2 ,1], which is the same as that of the quadratic sieve (§3.2.6). However, the elliptic curve algorithm is not as efﬁcient as the quadratic sieve in practice for such integers. 3.2.5 Random square factoring methods The basic idea behind the random square family of methods is the following. Supposex and yare integers such thatx2 ≡y2 (modn)butx̸≡±y(modn). Then ndivides x2−y2 =(x−y)(x+y)butndoes not divide either(x−y)or(x+y). Hence,gcd(x−y,n) must be a non-trivial factor ofn. This result is summarized next. 3.18 FactLetx,y, andnbe integers. Ifx2 ≡y2 (modn)butx̸≡±y(modn), thengcd(x− y,n)is a non-trivial factor ofn. The random square methods attempt to ﬁnd integersxand yat random so thatx2 ≡y2 (modn). Then, as shown in Fact 3.19, with probability at least1 2 it is the case thatx̸≡±y (modn), whencegcd(x−y,n)will yield a non-trivial factor ofn. 3.19 FactLetnbe an odd composite integer that is divisible bykdistinct odd primes. Ifa∈ Z∗ n, then the congruencex2 ≡ a2 (modn)has exactly2k solutions modulon, two of which arex=aand x=−a. 3.20 Example Letn=35. Then there are four solutions to the congruencex2 ≡4(mod35) , namely x=2,12,23, and33. □ A common strategy employed by the random square algorithms for ﬁndingxand yat random satisfyingx2 ≡y2 (modn)is the following. A set consisting of the ﬁrsttprimes S={p1,p2,...,p t}is chosen;Sis called thefactor base. Proceed to ﬁnd pairs of integers (ai,bi)satisfying (i)a2 i ≡bi (modn); and §3.2 The integer factorization problem 95 (ii)bi=∏t j=1 peij j ,eij ≥0; that is,biispt-smooth. Next ﬁnd a subset of thebi’s whose product is a perfect square. Knowing the factoriza- tions of thebi’s, this is possible by selecting a subset of thebi’s such that the power of each primepj appearing in their product is even. For this purpose, only the parity of the non-negative integer exponentseij needs to be considered. Thus, to simplify matters, for eachi, associate the binary vectorvi=(vi1,vi2,...,v it)with the integer exponent vector (ei1,ei2,...,e it)such thatvij =eij mod2.I ft+1 pairs(ai,bi)are obtained, then the t-dimensional vectorsv1,v2,...,v t+1 must be linearly dependent overZ2. That is, there must exist a non-empty subsetT⊆{1,2,...,t +1}such that∑ i∈Tvi=0 overZ2, and hence∏ i∈Tbiis a perfect square. The setTcan be found using ordinary linear algebra over Z2. Clearly,∏ i∈Ta2 i is also a perfect square. Thus settingx=∏ i∈Taiand yto be the integer square root of∏ i∈Tbiyields a pair of integers(x,y)satisfyingx2 ≡y2 (modn). If this pair also satisﬁesx̸≡±y(modn), thengcd(x−y,n)yields a non-trivial factor ofn. Otherwise, some of the(ai,bi)pairs may be replaced by some new such pairs, and the process is repeated. In practice, there will be several dependencies among the vectors v1,v2,...,v t+1, and with high probability at least one will yield an(x,y)pair satisfying x̸≡±y(modn); hence, this last step of generating new(ai,bi)pairs does not usually occur. This description of the random square methods is incomplete for two reasons. Firstly, the optimal choice oft, the size of the factor base, is not speciﬁed; this is addressed in Note 3.24. Secondly, a method for efﬁciently generating the pairs(ai,bi)is not speciﬁed. Several techniques have been proposed. In the simplest of these, calledDixon’ s algorithm, aiis chosen at random, andbi =a2 i modnis computed. Next, trial division by elements in the factor base is used to test whetherbiispt-smooth. If not, then another integeraiis chosen at random, and the procedure is repeated. The more efﬁcient techniques strategically select anaisuch thatbiis relatively small. Since the proportion ofpt-smooth integers in the interval[2,x]becomes larger asxde- creases, the probability of suchbi beingpt-smooth is higher. The most efﬁcient of such techniques is the quadratic sieve algorithm, which is described next. 3.2.6 Quadratic sieve factoring Suppose an integernis to be factored. Letm=⌊√ n⌋, and consider the polynomialq(x)= (x+m)2 −n. Note that q(x)= x2 +2mx+m2 −n ≈ x2 +2mx, (3.1) which is small (relative ton)i fxis small in absolute value. The quadratic sieve algorithm selectsai =( x+m)and tests whetherbi =( x+m)2 −nispt-smooth. Note that a2 i =(x+m)2 ≡bi (modn). Note also that if a primepdividesbithen(x+m)2 ≡n (modp), and hencenis a quadratic residue modulop. Thus the factor base need only contain those primespfor which the Legendre symbol (n p ) is1(Deﬁnition 2.145). Further- more, sincebimay be negative,−1is included in the factor base. The steps of the quadratic sieve algorithm are summarized in Algorithm 3.21. 96 Ch. 3 Number-Theoretic Reference Problems 3.21 AlgorithmQuadratic sieve algorithm for factoring integers INPUT: a composite integernthat is not a prime power. OUTPUT: a non-trivial factordofn. 1. Select the factor baseS={p1,p2,...,p t}, wherep1 =−1and pj (j≥2)i st h e (j−1)th primepfor whichnis a quadratic residue modulop. 2. Compute m=⌊√ n⌋. 3. (Collectt+1pairs(ai,bi). Thexvalues are chosen in the order0,±1,±2,... .) Seti←1. Whilei≤t+1do the following: 3.1 Computeb=q(x)=( x+m)2 −n, and test using trial division (cf. Note 3.23) by elements inSwhetherbispt-smooth. If not, pick a newxand repeat step 3.1. 3.2 Ifbispt-smooth, sayb=∏t j=1 peij j , then setai←(x+m),bi←b, andvi = (vi1,vi2,...,v it), wherevij =eij mod2 for1≤j≤t. 3.3 i←i+1. 4. Use linear algebra overZ2 to ﬁnd a non-empty subsetT ⊆{1,2,...,t +1}such that∑ i∈Tvi=0. 5. Compute x=∏ i∈Taimodn. 6. For eachj,1≤j≤t, computelj =(∑ i∈Teij)/2. 7. Compute y=∏t j=1 plj j modn. 8. Ifx≡±y(modn), then ﬁnd another non-empty subsetT⊆{1,2,...,t +1}such that∑ i∈Tvi =0, and go to step 5. (In the unlikely case such a subsetTdoes not exist, replace a few of the(ai,bi)pairs with new pairs (step 3), and go to step 4.) 9. Compute d=gcd(x−y,n)and return(d). 3.22 Example (quadratic sieve algorithm for ﬁnding a non-trivial factor ofn=24961) 1. Select the factor baseS={−1,2,3,5,13,23}of sizet=6.(7,11,17and 19are omitted fromSsince (n p ) =−1for these primes.) 2. Compute m=⌊ √ 24961⌋=157. 3. Following is the data collected for the ﬁrstt+1 values ofxfor whichq(x)is23- smooth. i x q(x) factorization ofq(x) ai vi 1 0 −312 −23 ·3 ·13 157 (1,1,1,0,1,0) 2 1 3 3 158 (0,0,1,0,0,0) 3 −1 −625 −54 156 (1,0,0,0,0,0) 4 2 320 26 ·5 159 (0,0,0,1,0,0) 5 −2 −936 −23 ·32 ·13 155 (1,1,0,0,1,0) 6 4 960 26 ·3 ·5 161 (0,0,1,1,0,0) 7 −6 −2160 −24 ·33 ·5 151 (1,0,1,1,0,0) 4. By inspection,v1 +v2 +v5 =0. (In the notation of Algorithm 3.21,T={1,2,5}.) 5. Compute x=(a1a2a5 modn)=936 . 6. Compute l1 =1,l2 =3,l3 =2,l4 =0,l5 =1,l6 =0. 7. Compute y=−23 ·32 ·13mod n=24025. 8. Since936≡−24025(mod n), another linear dependency must be found. 9. By inspection,v3 +v6 +v7 =0; thusT={3,6,7}. 10. Compute x=(a3a6a7 modn)=23405 . 11. Computel1 =1,l2 =5,l3 =2,l4 =3,l5 =0,l6 =0. §3.2 The integer factorization problem 97 12. Compute y=(−25 ·32 ·53 modn)=13922 . 13. Now,23405̸≡±13922(mod n), so computegcd(x−y,n)=gcd(9483 ,24961)= 109. Hence, two non-trivial factors of24961are109and 229. □ 3.23 Note (sieving) Instead of testing smoothness by trial division in step 3.1 of Algorithm 3.21, a more efﬁcient technique known assievingis employed in practice. Observe ﬁrst that ifp is an odd prime in the factor base andpdividesq(x), thenpalso dividesq(x+lp)for every integerl. Thus by solving the equationq(x)≡0(mod p)forx(for example, using the algorithms in§3.5.1), one knows either one or two (depending on the number of solutions to the quadratic equation) entire sequences of other valuesyfor whichpdividesq(y). The sieving processis the following. An arrayQ[] indexed byx,−M≤x≤M,i s created and thexth entry is initialized to⌊lg\|q(x)\|⌋. Letx1,x2 be the solutions toq(x)≡0 (modp), wherepis an odd prime in the factor base. Then the value⌊lgp⌋is subtracted from those entriesQ[x]in the array for whichx≡x1 orx2 (modp)and −M≤x≤M. This is repeated for each odd primepin the factor base. (The case ofp=2 and prime powers can be handled in a similar manner.) After the sieving, the array entriesQ[x]with values near0are most likely to bept-smooth (roundoff errors must be taken into account), and this can be veriﬁed by factoringq(x)by trial division. 3.24 Note (running time of the quadratic sieve) To optimize the running time of the quadratic sieve, the size of the factor base should be judiciously chosen. The optimal selection of t≈Ln[1 2 ,1 2 ](see Example 2.61) is derived from knowledge concerning the distribution of smooth integers close to√ n. With this choice, Algorithm 3.21 with sieving (Note 3.23) has an expected running time ofLn[1 2 ,1], independent of the size of the factors ofn. 3.25 Note (multiple polynomial variant) In order to collect a sufﬁcient number of(ai,bi)pairs, the sieving interval must be quite large. From equation (3.1) it can be seen that\|q(x)\|in- creases linearly with\|x\|, and consequently the probability of smoothness decreases. To overcome this problem, a variant (themultiple polynomial quadratic sieve) was proposed whereby many appropriately-chosenquadratic polynomials can be used instead of justq(x), each polynomial being sieved over an interval of much smaller length. This variant also has an expected running time ofLn[1 2 ,1], and is the method of choice in practice. 3.26 Note (parallelizing the quadratic sieve) The multiple polynomial variant of the quadratic sieve is well suited for parallelization. Each node of a parallel computer, or each computer in a network of computers, simply sieves through different collections of polynomials. Any (ai,bi)pair found is reported to a central processor. Once sufﬁcient pairs have been col- lected, the corresponding system of linear equations is solved on a single (possibly parallel) computer. 3.27 Note (quadratic sieve vs. elliptic curve factoring) The elliptic curve factoring algorithm (§3.2.4) has the same 4 expected (asymptotic) running time as the quadratic sieve factoring algorithm in the special case whennis the product of two primes of equal size. However, for such numbers, the quadratic sieve is superior in practice because the main steps in the algorithm are single precision operations, compared to the much more computationally in- tensive multi-precision elliptic curve operations required in the elliptic curve algorithm. 4This does not take into account the differento(1) terms in the two expressionsLn[ 1 2 ,1]. 98 Ch. 3 Number-Theoretic Reference Problems 3.2.7 Number ﬁeld sieve factoring For several years it was believed by some people that a running time ofLn[1 2 ,1]was, in fact, the best achievable by any integer factorization algorithm. This barrier was broken in 1990 with the discovery of thenumber ﬁeld sieve. Like the quadratic sieve, the number ﬁeld sieve is an algorithm in the random square family of methods (§3.2.5). That is, it attempts to ﬁnd integersxand ysuch thatx2 ≡y2 (modn)andx̸≡±y(modn). To achieve this goal, two factor bases are used, one consisting of all prime numbers less than some bound, and the other consisting of all prime ideals of norm less than some bound in the ring of integers of a suitably-chosen algebraic number ﬁeld. The details of the algorithm are quite complicated, and are beyond the scope of this book. A special version of the algorithm (thespecial number ﬁeld sieve) applies to integers of the formn=r e−sfor smallrand \|s\|, and has an expected running time ofLn[1 3 ,c], where c=(32/9)1/3 ≈1.526. The general version of the algorithm, sometimes called thegeneral number ﬁeld sieve, applies to all integers and has an expected running time ofLn[1 3 ,c], wherec=(64/9)1/3 ≈ 1.923. This is, asymptotically, the fastest algorithm known for integer factorization. The primary reason why the running time of the number ﬁeld sieve is smaller than that of the quadratic sieve is that the candidate smooth numbers in the former are much smaller than those in the latter. The general number ﬁeld sieve was at ﬁrst believed to be slower than the quadratic sieve for factoring integers having fewer than 150 decimal digits. However, experiments in 1994–1996 have indicated that the general number ﬁeld sieve is substantially faster than the quadratic sieve even for numbers in the 115 digit range. This implies that the crossover point between the effectiveness of the quadratic sieve vs. the general number ﬁeld sieve may be 110–120 digits. For this reason, the general number ﬁeld sieve is considered the current champion of all general-purpose factoring algorithms. 3.3 The RSA problem The intractability of the RSA problem forms the basis for the security of the RSA public-key encryption scheme (§8.2) and the RSA signature scheme (§11.3.1). 3.28 DeﬁnitionThe RSA problem(RSAP) is the following: given a positive integernthat is a product of two distinct odd primespand q, a positive integeresuch thatgcd(e,(p−1)(q− 1))=1 , and an integerc, ﬁnd an integermsuch thatm e≡c(modn). In other words, the RSA problem is that of ﬁndingeth roots modulo a composite integer n. The conditions imposed on the problem parametersnand eensure that for each integer c ∈{ 0,1,...,n −1}there is exactly onem ∈{ 0,1,...,n −1}such thatme ≡ c (modn). Equivalently, the functionf:Zn −→Zndeﬁned asf(m)= memodnis a permutation. 3.29 Remark (SQROOT vs. RSA problems) Sincep−1is even, it follows thateis odd. In particular,e̸=2, and hence the SQROOT problem (Deﬁnition 3.43) isnota special case of the RSA problem. §3.4 The quadratic residuosity problem 99 As is shown in§8.2.2(i), if the factors ofnare known then the RSA problem can be easily solved. This fact is stated next. 3.30 FactRSAP ≤P FACTORING. That is, the RSA problem polytime reduces to the integer factorization problem. It is widely believed that the RSA and the integer factorization problems are computa- tionally equivalent, although no proof of this is known. 3.4 The quadratic residuosity problem The security of the Goldwasser-Micali probabilistic public-key encryption scheme (§8.7) and the Blum-Blum-Shub pseudorandom bit generator (§5.5.2) are both based on the ap- parent intractability of the quadratic residuosity problem. Recall from§2.4.5 that ifn≥3is an odd integer, thenJnis the set of alla∈Z∗ n having Jacobi symbol 1. Recall also thatQnis the set of quadratic residues modulonand that the set of pseudosquares modulonis deﬁned by˜Qn=Jn−Qn. 3.31 DeﬁnitionThe quadratic residuosity problem(QRP) is the following: given an odd com- posite integernand a∈Jn, decide whether or notais a quadratic residue modulon. 3.32 Remark (QRP with a prime modulus)I fnis a prime, then it is easy to decide whether a∈Z∗ n is a quadratic residue modulonsince, by deﬁnition,a∈Qnif and only if (a n ) =1, and the Legendre symbol (a n ) can be efﬁciently calculated by Algorithm 2.149. Assume now thatnis a product of two distinct odd primespand q. It follows from Fact 2.137 that ifa∈Jn, thena∈Qnif and only if (a p ) =1. Thus, if the factorization of nis known, then QRP can be solved simply by computing the Legendre symbol (a p ) . This observation can be generalized to all integersnand leads to the following fact. 3.33 FactQRP ≤P FACTORING. That is, the QRP polytime reduces to the FACTORING problem. On the other hand, if the factorization ofnis unknown, then there is no efﬁcient pro- cedure known for solving QRP, other than by guessing the answer. Ifn= pq, then the probability of a correct guess is1 2 since\|Qn\|=\|˜Qn\|(Fact 2.155). It is believed that the QRP is as difﬁcult as the problem of factoring integers, although no proof of this is known. 3.5 Computing square roots inZn The operations of squaring modulo an integernand extracting square roots modulo an in- tegernare frequently used in cryptographic functions. The operation of computing square roots moduloncan be performed efﬁciently whennis a prime, but is difﬁcult whennis a composite integer whose prime factors are unknown. 100 Ch. 3 Number-Theoretic Reference Problems 3.5.1 Case (i):n prime Recall from Remark 3.32 that ifpis a prime, then it is easy to decide ifa∈Z∗ pis a quadratic residue modulop.I fais, in fact, a quadratic residue modulop, then the two square roots ofacan be efﬁciently computed, as demonstrated by Algorithm 3.34. 3.34 AlgorithmFinding square roots modulo a primep INPUT: an odd primepand an integera,1≤a≤p−1. OUTPUT: the two square roots ofamodulo p, providedais a quadratic residue modulop. 1. Compute the Legendre symbol (a p ) using Algorithm 2.149. If (a p ) =−1then return(a does not have a square root modulop) and terminate. 2. Select integersb,1≤b≤p−1, at random until one is found with (b p ) =−1.(bis a quadratic non-residue modulop.) 3. By repeated division by2, writep−1=2 st, wheretis odd. 4. Compute a−1 modpby the extended Euclidean algorithm (Algorithm 2.142). 5. Setc←btmodpand r←a(t+1)/2 modp(Algorithm 2.143). 6. Forifrom 1 tos−1do the following: 6.1 Compute d=(r2 ·a−1)2s−i−1 modp. 6.2 Ifd≡−1(mod p)then setr←r·cmodp. 6.3 Setc←c2 modp. 7. Return(r,−r). Algorithm 3.34 is a randomized algorithm because of the manner in which the quadratic non-residuebis selected in step 2. No deterministic polynomial-time algorithm for ﬁnding a quadratic non-residue modulo a primepis known (see Remark 2.151). 3.35 FactAlgorithm 3.34 has an expected running time ofO((lgp)4)bit operations. This running time is obtained by observing that the dominant step (step 6) is executed s−1times, each iteration involving a modular exponentiation and thus takingO((lgp)3)bit operations (Table 2.5). Since in the worst cases=O(lgp), the running time ofO((lgp)4) follows. Whensis small, the loop in step 6 is executed only a small number of times, and the running time of Algorithm 3.34 isO((lgp)3)bit operations. This point is demonstrated next for the special casess=1 and s=2. Specializing Algorithm 3.34 to the cases=1yields the following simple deterministic algorithm for ﬁnding square roots whenp≡3(mod4) . 3.36 AlgorithmFinding square roots modulo a primepwhere p≡3( m o d4 ) INPUT: an odd primepwhere p≡3(mod4) , and a squarea∈Qp. OUTPUT: the two square roots ofamodulo p. 1. Compute r=a(p+1)/4 modp(Algorithm 2.143). 2. Return(r,−r). Specializing Algorithm 3.34 to the cases=2, and using the fact that2is a quadratic non-residue modulopwhen p≡5(mod8) , yields the following simple deterministic al- gorithm for ﬁnding square roots whenp≡5(mod8) . §3.5 Computing square roots inZn 101 3.37 AlgorithmFinding square roots modulo a primepwhere p≡5( m o d8 ) INPUT: an odd primepwhere p≡5(mod8) , and a squarea∈Qp. OUTPUT: the two square roots ofamodulo p. 1. Compute d=a(p−1)/4 modp(Algorithm 2.143). 2. Ifd=1 then computer=a(p+3)/8 modp. 3. Ifd=p−1then computer=2a(4a)(p−5)/8 modp. 4. Return(r,−r). 3.38 FactAlgorithms 3.36 and 3.37 have running times ofO((lgp)3)bit operations. Algorithm 3.39 for ﬁnding square roots modulopis preferable to Algorithm 3.34 when p−1=2 stwithslarge. 3.39 AlgorithmFinding square roots modulo a primep INPUT: an odd primepand a squarea∈Qp. OUTPUT: the two square roots ofamodulo p. 1. Choose randomb ∈ Zp untilb2 −4ais a quadratic non-residue modulop, i.e.,(b2−4a p ) =−1. 2. Letfbe the polynomialx2 −bx+ainZp[x]. 3. Compute r=x(p+1)/2 modfusing Algorithm 2.227. (Note:rwill be an integer.) 4. Return(r,−r). 3.40 FactAlgorithm 3.39 has an expected running time ofO((lgp)3)bit operations. 3.41 Note (computing square roots in a ﬁnite ﬁeld) Algorithms 3.34, 3.36, 3.37, and 3.39 can be extended in a straightforward manner to ﬁnd square roots in any ﬁnite ﬁeldFqof odd order q=pm,pprime,m≥1. Square roots in ﬁnite ﬁelds of even order can also be computed efﬁciently via Fact 3.42. 3.42 FactEach elementa∈F2m has exactly one square root, namelya2m−1 . 3.5.2 Case (ii):n composite The discussion in this subsection is restricted to the case of computing square roots modulo n, wherenis a product of two distinct odd primespand q. However, all facts presented here generalize to the case wherenis an arbitrary composite integer. Unlike the case wherenis a prime, the problem of deciding whether a givena∈Z∗ n is a quadratic residue modulo a composite integern, is believed to be a difﬁcult problem. Certainly, if the Jacobi symbol (a n ) =−1, thenais a quadratic non-residue. On the other hand, if (a n ) =1 , then deciding whether or notais a quadratic residue is precisely the quadratic residuosity problem, considered in§3.4. 3.43 DeﬁnitionThe square root modulonproblem(SQROOT) is the following: given a com- posite integernand a quadratic residueamodulo n(i.e.a∈Qn), ﬁnd a square root ofa modulo n. 102 Ch. 3 Number-Theoretic Reference Problems If the factorspand qofnare known, then the SQROOT problem can be solved efﬁ- ciently by ﬁrst ﬁnding square roots ofamodulo pand moduloq, and then combining them using the Chinese remainder theorem (Fact 2.120) to obtain the square roots ofamodulo n. The steps are summarized in Algorithm 3.44, which, in fact, ﬁnds all of the four square roots ofamodulo n. 3.44 AlgorithmFinding square roots modulongiven its prime factorspand q INPUT: an integern, its prime factorspand q, anda∈Qn. OUTPUT: the four square roots ofamodulo n. 1. Use Algorithm 3.39 (or Algorithm 3.36 or 3.37, if applicable) to ﬁnd the two square rootsrand −rofamodulo p. 2. Use Algorithm 3.39 (or Algorithm 3.36 or 3.37, if applicable) to ﬁnd the two square rootssand −sofamodulo q. 3. Use the extended Euclidean algorithm (Algorithm 2.107) to ﬁnd integerscanddsuch thatcp+dq=1. 4. Setx←(rdq+scp)mod nand y←(rdq−scp)mod n. 5. Return(±xmodn,±ymodn). 3.45 FactAlgorithm 3.44 has an expected running time ofO((lgp)3)bit operations. Algorithm 3.44 shows that if one can factorn, then the SQROOT problem is easy. More precisely, SQROOT≤P FACTORING. The converse of this statement is also true, as stated in Fact 3.46. 3.46 FactFACTORING ≤P SQROOT. That is, the FACTORING problem polytime reduces to the SQROOT problem. Hence, since SQROOT≤P FACTORING, the FACTORING and SQROOT problems are computationally equivalent. Justiﬁcation.Suppose that one has a polynomial-time algorithmAfor solving the SQ- ROOT problem. This algorithm can then be used to factor a given composite integernas follows. Select an integerxat random withgcd(x,n)=1 , and computea=x2 modn. Next, algorithmAis run with inputsaand n, and a square rootyofamodulo nis returned. Ify≡±x(modn), then the trial fails, and the above procedure is repeated with a new xchosen at random. Otherwise, ify̸≡±x(modn), thengcd(x−y,n)is guaranteed to be a non-trivial factor ofn(Fact 3.18), namely,porq. Sinceahas four square roots mod- ulon(±xand ±zwith±z̸≡±x(modn)), the probability of success for each attempt is1 2 . Hence, the expected number of attempts before a factor ofnis obtained is two, and consequently the procedure runs in expected polynomial time. □ 3.47 Note (strengthening of Fact 3.46) The proof of Fact 3.46 can be easily modiﬁed to estab- lish the following stronger result. Letc≥1be any constant. If there is an algorithmA which, givenn, can ﬁnd a square root modulonin polynomial time for a1 (lg n)c fraction of all quadratic residuesa∈Qn, then the algorithmAcan be used to factornin expected polynomial time. The implication of this statement is that if the problem of factoringnis difﬁcult, then foralmost alla∈Qnit is difﬁcult to ﬁnd square roots modulon. The computational equivalence of the SQROOT and FACTORING problems was the basis of the ﬁrst “provably secure” public-key encryption and signature schemes, presented in§8.3. §3.6 The discrete logarithm problem 103 3.6 The discrete logarithm problem The security of many cryptographic techniques depends on the intractability of the discrete logarithm problem. A partial list of these includes Difﬁe-Hellman key agreement and its derivatives (§12.6), ElGamal encryption (§8.4), and the ElGamal signature scheme and its variants (§11.5). This section summarizes the current knowledge regarding algorithms for solving the discrete logarithm problem. Unless otherwise speciﬁed, algorithms in this section are described in the general set- ting of a (multiplicatively written) ﬁnite cyclic groupGof ordernwith generatorα(see Deﬁnition 2.167). For a more concrete approach, the reader may ﬁnd it convenient to think ofGas the multiplicative groupZ∗ pof orderp−1, where the group operation is simply multiplication modulop. 3.48 DeﬁnitionLetGbe a ﬁnite cyclic group of ordern. Letαbe a generator ofG, and let β∈G. Thediscrete logarithm ofβto the baseα, denotedlogαβ, is the unique integerx, 0≤x≤n−1, such thatβ=αx. 3.49 Example Letp=97. ThenZ∗ 97 is a cyclic group of ordern=96. A generator ofZ∗ 97 is α=5. Since532 ≡35(mod97) ,log5 35=32 inZ∗ 97 . □ The following are some elementary facts about logarithms. 3.50 FactLetαbe a generator of a cyclic groupGof ordern, and letβ,γ∈G. Letsbe an integer. Thenlogα(βγ)=(log αβ+logαγ)mod nand logα(βs)= slogαβmodn. The groups of most interest in cryptography are the multiplicative groupF∗ qof the ﬁnite ﬁeldFq(§2.6), including the particular cases of the multiplicative groupZ∗ pof the integers modulo a primep, and the multiplicative groupF∗ 2 m of the ﬁnite ﬁeldF2m of characteristic two. Also of interest are the group of unitsZ∗ nwhere nis a composite integer, the group of points on an elliptic curve deﬁned over a ﬁnite ﬁeld, and the jacobian of a hyperelliptic curve deﬁned over a ﬁnite ﬁeld. 3.51 DeﬁnitionThe discrete logarithm problem(DLP) is the following: given a primep,a generatorαofZ∗ p, and an elementβ∈Z∗ p , ﬁnd the integerx,0≤x≤p−2, such that αx≡β(modp). 3.52 DeﬁnitionThe generalized discrete logarithm problem(GDLP) is the following: given a ﬁnite cyclic groupGof ordern, a generatorαofG, and an elementβ∈G, ﬁnd the integer x,0≤x≤n−1, such thatαx=β. The discrete logarithm problem in elliptic curve groups and in the jacobians of hyper- elliptic curves are not explicitly considered in this section. The discrete logarithm problem inZ∗ nis discussed further in§3.8. 3.53 Note (difﬁculty of the GDLP is independent of generator) Letαand γbe two generators of a cyclic groupGof ordern, and letβ∈G. Letx=logαβ,y=logγβ, andz=logαγ. Then αx=β=γy =(αz)y. Consequentlyx=zymodn, and logγβ=(logαβ)(logαγ)−1 modn. This means that any algorithm which computes logarithms to the baseαcan be used to compute logarithms to any other baseγthat is also a generator ofG. 104 Ch. 3 Number-Theoretic Reference Problems 3.54 Note (generalization of GDLP) A more general formulation of the GDLP is the following: given a ﬁnite groupGand elementsα,β∈G, ﬁnd an integerxsuch thatαx=β, provided that such an integer exists. In this formulation, it is not required thatGbe a cyclic group, and, even if it is, it is not required thatαbe a generator ofG. This problem may be harder to solve, in general, than GDLP. However, in the case whereGis a cyclic group (for example ifGis the multiplicative group of a ﬁnite ﬁeld) and the order ofαis known, it can be easily recognized whether an integerxsatisfyingαx=βexists. This is because of the following fact: ifGis a cyclic group,αis an element of orderninG, andβ∈G, then there exists an integerxsuch thatαx=βif and only ifβn=1. 3.55 Note (solving the DLP in a cyclic groupGof ordernis in essence computing an isomor- phism betweenGand Zn) Even though any two cyclic groups of the same order areiso- morphic(that is, they have the same structure although the elements may be written in dif- ferent representations), an efﬁcient algorithm for computing logarithms in one group does not necessarily imply an efﬁcient algorithm for the other group. To see this, consider that every cyclic group of ordernis isomorphic to the additive cyclic groupZn, i.e., the set of integers{0,1,2,...,n −1}where the group operation is addition modulon. Moreover, the discrete logarithm problem in the latter group, namely, the problem of ﬁnding an inte- gerxsuch thatax≡b(modn)givena,b∈Z n, is easy as shown in the following. First note that there does not exist a solutionxifd=gcd(a,n)does not divideb(Fact 2.119). Otherwise, ifddividesb, the extended Euclidean algorithm (Algorithm 2.107) can be used to ﬁnd integerssand tsuch thatas+nt=d. Multiplying both sides of this equation by the integerb/dgivesa(sb/d)+ n(tb/d)= b. Reducing this equation modulonyields a(sb/d)≡b(modn)and hencex=(sb/d)mod nis the desired (and easily obtainable) solution. The known algorithms for the DLP can be categorized as follows: 1. algorithms which work in arbitrary groups, e.g., exhaustive search (§3.6.1), the baby- step giant-step algorithm (§3.6.2), Pollard’s rho algorithm (§3.6.3); 2. algorithms which work in arbitrary groups but are especially efﬁcient if the order of the group has only small prime factors, e.g., Pohlig-Hellman algorithm (§3.6.4); and 3. the index-calculus algorithms (§3.6.5) which are efﬁcient only in certain groups. 3.6.1 Exhaustive search The most obvious algorithm for GDLP (Deﬁnition 3.52) is to successively computeα0,α1, α2,... untilβis obtained. This method takesO(n)multiplications, wherenis the order ofα, and is therefore inefﬁcient ifnis large (i.e. in cases of cryptographic interest). 3.6.2 Baby-step giant-step algorithm Letm=⌈√ n⌉, wherenis the order ofα. The baby-step giant-step algorithm is a time- memory trade-off of the method of exhaustive search and is based on the following observa- tion. Ifβ=αx, then one can writex=im+j, where0≤i,j<m . Hence,αx=αimαj, which impliesβ(α−m)i=αj. This suggests the following algorithm for computingx. §3.6 The discrete logarithm problem 105 3.56 AlgorithmBaby-step giant-step algorithm for computing discrete logarithms INPUT: a generatorαof a cyclic groupGof ordern, and an elementβ∈G. OUTPUT: the discrete logarithmx=logαβ. 1. Setm←⌈√ n⌉. 2. Construct a table with entries(j,αj)for0 ≤ j<m . Sort this table by second component. (Alternatively, use conventional hashing on the second component to store the entries in a hash table; placing an entry, and searching for an entry in the table takes constant time.) 3. Compute α−mand setγ←β. 4. Forifrom 0 tom−1do the following: 4.1 Check ifγis the second component of some entry in the table. 4.2 Ifγ=αjthen return(x=im+j). 4.3 Setγ←γ·α−m. Algorithm 3.56 requires storage forO(√ n)group elements. The table takesO(√ n) multiplications to construct, andO(√ nlgn)comparisons to sort. Having constructed this table, step 4 takesO(√ n)multiplications andO(√ n)table look-ups. Under the assump- tion that a group multiplication takes more time thanlgncomparisons, the running time of Algorithm 3.56 can be stated more concisely as follows. 3.57 FactThe running time of the baby-step giant-step algorithm (Algorithm 3.56) isO(√ n) group multiplications. 3.58 Example (baby-step giant-step algorithm for logarithms inZ∗ 113) Letp=113. The ele- ment α=3 is a generator ofZ∗ 113 of ordern=112. Considerβ=57. Thenlog3 57is computed as follows. 1. Setm←⌈ √ 112⌉=11. 2. Construct a table whose entries are(j,αj modp)for0≤j< 11: j 0 1 2 3 4 5 6 7 8 9 10 3j mod 113 1 3 9 27 81 17 51 40 7 21 63 and sort the table by second component: j 0 1 8 2 5 9 3 7 6 10 4 3j mod 113 1 3 7 9 17 21 27 40 51 63 81 3. Using Algorithm 2.142, computeα−1 =3 −1 mod113 = 38and then compute α−m=3811 mod113=58 . 4. Next,γ = βα−mimod113 fori =0 ,1,2,... is computed until a value in the second row of the table is obtained. This yields: i 0 1 2 3 4 5 6 7 8 9 γ=5 7·58i mod 113 57 29 100 37 112 55 26 39 2 3 Finally, sinceβα−9m=3= α1,β=α100 and, therefore,log3 57=100 . □ 3.59 Note (restricted exponents) In order to improve performance, some cryptographic proto- cols which use exponentiation inZ∗ p select exponents of a special form, e.g. having small Hamming weight. (TheHamming weight of an integer is the number of ones in its binary representation.) Suppose thatpis ak-bit prime, and only exponents of Hamming weightt are used. The number of such exponents is (k t ) . Algorithm 3.56 can be modiﬁed to search the exponent space in roughly (k t/2 ) steps. The algorithm also applies to exponents that are restricted in certain other ways, and extends to all ﬁnite groups. 106 Ch. 3 Number-Theoretic Reference Problems 3.6.3 Pollard’s rho algorithm for logarithms Pollard’s rho algorithm (Algorithm 3.60) for computing discrete logarithms is a randomized algorithm with the same expected running time as the baby-step giant-step algorithm (Al- gorithm 3.56), but which requires a negligible amount of storage. For this reason, it is far preferable to Algorithm 3.56 for problems of practical interest. For simplicity, it is assumed in this subsection thatGis a cyclic group whose ordernis prime. The groupGis partitioned into three setsS 1,S2, andS3 of roughly equal size based on some easily testable property. Some care must be exercised in selecting the partition; for example,1̸∈S2. Deﬁne a sequence of group elementsx0,x1,x2,... by x0 =1 and xi+1 = f(xi) def =    β·xi, ifxi∈S1, x2 i, ifxi∈S2, α·xi, ifxi∈S3, (3.2) fori ≥ 0. This sequence of group elements in turn deﬁnes two sequences of integers a0,a1,a2,... and b0,b1,b2,... satisfyingxi=αai βbi fori≥0:a0 =0,b0 =0, and for i≥0, ai+1 =    ai, ifxi∈S1, 2aimodn, ifxi∈S2, ai+1mod n, ifxi∈S3, (3.3) and bi+1 =    bi+1mod n, ifxi∈S1, 2bimodn, ifxi∈S2, bi, ifxi∈S3. (3.4) Floyd’s cycle-ﬁnding algorithm (Note 3.8) can then be utilized to ﬁnd two group elements xi and x2i such thatxi = x2i. Hence αai βbi =αa2i βb2i , and soβbi−b2i =αa2i−ai . Taking logarithms to the baseαof both sides of this last equation yields (bi−b2i)·logαβ≡(a2i−ai)( m o dn). Providedbi̸≡b2i (modn)(note:bi≡b2ioccurs with negligible probability), this equa- tion can then be efﬁciently solved to determinelogαβ. 3.60 AlgorithmPollard’s rho algorithm for computing discrete logarithms INPUT: a generatorαof a cyclic groupGof prime ordern, and an elementβ∈G. OUTPUT: the discrete logarithmx=logαβ. 1. Setx0←1,a0←0,b0←0. 2. Fori=1,2,... do the following: 2.1 Using the quantitiesxi−1,ai−1,bi−1, andx2i−2,a2i−2,b2i−2 computed previ- ously, computexi,ai,biand x2i,a2i,b2iusing equations (3.2), (3.3), and (3.4). 2.2 Ifxi=x2i, then do the following: Setr←bi−b2imodn. Ifr =0 then terminate the algorithm with failure; otherwise, compute x=r−1(a2i−ai)mod nand return(x). In the rare case that Algorithm 3.60 terminates with failure, the procedure can be re- peated by selecting random integersa0,b0 in the interval[1,n−1], and starting withx0 = αa0 βb0 . Example 3.61 with artiﬁcially small parameters illustrates Pollard’s rho algorithm. §3.6 The discrete logarithm problem 107 3.61 Example (Pollard’ s rho algorithm for logarithms in a subgroup ofZ∗ 383) The elementα= 2is a generator of the subgroupGofZ∗ 383 of ordern=191. Supposeβ=228. Partition the elements ofGinto three subsets according to the rulex∈S1 ifx≡1(mod3) ,x∈S2 ifx≡0(mod3) , andx∈S3 ifx≡2(mod3) . Table 3.2 shows the values ofxi,ai,bi, x2i,a2i, andb2iat the end of each iteration of step 2 of Algorithm 3.60. Note thatx14 = x28 =144. Finally, computer=b14 −b28 mod191=125 ,r−1 =125−1 mod191= 136, andr−1(a28 −a14)mod191=110 . Hence,log2 228=110 . □ i xi ai bi x2i a2i b2i 1 228 0 1 279 0 2 2 279 0 2 184 1 4 3 92 0 4 14 1 6 4 184 1 4 256 2 7 5 205 1 5 304 3 8 6 14 1 6 121 6 18 7 28 2 6 144 12 38 8 256 2 7 235 48 152 9 152 2 8 72 48 154 10 304 3 8 14 96 118 11 372 3 9 256 97 119 12 121 6 18 304 98 120 13 12 6 19 121 5 51 14 144 12 38 144 10 104 Table 3.2:Intermediate steps of Pollard’ s rho algorithm in Example 3.61. 3.62 FactLetGbe a group of ordern, a prime. Assume that the functionf :G−→Gde- ﬁned by equation (3.2) behaves like a random function. Then the expected running time of Pollard’s rho algorithm for discrete logarithms inGisO(√ n)group operations. Moreover, the algorithm requires negligible storage. 3.6.4 Pohlig-Hellman algorithm Algorithm 3.63 for computing logarithms takes advantage of the factorization of the ordern of the groupG. Letn=pe1 1 pe2 2 ···per r be the prime factorization ofn.I fx=logαβ, then the approach is to determinexi=xmodpei i for1≤i≤r, and then use Gauss’s algorithm (Algorithm 2.121) to recoverxmodn. Each integerxi is determined by computing the digitsl0,l1,...,l ei−1 in turn of itspi-ary representation:xi=l0 +l1pi+···+lei−1pei−1 i , where 0≤lj ≤pi−1. To see that the output of Algorithm 3.63 is correct, observe ﬁrst that in step 2.3 the order of αisq. Next, at iterationjof step 2.4,γ=αl0+l1q+···+lj−1qj−1 . Hence, β =( β/γ)n/qj+1 =( αx−l0−l1q−···−lj−1qj−1 )n/qj+1 =( αn/qj+1 )xi−l0−l1q−···−lj−1qj−1 =( αn/qj+1 )ljqj +···+le−1qe−1 =( αn/q)lj+···+le−1qe−1−j =( α)lj , the last equality being true because αhas orderq. Hence,log α βis indeed equal tolj. 108 Ch. 3 Number-Theoretic Reference Problems 3.63 AlgorithmPohlig-Hellman algorithm for computing discrete logarithms INPUT: a generatorαof a cyclic groupGof ordern, and an elementβ∈G. OUTPUT: the discrete logarithmx=logαβ. 1. Find the prime factorization ofn:n=pe1 1 pe2 2 ···per r , whereei≥1. 2. Forifrom 1 tordo the following: (Compute xi=l0 +l1pi+··· +lei−1pei−1 i , wherexi=xmodpei i ) 2.1 (Simplify the notation) Setq←piand e←ei. 2.2 Setγ←1and l−1←0. 2.3 Compute α←αn/q. 2.4 (Compute thelj) Forjfrom 0toe−1do the following: Compute γ←γαlj−1qj−1 and β←(βγ−1)n/qj+1 . Compute lj←log α β(e.g., using Algorithm 3.56; see Note 3.67(iii)). 2.5 Setxi←l0 +l1q+··· +le−1qe−1. 3. Use Gauss’s algorithm (Algorithm 2.121) to compute the integerx,0≤x≤n−1, such thatx≡xi (modpei i )for1≤i≤r. 4. Return(x). Example 3.64 illustrates Algorithm 3.63 with artiﬁcially small parameters. 3.64 Example (Pohlig-Hellman algorithm for logarithms inZ∗ 251) Letp=251. The element α=71 is a generator ofZ∗ 251 of ordern=250. Considerβ=210. Thenx=log71 210 is computed as follows. 1. The prime factorization ofnis250=2 ·53. 2. (a) (Compute x1 =xmod2) Compute α=αn/2 modp=250 and β=βn/2 modp=250. Thenx1 = log250 250=1 . (b) (Computex2 =xmod53 =l0 +l15+l252) i. Compute α=αn/5 modp=20. ii. Computeγ =1 and β =(βγ−1)n/5 modp=149. Using exhaustive search,5 compute l0 =log20 149=2 . iii. Computeγ =γα2 modp=2 1and β =(βγ−1)n/25 modp= 113. Using exhaustive search, computel1 =log20 113=4 . iv. Computeγ=γα4·5 modp=115 and β=(βγ−1)(p−1)/125 modp= 149. Using exhaustive search, computel2 =log20 149=2 . Hence,x2 =2+4 ·5+2 ·52 =72. 3. Finally, solve the pair of congruencesx≡1(mod2) ,x≡72(mod125) to get x=log71 210=197 . □ 3.65 FactGiven the factorization ofn, the running time of the Pohlig-Hellman algorithm (Al- gorithm 3.63) isO(∑ r i=1 ei(lgn+√ pi))group multiplications. 3.66 Note (effectiveness of Pohlig-Hellman) Fact 3.65 implies that the Pohlig-Hellman algo- rithm is efﬁcient only if each prime divisorpiofnis relatively small; that is, ifnis a smooth 5Exhaustive search is preferable to Algorithm 3.56 when the group is very small (here the order of αis5). §3.6 The discrete logarithm problem 109 integer (Deﬁnition 3.13). An example of a group in which the Pohlig-Hellman algorithm is effective follows. Consider the multiplicative groupZ∗ pwhere pis the 107-digit prime: p=227088231986781039743145181950291021585250524967592855 96453269189798311427475159776411276642277139650833937. The order ofZ∗ p isn=p−1=2 4 ·1047298 ·2247378 ·3503774. Since the largest prime divisor ofp−1is only350377, it is relatively easy to compute logarithms in this group using the Pohlig-Hellman algorithm. 3.67 Note (miscellaneous) (i) Ifnis a prime, then Algorithm 3.63 (Pohlig-Hellman) is the same as baby-step giant- step (Algorithm 3.56). (ii) In step 1 of Algorithm 3.63, a factoring algorithm which ﬁnds small factors ﬁrst (e.g., Algorithm 3.9) should be employed; if the ordernis not a smooth integer, then Al- gorithm 3.63 is inefﬁcient anyway. (iii) The storage required for Algorithm 3.56 in step 2.4 can be eliminated by using instead Pollard’s rho algorithm (Algorithm 3.60). 3.6.5 Index-calculus algorithm The index-calculus algorithm is the most powerful method known for computing discrete logarithms. The technique employed does not apply to all groups, but when it does, it of- ten gives a subexponential-time algorithm. The algorithm is ﬁrst described in the general setting of a cyclic groupG(Algorithm 3.68). Two examples are then presented to illustrate how the index-calculus algorithm works in two kinds of groups that are used in practical applications, namelyZ∗ p(Example 3.69) andF∗ 2 m (Example 3.70). The index-calculus algorithm requires the selection of a relatively small subsetSof elements ofG, called thefactor base, in such a way that a signiﬁcant fraction of elements ofGcan be efﬁciently expressed as products of elements fromS. Algorithm 3.68 proceeds to precompute a database containing the logarithms of all the elements inS, and then reuses this database each time the logarithm of a particular group element is required. The description of Algorithm 3.68 is incomplete for two reasons. Firstly, a technique for selecting the factor baseSis not speciﬁed. Secondly, a method for efﬁciently generating relations of the form (3.5) and (3.7) is not speciﬁed. The factor baseSmust be a subset of Gthat is small (so that the system of equations to be solved in step 3 is not too large), but not too small (so that the expected number of trials to generate a relation (3.5) or (3.7) is not too large). Suitable factor bases and techniques for generating relations are known for some cyclic groups includingZ ∗ p(see§3.6.5(i)) andF∗ 2 m (see§3.6.5(ii)), and, moreover, the multiplicative groupF∗ q of a general ﬁnite ﬁeldFq. 3.68 AlgorithmIndex-calculus algorithm for discrete logarithms in cyclic groups INPUT: a generatorαof a cyclic groupGof ordern, and an elementβ∈G. OUTPUT: the discrete logarithmy=logαβ. 1. (Select a factor baseS) Choose a subsetS={p1,p2,...,p t}ofGsuch that a “sig- niﬁcant proportion” of all elements inGcan be efﬁciently expressed as a product of elements fromS. 2. (Collect linear relations involving logarithms of elements inS) 110 Ch. 3 Number-Theoretic Reference Problems 2.1 Select a random integerk,0≤k≤n−1, and computeαk. 2.2 Try to writeαkas a product of elements inS: αk= t∏ i=1 pci i ,c i≥0. (3.5) If successful, take logarithms of both sides of equation (3.5) to obtain a linear relation k≡ t∑ i=1 cilogαpi (modn). (3.6) 2.3 Repeat steps 2.1 and 2.2 untilt+crelations of the form (3.6) are obtained (c is a small positive integer, e.g.c=10, such that the system of equations given by thet+crelations has a unique solution with high probability). 3. (Find the logarithms of elements inS) Working modulon, solve the linear system oft+cequations (intunknowns) of the form (3.6) collected in step 2 to obtain the values oflogαpi,1≤i≤t. 4. (Compute y) 4.1 Select a random integerk,0≤k≤n−1, and computeβ·αk. 4.2 Try to writeβ·αkas a product of elements inS: β·αk = t∏ i=1 pdi i ,d i≥0. (3.7) If the attempt is unsuccessful then repeat step 4.1. Otherwise, taking logarithms of both sides of equation (3.7) yieldslogαβ=(∑t i=1 dilogαpi−k)mod n; thus, computey=(∑ t i=1 dilogαpi−k)mod nand return(y). (i) Index-calculus algorithm inZ∗ p For the ﬁeldZp,pa prime, the factor baseScan be chosen as the ﬁrsttprime numbers. A relation (3.5) is generated by computingαkmodpand then using trial division to check whether this integer is a product of primes inS. Example 3.69 illustrates Algorithm 3.68 inZ∗ pon a problem with artiﬁcially small parameters. 3.69 Example (Algorithm 3.68 for logarithms inZ∗ 229) Letp=229. The elementα=6 is a generator ofZ∗ 229 of ordern=228. Considerβ =13. Thenlog6 13is computed as follows, using the index-calculus technique. 1. The factor base is chosen to be the ﬁrst5primes:S={2,3,5,7,11}. 2. The following six relations involving elements of the factor base are obtained (un- successful attempts are not shown): 6100 mod229=180=2 2 ·32 ·5 618 mod229=176=2 4 ·11 612 mod229=165=3 ·5·11 662 mod229=154=2 ·7·11 6143 mod229=198=2 ·32 ·11 6206 mod229=210=2 ·3·5·7. §3.6 The discrete logarithm problem 111 These relations yield the following six equations involving the logarithms of ele- ments in the factor base: 100 ≡ 2log6 2+2log 6 3+log6 5 (mod228) 18 ≡ 4log6 2+log6 11 (mod228) 12 ≡ log6 3+log6 5+log6 11 (mod228) 62 ≡ log6 2+log6 7+log6 11 (mod228) 143 ≡ log6 2+2log 6 3+log6 11 (mod228) 206 ≡ log6 2+log6 3+log6 5+log6 7 (mod228). 3. Solving the linear system of six equations in ﬁve unknowns (the logarithmsxi = log6 pi) yields the solutionslog6 2=21 ,log6 3=208 ,log6 5=98 ,log6 7=107 , and log6 11=162 . 4. Suppose that the integerk=77 is selected. Sinceβ·αk =13 ·677 mod229= 147=3 ·72, it follows that log6 13 = (log6 3+2log 6 7−77)mod228 = 117. □ (ii) Index-calculus algorithm inF∗ 2m The elements of the ﬁnite ﬁeldF2m are represented as polynomials inZ2[x]of degree at most m−1, where multiplication is performed modulo a ﬁxed irreducible polynomialf(x) of degreeminZ2[x](see§2.6). The factor baseScan be chosen as the set of all irreducible polynomials inZ2[x]of degree at most some prescribed boundb. A relation (3.5) is gener- ated by computingαk modf(x)and then using trial division to check whether this poly- nomial is a product of polynomials inS. Example 3.70 illustrates Algorithm 3.68 inF∗ 2m on a problem with artiﬁcially small parameters. 3.70 Example (Algorithm 3.68 for logarithms inF∗ 2 7 ) The polynomialf(x)= x7 +x+1is irreducible overZ2. Hence, the elements of the ﬁnite ﬁeldF27 of order128can be repre- sented as the set of all polynomials inZ2[x]of degree at most6, where multiplication is performed modulof(x). The order ofF∗ 27 isn=27 −1=127 , andα=xis a generator ofF∗ 2 7 . Supposeβ=x4 +x3 +x2 +x+1. Theny=logxβcan be computed as follows, using the index-calculus technique. 1. The factor base is chosen to be the set of all irreducible polynomials inZ2[x]of degree at most3:S={x,x+1,x2 +x+1,x3 +x+1,x3 +x2 +1}. 2. The following ﬁve relations involving elements of the factor base are obtained (un- successful attempts are not shown): x18 modf(x)= x6 +x4 =x4(x+1)2 x105 modf(x)= x6 +x5 +x4 +x =x(x+1)2(x3 +x2 +1) x72 modf(x)= x6 +x5 +x3 +x2 =x2(x+1)2(x2 +x+1) x45 modf(x)= x5 +x2 +x+1 =( x+1)2(x3 +x+1) x121 modf(x)= x6 +x5 +x4 +x3 +x2 +x+1=( x3 +x+1)(x3 +x2 +1). These relations yield the following ﬁve equations involving the logarithms of ele- ments in the factor base (for convenience of notation, letp1 =logxx,p2 =logx(x+ 112 Ch. 3 Number-Theoretic Reference Problems 1),p3 =logx(x2 +x+1),p4 =logx(x3 +x+1), andp5 =logx(x3 +x2 +1)): 18 ≡ 4p1 +2p2 (mod127) 105 ≡ p1 +2p2 +p5 (mod127) 72 ≡ 2p1 +2p2 +p3 (mod127) 45 ≡ 2p2 +p4 (mod127) 121 ≡ p4 +p5 (mod127). 3. Solving the linear system of ﬁve equations in ﬁve unknowns yields the valuesp1 =1, p2 =7,p3 =56,p4 =31, andp5 =90. 4. Supposek=66 is selected. Since βαk=(x4 +x3 +x2 +x+1)x66 modf(x)= x5 +x3 +x=x(x2 +x+1)2, it follows that logx(x4 +x3 +x2 +x+1)=( p1 +2p3 −66)mod127 = 47. □ 3.71 Note (running time of Algorithm 3.68) To optimize the running time of the index-calculus algorithm, the sizetof the factor base should be judiciously chosen. The optimal selection relies on knowledge concerning the distribution of smooth integers in the interval[1,p−1] for the case ofZ∗ p, and for the case ofF∗ 2 m on the distribution ofsmooth polynomials(that is, polynomials all of whose irreducible factors have relatively small degrees) among poly- nomials inF2[x]of degree less thanm. With an optimal choice oft, the index-calculus al- gorithm as described above forZ∗ pand F∗ 2 m has an expected running time ofLq[1 2 ,c]where q=porq=2m, andc> 0is a constant. 3.72 Note (fastest algorithms known for discrete logarithms inZ∗ p and F∗ 2 m ) Currently, the best algorithm known for computing logarithms inF∗ 2 m is a variation of the index-calculus algo- rithm calledCoppersmith’ s algorithm, with an expected running time ofL2m [1 3 ,c]for some constantc< 1.587. The best algorithm known for computing logarithms inZ∗ p is a varia- tion of the index-calculus algorithm called thenumber ﬁeld sieve, with an expected running time ofLp[1 3 ,1.923]. The latest efforts in these directions are surveyed in the Notes section (§3.12). 3.73 Note (parallelization of the index-calculus algorithm) (i) For the optimal choice of parameters, the most time-consuming phase of the index- calculus algorithm is usually the generation of relations involving factor base loga- rithms (step 2 of Algorithm 3.68). The work for this stage can be easily distributed among a network of processors by simply having the processors search for relations independently of each other. The relations generated are collected by a central pro- cessor. When enough relations have been generated, the corresponding system of lin- ear equations can be solved (step 3 of Algorithm 3.68) on a single (possibly parallel) computer. (ii) The database of factor base logarithms need only be computed once for a given ﬁ- nite ﬁeld. Relative to this, the computation of individual logarithms (step 4 of Algo- rithm 3.68) is considerably faster. §3.7 The Difﬁe-Hellman problem 113 3.6.6 Discrete logarithm problem in subgroups ofZ∗ p The discrete logarithm problem in subgroups ofZ∗ phas special interest because its presumed intractability is the basis for the security of the U.S. Government NIST Digital Signature Algorithm (§11.5.1), among other cryptographic techniques. Letpbe a prime andqa prime divisor ofp−1. LetGbe the unique cyclic subgroup ofZ∗ pof orderq, and letαbe a generator ofG. Then the discrete logarithm problem inGis the following: givenp,q,α, andβ∈G, ﬁnd the unique integerx,0≤x≤q−1, such that αx≡β(modp). The powerful index-calculus algorithms do not appear to apply directly inG. That is, one needs to apply the index-calculus algorithm in the groupZ∗ pitself in order to compute logarithms in the smaller groupG. Consequently, there are two approaches one could take to computing logarithms inG: 1. Use a “square-root” algorithm directly inG, such as Pollard’s rho algorithm (Algo- rithm 3.60). The running time of this approach isO(√ q). 2. Letγbe a generator ofZ∗ p , and letl=(p−1)/q. Use an index-calculus algorithm inZ∗ p to ﬁnd integersyand zsuch thatα=γy and β=γz. Thenx=logαβ= (z/l)(y/l)−1 modq. (Sinceyand zare both divisible byl,y/land z/lare indeed integers.) The running time of this approach isLp[1 3 ,c]if the number ﬁeld sieve is used. Which of the two approaches is faster depends on the relative size of√ qand Lp[1 3 ,c]. 3.7 The Difﬁe-Hellman problem The Difﬁe-Hellman problem is closely related to the well-studied discrete logarithm prob- lem (DLP) of§3.6. It is of signiﬁcance to public-key cryptography because its apparent in- tractability forms the basis for the security of many cryptographic schemes including Difﬁe- Hellman key agreement and its derivatives (§12.6), and ElGamal public-key encryption (§8.4). 3.74 DeﬁnitionThe Difﬁe-Hellman problem (DHP)is the following: given a primep, a gen- eratorαofZ∗ p, and elementsαamodpand αbmodp, ﬁndαabmodp. 3.75 DeﬁnitionThe generalized Difﬁe-Hellman problem (GDHP)is the following: given a ﬁ- nite cyclic groupG, a generatorαofG, and group elementsαaand αb, ﬁndαab. Suppose that the discrete logarithm problem inZ∗ p could be efﬁciently solved. Then givenα,p,αamodpand αbmodp, one could ﬁrst ﬁndafrom α,p, andαamodpby solving a discrete logarithm problem, and then compute(αb)a =αabmodp. This estab- lishes the following relation between the Difﬁe-Hellman problem and the discrete logarithm problem. 3.76 FactDHP ≤P DLP. That is, DHP polytime reduces to the DLP. More generally, GDHP ≤P GDLP. The question then remains whether the GDLP and GDHP are computationally equiv- alent. This remains unknown; however, some recent progress in this regard is summarized in Fact 3.77. Recall thatφis the Euler phi function (Deﬁnition 2.100), and an integer is B-smooth if all its prime factors are≤B(Deﬁnition 3.13). 114 Ch. 3 Number-Theoretic Reference Problems 3.77 Fact(known equivalences between GDHP and GDLP) (i) Letpbe a prime where the factorization ofp−1is known. Suppose also thatφ(p−1) isB-smooth, whereB=O((lnp)c)for some constantc. Then the DHP and DLP in Z∗ pare computationally equivalent. (ii) More generally, letGbe a ﬁnite cyclic group of ordernwhere the factorization of nis known. Suppose also thatφ(n)isB-smooth, whereB=O((lnn)c)for some constantc. Then the GDHP and GDLP inGare computationally equivalent. (iii) LetGbe a ﬁnite cyclic group of ordernwhere the factorization ofnis known. If for each prime divisorpofneitherp−1orp+1isB-smooth, whereB=O((lnn)c) for some constantc, then the GDHP and GDLP inGare computationally equivalent. 3.8 Composite moduli The group of units ofZn, namelyZ∗ n , has been proposed for use in several cryptographic mechanisms, including the key agreement protocols of Yacobi and McCurley (see§12.6 notes on page 538) and the identiﬁcation scheme of Girault (see§10.4 notes on page 423). There are connections of cryptographic interest between the discrete logarithm and Difﬁe- Hellman problems in (cyclic subgroups of)Z∗ n, and the problem of factoringn. This section summarizes the results known along these lines. 3.78 FactLetnbe a composite integer. If the discrete logarithm problem inZ∗ ncan be solved in polynomial time, thenncan be factored in expected polynomial time. In other words, the discrete logarithm problem inZ∗ n is at least as difﬁcult as the prob- lem of factoringn. Fact 3.79 is a partial converse to Fact 3.78 and states that the discrete logarithm inZ∗ nis no harder than the combination of the problems of factoringnand com- puting discrete logarithms inZ∗ pfor each prime factorpofn. 3.79 FactLetnbe a composite integer. The discrete logarithm problem inZ∗ n polytime reduces to the combination of the integer factorization problem and the discrete logarithm problem inZ∗ pfor each prime factorpofn. Fact 3.80 states that the Difﬁe-Hellman problem inZ∗ n is at least as difﬁcult as the prob- lem of factoringn. 3.80 FactLetn=pqwhere pand qare odd primes. If the Difﬁe-Hellman problem inZ∗ ncan be solved in polynomial time for a non-negligible proportion of all basesα∈Z∗ n, thenn can be factored in expected polynomial time. 3.9 Computing individual bits While the discrete logarithm problem inZ∗ p(§3.6), the RSA problem (§3.3), and the problem of computing square roots modulo a composite integern(§3.5.2) appear to be intractable, when the problem parameters are carefully selected, it remains possible that it is much eas- ier to compute some partial information about the solution, for example, its least signiﬁ- cant bit. It turns out that while some bits of the solution to these problems are indeed easy §3.9 Computing individual bits 115 to compute, other bits are equally difﬁcult to compute as the entire solution. This section summarizes the results known along these lines. The results have applications to the con- struction of probabilistic public-key encryption schemes (§8.7) and pseudorandom bit gen- eration (§5.5). Recall (Deﬁnition 1.12) that a functionfis called a one-way function iff(x)is easy to compute for allxin its domain, but for essentially allyin the range off, it is computa- tionally infeasible to ﬁnd anyxsuch thatf(x)= y. Three (candidate) one-way functions Although no proof is known for the existence of a one-way function, it is widely believed that one-way functions do exist (cf. Remark 9.12). The following are candidate one-way functions (in fact, one-way permutations) since they are easy to compute, but their inver- sion requires the solution of the discrete logarithm problem inZ ∗ p, the RSA problem, or the problem of computing square roots modulon, respectively: 1. exponentiation modulop. Letpbe a prime and letαbe a generator ofZ∗ p. The func- tion isf:Z∗ p−→Z∗ p deﬁned asf(x)= αxmodp. 2. RSA function. Letpand qbe distinct odd primes,n=pq, and letebe an integer such thatgcd(e,(p−1)(q−1))=1 . The function isf :Zn −→Zndeﬁned as f(x)= xemodn. 3. Rabin function. Letn=pq, wherepand qare distinct primes each congruent to 3modulo 4. The function isf :Qn −→Qndeﬁned asf(x)= x2 modn. (Re- call from Fact 2.160 thatfis a permutation, and from Fact 3.46 that invertingf, i.e., computing principal square roots, is difﬁcult assuming integer factorization is intractable.) The following deﬁnitions are used in§3.9.1, 3.9.2, and 3.9.3. 3.81 DeﬁnitionLetf :S−→Sbe a one-way function, whereSis a ﬁnite set. A Boolean predicateB:S−→{0,1}is said to be ahard predicateforfif: (i)B(x)is easy to compute givenx∈S; and (ii) an oracle which computesB(x)correctly with non-negligible advantage6 given only f(x)(wherex∈S) can be used to invertfeasily. Informally,Bis a hard predicate for the one-way functionfif determining the single bitB(x)of information aboutx, given onlyf(x), is as difﬁcult as invertingfitself. 3.82 DeﬁnitionLetf:S−→Sbe a one-way function, whereSis a ﬁnite set. Ak-bit predi- cateB(k) :S−→{0,1}kis said to be ahardk-bit predicateforfif: (i)B(k)(x)is easy to compute givenx∈S; and (ii) for every Boolean predicateB : {0,1}k −→ {0,1}, an oracle which computes B(B(k)(x))correctly with non-negligible advantage given onlyf(x)(wherex∈S) can be used to invertfeasily. If such aB(k) exists, thenfis said tohidekbits, or thekbits are said to besimultaneously secure. Informally,B(k) is a hardk-bit predicate for the one-way functionfif determining any partial information whatsoever aboutB(k)(x), given onlyf(x), is as difﬁcult as inverting fitself. 6In Deﬁnitions 3.81 and 3.82, the probability is taken over all choices ofx∈Sand random coin tosses of the oracle. 116 Ch. 3 Number-Theoretic Reference Problems 3.9.1 The discrete logarithm problem inZ∗ p— individual bits Letpbe an odd prime andαa generator ofZ∗ p. Assume that the discrete logarithm problem inZ∗ p is intractable. Letβ ∈Z∗ p , and letx=l o gαβ. Recall from Fact 2.135 thatβis a quadratic residue modulopif and only ifxis even. Hence, the least signiﬁcant bit of xis equal to(1− (β p ) )/2, where the Legendre symbol (β p ) can be efﬁciently computed (Algorithm 2.149). More generally, the following is true. 3.83 FactLetpbe an odd prime, and letαbe a generator ofZ∗ p . Suppose thatp−1=2 st, where tis odd. Then there is an efﬁcient algorithm which, givenβ∈Z∗ p , computes thes least signiﬁcant bits ofx=logαβ. 3.84 FactLetpbe a prime andαa generator ofZ∗ p . Deﬁne the predicateB:Z∗ p −→{0,1}by B(x)= { 0, if1≤x≤(p−1)/2, 1, if(p−1)/2<x ≤p−1. Then Bis a hard predicate for the function of exponentiation modulop. In other words, givenp,α, andβ, computing the single bitB(x)of the discrete logarithmx=logαβis as difﬁcult as computing the entire discrete logarithm. 3.85 FactLetpbe a prime andαa generator ofZ∗ p . Letk=O(lglgp)be an integer. Let the interval[1,p−1]be partitioned into2kintervalsI0,I1,...,I 2k−1 of roughly equal lengths. Deﬁne thek-bit predicateB(k) :Z∗ p −→{0,1}kby B(k)(x)= jifx∈Ij. ThenB(k) is a hardk-bit predicate for the function of exponentiation modulop. 3.9.2 The RSA problem — individual bits Let nbe a product of two distinct odd primespand q, and letebe an integer such that gcd(e,(p−1)(q−1))=1 . Givenn,e, andc=xemodn(for somex∈Zn), some information aboutxis easily obtainable. For example, sinceeis an odd integer, (c n ) = (xe n ) = (x n )e = (x n ) , and hence the single bit of information (x n ) can be obtained simply by computing the Jacobi symbol (c n ) (Algorithm 2.149). There are, however, other bits of information aboutxthat are difﬁcult to compute, as the next two results show. 3.86 FactDeﬁne the predicateB:Zn −→ {0,1}by B(x)= xmod2; that is,B(x)is the least signiﬁcant bit ofx. ThenBis a hard predicate for the RSA function (see page 115). 3.87 FactLetk=O(lglgn)be an integer. Deﬁne thek-bit predicateB(k) :Zn −→ {0,1}k by B(k)(x)= xmod2k. That is,B(k)(x)consists of thekleast signiﬁcant bits ofx. Then B(k) is a hardk-bit predicate for the RSA function. Thus the RSA function haslglgnsimultaneously secure bits. §3.10 The subset sum problem 117 3.9.3 The Rabin problem — individual bits Letn=pq, wherepand qare distinct primes each congruent to3modulo 4. 3.88 FactDeﬁne the predicateB:Qn −→ {0,1}by B(x)= xmod2; that is,B(x)is the least signiﬁcant bit of the quadratic residuex. Then Bis a hard predicate for the Rabin function (see page 115). 3.89 FactLetk=O(lglgn)be an integer. Deﬁne thek-bit predicateB(k) :Qn−→{0,1}k by B(k)(x)= xmod2k. That is,B(k)(x)consists of thekleast signiﬁcant bits of the quadratic residuex. ThenB(k) is a hardk-bit predicate for the Rabin function. Thus the Rabin function haslglgnsimultaneously secure bits. 3.10 The subset sum problem The difﬁculty of the subset sum problem was the basis for the (presumed) security of the ﬁrst public-key encryption scheme, called the Merkle-Hellman knapsack scheme (§8.6.1). 3.90 DeﬁnitionThe subset sum problem(SUBSET-SUM) is the following: given a set{a1,a2, ...,a n}of positive integers, called aknapsack set, and a positive integers, determine whether or not there is a subset of theaj that sum tos. Equivalently, determine whether or not there existxi∈{0,1},1≤i≤n, such that∑ n i=1 aixi=s. The subset sum problem above is stated as a decision problem. It can be shown that the problem is computationally equivalent to its computational version which is to actually determine thexisuch that∑ n i=1 aixi=s, provided that suchxiexist. Fact 3.91 provides evidence of the intractability of the subset sum problem. 3.91 FactThe subset sum problem isNP -complete. The computational version of the subset sum problem isNP -hard (see Example 2.74). Algorithms 3.92 and 3.94 give two methods for solving the computational version of the subset sum problem; both are exponential-time algorithms. Algorithm 3.94 is the fastest method known for the general subset sum problem. 3.92 AlgorithmNaive algorithm for subset sum problem INPUT: a set of positive integers{a1,a2,...,a n}and a positive integers. OUTPUT: xi∈{0,1},1≤i≤n, such that∑ n i=1 aixi=s, provided suchxiexist. 1. For each possible vector(x1,x2,...,x n)∈(Z2)ndo the following: 1.1 Compute l=∑ n i=1 aixi. 1.2 Ifl=sthen return(a solution is(x1,x2,...,x n)). 2. Return(no solution exists). 3.93 FactAlgorithm 3.92 takesO(2n)steps and, hence, is inefﬁcient. 118 Ch. 3 Number-Theoretic Reference Problems 3.94 AlgorithmMeet-in-the-middle algorithm for subset sum problem INPUT: a set of positive integers{a1,a2,...,a n}and a positive integers. OUTPUT: xi∈{0,1},1≤i≤n, such that∑ n i=1 aixi=s, provided suchxiexist. 1. Sett←⌊n/2⌋. 2. Construct a table with entries(∑ t i=1 aixi,(x1,x2,...,x t))for(x1,x2,...,x t)∈ (Z2)t. Sort this table by ﬁrst component. 3. For each(xt+1,xt+2,...,x n)∈(Z2)n−t, do the following: 3.1 Compute l=s−∑n i=t+1 aixiand check, using a binary search, whetherlis the ﬁrst component of some entry in the table. 3.2 Ifl=∑ t i=1 aixithen return(a solution is(x1,x2,...,x n)). 4. Return(no solution exists). 3.95 FactAlgorithm 3.94 takesO(n2n/2)steps and, hence, is inefﬁcient. 3.10.1 TheL3-lattice basis reduction algorithm The L3-lattice basis reduction algorithm is a crucial component in many number-theoretic algorithms. It is useful for solving certain subset sum problems, and has been used for crypt- analyzing public-key encryption schemes which are based on the subset sum problem. 3.96 DeﬁnitionLetx=(x1,x2,...,x n)andy=(y1,y2,...,y n)be two vectors inRn. The inner productofxand yis the real number <x,y> = x1y1 +x2y2 +··· +xnyn. 3.97 DeﬁnitionLety=(y1,y2,...,y n)be a vector inRn. Thelengthofyis the real number ∥y∥= √ <y,y> = √ y2 1 +y2 2 +··· +y2n. 3.98 DeﬁnitionLetB={b1,b2,...,b m}be a set of linearly independent vectors inRn(so thatm≤n). The setLof all integer linear combinations ofb1,b2,...,b mis called alattice ofdimensionm; that is,L=Zb1 +Zb2 +··· +Zbm. The setBis called abasisfor the latticeL. A lattice can have many different bases. A basis consisting of vectors of relatively small lengths is calledreduced. The following deﬁnition provides a useful notion of a re- duced basis, and is based on the Gram-Schmidt orthogonalization process. 3.99 DeﬁnitionLetB={b1,b2,...,b n}be a basis for a latticeL⊂Rn. Deﬁne the vectors b∗ i (1≤i≤n) and the real numbersµi,j(1≤j<i ≤n) inductively by µi,j = <bi,b∗ j > <b∗ j ,b∗ j >, 1≤j<i ≤n, (3.8) b∗ i = bi− i−1∑ j=1 µi,jb∗ j , 1≤i≤n. (3.9) The basisBis said to bereduced(more precisely,Lov´asz-reduced)i f \|µi,j\|≤ 1 2, for1≤j<i ≤n §3.10 The subset sum problem 119 (where\|µi,j\|denotes the absolute value ofµi,j), and ∥b∗ i∥2 ≥ (3 4−µ2 i,i−1 ) ∥b∗ i−1 ∥2, for1<i ≤n. (3.10) Fact 3.100 explains the sense in which the vectors in a reduced basis are relatively short. 3.100 FactLetL⊂Rnbe a lattice with a reduced basis{b1,b2,...,b n}. (i) For every non-zerox∈L,∥b1∥≤2(n−1)/2∥x∥. (ii) More generally, for any set{a1,a2,...,a t}of linearly independent vectors inL, ∥bj∥≤2(n−1)/2 max(∥a1∥,∥a2∥,..., ∥at∥), for1≤j≤t. The L3-lattice basis reduction algorithm (Algorithm 3.101) is a polynomial-time algo- rithm (Fact 3.103) for ﬁnding a reduced basis, given a basis for a lattice. 3.101 AlgorithmL3-lattice basis reduction algorithm INPUT: a basis(b1,b2,...,b n)for a latticeLinRm,m≥n. OUTPUT: a reduced basis forL. 1. b∗ 1←b1,B1←<b∗ 1 ,b∗ 1 >. 2. Forifrom 2tondo the following: 2.1 b∗ i←bi. 2.2 Forjfrom 1toi−1, setµi,j←<bi,b∗ j >/Bjand b∗ i ←b∗ i −µi,jb∗ j . 2.3 Bi←<b∗ i,b∗ i >. 3. k←2. 4. Execute subroutine RED(k,k−1) to possibly update someµi,j. 5. IfBk<(3 4 −µ2 k,k−1)Bk−1 then do the following: 5.1 Setµ←µk,k−1, B←Bk+µ2Bk−1, µk,k−1←µBk−1/B, Bk←Bk−1Bk/B, and Bk−1←B. 5.2 Exchangebkand bk−1. 5.3 Ifk> 2then exchangeµk,j and µk−1,jforj=1,2,...,k −2. 5.4 Fori=k+1,k+2,...,n : Sett←µi,k, µi,k←µi,k−1 −µt, andµi,k−1←t+µk,k−1µi,k. 5.5 k←max(2,k−1). 5.6 Go to step 4. Otherwise, forl=k−2,k−3,..., 1, execute RED(k,l), and ﬁnally setk←k+1. 6. Ifk≤nthen go to step 4. Otherwise, return(b1,b2,...,b n). RED (k,l)I f\|µk,l\|>1 2 then do the following: 1. r←⌊0.5+µk,l⌋,bk←bk−rbl. 2. Forjfrom 1 tol−1, setµk,j←µk,j−rµl,j. 3. µk,l←µk,l−r. 3.102 Note (explanation of selected steps of Algorithm 3.101) (i) Steps 1 and 2 initialize the algorithm by computingb∗ i (1≤i≤n) andµi,j(1≤j< i≤n) as deﬁned in equations (3.9) and (3.8), and alsoBi=<b∗ i ,b∗ i >(1≤i≤n). (ii)kis a variable such that the vectorsb1,b2,...,b k−1 are reduced (initiallyk=2 in step 3). The algorithm then attempts to modifybk, so thatb1,b2,...,b kare reduced. 120 Ch. 3 Number-Theoretic Reference Problems (iii) In step 4, the vectorbkis modiﬁed appropriately so that\|µk,k−1\|≤ 1 2 , and theµk,j are updated for1≤j<k −1. (iv) In step 5, if the condition of equation (3.10) is violated fori=k, then vectorsbk and bk−1 are exchanged and their corresponding parameters are updated. Also,kis decremented by 1 since then it is only guaranteed thatb1,b2,...,b k−2 are reduced. Otherwise,bkis modiﬁed appropriately so that\|µk,j\|≤ 1 2 forj=1,2,...,k −2, while keeping (3.10) satisﬁed.kis then incremented because nowb1,b2,...,b kare reduced. It can be proven that theL3-algorithm terminates after a ﬁnite number of iterations. Note that ifLis an integer lattice, i.e.L⊂Zn, then theL3-algorithm only operates on rational numbers. The precise running time is given next. 3.103 FactLetL⊂Znbe a lattice with basis{b1,b2,...,b n}, and letC∈R,C≥2, be such that∥bi∥2 ≤Cfori=1,2,...,n . Then the number of arithmetic operations needed by Algorithm 3.101 isO(n4 logC), on integers of sizeO(nlogC)bits. 3.10.2 Solving subset sum problems of low density The density of a knapsack set, as deﬁned below, provides a measure of the size of the knap- sack elements. 3.104 DeﬁnitionLetS={a1,a2,...,a n}be a knapsack set. ThedensityofSis deﬁned to be d= n max{lgai\|1≤i≤n}. Algorithm 3.105 reduces the subset sum problem to one of ﬁnding a particular short vector in a lattice. By Fact 3.100, the reduced basis produced by theL3-algorithm includes a vector of length which is guaranteed to be within a factor of2(n−1)/2 of the shortest non- zero vector of the lattice. In practice, however, theL3-algorithm usually ﬁnds a vector which is much shorter than what is guaranteed by Fact 3.100. Hence, theL3-algorithm can be expected to ﬁnd the short vector which yields a solution to the subset sum problem, provided that this vector is shorter than most of the non-zero vectors in the lattice. 3.105 AlgorithmSolving subset sum problems usingL3-algorithm INPUT: a set of positive integers{a1,a2,...,a n}and an integers. OUTPUT: xi∈{0,1},1≤i≤n, such that∑ n i=1 aixi=s, provided suchxiexist. 1. Letm=⌈1 2 √ n⌉. 2. Form an(n+1)-dimensional latticeLwith basis consisting of the rows of the matrix A=      100 ··· 0 ma 1 010 ··· 0 ma2 001 ··· 0 ma3 ... ... ... ... ... ... 000 ··· 1 man 1 2 1 2 1 2 ··· 1 2 ms      3. Find a reduced basisBofL(use Algorithm 3.101). 4. For each vectory=(y 1,y2,...,y n+1)inB, do the following: §3.10 The subset sum problem 121 4.1 Ifyn+1 =0 and yi∈{−1 2 ,1 2 }for alli=1,2,...,n , then do the following: For i=1,2,...,n , setxi←yi+1 2 . If∑ n i=1 aixi=s, then return(a solution is(x1,x2,...,x n)). For i=1,2,...,n , setxi←−yi+1 2 . If∑ n i=1 aixi=s, then return(a solution is(x1,x2,...,x n)). 5. Return(FAILURE). (Either no solution exists, or the algorithm has failed to ﬁnd one.) Justiﬁcation.Let the rows of the matrixAbe b1,b2,...,b n+1, and letLbe the(n+1)- dimensional lattice generated by these vectors. If(x1,x2,...,x n)is a solution to the subset sum problem, the vectory = ∑ n i=1 xibi −bn+1 is inL. Note thatyi ∈{ −1 2 ,1 2 }for i=1,2,...,n and yn+1 =0. Since∥y∥= √ y2 1 +y2 2 +··· +y2 n+1 the vectoryis a vector of short length inL. If the density of the knapsack set is small, i.e. theaiare large, then most vectors inLwill have relatively large lengths, and henceymay be the unique shortest non-zero vector inL. If this is indeed the case, then there is good possibility of the L3-algorithm ﬁnding a basis which includes this vector. Algorithm 3.105 is not guaranteed to succeed. Assuming that theL3-algorithm always produces a basis which includes the shortest non-zero lattice vector, Algorithm 3.105 suc- ceeds with high probability if the density of the knapsack set is less than0.9408. 3.10.3 Simultaneous diophantine approximation Simultaneous diophantine approximation is concerned with approximating a vector(q1 q,q2 q, ..., qn q )of rational numbers (more generally, a vector(α1,α2,...,α n)of real numbers) by a vector(p1 p,p2 p,..., pn p )of rational numbers with a smaller denominatorp. Algorithms for ﬁnding simultaneous diophantine approximation have been used to break some knap- sack public-key encryption schemes (§8.6). 3.106 DeﬁnitionLetδbe a real number. The vector(p1 p,p2 p,..., pn p )of rational numbers is said to be asimultaneous diophantine approximation ofδ-qualityto the vector(q1 q,q2 q,..., qn q ) of rational numbers ifp<q and ⏐⏐ ⏐ ⏐pq i q −pi ⏐ ⏐ ⏐ ⏐≤q −δfori=1,2,...,n. (The largerδis, the better is the approximation.) Furthermore, it is anunusually good si- multaneous diophantine approximation(UGSDA) if δ> 1 n. Fact 3.107 shows that an UGSDA is indeed unusual. 3.107 FactFor n≥2, the set Sn(q)= {(q1 q,q2 q,..., qn q ) \|0≤qi<q, gcd(q1,q2,...,q n,q)=1 } has at least1 2 qnmembers. Of these, at mostO(qn(1−δ)+1)members have at least oneδ- quality simultaneous diophantine approximation. Hence, for any ﬁxedδ> 1 n, the fraction of members ofSn(q)having at least one UGSDA approaches0asq→∞. Algorithm 3.108 reduces the problem of ﬁnding aδ-quality simultaneous diophantine approximation, and hence also a UGSDA, to the problem of ﬁnding a short vector in a lat- tice. The latter problem can (usually) be solved using theL3-lattice basis reduction. 122 Ch. 3 Number-Theoretic Reference Problems 3.108 AlgorithmFinding aδ-quality simultaneous diophantine approximation INPUT: a vectorw=(q1 q,q2 q,..., qn q )of rational numbers, and a rational numberδ> 0. OUTPUT: a δ-quality simultaneous diophantine approximation(p1 p,p2 p,..., pn p )ofw. 1. Choose an integerλ≈qδ. 2. Use Algorithm 3.101 to ﬁnd a reduced basisBfor the(n+1)-dimensional latticeL which is generated by the rows of the matrix A=      λq 00 ··· 00 0 λq 0 ··· 00 00 λq ··· 00 ... ... ... ... ... ... 000 ··· λq 0 −λq 1 −λq2 −λq3 ··· −λqn 1      3. For eachv=(v 1,v2,...,v n,vn+1)inBsuch thatvn+1 ̸=q, do the following: 3.1 p←vn+1. 3.2 Forifrom 1 ton, setpi←1 q (vi λ +pqi ) . 3.3 If\|pqi q −pi\|≤ q−δfor eachi,1≤i≤n, then return(p1 p,p2 p,..., pn p ). 4. Return(FAILURE). (Either noδ-quality simultaneous diophantine approximation ex- ists, or the algorithm has failed to ﬁnd one.) Justiﬁcation.Let the rows of the matrixAbe denoted byb1,b2,...,b n+1. Suppose that (q1 q,q2 q,..., qn q )has aδ-quality approximation(p1 p,p2 p,..., pn p ). Then the vector x = p1b1 +p2b2 +··· +pnbn+pbn+1 =( λ(p1q−pq1),λ(p2q−pq2),...,λ (pnq−pqn),p) is inLand has length less than approximately(√ n+1)q. Thusxis short compared to the original basis vectors, which are of length roughlyq1+δ. Also, ifv=(v1,v2,...,v n+1)is a vector inLof length less thanq, then the vector(p1 p,p2 p,..., pn p )deﬁned in step 3 is aδ- quality approximation. Hence there is a good possibility that theL3-algorithm will produce a reduced basis which includes a vectorvthat corresponds to aδ-quality approximation. 3.11 Factoring polynomials over ﬁnite ﬁelds The problem considered in this section is the following: given a polynomialf(x)∈Fq[x], withq=pm, ﬁnd its factorizationf(x)= f1(x)e1 f2(x)e2 ···ft(x)et , where eachfi(x)is an irreducible polynomial inFq[x]and eachei≥1.(eiis called themultiplicityof the fac- torfi(x).) Several situations call for the factoring of polynomials over ﬁnite ﬁelds, such as index-calculus algorithms inF∗ 2m (Example 3.70) and Chor-Rivest public-key encryption (§8.6.2). This section presents an algorithm for square-free factorization, and Berlekamp’s classical deterministic algorithm for factoring polynomials which is efﬁcient if the under- lying ﬁeld is small. Efﬁcient randomized algorithms are known for the case of largeq; ref- erences are provided on page 132. §3.11 Factoring polynomials over ﬁnite ﬁelds 123 3.11.1 Square-free factorization Observe ﬁrst thatf(x)may be divided by its leading coefﬁcient. Thus, it may be assumed thatf(x)is monic (see Deﬁnition 2.187). This section shows how the problem of factoring a monic polynomialf(x)may then be reduced to the problem of factoring one or more monic square-free polynomials. 3.109 DeﬁnitionLetf(x)∈Fq[x]. Thenf(x)issquare-freeif it has no repeated factors, i.e., there is no polynomialg(x)withdegg(x)≥1such thatg(x)2 dividesf(x). Thesquare- free factorizationoff(x)isf(x)= ∏k i=1 fi(x)i, where eachfi(x)is a square-free poly- nomial andgcd(fi(x),fj(x))=1 fori̸=j. (Some of thefi(x)in the square-free factor- ization off(x)may be 1.) Letf(x)= ∑ n i=0 aixibe a polynomial of degreen≥1. The (formal)derivativeof f(x)is the polynomialf′(x)= ∑ n−1 i=0 ai+1(i+1)xi.I ff′(x)=0 , then, becausepis the characteristic ofFq, in each termaixioff(x)for whichai ̸=0, the exponent ofxmust be a multiple ofp. Hence,f(x)has the formf(x)= a(x)p, wherea(x)= ∑ n/p i=0 aq/p ip xi, and the problem of ﬁnding the square-free factorization off(x)is reduced to ﬁnding that ofa(x). Now, it is possible thata′(x)=0 , but repeating this process as necessary, it may be assumed thatf′(x)̸=0. Next, letg(x)=gcd( f(x),f′(x)). Noting that an irreducible factor of multiplicityk inf(x)will have multiplicityk−1inf′(x)ifgcd(k,p)=1 , and will retain multiplicity kinf′(x)otherwise, the following conclusions may be drawn. Ifg(x)=1 , thenf(x) has no repeated factors; and ifg(x)has positive degree, theng(x)is a non-trivial factor of f(x), andf(x)/g(x)has no repeated factors. Note, however, the possibility ofg(x) having repeated factors, and, indeed, the possibility thatg′(x)=0 . Nonetheless,g(x)can be reﬁned further as above. The steps are summarized in Algorithm 3.110. In the algorithm, Fdenotes the square-free factorization of a factor off(x)in factored form. 3.110 AlgorithmSquare-free factorization SQUARE-FREE( f(x)) INPUT: a monic polynomialf(x)∈Fq[x]of degree≥1, whereFqhas characteristicp. OUTPUT: the square-free factorization off(x). 1. Seti←1,F←1, and computef′(x). 2. Iff′(x)=0 then setf(x)←f(x)1/pand F←(SQUARE-FREE (f(x)))p. Otherwise (i.e.f′(x)̸=0) do the following: 2.1 Compute g(x)←gcd(f(x),f′(x))and h(x)←f(x)/g(x). 2.2 Whileh(x)̸=1do the following: Compute h(x)←gcd(h(x),g(x)) and l(x)←h(x)/ h(x). SetF←F·l(x)i, i←i+1, h(x)← h(x), andg(x)←g(x)/ h(x). 2.3 Ifg(x)̸=1then setg(x)←g(x)1/pand F←F·(SQUARE-FREE (g(x)))p. 3. Return(F). Once the square-free factorizationf(x)= ∏k i=1 fi(x)iis found, the square-free poly- nomialsf1(x),f2(x),...,f k(x)need to be factored in order to obtain the complete fac- torization off(x). 124 Ch. 3 Number-Theoretic Reference Problems 3.11.2 Berlekamp’sQ-matrix algorithm Letf(x)= ∏t i=1 fi(x)be a monic polynomial inFq[x]of degreenhaving distinct irre- ducible factorsfi(x),1≤i≤t. Berlekamp’sQ-matrix algorithm (Algorithm 3.111) for factoringf(x)is based on the following facts. The set of polynomials B={b(x)∈Fq[x]/(f(x))\|b(x)q ≡b(x)( m o df(x))} is a vector space of dimensiontoverFq. Bconsists of precisely those vectors in the null space of the matrixQ−In, whereQis then×nmatrix with(i,j)-entryqijspeciﬁed by xiq modf(x)= n−1∑ j=0 qijxj, 0≤i≤n−1, and whereIn is then×nidentity matrix. A basisB ={v1(x),v2(x),...,v t(x)}for Bcan thus be found by standard techniques from linear algebra. Finally, for each pair of distinct factorsfi(x)and fj(x)off(x)there exists somevk(x)∈Band someα∈Fq such thatfi(x)dividesvk(x)−αbutfj(x)does not dividevk(x)−α; these two factors can thus be split by computinggcd(f(x),vk(x)−α). In Algorithm 3.111, a vectorw= (w0,w1,...,w n−1)is identiﬁed with the polynomialw(x)= ∑n−1 i=0 wixi. 3.111 AlgorithmBerlekamp’sQ-matrix algorithm for factoring polynomials over ﬁnite ﬁelds INPUT: a square-free monic polynomialf(x)of degreeninFq[x]. OUTPUT: the factorization off(x)into monic irreducible polynomials. 1. For eachi,0≤i≤n−1, compute the polynomial xiq modf(x)= n−1∑ j=0 qijxj. Note that eachqijis an element ofFq. 2. Form then×nmatrixQwhose (i,j)-entry isqij. 3. Determine a basisv1,v2,...,v tfor the null space of the matrix(Q−In), whereIn is then×nidentity matrix. The number of irreducible factors off(x)is preciselyt. 4. SetF←{f(x)}.(Fis the set of factors off(x)found so far; their product is equal tof(x).) 5. Forifrom 1 totdo the following: 5.1 For each polynomialh(x)∈Fsuch thatdegh(x)>1do the following: com- putegcd(h(x),vi(x)−α)for eachα∈Fq, and replaceh(x)inFby all those polynomials in the gcd computations whose degrees are≥1. 6. Return(the polynomials inFare the irreducible factors off(x)). 3.112 FactThe running time of Algorithm 3.111 for factoring a square-free polynomial of degree noverFq isO(n3 +tqn2)Fq-operations, wheretis the number of irreducible factors of f(x). The method is efﬁcient only whenqis small. §3.12 Notes and further references 125 3.12 Notes and further references §3.1 Many of the topics discussed in this chapter lie in the realm of algorithmic number the- ory. Excellent references on this subject include the books by Bach and Shallit [70], Cohen [263], and Pomerance [993]. Adleman and McCurley [15] give an extensive survey of the important open problems in algorithmic number theory. Two other recommended surveys are by Bach [65] and Lenstra and Lenstra [748]. Woll [1253] gives an overview of the re- ductions among thirteen of these problems. §3.2 A survey of the integer factorization problem is given by Pomerance [994]. See also Chap- ters 8 and 10 of Cohen [263], and the books by Bressoud [198] and Koblitz [697]. Brillhart et al. [211] provide extensive listings of factorizations of integers of the formb n±1for “small”nand b=2,3,5,6,7,10,11,12. Bach and Sorenson [71] presented some algorithms for recognizing perfect powers (cf. Note 3.6), one having a worst-case running time ofO(lg3 n)bit operations, and a sec- ond having an average-case running time ofO(lg2 n)bit operations. A more recent algo- rithm of Bernstein [121] runs in essentially linear timeO((lgn)1+o(1)). Fact 3.7 is from Knuth [692]. Pages 367–369 of this reference contain explicit formulas regarding the ex- pected sizes of the largest and second largest prime factors, and the expected total number of prime factors, of a randomly chosen positive integer. For further results, see Knuth and Trabb Pardo [694], who prove that the average number of bits in thek th largest prime fac- tor of a randomm-bit number is asymptotically equivalent to the average length of thekth longest cycle in a permutation onmobjects. Floyd’s cycle-ﬁnding algorithm (Note 3.8) is described by Knuth [692, p.7]. Sedgewick, Szymanski, and Yao [1106] showed that by saving a small number of values from thexi sequence, a collision can be found by doing roughly one-third the work as in Floyd’s cycle- ﬁnding algorithm. Pollard’s rho algorithm for factoring (Algorithm 3.9) is due to Pollard [985]. Regarding Note 3.12, Cohen [263, p.422] provides an explanation for the restriction c ̸=0,−2. Brent [196] presented a cycle-ﬁnding algorithm which is better on average than Floyd’s cycle-ﬁnding algorithm, and applied it to yield a factorization algorithm which is similar to Pollard’s but about 24 percent faster. Brent and Pollard [197] later modiﬁed this algorithm to factor the eighth Fermat numberF 8 =228 +1. Using techniques from algebraic geometry, Bach [67] obtained the ﬁrst rigorously proven result concerning the expected running time of Pollard’s rho algorithm: for ﬁxedk, the probability that a prime factorpis discovered before stepkis at least (k 2 ) /p+O(p−3/2)asp→∞. The p−1algorithm (Algorithm 3.14) is due to Pollard [984]. Several practical improve- ments have been proposed for thep−1algorithm, including those by Montgomery [894] and Montgomery and Silverman [895], the latter using fast Fourier transform techniques. Williams [1247] presented an algorithm for factoringnwhich is efﬁcient ifnhas a prime factorpsuch thatp+1is smooth. These methods were generalized by Bach and Shallit [69] to techniques that factornefﬁciently providednhas a prime factorpsuch that thek th cy- clotomic polynomialΦk(p)is smooth. The ﬁrst few cyclotomic polynomials areΦ1(p)= p−1,Φ2(p)= p+1,Φ3(p)= p2 +p+1,Φ4(p)= p2 +1,Φ5(p)= p4 +p3 +p2 +p+1, and Φ6(p)= p2 −p+1. The elliptic curve factoring algorithm (ECA) of§3.2.4 was invented by Lenstra [756]. Montgomery [894] gave several practical improvements to the ECA. Silverman and 126 Ch. 3 Number-Theoretic Reference Problems Wagstaff [1136] gave a practical analysis of the complexity of the ECA, and suggested op- timal parameter selection and running-time guidelines. Lenstra and Manasse [753] imple- mented the ECA on a network of MicroV AX computers, and were successful in ﬁnding 35- decimal digit prime factors of large (at least 85 digit) composite integers. Later, Dixon and Lenstra [350] implemented the ECA on a 16K MasPar (massively parallel) SIMD (single instruction, multiple data) machine. The largest factor they found was a 40-decimal digit prime factor of an 89-digit composite integer. On November 26 1995, Peter Montgomery reported ﬁnding a 47-decimal digit prime factor of the 99-digit composite integer5 256 +1 with the ECA. Hafner and McCurley [536] estimated the number of integersn≤xthat can be factored with probability at least1 2 using at mosttarithmetic operations, by trial division and the elliptic curve algorithm. Pomerance and Sorenson [997] provided the analogous estimates for Pollard’sp−1algorithm and Williams’p+1algorithm. They conclude that for a given running time bound, both Pollard’sp−1and Williams’p+1algorithms factor more integers than trial division, but fewer than the elliptic curve algorithm. Pomerance [994] credits the idea of multiplying congruences to produce a solution tox2 ≡ y2 (modn)for the purpose of factoringn(§3.2.5) to some old work of Kraitchik circa 1926-1929. Thecontinued fraction factoring algorithm, ﬁrst introduced by Lehmer and Powers [744] in 1931, and reﬁned more than 40 years later by Morrison and Brillhart [908], was the ﬁrst realization of a random square method to result in a subexponential-time al- gorithm. The algorithm was later analyzed by Pomerance [989] and conjectured to have an expected running time ofLn[1 2 , √ 2]. If the smoothness testing in the algorithm is done with the elliptic curve method, then the expected running time drops toLn[1 2 ,1]. Morrison and Brillhart were also the ﬁrst to use the idea of a factor base to test for good(ai,bi)pairs. The continued fraction algorithm was the champion of factoring algorithms from the mid 1970s until the early 1980s, when it was surpassed by the quadratic sieve algorithm. The quadratic sieve (QS) (§3.2.6) was discovered by Pomerance [989, 990]. The multiple polynomial variant of the quadratic sieve (Note 3.25) is due to P. Montgomery, and is de- scribed by Pomerance [990]; see also Silverman [1135]. A detailed practical analysis of the QS is given by van Oorschot [1203]. Several practical improvements to the original algorithms have subsequently been proposed and successfully implemented. The ﬁrst seri- ous implementation of the QS was by Gerver [448] who factored a 47-decimal digit num- ber. In 1984, Davis, Holdridge, and Simmons [311] factored a 71-decimal digit number with the QS. In 1988, Lenstra and Manasse [753] used the QS to factor a 106-decimal digit number by distributing the computations to hundreds of computers by electronic mail; see also Lenstra and Manasse [754]. In 1993, the QS was used by Denny et al. [333] to factor a 120-decimal digit number. In 1994, the 129-decimal digit (425 bit) RSA-129 challenge number (see Gardner [440]), was factored by Atkins et al. [59] by enlisting the help of about 1600 computers around the world. The factorization was carried out in 8 months. Table 3.3 shows the estimated time taken, in mips years, for the above factorizations. Amips yearis equivalent to the computational power of a computer that is rated at 1 mips (million instruc- tions per second) and utilized for one year, or, equivalently, about3·10 13 instructions. The number ﬁeld sieve was ﬁrst proposed by Pollard [987] and reﬁned by others. Lenstra et al. [752] described the special number ﬁeld sieve (SNFS) for factoring integers of the form re−sfor small positiverand \|s\|. A readable introduction to the algorithm is provided by Pomerance [995]. A detailed report of an SNFS implementation is given by Lenstra et al. [751]. This implementation was used to factor the ninth Fermat numberF9 =2512 +1, which is the product of three prime factors having 7, 49, and 99 decimal digits. The gen- eral number ﬁeld sieve (GNFS) was introduced by Buhler, Lenstra, and Pomerance [219]. §3.12 Notes and further references 127 Year Number of digits mips years 1984 71 0.1 1988 106 140 1993 120 825 1994 129 5000 Table 3.3:Running time estimates for numbers factored with QS. Coppersmith [269] proposed modiﬁcations to the GNFS which improve its running time toLn[1 3 ,1.902], however, the method is not practical; another modiﬁcation (also imprac- tical) allows a precomputation takingLn[1 3 ,2.007]time andLn[1 3 ,1.639]storage, follow- ing which all integers in a large range of values can be factored inLn[1 3 ,1.639]time. A detailed report of a GNFS implementation on a massively parallel computer with 16384 processors is given by Bernstein and Lenstra [122]. See also Buchmann, Loho, and Za- yer [217], and Golliver, Lenstra, and McCurley [493]. More recently, Dodson and Lenstra [356] reported on their GNFS implementation which was successful in factoring a 119- decimal digit number using about 250 mips years of computing power. They estimated that this factorization completed about 2.5 times faster than it would with the quadratic sieve. Most recently, Lenstra [746] announced the factorization of the 130-decimal digit RSA- 130 challenge number using the GNFS. This number is the product of two 65-decimal digit primes. The factorization was estimated to have taken about 500 mips years of computing power (compare with Table 3.3). The book edited by Lenstra and Lenstra [749] contains several other articles related to the number ﬁeld sieve. The ECA, continued fraction algorithm, quadratic sieve, special number ﬁeld sieve, and general number ﬁeld sieve haveheuristic(orconjectured) rather thanprovenrunning times because the analyses make (reasonable) assumptions about the proportion of integers gen- erated that are smooth. See Canﬁeld, Erd¨os, and Pomerance [231] for bounds on the pro- portion ofy-smooth integers in the interval[2,x]. Dixon’s algorithm [351] was the ﬁrst rigorously analyzed subexponential-time algorithm for factoring integers. The fastest rig- orously analyzed algorithm currently known is due to Lenstra and Pomerance [759] with an expected running time ofL n[1 2 ,1]. These algorithms are of theoretical interest only, as they do not appear to be practical. §3.3 The RSA problem was introduced in the landmark 1977 paper by Rivest, Shamir, and Adle- man [1060]. §3.4 The quadratic residuosity problem is of much historical interest, and was one of the main algorithmic problems discussed by Gauss [444]. §3.5 An extensive treatment of the problem of ﬁnding square roots modulo a primep, or more generally, the problem of ﬁndingdth roots in a ﬁnite ﬁeld, can be found in Bach and Shallit [70]. The presentation of Algorithm 3.34 for ﬁnding square roots modulo a prime is de- rived from Koblitz [697, pp.48-49]; a proof of correctness can be found there. Bach and Shallit attribute the essential ideas of Algorithm 3.34 to an 1891 paper by A. Tonelli. Al- gorithm 3.39 is from Bach and Shallit [70], who attribute it to a 1903 paper of M. Cipolla. The computational equivalence of computing square roots modulo a compositenand fac- toringn(Fact 3.46 and Note 3.47) was ﬁrst discovered by Rabin [1023]. 128 Ch. 3 Number-Theoretic Reference Problems §3.6 A survey of the discrete logarithm problem is given by McCurley [827]. See also Odlyzko [942] for a survey of recent advances. Knuth [693] attributes the baby-step giant-step algorithm (Algorithm 3.56) to D. Shanks. The baby-step giant-step algorithms for searching restricted exponent spaces (cf. Note 3.59) are described by Heiman [546]. Suppose thatpis ak-bit prime, and that only exponents of Hamming weight tare used. Coppersmith (personal communication, July 1995) observed that this exponent space can be searched ink· (k/2 t/2 ) steps by dividing the exponent into two equal pieces so that the Hamming weight of each piece ist/2;i fkis much smaller than2t/2, this is an improvement over Note 3.59. Pollard’s rho algorithm for logarithms (Algorithm 3.60) is due to Pollard [986]. Pollard also presented alambda methodfor computing discrete logarithms which is applicable whenx, the logarithm sought, is known to lie in a certain interval. More speciﬁcally, if the interval is of widthw, the method is expected to takeO(√ w)group operations and requires storage for onlyO(lgw)group elements. Van Oorschot and Wiener [1207] showed how Pollard’s rho algorithm can be parallelized so that usingmprocessors results in a speedup by a factor of m. This has particular signiﬁcance to cyclic groups such as elliptic curve groups, for which no subexponential-time discrete logarithm algorithm is known. The Pohlig-Hellman algorithm (Algorithm 3.63) was discovered by Pohlig and Hellman [982]. A variation which represents the logarithm in a mixed-radix notation and does not use the Chinese remainder theorem was given by Thiong Ly [1190]. According to McCurley [827], the basic ideas behind the index-calculus algorithm (Algo- rithm 3.68) ﬁrst appeared in the work of Kraitchik (circa 1922-1924) and of Cunningham (see Western and Miller [1236]), and was rediscovered by several authors. Adleman [8] de- scribed the method for the groupZ∗ pand analyzed the complexity of the algorithm. Hellman and Reyneri [555] gave the ﬁrst description of an index-calculus algorithm for extension ﬁeldsFpm withpﬁxed. Coppersmith, Odlyzko, and Schroeppel [280] presented three variants of the index-calculus method for computing logarithms inZ∗ p: thelinear sieve, theresidue list sieve, and the Gaussian integer method. Each has a heuristic expected running time ofLp[1 2 ,1](cf. Note 3.71). The Gaussian integer method, which is related to the method of ElGamal [369], was implemented in 1990 by LaMacchia and Odlyzko [736] and was successful in comput- ing logarithms inZ∗ pwithpa 192-bit prime. The paper concludes that it should be feasible to compute discrete logarithms modulo primes of about 332 bits (100 decimal digits) using the Gaussian integer method. Gordon [510] adapted the number ﬁeld sieve for factoring in- tegers to the problem of computing logarithms inZ∗ p; his algorithm has a heuristic expected running time ofLp[1 3 ,c], wherec=32/3 ≈2.080. Schirokauer [1092] subsequently pre- sented a modiﬁcation of Gordon’s algorithm that has a heuristic expected running time of Lp[1 3 ,c], wherec =( 6 4/9)1/3 ≈1.923(Note 3.72). This is the same running time as conjectured for the number ﬁeld sieve for factoring integers (see§3.2.7). Recently, Weber [1232] implemented the algorithms of Gordon and Schirokauer and was successful in com- puting logarithms inZ ∗ p, wherepis a 40-decimal digit prime such thatp−1is divisible by a 38-decimal digit (127-bit) prime. More recently, Weber, Denny, and Zayer (personal com- munication, April 1996) announced the solution of a discrete logarithm problem modulo a 75-decimal digit (248-bit) primepwith(p−1)/2prime. Blake et al. [145] made improvements to the index-calculus technique forF ∗ 2m and com- puted logarithms inF∗ 2127 . Coppersmith [266] dramatically improved the algorithm and showed that under reasonable assumptions the expected running time of his improved al- §3.12 Notes and further references 129 gorithm isL2m [1 3 ,c]for some constantc< 1.587(Note 3.72). Later, Odlyzko [940] gave several reﬁnements to Coppersmith’s algorithm, and a detailed practical analysis; this pa- per provides the most extensive account to date of the discrete logarithm problem inF∗ 2m . A similar practical analysis was also given by van Oorschot [1203]. Most recently in 1992, Gordon and McCurley [511] reported on their massively parallel implementation of Cop- persmith’s algorithm, combined with their own improvements. Using primarily a 1024 pro- cessor nCUBE-2 machine with 4 megabytes of memory per processor, they completed the precomputation of logarithms of factor base elements (which is the dominant step of the algorithm) required to compute logarithms inF ∗ 2227 ,F∗ 2 313 , andF∗ 2 401 . The calculations for F∗ 2401 were estimated to take 5 days. Gordon and McCurley also completed most of the pre- computations required for computing logarithms inF∗ 2 503 ; the amount of time to complete this task on the 1024 processor nCUBE-2 was estimated to be 44 days. They concluded that computing logarithms in the multiplicative groups of ﬁelds as large asF2593 still seems to be out of their reach, but might be possible in the near future with a concerted effort. It was not until 1992 that a subexponential-time algorithm for computing discrete loga- rithms over all ﬁnite ﬁeldsFq was discovered by Adleman and DeMarrais [11]. The ex- pected running time of the algorithm is conjectured to beLq[1 2 ,c]for some constantc. Adle- man [9] generalized the number ﬁeld sieve from algebraic number ﬁelds to algebraic func- tion ﬁelds which resulted in an algorithm, called thefunction ﬁeld sieve, for computing dis- crete logarithms inF∗ pm ; the algorithm has a heuristic expected running time ofLpm [1 3 ,c] for some constantc> 0when logp ≤ mg(m), and wheregis any function such that 0<g(m)<0.98and limm→∞g(m)=0 . The practicality of the function ﬁeld sieve has not yet been determined. It remains an open problem to ﬁnd an algorithm with a heuristic expected running time ofLq[1 3 ,c]forallﬁnite ﬁeldsFq. The algorithms mentioned in the previous three paragraphs haveheuristic(orconjectured) rather thanprovenrunning times because the analyses make some (reasonable) assump- tions about the proportion of integers or polynomials generated that are smooth, and also because it is not clear when the system of linear equations generated has full rank, i.e., yields a unique solution. The best rigorously analyzed algorithms known for the discrete loga- rithm problem inZ ∗ pand F∗ 2 m are due to Pomerance [991] with expected running times of Lp[1 2 , √ 2]and L2m [1 2 , √ 2], respectively. Lovorn [773] obtained rigorously analyzed algo- rithms for the ﬁeldsFp2 and Fpm withlogp<m 0.98, having expected running times of Lp2 [1 2 ,3 2 ]and Lpm [1 2 , √ 2], respectively. The linear system of equations collected in the quadratic sieve and number ﬁeld sieve fac- toring algorithms, and the index-calculus algorithms for computing discrete logarithms in Z∗ pand F∗ 2 m , are very large. For the problem sizes currently under consideration, these sys- tems cannot be solved using ordinary linear algebra techniques, due to both time and space constraints. However, the equations generated are extremelysparse, typically with at most 50 non-zero coefﬁcients per equation. The technique ofstructuredor so-calledintelligent Gaussian elimination (see Odlyzko [940]) can be used to reduce the original sparse system to a much smaller system that is still fairly sparse. The resulting system can be solved us- ing either ordinary Gaussian elimination, or one of theconjugate gradient,Lanczos(Cop- persmith, Odlyzko, and Schroeppel [280]), orWiedemann algorithms [1239] which were also designed to handle sparse systems. LaMacchia and Odlyzko [737] have implemented some of these algorithms and concluded that the linear algebra stages arising in both integer factorization and the discrete logarithm problem are not running-time bottlenecks in prac- tice. Recently, Coppersmith [272] proposed a modiﬁcation of the Wiedemann algorithm which allows parallelization of the algorithm; for an analysis of Coppersmith’s algorithm, see Kaltofen [657]. Coppersmith [270] (see also Montgomery [896]) presented a modiﬁ- 130 Ch. 3 Number-Theoretic Reference Problems cation of the Lanczos algorithm for solving sparse linear equations overF2; this variant appears to be the most efﬁcient in practice. As an example of the numbers involved, Gordon and McCurley’s [511] implementation for computing logarithms inF∗ 2401 produced a total of117164equations from a factor base con- sisting of the58636irreducible polynomials inF2[x]of degree at most 19. The system of equations had2068707non-zero entries. Structured Gaussian elimination was then applied to this system, the result being a16139×16139system of equations having1203414non- zero entries, which was then solved using the conjugate gradient method. Another example is from the recent factorization of the RSA-129 number (see Atkins et al. [59]). The sieving step produced a sparse matrix of569466rows and524339columns. Structured Gaussian elimination was used to reduce this to a dense188614×188160system, which was then solved using ordinary Gaussian elimination. There are many ways of representing a ﬁnite ﬁeld, although any two ﬁnite ﬁelds of the same order are isomorphic (see also Note 3.55). Lenstra [757] showed how to compute an iso- morphism between any two explicitly given representations of a ﬁnite ﬁeld in deterministic polynomial time. Thus, it is sufﬁcient to ﬁnd an algorithm for computing discrete loga- rithms in one representation of a given ﬁeld; this algorithm can then be used, together with the isomorphism obtained by Lenstra’s algorithm, to compute logarithms in any other rep- resentation of the same ﬁeld. Menezes, Okamoto, and Vanstone [843] showed how the discrete logarithm problem for an elliptic curve over a ﬁnite ﬁeldF qcan be reduced to the discrete logarithm problem in some extension ﬁeldFqk . For the special class ofsupersingular curves,kis at most6, thus pro- viding a subexponential-time algorithm for the former problem. This work was extended by Frey and R¨uck [422]. No subexponential-time algorithm is known for the discrete log- arithm problem in the more general class ofnon-supersingularelliptic curves. Adleman, DeMarrais, and Huang [12] presented a subexponential-time algorithm for ﬁnd- ing logarithms in the jacobian of large genus hyperelliptic curves over ﬁnite ﬁelds. More precisely, there exists a numberc,0<c ≤2.181, such that for all sufﬁciently largeg≥1 and all odd primespwithlogp≤(2g+1) 0.98, the expected running time of the algo- rithm for computing logarithms in the jacobian of a genusghyperelliptic curve overZpis conjectured to beLp2g+1 [1 2 ,c]. McCurley [826] invented a subexponential-time algorithm for the discrete logarithm prob- lem in the class group of an imaginary quadratic number ﬁeld. See also Hafner and Mc- Curley [537] for further details, and Buchmann and D¨ullmann [216] for an implementation report. In 1994, Shor [1128] conceived randomized polynomial-time algorithms for computing dis- crete logarithms and factoring integers on aquantum computer, a computational device based on quantum mechanical principles; presently it is not known how to build a quantum computer, nor if this is even possible. Also recently, Adleman [10] demonstrated the feasi- bility of using tools from molecular biology to solve an instance of the directed Hamiltonian path problem, which isNP -complete. The problem instance was encoded in molecules of DNA, and the steps of the computation were performed with standard protocols and en- zymes. Adleman notes that while the currently available fastest supercomputers can exe- cute approximately10 12 operations per second, it is plausible for a DNA computer to ex- ecute1020 or more operations per second. Moreover such a DNA computer would be far more energy-efﬁcient than existing supercomputers. It is not clear at present whether it is feasible to build a DNA computer with such performance. However, should either quantum computers or DNA computers ever become practical, they would have a very signiﬁcant §3.12 Notes and further references 131 impact on public-key cryptography. §3.7 Fact 3.77(i) is due to den Boer [323]. Fact 3.77(iii) was proven by Maurer [817], who also proved more generally that the GDHP and GDLP in a groupGof ordernare computation- ally equivalent when certain extra information of lengthO(lgn)bits is given. The extra information depends only onnand not on the deﬁnition ofG, and consists of parameters that deﬁne cyclic elliptic curves of smooth order over the ﬁeldsZpi where thepi are the prime divisors ofn. Waldvogel and Massey [1228] proved that ifaand bare chosen uniformly and randomly from the interval{0,1,...,p −1}, the valuesαabmodpare roughly uniformly distributed (see page 537). §3.8 Facts 3.78 and 3.79 are due to Bach [62]. Fact 3.80 is due to Shmuely [1127]. McCurley [825] reﬁned this result to prove that for specially chosen compositen, the ability to solve the Difﬁe-Hellman problem inZ∗ nfor theﬁxed baseα=16 implies the ability to factorn. §3.9 The notion of a hard Boolean predicate (Deﬁnition 3.81) was introduced by Blum and Mi- cali [166], who also proved Fact 3.84. The notion of a hardk-bit predicate (Deﬁnition 3.82) was introduced by Long and Wigderson [772], who also proved Fact 3.85; see also Peralta [968]. Fact 3.83 is due to Peralta [968]. The results on hard predicates andk-bit predicates for the RSA functions (Facts 3.86 and 3.87) are due to Alexi et al. [23]. Facts 3.88 and 3.89 are due to Vazirani and Vazirani [1218]. Yao [1258] showed how any one-way length-preserving permutation can be transformed into a more complicated one-way length-preserving permutation which has a hard predi- cate. Subsequently, Goldreich and Levin [471] showed how any one-way functionfcan be transformed into a one-way functiongwhich has a hard predicate. Their construction is as follows. Deﬁne the functiongby g(p,x)=( p,f(x)), wherepis a binary string of the same length asx, sayn. Thengis also a one-way function andB(p,x)= ∑ n i=1 piximod2 is a hard predicate forg. H˚astad, Schrift, and Shamir [543] considered the one-way functionf(x)= αxmodn, where nis a Blum integer andα∈Z∗ n. Under the assumption that factoring Blum integers is intractable, they proved that all the bits of this function are individually hard. Moreover, the lower half as well as the upper half of the bits are simultaneously secure. §3.10 The subset sum problem (Deﬁnition 3.90) is sometimes confused with theknapsack prob- lem which is the following: given two sets{a1,a2,...,a n}and {b1,b2,...,b n}of pos- itive integers, and given two positive integerssand t, determine whether or not there is a subsetSof{1,2,...,n }such that∑ i∈Sai≤sand ∑ i∈Sbi ≥t. The subset sum prob- lem is actually a special case of the knapsack problem whenai =bifori=1,2,...,n and s=t. Algorithm 3.94 is described by Odlyzko [941]. The L3-lattice basis reduction algorithm (Algorithm 3.101) and Fact 3.103 are both due to Lenstra, Lenstra, and Lov´asz [750]. Improved algorithms have been given for lattice basis reduction, for example, by Schnorr and Euchner [1099]; consult also Section 2.6 of Cohen [263]. Algorithm 3.105 for solving the subset sum problem involving knapsacks sets of low density is from Coster et al. [283]. Unusually good simultaneous diophantine approxima- tions were ﬁrst introduced and studied by Lagarias [723]; Fact 3.107 and Algorithm 3.108 are from this paper. 132 Ch. 3 Number-Theoretic Reference Problems §3.11 A readable introduction to polynomial factorization algorithms is given by Lidl and Nieder- reiter [764, Chapter 4]. Algorithm 3.110 for square-free factorization is from Geddes, Cza- por, and Labahn [445]. Yun [1261] presented an algorithm that is more efﬁcient than Algo- rithm 3.110 for ﬁnding the square-free factorization of a polynomial. The running time of the algorithm is onlyO(n 2)Zp-operations whenf(x)is a polynomial of degreeninZp[x]. A lucid presentation of Yun’s algorithm is provided by Bach and Shallit [70]. Berlekamp’s Q-matrix algorithm (Algorithm 3.111) was ﬁrst discovered by Prange [999] for the purpose of factoring polynomials of the formx n−1over ﬁnite ﬁelds. The algorithm was later and independently discovered by Berlekamp [117] who improved it for factoring general poly- nomials over ﬁnite ﬁelds. There is no deterministic polynomial-time algorithm known for the problem of factoring polynomials over ﬁnite ﬁelds. There are, however, many efﬁcient randomized algorithms that work well even when the underlying ﬁeld is very large, such as the algorithms given by Ben-Or [109], Berlekamp [119], Cantor and Zassenhaus [232], and Rabin [1025]. For recent work along these lines, see von zur Gathen and Shoup [1224], as well as Kaltofen and Shoup [658]. Chapter /4 Public-Key Parameters Contents in Brief 4.1 Introduction ............................. 133 4.2 Probabilistic primality tests..................... 135 4.3 (True) Primality tests........................ 142 4.4 Prime number generation ...................... 145 4.5 Irreducible polynomials overZp .................. 154 4.6 Generators and elements of high order............... 160 4.7 Notes and further references.................... 165 4.1 Introduction The efﬁcient generation of public-key parameters is a prerequisite in public-key systems. A speciﬁc example is the requirement of a prime numberpto deﬁne a ﬁnite ﬁeldZp for use in the Difﬁe-Hellman key agreement protocol and its derivatives (§12.6). In this case, an element of high order inZ∗ p is also required. Another example is the requirement of primespand qfor an RSA modulusn = pq(§8.2). In this case, the prime must be of sufﬁcient size, and be “random” in the sense that the probability of any particular prime being selected must be sufﬁciently small to preclude an adversary from gaining advantage through optimizing a search strategy based on such probability. Prime numbers may be required to have certain additional properties, in order that they do not make the associated cryptosystems susceptible to specialized attacks. A third example is the requirement of an irreducible polynomialf(x) of degreemover the ﬁnite ﬁeldZ pfor constructing the ﬁnite ﬁeldFpm . In this case, an element of high order inF∗ pm is also required. Chapter outline The remainder of§4.1 introduces basic concepts relevant to prime number generation and summarizes some results on the distribution of prime numbers. Probabilistic primality tests, the most important of which is the Miller-Rabin test, are presented in§4.2. True primality tests by which arbitrary integers can be proven to be prime are the topic of§4.3; since these tests are generally more computationally intensive than probabilistic primality tests, they are not described in detail.§4.4 presents four algorithms for generating prime numbers, strong primes, and provable primes.§4.5 describes techniques for constructing irreducible and primitive polynomials, while§4.6 considers the production of generators and elements of high orders in groups.§4.7 concludes with chapter notes and references. 133 134 Ch. 4 Public-Key Parameters 4.1.1 Approaches to generating large prime numbers To motivate the organization of this chapter and introduce many of the relevant concepts, the problem of generating large prime numbers is ﬁrst considered. The most natural method is to generate a random numbernof appropriate size, and check if it is prime. This can be done by checking whethernis divisible by any of the prime numbers≤ √ n. While more efﬁcient methods are required in practice, to motivate further discussion consider the following approach: 1. Generate ascandidatea random odd numbernof appropriate size. 2. Testnfor primality. 3. Ifnis composite, return to the ﬁrst step. A slight modiﬁcation is to consider candidates restricted to somesearch sequencestart- ing fromn; a trivial search sequence which may be used isn,n+2 ,n+4 ,n+6 ,... . Us- ing speciﬁc search sequences may allow one to increase the expectation that a candidate is prime, and to ﬁnd primes possessing certain additional desirable propertiesa priori. In step 2, the test for primality might be either a test whichprovesthat the candidate is prime (in which case the outcome of the generator is called aprovable prime), or a test which establishes a weaker result, such as thatnis “probably prime” (in which case the out- come of the generator is called aprobable prime). In the latter case, careful consideration must be given to the exact meaning of this expression. Most so-calledprobabilistic primal- ity testsare absolutely correct when they declare candidatesnto be composite, but do not provide a mathematical proof thatnis prime in the case when such a number is declared to be “probably” so. In the latter case, however, when used properly one may often be able to draw conclusions more than adequate for the purpose at hand. For this reason, such tests are more properly calledcompositeness teststhan probabilistic primality tests. True primality tests, which allow one to conclude with mathematical certainty that a number is prime, also exist, but generally require considerably greater computational resources. While (true) primality tests can determine (with mathematical certainty) whether a typ- ically random candidate number is prime, other techniques exist whereby candidatesnare specially constructed such that it can be established by mathematical reasoning whether a candidate actually is prime. These are calledconstructive prime generationtechniques. A ﬁnal distinction between different techniques for prime number generation is the use of randomness. Candidates are typically generated as a function of a random input. The technique used to judge the primality of the candidate, however, may or may not itself use random numbers. If it does not, the technique isdeterministic, and the result is reproducible; if it does, the technique is said to berandomized. Both deterministic and randomized prob- abilistic primality tests exist. In some cases, prime numbers are required which have additional properties. For ex- ample, to make the extraction of discrete logarithms inZ ∗ presistant to an algorithm due to Pohlig and Hellman (§3.6.4), it is a requirement thatp−1 have a large prime divisor. Thus techniques for generating public-key parameters, such as prime numbers, of special form need to be considered. 4.1.2 Distribution of prime numbers Let π(x) denote the number of primes in the interval[2,x]. The prime number theorem (Fact 2.95) states thatπ(x) ∼ x ln x.1 In other words, the number of primes in the interval 1Iff(x) and g(x) are two functions, thenf(x) ∼g(x) means thatlimx→∞ f(x) g(x) =1 . §4.2 Probabilistic primality tests 135 [2,x] is approximately equal tox ln x. The prime numbers are quite uniformly distributed, as the following three results illustrate. 4.1 Fact(Dirichlet theorem)I fgcd(a,n)=1 , then there are inﬁnitely many primes congruent toamodulo n. A more explicit version of Dirichlet’s theorem is the following. 4.2 FactLetπ(x,n,a) denote the number of primes in the interval[2,x] which are congruent toamodulo n, wheregcd(a,n)=1 . Then π(x,n,a) ∼ x φ(n)l nx. In other words, the prime numbers are roughly uniformly distributed among theφ(n) con- gruence classes inZ∗ n, for any value ofn. 4.3 Fact(approximation for thenth prime number) Letpndenote thenth prime number. Then pn∼nln n. More explicitly, nln n<p n <n (ln n+l nl nn) forn≥6. 4.2 Probabilistic primality tests The algorithms in this section are methods by which arbitrary positive integers are tested to provide partial information regarding their primality. More speciﬁcally, probabilistic pri- mality tests have the following framework. For each odd positive integern, a setW(n) ⊂ Znis deﬁned such that the following properties hold: (i) givena∈Zn, it can be checked in deterministic polynomial time whethera∈W(n); (ii) ifnis prime, thenW(n)= ∅(the empty set); and (iii) ifnis composite, then#W(n) ≥n 2 . 4.4 DeﬁnitionIfnis composite, the elements ofW(n) are calledwitnessesto the compos- iteness ofn, and the elements of the complementary setL(n)= Zn−W(n) are called liars. A probabilistic primality test utilizes these properties of the setsW(n) in the following manner. Suppose thatnis an integer whose primality is to be determined. An integera∈ Znis chosen at random, and it is checked ifa∈W(n). The test outputs “composite” if a∈W(n), and outputs “prime” ifa̸∈W(n). If indeeda∈W(n), thennis said tofail the primality test for the basea; in this case,nis surely composite. Ifa̸∈W(n), thennis said topass the primality test for the basea; in this case, no conclusion with absolute certainty can be drawn about the primality ofn, and the declaration “prime” may be incorrect.2 Any single execution of this test which declares “composite” establishes this with cer- tainty. On the other hand, successive independent runs of the test all of which return the an- swer “prime” allow the conﬁdence that the input is indeed prime to be increased to whatever level is desired — the cumulative probability of error is multiplicative over independent tri- als. If the test is runttimes independently on the composite numbern, the probability that nis declared “prime” allttimes (i.e., the probability of error) is at most( 1 2 )t. 2This discussion illustrates why a probabilistic primality test is more properly called acompositenesstest. 136 Ch. 4 Public-Key Parameters 4.5 DeﬁnitionAn integernwhich is believed to be prime on the basis of a probabilistic pri- mality test is called aprobable prime. Two probabilistic primality tests are covered in this section: the Solovay-Strassen test (§4.2.2) and the Miller-Rabin test (§4.2.3). For historical reasons, the Fermat test is ﬁrst discussed in§4.2.1; this test is not truly a probabilistic primality test since it usually fails to distinguish between prime numbers and special composite integers called Carmichael numbers. 4.2.1 Fermat’s test Fermat’s theorem (Fact 2.127) asserts that ifnis a prime andais any integer,1 ≤a≤n−1, thenan−1 ≡1( m o dn). Therefore, given an integernwhose primality is under question, ﬁnding any integerain this interval such that this equivalence is not true sufﬁces to prove thatnis composite. 4.6 DeﬁnitionLetnbe an odd composite integer. An integera,1 ≤a≤n−1, such that an−1 ̸≡1( m o dn) is called aFermat witness(to compositeness) forn. Conversely, ﬁnding an integerabetween1 and n−1 such thatan−1 ≡1( m o dn) makes nappear to be a prime in the sense that it satisﬁes Fermat’s theorem for the basea. This motivates the following deﬁnition and Algorithm 4.9. 4.7 DeﬁnitionLetnbe an odd composite integer and letabe an integer,1 ≤a≤n−1. Then nis said to be apseudoprime to the baseaifan−1 ≡1( m o dn). The integerais called aFermat liar(to primality) forn. 4.8 Example (pseudoprime) The composite integern= 341(=1 1×31) is a pseudoprime to the base2 since2340 ≡1 (mod 341). □ 4.9 AlgorithmFermat primality test FERMAT( n,t) INPUT: an odd integern≥3 and security parametert≥1. OUTPUT: an answer “prime” or “composite” to the question: “Isnprime?” 1. Forifrom 1 totdo the following: 1.1 Choose a random integera,2 ≤a≤n−2. 1.2 Compute r= an−1 mod nusing Algorithm 2.143. 1.3 Ifr̸=1then return(“composite”). 2. Return(“prime”). If Algorithm 4.9 declares “composite”, thennis certainly composite. On the other hand, if the algorithm declares “prime” then no proof is provided thatnis indeed prime. Nonetheless, since pseudoprimes for a given baseaare known to be rare, Fermat’s test provides a correct answer onmost inputs; this, however, is quite distinct from providing a correct answer most of the time (e.g., if run with different bases) oneveryinput. In fact, it does not do the latter because there are (even rarer) composite numbers which are pseu- doprimes toeverybaseafor whichgcd(a,n)=1 . §4.2 Probabilistic primality tests 137 4.10 DeﬁnitionA Carmichael numbernis a composite integer such thatan−1 ≡1( m o dn) for all integersawhich satisfygcd(a,n)=1 . Ifnis a Carmichael number, then the only Fermat witnesses fornare those integers a,1 ≤a≤n−1, for whichgcd(a,n) >1. Thus, if the prime factors ofnare all large, then with high probability the Fermat test declares thatnis “prime”, even if the number of iterationstis large. This deﬁciency in the Fermat test is removed in the Solovay-Strassen and Miller-Rabin probabilistic primality tests by relying on criteria which are stronger than Fermat’s theorem. This subsection is concluded with some facts about Carmichael numbers. If the prime factorization ofnis known, then Fact 4.11 can be used to easily determine whethernis a Carmichael number. 4.11 Fact(necessary and sufﬁcient conditions for Carmichael numbers) A composite integer nis a Carmichael number if and only if the following two conditions are satisﬁed: (i)nis square-free, i.e.,nis not divisible by the square of any prime; and (ii)p−1 dividesn−1 for every prime divisorpofn. A consequence of Fact 4.11 is the following. 4.12 FactEvery Carmichael number is the product of at least three distinct primes. 4.13 Fact(bounds for the number of Carmichael numbers) (i) There are an inﬁnite number of Carmichael numbers. In fact, there are more than n2/7 Carmichael numbers in the interval[2,n], oncenis sufﬁciently large. (ii) The best upper bound known forC(n), the number of Carmichael numbers≤n, is: C(n) ≤n1−{1+o(1)}ln ln lnn/ln lnn forn→∞. The smallest Carmichael number isn= 561 = 3×11 ×17. Carmichael numbers are relatively scarce; there are only105212 Carmichael numbers≤1015. 4.2.2 Solovay-Strassen test The Solovay-Strassen probabilistic primality test was the ﬁrst such test popularized by the advent of public-key cryptography, in particular the RSA cryptosystem. There is no longer any reason to use this test, because an alternative is available (the Miller-Rabin test) which is both more efﬁcient and always at least as correct (see Note 4.33). Discussion is nonethe- less included for historical completeness and to clarify this exact point, since many people continue to reference this test. Recall (§2.4.5) that ( a n ) denotes the Jacobi symbol, and is equivalent to the Legendre symbol ifnis prime. The Solovay-Strassen test is based on the following fact. 4.14 Fact(Euler’ s criterion) Letnbe an odd prime. Thena(n−1)/2 ≡ (a n ) (mod n) for all integersawhich satisfygcd(a,n)=1 . Fact 4.14 motivates the following deﬁnitions. 4.15 DeﬁnitionLetnbe an odd composite integer and letabe an integer,1 ≤a≤n−1. (i) If eithergcd(a,n) >1 ora(n−1)/2 ̸≡ (a n ) (mod n), thenais called anEuler witness (to compositeness) forn. 138 Ch. 4 Public-Key Parameters (ii) Otherwise, i.e., ifgcd(a,n)=1 and a(n−1)/2 ≡ (a n ) (mod n), thennis said to be an Euler pseudoprime to the basea. (That is,nacts like a prime in that it satisﬁes Euler’s criterion for the particular basea.) The integerais called anEuler liar(to primality) forn. 4.16 Example (Euler pseudoprime) The composite integer91 (=7 ×13) is an Euler pseudo- prime to the base9 since945 ≡1 (mod 91)and (9 91 ) =1 . □ Euler’s criterion (Fact 4.14) can be used as a basis for a probabilistic primality test be- cause of the following result. 4.17 FactLetnbe an odd composite integer. Then at mostφ(n)/2 of all the numbersa,1 ≤ a≤n−1, are Euler liars forn(Deﬁnition 4.15). Here,φis the Euler phi function (Deﬁ- nition 2.100). 4.18 AlgorithmSolovay-Strassen probabilistic primality test SOLOV AY-STRASSEN( n,t) INPUT: an odd integern≥3 and security parametert≥1. OUTPUT: an answer “prime” or “composite” to the question: “Isnprime?” 1. Forifrom 1 totdo the following: 1.1 Choose a random integera,2 ≤a≤n−2. 1.2 Compute r= a(n−1)/2 mod nusing Algorithm 2.143. 1.3 Ifr̸=1and r̸=n−1 then return(“composite”). 1.4 Compute the Jacobi symbols= (a n ) using Algorithm 2.149. 1.5 Ifr̸≡s(mod n) then return (“composite”). 2. Return(“prime”). Ifgcd(a,n)= d, thendis a divisor ofr= a(n−1)/2 mod n. Hence, testing whether r ̸=1is step 1.3, eliminates the necessity of testing whethergcd(a,n) ̸=1. If Algo- rithm 4.18 declares “composite”, thennis certainly composite because prime numbers do not violate Euler’s criterion (Fact 4.14). Equivalently, ifnis actually prime, then the algo- rithm always declares “prime”. On the other hand, ifnis actually composite, then since the basesain step 1.1 are chosen independently during each iteration of step 1, Fact 4.17 can be used to deduce the following probability of the algorithm erroneously declaring “prime”. 4.19 Fact(Solovay-Strassen error-probability bound) Letnbe an odd composite integer. The probability that SOLOV AY-STRASSEN(n,t) declaresnto be “prime” is less than(1 2 )t. 4.2.3 Miller-Rabin test The probabilistic primality test used most in practice is the Miller-Rabin test, also known as thestrong pseudoprime test. The test is based on the following fact. 4.20 FactLet nbe an odd prime, and letn−1=2 srwhere ris odd. Letabe any integer such thatgcd(a,n)=1 . Then eitherar ≡1( m o dn) ora2jr ≡−1( m o dn) for some j,0 ≤j≤s−1. Fact 4.20 motivates the following deﬁnitions. §4.2 Probabilistic primality tests 139 4.21 DeﬁnitionLetnbe an odd composite integer and letn−1=2 srwhere ris odd. Leta be an integer in the interval[1,n−1]. (i) Ifar ̸≡1( m o dn) and ifa2jr ̸≡−1( m o dn) for allj,0 ≤j≤s−1, thenais called astrong witness(to compositeness) forn. (ii) Otherwise, i.e., if eitherar ≡1( m o dn) ora2jr ≡−1( m o dn) for somej,0 ≤ j≤s−1, thennis said to be astrong pseudoprime to the basea. (That is,nacts like a prime in that it satisﬁes Fact 4.20 for the particular basea.) The integerais called astrong liar(to primality) forn. 4.22 Example (strong pseudoprime) Consider the composite integern=9 1(=7 ×13). Since 91 −1=9 0=2 ×45,s=1 and r=4 5. Since9r =9 45 ≡1 (mod 91),91 is a strong pseudoprime to the base9. The set of all strong liars for91 is: {1,9,10,12,16,17,22,29,38,53,62,69,74,75,79,81,82,90}. Notice that the number of strong liars for91 is18 = φ(91)/4, whereφis the Euler phi function (cf. Fact 4.23). □ Fact 4.20 can be used as a basis for a probabilistic primality test due to the following result. 4.23 FactIfnis an odd composite integer, then at most1 4 of all the numbersa,1 ≤a≤n−1, are strong liars forn. In fact, ifn̸=9, the number of strong liars fornis at mostφ(n)/4, where φis the Euler phi function (Deﬁnition 2.100). 4.24 AlgorithmMiller-Rabin probabilistic primality test MILLER-RABIN( n,t) INPUT: an odd integern≥3 and security parametert≥1. OUTPUT: an answer “prime” or “composite” to the question: “Isnprime?” 1. Writen−1=2 srsuch thatris odd. 2. Forifrom 1 totdo the following: 2.1 Choose a random integera,2 ≤a≤n−2. 2.2 Compute y= armod nusing Algorithm 2.143. 2.3 Ify̸=1and y̸=n−1 then do the following: j←1. While j≤s−1 and y̸=n−1 do the following: Compute y←y2 mod n. Ify=1 then return(“composite”). j←j+1 . Ify̸=n−1 then return (“composite”). 3. Return(“prime”). Algorithm 4.24 tests whether each baseasatisﬁes the conditions of Deﬁnition 4.21(i). In the ﬁfth line of step 2.3, ify=1 , thena2jr ≡1( m o dn). Since it is also the case that a2j−1r̸≡±1( m o dn), it follows from Fact 3.18 thatnis composite (in factgcd(a2j−1r− 1,n) is a non-trivial factor ofn). In the seventh line of step 2.3, ify̸=n−1, thenais a strong witness forn. If Algorithm 4.24 declares “composite”, thennis certainly compos- ite because prime numbers do not violate Fact 4.20. Equivalently, ifnis actually prime, then the algorithm always declares “prime”. On the other hand, ifnis actually composite, then Fact 4.23 can be used to deduce the following probability of the algorithm erroneously declaring “prime”. 140 Ch. 4 Public-Key Parameters 4.25 Fact(Miller-Rabin error-probability bound) For any odd composite integern, the proba- bility that MILLER-RABIN(n,t) declaresnto be “prime” is less than(1 4 )t. 4.26 Remark (number of strong liars) For most composite integersn, the number of strong liars fornis actually much smaller than the upper bound ofφ(n)/4 given in Fact 4.23. Consequently, the Miller-Rabin error-probability bound is much smaller than( 1 4 )tfor most positive integersn. 4.27 Example (some composite integers have very few strong liars) The only strong liars for the composite integern= 105(=3 ×5 ×7) are1 and 104. More generally, ifk≥2 and nis the product of the ﬁrstkodd primes, there are only2 strong liars forn, namely1 and n−1. □ 4.28 Remark (ﬁxed bases in Miller-Rabin)I fa1 and a2 are strong liars forn, their product a1a2 is very likely, but not certain, to also be a strong liar forn. A strategy that is some- times employed is to ﬁx the basesain the Miller-Rabin algorithm to be the ﬁrst few primes (composite bases are ignored because of the preceding statement), instead of choosing them at random. 4.29 DeﬁnitionLetp1,p2,...,p tdenote the ﬁrsttprimes. Thenψtis deﬁned to be the small- est positive composite integer which is a strong pseudoprime to all the basesp1,p2,...,p t. The numbersψt can be interpreted as follows: to determine the primality of any integer n<ψ t, it is sufﬁcient to apply the Miller-Rabin algorithm tonwith the basesabeing the ﬁrsttprime numbers. With this choice of bases, the answer returned by Miller-Rabin is always correct. Table 4.1 gives the value ofψtfor1 ≤t≤8. t ψt 1 2047 2 1373653 3 25326001 4 3215031751 5 2152302898747 6 3474749660383 7 341550071728321 8 341550071728321 Table 4.1:Smallest strong pseudoprimes. The table lists values ofψt, the smallest positive composite integer that is a strong pseudoprime to each of the ﬁrsttprime bases, for1 ≤t≤8. 4.2.4 Comparison: Fermat, Solovay-Strassen, and Miller-Rabin Fact 4.30 describes the relationships between Fermat liars, Euler liars, and strong liars (see Deﬁnitions 4.7, 4.15, and 4.21). 4.30 FactLetnbe an odd composite integer. (i) Ifais an Euler liar forn, then it is also a Fermat liar forn. (ii) Ifais a strong liar forn, then it is also an Euler liar forn. §4.2 Probabilistic primality tests 141 4.31 Example (Fermat, Euler, strong liars) Consider the composite integern=6 5 (=5 × 13). The Fermat liars for65 are{1,8,12,14,18,21,27,31,34,38,44,47,51,53,57,64}. The Euler liars for65 are{1,8,14,18,47,51,57,64}, while the strong liars for65 are {1,8,18,47,57,64}. □ For a ﬁxed composite candidaten, the situation is depicted in Figure 4.1. This set- strong liars forn Fermat liars forn Euler liars forn Figure 4.1:Relationships between Fermat, Euler, and strong liars for a composite integern. tles the question of the relative accuracy of the Fermat, Solovay-Strassen, and Miller-Rabin tests, not only in the sense of the relative correctness of each test on a ﬁxed candidaten, but also in the sense that givenn, the speciﬁed containments hold foreach randomly chosen basea. Thus, from a correctness point of view, the Miller-Rabin test is never worse than the Solovay-Strassen test, which in turn is never worse than the Fermat test. As the following result shows, there are, however, some composite integersnfor which the Solovay-Strassen and Miller-Rabin tests are equally good. 4.32 FactIfn≡3( m o d4 ), thenais an Euler liar fornif and only if it is a strong liar forn. What remains is a comparison of the computational costs. While the Miller-Rabin test may appear more complex, it actually requires, at worst, the same amount of computation as Fermat’s test in terms of modular multiplications; thus the Miller-Rabin test is better than Fermat’s test in all regards. At worst, the sequence of computations deﬁned in MILLER- RABIN( n,1) requires the equivalent of computinga (n−1)/2 mod n. It is also the case that MILLER-RABIN( n,1) requires less computation than SOLOV AY-STRASSEN(n,1), the latter requiring the computation ofa(n−1)/2 mod nand possibly a further Jacobi symbol computation. For this reason, the Solovay-Strassen test is both computationally and con- ceptually more complex. 4.33 Note (Miller-Rabin is better than Solovay-Strassen) In summary, both the Miller-Rabin and Solovay-Strassen tests are correct in the event that either their input is actually prime, or that they declare their input composite. There is, however, no reason to use the Solovay- Strassen test (nor the Fermat test) over the Miller-Rabin test. The reasons for this are sum- marized below. (i) The Solovay-Strassen test is computationally more expensive. (ii) The Solovay-Strassen test is harder to implement since it also involves Jacobi symbol computations. (iii) The error probability for Solovay-Strassen is bounded above by( 1 2 )t, while the error probability for Miller-Rabin is bounded above by( 1 4 )t. 142 Ch. 4 Public-Key Parameters (iv) Any strong liar fornis also an Euler liar forn. Hence, from a correctness point of view, the Miller-Rabin test is never worse than the Solovay-Strassen test. 4.3 (True) Primality tests The primality tests in this section are methods by which positive integers can beproven to be prime, and are often referred to asprimality proving algorithms. These primality tests are generally more computationally intensive than the probabilistic primality tests of §4.2. Consequently, before applying one of these tests to a candidate primen, the candidate should be subjected to a probabilistic primality test such as Miller-Rabin (Algorithm 4.24). 4.34 DeﬁnitionAn integernwhich is determined to be prime on the basis of a primality prov- ing algorithm is called aprovable prime. 4.3.1 Testing Mersenne numbers Efﬁcient algorithms are known for testing primality of some special classes of numbers, such as Mersenne numbers and Fermat numbers. Mersenne primesnare useful because the arithmetic in the ﬁeldZnfor suchncan be implemented very efﬁciently (see§14.3.4). The Lucas-Lehmer test for Mersenne numbers (Algorithm 4.37) is such an algorithm. 4.35 DeﬁnitionLets≥2 be an integer. AMersenne numberis an integer of the form2s−1. If2s−1 is prime, then it is called aMersenne prime. The following are necessary and sufﬁcient conditions for a Mersenne number to be prime. 4.36 FactLets≥3. The Mersenne numbern=2 s−1 is prime if and only if the following two conditions are satisﬁed: (i)sis prime; and (ii) the sequence of integers deﬁned byu0 =4 and uk+1 =( u2 k−2) modnfork≥0 satisﬁesus−2 =0 . Fact 4.36 leads to the following deterministic polynomial-time algorithm for determin- ing (with certainty) whether a Mersenne number is prime. 4.37 AlgorithmLucas-Lehmer primality test for Mersenne numbers INPUT: a Mersenne numbern=2 s−1 withs≥3. OUTPUT: an answer “prime” or “composite” to the question: “Isnprime?” 1. Use trial division to check ifshas any factors between2 and ⌊√ s⌋. If it does, then return(“composite”). 2. Setu←4. 3. Forkfrom 1 tos−2 do the following: computeu←(u2 −2) modn. 4. Ifu=0 then return(“prime”). Otherwise, return(“composite”). It is unknown whether there are inﬁnitely many Mersenne primes. Table 4.2 lists the 33 known Mersenne primes. §4.3 (True) Primality tests 143 Index Mj decimal j digits 1 2 1 2 3 1 3 5 2 4 7 3 5 13 4 6 17 6 7 19 6 8 31 10 9 61 19 10 89 27 11 107 33 12 127 39 13 521 157 14 607 183 15 1279 386 16 2203 664 17 2281 687 Index Mj decimal j digits 18 3217 969 19 4253 1281 20 4423 1332 21 9689 2917 22 9941 2993 23 11213 3376 24 19937 6002 25 21701 6533 26 23209 6987 27 44497 13395 28 86243 25962 29 110503 33265 30 132049 39751 31 216091 65050 32? 756839 227832 33? 859433 258716 Table 4.2:Known Mersenne primes. The table shows the33 known exponentsMj,1 ≤j≤33, for which2Mj −1 is a Mersenne prime, and also the number of decimal digits in2Mj −1. The question marks afterj=3 2and j=3 3indicate that it is not known whether there are any other exponentss betweenM31 and these numbers for which2s−1 is prime. 4.3.2 Primality testing using the factorization ofn −1 This section presents results which can be used to prove that an integernis prime, provided that the factorization or a partial factorization ofn−1 is known. It may seem odd to consider a technique which requires the factorization ofn−1 as a subproblem — if integers of this size can be factored, the primality ofnitself could be determined by factoringn. However, the factorization ofn−1 may be easier to compute ifnhas a special form, such as aFermat number n=2 2k +1 . Another situation where the factorization ofn−1 may be easy to compute is when the candidatenis “constructed” by speciﬁc methods (see§4.4.4). 4.38 FactLet n ≥ 3 be an integer. Thennis prime if and only if there exists an integera satisfying: (i)an−1 ≡1( m o dn); and (ii)a(n−1)/q ̸≡1( m o dn) for each prime divisorqofn−1. This result follows from the fact thatZ∗ nhas an element of ordern−1 (Deﬁnition 2.128) if and only ifnis prime; an elementasatisfying conditions (i) and (ii) has ordern−1. 4.39 Note (primality test based on Fact 4.38)I fnis a prime, the number of elements of order n−1 is preciselyφ(n−1). Hence, to prove a candidatenprime, one may simply choose an integera∈Znat random and uses Fact 4.38 to check ifahas ordern−1. If this is the case, thennis certainly prime. Otherwise, anothera∈Znis selected and the test is repeated. Ifnis indeed prime, the expected number of iterations before an elementaof ordern−1 is selected isO(ln lnn); this follows since(n−1)/φ(n−1) <6l nl nnfor 144 Ch. 4 Public-Key Parameters n≥5 (Fact 2.102). Thus, if such anais not found after a “reasonable” number (for ex- ample,12 ln lnn) of iterations, thennis probably composite and should again be subjected to a probabilistic primality test such as Miller-Rabin (Algorithm 4.24).3 This method is, in effect, a probabilistic compositeness test. The next result gives a method for proving primality which requires knowledge of only a partialfactorization ofn−1. 4.40 Fact(Pocklington’ s theorem) Letn≥3 be an integer, and letn= RF+1 (i.e.Fdivides n−1) where the prime factorization ofFisF = ∏t j=1 qej j . If there exists an integera satisfying: (i)an−1 ≡1( m o dn); and (ii)gcd(a(n−1)/qj −1,n)=1 for eachj,1 ≤j≤t, then every prime divisorpofnis congruent to 1 moduloF. It follows that ifF> √ n−1, thennis prime. Ifnis indeed prime, then the following result establishes that most integersasatisfy conditions (i) and (ii) of Fact 4.40, provided that the prime divisors ofF> √ n−1 are sufﬁciently large. 4.41 FactLetn= RF+1 be an odd prime withF> √ n−1 and gcd(R,F)=1 . Let the distinct prime factors ofFbe q1,q2,...,q t. Then the probability that a randomly selected basea,1 ≤a≤n−1, satisﬁes both: (i)an−1 ≡1( m o dn); and (ii)gcd(a(n−1)/qj − 1,n)=1 for eachj,1 ≤j≤t,i s∏t j=1 (1 −1/qj) ≥1 −∑t j=1 1/qj. Thus, if the factorization of a divisorF> √ n−1 ofn−1 is known then to testnfor primality, one may simply choose random integersain the interval[2,n−2] until one is found satisfying conditions (i) and (ii) of Fact 4.40, implying thatnis prime. If such ana is not found after a “reasonable” number of iterations,4 thennis probably composite and this could be established by subjecting it to a probabilistic primality test (footnote 3 also applies here). This method is, in effect, a probabilistic compositeness test. The next result gives a method for proving primality which only requires the factoriza- tion of a divisorFofn−1 that is greater than3√ n. For an example of the use of Fact 4.42, see Note 4.63. 4.42 FactLetn≥3 be an odd integer. Letn=2 RF+1 , and suppose that there exists an integerasatisfying both: (i)an−1 ≡1( m o dn); and (ii)gcd(a(n−1)/q−1,n)=1 for each prime divisorqofF. Letx≥0 and ybe deﬁned by2R= xF+ yand 0 ≤y<F . IfF≥ 3√ nand ify2 −4xis neither0 nor a perfect square, thennis prime. 4.3.3 Jacobi sum test The Jacobi sum testis another true primality test. The basic idea is to test a set of con- gruences which are analogues of Fermat’s theorem (Fact 2.127(i)) in certaincyclotomic rings. The running time of the Jacobi sum test for determining the primality of an integer nisO((ln n)cln ln lnn) bit operations for some constantc. This is “almost” a polynomial- time algorithm since the exponentln ln lnnacts like a constant for the range of values for 3 Another approach is to run both algorithms in parallel (with an unlimited number of iterations), until one of them stops with a deﬁnite conclusion “prime” or “composite”. 4The number of iterations may be taken to beT where PT ≤( 1 2 )100, and whereP =1 −∏t j=1(1−1/qj). §4.4 Prime number generation 145 nof interest. For example, ifn≤2512, thenln ln lnn< 1.78. The version of the Ja- cobi sum primality test used in practice is a randomized algorithm which terminates within O(k(ln n)cln ln lnn) steps with probability at least1 −(1 2 )k for everyk≥1, and always gives a correct answer. One drawback of the algorithm is that it does not produce a “certiﬁ- cate” which would enable the answer to be veriﬁed in much shorter time than running the algorithm itself. The Jacobi sum test is, indeed, practical in the sense that the primality of numbers that are several hundred decimal digits long can be handled in just a few minutes on a com- puter. However, the test is not as easy to program as the probabilistic Miller-Rabin test (Algorithm 4.24), and the resulting code is not as compact. The details of the algorithm are complicated and are not given here; pointers to the literature are given in the chapter notes on page 166. 4.3.4 Tests using elliptic curves Elliptic curve primality proving algorithms are based on an elliptic curve analogue of Pock- lington’s theorem (Fact 4.40). The version of the algorithm used in practice is usually re- ferred to asAtkin’ s testor theElliptic Curve Primality Proving algorithm(ECPP). Under heuristic arguments, the expected running time of this algorithm for proving the primality of an integernhas been shown to beO((ln n)6+ϵ) bit operations for anyϵ>0. Atkin’s test has the advantage over the Jacobi sum test (§4.3.3) that it produces a shortcertiﬁcate of primalitywhich can be used to efﬁciently verify the primality of the number. Atkin’s test has been used to prove the primality of numbers more than 1000 decimal digits long. The details of the algorithm are complicated and are not presented here; pointers to the literature are given in the chapter notes on page 166. 4.4 Prime number generation This section considers algorithms for the generation of prime numbers for cryptographic purposes. Four algorithms are presented: Algorithm 4.44 for generatingprobableprimes (see Deﬁnition 4.5), Algorithm 4.53 for generatingstrongprimes (see Deﬁnition 4.52), Al- gorithm 4.56 for generatingprobableprimespandqsuitable for use in the Digital Signature Algorithm (DSA), and Algorithm 4.62 for generatingprovableprimes (see Deﬁnition 4.34). 4.43 Note (prime generation vs. primality testing) Prime numbergenerationdiffers from pri- malitytestingas described in§4.2 and§4.3, but may and typically does involve the latter. The former allows the construction of candidates of a ﬁxed form which may lead to more efﬁcient testing than possible for random candidates. 4.4.1 Random search for probable primes By the prime number theorem (Fact 2.95), the proportion of (positive) integers≤xthat are prime is approximately1/ln x. Since half of all integers≤xare even, the proportion ofodd integers≤xthat are prime is approximately2/ln x. For instance, the proportion of all odd integers≤2512 that are prime is approximately2/(512 ·ln(2)) ≈1/177. This suggests that a reasonable strategy for selecting a randomk-bit (probable) prime is to re- peatedly pick randomk-bit odd integersnuntil one is found that is declared to be “prime” 146 Ch. 4 Public-Key Parameters by MILLER-RABIN( n,t) (Algorithm 4.24) for an appropriate value of the security param- etert(discussed below). If a randomk-bit odd integernis divisible by a small prime, it is less computationally expensive to rule out the candidatenby trial division than by using the Miller-Rabin test. Since the probability that a random integernhas a small prime divisor is relatively large, before applying the Miller-Rabin test, the candidatenshould be tested for small divisors below a pre-determined boundB. This can be done by dividingnby all the primes below B, or by computing greatest common divisors ofnand (pre-computed) products of several of the primes≤B. The proportion of candidate odd integersnnot ruled out by this trial division is∏ 3≤p≤B(1−1 p) which, by Mertens’s theorem, is approximately1.12/lnB(here pranges over prime values). For example, ifB= 256, then only 20% of candidate odd integersnpass the trial division stage, i.e., 80% are discarded before the more costly Miller- Rabin test is performed. 4.44 AlgorithmRandom search for a prime using the Miller-Rabin test RANDOM-SEARCH( k,t) INPUT: an integerk, and a security parametert(cf. Note 4.49). OUTPUT: a random k-bit probable prime. 1. Generate an oddk-bit integernat random. 2. Use trial division to determine whethernis divisible by any odd prime≤B(see Note 4.45 for guidance on selectingB). If it is then go to step 1. 3. If MILLER-RABIN( n,t) (Algorithm 4.24) outputs “prime” then return(n). Otherwise, go to step 1. 4.45 Note (optimal trial division boundB) LetEdenote the time for a fullk-bit modular ex- ponentiation, and letDdenote the time required for ruling out one small prime as divisor of ak-bit integer. (The valuesEand Ddepend on the particular implementation of long- integer arithmetic.) Then the trial division boundBthat minimizes the expected running time of Algorithm 4.44 for generating ak-bit prime is roughlyB= E/D. A more accurate estimate of the optimum choice forBcan be obtained experimentally. The odd primes up toBcan be precomputed and stored in a table. If memory is scarce, a value ofBthat is smaller than the optimum value may be used. Since the Miller-Rabin test does not provide a mathematical proof that a number is in- deed prime, the numbernreturned by Algorithm 4.44 is a probable prime (Deﬁnition 4.5). It is important, therefore, to have an estimate of the probability thatnis in fact composite. 4.46 DeﬁnitionThe probability that RANDOM-SEARCH( k,t) (Algorithm 4.44) returns a composite number is denoted bypk,t. 4.47 Note (remarks on estimatingpk,t) It is tempting to conclude directly from Fact 4.25 that pk,t≤(1 4 )t. This reasoning is ﬂawed (although typically the conclusion will be correct in practice) since it does not take into account the distribution of the primes. (For example, if all candidatesnwere chosen from a setSof composite numbers, the probability of error is 1.) The following discussion elaborates on this point. LetXrepresent the event thatnis composite, and letYtdenote the event than MILLER-RABIN(n,t) declaresnto be prime. Then Fact 4.25 states thatP(Yt\|X) ≤(1 4 )t. What is relevant, however, to the estimation of pk,tis the quantityP(X\|Yt). Suppose that candidatesnare drawn uniformly and randomly §4.4 Prime number generation 147 from a setSof odd numbers, and supposepis the probability thatnis prime (this depends on the candidate setS). Assume also that0 <p< 1. Then by Bayes’ theorem (Fact 2.10): P(X\|Yt)= P(X)P(Yt\|X) P(Yt) ≤ P(Yt\|X) P(Yt) ≤ 1 p (1 4 )t , sinceP(Yt) ≥p. Thus the probabilityP(X\|Yt) may be considerably larger than( 1 4 )tifpis small. However, the error-probability of Miller-Rabin is usually far smaller than( 1 4 )t(see Remark 4.26). Using better estimates forP(Yt\|X) and estimates on the number ofk-bit prime numbers, it has been shown thatpk,tis, in fact, smaller than(1 4 )tfor all sufﬁciently largek. A more concrete result is the following: if candidatesnare chosen at random from the set of odd numbers in the interval[3,x], thenP(X\|Yt) ≤(1 4 )tfor allx≥1060. Further reﬁnements forP(Yt\|X) allow the following explicit upper bounds onpk,tfor various values ofkand t.5 4.48 Fact(some upper bounds onpk,tin Algorithm 4.44) (i)pk,1 <k242− √ kfork≥2. (ii)pk,t<k3/22tt−1/242− √ tkfor (t=2 ,k≥88)o r(3 ≤t≤k/9,k≥21). (iii)pk,t< 7 20 k2−5t+ 1 7 k15/42−k/2−2t+1 2k2−k/4−3tfork/9 ≤t≤k/4,k≥21. (iv)pk,t<1 7 k15/42−k/2−2tfort≥k/4,k≥21. For example, ifk= 512and t=6 , then Fact 4.48(ii) givesp512,6 ≤(1 2 )88. In other words, the probability that RANDOM-SEARCH(512,6) returns a 512-bit composite integer is less than(1 2 )88. Using more advanced techniques, the upper bounds onpk,t given by Fact 4.48 have been improved. These upper bounds arise from complicated formulae which are not given here. Table 4.3 lists some improved upper bounds onpk,tfor some sample values ofkand t. As an example, the probability that RANDOM-SEARCH(500,6) returns a composite number is≤ (1 2 )92. Notice that the values ofpk,t implied by the table are considerably smaller than(1 4 )t=( 1 2 )2t. t k 1 2 3 456789 1 0 100 5 1 4 2 0 2 52 93 33 63 94 14 4 150 8 2 0 2 8 3 43 94 34 75 15 45 7 200 11 25 34 41 47 52 57 61 65 69 250 14 29 39 47 54 60 65 70 75 79 300 19 33 44 53 60 67 73 78 83 88 350 28 38 48 58 66 73 80 86 91 97 400 37 46 55 63 72 80 87 93 99 105 450 46 54 62 70 78 85 93 100 106 112 500 56 63 70 78 85 92 99 106 113 119 550 65 72 79 86 93 100 107 113 119 126 600 75 82 88 95 102 108 115 121 127 133 Table 4.3:Upper bounds onpk,t for sample values ofkand t. An entryjcorresponding tokand t impliespk,t ≤( 1 2 )j. 5The estimates ofpk,t presented in the remainder of this subsection were derived for the situation where Al- gorithm 4.44 does not use trial division by small primes to rule out some candidatesn. Since trial division never rules out a prime, it can only give a better chance of rejecting composites. Thus the error probabilitypk,t might actually be even smaller than the estimates given here. 148 Ch. 4 Public-Key Parameters 4.49 Note (controlling the error probability) In practice, one is usually willing to tolerate an er- ror probability of( 1 2 )80 when using Algorithm 4.44 to generate probable primes. For sam- ple values ofk, Table 4.4 lists the smallest value oftthat can be derived from Fact 4.48 for whichpk,t ≤(1 2 )80. For example, when generating 1000-bit probable primes, Miller- Rabin witht=3 repetitions sufﬁces. Algorithm 4.44 rules out most candidatesneither by trial division (in step 2) or by performing just one iteration of the Miller-Rabin test (in step 3). For this reason, the only effect of selecting a larger security parameterton the run- ning time of the algorithm will likely be to increase the time required in the ﬁnal stage when the (probable) prime is chosen. k t 100 27 150 18 200 15 250 12 300 9 350 8 400 7 450 6 k t 500 6 550 5 600 5 650 4 700 4 750 4 800 4 850 3 k t 900 3 950 3 1000 3 1050 3 1100 3 1150 3 1200 3 1250 3 k t 1300 2 1350 2 1400 2 1450 2 1500 2 1550 2 1600 2 1650 2 k t 1700 2 1750 2 1800 2 1850 2 1900 2 1950 2 2000 2 2050 2 Table 4.4:For samplek, the smallesttfrom Fact 4.48 is given for whichpk,t ≤( 1 2 )80. 4.50 Remark (Miller-Rabin test with basea=2 ) The Miller-Rabin test involves exponenti- ating the basea; this may be performed using the repeated square-and-multiply algorithm (Algorithm 2.143). Ifa=2 , then multiplication byais a simple procedure relative to mul- tiplying byain general. One optimization of Algorithm 4.44 is, therefore, to ﬁx the base a=2 when ﬁrst performing the Miller-Rabin test in step 3. Since most composite numbers will fail the Miller-Rabin test with basea=2 , this modiﬁcation will lower the expected running time of Algorithm 4.44. 4.51 Note (incremental search) (i) An alternative technique to generating candidatesnat random in step 1 of Algo- rithm 4.44 is to ﬁrst select a randomk-bit odd numbern0, and then test thesnumbers n= n0,n0 +2 ,n0 +4 ,...,n 0 +2 (s−1) for primality. If all thesescandidates are found to be composite, the algorithm is said to havefailed.I fs= c·ln 2kwherecis a constant, the probabilityqk,t,sthat this incremental search variant of Algorithm 4.44 returns a composite number has been shown to be less thanδk32− √ kfor some con- stantδ. Table 4.5 gives some explicit bounds on this error probability fork= 500and t≤10. Under reasonable number-theoretic assumptions, the probability of the algo- rithm failing has been shown to be less than2e−2cfor largek(here,e≈2.71828). (ii) Incremental search has the advantage that fewer random bits are required. Further- more, the trial division by small primes in step 2 of Algorithm 4.44 can be accom- plished very efﬁciently as follows. First the valuesR[p]= n 0 mod pare computed for each odd primep≤B. Each time2 is added to the current candidate, the values in the tableRare updated asR[p]←(R[p]+2)mod p. The candidate passes the trial division stage if and only if none of theR[p] values equal0. (iii) IfBis large, an alternative method for doing the trial division is to initialize a table S[i]←0 for0 ≤i≤(s−1); the entryS[i] corresponds to the candidaten0 +2 i. For each odd primep≤B,n0 mod pis computed. Letjbe the smallest index for §4.4 Prime number generation 149 t c 12345678 91 0 1 17 37 51 63 72 81 89 96 103 110 5 13 32 46 58 68 77 85 92 99 105 10 11 30 44 56 66 75 83 90 97 103 Table 4.5:Upper bounds on the error probability of incremental search (Note 4.51) fork= 500 and sample values ofcand t. An entryjcorresponding tocand timpliesq500,t,s ≤( 1 2 )j, where s= c·ln 2500. which (n0 +2 j) ≡0( m o dp). ThenS[j] and eachpth entry after it are set to1.A candidaten0 +2 ithen passes the trial division stage if and only ifS[i]=0 . Note that the estimate for the optimal trial division boundBgiven in Note 4.45 does not apply here (nor in (ii)) since the cost of division is amortized over all candidates. 4.4.2 Strong primes The RSA cryptosystem (§8.2) uses a modulus of the formn= pq, wherepand qare dis- tinct odd primes. The primespand qmust be of sufﬁcient size that factorization of their product is beyond computational reach. Moreover, they should be random primes in the sense that they be chosen as a function of a random input through a process deﬁning a pool of candidates of sufﬁcient cardinality that an exhaustive attack is infeasible. In practice, the resulting primes must also be of a pre-determined bitlength, to meet system speciﬁcations. The discovery of the RSA cryptosystem led to the consideration of several additional con- straints on the choice ofpandqwhich are necessary to ensure the resulting RSA system safe from cryptanalytic attack, and the notion of a strong prime (Deﬁnition 4.52) was deﬁned. These attacks are described at length in Note 8.8(iii); as noted there, it is now believed that strong primes offer little protection beyond that offered by random primes, since randomly selected primes of the sizes typically used in RSA moduli today will satisfy the constraints with high probability. On the other hand, they are no less secure, and require only minimal additional running time to compute; thus, there is little real additional cost in using them. 4.52 DeﬁnitionA prime numberpis said to be astrong primeif integersr,s, andtexist such that the following three conditions are satisﬁed: (i)p−1 has a large prime factor, denotedr; (ii)p+1 has a large prime factor, denoteds; and (iii)r−1 has a large prime factor, denotedt. In Deﬁnition 4.52, a precise qualiﬁcation of “large” depends on speciﬁc attacks that should be guarded against; for further details, see Note 8.8(iii). 150 Ch. 4 Public-Key Parameters 4.53 AlgorithmGordon’s algorithm for generating a strong prime SUMMARY: a strong primepis generated. 1. Generate two large random primessand tof roughly equal bitlength (see Note 4.54). 2. Select an integeri0. Find the ﬁrst prime in the sequence2it+1 , fori= i0,i0 + 1,i0 +2 ,... (see Note 4.54). Denote this prime byr=2 it+1 . 3. Compute p0 =2 (sr−2 mod r)s−1. 4. Select an integerj0. Find the ﬁrst prime in the sequencep0 +2 jrs, forj= j0,j0 + 1,j0 +2 ,... (see Note 4.54). Denote this prime byp= p0 +2 jrs. 5. Return(p). Justiﬁcation.To see that the primepreturned by Gordon’s algorithm is indeed a strong prime, observe ﬁrst (assumingr̸=s) thatsr−1 ≡1( m o dr); this follows from Fermat’s theorem (Fact 2.127). Hence,p0 ≡1( m o dr) and p0 ≡−1( m o ds). Finally (cf. Deﬁ- nition 4.52), (i)p−1= p0 +2 jrs−1 ≡0( m o dr), and hencep−1 has the prime factorr; (ii)p+1= p0 +2 jrs+1 ≡0( m o ds), and hencep+1 has the prime factors; and (iii)r−1=2 it≡0( m o dt), and hencer−1 has the prime factort. 4.54 Note (implementing Gordon’ s algorithm) (i) The primessand trequired in step 1 can be probable primes generated by Algo- rithm 4.44. The Miller-Rabin test (Algorithm 4.24) can be used to test each candidate for primality in steps 2 and 4, after ruling out candidates that are divisible by a small prime less than some boundB. See Note 4.45 for guidance on selectingB. Since the Miller-Rabin test is a probabilistic primality test, the output of this implementation of Gordon’s algorithm is a probable prime. (ii) By carefully choosing the sizes of primess,tand parametersi0,j0, one can control the exact bitlength of the resulting primep. Note that the bitlengths ofrand swill be about half that ofp, while the bitlength oftwill be slightly less than that ofr. 4.55 Fact(running time of Gordon’ s algorithm) If the Miller-Rabin test is the primality test used in steps 1, 2, and 4, the expected time Gordon’s algorithm takes to ﬁnd a strong prime is only about 19% more than the expected time Algorithm 4.44 takes to ﬁnd a random prime. 4.4.3 NIST method for generating DSA primes Some public-key schemes require primes satisfying various speciﬁc conditions. For exam- ple, the NIST Digital Signature Algorithm (DSA of§11.5.1) requires two primespand q satisfying the following three conditions: (i)2 159 <q< 2160; that is,qis a160-bit prime; (ii)2L−1 <p< 2Lfor a speciﬁedL, whereL= 512 + 64lfor some0 ≤l≤8; and (iii)qdividesp−1. This section presents an algorithm for generating such primespand q. In the following, Hdenotes the SHA-1 hash function (Algorithm 9.53) which maps bitstrings of bitlength <264 to160-bit hash-codes. Where required, an integerxin the range0 ≤x< 2gwhose binary representation isx= xg−12g−1 + xg−22g−2 + ··· + x222 + x12+ x0 should be converted to theg-bit sequence(xg−1xg−2 ···x2x1x0), and vice versa. §4.4 Prime number generation 151 4.56 AlgorithmNIST method for generating DSA primes INPUT: an integerl,0 ≤l≤8. OUTPUT: a 160-bit primeqand anL-bit primep, whereL= 512 + 64land q\|(p−1). 1. Compute L= 512 + 64l. Using long division of(L−1) by 160, ﬁndn,bsuch that L−1 = 160n+ b, where0 ≤b< 160. 2. Repeat the following: 2.1 Choose a random seeds(not necessarily secret) of bitlengthg≥160. 2.2 Compute U= H(s)⊕H((s+1 )m o d2g). 2.3 Form qfrom Uby setting to1 the most signiﬁcant and least signiﬁcant bits of U. (Note thatqis a160-bit odd integer.) 2.4 Testqfor primality using MILLER-RABIN(q,t) fort≥18 (see Note 4.57). Untilqis found to be a (probable) prime. 3. Seti←0,j←2. 4. Whilei< 4096 do the following: 4.1 Forkfrom 0 tondo the following: setVk←H((s+ j+ k)m o d2g). 4.2 For the integerWdeﬁned below, letX= W+2 L−1.(Xis anL-bit integer.) W= V0 + V12160 + V22320 + ··· + Vn−12160(n−1) +( Vnmod 2b)2160n. 4.3 Computec= Xmod 2qand setp= X−(c−1). (Note thatp≡1( m o d2q).) 4.4 Ifp≥2L−1 then do the following: Testpfor primality using MILLER-RABIN(p,t) fort≥5 (see Note 4.57). Ifpis a (probable) prime then return(q,p). 4.5 Seti←i+1 ,j←j+ n+1 . 5. Go to step 2. 4.57 Note (choice of primality test in Algorithm 4.56) (i) The FIPS 186 document where Algorithm 4.56 was originally described only speci- ﬁes that arobustprimality test be used in steps 2.4 and 4.4, i.e., a primality test where the probability of a composite integer being declared prime is at most( 1 2 )80. If the heuristic assumption is made thatqis a randomly chosen160-bit integer then, by Ta- ble 4.4, MILLER-RABIN(q,18) is a robust test for the primality ofq.I fpis assumed to be a randomly chosenL-bit integer, then by Table 4.4, MILLER-RABIN(p,5)i s a robust test for the primality ofp. Since the Miller-Rabin test is a probabilistic pri- mality test, the output of Algorithm 4.56 is a probable prime. (ii) To improve performance, candidate primesqand pshould be subjected to trial divi- sion by all odd primes less than some boundBbefore invoking the Miller-Rabin test. See Note 4.45 for guidance on selectingB. 4.58 Note (“weak” primes cannot be intentionally constructed) Algorithm 4.56 has the feature that the random seedsis not input to the prime number generation portion of the algorithm itself, but rather to an unpredictable and uncontrollable randomization process (steps 2.2 and 4.1), the output of which is used as the actual random seed. This precludes manipulation of the input seed to the prime number generation. If the seedsand counteriare made public, then anyone can verify thatqandpwere generated using the approved method. This feature prevents a central authority who generatespand qas system-wide parameters for use in the DSA from intentionally constructing “weak” primesqand pwhich it could subsequently exploit to recover other entities’ private keys. 152 Ch. 4 Public-Key Parameters 4.4.4 Constructive techniques for provableprimes Maurer’s algorithm (Algorithm 4.62) generates randomprovableprimes that are almost uniformly distributed over the set of all primes of a speciﬁed size. The expected time for generating a prime is only slightly greater than that for generating a probable prime of equal size using Algorithm 4.44 with security parametert=1 . (In practice, one may wish to chooset> 1 in Algorithm 4.44; cf. Note 4.49.) The main idea behind Algorithm 4.62 is Fact 4.59, which is a slight modiﬁcation of Pocklington’s theorem (Fact 4.40) and Fact 4.41. 4.59 FactLetn≥3 be an odd integer, and suppose thatn=1+2 Rqwhere qis an odd prime. Suppose further thatq>R . (i) If there exists an integerasatisfyinga n−1 ≡1( m o dn) and gcd(a2R−1,n)=1 , thennis prime. (ii) Ifnis prime, the probability that a randomly selected basea,1 ≤a≤n−1, satisﬁes an−1 ≡1( m o dn) and gcd(a2R−1,n)=1 is(1 −1/q). Algorithm 4.62 recursively generates an odd primeq, and then chooses random integersR, R<q , untiln=2 Rq+1 can be proven prime using Fact 4.59(i) for some basea.B y Fact 4.59(ii) the proportion of such bases is1 −1/qfor primen. On the other hand, ifnis composite, then most basesawill fail to satisfy the conditionan−1 ≡1( m o dn). 4.60 Note (description of constantscand min Algorithm 4.62) (i) The optimal value of the constantcdeﬁning the trial division boundB = ck2 in step 2 depends on the implementation of long-integer arithmetic, and is best deter- mined experimentally (cf. Note 4.45). (ii) The constantm=2 0 ensures thatIis at least20 bits long and hence the interval from whichRis selected, namely[I+1 ,2I], is sufﬁciently large (for the values of kof practical interest) that it most likely contains at least one valueRfor whichn= 2Rq+1 is prime. 4.61 Note (relative sizerofqwith respect tonin Algorithm 4.62) Therelative sizerofqwith respect tonis deﬁned to ber=l gq/lg n. In order to assure that the generated primenis chosen randomly with essentially uniform distribution from the set of allk-bit primes, the size of the prime factorqofn−1 must be chosen according to the probability distribution of the largest prime factor of a randomly selectedk-bit integer. Sinceqmust be greater than Rin order for Fact 4.59 to apply, the relative sizerofqis restricted to being in the interval [ 1 2 ,1]. It can be deduced from Fact 3.7(i) that the cumulative probability distribution of the relative sizerof the largest prime factor of a large random integer, given thatris at least 1 2 ,i s(1 + lgr) for1 2 ≤r≤1. In step 4 of Algorithm 4.62, the relative sizeris generated according to this distribution by selecting a random numbers∈[0,1] and then settingr= 2s−1.I fk≤2mthenris chosen to be the smallest permissible value, namely1 2 , in order to ensure that the interval from whichRis selected is sufﬁciently large (cf. Note 4.60(ii)). §4.4 Prime number generation 153 4.62 AlgorithmMaurer’s algorithm for generating provable primes PROV ABLE PRIME( k) INPUT: a positive integerk. OUTPUT: a k-bit prime numbern. 1. (Ifkis small, then test random integers by trial division. A table of small primes may be precomputed for this purpose.) Ifk≤20 then repeatedly do the following: 1.1 Select a randomk-bit odd integern. 1.2 Use trial division by all primes less than√ nto determine whethernis prime. 1.3 Ifnis prime then return(n). 2. Setc←0.1 and m←20 (see Note 4.60). 3. (Trial division bound) SetB←c·k2 (see Note 4.60). 4. (Generater, the size ofqrelative ton— see Note 4.61)I fk> 2mthen repeatedly do the following: select a random numbersin the interval[0,1], setr←2s−1, until (k−rk) >m. Otherwise (i.e.k≤2m), setr←0.5. 5. Compute q←PROVABLE PRIME(⌊r·k⌋+1 ). 6. SetI←⌊2k−1/(2q)⌋. 7. success←0. 8. While (success=0 ) do the following: 8.1 (select a candidate integern) Select a random integerRin the interval[I+ 1,2I] and setn←2Rq+1 . 8.2 Use trial division to determine whethernis divisible by any prime number<B. If it is not then do the following: Select a random integerain the interval[2,n−2]. Compute b←an−1 mod n. Ifb=1 then do the following: Compute b←a2Rmod nand d←gcd(b−1,n). Ifd=1 then success←1. 9. Return(n). 4.63 Note (improvements to Algorithm 4.62) (i) A speedup can be achieved by using Fact 4.42 instead of Fact 4.59(i) for proving n=2 Rq+1 prime in step 8.2 of Maurer’s algorithm — Fact 4.42 only requires that qbe greater than3√ n. (ii) If a candidatenpasses the trial division (in step 8.2), then a Miller-Rabin test (Algo- rithm 4.24) with the single basea=2 should be performed onn; only ifnpasses this test should the attempt to prove its primality (the remainder of step 8.2) be under- taken. This leads to a faster implementation due to the efﬁciency of the Miller-Rabin test with a single basea=2 (cf. Remark 4.50). (iii) Step 4 requires the use of real number arithmetic when computing2s−1. To avoid these computations, one can precompute and store a list of such values for a selection of random numberss∈[0,1]. 4.64 Note (provable primes vs. probable primes) Probable primes are advantageous over prov- able primes in that Algorithm 4.44 for generating probable primes witht=1 is slightly faster than Maurer’s algorithm. Moreover, the latter requires more run-time memory due 154 Ch. 4 Public-Key Parameters to its recursive nature. Provable primes are preferable to probable primes in the sense that the former have zero error probability. In any cryptographic application, however, there is always a non-zero error probability of some catastrophic failure, such as the adversary guessing a secret key or hardware failure. Since the error probability of probable primes can be efﬁciently brought down to acceptably low levels (see Note 4.49 but note the depen- dence ont), there appears to be no reason for mandating the use of provable primes over probable primes. 4.5 Irreducible polynomials overZp Recall (Deﬁnition 2.190) that a polynomialf(x) ∈Zp[x] of degreem≥1 is said to be irreducible overZpif it cannot be written as a product of two polynomials inZp[x] each having degree less thanm. Such a polynomialf(x) can be used to represent the elements of the ﬁnite ﬁeldFpm asFpm = Zp[x]/(f(x)), the set of all polynomials inZp[x] of de- gree less thanmwhere the addition and multiplication of polynomials is performed modulo f(x) (see§2.6.3). This section presents techniques for constructing irreducible polynomials overZp, wherepis a prime. The characteristic two ﬁnite ﬁeldsF2m are of particular inter- est for cryptographic applications because the arithmetic in these ﬁelds can be efﬁciently performed both in software and in hardware. For this reason, additional attention is given to the special case of irreducible polynomials overZ 2. The arithmetic in ﬁnite ﬁelds can usually be implemented more efﬁciently if the irre- ducible polynomial chosen has few non-zero terms. Irreducibletrinomials, i.e., irreducible polynomials having exactly three non-zero terms, are considered in§4.5.2.Primitivepoly- nomials, i.e., irreducible polynomialsf(x) of degreeminZp[x] for whichxis a generator ofF∗ pm , the multiplicative group of the ﬁnite ﬁeldFpm = Zp[x]/(f(x)) (Deﬁnition 2.228), are the topic of§4.5.3. Primitive polynomials are also used in the generation of linear feed- back shift register sequences having the maximum possible period (Fact 6.12). 4.5.1 Irreducible polynomials Iff(x) ∈Zp[x] is irreducible overZpandais a non-zero element inZp, thena·f(x) is also irreducible overZp. Hence it sufﬁces to restrict attention tomonic polynomials inZp[x], i.e., polynomials whose leading coefﬁcient is 1. Observe also that iff(x) is an irreducible polynomial, then its constant term must be non-zero. In particular, iff(x) ∈Z2[x], then its constant term must be 1. There is a formula for computing exactly the number of monic irreducible polynomi- als inZp[x] of a ﬁxed degree. The M¨obius function, which is deﬁned next, is used in this formula. 4.65 DeﬁnitionLetmbe a positive integer. TheM¨obius functionµis deﬁned by µ(m)=    1, ifm=1 , 0, ifmis divisible by the square of a prime, (−1)k, ifmis the product ofkdistinct primes. 4.66 Example (M¨obius function) The following table gives the values of the M¨obius function µ(m) for the ﬁrst 10 values ofm: §4.5 Irreducible polynomials overZp 155 m 1 2 3 4 5 6 7 8 9 10 µ(m) 1 −1 −1 0 −1 1 −1 0 0 1 □ 4.67 Fact(number of monic irreducible polynomials) Letpbe a prime andma positive integer. (i) The numberNp(m) of monic irreducible polynomials of degreeminZp[x] is given by the following formula: Np(m)= 1 m ∑ d\|m µ(d)pm/d, where the summation ranges over all positive divisorsdofm. (ii) The probability of a random monic polynomial of degreeminZp[x] being irreducible overZpis roughly1 m. More speciﬁcally, the numberNp(m) satisﬁes 1 2m ≤ Np(m) pm ≈ 1 m. Testing irreducibility of polynomials inZp[x] is signiﬁcantly simpler than testing pri- mality of integers. A polynomial can be tested for irreducibility by verifying that it has no irreducible factors of degree≤⌊m 2 ⌋. The following result leads to an efﬁcient method (Al- gorithm 4.69) for accomplishing this. 4.68 FactLetpbe a prime and letkbe a positive integer. (i) The product of all monic irreducible polynomials inZp[x] of degree dividingkis equal toxpk −x. (ii) Letf(x) be a polynomial of degreeminZp[x]. Thenf(x) is irreducible overZpif and only ifgcd(f(x),xpi −x)=1 for eachi,1 ≤i≤⌊m 2 ⌋. 4.69 AlgorithmTesting a polynomial for irreducibility INPUT: a primepand a monic polynomialf(x) of degreeminZp[x]. OUTPUT: an answer to the question: “Isf(x) irreducible overZp?” 1. Setu(x)←x. 2. Forifrom 1 to⌊m 2 ⌋do the following: 2.1 Computeu(x)←u(x)pmod f(x) using Algorithm 2.227. (Note thatu(x) is a polynomial inZp[x] of degree less thanm.) 2.2 Compute d(x)=g c d (f(x),u(x) −x) (using Algorithm 2.218). 2.3 Ifd(x) ̸=1then return(“reducible”). 3. Return(“irreducible”). Fact 4.67 suggests that one method for ﬁnding an irreducible polynomial of degreem inZp[x] is to generate a random monic polynomial of degreeminZp[x], test it for irre- ducibility, and continue until an irreducible one is found (Algorithm 4.70). The expected number of polynomials to be tried before an irreducible one is found is approximatelym. 156 Ch. 4 Public-Key Parameters 4.70 AlgorithmGenerating a random monic irreducible polynomial overZp INPUT: a primepand a positive integerm. OUTPUT: a monic irreducible polynomialf(x) of degreeminZp[x]. 1. Repeat the following: 1.1 (Generate a random monic polynomial of degreeminZp[x]) Randomly select integersa0,a1,a2,...,a m−1 between0 and p−1 witha0 ̸= 0. Letf(x) be the polynomialf(x)= xm+am−1xm−1+···+a2x2+a1x+a0. 1.2 Use Algorithm 4.69 to test whetherf(x) is irreducible overZp. Untilf(x) is irreducible. 2. Return(f(x)). It is known that the expected degree of the irreducible factor of least degree of a random polynomial of degreeminZp[x] isO(lg m). Hence for each choice off(x), the expected number of times steps 2.1 – 2.3 of Algorithm 4.69 are iterated isO(lg m). Each iteration takesO((lg p)m2) Zp-operations. These observations, together with Fact 4.67(ii), deter- mine the running time for Algorithm 4.70. 4.71 FactAlgorithm 4.70 has an expected running time ofO(m3(lg m)(lg p)) Zp-operations. Given one irreducible polynomial of degreemoverZp, Note 4.74 describes a method, which is more efﬁcient than Algorithm 4.70, for randomly generating additional such poly- nomials. 4.72 DeﬁnitionLetFqbe a ﬁnite ﬁeld of characteristicp, and letα∈Fq.A minimum polyno- mialofαoverZpis a monic polynomial of least degree inZp[x] havingαas a root. 4.73 FactLetFqbe a ﬁnite ﬁeld of orderq= pm, and letα∈Fq. (i) The minimum polynomial ofαoverZp, denotedmα(x), is unique. (ii)mα(x) is irreducible overZp. (iii) The degree ofmα(x) is a divisor ofm. (iv) Lettbe the smallest positive integer such thatαpt = α. (Note that such atexists since, by Fact 2.213,αpm = α.) Then mα(x)= t−1∏ i=0 (x−αpi ). (4.1) 4.74 Note (generating new irreducible polynomials from a given one) Suppose thatf(y) is a given irreducible polynomial of degreemoverZp. The ﬁnite ﬁeldFpm can then be repre- sented asFpm = Zp[y]/(f(y)). A random monic irreducible polynomial of degreemover Zpcan be efﬁciently generated as follows. First generate a random elementα∈Fpm and then, by repeated exponentiation byp, determine the smallest positive integertfor which αpt = α.I ft<m , then generate a new random elementα∈Fpm and repeat; the probabil- ity thatt<m is known to be at most(lg m)/qm/2. If indeedt= m, then computemα(x) using the formula (4.1). Thenmα(x) is a random monic irreducible polynomial of degree minZp[x]. This method has an expected running time ofO(m3(lg p)) Zp-operations (com- pare with Fact 4.71). §4.5 Irreducible polynomials overZp 157 4.5.2 Irreducible trinomials If a polynomialf(x) inZ2[x] has an even number of non-zero terms, thenf(1) = 0, whence (x+1 )is a factor off(x). Hence, the smallest number of non-zero terms an irreducible polynomial of degree≥2 inZ2[x] can have is three. An irreducibletrinomialof degreem inZ2[x] must be of the formxm+ xk+1 , where1 ≤k≤m−1. Choosing an irreducible trinomialf(x) ∈Z2[x] of degreemto represent the elements of the ﬁnite ﬁeldF2m = Z2[x]/(f(x)) can lead to a faster implementation of the ﬁeld arithmetic. The following facts are sometimes of use when searching for irreducible trinomials. 4.75 FactLetmbe a positive integer, and letkdenote an integer in the interval[1,m−1]. (i) If the trinomialxm+ xk+1 is irreducible overZ2 then so isxm+ xm−k+1 . (ii) Ifm≡0( m o d8 ), there is no irreducible trinomial of degreeminZ2[x]. (iii) Suppose that eitherm≡3( m o d8 )orm≡5( m o d8 ). Then a necessary condition forxm+ xk+1 to be irreducible overZ2 is that eitherkorm−kmust be of the form 2dfor some positive divisordofm. Tables 4.6 and 4.7 list an irreducible trinomial of degreemoverZ2 for eachm≤1478 for which such a trinomial exists. 4.5.3 Primitive polynomials Primitive polynomials were introduced at the beginning of§4.5. Letf(x) ∈Zp[x] be an irreducible polynomial of degreem. If the factorization of the integerpm−1 is known, then Fact 4.76 yields an efﬁcient algorithm (Algorithm 4.77) for testing whether or notf(x) is a primitive polynomial. If the factorization ofpm −1 is unknown, there is no efﬁcient algorithm known for performing this test. 4.76 FactLetpbe a prime and let the distinct prime factors ofpm−1 be r1,r2,...,r t. Then an irreducible polynomialf(x) ∈Zp[x] is primitive if and only if for eachi,1 ≤i≤t: x(pm−1)/ri ̸≡1( m o df(x)). (That is,xis an element of orderpm−1 in the ﬁeldZp[x]/(f(x)).) 4.77 AlgorithmTesting whether an irreducible polynomial is primitive INPUT: a primep, a positive integerm, the distinct prime factorsr1,r2,...,r tofpm−1, and a monic irreducible polynomialf(x) of degreeminZp[x]. OUTPUT: an answer to the question: “Isf(x) a primitive polynomial?” 1. Forifrom 1 totdo the following: 1.1 Compute l(x)= x(pm−1)/ri mod f(x) (using Algorithm 2.227). 1.2 Ifl(x)=1 then return(“not primitive”). 2. Return(“primitive”). There are preciselyφ(pm−1)/mmonic primitive polynomials of degreeminZp[x] (Fact 2.230), whereφis the Euler phi function (Deﬁnition 2.100). Since the number of monic irreducible polynomials of degreeminZp[x] is roughlypm/m(Fact 4.67(ii)), it fol- lows that the probability of a random monic irreducible polynomial of degreeminZp[x] 158 Ch. 4 Public-Key Parameters m k m k m k m k m k m k m k 2 1 93 2 193 15 295 48 402 171 508 9 618 295 3 1 94 21 194 87 297 5 404 65 510 69 620 9 4 1 95 11 196 3 300 5 406 141 511 10 622 297 5 2 97 6 198 9 302 41 407 71 513 26 623 68 6 1 98 11 199 34 303 1 409 87 514 67 625 133 7 1 100 15 201 14 305 102 412 147 516 21 626 251 9 1 102 29 202 55 308 15 414 13 518 33 628 223 10 3 103 9 204 27 310 93 415 102 519 79 631 307 11 2 105 4 207 43 313 79 417 107 521 32 633 101 12 3 106 15 209 6 314 15 418 199 522 39 634 39 14 5 108 17 210 7 316 63 420 7 524 167 636 217 15 1 110 33 212 105 318 45 422 149 526 97 639 16 17 3 111 10 214 73 319 36 423 25 527 47 641 11 18 3 113 9 215 23 321 31 425 12 529 42 642 119 20 3 118 33 217 45 322 67 426 63 532 1 646 249 21 2 119 8 218 11 324 51 428 105 534 161 647 5 22 1 121 18 220 7 327 34 431 120 537 94 649 37 23 5 123 2 223 33 329 50 433 33 538 195 650 3 25 3 124 19 225 32 330 99 436 165 540 9 651 14 28 1 126 21 228 113 332 89 438 65 543 16 652 93 29 2 127 1 231 26 333 2 439 49 545 122 654 33 30 1 129 5 233 74 337 55 441 7 550 193 655 88 31 3 130 3 234 31 340 45 444 81 551 135 657 38 33 10 132 17 236 5 342 125 446 105 553 39 658 55 34 7 134 57 238 73 343 75 447 73 556 153 660 11 35 2 135 11 239 36 345 22 449 134 558 73 662 21 36 9 137 21 241 70 346 63 450 47 559 34 663 107 39 4 140 15 242 95 348 103 455 38 561 71 665 33 41 3 142 21 244 111 350 53 457 16 564 163 668 147 42 7 145 52 247 82 351 34 458 203 566 153 670 153 44 5 146 71 249 35 353 69 460 19 567 28 671 15 46 1 147 14 250 103 354 99 462 73 569 77 673 28 47 5 148 27 252 15 358 57 463 93 570 67 676 31 49 9 150 53 253 46 359 68 465 31 574 13 679 66 52 3 151 3 255 52 362 63 468 27 575 146 682 171 54 9 153 1 257 12 364 9 470 9 577 25 684 209 55 7 154 15 258 71 366 29 471 1 580 237 686 197 57 4 155 62 260 15 367 21 473 200 582 85 687 13 58 19 156 9 263 93 369 91 474 191 583 130 689 14 60 1 159 31 265 42 370 139 476 9 585 88 690 79 62 29 161 18 266 47 372 111 478 121 588 35 692 299 63 1 162 27 268 25 375 16 479 104 590 93 694 169 65 18 166 37 270 53 377 41 481 138 593 86 695 177 66 3 167 6 271 58 378 43 484 105 594 19 697 267 68 9 169 34 273 23 380 47 486 81 596 273 698 215 71 6 170 11 274 67 382 81 487 94 599 30 700 75 73 25 172 1 276 63 383 90 489 83 601 201 702 37 74 35 174 13 278 5 385 6 490 219 602 215 705 17 76 21 175 6 279 5 386 83 492 7 604 105 708 15 79 9 177 8 281 93 388 159 494 17 606 165 711 92 81 4 178 31 282 35 390 9 495 76 607 105 713 41 84 5 180 3 284 53 391 28 497 78 609 31 714 23 86 21 182 81 286 69 393 7 498 155 610 127 716 183 87 13 183 56 287 71 394 135 500 27 612 81 718 165 89 38 185 24 289 21 396 25 503 3 614 45 719 150 90 27 186 11 292 37 399 26 505 156 615 211 721 9 92 21 191 9 294 33 401 152 506 23 617 200 722 231 Table 4.6:Irreducible trinomialsxm + xk +1 overZ2. For eachm,1 ≤m≤722, for which an irreducible trinomial of degreeminZ2[x] exists, the table lists the smallestkfor whichxm + xk +1 is irreducible overZ2. §4.5 Irreducible polynomials overZp 159 m k m k m k m k m k m k m k 724 207 831 49 937 217 1050 159 1159 66 1265 119 1374 609 726 5 833 149 938 207 1052 291 1161 365 1266 7 1375 52 727 180 834 15 942 45 1054 105 1164 19 1268 345 1377 100 729 58 838 61 943 24 1055 24 1166 189 1270 333 1380 183 730 147 839 54 945 77 1057 198 1167 133 1271 17 1383 130 732 343 841 144 948 189 1058 27 1169 114 1273 168 1385 12 735 44 842 47 951 260 1060 439 1170 27 1276 217 1386 219 737 5 844 105 953 168 1062 49 1174 133 1278 189 1388 11 738 347 845 2 954 131 1063 168 1175 476 1279 216 1390 129 740 135 846 105 956 305 1065 463 1177 16 1281 229 1391 3 742 85 847 136 959 143 1071 7 1178 375 1282 231 1393 300 743 90 849 253 961 18 1078 361 1180 25 1284 223 1396 97 745 258 850 111 964 103 1079 230 1182 77 1286 153 1398 601 746 351 852 159 966 201 1081 24 1183 87 1287 470 1399 55 748 19 855 29 967 36 1082 407 1185 134 1289 99 1401 92 750 309 857 119 969 31 1084 189 1186 171 1294 201 1402 127 751 18 858 207 972 7 1085 62 1188 75 1295 38 1404 81 753 158 860 35 975 19 1086 189 1190 233 1297 198 1407 47 754 19 861 14 977 15 1087 112 1191 196 1298 399 1409 194 756 45 862 349 979 178 1089 91 1193 173 1300 75 1410 383 758 233 865 1 982 177 1090 79 1196 281 1302 77 1412 125 759 98 866 75 983 230 1092 23 1198 405 1305 326 1414 429 761 3 868 145 985 222 1094 57 1199 114 1306 39 1415 282 762 83 870 301 986 3 1095 139 1201 171 1308 495 1417 342 767 168 871 378 988 121 1097 14 1202 287 1310 333 1420 33 769 120 873 352 990 161 1098 83 1204 43 1311 476 1422 49 772 7 876 149 991 39 1100 35 1206 513 1313 164 1423 15 774 185 879 11 993 62 1102 117 1207 273 1314 19 1425 28 775 93 881 78 994 223 1103 65 1209 118 1319 129 1426 103 777 29 882 99 996 65 1105 21 1210 243 1321 52 1428 27 778 375 884 173 998 101 1106 195 1212 203 1324 337 1430 33 780 13 887 147 999 59 1108 327 1214 257 1326 397 1431 17 782 329 889 127 1001 17 1110 417 1215 302 1327 277 1433 387 783 68 890 183 1007 75 1111 13 1217 393 1329 73 1434 363 785 92 892 31 1009 55 1113 107 1218 91 1332 95 1436 83 791 30 894 173 1010 99 1116 59 1220 413 1334 617 1438 357 793 253 895 12 1012 115 1119 283 1223 255 1335 392 1441 322 794 143 897 113 1014 385 1121 62 1225 234 1337 75 1442 395 798 53 898 207 1015 186 1122 427 1226 167 1338 315 1444 595 799 25 900 1 1020 135 1126 105 1228 27 1340 125 1446 421 801 217 902 21 1022 317 1127 27 1230 433 1343 348 1447 195 804 75 903 35 1023 7 1129 103 1231 105 1345 553 1449 13 806 21 905 117 1025 294 1130 551 1233 151 1348 553 1452 315 807 7 906 123 1026 35 1134 129 1234 427 1350 237 1454 297 809 15 908 143 1028 119 1135 9 1236 49 1351 39 1455 52 810 159 911 204 1029 98 1137 277 1238 153 1353 371 1457 314 812 29 913 91 1030 93 1138 31 1239 4 1354 255 1458 243 814 21 916 183 1031 68 1140 141 1241 54 1356 131 1460 185 815 333 918 77 1033 108 1142 357 1242 203 1358 117 1463 575 817 52 919 36 1034 75 1145 227 1246 25 1359 98 1465 39 818 119 921 221 1036 411 1146 131 1247 14 1361 56 1466 311 820 123 924 31 1039 21 1148 23 1249 187 1362 655 1468 181 822 17 926 365 1041 412 1151 90 1252 97 1364 239 1470 49 823 9 927 403 1042 439 1153 241 1255 589 1366 1 1471 25 825 38 930 31 1044 41 1154 75 1257 289 1367 134 1473 77 826 255 932 177 1047 10 1156 307 1260 21 1369 88 1476 21 828 189 935 417 1049 141 1158 245 1263 77 1372 181 1478 69 Table 4.7:Irreducible trinomialsxm+xk+1 overZ2. For eachm,723 ≤m≤1478, for which an irreducible trinomial of degreeminZ2[x] exists, the table gives the smallestkfor whichxm +xk +1 is irreducible overZ2. 160 Ch. 4 Public-Key Parameters being primitive is approximatelyφ(pm−1)/pm. Using the lower bound for the Euler phi function (Fact 2.102), this probability can be seen to be at least1/(6 ln lnpm). This sug- gests the following algorithm for generating primitive polynomials. 4.78 AlgorithmGenerating a random monic primitive polynomial overZp INPUT: a primep, integerm≥1, and the distinct prime factorsr1,r2,...,r tofpm−1. OUTPUT: a monic primitive polynomialf(x) of degreeminZp[x]. 1. Repeat the following: 1.1 Use Algorithm 4.70 to generate a random monic irreducible polynomialf(x) of degreeminZp[x]. 1.2 Use Algorithm 4.77 to test whetherf(x) is primitive. Untilf(x) is primitive. 2. Return(f(x)). For eachm,1 ≤m≤229, Table 4.8 lists a polynomial of degreemthat is primitive overZ2. If there exists a primitive trinomialf(x)= xm+ xk+1 , then the trinomial with the smallestkis listed. If no primitive trinomial exists, then a primitive pentanomial of the form f(x)= xm+ xk1 + xk2 + xk3 +1 is listed. Ifpm −1 is prime, then Fact 4.76 implies that every irreducible polynomial of de- greeminZp[x] is also primitive. Table 4.9 gives either a primitive trinomial or a primitive pentanomial of degreemoverZ2 where mis an exponent of one of the ﬁrst 27 Mersenne primes (Deﬁnition 4.35). 4.6 Generators and elements of high order Recall (Deﬁnition 2.169) that ifGis a (multiplicative) ﬁnite group, theorderof an element a∈Gis the least positive integertsuch thatat =1 . If there arenelements inG, and if a∈Gis an element of ordern, thenGis said to becyclicand ais called ageneratoror a primitive elementofG(Deﬁnition 2.167). Of special interest for cryptographic applications are the multiplicative groupZ∗ p of the integers modulo a primep, and the multiplicative groupF∗ 2 m of the ﬁnite ﬁeldF2m of characteristic two; these groups are cyclic (Fact 2.213). Also of interest is the groupZ∗ n (Deﬁnition 2.124), wherenis the product of two distinct odd primes. This section deals with the problem of ﬁnding generators and other elements of high order inZ∗ p,F∗ 2 m , andZ∗ n . See§2.5.1 for background in group theory and§2.6 for background in ﬁnite ﬁelds. Algorithm 4.79 is an efﬁcient method for determining the order of a group element, given the prime factorization of the group ordern. The correctness of the algorithm follows from the fact that the order of an element must dividen(Fact 2.171). §4.6 Generators and elements of high order 161 k or k or k or k or m (k1,k 2,k 3) m (k1,k 2,k 3) m (k1,k 2,k 3) m (k1,k 2,k 3) 2 1 59 22, 21, 1 116 71, 70, 1 173 100, 99, 1 3 1 60 1 117 20, 18, 2 174 13 4 1 61 16, 15, 1 118 33 175 6 5 2 62 57, 56, 1 119 8 176 119, 118, 1 6 1 63 1 120 118, 111, 7 177 8 7 1 64 4, 3, 1 121 18 178 87 8 6, 5, 1 65 18 122 60, 59, 1 179 34, 33, 1 9 4 66 10, 9, 1 123 2 180 37, 36, 1 10 3 67 10, 9, 1 124 37 181 7, 6, 1 11 2 68 9 125 108, 107, 1 182 128, 127, 1 12 7, 4, 3 69 29, 27, 2 126 37, 36, 1 183 56 13 4, 3, 1 70 16, 15, 1 127 1 184 102, 101, 1 14 12, 11, 1 71 6 128 29, 27, 2 185 24 15 1 72 53, 47, 6 129 5 186 23, 22, 1 16 5, 3, 2 73 25 130 3 187 58, 57, 1 17 3 74 16, 15, 1 131 48, 47, 1 188 74, 73, 1 18 7 75 11, 10, 1 132 29 189 127, 126, 1 19 6, 5, 1 76 36, 35, 1 133 52, 51, 1 190 18, 17, 1 20 3 77 31, 30, 1 134 57 191 9 21 2 78 20, 19, 1 135 11 192 28, 27, 1 22 1 79 9 136 126, 125, 1 193 15 23 5 80 38, 37, 1 137 21 194 87 24 4, 3, 1 81 4 138 8, 7, 1 195 10, 9, 1 25 3 82 38, 35, 3 139 8, 5, 3 196 66, 65, 1 26 8, 7, 1 83 46, 45, 1 140 29 197 62, 61, 1 27 8, 7, 1 84 13 141 32, 31, 1 198 65 28 3 85 28, 27, 1 142 21 199 34 29 2 86 13, 12, 1 143 21, 20, 1 200 42, 41, 1 30 16, 15, 1 87 13 144 70, 69, 1 201 14 31 3 88 72, 71, 1 145 52 202 55 32 28, 27, 1 89 38 146 60, 59, 1 203 8, 7, 1 33 13 90 19, 18, 1 147 38, 37, 1 204 74, 73, 1 34 15, 14, 1 91 84, 83, 1 148 27 205 30, 29, 1 35 2 92 13, 12, 1 149 110, 109, 1 206 29, 28, 1 36 11 93 2 150 53 207 43 37 12, 10, 2 94 21 151 3 208 62, 59, 3 38 6, 5, 1 95 11 152 66, 65, 1 209 6 39 4 96 49, 47, 2 153 1 210 35, 32, 3 40 21, 19, 2 97 6 154 129, 127, 2 211 46, 45, 1 41 3 98 11 155 32, 31, 1 212 105 42 23, 22, 1 99 47, 45, 2 156 116, 115, 1 213 8, 7, 1 43 6, 5, 1 100 37 157 27, 26, 1 214 49, 48, 1 44 27, 26, 1 101 7, 6, 1 158 27, 26, 1 215 23 45 4, 3, 1 102 77, 76, 1 159 31 216 196, 195, 1 46 21, 20, 1 103 9 160 19, 18, 1 217 45 47 5 104 11, 10, 1 161 18 218 11 48 28, 27, 1 105 16 162 88, 87, 1 219 19, 18, 1 49 9 106 15 163 60, 59, 1 220 15, 14, 1 50 27, 26, 1 107 65, 63, 2 164 14, 13, 1 221 35, 34, 1 51 16, 15, 1 108 31 165 31, 30, 1 222 92, 91, 1 52 3 109 7, 6, 1 166 39, 38, 1 223 33 53 16, 15, 1 110 13, 12, 1 167 6 224 31, 30, 1 54 37, 36, 1 111 10 168 17, 15, 2 225 32 55 24 112 45, 43, 2 169 34 226 58, 57, 1 56 22, 21, 1 113 9 170 23 227 46, 45, 1 57 7 114 82, 81, 1 171 19, 18, 1 228 148, 147, 1 58 19 115 15, 14, 1 172 7 229 64, 63, 1 Table 4.8:Primitive polynomials overZ2. For eachm,1 ≤m≤229, an exponentkis given for which the trinomialxm +xk +1 is primitive overZ2. If no such trinomial exists, a triple of exponents (k1,k2,k3) is given for which the pentanomialxm + xk1 + xk2 + xk3 +1 is primitive overZ2. 162 Ch. 4 Public-Key Parameters j m k(k1,k2,k3) 1 2 1 2 3 1 3 5 2 4 7 1, 3 5 13 none (4,3,1) 6 17 3, 5, 6 7 19 none (5,2,1) 8 31 3, 6, 7, 13 9 61 none (43,26,14) 10 89 38 11 107 none (82,57,31) 12 127 1, 7, 15, 30, 63 13 521 32, 48, 158, 168 14 607 105, 147, 273 15 1279 216, 418 16 2203 none (1656,1197,585) 17 2281 715, 915, 1029 18 3217 67, 576 19 4253 none (3297,2254,1093) 20 4423 271, 369, 370, 649, 1393, 1419, 2098 21 9689 84, 471, 1836, 2444, 4187 22 9941 none (7449,4964,2475) 23 11213 none (8218,6181,2304) 24 19937 881, 7083, 9842 25 21701 none (15986,11393,5073) 26 23209 1530, 6619, 9739 27 44497 8575, 21034 Table 4.9:Primitive polynomials of degreemoverZ2,2m−1 a Mersenne prime. For each exponent m= Mj of the ﬁrst 27 Mersenne primes, the table lists all values ofk,1 ≤k≤m/2, for which the trinomialxm + xk +1 is irreducible overZ2. If no such trinomial exists, a triple of exponents (k1,k2,k3) is listed such that the pentanomialxm + xk1 + xk2 + xk3 +1 is irreducible overZ2. 4.79 AlgorithmDetermining the order of a group element INPUT: a (multiplicative) ﬁnite groupGof ordern, an elementa∈G, and the prime fac- torizationn= pe1 1 pe2 2 ···pek k . OUTPUT: the ordertofa. 1. Sett←n. 2. Forifrom 1 tokdo the following: 2.1 Sett←t/pei i . 2.2 Compute a1←at. 2.3 Whilea1 ̸=1do the following: computea1←api 1 and sett←t·pi. 3. Return(t). Suppose now thatGis a cyclic group of ordern. Then for any divisordofnthe number of elements of orderdinGis exactlyφ(d) (Fact 2.173(ii)), whereφis the Euler phi function (Deﬁnition 2.100). In particular,Ghas exactlyφ(n) generators, and hence the probability of a random element inGbeing a generator isφ(n)/n. Using the lower bound for the Eu- ler phi function (Fact 2.102), this probability can be seen to be at least1/(6 ln lnn). This §4.6 Generators and elements of high order 163 suggests the following efﬁcient randomized algorithm for ﬁnding a generator of a cyclic group. 4.80 AlgorithmFinding a generator of a cyclic group INPUT: a cyclic groupGof ordern, and the prime factorizationn= pe1 1 pe2 2 ···pek k . OUTPUT: a generatorαofG. 1. Choose a random elementαinG. 2. Forifrom 1 tokdo the following: 2.1 Compute b←αn/pi . 2.2 Ifb=1 then go to step 1. 3. Return(α). 4.81 Note (group elements of high order) In some situations it may be desirable to have an el- ement of high order, and not a generator. Given a generatorαin a cyclic groupGof order n, and given a divisordofn, an elementβof orderdinGcan be efﬁciently obtained as follows:β= αn/d.I fqis a prime divisor of the ordernof a cyclic groupG, then the fol- lowing method ﬁnds an elementβ∈Gof orderqwithout ﬁrst having to ﬁnd a generator ofG: select a random elementg∈Gand computeβ= gn/q; repeat untilβ̸=1. 4.82 Note (generators ofF∗ 2m ) There are two basic approaches to ﬁnding a generator ofF∗ 2 m . Both techniques require the factorization of the order ofF∗ 2 m , namely2m−1. (i) Generate a monic primitive polynomialf(x) of degreemoverZ2 (Algorithm 4.78). The ﬁnite ﬁeldF2m can then be represented asZ2[x]/(f(x)), the set of all polyno- mials overZ2 modulo f(x), and the elementα= xis a generator. (ii) Select the method for representing elements ofF2m ﬁrst. Then use Algorithm 4.80 withG= F∗ 2 m and n=2 m−1 to ﬁnd a generatorαofF∗ 2 m . Ifn= pq, wherepandqare distinct odd primes, thenZ∗ n is a non-cyclic group of order φ(n)=( p−1)(q−1). The maximum order of an element inZ∗ n islcm(p−1,q−1). Algorithm 4.83 is a method for generating such an element which requires the factorizations ofp−1 and q−1. 4.83 AlgorithmSelecting an element of maximum order inZ∗ n, wheren= pq INPUT: two distinct odd primes,p,q, and the factorizations ofp−1 and q−1. OUTPUT: an elementαof maximum orderlcm(p−1,q−1) inZ∗ n, wheren= pq. 1. Use Algorithm 4.80 withG= Z∗ p and n= p−1 to ﬁnd a generatoraofZ∗ p . 2. Use Algorithm 4.80 withG= Z∗ qand n= q−1 to ﬁnd a generatorbofZ∗ q . 3. Use Gauss’s algorithm (Algorithm 2.121) to ﬁnd an integerα,1 ≤ α ≤ n−1, satisfyingα≡a(mod p) and α≡b(mod q). 4. Return(α). 164 Ch. 4 Public-Key Parameters 4.6.1 Selecting a primep and generator ofZ∗ p In cryptographic applications for which a generator ofZ∗ pis required, one usually has the ﬂexibility of selecting the primep. To guard against the Pohlig-Hellman algorithm for com- puting discrete logarithms (Algorithm 3.63), a security requirement is thatp−1 should con- tain a “large” prime factorq. In this context, “large” means that the quantity√ qrepresents an infeasible amount of computation; for example,q≥2160. This suggests the following algorithm for selecting appropriate parameters(p,α). 4.84 AlgorithmSelecting ak-bit primepand a generatorαofZ∗ p INPUT: the required bitlengthkof the prime and a security parametert. OUTPUT: a k-bit primepsuch thatp−1 has a prime factor≥t, and a generatorαofZ∗ p. 1. Repeat the following: 1.1 Select a randomk-bit primep(for example, using Algorithm 4.44). 1.2 Factorp−1. Untilp−1 has a prime factor≥t. 2. Use Algorithm 4.80 withG= Z∗ p and n= p−1 to ﬁnd a generatorαofZ∗ p . 3. Return(p,α). Algorithm 4.84 is relatively inefﬁcient as it requires the use of an integer factorization algorithm in step 1.2. An alternative approach is to generate the primepby ﬁrst choosing a large primeqand then selecting relatively small integersRat random untilp=2 Rq+1 is prime. Sincep−1=2 Rq, the factorization ofp−1 can be obtained by factoringR.A particularly convenient situation occurs by imposing the conditionR=1 . In this case the factorization ofp−1 is simply2q. Furthermore, sinceφ(p−1) =φ(2q)= φ(2)φ(q)= q−1, the probability that a randomly selected elementα∈Z∗ p is a generator isq−1 2q ≈1 2 . 4.85 DeﬁnitionA safe primepis a prime of the formp=2 q+1 where qis prime. Algorithm 4.86 generates a safe (probable) primepand a generator ofZ∗ p . 4.86 AlgorithmSelecting ak-bit safe primepand a generatorαofZ∗ p INPUT: the required bitlengthkof the prime. OUTPUT: a k-bit safe primepand a generatorαofZ∗ p. 1. Do the following: 1.1 Select a random(k−1)-bit primeq(for example, using Algorithm 4.44). 1.2 Computep←2q+1 , and test whetherpis prime (for example, using trial divi- sion by small primes and Algorithm 4.24). Untilpis prime. 2. Use Algorithm 4.80 to ﬁnd a generatorαofZ∗ p. 3. Return(p,α). §4.7 Notes and further references 165 4.7 Notes and further references §4.1 Several books provide extensive treatments of primality testing including those by Bres- soud [198], Bach and Shallit [70], and Koblitz [697]. The book by Kranakis [710] offers a more theoretical approach. Cohen [263] gives a comprehensive treatment of modern pri- mality tests. See also the survey articles by A. Lenstra [747] and A. Lenstra and H. Lenstra [748]. Facts 4.1 and 4.2 were proven in 1837 by Dirichlet. For proofs of these results, see Chapter 16 of Ireland and Rosen [572]. Fact 4.3 is due to Rosser and Schoenfeld [1070]. Bach and Shallit [70] have further results on the distribution of prime numbers. §4.2 Fact 4.13(i) was proven by Alford, Granville, and Pomerance [24]; see also Granville [521]. Fact 4.13(ii) is due to Pomerance, Selfridge, and Wagstaff [996]. Pinch [974] showed that there are105212 Carmichael numbers up to10 15. The Solovay-Strassen probabilistic primality test (Algorithm 4.18) is due to Solovay and Strassen [1163], as modiﬁed by Atkin and Larson [57]. Fact 4.23 was proven independently by Monier [892] and Rabin [1024]. The Miller-Rabin test (Algorithm 4.24) originated in the work of Miller [876] who presented it as a non- probabilistic polynomial-time algorithm assuming the correctness of the Extended Riemann Hypothesis (ERH). Rabin [1021, 1024] rephrased Miller’s algorithm as a probabilistic pri- mality test. Rabin’s algorithm required a small number of gcd computations. The Miller- Rabin test (Algorithm 4.24) is a simpliﬁcation of Rabin’s algorithm which does not require any gcd computations, and is due to Knuth [692, p.379]. Arazi [55], making use of Mont- gomery modular multiplication (§14.3.2), showed how the Miller-Rabin test can be imple- mented by “divisionless modular exponentiations” only, yielding a probabilistic primality test which does not use any division operations. Miller [876], appealing to the work of Ankeny [32], proved under assumption of the Ex- tended Riemann Hypothesis that, ifnis an odd composite integer, then its least strong wit- ness is less thanc(ln n) 2, wherecis some constant. Bach [63] proved that this constant may be taken to bec=2 ; see also Bach [64]. As a consequence, one can testnfor pri- mality inO((lg n)5) bit operations by executing the Miller-Rabin algorithm for all bases a≤2(ln n)2. This gives a deterministic polynomial-time algorithm for primality testing, under the assumption that the ERH is true. Table 4.1 is from Jaeschke [630], building on earlier work of Pomerance, Selfridge, and Wagstaff [996]. Arnault [56] found the following46-digit composite integer n= 1195068768795265792518361315725116351898245581 that is a strong pseudoprime to all the11 prime bases up to31. Arnault also found a337- digit composite integer which is a strong pseudoprime to all46 prime bases up to199. The Miller-Rabin test (Algorithm 4.24) randomly generatestindependent basesaand tests to see if each is a strong witness forn. Letnbe an odd composite integer and lett = ⌈1 2 lg n⌉. In situations where random bits are scarce, one may choose instead to generate a single random baseaand use the basesa,a+1 ,...,a + t−1. Bach [66] proved that for a randomly chosen integera, the probability thata,a+1 ,...,a + t−1 are all strong liars fornis bounded above byn−1/4+o(1); in other words, the probability that the Miller- Rabin algorithm using these bases mistakenly declares an odd composite integer “prime” is at mostn−1/4+o(1). Peralta and Shoup [969] later improved this bound ton−1/2+o(1). 166 Ch. 4 Public-Key Parameters Monier [892] gave exact formulas for the number of Fermat liars, Euler liars, and strong liars for composite integers. One consequence of Monier’s formulas is the following im- provement (in the case wherenis not a prime power) of Fact 4.17 (see Kranakis [710, p.68]). Ifn≥3 is an odd composite integer havingrdistinct prime factors, and ifn≡3 (mod 4), then there are at mostφ(n)/2r−1 Euler liars forn. Another consequence is the following improvement (in the case wherenhas at least three distinct prime factors) of Fact 4.23. Ifn≥3 is an odd composite integer havingrdistinct prime factors, then there are at mostφ(n)/2r−1 strong liars forn. Erd¨os and Pomerance [373] estimated the average number of Fermat liars, Euler liars, and strong liars for composite integers. Fact 4.30(ii) was proven independently by Atkin and Larson [57], Monier [892], and Pomerance, Selfridge, and Wagstaff [996]. Pinch [975] reviewed the probabilistic primality tests used in theMathematica,Maple V, Axiom, andPari/GP computer algebra systems. Some of these systems use a probabilistic primality test known as theLucas test; a description of this test is provided by Pomerance, Selfridge, and Wagstaff [996]. §4.3 If a numbernis composite, providing a non-trivial divisor ofnis evidence of its composite- ness that can be veriﬁed in polynomial time (by long division). In other words, the decision problem “isncomposite?” belongs to the complexity classNP (cf. Example 2.65). Pratt [1000] used Fact 4.38 to show that this decision problem is also inco-NP. That is, ifnis prime there exists some evidence of this (called acertiﬁcate of primality) that can be veri- ﬁed in polynomial time. Note that the issue here is not in ﬁnding such evidence, but rather in determining whether such evidence exists which, if found, allows efﬁcient veriﬁcation. Pomerance [992] improved Pratt’s results and showed that every primenhas a certiﬁcate of primality which requiresO(ln n) multiplications modulonfor its veriﬁcation. Primality of theFermat numberF k =2 2k +1 can be determined in deterministic polyno- mial time byPepin’ s test: fork≥2,Fkis prime if and only if5(Fk−1)/2 ≡−1( m o dFk). For the history behind Pepin’s test and the Lucas-Lehmer test (Algorithm 4.37), see Bach and Shallit [70]. In Fact 4.38, the integeradoes not have to be the same for allq. More precisely, Brillhart and Selfridge [212] showed that Fact 4.38 can be reﬁned as follows: an integern≥3 is prime if and only if for each prime divisorqofn−1, there exists an integera qsuch that an−1 q ≡1( m o dn) and a(n−1)/q q ̸≡1( m o dn). The same is true of Fact 4.40, which is due to Pocklington [981]. For a proof of Fact 4.41, see Maurer [818]. Fact 4.42 is due to Brillhart, Lehmer, and Selfridge [210]; a simpliﬁed proof is given by Maurer [818]. The original Jacobi sum test was discovered by Adleman, Pomerance, and Rumely [16]. The algorithm was simpliﬁed, both theoretically and algorithmically, by Cohen and H. Lenstra [265]. Cohen and A. Lenstra [264] give an implementation report of the Cohen- Lenstra Jacobi sum test; see also Chapter 9 of Cohen [263]. Further improvements of the Jacobi sum test are reported by Bosma and van der Hulst [174]. Elliptic curves were ﬁrst used for primality proving by Goldwasser and Kilian [477], who presented a randomized algorithm which has an expected running time ofO((ln n) 11) bit operations formostinputsn. Subsequently, Adleman and Huang [13] designed a primality proving algorithm using hyperelliptic curves of genus two whose expected running time is polynomial forallinputsn. This established that the decision problem “isnprime?” is in the complexity classRP (Deﬁnition 2.77(ii)). The Goldwasser-Kilian and Adleman- Huang algorithms are inefﬁcient in practice. Atkin’s test, and an implementation of it, is extensively described by Atkin and Morain [58]; see also Chapter 9 of Cohen [263]. The §4.7 Notes and further references 167 largest number proven prime as of 1996 by a general purpose primality proving algorithm is a 1505-decimal digit number, accomplished by Morain [903] using Atkin’s test. The total time for the computation was estimated to be 4 years of CPU time distributed among 21 SUN 3/60 workstations. See also Morain [902] for an implementation report on Atkin’s test which was used to prove the primality of the 1065-decimal digit number(23539 +1)/3. §4.4 A proof of Mertens’s theorem can be found in Hardy and Wright [540]. The optimal trial division bound (Note 4.45) was derived by Maurer [818]. The discussion (Note 4.47) on the probabilityP(X\|Yt) is from Beauchemin et al. [81]; the result mentioned in the last sen- tence of this note is due to Kim and Pomerance [673]. Fact 4.48 was derived by Damg˚ard, Landrock, and Pomerance [300], building on earlier work of Erd¨os and Pomerance [373], Kim and Pomerance [673], and Damg˚ard and Landrock [299]. Table 4.3 is Table 2 of Dam- g˚ard, Landrock, and Pomerance [300]. The suggestions to ﬁrst do a Miller-Rabin test with basea=2 (Remark 4.50) and to do an incremental search (Note 4.51) in Algorithm 4.44 were made by Brandt, Damg˚ard, and Landrock [187]. The error and failure probabilities for incremental search (Note 4.51(i)) were obtained by Brandt and Damg˚ard [186]; consult this paper for more concrete estimates of these probabilities. Algorithm 4.53 for generating strong primes is due to Gordon [514, 513]. Gordon originally proposed computingp0 =( sr−1 −rs−1)m o drsin step 3. Kaliski (personal communica- tion, April 1996) proposed the modiﬁed formulap0 =( 2sr−2 mod r)s−1 which can be computed more efﬁciently. Williams and Schmid [1249] proposed an algorithm for gener- ating strong primespwith the additional constraint thatp−1=2 qwhere qis prime; this algorithm is not as efﬁcient as Gordon’s algorithm. Hellman and Bach [550] recommended an additional constraint on strong primes, specifying thats−1 (wheresis a large prime factor ofp+1 ) must have a large prime factor (see§15.2.3(v)); this thwarts cycling attacks based on Lucas sequences. The NIST method for prime generation (Algorithm 4.56) is that recommended by the NIST Federal Information Processing Standards Publication (FIPS) 186 [406]. Fact 4.59 and Algorithm 4.62 for provable prime generation are derived from Maurer [818]. Algorithm 4.62 is based on that of Shawe-Taylor [1123]. Maurer notes that the total diver- sity of reachable primes using the original version of his algorithm is roughly 10% of all primes. Maurer also presents a more complicated algorithm for generating provable primes with a better diversity than Algorithm 4.62, and provides extensive implementation details and analysis of the expected running time. Maurer [812] provides heuristic justiﬁcation that Algorithm 4.62 generates primes with virtually uniform distribution. Mihailescu [870] ob- served that Maurer’s algorithm can be improved by using the Eratosthenes sieve method for trial division (in step 8.2 of Algorithm 4.62) and by searching for a primenin an appro- priate interval of the arithmetic progression2q+1,4q+1,6q+1,... instead of generating R’s at random untiln=2 Rq+1 is prime. The second improvement comes at the expense of a reduction of the set of primes which may be produced by the algorithm. Mihailescu’s paper includes extensive analysis and an implementation report. §4.5 Lidl and Niederreiter [764] provide a comprehensive treatment of irreducible polynomials; proofs of Facts 4.67 and 4.68 can be found there. Algorithm 4.69 for testing a polynomial for irreducibility is due to Ben-Or [109]. The fast- est algorithm known for generating irreducible polynomials is due to Shoup [1131] and has an expected running time ofO(m 3 lg m+ m2 lg p) Zp-operations. There is nodeterminis- ticpolynomial-time algorithm known for ﬁnding an irreducible polynomial of a speciﬁed 168 Ch. 4 Public-Key Parameters degreeminZp[x]. Adleman and Lenstra [14] give a deterministic algorithm that runs in polynomial time under the assumption that the ERH is true. The best deterministic algo- rithm known is due to Shoup [1129] and takesO(m4√ p) Zp-operations, ignoring powers oflog mand log p. Gordon [512] presents an improved method for computing minimum polynomials of elements inF2m . Zierler and Brillhart [1271] provide a table of all irreducible trinomials of degree≤1000 inZ2[x]. Blake, Gao, and Lambert [146] extended this list to all irreducible trinomials of degree≤2000 inZ2[x]. Fact 4.75 is from their paper. Table 4.8 extends a similar table by Stahnke [1168]. The primitive pentanomialsxm + xk1 + xk2 + xk3 +1 listed in Table 4.8 have the following properties: (i)k1 = k2 + k3; (ii)k2 >k3; and (iii)k3 is as small as possible, and for this particular value ofk3,k2 is as small as possible. The rational behind this form is explained in Stahnke’s paper. For eachm< 5000 for which the factorization of2m−1 is known,ˇZivkovi´c [1275, 1276] gives a primitive trinomial inZ2[x], one primitive polynomial inZ2[x] having ﬁve non- zero terms, and one primitive polynomial inZ2[x] having seven non-zero terms, provided that such polynomials exist. The factorizations of2m−1 are known for allm≤510 and for some additionalm≤5000. A list of such factorizations can be found in Brillhart et al. [211] and updates of the list are available by anonymous ftp fromsable.ox.ac.uk in the/pub/math/cunningham/directory. Hansen and Mullen [538] describe some improvements to Algorithm 4.78 for generating primitive polynomials. They also give ta- bles of primitive polynomials of degreeminZ p[x] for each prime powerpm≤1050 with p≤97. Moreover, for each suchpand m, the primitive polynomial of degreemoverZp listed has the smallest number of non-zero coefﬁcients among all such polynomials. The entries of Table 4.9 were obtained from Zierler [1270] for Mersenne exponentsMj, 1 ≤j≤23, and from Kurita and Matsumoto [719] for Mersenne exponentsMj,24 ≤j≤ 27. Letf(x) ∈Zp[x] be an irreducible polynomial of degreem, and consider the ﬁnite ﬁeld Fpm = Zp[x]/(f(x)). Thenf(x) is called anormal polynomialif the set{x,xp,xp2 ,... , xpm−1 }forms a basis forFpm overZp; such a basis is called anormal basis. Mullin et al. [911] introduced the concept of anoptimal normal basisin order to reduce the hardware complexity of multiplying ﬁeld elements in the ﬁnite ﬁeldF2m . A VLSI implementation of the arithmetic inF2m which uses optimal normal bases is described by Agnew et al. [18]. A normal polynomial which is also primitive is called aprimitive normal polynomial. Dav- enport [301] proved that for any primepand positive integermthere exists a primitive normal polynomial of degreeminZp[x]. See also Lenstra and Schoof [760] who general- ized this result from prime ﬁeldsZpto prime power ﬁeldsFq. Morgan and Mullen [905] give a primitive normal polynomial of degreemoverZpfor each prime powerpm≤1050 withp≤97. Moreover, each polynomial has the smallest number of non-zero coefﬁcients among all primitive normal polynomials of degreemoverZp; in fact, each polynomial has at most ﬁve non-zero terms. §4.6 No polynomial-time algorithm is known for ﬁnding generators, or even for testing whether an element is a generator, of a ﬁnite ﬁeldFqif the factorization ofq−1 is unknown. Shoup [1130] considered the problem of deterministically generating in polynomial time a subset ofFqthat contains a generator, and presented a solution to the problem for the case where the characteristicpofFqis small (e.g.p=2 ). Maurer [818] discusses how his algorithm (Algorithm 4.62) can be used to generate the parameters(p,α), wherepis aprovableprime and αis a generator ofZ∗ p. Chapter /5 Pseudorandom Bits and Sequences Contents in Brief 5.1 Introduction ............................. 169 5.2 Random bit generation ....................... 171 5.3 Pseudorandom bit generation.................... 173 5.4 Statistical tests........................... 175 5.5 Cryptographically secure pseudorandom bit generation...... 185 5.6 Notes and further references.................... 187 5.1 Introduction The security of many cryptographic systems depends upon the generation of unpredictable quantities. Examples include the keystream in the one-time pad (§1.5.4), the secret key in the DES encryption algorithm (§7.4.2), the primesp,q in the RSA encryption (§8.2) and digital signature (§11.3.1) schemes, the private keyain the DSA (§11.5.1), and the chal- lenges used in challenge-response identiﬁcation systems (§10.3). In all these cases, the quantities generated must be of sufﬁcient size and be “random” in the sense that the proba- bility of any particular value being selected must be sufﬁciently small to preclude an adver- sary from gaining advantage through optimizing a search strategy based on such probability. For example, the key space for DES has size256. If a secret keykwere selected using a true random generator, an adversary would on average have to try255 possible keys before guessing the correct keyk. If, on the other hand, a keykwere selected by ﬁrst choosing a 16-bit random secrets, and then expanding it into a 56-bit keykusing a complicated but publicly known functionf, the adversary would on average only need to try215 possible keys (obtained by running every possible value forsthrough the functionf). This chapter considers techniques for the generation of random and pseudorandom bits and numbers. Related techniques for pseudorandom bit generation that are generally discussed in the literature in the context of stream ciphers, including linear and nonlinear feedback shift registers (Chapter 6) and the output feedback mode (OFB) of block ciphers (Chapter 7), are addressed elsewhere in this book. Chapter outline The remainder of§5.1 introduces basic concepts relevant to random and pseudorandom bit generation.§5.2 considers techniques for random bit generation, while§5.3 considers some techniques for pseudorandom bit generation.§5.4 describes statistical tests designed 169 170 Ch. 5 Pseudorandom Bits and Sequences to measure the quality of a random bit generator. Cryptographically secure pseudorandom bit generators are the topic of§5.5.§5.6 concludes with references and further chapter notes. 5.1.1 Background and Classiﬁcation 5.1 DeﬁnitionA random bit generatoris a device or algorithm which outputs a sequence of statistically independent and unbiased binary digits. 5.2 Remark (random bits vs. random numbers) A random bit generator can be used to gener- ate (uniformly distributed) random numbers. For example, a random integer in the interval [0,n]can be obtained by generating a random bit sequence of length⌊lgn⌋+1, and con- verting it to an integer; if the resulting integer exceedsn, one option is to discard it and generate a new random bit sequence. §5.2 outlines some physical sources of random bits that are used in practice. Ideally, secrets required in cryptographic algorithms and protocols should be generated with a (true) random bit generator. However, the generation of random bits is an inefﬁcient procedure in most practical environments. Moreover, it may be impractical to securely store and transmit a large number of random bits if these are required in applications such as the one-time pad (§6.1.1). In such situations, the problem can be ameliorated by substituting a random bit generator with a pseudorandom bit generator. 5.3 DeﬁnitionA pseudorandom bit generator(PRBG) is a deterministic 1 algorithm which, given a truly random binary sequence of lengthk, outputs a binary sequence of lengthl≫k which “appears” to be random. The input to the PRBG is called theseed, while the output of the PRBG is called apseudorandom bit sequence. The output of a PRBG isnotrandom; in fact, the number of possible output sequences is at most a small fraction, namely2k/2l, of all possible binary sequences of lengthl. The intent is to take a small truly random sequence and expand it to a sequence of much larger length, in such a way that an adversary cannot efﬁciently distinguish between output sequences of the PRBG and truly random sequences of lengthl. §5.3 discusses ad-hoc techniques for pseudorandom bit generation. In order to gain conﬁdence that such generators are secure, they should be subjected to a variety of statistical tests designed to detect the speciﬁc char- acteristics expected of random sequences. A collection of such tests is given in§5.4. As the following example demonstrates, passing these statistical tests is anecessarybut not sufﬁcientcondition for a generator to be secure. 5.4 Example (linear congruential generators)A linear congruential generatorproduces a pseudorandom sequence of numbersx 1,x2,x3,... according to the linear recurrence xn = axn−1 +bmodm, n ≥1; integersa,b, andmareparameterswhich characterize the generator, whilex0 is the (secret) seed. While such generators are commonly used for simulation purposes and probabilistic algorithms, and pass the statistical tests of§5.4, they arepredictableand hence entirely in- secure for cryptographic purposes: given a partial output sequence, the remainder of the sequence can be reconstructed even if the parametersa,b, andmare unknown. □ 1Deterministichere means that given the same initial seed, the generator will always produce the same output sequence. §5.2 Random bit generation 171 A minimum security requirement for a pseudorandom bit generator is that the length kof the random seed should be sufﬁciently large so that a search over2k elements (the total number of possible seeds) is infeasible for the adversary. Two general requirements are that the output sequences of a PRBG should be statistically indistinguishable from truly random sequences, and the output bits should be unpredictable to an adversary with limited computational resources; these requirements are captured in Deﬁnitions 5.5 and 5.6. 5.5 DeﬁnitionA pseudorandom bit generator is said to pass allpolynomial-time2 statistical testsif no polynomial-time algorithm can correctly distinguish between an output sequence of the generator and a truly random sequence of the same length with probability signiﬁ- cantly greater that1 2 . 5.6 DeﬁnitionA pseudorandom bit generator is said to pass thenext-bit testif there is no polynomial-time algorithm which, on input of the ﬁrstlbits of an output sequences, can predict the(l+1)st bit ofswith probability signiﬁcantly greater than1 2 . Although Deﬁnition 5.5 appears to impose a more stringent security requirement on pseudorandom bit generators than Deﬁnition 5.6 does, the next result asserts that they are, in fact, equivalent. 5.7 Fact(universality of the next-bit test) A pseudorandom bit generator passes the next-bit test if and only if it passes all polynomial-time statistical tests. 5.8 DeﬁnitionA PRBG that passes the next-bit test (possibly under some plausible but un- proved mathematical assumption such as the intractability of factoring integers) is called a cryptographically secure pseudorandom bit generator(CSPRBG). 5.9 Remark (asymptotic nature of Deﬁnitions 5.5, 5.6, and 5.8) Each of the three deﬁnitions above are given in complexity-theoretic terms and are asymptotic in nature because the no- tion of “polynomial-time” is meaningful for asymptotically large inputs only; the resulting notions of security are relative in the same sense. To be more precise in Deﬁnitions 5.5, 5.6, 5.8, and Fact 5.7, a pseudorandom bit generator is actually afamilyof such PRBGs. Thus the theoretical security results for a family of PRBGs are only an indirect indication about the security of individual members. Two cryptographically secure pseudorandom bit generators are presented in§5.5. 5.2 Random bit generation A (true) random bit generator requires a naturally occurring source of randomness. De- signing a hardware device or software program to exploit this randomness and produce a bit sequence that is free of biases and correlations is a difﬁcult task. Additionally, for most cryptographic applications, the generator must not be subject to observation or manipula- tion by an adversary. This section surveys some potential sources of random bits. Random bit generators based on natural sources of randomness are subject to inﬂuence by external factors, and also to malfunction. It is imperative that such devices be tested periodically, for example by using the statistical tests of§5.4. 2The running time of the test is bounded by a polynomial in the lengthlof the output sequence. 172 Ch. 5 Pseudorandom Bits and Sequences (i) Hardware-based generators Hardware-based random bit generators exploit the randomness which occurs in some phys- ical phenomena. Such physical processes may produce bits that are biased or correlated, in which case they should be subjected to de-skewing techniques mentioned in (iii) below. Examples of such physical phenomena include: 1. elapsed time between emission of particles during radioactive decay; 2. thermal noise from a semiconductor diode or resistor; 3. the frequency instability of a free running oscillator; 4. the amount a metal insulator semiconductor capacitor is charged during a ﬁxed period of time; 5. air turbulence within a sealed disk drive which causes random ﬂuctuations in disk drive sector read latency times; and 6. sound from a microphone or video input from a camera. Generators based on the ﬁrst two phenomena would, in general, have to be built externally to the device using the random bits, and hence may be subject to observation or manipula- tion by an adversary. Generators based on oscillators and capacitors can be built on VLSI devices; they can be enclosed in tamper-resistant hardware, and hence shielded from active adversaries. (ii) Software-based generators Designing a random bit generator in software is even more difﬁcult than doing so in hard- ware. Processes upon which software random bit generators may be based include: 1. the system clock; 2. elapsed time between keystrokes or mouse movement; 3. content of input/output buffers; 4. user input; and 5. operating system values such as system load and network statistics. The behavior of such processes can vary considerably depending on various factors, such as the computer platform. It may also be difﬁcult to prevent an adversary from observing or manipulating these processes. For instance, if the adversary has a rough idea of when a ran- dom sequence was generated, she can guess the content of the system clock at that time with a high degree of accuracy. A well-designed software random bit generator should utilize as many good sources of randomness as are available. Using many sources guards against the possibility of a few of the sources failing, or being observed or manipulated by an adver- sary. Each source should be sampled, and the sampled sequences should be combined using a complexmixing function; one recommended technique for accomplishing this is to apply a cryptographic hash function such as SHA-1 (Algorithm 9.53) or MD5 (Algorithm 9.51) to a concatenation of the sampled sequences. The purpose of the mixing function is to distill the (true) random bits from the sampled sequences. (iii) De-skewing A natural source of random bits may be defective in that the output bits may bebiased(the probability of the source emitting a1is not equal to 1 2 )o rcorrelated(the probability of the source emitting a1depends on previous bits emitted). There are various techniques for generating truly random bit sequences from the output bits of such a defective generator; such techniques are calledde-skewing techniques. §5.3 Pseudorandom bit generation 173 5.10 Example (removing biases in output bits) Suppose that a generator produces biased but uncorrelated bits. Suppose that the probability of a1isp, and the probability of a0is1−p, where pis unknown but ﬁxed,0<p< 1. If the output sequence of such a generator is grouped into pairs of bits, with a10pair transformed to a1,a01pair transformed to a0, and 00and 11pairs discarded, then the resulting sequence is both unbiased and uncorrelated.□ A practical (although not provable) de-skewing technique is to pass sequences whose bits are biased or correlated through a cryptographic hash function such as SHA-1 or MD5. 5.3 Pseudorandom bit generation A one-way functionf(Deﬁnition 1.12) can be utilized to generate pseudorandom bit se- quences (Deﬁnition 5.3) by ﬁrst selecting a random seeds, and then applying the function to the sequence of valuess,s+1,s+2,... ; the output sequence isf(s),f(s+1),f(s+2),... . Depending on the properties of the one-way function used, it may be necessary to only keep a few bits of the output valuesf(s+i)in order to remove possible correlations between successive values. Examples of suitable one-way functionsfinclude a cryptographic hash function such as SHA-1 (Algorithm 9.53), or a block cipher such as DES (§7.4) with secret key k. Although such ad-hoc methods have not been proven to be cryptographically secure, they appear sufﬁcient for most applications. Two such methods for pseudorandom bit and number generation which have been standardized are presented in§5.3.1 and§5.3.2. Tech- niques for the cryptographically secure generation of pseudorandom bits are given in§5.5. 5.3.1 ANSI X9.17 generator Algorithm 5.11 is a U.S. Federal Information Processing Standard (FIPS) approved method from the ANSI X9.17 standard for the purpose of pseudorandomly generating keys and initialization vectors for use with DES.Ekdenotes DES E-D-E two-key triple-encryption (Deﬁnition 7.32) under a keyk; the keykshould be reserved exclusively for use in this algorithm. 5.11 AlgorithmANSI X9.17 pseudorandom bit generator INPUT: a random (and secret) 64-bit seeds, integerm, and DES E-D-E encryption keyk. OUTPUT: mpseudorandom 64-bit stringsx1,x2,...,x m. 1. Compute the intermediate valueI=Ek(D), whereDis a 64-bit representation of the date/time to as ﬁne a resolution as is available. 2. Forifrom 1tomdo the following: 2.1 xi←Ek(I⊕s). 2.2 s←Ek(xi⊕I). 3. Return(x1,x2,...,x m). Each output bitstringximay be used as an initialization vector (IV) for one of the DES modes of operation (§7.2.2). To obtain a DES key fromxi, every eighth bit ofxishould be reset to odd parity (cf.§7.4.2). 174 Ch. 5 Pseudorandom Bits and Sequences 5.3.2 FIPS 186 generator The algorithms presented in this subsection are FIPS-approved methods for pseudorandom- ly generating the secret parameters for the DSA (§11.5.1). Algorithm 5.12 generates DSA private keysa, while Algorithm 5.14 generates the per-message secretskto be used in sign- ing messages. Both algorithms use a secret seedswhich should be randomly generated, and utilize a one-way function constructed by using either SHA-1 (Algorithm 9.53) or DES (Al- gorithm 7.82), respectively described in Algorithms 5.15 and 5.16. 5.12 AlgorithmFIPS 186 pseudorandom number generator for DSA private keys INPUT: an integermand a 160-bit prime numberq. OUTPUT: mpseudorandom numbersa1,a2,...,a min the interval[0,q−1]which may be used as DSA private keys. 1. If Algorithm 5.15 is to be used in step 4.3 then select an arbitrary integerb,160≤ b≤512; if Algorithm 5.16 is to be used then setb←160. 2. Generate a random (and secret)b-bit seeds. 3. Deﬁne the 160-bit stringt=67452301 efcdab89 98badcfe 10325476 c3d2e1f0(in hexadecimal). 4. Forifrom 1tomdo the following: 4.1 (optional user input) Either select ab-bit stringyi,o rs e tyi←0. 4.2 zi←(s+yi)mod2 b. 4.3 ai←G(t,zi)mod q.(Gis either that deﬁned in Algorithm 5.15 or 5.16.) 4.4 s←(1+s+ai)mod2 b. 5. Return(a1,a2,...,a m). 5.13 Note (optional user input) Algorithm 5.12 permits a user to augment the seedswith ran- dom or pseudorandom strings derived from alternate sources. The user may desire to do this if she does not trust the quality or integrity of the random bit generator which may be built into a cryptographic module implementing the algorithm. 5.14 AlgorithmFIPS 186 pseudorandom number generator for DSA per-message secrets INPUT: an integermand a 160-bit prime numberq. OUTPUT: mpseudorandom numbersk1,k2,...,k min the interval[0,q−1]which may be used as the per-message secret numberskin the DSA. 1. If Algorithm 5.15 is to be used in step 4.1 then select an integerb,160≤b≤512; if Algorithm 5.16 is to be used then setb←160. 2. Generate a random (and secret)b-bit seeds. 3. Deﬁne the 160-bit stringt=efcdab89 98badcfe 10325476 c3d2e1f0 67452301(in hexadecimal). 4. Forifrom 1 tomdo the following: 4.1 ki←G(t,s)mod q.(Gis either that deﬁned in Algorithm 5.15 or 5.16.) 4.2 s←(1+s+ki)mod2 b. 5. Return(k1,k2,...,k m). §5.4 Statistical tests 175 5.15 AlgorithmFIPS 186 one-way function using SHA-1 INPUT: a 160-bit stringtand ab-bit stringc,160≤b≤512. OUTPUT: a 160-bit string denotedG(t,c). 1. Break uptinto ﬁve 32-bit blocks:t=H1∥H2∥H3∥H4∥H5. 2. Padcwith0’s to obtain a 512-bit message block:X←c∥0512−b. 3. DivideXinto 16 32-bit words:x0x1 ...x15, and setm←1. 4. Execute step 4 of SHA-1 (Algorithm 9.53). (This alters theHi’s.) 5. The output is the concatenation:G(t,c)= H1∥H2∥H3∥H4∥H5. 5.16 AlgorithmFIPS 186 one-way function using DES INPUT: two 160-bit stringstand c. OUTPUT: a 160-bit string denotedG(t,c). 1. Break uptinto ﬁve 32-bit blocks:t=t0∥t1∥t2∥t3∥t4. 2. Break upcinto ﬁve 32-bit blocks:c=c0∥c1∥c2∥c3∥c4. 3. Forifrom 0to4do the following:xi←ti⊕ci. 4. Forifrom 0to4do the following: 4.1 b1←c(i+4)mod5 , b2←c(i+3)mod5 . 4.2 a1←xi, a2←x(i+1)mod5 ⊕x(i+4)mod5 . 4.3 A←a1∥a2, B←b′ 1∥b2, whereb′ 1 denotes the 24 least signiﬁcant bits ofb1. 4.4 Use DES with keyBto encryptA:yi←DES B(A). 4.5 Break upyiinto two 32-bit blocks:yi=Li∥Ri. 5. Forifrom 0to4do the following:zi←Li⊕R(i+2)mod5 ⊕L(i+3)mod5 . 6. The output is the concatenation:G(t,c)= z0∥z1∥z2∥z3∥z4. 5.4 Statistical tests This section presents some tests designed to measure the quality of a generator purported to be a random bit generator (Deﬁnition 5.1). While it is impossible to give a mathematical proof that a generator is indeed a random bit generator, the tests described here help detect certain kinds of weaknesses the generator may have. This is accomplished by taking a sam- ple output sequence of the generator and subjecting it to various statistical tests. Each statis- tical test determines whether the sequence possesses a certain attribute that a truly random sequence would be likely to exhibit; the conclusion of each test is not deﬁnite, but rather probabilistic. An example of such an attribute is that the sequence should have roughly the same number of0’s as1’s. If the sequence is deemed to have failed any one of the statistical tests, the generator may berejectedas being non-random; alternatively, the generator may be subjected to further testing. On the other hand, if the sequence passes all of the statisti- cal tests, the generator isacceptedas being random. More precisely, the term “accepted” should be replaced by “not rejected”, since passing the tests merely provides probabilistic evidence that the generator produces sequences which have certain characteristics of ran- dom sequences. §5.4.1 and§5.4.2 provide some relevant background in statistics.§5.4.3 establishes some notation and lists Golomb’s randomness postulates. Speciﬁc statistical tests for ran- domness are described in§5.4.4 and§5.4.5. 176 Ch. 5 Pseudorandom Bits and Sequences 5.4.1 The normal and chi-square distributions The normal andχ2 distributions are widely used in statistical applications. 5.17 DeﬁnitionIf the resultXof an experiment can be any real number, thenXis said to be a continuousrandom variable. 5.18 DeﬁnitionA probability density functionof a continuous random variableXis a function f(x)which can be integrated and satisﬁes: (i)f(x)≥0for allx∈R; (ii) ∫∞ −∞f(x)dx=1; and (iii) for alla,b∈R,P(a<X ≤b)= ∫b af(x)dx. (i) The normal distribution The normal distribution arises in practice when a large number of independent random vari- ables having the same mean and variance are summed. 5.19 DeﬁnitionA (continuous) random variableXhas anormal distributionwithmean µand varianceσ2 if its probability density function is deﬁned by f(x)= 1 σ √ 2πexp {−(x−µ)2 2σ2 } , −∞<x< ∞. Notation:Xis said to beN(µ,σ2).I fXisN(0,1), thenXis said to have astandard normal distribution. A graph of theN(0,1)distribution is given in Figure 5.1. The graph is symmetric 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 -3 -2 -1 0 1 2 3x f(x) Figure 5.1:The normal distributionN(0,1). about the vertical axis, and henceP(X>x )= P(X< −x)for anyx. Table 5.1 gives some percentiles for the standard normal distribution. For example, the entry (α=0.05, x=1.6449) means that ifXisN(0,1), thenXexceeds1.6449about5% of the time. Fact 5.20 can be used to reduce questions about a normal distribution to questions about the standard normal distribution. §5.4 Statistical tests 177 α 0.1 0.05 0.025 0.01 0.005 0.0025 0.001 0.0005 x 1.2816 1.6449 1.9600 2.3263 2.5758 2.8070 3.0902 3.2905 Table 5.1:Selected percentiles of the standard normal distribution. IfXis a random variable having a standard normal distribution, thenP(X>x )= α. 5.20 FactIf the random variableXisN(µ,σ2), then the random variableZ=(X−µ)/σis N(0,1). (ii) Theχ2 distribution The χ2 distribution can be used to compare thegoodness-of-ﬁtof the observed frequencies of events to their expected frequencies under a hypothesized distribution. Theχ2 distribu- tion withvdegrees of freedom arises in practice when the squares ofvindependent random variables having standard normal distributions are summed. 5.21 DeﬁnitionLetv≥1be an integer. A (continuous) random variableXhas aχ2 (chi-squ- are) distributionwithvdegrees of freedomif its probability density function is deﬁned by f(x)=    1 Γ(v/2)2v/2 x(v/2)−1e−x/2, 0≤x< ∞, 0,x < 0, where Γis the gamma function.3 The mean and varianceof this distribution areµ=v, and σ2 =2v. A graph of theχ2 distribution withv=7 degrees of freedom is given in Figure 5.2. Table 5.2 gives some percentiles of theχ2 distribution for various degrees of freedom. For 0 0.02 0.04 0.06 0.08 0.1 0.12 0 5 10 15 20x f(x) Figure 5.2:The χ2 (chi-square) distribution withv=7 degrees of freedom. example, the entry in rowv=5 and columnα=0.05isx=11.0705; this means that if Xhas aχ2 distribution with5degrees of freedom, thenXexceeds11.0705about5%o f the time. 3The gamma functionis deﬁned byΓ(t)= ∫ ∞ 0 xt−1e−xdx, fort> 0. 178 Ch. 5 Pseudorandom Bits and Sequences α v 0.100 0.050 0.025 0.010 0.005 0.001 1 2.7055 3.8415 5.0239 6.6349 7.8794 10.8276 2 4.6052 5.9915 7.3778 9.2103 10.5966 13.8155 3 6.2514 7.8147 9.3484 11.3449 12.8382 16.2662 4 7.7794 9.4877 11.1433 13.2767 14.8603 18.4668 5 9.2364 11.0705 12.8325 15.0863 16.7496 20.5150 6 10.6446 12.5916 14.4494 16.8119 18.5476 22.4577 7 12.0170 14.0671 16.0128 18.4753 20.2777 24.3219 8 13.3616 15.5073 17.5345 20.0902 21.9550 26.1245 9 14.6837 16.9190 19.0228 21.6660 23.5894 27.8772 10 15.9872 18.3070 20.4832 23.2093 25.1882 29.5883 11 17.2750 19.6751 21.9200 24.7250 26.7568 31.2641 12 18.5493 21.0261 23.3367 26.2170 28.2995 32.9095 13 19.8119 22.3620 24.7356 27.6882 29.8195 34.5282 14 21.0641 23.6848 26.1189 29.1412 31.3193 36.1233 15 22.3071 24.9958 27.4884 30.5779 32.8013 37.6973 16 23.5418 26.2962 28.8454 31.9999 34.2672 39.2524 17 24.7690 27.5871 30.1910 33.4087 35.7185 40.7902 18 25.9894 28.8693 31.5264 34.8053 37.1565 42.3124 19 27.2036 30.1435 32.8523 36.1909 38.5823 43.8202 20 28.4120 31.4104 34.1696 37.5662 39.9968 45.3147 21 29.6151 32.6706 35.4789 38.9322 41.4011 46.7970 22 30.8133 33.9244 36.7807 40.2894 42.7957 48.2679 23 32.0069 35.1725 38.0756 41.6384 44.1813 49.7282 24 33.1962 36.4150 39.3641 42.9798 45.5585 51.1786 25 34.3816 37.6525 40.6465 44.3141 46.9279 52.6197 26 35.5632 38.8851 41.9232 45.6417 48.2899 54.0520 27 36.7412 40.1133 43.1945 46.9629 49.6449 55.4760 28 37.9159 41.3371 44.4608 48.2782 50.9934 56.8923 29 39.0875 42.5570 45.7223 49.5879 52.3356 58.3012 30 40.2560 43.7730 46.9792 50.8922 53.6720 59.7031 31 41.4217 44.9853 48.2319 52.1914 55.0027 61.0983 63 77.7454 82.5287 86.8296 92.0100 95.6493 103.4424 127 147.8048 154.3015 160.0858 166.9874 171.7961 181.9930 255 284.3359 293.2478 301.1250 310.4574 316.9194 330.5197 511 552.3739 564.6961 575.5298 588.2978 597.0978 615.5149 1023 1081.3794 1098.5208 1113.5334 1131.1587 1143.2653 1168.4972 Table 5.2:Selected percentiles of theχ2 (chi-square) distribution. A(v,α)-entry ofxin the table has the following meaning: ifXis a random variable having aχ2 distribution withvdegrees of freedom, thenP(X>x )= α. §5.4 Statistical tests 179 Fact 5.22 relates the normal distribution to theχ2 distribution. 5.22 FactIf the random variableXisN(µ,σ2),σ2 >0, then the random variableZ=(X− µ)2/σ2 has aχ2 distribution with 1 degree of freedom. In particular, ifXisN(0,1), then Z=X2 has aχ2 distribution with 1 degree of freedom. 5.4.2 Hypothesis testing A statistical hypothesis, denotedH0, is an assertion about a distribution of one or more ran- dom variables. Atestof a statistical hypothesis is a procedure, based upon observed values of the random variables, that leads to the acceptance or rejection of the hypothesisH0. The test only provides a measure of the strength of the evidence provided by the data against the hypothesis; hence, the conclusion of the test is not deﬁnite, but rather probabilistic. 5.23 DeﬁnitionThe signiﬁcance levelαof the test of a statistical hypothesisH0 is the proba- bility of rejectingH0 when it is true. In this section,H0 will be the hypothesis that a given binary sequence was produced by a random bit generator. If the signiﬁcance levelαof a test ofH0 is too high, then the test may reject sequences that were, in fact, produced by a random bit generator (such an error is called aType I error). On the other hand, if the signiﬁcance level of a test ofH 0 is too low, then there is the danger that the test may accept sequences even though they were not produced by a random bit generator (such an error is called aType II error). 4 It is, therefore, important that the test be carefully designed to have a signiﬁcance level that is appropriate for the purpose at hand; a signiﬁcance levelαbetween0.001and 0.05might be employed in practice. A statistical test is implemented by specifying astatisticon the random sample.5 Statis- tics are generally chosen so that they can be efﬁciently computed, and so that they (approxi- mately) follow anN(0,1)or aχ2 distribution (see§5.4.1). The value of the statistic for the sample output sequence is computed and compared with the value expected for a random sequence as described below. 1. Suppose that a statisticXfor a random sequence follows aχ 2 distribution withv degrees of freedom, and suppose that the statistic can be expected to take on larger values for nonrandom sequences. To achieve a signiﬁcance level ofα,a threshold valuex αis chosen (using Table 5.2) so thatP(X>x α)= α. If the valueXsof the statistic for the sample output sequence satisﬁesXs>xα, then the sequencefailsthe test; otherwise, itpassesthe test. Such a test is called aone-sidedtest. For example, ifv=5 and α=0.025, thenxα=12.8325, and one expects a random sequence to fail the test only2.5%of the time. 2. Suppose that a statisticXfor a random sequence follows anN(0,1)distribution, and suppose that the statistic can be expected to take on both larger and smaller values for nonrandom sequences. To achieve a signiﬁcance level ofα,a thresholdvaluexαis chosen (using Table 5.1) so thatP(X>x α)= P(X< −xα)= α/2. If the value 4Actually, the probabilityβof a Type II error may be completely independent ofα. If the generator is not a random bit generator, the probabilityβdepends on the nature of the defects of the generator, and is usually difﬁcult to determine in practice. For this reason, assuming that the probability of a Type II error is proportional toαis a useful intuitive guide when selecting an appropriate signiﬁcance level for a test. 5A statisticis a function of the elements of a random sample; for example, the number of0’s in a binary se- quence is a statistic. 180 Ch. 5 Pseudorandom Bits and Sequences Xsof the statistic for the sample output sequence satisﬁesXs>xαorXs<−xα, then the sequencefailsthe test; otherwise, itpassesthe test. Such a test is called a two-sidedtest. For example, ifα=0.05, thenxα=1.96, and one expects a random sequence to fail the test only5%of the time. 5.4.3 Golomb’s randomness postulates Golomb’s randomness postulates (Deﬁnition 5.28) are presented here for historical reasons – they were one of the ﬁrst attempts to establish somenecessaryconditions for a periodic pseudorandom sequence to look random. It is emphasized that these conditions are far from beingsufﬁcientfor such sequences to be considered random. Unless otherwise stated, all sequences are binary sequences. 5.24 DeﬁnitionLets=s0,s1,s2,... be an inﬁnite sequence. The subsequence consisting of the ﬁrstnterms ofsis denoted bysn=s0,s1,...,s n−1. 5.25 DeﬁnitionThe sequences=s0,s1,s2,... is said to beN-periodicifsi =si+N for alli≥0. The sequencesisperiodicif it isN-periodic for some positive integerN. The periodof a periodic sequencesis the smallest positive integerNfor whichsisN-periodic. Ifsis a periodic sequence of periodN, then thecycleofsis the subsequencesN. 5.26 DeﬁnitionLetsbe a sequence. Arunofsis a subsequence ofsconsisting of consecutive 0’s or consecutive1’s which is neither preceded nor succeeded by the same symbol. A run of0’s is called agap, while a run of1’s is called ablock. 5.27 DeﬁnitionLets=s0,s1,s2,... be a periodic sequence of periodN. Theautocorrela- tion functionofsis the integer-valued functionC(t)deﬁned as C(t)= 1 N N−1∑ i=0 (2si−1)·(2si+t−1), for0≤t≤N−1. The autocorrelation functionC(t)measures the amount of similarity between the se- quencesand a shift ofsby tpositions. Ifsis a random periodic sequence of periodN, then\|N·C(t)\|can be expected to be quite small for all values oft,0<t<N . 5.28 DeﬁnitionLet sbe a periodic sequence of periodN. Golomb’ s randomness postulates are the following. R1: In the cyclesN ofs, the number of1’s differs from the number of0’s by at most1. R2: In the cyclesN, at least half the runs have length1, at least one-fourth have length 2, at least one-eighth have length3, etc., as long as the number of runs so indicated exceeds1. Moreover, for each of these lengths, there are (almost) equally many gaps and blocks.6 R3: The autocorrelation functionC(t)is two-valued. That is for some integerK, N·C(t)= N−1∑ i=0 (2si−1)·(2si+t−1)= { N, ift=0, K, if1≤t≤N−1. 6Postulate R2 implies postulate R1. §5.4 Statistical tests 181 5.29 DeﬁnitionA binary sequence which satisﬁes Golomb’s randomness postulates is called a pseudo-noise sequenceor apn-sequence. Pseudo-noise sequences arise in practice as output sequences of maximum-length lin- ear feedback shift registers (cf. Fact 6.14). 5.30 Example (pn-sequence) Consider the periodic sequencesof periodN=15 with cycle s15 =0,1,1,0,0,1,0,0,0,1,1,1,1,0,1. The following shows that the sequencessatisﬁes Golomb’s randomness postulates. R1: The number of0’s ins15 is7, while the number of1’s is8. R2: s15 has8runs. There are4runs of length1(2gaps and2blocks),2runs of length2 (1gap and1block),1run of length3(1gap), and1run of length4(1block). R3: The autocorrelation functionC(t)takes on two values:C(0)=1 and C(t)= −1 15 for1≤t≤14. Hence,sis a pn-sequence. □ 5.4.4 Five basic tests Lets=s0,s1,s2,...,s n−1 be a binary sequence of lengthn. This subsection presents ﬁve statistical tests that are commonly used for determining whether the binary sequence spossesses some speciﬁc characteristics that a truly random sequence would be likely to exhibit. It is emphasized again that the outcome of each test is not deﬁnite, but rather prob- abilistic. If a sequence passes all ﬁve tests, there is no guarantee that it was indeed produced by a random bit generator (cf. Example 5.4). (i) Frequency test (monobit test) The purpose of this test is to determine whether the number of0’s and1’s insare approxi- mately the same, as would be expected for a random sequence. Letn 0,n1 denote the num- ber of0’s and1’s ins, respectively. The statistic used is X1 = (n0 −n1)2 n (5.1) which approximately follows aχ2 distribution with1degree of freedom ifn≥10.7 (ii) Serial test (two-bit test) The purpose of this test is to determine whether the number of occurrences of00,01,10, and11as subsequences ofsare approximately the same, as would be expected for a random sequence. Letn 0,n1 denote the number of0’s and1’s ins, respectively, and letn00,n01, n10,n11 denote the number of occurrences of00,01,10,11ins, respectively. Note that n00 +n01 +n10 +n11 =(n−1)since the subsequences are allowed to overlap. The statistic used is X2 = 4 n−1 ( n2 00 +n2 01 +n2 10 +n2 11 ) −2 n ( n2 0 +n2 1 ) +1 (5.2) which approximately follows aχ2 distribution with2degrees of freedom ifn≥21. 7In practice, it is recommended that the lengthnof the sample output sequence be much larger (for example, n≫ 10000) than the minimum speciﬁed for each test in this subsection. 182 Ch. 5 Pseudorandom Bits and Sequences (iii) Poker test Letmbe a positive integer such that⌊n m⌋≥5·(2m), and letk=⌊n m⌋. Divide the sequence sintoknon-overlapping parts each of lengthm, and letnibe the number of occurrences of theith type of sequence of lengthm,1≤i≤2m. The poker test determines whether the sequences of lengthmeach appear approximately the same number of times ins, as would be expected for a random sequence. The statistic used is X3 = 2m k (2m ∑ i=1 n2 i ) −k (5.3) which approximately follows aχ2 distribution with2m−1degrees of freedom. Note that the poker test is a generalization of the frequency test: settingm=1in the poker test yields the frequency test. (iv) Runs test The purpose of the runs test is to determine whether the number of runs (of either zeros or ones; see Deﬁnition 5.26) of various lengths in the sequencesis as expected for a random sequence. The expected number of gaps (or blocks) of lengthiin a random sequence of lengthnisei=(n−i+3)/2i+2. Letkbe equal to the largest integerifor whichei≥5. Let Bi,Gibe the number of blocks and gaps, respectively, of lengthiinsfor eachi,1≤i≤k. The statistic used is X4 = k∑ i=1 (Bi−ei)2 ei + k∑ i=1 (Gi−ei)2 ei (5.4) which approximately follows aχ2 distribution with2k−2degrees of freedom. (v) Autocorrelation test The purpose of this test is to check for correlations between the sequencesand (non-cyclic) shifted versions of it. Letdbe a ﬁxed integer,1≤d≤⌊n/2⌋. The number of bits insnot equal to theird-shifts isA(d)= ∑ n−d−1 i=0 si⊕si+d, where⊕denotes the XOR operator. The statistic used is X5 =2 ( A(d)−n−d 2 ) / √ n−d (5.5) which approximately follows anN(0,1)distribution ifn−d≥10. Since small values of A(d)are as unexpected as large values ofA(d), a two-sided test should be used. 5.31 Example (basic statistical tests) Consider the (non-random) sequencesof lengthn = 160obtained by replicating the following sequence four times: 11100 01100 01000 10100 11101 11100 10010 01001. (i) (frequency test)n0 =84,n1 =76, and the value of the statisticX1 is0.4. (ii) (serial test)n00 =44,n01 =40,n10 =40,n11 =35, and the value of the statistic X2 is0.6252. (iii) (poker test) Herem=3 and k=53. The blocks000,001,010,011,100,101,110, 111appear5,10,6,4,12,3,6, and7times, respectively, and the value of the statistic X3 is9.6415. (iv) (runs test) Heree1 =20.25,e2 =10.0625,e3 =5, andk=3. There are25,4,5 blocks of lengths1,2,3, respectively, and8,20,12gaps of lengths1,2,3, respec- tively. The value of the statisticX4 is31.7913. §5.4 Statistical tests 183 (v) (autocorrelation test)I fd=8, thenA(8)=100 . The value of the statisticX5 is 3.8933. For a signiﬁcance level ofα=0.05, the threshold values forX1,X2,X3,X4, andX5 are 3.8415,5.9915,14.0671,9.4877, and1.96, respectively (see Tables 5.1 and 5.2). Hence, the given sequencespasses the frequency, serial, and poker tests, but fails the runs and autocorrelation tests. □ 5.32 Note (FIPS 140-1 statistical tests for randomness) FIPS 140-1 speciﬁes four statistical tests for randomness. Instead of making the user select appropriate signiﬁcance levels for these tests, explicit bounds are provided that the computed value of a statistic must satisfy. A single bitstringsof length20000bits, output from a generator, is subjected to each of the following tests. If any of the tests fail, then the generator fails the test. (i)monobit test. The numbern1 of1’s insshould satisfy9654<n1 <10346. (ii)poker test. The statisticX3 deﬁned by equation (5.3) is computed form=4. The poker test is passed if1.03<X3 <57.4. (iii)runs test. The numberBiand Giof blocks and gaps, respectively, of lengthiinsare counted for eachi,1≤i≤6. (For the purpose of this test, runs of length greater than6are considered to be of length6.) The runs test is passed if the12countsBi, Gi,1≤i≤6, are each within the corresponding interval speciﬁed by the following table. Length of run Required interval 1 2267 −2733 2 1079 −1421 3 502 −748 4 223 −402 5 90 −223 6 90 −223 (iv)long run test. The long run test is passed if there are no runs of length34or more. For high security applications, FIPS 140-1 mandates that the four tests be performed each time the random bit generator is powered up. FIPS 140-1 allows these tests to be substituted by alternative tests which provide equivalent or superior randomness checking. 5.4.5 Maurer’s universal statistical test The basic idea behind Maurer’s universal statistical test is that it should not be possible to signiﬁcantly compress (without loss of information) the output sequence of a random bit generator. Thus, if a sample output sequencesof a bit generator can be signiﬁcantly com- pressed, the generator should be rejected as being defective. Instead of actually compress- ing the sequences, the universal statistical test computes a quantity that is related to the length of the compressed sequence. The universalityof Maurer’s universal statistical test arises because it is able to detect any one of a very general class of possible defects a bit generator might have. This class includes the ﬁve defects that are detectable by the basic tests of§5.4.4. A drawback of the universal statistical test over the ﬁve basic tests is that it requires a much longer sample output sequence in order to be effective. Provided that the required output sequence can be efﬁciently generated, this drawback is not a practical concern since the universal statistical test itself is very efﬁcient. Algorithm 5.33 computes the statisticX ufor a sample output sequences=s0,s1,..., sn−1 to be used in the universal statistical test. The parameterLis ﬁrst chosen from the 184 Ch. 5 Pseudorandom Bits and Sequences L µ σ2 1 1 0.7326495 0.690 2 1.5374383 1.338 3 2.4016068 1.901 4 3.3112247 2.358 5 4.2534266 2.705 6 5.2177052 2.954 7 6.1962507 3.125 8 7.1836656 3.238 L µ σ2 1 9 8.1764248 3.311 10 9.1723243 3.356 11 10.170032 3.384 12 11.168765 3.401 13 12.168070 3.410 14 13.167693 3.416 15 14.167488 3.419 16 15.167379 3.421 Table 5.3:Mean µand varianceσ2 of the statisticXu for random sequences, with parametersL, Kas Q→∞. The variance ofXu isσ2 = c(L,K)2 ·σ2 1/K, wherec(L,K) ≈0.7 −(0.8/L)+ (1.6+( 1 2.8/L)) ·K−4/L forK≥2L. interval[6,16]. The sequencesis then partitioned into non-overlappingL-bit blocks, with any leftover bits discarded; the total number of blocks isQ+K, whereQandKare deﬁned below. For eachi,1≤i≤Q+K, letbibe the integer whose binary representation is theith block. The blocks are scanned in order. A tableTis maintained so that at each stageT[j]is the position of the last occurrence of the block corresponding to integerj,0≤j≤2L−1. The ﬁrstQblocks ofsare used to initialize tableT;Qshould be chosen to be at least10·2L in order to have a high likelihood that each of the2LL-bit blocks occurs at least once in the ﬁrstQblocks. The remainingKblocks are used to deﬁne the statisticXuas follows. For eachi,Q+1 ≤i≤Q+K, letAi =i−T[bi];Aiis the number of positions since the last occurrence of blockbi. Then Xu = 1 K Q+K∑ i=Q+1 lgAi. (5.6) Kshould be at least1000·2L(and, hence, the sample sequencesshould be at least(1010· 2L·L)bits in length). Table 5.3 lists the meanµand varianceσ2 ofXufor random se- quences for some sample choices ofLasQ→∞. 5.33 AlgorithmComputing the statisticXu for Maurer’s universal statistical test INPUT: a binary sequences=s0,s1,...,s n−1 of lengthn, and parametersL,Q,K. OUTPUT: the value of the statisticXufor the sequences. 1. Zero the tableT. Forjfrom 0to2L−1do the following:T[j]←0. 2. Initialize the tableT. Forifrom 1toQdo the following:T[bi]←i. 3. sum←0. 4. Forifrom Q+1toQ+Kdo the following: 4.1 sum←sum +lg(i−T[bi]). 4.2 T[bi]←i. 5. Xu←sum /K. 6. Return(Xu). Maurer’s universal statistical test uses the computed value ofXufor the sample output sequencesin the manner prescribed by Fact 5.34. To test the sequences, a two-sided test should be used with a signiﬁcance levelαbetween0.001and 0.01(see§5.4.2). §5.5 Cryptographically secure pseudorandom bit generation 185 5.34 FactLetXube the statistic deﬁned in (5.6) having meanµand varianceσ2 as given in Table 5.3. Then, for random sequences, the statisticZu =( Xu −µ)/σapproximately follows anN(0,1)distribution. 5.5 Cryptographically secure pseudorandom bit generation Two cryptographically secure pseudorandom bit generators (CSPRBG – see Deﬁnition 5.8) are presented in this section. The security of each generator relies on the presumed in- tractability of an underlying number-theoretic problem. The modular multiplications that these generators use make them relatively slow compared to the (ad-hoc) pseudorandom bit generators of§5.3. Nevertheless they may be useful in some circumstances, for exam- ple, generating pseudorandom bits on hardware devices which already have the circuitry for performing modular multiplications. Efﬁcient techniques for implementing modular mul- tiplication are presented in§14.3. 5.5.1 RSA pseudorandom bit generator The RSA pseudorandom bit generator is a CSPRBG under the assumption that the RSA problem is intractable (§3.3; see also§3.9.2). 5.35 AlgorithmRSA pseudorandom bit generator SUMMARY: a pseudorandom bit sequencez1,z2,...,z lof lengthlis generated. 1. Setup. Generate two secret RSA-like primespandq(cf. Note 8.8), and computen= pqand φ =( p−1)(q−1). Select a random integere,1 <e<φ , such that gcd(e,φ)=1 . 2. Select a random integerx0 (theseed) in the interval[1,n−1]. 3. Forifrom 1 toldo the following: 3.1 xi←xe i−1 modn. 3.2 zi←the least signiﬁcant bit ofxi. 4. The output sequence isz1,z2,...,z l. 5.36 Note (efﬁciency of the RSA PRBG)I fe=3 is chosen (cf. Note 8.9(ii)), then generating each pseudorandom bitzirequires one modular multiplication and one modular squaring. The efﬁciency of the generator can be improved by extracting thejleast signiﬁcant bits of xi in step 3.2, wherej =clglgnand cis a constant. Provided thatnis sufﬁciently large, this modiﬁed generator is also cryptographically secure (cf. Fact 3.87). For a mod- ulusnof a ﬁxed bitlength (e.g., 1024 bits), an explicit range of values ofcfor which the resulting generator remains cryptographically secure (cf. Remark 5.9) under the intractabil- ity assumption of the RSA problem has not been determined. The following modiﬁcation improves the efﬁciency of the RSA PRBG. 186 Ch. 5 Pseudorandom Bits and Sequences 5.37 AlgorithmMicali-Schnorr pseudorandom bit generator SUMMARY: a pseudorandom bit sequence is generated. 1. Setup. Generate two secret RSA-like primespandq(cf. Note 8.8), and computen= pqand φ=(p−1)(q−1). LetN=⌊lgn⌋+1(the bitlength ofn). Select an integer e,1<e<φ , such thatgcd(e,φ)=1 and 80e≤N. Letk=⌊N(1−2 e)⌋and r=N−k. 2. Select a random sequencex0 (theseed) of bitlengthr. 3. Generate a pseudorandom sequence of lengthk·l. Forifrom 1 toldo the following: 3.1 yi←xe i−1 modn. 3.2 xi←thermost signiﬁcant bits ofyi. 3.3 zi←thekleast signiﬁcant bits ofyi. 4. The output sequence isz1 ∥z2 ∥···∥zl, where∥denotes concatenation. 5.38 Note (efﬁciency of the Micali-Schnorr PRBG) Algorithm 5.37 is more efﬁcient than the RSA PRBG since ⌊N(1−2 e)⌋bits are generated per exponentiation bye. For example, ife=3 and N=1024, thenk=341 bits are generated per exponentiation. Moreover, each exponentiation requires only one modular squaring of anr=683-bit number, and one modular multiplication. 5.39 Note (security of the Micali-Schnorr PRBG) Algorithm 5.37 is cryptographically secure under the assumption that the following is true: the distributionxemodnfor randomr-bit sequencesxis indistinguishable by all polynomial-time statistical tests from the uniform distribution of integers in the interval[0,n−1]. This assumption is stronger than requiring that the RSA problem be intractable. 5.5.2 Blum-Blum-Shub pseudorandom bit generator The Blum-Blum-Shub pseudorandom bit generator (also known as thex2 modngenera- tor or theBBS generator) is a CSPRBG under the assumption that integer factorization is intractable (§3.2). It forms the basis for the Blum-Goldwasser probabilistic public-key en- cryption scheme (Algorithm 8.56). 5.40 AlgorithmBlum-Blum-Shub pseudorandom bit generator SUMMARY: a pseudorandom bit sequencez1,z2,...,z lof lengthlis generated. 1. Setup. Generate two large secret random (and distinct) primespand q(cf. Note 8.8), each congruent to3modulo 4, and computen=pq. 2. Select a random integers(theseed) in the interval[1,n−1]such thatgcd(s,n)=1 , and computex0←s2 modn. 3. Forifrom 1 toldo the following: 3.1 xi←x2 i−1 modn. 3.2 zi←the least signiﬁcant bit ofxi. 4. The output sequence isz1,z2,...,z l. §5.6 Notes and further references 187 5.41 Note (efﬁciency of the Blum-Blum-Shub PRBG) Generating each pseudorandom bitzire- quires one modular squaring. The efﬁciency of the generator can be improved by extracting thejleast signiﬁcant bits ofxiin step 3.2, wherej=clglgnand cis a constant. Provided thatnis sufﬁciently large, this modiﬁed generator is also cryptographically secure. For a modulus nof a ﬁxed bitlength (eg. 1024 bits), an explicit range of values ofcfor which the resulting generator is cryptographically secure (cf. Remark 5.9) under the intractability assumption of the integer factorization problem has not been determined. 5.6 Notes and further references §5.1 Chapter 3 of Knuth [692] is the deﬁnitive reference for the classic (non-cryptographic) gen- eration of pseudorandom numbers. Knuth [692, pp.142-166] contains an extensive discus- sion of what it means for a sequence to be random. Lagarias [724] gives a survey of theo- retical results on pseudorandom number generators. Luby [774] provides a comprehensive and rigorous overview of pseudorandom generators. For a study of linear congruential generators (Example 5.4), see Knuth [692, pp.9-25]. Plumstead/Boyar [979, 980] showed how to predict the output of a linear congruential gen- erator given only a few elements of the output sequence, and when the parametersa,b, and mof the generator are unknown. Boyar [180] extended her method and showed that linearmultivariatecongruential generators (having recurrence equationx n =a1xn−1 + a2xn−2 +···+alxn−l+bmodm), andquadraticcongruential generators (having recur- rence equationxn=ax2 n−1 +bxn−1 +cmodm) are cryptographically insecure. Finally, Krawczyk [713] generalized these results and showed how the output of anymultivariate polynomialcongruential generator can be efﬁciently predicted. Atruncatedlinear congru- ential generator is one where a fraction of the least signiﬁcant bits of thexiare discarded. Frieze et al. [427] showed that these generators can be efﬁciently predicted if the genera- tor parametersa,b, andmare known. Stern [1173] extended this method to the case where onlymis known. Boyar [179] presented an efﬁcient algorithm for predicting linear congru- ential generators whenO(loglogm)bits are discarded, and when the parametersa,b, and mare unknown. No efﬁcient prediction algorithms are known for truncated multivariate polynomial congruential generators. For a summary of cryptanalytic attacks on congruen- tial generators, see Brickell and Odlyzko [209, pp.523-526]. For a formal deﬁnition of a statistical test (Deﬁnition 5.5), see Yao [1258]. Fact 5.7 on the universality of the next-bit test is due to Yao [1258]. For a proof of Yao’s result, see Kranakis [710] and§12.2 of Stinson [1178]. A proof of a generalization of Yao’s result is given by Goldreich, Goldwasser, and Micali [468]. The notion of a cryptographically secure pseudorandom bit generator (Deﬁnition 5.8) was introduced by Blum and Micali [166]. Blum and Micali also gave a formal description of the next-bit test (Deﬁnition 5.6), and presented the ﬁrst cryptographically secure pseudorandom bit generator whose security is based on the discrete logarithm problem (see page 189). Universal tests were presented by Schrift and Shamir [1103] for verifying the assumed properties of a pseudorandom gen- erator whose output sequences are not necessarily uniformly distributed. The ﬁrst provably secure pseudorandomnumber generator was proposed by Shamir [1112]. Shamir proved that predicting the next number of an output sequence of this generator is equivalent to inverting the RSA function. However, even though the numbers as a whole may be unpredictable, certain parts of the number (for example, its least signiﬁcant bit) may 188 Ch. 5 Pseudorandom Bits and Sequences be biased or predictable. Hence, Shamir’s generator is not cryptographically secure in the sense of Deﬁnition 5.8. §5.2 Agnew [17] proposed a VLSI implementation of a random bit generator consisting of two identical metal insulator semiconductor capacitors close to each other. The cells are charged over the same period of time, and then a1or0is assigned depending on which cell has a greater charge. Fairﬁeld, Mortenson, and Coulthart [382] described an LSI random bit generator based on the frequency instability of a free running oscillator. Davis, Ihaka, and Fenstermacher [309] used the unpredictability of air turbulence occurring in a sealed disk drive as a random bit generator. The bits are extracted by measuring the variations in the time to access disk blocks. Fast Fourier Transform (FFT) techniques are then used to re- move possible biases and correlations. A sample implementation generated 100 random bits per minute. For further guidance on hardware and software-based techniques for gen- erating random bits, see RFC 1750 [1043]. The de-skewing technique of Example 5.10 is due to von Neumann [1223]. Elias [370] generalized von Neumann’s technique to a more efﬁcient scheme (one where fewer bits are discarded). Fast Fourier Transform techniques for removing biases and correlations are described by Brillinger [213]. For further ways of removing correlations, see Blum [161], Santha and Vazirani [1091], Vazirani [1217], and Chor and Goldreich [258]. §5.3 The idea of using a one-way functionffor generating pseudorandom bit sequences is due to Shamir [1112]. Shamir illustrated why it is difﬁcult to prove that such ad-hoc generators are cryptographically secure without imposing some further assumptions onf. Algorithm 5.11 is from Appendix C of the ANSI X9.17 standard [37]; it is one of the approved methods for pseudorandom bit generation listed in FIPS 186 [406]. Meyer and Matyas [859, pp.316- 317] describe another DES-based pseudorandom bit generator whose output is intended for use as data-encrypting keys. The four algorithms of§5.3.2 for generating DSA parameters are from FIPS 186. §5.4 Standard references on statistics include Hogg and Tanis [559] and Wackerly, Mendenhall, and Scheaffer [1226]. Tables 5.1 and 5.2 were generated using the Maple symbolic algebra system [240]. Golomb’s randomness postulates (§5.4.3) were proposed by Golomb [498]. The ﬁve statistical tests for local randomness outlined in§5.4.4 are from Beker and Piper [84]. The serial test (§5.4.4(ii)) is due to Good [508]. It was generalized to subsequences of length greater than2by Marsaglia [782] who called it theoverlappingm-tuple test, and later by Kimberley [674] who called it thegeneralized serial test. The underlying distribution theories of the serial test and the runs test (§5.4.4(iv)) were analyzed by Good [507] and Mood [897], respectively. Gustafson [531] considered alternative statistics for the runs test and the autocorrelation test (§5.4.4(v)). There are numerous other statistical tests of local randomness. Many of these tests, includ- ing the gap test, coupon collector’s test, permutation test, run test, maximum-of-ttest, col- lision test, serial test, correlation test, and spectral test are described by Knuth [692]. The poker test as formulated by Knuth [692, p.62] is quite different from that of§5.4.4(iii). In the former, a sample sequence is divided intom-bit blocks, each of which is further subdi- vided intol-bit sub-blocks (for some divisorlofm). The number ofm-bit blocks havingr distinctl-bit sub-blocks (1≤r≤m/l) is counted and compared to the corresponding ex- pected numbers for random sequences. Erdmann [372] gives a detailed exposition of many §5.6 Notes and further references 189 of these tests, and applies them to sample output sequences of six pseudorandom bit gener- ators. Gustafson et al. [533] describe a computer package which implements various statis- tical tests for assessing the strength of a pseudorandom bit generator. Gustafson, Dawson, and Goli´c [532] proposed a newrepetition testwhich measures the number of repetitions of l-bit blocks. The test requires a count of the number of patterns repeated, but does not re- quire the frequency of each pattern. For this reason, it is feasible to apply this test for larger values ofl(e.g.l=64) than would be permissible by the poker test or Maurer’s universal statistical test (Algorithm 5.33). Two spectral tests have been developed, one based on the discrete Fourier transform by Gait [437], and one based on the Walsh transform by Yuen [1260]. For extensions of these spectral tests, see Erdmann [372] and Feldman [389]. FIPS 140-1 [401] speciﬁes security requirements for the design and implementation of cryptographic modules, including random and pseudorandom bit generators, for protecting (U.S. government) unclassiﬁed information. The universal statistical test (Algorithm 5.33) is due to Maurer [813] and was motivated by source coding algorithms of Elias [371] and Willems [1245]. The class of defects that the test is able to detect consists of those that can be modeled by an ergodic stationary source with limited memory; Maurer argues that this class includes the possible defects that could occur in a practical implementation of a random bit generator. Table 5.3 is due to Maurer [813], who provides derivations of formulae for the mean and variance of the statisticX u. §5.5 Blum and Micali [166] presented the following general construction for CSPRBGs. LetD be a ﬁnite set, and letf:D→Dbe a permutation that can be efﬁciently computed. Let B:D →{0,1}be a Boolean predicate with the property thatB(x)is hard to compute given onlyx∈D, however,B(x)can be efﬁciently computed giveny=f−1(x). The output sequencez1,z2,...,z lcorresponding to a seedx0 ∈Dis obtained by computing xi = f(xi−1),zi = B(xi), for1 ≤ i ≤ l. This generator can be shown to pass the next-bit test (Deﬁnition 5.6). Blum and Micali [166] proposed the ﬁrst concrete instance of a CSPRBG, called theBlum-Micali generator. Using the notation introduced above, their method can be described as follows. Letpbe a large prime, andαa generator ofZ ∗ p. Deﬁne D=Z∗ p ={1,2,...,p −1}. The functionf:D→Dis deﬁned byf(x)= αxmodp. The functionB:D→{0,1}is deﬁned byB(x)=1 if0≤logαx≤(p−1)/2, and B(x)=0 iflogαx> (p−1)/2. Assuming the intractability of the discrete logarithm prob- lem inZ∗ p(§3.6; see also§3.9.1), the Blum-Micali generator was proven to satisfy the next- bit test. Long and Wigderson [772] improved the efﬁciency of the Blum-Micali generator by simultaneously extractingO(lglgp)bits (cf.§3.9.1) from eachxi. Kaliski [650, 651] modiﬁed the Blum-Micali generator so that the security depends on the discrete logarithm problem in the group of points on an elliptic curve deﬁned over a ﬁnite ﬁeld. The RSA pseudorandom bit generator (Algorithm 5.35) and the improvement mentioned in Note 5.36 are due to Alexi et al. [23]. The Micali-Schnorr improvement of the RSA PRBG (Algorithm 5.37) is due to Micali and Schnorr [867], who also described a method that transforms any CSPRBG into one that can be accelerated by parallel evaluation. The method of parallelization isperfect:mparallel processors speed the generation of pseudo- random bits by a factor ofm. Algorithm 5.40 is due to Blum, Blum, and Shub [160], who showed that their pseudoran- dom bit generator is cryptographically secure assuming the intractability of the quadratic residuosity problem (§3.4). Vazirani and Vazirani [1218] established a stronger result re- garding the security of this generator by proving it cryptographically secure under the weaker assumption that integer factorization is intractable. The improvement mentioned in 190 Ch. 5 Pseudorandom Bits and Sequences Note 5.41 is due to Vazirani and Vazirani. Alexi et al. [23] proved analogous results for the modiﬁed-Rabin generator, which differs as follows from the Blum-Blum-Shub generator: in step 3.1 of Algorithm 5.40, let x=x2 i−1 modn;i f x<n/ 2, thenxi = x; otherwise, xi=n− x. Impagliazzo and Naor [569] devised efﬁcient constructions for a CSPRBG and for a univer- sal one-way hash function which are provably as secure as the subset sum problem. Fischer and Stern [411] presented a simple and efﬁcient CSPRBG which is provably as secure as thesyndrome decoding problem. Yao [1258] showed how to obtain a CSPRBG using any one-way permutation. Levin [761] generalized this result and showed how to obtain a CSPRBG using any one-way function. For further reﬁnements, see Goldreich, Krawczyk, and Luby [470], Impagliazzo, Levin, and Luby [568], and H˚astad [545]. A random functionf:{0,1} n→{0,1}nis a function which assigns independent and ran- dom valuesf(x)∈{0,1}nto all argumentsx∈{0,1}n. Goldreich, Goldwasser, and Micali [468] introduced a computational complexity measure of the randomness of func- tions. They deﬁned a function to bepoly-randomif no polynomial-time algorithm can dis- tinguish between values of the function and true random strings, even when the algorithm is permitted to select the arguments to the function. Goldreich, Goldwasser, and Micali presented an algorithm for constructing poly-random functions assuming the existence of one-way functions. This theory was applied by Goldreich, Goldwasser, and Micali [467] to develop provably secure protocols for the (essentially) storageless distribution of secret identiﬁcation numbers, message authentication with timestamping, dynamic hashing, and identify friend or foe systems. Luby and Rackoff [776] showed how poly-random permu- tations can be efﬁciently constructed from poly-random functions. This result was used, together with some of the design principles of DES, to show how any CSPRBG can be used to construct a symmetric-key block cipher which is provably secure against chosen- plaintext attack. A simpliﬁed and generalized treatment of Luby and Rackoff’s construction was given by Maurer [816]. Schnorr [1096] used Luby and Rackoff’s poly-random permutation generator to construct a pseudorandom bit generator that was claimed to pass all statistical tests depending only on a small fraction of the output sequence, even when inﬁnite computational resources are available. Rueppel [1079] showed that this claim is erroneous, and demonstrated that the generator can be distinguished from a truly random bit generator using only a small num- ber of output bits. Maurer and Massey [821] extended Schnorr’s work, and proved the ex- istence of pseudorandom bit generators that pass all statistical tests depending only on a small fraction of the output sequence, even when inﬁnite computational resources are avail- able. The security of the generators does not rely on any unproved hypothesis, but rather on the assumption that the adversary can access only a limited number of bits of the gener- ated sequence. This work is primarily of theoretical interest since no such polynomial-time generators are known. Chapter /6 Stream Ciphers Contents in Brief 6.1 Introduction ............................. 191 6.2 Feedback shift registers....................... 195 6.3 Stream ciphers based on LFSRs .................. 203 6.4 Other stream ciphers ........................ 212 6.5 Notes and further references.................... 216 6.1 Introduction Stream ciphersare an important class of encryption algorithms. They encrypt individual characters (usually binary digits) of a plaintext message one at a time, using an encryp- tion transformation which varies with time. By contrast,block ciphers(Chapter 7) tend to simultaneously encrypt groups of characters of a plaintext message using a ﬁxed encryp- tion transformation. Stream ciphers are generally faster than block ciphers in hardware, and have less complex hardware circuitry. They are also more appropriate, and in some cases mandatory (e.g., in some telecommunications applications), when buffering is lim- ited or when characters must be individually processed as they are received. Because they have limited or no error propagation, stream ciphers may also be advantageous in situations where transmission errors are highly probable. There is a vast body of theoretical knowledge on stream ciphers, and various design principles for stream ciphers have been proposed and extensively analyzed. However, there are relatively few fully-speciﬁed stream cipher algorithms in the open literature. This un- fortunate state of affairs can partially be explained by the fact that most stream ciphers used in practice tend to be proprietary and conﬁdential. By contrast, numerous concrete block cipher proposals have been published, some of which have been standardized or placed in the public domain. Nevertheless, because of their signiﬁcant advantages, stream ciphers are widely used today, and one can expect increasingly more concrete proposals in the coming years. Chapter outline The remainder of§6.1 introduces basic concepts relevant to stream ciphers. Feedback shift registers, in particular linear feedback shift registers (LFSRs), are the basic building block in most stream ciphers that have been proposed; they are studied in§6.2. Three general tech- niques for utilizing LFSRs in the construction of stream ciphers are presented in§6.3: using 191 192 Ch. 6 Stream Ciphers a nonlinear combining function on the outputs of several LFSRs (§6.3.1), using a nonlin- ear ﬁltering function on the contents of a single LFSR (§6.3.2), and using the output of one (or more) LFSRs to control the clock of one (or more) other LFSRs (§6.3.3). Two concrete proposals for clock-controlled generators, the alternating step generator and the shrinking generator are presented in§6.3.3.§6.4 presents a stream cipher not based on LFSRs, namely SEAL. §6.5 concludes with references and further chapter notes. 6.1.1 Classiﬁcation Stream ciphers can be either symmetric-key or public-key. The focus of this chapter is symmetric-key stream ciphers; the Blum-Goldwasser probabilistic public-key encryption scheme (§8.7.2) is an example of a public-key stream cipher. 6.1 Note (block vs. stream ciphers) Block ciphers process plaintext in relatively large blocks (e.g.,n≥64 bits). The same function is used to encrypt successive blocks; thus (pure) block ciphers arememoryless. In contrast, stream ciphers process plaintext in blocks as small as a single bit, and the encryption function may vary as plaintext is processed; thus stream ciphers are said to have memory. They are sometimes calledstate cipherssince encryption depends on not only the key and plaintext, but also on the current state. This distinction between block and stream ciphers is not deﬁnitive (see Remark 7.25); adding a small amount of memory to a block cipher (as in the CBC mode) results in a stream cipher with large blocks. (i) The one-time pad Recall (Deﬁnition 1.39) that aVernam cipherover the binary alphabet is deﬁned by c i= mi⊕ki fori=1 ,2,3 ..., where m1,m2,m3,... are the plaintext digits,k1,k2,k3,... (thekeystream) are the key digits,c1,c2,c3,... are the ciphertext digits, and⊕is the XOR function (bitwise addition modulo 2). Decryption is deﬁned bymi = ci⊕ki. If the keystream digits are generated independently and randomly, the Vernam cipher is called aone-time pad, and is uncondi- tionally secure (§1.13.3(i)) against a ciphertext-only attack. More precisely, ifM,C, and Kare random variables respectively denoting the plaintext, ciphertext, and secret key, and ifH() denotes the entropy function (Deﬁnition 2.39), thenH(M\|C)= H(M). Equiva- lently,I(M; C)=0 (see Deﬁnition 2.45): the ciphertext contributes no information about the plaintext. Shannon proved that a necessary condition for a symmetric-key encryption scheme to be unconditionally secure is thatH(K) ≥H(M). That is, the uncertainty of the secret key must be at least as great as the uncertainty of the plaintext. If the key has bitlengthk, and the key bits are chosen randomly and independently, thenH(K)= k, and Shannon’s necessary condition for unconditional security becomesk≥H(M). The one-time pad is unconditionally secure regardless of the statistical distribution of the plaintext, and is op- timal in the sense that its key is the smallest possible among all symmetric-key encryption schemes having this property. An obvious drawback of the one-time pad is that the key should be as long as the plain- text, which increases the difﬁculty of key distribution and key management. This moti- vates the design of stream ciphers where the keystream ispseudorandomlygenerated from a smaller secret key, with the intent that the keystream appears random to a computation- ally bounded adversary. Such stream ciphers do not offer unconditional security (since H(K) ≪H(M)), but the hope is that they are computationally secure (§1.13.3(iv)). §6.1 Introduction 193 Stream ciphers are commonly classiﬁed as beingsynchronousorself-synchronizing. (ii) Synchronous stream ciphers 6.2 DeﬁnitionA synchronousstream cipher is one in which the keystream is generated inde- pendently of the plaintext message and of the ciphertext. The encryption process of a synchronous stream cipher can be described by the equations σi+1 = f(σi,k), zi = g(σi,k), ci = h(zi,mi), where σ0 is theinitial stateand may be determined from the keyk,fis thenext-state function,gis the function which produces thekeystreamzi, andhis theoutput function which combines the keystream and plaintextmito produce ciphertextci. The encryption and decryption processes are depicted in Figure 6.1. The OFB mode of a block cipher (see §7.2.2(iv)) is an example of a synchronous stream cipher. zi f k zi k σi+1 (ii) Decryption(i) Encryption Plaintextmi Ciphertextci Key k Keystreamzi Stateσiσi+1 gh σi mi ci ci mih−1g f σi Figure 6.1:General model of a synchronous stream cipher. 6.3 Note (properties of synchronous stream ciphers) (i)synchronization requirements.In a synchronous stream cipher, both the sender and receiver must besynchronized– using the same key and operating at the same posi- tion (state) within that key – to allow for proper decryption. If synchronization is lost due to ciphertext digits being inserted or deleted during transmission, then decryption fails and can only be restored through additional techniques for re-synchronization. Techniques for re-synchronization include re-initialization, placing special markers at regular intervals in the ciphertext, or, if the plaintext contains enough redundancy, trying all possible keystream offsets. (ii)no error propagation.A ciphertext digit that is modiﬁed (but not deleted) during transmission does not affect the decryption of other ciphertext digits. (iii)active attacks.As a consequence of property (i), the insertion, deletion, or replay of ciphertext digits by an active adversary causes immediate loss of synchronization, and hence might possibly be detected by the decryptor. As a consequence of property (ii), an active adversary might possibly be able to make changes to selected ciphertext digits, and know exactly what affect these changes have on the plaintext. This illus- trates that additional mechanisms must be employed in order to provide data origin authentication and data integrity guarantees (see§9.5.4). Most of the stream ciphers that have been proposed to date in the literature are additive stream ciphers, which are deﬁned below. 194 Ch. 6 Stream Ciphers 6.4 DeﬁnitionA binary additive stream cipheris a synchronous stream cipher in which the keystream, plaintext, and ciphertext digits are binary digits, and the output functionhis the XOR function. Binary additive stream ciphers are depicted in Figure 6.2. Referring to Figure 6.2, the keystream generatoris composed of the next-state functionfand the functiong(see Fig- ure 6.1), and is also known as therunning key generator. Generator Keystream mi zi cimi ci Plaintextmi Ciphertextci Key k Keystreamzi zi kk Keystream Generator (ii) Decryption(i) Encryption Figure 6.2:General model of a binary additive stream cipher. (iii) Self-synchronizing stream ciphers 6.5 DeﬁnitionA self-synchronizingorasynchronousstream cipher is one in which the key- stream is generated as a function of the key and a ﬁxed number of previous ciphertext digits. The encryption function of a self-synchronizing stream cipher can be described by the equations σi =( ci−t,ci−t+1,...,c i−1), zi = g(σi,k), ci = h(zi,mi), where σ0 =( c−t,c−t+1,...,c −1) is the (non-secret)initial state,kis thekey,gis the function which produces thekeystreamzi, andhis theoutput functionwhich combines the keystream and plaintextmi to produce ciphertextci. The encryption and decryption processes are depicted in Figure 6.3. The most common presently-used self-synchronizing stream ciphers are based on block ciphers in 1-bit cipher feedback mode (see§7.2.2(iii)). hk zi ci (i) Encryption gk zi mi (ii) Decryption g h−1 cimi Figure 6.3:General model of a self-synchronizing stream cipher. §6.2 Feedback shift registers 195 6.6 Note (properties of self-synchronizing stream ciphers) (i)self-synchronization.Self-synchronization is possible if ciphertext digits are deleted or inserted, because the decryption mapping depends only on a ﬁxed number of pre- ceding ciphertext characters. Such ciphers are capable of re-establishing proper de- cryption automatically after loss of synchronization, with only a ﬁxed number of plaintext characters unrecoverable. (ii)limited error propagation.Suppose that the state of a self-synchronization stream ci- pher depends ontprevious ciphertext digits. If a single ciphertext digit is modiﬁed (or even deleted or inserted) during transmission, then decryption of up totsubse- quent ciphertext digits may be incorrect, after which correct decryption resumes. (iii)active attacks.Property (ii) implies that any modiﬁcation of ciphertext digits by an active adversary causes several other ciphertext digits to be decrypted incorrectly, thereby improving (compared to synchronous stream ciphers) the likelihood of being detected by the decryptor. As a consequence of property (i), it is more difﬁcult (than for synchronous stream ciphers) to detect insertion, deletion, or replay of ciphertext digits by an active adversary. This illustrates that additional mechanisms must be employed in order to provide data origin authentication and data integrity guarantees (see§9.5.4). (iv)diffusion of plaintext statistics.Since each plaintext digit inﬂuences the entire fol- lowing ciphertext, the statistical properties of the plaintext are dispersed through the ciphertext. Hence, self-synchronizing stream ciphers may be more resistant than syn- chronous stream ciphers against attacks based on plaintext redundancy. 6.2 Feedback shift registers Feedback shift registers, in particular linear feedback shift registers, are the basic compo- nents of many keystream generators.§6.2.1 introduces linear feedback shift registers. The linear complexity of binary sequences is studied in§6.2.2, while the Berlekamp-Massey al- gorithm for computing it is presented in§6.2.3. Finally, nonlinear feedback shift registers are discussed in§6.2.4. 6.2.1 Linear feedback shift registers Linear feedback shift registers (LFSRs) are used in many of the keystream generators that have been proposed in the literature. There are several reasons for this: 1. LFSRs are well-suited to hardware implementation; 2. they can produce sequences of large period (Fact 6.12); 3. they can produce sequences with good statistical properties (Fact 6.14); and 4. because of their structure, they can be readily analyzed using algebraic techniques. 6.7 DeﬁnitionA linear feedback shift register(LFSR) of lengthLconsists ofLstages(or delay elements) numbered0,1,...,L −1, each capable of storing one bit and having one input and one output; and a clock which controls the movement of data. During each unit of time the following operations are performed: (i) the content of stage0 is output and forms part of theoutput sequence; 196 Ch. 6 Stream Ciphers (ii) the content of stageiis moved to stagei−1 for eachi,1 ≤i≤L−1; and (iii) the new content of stageL−1 is thefeedback bitsj which is calculated by adding together modulo2 the previous contents of a ﬁxed subset of stages0,1,...,L −1. Figure 6.4 depicts an LFSR. Referring to the ﬁgure, eachciis either0 or1; the closed semi-circles are AND gates; and the feedback bitsjis the modulo2 sum of the contents of those stagesi,0 ≤i≤L−1, for whichcL−i=1 . Stage Stage L-2 sj L-1 c2c1 cL−1 cL output0 StageStage 1 Figure 6.4:A linear feedback shift register (LFSR) of lengthL. 6.8 DeﬁnitionThe LFSR of Figure 6.4 is denoted⟨L,C(D)⟩, whereC(D)=1 + c1D+ c2D2 + ··· + cLDL ∈Z2[D] is theconnection polynomial. The LFSR is said to benon- singularif the degree ofC(D) isL(that is,cL =1 ). If the initial content of stageiis si ∈{0,1}for eachi,0 ≤i≤L−1, then[sL−1,...,s 1,s0] is called theinitial stateof the LFSR. 6.9 FactIf the initial state of the LFSR in Figure 6.4 is[sL−1,...,s 1,s0], then the output sequences= s0,s1,s2,... is uniquely determined by the following recursion: sj =( c1sj−1 + c2sj−2 + ··· + cLsj−L)m o d2forj≥L. 6.10 Example (output sequence of an LFSR) Consider the LFSR⟨4,1+ D+ D4⟩depicted in Figure 6.5. If the initial state of the LFSR is[0,0,0,0], the output sequence is the zero sequence. The following tables show the contents of the stagesD3,D2,D1,D0 at the end of each unit of timetwhen the initial state is[0,1,1,0]. t D3 D2 D1 D0 0 0 1 1 0 1 0 0 1 1 2 1 0 0 1 3 0 1 0 0 4 0 0 1 0 5 0 0 0 1 6 1 0 0 0 7 1 1 0 0 t D3 D2 D1 D0 8 1 1 1 0 9 1 1 1 1 10 0 1 1 1 11 1 0 1 1 12 0 1 0 1 13 1 0 1 0 14 1 1 0 1 15 0 1 1 0 The output sequence iss=0 ,1,1,0,0,1,0,0,0,1,1,1,1,0,1,... , and is periodic with period15 (see Deﬁnition 5.25). □ The signiﬁcance of an LFSR being non-singular is explained by Fact 6.11. §6.2 Feedback shift registers 197 Stage 3 Stage 1 Stage Stage 20 output D3 D2 D1 D0 Figure 6.5:The LFSR ⟨4, 1+ D + D4⟩of Example 6.10. 6.11 FactEvery output sequence (i.e., for all possible initial states) of an LFSR⟨L,C(D)⟩is periodic if and only if the connection polynomialC(D) has degreeL. If an LFSR⟨L,C(D)⟩issingular(i.e.,C(D) has degree less thanL), then not all out- put sequences are periodic. However, the output sequences areultimately periodic; that is, the sequences obtained by ignoring a certain ﬁnite number of terms at the beginning are periodic. For the remainder of this chapter, it will be assumed that all LFSRs are non- singular. Fact 6.12 determines the periods of the output sequences of some special types of non-singular LFSRs. 6.12 Fact(periods of LFSR output sequences) LetC(D) ∈Z 2[D] be a connection polynomial of degreeL. (i) IfC(D) is irreducible overZ2 (see Deﬁnition 2.190), then each of the2L−1 non- zero initial states of the non-singular LFSR⟨L,C(D)⟩produces an output sequence with period equal to the least positive integerNsuch thatC(D) divides1+ DN in Z2[D]. (Note: it is always the case that thisNis a divisor of2L−1.) (ii) IfC(D) is a primitive polynomial (see Deﬁnition 2.228), then each of the2L−1 non- zero initial states of the non-singular LFSR⟨L,C(D)⟩produces an output sequence with maximum possible period2L−1. A method for generating primitive polynomials overZ2 uniformly at random is given in Algorithm 4.78. Table 4.8 lists a primitive polynomial of degreemoverZ2 for eachm, 1 ≤m≤229. Fact 6.12(ii) motivates the following deﬁnition. 6.13 DeﬁnitionIfC(D) ∈Z2[D] is a primitive polynomial of degreeL, then⟨L,C(D)⟩is called amaximum-lengthLFSR. The output of a maximum-length LFSR with non-zero ini- tial state is called anm-sequence. Fact 6.14 demonstrates that the output sequences of maximum-length LFSRs have good statistical properties. 6.14 Fact(statistical properties ofm-sequences) Letsbe anm-sequence that is generated by a maximum-length LFSR of lengthL. (i) Letkbe an integer,1 ≤k≤L, and let sbe any subsequence ofsof length2L+ k−2. Then each non-zero sequence of lengthkappears exactly2L−k times as a subsequence of s. Furthermore, the zero sequence of lengthkappears exactly2L−k− 1 times as a subsequence of s. In other words, the distribution of patterns having ﬁxed length of at mostLis almost uniform. (ii)ssatisﬁes Golomb’s randomness postulates (§5.4.3). That is, everym-sequence is also a pn-sequence (see Deﬁnition 5.29). 198 Ch. 6 Stream Ciphers 6.15 Example (m-sequence) SinceC(D)=1+ D+ D4 is a primitive polynomial overZ2, the LFSR⟨4,1+ D+ D4⟩is a maximum-length LFSR. Hence, the output sequence of this LFSR is anm-sequence of maximum possible periodN=2 4 −1=1 5 (cf. Example 6.10). Example 5.30 veriﬁes that this output sequence satisﬁes Golomb’s randomness properties. □ 6.2.2 Linear complexity This subsection summarizes selected results about the linear complexity of sequences. All sequences are assumed to be binary sequences. Notation:sdenotes an inﬁnite sequence whose terms ares0,s1,s2,... ;sndenotes a ﬁnite sequence of lengthnwhose terms are s0,s1,...,s n−1 (see Deﬁnition 5.24). 6.16 DeﬁnitionAn LFSR is said togeneratea sequencesif there is some initial state for which the output sequence of the LFSR iss. Similarly, an LFSR is said togeneratea ﬁnite se- quencesnif there is some initial state for which the output sequence of the LFSR hassn as its ﬁrstnterms. 6.17 DeﬁnitionThe linear complexityof an inﬁnite binary sequences, denotedL(s), is deﬁned as follows: (i) ifsis the zero sequences=0 ,0,0,... , thenL(s)=0 ; (ii) if no LFSR generatess, thenL(s)= ∞; (iii) otherwise,L(s) is the length of the shortest LFSR that generatess. 6.18 DeﬁnitionThe linear complexityof a ﬁnite binary sequencesn, denotedL(sn), is the length of the shortest LFSR that generates a sequence havingsnas its ﬁrstnterms. Facts 6.19 – 6.22 summarize some basic results about linear complexity. 6.19 Fact(properties of linear complexity) Letsand tbe binary sequences. (i) For anyn≥1, the linear complexity of the subsequencesnsatisﬁes0 ≤L(sn) ≤n. (ii)L(sn)=0 if and only ifsnis the zero sequence of lengthn. (iii)L(sn)= nif and only ifsn=0 ,0,0,..., 0,1. (iv) Ifsis periodic with periodN, thenL(s) ≤N. (v) L(s⊕t) ≤L(s)+ L(t), wheres⊕tdenotes the bitwise XOR ofsand t. 6.20 FactIf the polynomialC(D) ∈Z2[D] is irreducible overZ2 and has degreeL, then each of the2L−1 non-zero initial states of the non-singular LFSR⟨L,C(D)⟩produces an output sequence with linear complexityL. 6.21 Fact(expectation and variance of the linear complexity of a random sequence) Letsnbe chosen uniformly at random from the set of all binary sequences of lengthn, and letL(sn) be the linear complexity ofsn. LetB(n) denote the parity function:B(n)=0 ifnis even; B(n)=1 ifnis odd. (i) The expected linear complexity ofsnis E(L(sn)) = n 2 + 4+ B(n) 18 − 1 2n (n 3 + 2 9 ) . Hence, for moderately largen,E(L(sn)) ≈ n 2 + 2 9 ifnis even, andE(L(sn)) ≈ n 2 + 5 18 ifnis odd. §6.2 Feedback shift registers 199 (ii) The variance of the linear complexity ofsnisVar(L(sn)) = 86 81 − 1 2n (14 −B(n) 27 n+ 82 −2B(n) 81 ) − 1 22n (1 9 n2 + 4 27n+ 4 81 ) . Hence,Var(L(sn)) ≈86 81 for moderately largen. 6.22 Fact(expectation of the linear complexity of a random periodic sequence) Letsnbe cho- sen uniformly at random from the set of all binary sequences of lengthn, wheren=2 tfor some ﬁxed t≥1, and letsbe then-periodic inﬁnite sequence obtained by repeating the sequencesn. Then the expected linear complexity ofsisE(L(sn)) =n−1+2 −n. The linear complexity proﬁle of a binary sequence is introduced next. 6.23 DeﬁnitionLets= s0,s1,... be a binary sequence, and letLN denote the linear com- plexity of the subsequencesN = s0,s1,...,s N−1,N ≥ 0. The sequenceL1,L2,... is called thelinear complexity proﬁle ofs. Similarly, ifsn = s0,s1,...,s n−1 is a ﬁnite binary sequence, the sequenceL1,L2,...,L nis called thelinear complexity proﬁle ofsn. The linear complexity proﬁle of a sequence can be computed using the Berlekamp- Massey algorithm (Algorithm 6.30); see also Note 6.31. The following properties of the linear complexity proﬁle can be deduced from Fact 6.29. 6.24 Fact(properties of linear complexity proﬁle) LetL1,L2,... be the linear complexity pro- ﬁle of a sequences= s0,s1,... . (i) Ifj>i , thenLj ≥Li. (ii)LN+1 >LN is possible only ifLN ≤N/2. (iii) IfLN+1 >LN, thenLN+1 + LN = N+1 . The linear complexity proﬁle of a sequencescan be graphed by plotting the points (N,LN),N ≥1, in theN×Lplane and joining successive points by a horizontal line followed by a vertical line, if necessary (see Figure 6.6). Fact 6.24 can then be interpreted as saying that the graph of a linear complexity proﬁle is non-decreasing. Moreover, a (vertical) jump in the graph can only occur from below the lineL= N/2; if a jump occurs, then it is symmetric about this line. Fact 6.25 shows that the expected linear complexity of a random sequence should closely follow the lineL= N/2. 6.25 Fact(expected linear complexity proﬁle of a random sequence) Lets= s0,s1,... be a random sequence, and letLNbe the linear complexity of the subsequencesN = s0,s1,..., sN−1 for eachN ≥1. For any ﬁxed indexN ≥ 1, the expected smallestjfor which LN+j >LN is2 ifLN ≤N/2,o r2+2 LN−NifLN >N/2. Moreover, the expected increase in linear complexity is2 ifLN ≥N/2,o rN−2LN+2 ifLN <N/2. 6.26 Example (linear complexity proﬁle) Consider the20-periodic sequenceswith cycle s20 =1 ,0,0,1,0,0,1,1,1,1,0,0,0,1,0,0,1,1,1,0. The linear complexity proﬁle ofsis1,1,1,3,3,3,3,5,5,5,6,6,6,8,8,8,9,9,10,10,11, 11,11,11,14,14,14,14,15,15,15,17,17,17,18,18,19,19,19,19,... . Figure 6.6 shows the graph of the linear complexity proﬁle ofs. □ 200 Ch. 6 Stream Ciphers 10 30 20 15 10 5 20 40 L = L(sN) N L= N/2 line Figure 6.6:Linear complexity proﬁle of the20-periodic sequence of Example 6.26. As is the case with all statistical tests for randomness (cf.§5.4), the condition that a se- quenceshave a linear complexity proﬁle that closely resembles that of a random sequence isnecessarybut notsufﬁcientforsto be considered random. This point is illustrated in the following example. 6.27 Example (limitations of the linear complexity proﬁle) The linear complexity proﬁle of the sequencesdeﬁned as si = { 1, ifi=2 j−1 for somej≥0, 0, otherwise, follows the lineL= N/2 as closely as possible. That is,L(sN)= ⌊(N+1 )/2⌋for all N≥1. However, the sequencesis clearly non-random. □ 6.2.3 Berlekamp-Massey algorithm The Berlekamp-Massey algorithm (Algorithm 6.30) is an efﬁcient algorithm for determin- ing the linear complexity of a ﬁnite binary sequencesnof lengthn(see Deﬁnition 6.18). The algorithm takesniterations, with theNth iteration computing the linear complexity of the subsequencesN consisting of the ﬁrstNterms ofsn. The theoretical basis for the algorithm is Fact 6.29. 6.28 DeﬁnitionConsider the ﬁnite binary sequencesN+1 = s0,s1,...,s N−1,sN. ForC(D) =1 +c1D+ ···+ cLDL, let⟨L,C(D)⟩be an LFSR that generates the subsequencesN = s0,s1,...,s N−1. Thenext discrepancydNis the difference betweensNand the(N+1)st term generated by the LFSR:dN =( sN + ∑ L i=1 cisN−i)m o d2. 6.29 FactLet sN = s0,s1,...,s N−1 be a ﬁnite binary sequence of linear complexityL= L(sN), and let⟨L,C(D)⟩be an LFSR which generatessN. §6.2 Feedback shift registers 201 (i) The LFSR⟨L,C(D)⟩also generatessN+1 = s0,s1,...,s N−1,sNif and only if the next discrepancydN is equal to 0. (ii) IfdN =0 , thenL(sN+1)= L. (iii) SupposedN =1 . Letmthe largest integer<N such thatL(sm) <L(sN), and let ⟨L(sm),B(D)⟩be an LFSR of lengthL(sm) which generatessm. Then⟨L′,C′(D)⟩ is an LFSR of smallest length which generatessN+1, where L′= { L, ifL>N/ 2, N+1 −L, ifL≤N/2, and C′(D)= C(D)+ B(D) ·DN−m. 6.30 AlgorithmBerlekamp-Massey algorithm INPUT: a binary sequencesn= s0,s1,s2,...,s n−1 of lengthn. OUTPUT: the linear complexityL(sn) ofsn,0 ≤L(sn) ≤n. 1. Initialization.C(D)←1, L←0, m←−1, B(D)←1, N←0. 2. While(N<n ) do the following: 2.1 Compute the next discrepancyd.d←(sN + ∑ L i=1 cisN−i)m o d2. 2.2 Ifd=1 then do the following: T(D)←C(D), C(D)←C(D)+ B(D) ·DN−m. IfL≤N/2 thenL←N+1 −L, m←N, B(D)←T(D). 2.3 N←N+1 . 3. Return(L). 6.31 Note (intermediate results in Berlekamp-Massey algorithm) At the end of each iteration of step 2,⟨L,C(D)⟩is an LFSR of smallest length which generatessN. Hence, Algo- rithm 6.30 can also be used to compute the linear complexity proﬁle (Deﬁnition 6.23) of a ﬁnite sequence. 6.32 FactThe running time of the Berlekamp-Massey algorithm (Algorithm 6.30) for deter- mining the linear complexity of a binary sequence of bitlengthnisO(n2) bit operations. 6.33 Example (Berlekamp-Massey algorithm) Table 6.1 shows the steps of Algorithm 6.30 for computing the linear complexity of the binary sequencesn=0 ,0,1,1,0,1,1,1,0of length n=9 . This sequence is found to have linear complexity5, and an LFSR which generates it is⟨5,1+ D3 + D5⟩. □ 6.34 FactLetsnbe a ﬁnite binary sequence of lengthn, and let the linear complexity ofsnbe L. Then there is a unique LFSR of lengthLwhich generatessnif and only ifL≤n 2 . An important consequence of Fact 6.34 and Fact 6.24(iii) is the following. 6.35 FactLetsbe an (inﬁnite) binary sequence of linear complexityL, and lettbe a (ﬁnite) subsequence ofsof length at least2L. Then the Berlekamp-Massey algorithm (with step 3 modiﬁed to return bothLand C(D)) on inputtdetermines an LFSR of lengthLwhich generatess. 202 Ch. 6 Stream Ciphers sN d T(D) C(D) L m B(D) N − − − 1 0 −1 1 0 0 0 − 1 0 −1 1 1 0 0 − 1 0 −1 1 2 1 1 1 1+ D3 3 2 1 3 1 1 1+ D3 1+ D+ D3 3 2 1 4 0 1 1+ D+ D3 1+ D+ D2 + D3 3 2 1 5 1 1 1+ D+ D2 + D3 1+ D+ D2 3 2 1 6 1 0 1+ D+ D2 + D3 1+ D+ D2 3 2 1 7 1 1 1+ D+ D2 1+ D+ D2 + D5 5 7 1+ D+ D2 8 0 1 1+ D+ D2 + D5 1+ D3 + D5 5 7 1+ D+ D2 9 Table 6.1:Steps of the Berlekamp-Massey algorithm of Example 6.33. 6.2.4 Nonlinear feedback shift registers This subsection summarizes selected results about nonlinear feedback shift registers. A function withnbinary inputs and one binary output is called aBoolean functionofnvari- ables; there are22n different Boolean functions ofnvariables. 6.36 DeﬁnitionA (general)feedback shift register(FSR) of lengthLconsists ofLstages(or delay elements) numbered0,1,...,L −1, each capable of storing one bit and having one input and one output, and a clock which controls the movement of data. During each unit of time the following operations are performed: (i) the content of stage0 is output and forms part of theoutput sequence; (ii) the content of stageiis moved to stagei−1 for eachi,1 ≤i≤L−1; and (iii) the new content of stageL−1 is thefeedback bitsj = f(sj−1,sj−2,...,s j−L), where thefeedback functionfis a Boolean function andsj−iis the previous content of stageL−i,1 ≤i≤L. If the initial content of stageiissi∈{0,1}for each0 ≤i≤L−1, then[sL−1,...,s 1,s0] is called theinitial stateof the FSR. Figure 6.7 depicts an FSR. Note that if the feedback functionfis a linear function, then the FSR is an LFSR (Deﬁnition 6.7). Otherwise, the FSR is called anonlinearFSR. Stage sj Stage L-1 L-2 1 0 Stage Stage sj−L+1sj−1 sj−2 sj−L f(sj−1,s j−2,... ,s j−L) output Figure 6.7:A feedback shift register (FSR) of lengthL. 6.37 FactIf the initial state of the FSR in Figure 6.7 is[sL−1,...,s 1,s0], then the output se- quences= s0,s1,s2,... is uniquely determined by the following recursion: sj = f(sj−1,sj−2,...,s j−L) forj≥L. §6.3 Stream ciphers based on LFSRs 203 6.38 DeﬁnitionAn FSR is said to benon-singularif and only if every output sequence of the FSR (i.e., for all possible initial states) is periodic. 6.39 FactAn FSR with feedback functionf(sj−1,sj−2,...,s j−L) is non-singular if and only iffis of the formf= sj−L⊕g(sj−1,sj−2,...,s j−L+1) for some Boolean functiong. The period of the output sequence of a non-singular FSR of lengthLis at most2L. 6.40 DeﬁnitionIf the period of the output sequence (for any initial state) of a non-singular FSR of lengthLis2L, then the FSR is called ade Bruijn FSR, and the output sequence is called a de Bruijn sequence. 6.41 Example (de Bruijn sequence) Consider the FSR of length3 with nonlinear feedback functionf(x1,x2,x3)=1 ⊕x2⊕x3⊕x1x2. The following tables show the contents of the 3 stages of the FSR at the end of each unit of timetwhen the initial state is[0,0,0]. t Stage 2 Stage 1 Stage 0 0 0 0 0 1 1 0 0 2 1 1 0 3 1 1 1 t Stage 2 Stage 1 Stage 0 4 0 1 1 5 1 0 1 6 0 1 0 7 0 0 1 The output sequence is the de Bruijn sequence with cycle0,0,0,1,1,1,0,1. □ Fact 6.42 demonstrates that the output sequence of de Bruijn FSRs have good statistical properties (compare with Fact 6.14(i)). 6.42 Fact(statistical properties of de Bruijn sequences) Letsbe a de Bruijn sequence that is generated by a de Bruijn FSR of lengthL. Letkbe an integer,1 ≤k≤L, and let sbe any subsequence ofsof length2L+ k−1. Then each sequence of lengthkappears exactly 2L−ktimes as a subsequence of s. In other words, the distribution of patterns having ﬁxed length of at mostLis uniform. 6.43 Note (converting a maximum-length LFSR to a de Bruijn FSR) LetR1 be a maximum- length LFSR of lengthLwith (linear) feedback functionf(sj−1,sj−2,...,s j−L). Then the FSRR2 with feedback functiong(sj−1,sj−2,...,s j−L)= f⊕ sj−1 sj−2 ··· sj−L+1 is a de Bruijn FSR. Here, sidenotes the complement ofsi. The output sequence ofR2 is obtained from that ofR1 by simply adding a0 to the end of each subsequence ofL−10 ’s occurring in the output sequence ofR1. 6.3 Stream ciphers based on LFSRs As mentioned in the beginning of§6.2.1, linear feedback shift registers are widely used in keystream generators because they are well-suited for hardware implementation, pro- duce sequences having large periods and good statistical properties, and are readily ana- lyzed using algebraic techniques. Unfortunately, the output sequences of LFSRs are also easily predictable, as the following argument shows. Suppose that the output sequencesof an LFSR has linear complexityL. The connection polynomialC(D) of an LFSR of length Lwhich generatesscan be efﬁciently determined using the Berlekamp-Massey algorithm 204 Ch. 6 Stream Ciphers (Algorithm 6.30) from any (short) subsequencetofshaving length at leastn=2 L(cf. Fact 6.35). Having determinedC(D), the LFSR⟨L,C(D)⟩can then be initialized with any substring ofthaving lengthL, and used to generate the remainder of the sequences. An adversary may obtain the required subsequencetofsby mounting a known or chosen- plaintext attack (§1.13.1) on the stream cipher: if the adversary knows the plaintext subse- quencem1,m2,...,m ncorresponding to a ciphertext sequencec1,c2,...,c n, the corre- sponding keystream bits are obtained asmi⊕ci,1 ≤i≤n. 6.44 Note (use of LFSRs in keystream generators) Since a well-designed system should be se- cure against known-plaintext attacks, an LFSR should never be used by itself as a keystream generator. Nevertheless, LFSRs are desirable because of their very low implementation costs. Three general methodologies for destroying the linearity properties of LFSRs are discussed in this section: (i) using a nonlinear combining function on the outputs of several LFSRs (§6.3.1); (ii) using a nonlinear ﬁltering function on the contents of a single LFSR (§6.3.2); and (iii) using the output of one (or more) LFSRs to control the clock of one (or more) other LFSRs (§6.3.3). Desirable properties of LFSR-based keystream generators For essentially all possible secret keys, the output sequence of an LFSR-based keystream generator should have the following properties: 1. large period; 2. large linear complexity; and 3. good statistical properties (e.g., as described in Fact 6.14). It is emphasized that these properties are onlynecessaryconditions for a keystream gen- erator to be considered cryptographically secure. Since mathematical proofs of security of such generators are not known, such generators can only be deemedcomputationally secure (§1.13.3(iv)) after having withstood sufﬁcient public scrutiny. 6.45 Note (connection polynomial) Since a desirable property of a keystream generator is that its output sequences have large periods, component LFSRs should always be chosen to be maximum-length LFSRs, i.e., the LFSRs should be of the form⟨L,C(D)⟩where C(D) ∈ Z 2[D] is a primitive polynomial of degreeL(see Deﬁnition 6.13 and Fact 6.12(ii)). 6.46 Note (known vs. secret connection polynomial) The LFSRs in an LFSR-based keystream generator may haveknown orsecretconnection polynomials. For known connections, the secret key generally consists of the initial contents of the component LFSRs. For secret connections, the secret key for the keystream generator generally consists of both the initial contents and the connections. For LFSRs of lengthLwith secret connections, the connection polynomials should be selected uniformly at random from the set of all primitive polynomials of degreeLoverZ 2. Secret connections are generally recommended over known connections as the former are more resistant to certain attacks which use precomputation for analyzing the particular con- nection, and because the former are more amenable to statistical analysis. Secret connection LFSRs have the drawback of requiring extra circuitry to implement in hardware. However, because of the extra security possible with secret connections, this cost may sometimes be compensated for by choosing shorter LFSRs. §6.3 Stream ciphers based on LFSRs 205 6.47 Note (sparse vs. dense connection polynomial) For implementation purposes, it is advan- tageous to choose an LFSR that issparse; i.e., only a few of the coefﬁcients of the con- nection polynomial are non-zero. Then only a small number of connections must be made between the stages of the LFSR in order to compute the feedback bit. For example, the con- nection polynomial might be chosen to be a primitive trinomial (cf. Table 4.8). However, in some LFSR-based keystream generators, special attacks can be mounted if sparse connec- tion polynomials are used. Hence, it is generally recommended not to use sparse connection polynomials in LFSR-based keystream generators. 6.3.1 Nonlinear combination generators One general technique for destroying the linearity inherent in LFSRs is to use several LF- SRs in parallel. The keystream is generated as a nonlinear functionfof the outputs of the component LFSRs; this construction is illustrated in Figure 6.8. Such keystream generators are callednonlinear combination generators, andfis called thecombining function. The remainder of this subsection demonstrates that the functionfmust satisfy several criteria in order to withstand certain particular cryptographic attacks. LFSR 1 LFSR 2 LFSR n f keystream Figure 6.8:A nonlinear combination generator.f is a nonlinear combining function. 6.48 DeﬁnitionA product ofmdistinct variables is called anmth order productof the vari- ables. Every Boolean functionf(x1,x2,...,x n) can be written as a modulo2 sum of dis- tinctmth order products of its variables,0 ≤m≤n; this expression is called thealgebraic normal formoff. Thenonlinear orderoffis the maximum of the order of the terms ap- pearing in its algebraic normal form. For example, the Boolean functionf(x1,x2,x3,x4,x5)=1 ⊕x2 ⊕x3 ⊕x4x5 ⊕ x1x3x4x5 has nonlinear order4. Note that the maximum possible nonlinear order of a Boolean function innvariables isn. Fact 6.49 demonstrates that the output sequence of a nonlinear combination generator has high linear complexity, provided that a combining functionfof high nonlinear order is employed. 6.49 FactSuppose thatnmaximum-length LFSRs, whose lengthsL1,L2,...,L nare pairwise distinct and greater than2, are combined by a nonlinear functionf(x1,x2,...,x n) (as in Figure 6.8) which is expressed in algebraic normal form. Then the linear complexity of the keystream isf(L 1,L2,...,L n). (The expressionf(L1,L2,...,L n) is evaluated over the integers rather than overZ2.) 206 Ch. 6 Stream Ciphers 6.50 Example (Geffe generator) The Geffe generator, as depicted in Figure 6.9, is deﬁned by three maximum-length LFSRs whose lengthsL1,L2,L3 are pairwise relatively prime, with nonlinear combining function f(x1,x2,x3)= x1x2 ⊕(1 +x2)x3 = x1x2 ⊕x2x3 ⊕x3. The keystream generated has period(2L1 −1) ·(2L2 −1) ·(2L3 −1) and linear complexity L= L1L2 + L2L3 + L3. keystream x1 x2 x3 LFSR 3 LFSR 2 LFSR 1 Figure 6.9:The Geffe generator. The Geffe generator is cryptographically weak because information about the states of LFSR 1 and LFSR 3 leaks into the output sequence. To see this, letx1(t),x2(t),x3(t),z(t) denote thetth output bits of LFSRs 1, 2, 3 and the keystream, respectively. Then thecor- relation probabilityof the sequencex1(t) to the output sequencez(t) is P(z(t)= x1(t)) = P(x2(t)=1 )+ P(x2(t)=0 ) ·P(x3(t)= x1(t)) = 1 2 + 1 2 ·1 2 = 3 4 . Similarly,P(z(t)= x3(t)) = 3 4 . For this reason, despite having high period and mod- erately high linear complexity, the Geffe generator succumbs to correlation attacks, as de- scribed in Note 6.51. □ 6.51 Note (correlation attacks) Suppose thatnmaximum-length LFSRs R1,R2,...,R n of lengthsL1,L2,...,L nare employed in a nonlinear combination generator. If the connec- tion polynomials of the LFSRs and the combining functionfare public knowledge, then the number of different keys of the generator is∏ n i=1(2Li −1). (A key consists of the ini- tial states of the LFSRs.) Suppose that there is a correlation between the keystream and the output sequence ofR1, with correlation probabilityp> 1 2 . If a sufﬁciently long seg- ment of the keystream is known (e.g., as is possible under a known-plaintext attack on a binary additive stream cipher), the initial state ofR1 can be deduced by counting the num- ber of coincidences between the keystream and all possible shifts of the output sequence ofR1, until this number agrees with the correlation probabilityp. Under these conditions, ﬁnding the initial state ofR1 will take at most2L1 −1 trials. In the case where there is a correlation between the keystream and the output sequences of each ofR1,R2,...,R n, the (secret) initial state of each LFSR can be determined independently in a total of about∑ n i=1(2Li −1) trials; this number is far smaller than the total number of different keys. In a similar manner, correlations between the output sequences of particular subsets of the LFSRs and the keystream can be exploited. In view of Note 6.51, the combining functionfshould be carefully selected so that there is no statistical dependence between any small subset of thenLFSR sequences and §6.3 Stream ciphers based on LFSRs 207 the keystream. This condition can be satisﬁed iffis chosen to bemth-order correlation immune. 6.52 DeﬁnitionLetX1,X2,...,X nbe independent binary variables, each taking on the val- ues0 or1 with probability1 2 . A Boolean functionf(x1,x2,...,x n) ismth-order corre- lation immuneif for each subset ofmrandom variablesXi1 ,Xi2 ,...,X im with1 ≤i1 < i2 <··· <im≤n, the random variableZ= f(X1,X2,...,X n) is statistically indepen- dent of the random vector(Xi1 ,Xi2,...,X im); equivalently,I(Z; Xi1 ,Xi2 ,...,X im )= 0 (see Deﬁnition 2.45). For example, the functionf(x1,x2,...,x n)= x1 ⊕x2 ⊕···⊕ xn is(n−1)th- order correlation immune. In light of Fact 6.49, the following shows that there is a tradeoff between achieving high linear complexity and high correlation immunity with a combining function. 6.53 FactIf a Boolean functionf(x1,x2,...,x n) ismth-order correlation immune, where1 ≤ m<n , then the nonlinear order offis at mostn−m. Moreover, iffisbalanced(i.e., exactly half of the output values offare0) then the nonlinear order offis at mostn−m−1 for1 ≤m≤n−2. The tradeoff between high linear complexity and high correlation immunity can be avoided by permittingmemory in the nonlinear combination functionf. This point is il- lustrated by the summation generator. 6.54 Example (summation generator) The combining function in the summation generator is based on the fact that integer addition, when viewed overZ2, is a nonlinear function with memory whose correlation immunity is maximum. To see this in the casen=2 , leta= am−12m−1+···+a12+a0 andb= bm−12m−1+···+b12+b0 be the binary representations of integersaand b. Then the bits ofz= a+ bare given by the recursive formula: zj = f1(aj,bj,cj−1)= aj⊕bj⊕cj−1 0 ≤j≤m, cj = f2(aj,bj,cj−1)= ajbj⊕(aj⊕bj)cj−1, 0 ≤j≤m−1, where cj is the carry bit, andc−1 = am = bm =0 . Note thatf1 is2nd-order corre- lation immune, whilef2 is amemorylessnonlinear function. The carry bitcj−1 carries all the nonlinear inﬂuence of less signiﬁcant bits ofaand b(namely,aj−1,...,a 1,a0 and bj−1,...,b 1,b0). The summation generator, as depicted in Figure 6.10, is deﬁned bynmaximum-length LFSRs whose lengthsL1,L2,...,L nare pairwise relatively prime. The secret key con- keystream x1 x2 xn LFSR 1 LFSR 2 LFSR n Carry Figure 6.10:The summation generator. 208 Ch. 6 Stream Ciphers sists of the initial states of the LFSRs, and an initial (integer) carryC0. The keystream is generated as follows. At timej(j≥1), the LFSRs are stepped producing output bits x1,x2,...,x n, and theintegersum Sj = ∑ n i=1 xi+ Cj−1 is computed. The keystream bit isSj mod 2(the least signiﬁcant bit ofSj), while the new carry is computed asCj = ⌊Sj/2⌋(the remaining bits ofSj). The period of the keystream is∏ n i=1 (2Li −1), while its linear complexity is close to this number. Even though the summation generator has high period, linear complexity, and corre- lation immunity, it is vulnerable to certain correlation attacks and a known-plaintext attack based on its2-adic span (see page 218). □ 6.3.2 Nonlinear ﬁlter generators Another general technique for destroying the linearity inherent in LFSRs is to generate the keystream as some nonlinear function of the stages of a single LFSR; this construction is illustrated in Figure 6.11. Such keystream generators are callednonlinear ﬁlter generators, and fis called theﬁltering function. Stage Stage L-2 sj L-1 1 c2c1 cL−1 cL f keystream Stage 0 Stage Figure 6.11:A nonlinear ﬁlter generator.f is a nonlinear Boolean ﬁltering function. Fact 6.55 describes the linear complexity of the output sequence of a nonlinear ﬁlter generator. 6.55 FactSuppose that a nonlinear ﬁlter generator is constructed using a maximum-length LFSR of lengthLand a ﬁltering functionfof nonlinear orderm(as in Figure 6.11). (i) (Key’ s bound) The linear complexity of the keystream is at mostLm= ∑ m i=1 (L i ) . (ii) For a ﬁxed maximum-length LFSR of prime lengthL, the fraction of Boolean func- tionsfof nonlinear ordermwhich produce sequences of maximum linear complex- ityLmis Pm ≈ exp(−Lm/(L·2L)) >e −1/L. Therefore, for largeL, most of the generators produce sequences whose linear com- plexity meets the upper bound in (i). The nonlinear functionfselected for a ﬁlter generator should include many terms of each order up to the nonlinear order off. §6.3 Stream ciphers based on LFSRs 209 6.56 Example (knapsack generator) The knapsack keystream generator is deﬁned by a maxim- um-length LFSR⟨L,C(D)⟩and a modulusQ=2 L. The secret key consists ofLknapsack integer weightsa1,a2,...,a Leach of bitlengthL, and the initial state of the LFSR. Re- call that the subset sum problem (§3.10) is to determine a subset of the knapsack weights which add up to a given integers, provided that such a subset exists; this problem isNP - hard (Fact 3.91). The keystream is generated as follows: at timej, the LFSR is stepped and the knapsack sumSj = ∑ L i=1 xiaimod Qis computed, where[xL,...,x 2,x1] is the state of the LFSR at timej. Finally, selected bits ofSj (afterSj is converted to its binary representation) are extracted to form part of the keystream (the⌈lg L⌉least signiﬁcant bits ofSjshould be discarded). The linear complexity of the keystream is then virtually certain to beL(2L−1). Since the state of an LFSR is a binary vector, the function which maps the LFSR state to the knapsack sumSj is indeed nonlinear. Explicitly, let the functionfbe deﬁned by f(x)= ∑ L i=1 xiaimod Q, wherex =[ xL,...,x 2,x1] is a state. Ifxand yare two states then, in general,f(x⊕y) ̸=f(x)+ f(y). □ 6.3.3 Clock-controlled generators In nonlinear combination generators and nonlinear ﬁlter generators, the component LFSRs are clocked regularly; i.e., the movement of data in all the LFSRs is controlled by the same clock. The main idea behind aclock-controlled generatoris to introduce nonlinearity into LFSR-based keystream generators by having the output of one LFSR control theclocking (i.e., stepping) of a second LFSR. Since the second LFSR is clocked in an irregular manner, the hope is that attacks based on the regular motion of LFSRs can be foiled. Two clock- controlled generators are described in this subsection: (i) the alternating step generator and (ii) the shrinking generator. (i) The alternating step generator The alternating step generator uses an LFSRR 1 to control the stepping of two LFSRs,R2 and R3. The keystream produced is the XOR of the output sequences ofR2 and R3. 6.57 AlgorithmAlternating step generator SUMMARY: a control LFSR R1 is used to selectively step two other LFSRs,R2 and R3. OUTPUT: a sequence which is the bitwise XOR of the output sequences ofR2 and R3. The following steps are repeated until a keystream of desired length is produced. 1. RegisterR1 is clocked. 2. If the output ofR1 is1 then: R2 is clocked;R3 is not clocked but its previous output bit is repeated. (For the ﬁrst clock cycle, the “previous output bit” ofR3 is taken to be0.) 3. If the output ofR1 is0 then: R3 is clocked;R2 is not clocked but its previous output bit is repeated. (For the ﬁrst clock cycle, the “previous output bit” ofR2 is taken to be0.) 4. The output bits ofR2 and R3 are XORed; the resulting bit is part of the keystream. More formally, let the output sequences of LFSRsR1,R2, andR3 be a0,a1,a2,... , b0,b1,b2,... , andc0,c1,c2 ..., respectively. Deﬁneb−1 = c−1 =0 . Then the keystream produced by the alternating step generator isx0,x1,x2,... , wherexj = bt(j) ⊕cj−t(j)−1 210 Ch. 6 Stream Ciphers and t(j)=( ∑ j i=0 ai) −1 for allj ≥ 0. The alternating step generator is depicted in Figure 6.12. LFSR R2 LFSR R3 LFSR R1 outputclock Figure 6.12:The alternating step generator. 6.58 Example (alternating step generator with artiﬁcially small parameters) Consider an al- ternating step generator with component LFSRsR1 = ⟨3,1+ D2 + D3⟩,R2 = ⟨4,1+ D3 + D4⟩, andR3 = ⟨5,1+ D+ D3 + D4 + D5⟩. Suppose that the initial states ofR1, R2, andR3 are[0,0,1],[1,0,1,1], and[0,1,0,0,1], respectively. The output sequence of R1 is the7-periodic sequence with cycle a7 =1 ,0,0,1,0,1,1. The output sequence ofR2 is the15-periodic sequence with cycle b15 =1 ,1,0,1,0,1,1,1,1,0,0,0,1,0,0. The output sequence ofR3 is the31-periodic sequence with cycle c31 =1 ,0,0,1,0,1,0,1,1,0,0,0,0,1,1,1,0,0,1,1,0,1,1,1,1,1,0,1,0,0,0. The keystream generated is x =1 ,0,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0,1,1,1,1,0,1,1,0,0,0,1,1,1,0,.... □ Fact 6.59 establishes, under the assumption thatR1 produces a de Bruijn sequence (see Deﬁnition 6.40), that the output sequence of an alternating step generator satisﬁes the basic requirements of high period, high linear complexity, and good statistical properties. 6.59 Fact(properties of the alternating step generator) Suppose thatR1 produces a de Bruijn sequence of period2L1. Furthermore, suppose thatR2 andR3 are maximum-length LFSRs of lengthsL2 andL3, respectively, such thatgcd(L2,L3)=1 . Letxbe the output sequence of the alternating step generator formed byR1,R2, andR3. (i) The sequencexhas period2L1 ·(2L2 −1) ·(2L3 −1). (ii) The linear complexityL(x) ofxsatisﬁes (L2 + L3) ·2L1−1 <L (x) ≤ (L2 + L3) ·2L1. (iii) The distribution of patterns inxis almost uniform. More precisely, letPbe any bi- nary string of lengthtbits, wheret≤min(L2,L3).I fx(t) denotes anytconsecutive bits inx, then the probability thatx(t)= Pis (1 2 )t + O(1/2L2−t)+ O(1/2L3−t). Since a de Bruijn sequence can be obtained from the output sequencesof a maximum- length LFSR (of lengthL) by simply adding a0 to the end of each subsequence ofL−10 ’s occurring ins(see Note 6.43), it is reasonable to expect that the assertions of high period, §6.3 Stream ciphers based on LFSRs 211 high linear complexity, and good statistical properties in Fact 6.59 also hold whenR1 is a maximum-length LFSR. Note, however, that this has not yet been proven. 6.60 Note (security of the alternating step generator) The LFSRsR1,R2,R3 should be cho- sen to be maximum-length LFSRs whose lengthsL1,L2,L3 are pairwise relatively prime: gcd(L1,L2)=1 ,gcd(L2,L3)=1 ,gcd(L1,L3)=1 . Moreover, the lengths should be about the same. IfL1 ≈l,L2 ≈l, andL3 ≈l, the best known attack on the alternating step generator is a divide-and-conquer attack on the control registerR1 which takes ap- proximately2lsteps. Thus, ifl≈128, the generator is secure against all presently known attacks. (ii) The shrinking generator The shrinking generator is a relatively new keystream generator, having been proposed in 1993. Nevertheless, due to its simplicity and provable properties, it is a promising candi- date for high-speed encryption applications. In the shrinking generator, a control LFSRR 1 is used to select a portion of the output sequence of a second LFSRR2. The keystream produced is, therefore, ashrunkenversion (also known as anirregularly decimated subse- quence) of the output sequence ofR2, as speciﬁed in Algorithm 6.61 and depicted in Fig- ure 6.13. 6.61 AlgorithmShrinking generator SUMMARY: a control LFSR R1 is used to control the output of a second LFSRR2. The following steps are repeated until a keystream of desired length is produced. 1. RegistersR1 and R2 are clocked. 2. If the output ofR1 is1, the output bit ofR2 forms part of the keystream. 3. If the output ofR1 is0, the output bit ofR2 is discarded. More formally, let the output sequences of LFSRsR1 and R2 be a0,a1,a2,... and b0,b1,b2,... , respectively. Then the keystream produced by the shrinking generator is x0,x1,x2,... , wherexj = bij , and, for eachj≥0,ij is the position of thejth 1 in the sequencea0,a1,a2,... . ai =0 outputbi discardbi ai =1 aiLFSR R1 LFSR R2 clock bi Figure 6.13:The shrinking generator. 6.62 Example (shrinking generator with artiﬁcially small parameters) Consider a shrinking generator with component LFSRsR1 = ⟨3,1+ D+ D3⟩and R2 = ⟨5,1+ D3 + D5⟩. Suppose that the initial states ofR1 and R2 are[1,0,0] and [0,0,1,0,1], respectively. The output sequence ofR1 is the7-periodic sequence with cycle a7 =0 ,0,1,1,1,0,1, 212 Ch. 6 Stream Ciphers while the output sequence ofR2 is the31-periodic sequence with cycle b31 =1 ,0,1,0,0,0,0,1,0,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,1,1,0,1,1,1,0. The keystream generated is x =1 ,0,0,0,0,1,0,1,1,1,1,1,0,1,1,1,0,.... □ Fact 6.63 establishes that the output sequence of a shrinking generator satisﬁes the basic requirements of high period, high linear complexity, and good statistical properties. 6.63 Fact(properties of the shrinking generator) LetR1 andR2 be maximum-length LFSRs of lengthsL1 and L2, respectively, and letxbe an output sequence of the shrinking generator formed byR1 and R2. (i) Ifgcd(L1,L2)=1 , thenxhas period(2L2 −1) ·2L1−1. (ii) The linear complexityL(x) ofxsatisﬁes L2 ·2L1−2 <L (x) ≤ L2 ·2L1−1. (iii) Suppose that the connection polynomials forR1 and R2 are chosen uniformly at ran- dom from the set of all primitive polynomials of degreesL1 and L2 overZ2. Then the distribution of patterns inxis almost uniform. More precisely, ifPis any binary string of lengthtbits andx(t) denotes anytconsecutive bits inx, then the probability thatx(t)= Pis(1 2 )t+ O(t/2L2). 6.64 Note (security of the shrinking generator) Suppose that the component LFSRsR1 and R2 of the shrinking generator have lengthsL1 and L2, respectively. If the connection polyno- mials forR1 and R2 are known (but not the initial contents ofR1 and R2), the best attack known for recovering the secret key takesO(2L1 ·L3 2) steps. On the other hand, if secret (and variable) connection polynomials are used, the best attack known takesO(22L1 ·L1 · L2) steps. There is also an attack through the linear complexity of the shrinking generator which takesO(2L1 ·L2 2 ) steps (regardless of whether the connections are known or secret), but this attack requires2L1 ·L2 consecutive bits from the output sequence and is, therefore, infeasible for moderately largeL1 and L2. For maximum security,R1 and R2 should be maximum-length LFSRs, and their lengths should satisfygcd(L1,L2)=1 . Moreover, se- cret connections should be used. Subject to these constraints, ifL1 ≈land L2 ≈l, the shrinking generator has a security level approximately equal to22l. Thus, ifL1 ≈64 and L2 ≈64, the generator appears to be secure against all presently known attacks. 6.4 Other stream ciphers While the LFSR-based stream ciphers discussed in§6.3 are well-suited to hardware im- plementation, they are not especially amenable to software implementation. This has led to several recent proposals for stream ciphers designed particularly for fast software imple- mentation. Most of these proposals are either proprietary, or are relatively new and have not received sufﬁcient scrutiny from the cryptographic community; for this reason, they are not presented in this section, and instead only mentioned in the chapter notes on page 222. Two promising stream ciphers speciﬁcally designed for fast software implementation are SEAL and RC4. SEAL is presented in§6.4.1. RC4 is used in commercial products, and has a variable key-size, but it remains proprietary and is not presented here. Two §6.4 Other stream ciphers 213 other widely used stream ciphers not based on LFSRs are the Output Feedback (OFB; see §7.2.2(iv)) and Cipher Feedback (CFB; see§7.2.2(iii)) modes of block ciphers. Another class of keystream generators not based on LFSRs are those whose security relies on the intractability of an underlying number-theoretic problem; these generators are much slower than those based on LFSRs and are discussed in§5.5. 6.4.1 SEAL SEAL (Software-optimized Encryption Algorithm) is a binary additive stream cipher (see Deﬁnition 6.4) that was proposed in 1993. Since it is relatively new, it has not yet received much scrutiny from the cryptographic community. However, it is presented here because it is one of the few stream ciphers that was speciﬁcally designed for efﬁcient software im- plementation and, in particular, for 32-bit processors. SEAL is a length-increasing pseudorandom function which maps a 32-bitsequence number nto anL-bit keystream under control of a 160-bit secret keya. In the preprocess- ing stage (step 1 of Algorithm 6.68), the key is stretched into larger tables using the table- generation functionG a speciﬁed in Algorithm 6.67; this function is based on the Secure Hash Algorithm SHA-1 (Algorithm 9.53). Subsequent to this preprocessing, keystream generation requires about 5 machine instructions per byte, and is an order of magnitude faster than DES (Algorithm 7.82). The following notation is used in SEAL for 32-bit quantitiesA,B,C,D,Xi, andYj: • A: bitwise complement ofA •A∧B,A∨B,A⊕B: bitwise AND, inclusive-OR, exclusive-OR •“A←↩s”:32-bit result of rotatingAleft throughspositions •“A↪→s”:32-bit result of rotatingAright throughspositions •A+ B: mod232 sum of the unsigned integersAand B •f(B,C,D) def =( B∧C)∨( B∧D); g(B,C,D) def =( B∧C)∨(B∧D)∨(C∧D); h(B,C,D) def = B⊕C⊕D •A∥B: concatenation ofAand B •(X1,...,X j)←(Y1,...,Y j): simultaneous assignments(Xi←Yi), where (Y1,...,Y j) is evaluated prior to any assignments. 6.65 Note (SEAL 1.0 vs. SEAL 2.0) The table-generation function (Algorithm 6.67) for the ﬁrst version of SEAL (SEAL 1.0) was based on the Secure Hash Algorithm (SHA). SEAL 2.0 differs from SEAL 1.0 in that the table-generation function for the former is based on the modiﬁed Secure Hash Algorithm SHA-1 (Algorithm 9.53). 6.66 Note (tables) The table generation (step 1 of Algorithm 6.68) uses the compression func- tion of SHA-1 to expand the secret keyainto larger tablesT,S, andR. These tables can be precomputed, but only after the secret keyahas been established. TablesTand Sare 2K bytes and 1K byte in size, respectively. The size of tableRdepends on the desired bitlengthLof the keystream — each 1K byte of keystream requires 16 bytes ofR. 214 Ch. 6 Stream Ciphers 6.67 AlgorithmTable-generation function for SEAL 2.0 Ga(i) INPUT: a 160-bit stringaand an integeri,0 ≤i< 232. OUTPUT: a 160-bit string, denotedGa(i). 1. Deﬁnition of constants. Deﬁne four 32-bit constants (in hex):y1 = 0x5a827999, y2 = 0x6ed9eba1,y3 = 0x8f1bbcdc,y4 = 0xca62c1d6. 2. Table-generation function. (initialize80 32-bit wordsX0,X1,...,X 79) SetX0 ←i. Forjfrom 1 to15 do:Xj←0x00000000. For jfrom 16 to79 do:Xj ←((Xj−3⊕Xj−8⊕Xj−14⊕Xj−16) ←↩1). (initialize working variables) Break up the 160-bit stringainto ﬁve 32-bit words:a= H0H1H2H3H4. (A,B,C,D,E ) ←(H0,H1,H2,H3,H4). (execute four rounds of 20 steps, then update;tis a temporary variable) (Round 1) Forjfrom 0 to19 do the following: t ←((A←↩5) +f(B,C,D)+ E+ Xj+ y1), (A,B,C,D,E ) ←(t,A,B ←↩30,C,D). (Round 2) Forjfrom 20 to39 do the following: t ←((A←↩5) +h(B,C,D)+ E+ Xj+ y2), (A,B,C,D,E ) ←(t,A,B ←↩30,C,D). (Round 3) Forjfrom 40 to59 do the following: t ←((A←↩5) +g(B,C,D)+ E+ Xj+ y3), (A,B,C,D,E ) ←(t,A,B ←↩30,C,D). (Round 4) Forjfrom 60 to79 do the following: t ←((A←↩5) +h(B,C,D)+ E+ Xj+ y4), (A,B,C,D,E ) ←(t,A,B ←↩30,C,D). (update chaining values) (H0,H1,H2,H3,H4) ←(H0 + A,H1 + B,H2 + C,H3 + D,H4 + E). (completion) The value ofGa(i) is the 160-bit stringH0∥H1∥H2∥H3∥H4. 6.68 AlgorithmKeystream generator for SEAL 2.0 SEAL( a,n) INPUT: a 160-bit stringa(the secret key), a (non-secret) integern,0 ≤ n< 232 (the sequence number), and the desired bitlengthLof the keystream. OUTPUT: keystream yof bitlengthL′, whereL′is the least multiple of 128 which is≥L. 1. Table generation. Generate the tablesT,S, andR, whose entries are 32-bit words. The functionFused below is deﬁned byFa(i)= Hi imod5 , whereHi 0Hi 1Hi 2Hi 3Hi 4 = Ga(⌊i/5⌋), and where the functionGais deﬁned in Algorithm 6.67. 1.1 Forifrom 0 to511 do the following:T[i]←Fa(i). 1.2 Forjfrom 0 to255 do the following:S[j]←Fa(0x00001000 + j). 1.3 Forkfrom 0 to4 ·⌈(L−1)/8192⌉−1 do:R[k]←Fa(0x00002000 + k). 2. Initialization procedure.The following is a description of the subroutine Initialize(n,l,A,B,C,D,n 1,n2,n3,n4) which takes as input a 32-bit wordn and an integerl, and outputs eight 32-bit wordsA,B,C,D,n1,n2,n3, andn4. This subroutine is used in step 4. A←n⊕R[4l], B←(n↪→8)⊕R[4l+1 ], C←(n↪→16)⊕R[4l+2 ], D←(n↪→24)⊕R[4l+3 ]. §6.4 Other stream ciphers 215 For jfrom 1 to2 do the following: P←A∧0x000007fc,B←B+ T[P/4], A←(A↪→9), P←B∧0x000007fc,C←C+ T[P/4], B←(B↪→9), P←C∧0x000007fc,D←D+ T[P/4], C←(C↪→9), P←D∧0x000007fc,A←A+ T[P/4], D←(D↪→9). (n1,n2,n3,n4)←(D,B,A,C ). P←A∧0x000007fc,B←B+ T[P/4], A←(A↪→9). P←B∧0x000007fc,C←C+ T[P/4], B←(B↪→9). P←C∧0x000007fc,D←D+ T[P/4], C←(C↪→9). P←D∧0x000007fc,A←A+ T[P/4], D←(D↪→9). 3. Initializeyto be the empty string, andl←0. 4. Repeat the following: 4.1 Execute the procedureInitialize(n,l,A,B,C,D,n 1,n2,n3,n4). 4.2 Forifrom 1 to64 do the following: P←A∧0x000007fc,B←B+ T[P/4], A←(A↪→9), B←B⊕A, Q←B∧0x000007fc,C←C⊕T[Q/4], B←(B↪→9), C←C+ B, P←(P+ C)∧0x000007fc,D←D+ T[P/4], C←(C↪→9), D←D⊕C, Q←(Q+ D)∧0x000007fc,A←A⊕T[Q/4], D←(D↪→9), A←A+ D, P←(P+ A)∧0x000007fc,B←B⊕T[P/4], A←(A↪→9), Q←(Q+ B)∧0x000007fc,C←C+ T[Q/4], B←(B↪→9), P←(P+ C)∧0x000007fc,D←D⊕T[P/4], C←(C↪→9), Q←(Q+ D)∧0x000007fc,A←A+ T[Q/4], D←(D↪→9), y←y∥(B+ S[4i−4]) ∥(C⊕S[4i−3]) ∥(D+ S[4i−2]) ∥(A⊕S[4i−1]). Ifyis≥Lbits in length then return(y) and stop. Ifiis odd, set(A,C)←(A+n1,C+n2). Otherwise,(A,C)←(A+n3,C+n4). 4.3 Setl←l+1 . 6.69 Note (choice of parameterL) In most applications of SEAL 2.0 it is expected thatL≤ 219; larger values ofLare permissible, but come at the expense of a larger tableR.A preferred method for generating a longer keystream without requiring a larger tableRis to compute the concatenation of the keystreams SEAL(a,0), SEAL(a,1), SEAL(a,2),.... Since the sequence number isn< 232, a keystream of length up to251 bits can be obtained in this manner withL=2 19. 6.70 Example (test vectors for SEAL 2.0) Suppose the keyais the 160-bit (hexadecimal) string 67452301 efcdab89 98badcfe 10325476 c3d2e1f0, n= 0x013577af, andL= 32768bits. TableRconsists of wordsR[0],R[1],...,R [15]: 5021758d ce577c11 fa5bd5dd 366d1b93 182cff72 ac06d7c6 2683ead8 fabe3573 82a10c96 48c483bd ca92285c 71fe84c0 bd76b700 6fdcc20c 8dada151 4506dd64 The tableTconsists of wordsT[0],T[1],...,R [511]: 92b404e5 56588ced 6c1acd4e bf053f68 09f73a93 cd5f176a b863f14e 2b014a2f 4407e646 38665610 222d2f91 4d941a21 ........ ........ ........ ........ ........ ........ 3af3a4bf 021e4080 2a677d95 405c7db0 338e4b1e 19ccf158 216 Ch. 6 Stream Ciphers The tableSconsists of wordsS[0],S[1],...,S [255]: 907c1e3d ce71ef0a 48f559ef 2b7ab8bc 4557f4b8 033e9b05 4fde0efa 1a845f94 38512c3b d4b44591 53765dce 469efa02 ........ ........ ........ ........ ........ ........ bd7dea87 fd036d87 53aa3013 ec60e282 1eaef8f9 0b5a0949 The outputyof Algorithm 6.68 consists of 1024 wordsy[0],y[1],...,y [1023]: 37a00595 9b84c49c a4be1e05 0673530f 0ac8389d c5878ec8 da6666d0 6da71328 1419bdf2 d258bebb b6a42a4d 8a311a72 ........ ........ ........ ........ ........ ........ 547dfde9 668d50b5 ba9e2567 413403c5 43120b5a ecf9d062 The XOR of the 1024 words ofyis 0x098045fc. □ 6.5 Notes and further references §6.1 Although now dated, Rueppel [1075] provides a solid introduction to the analysis and design of stream ciphers. For an updated and more comprehensive survey, see Rueppel [1081]. Another recommended survey is that of Robshaw [1063]. The concept of unconditional security was introduced in the seminal paper by Shannon [1120]. Maurer [819] surveys the role of information theory in cryptography and, in partic- ular, secrecy, authentication, and secret sharing schemes. Maurer [811] devised arandom- ized stream cipherthat is unconditionally secure “with high probability”. More precisely, an adversary is unable to obtain any information whatsoever about the plaintext with prob- ability arbitrarily close to1, unless the adversary can perform an infeasible computation. The cipher utilizes a publicly-accessible source of random bits whose length is much greater than that of all the plaintext to be encrypted, and can conceivably be made practical. Mau- rer’s cipher is based on the impracticalRip van Winkle cipherof Massey and Ingermarsson [789], which is described by Rueppel [1081]. One technique for solving the re-synchronization problem with synchronous stream ciphers is to have the receiver send a resynchronization request to the sender, whereby a new inter- nal state is computed as a (public) function of the original internal state (or key) and some public information (such as the time at the moment of the request). Daemen, Govaerts, and Vandewalle [291] showed that this approach can result in a total loss of security for some published stream cipher proposals. Proctor [1011] considered the trade-off between the security and error propagation problems that arise by varying the number of feedback ciphertext digits. Maurer [808] presented various design approaches for self-synchronizing stream ciphers that are potentially superior to designs based on block ciphers, both with re- spect to encryption speed and security. §6.2 An excellent introduction to the theory of both linear and nonlinear shift registers is the book by Golomb [498]; see also Selmer [1107], Chapters 5 and 6 of Beker and Piper [84], and Chapter 8 of Lidl and Niederreiter [764]. A lucid treatment ofm-sequences can be found in Chapter 10 of McEliece [830]. While the discussion in this chapter has been restricted to se- quences and feedback shift registers over the binary ﬁeldZ 2, many of the results presented can be generalized to sequences and feedback shift registers over any ﬁnite ﬁeldFq. §6.5 Notes and further references 217 The results on the expected linear complexity and linear complexity proﬁle of random se- quences (Facts 6.21, 6.22, 6.24, and 6.25) are from Chapter 4 of Rueppel [1075]; they also appear in Rueppel [1077]. Dai and Yang [294] extended Fact 6.22 and obtained bounds for the expected linear complexity of ann-periodic sequence for each possible value ofn. The bounds imply that the expected linear complexity of a random periodic sequence is close to the period of the sequence. The linear complexity proﬁle of the sequence deﬁned in Example 6.27 was established by Dai [293]. For further theoretical analysis of the linear complexity proﬁle, consult the work of Niederreiter [927, 928, 929, 930]. Facts 6.29 and 6.34 are due to Massey [784]. The Berlekamp-Massey algorithm (Algo- rithm 6.30) is due to Massey [784], and is based on an earlier algorithm of Berlekamp [118] for decoding BCH codes. While the algorithm in§6.2.3 is only described for binary se- quences, it can be generalized to ﬁnd the linear complexity of sequences over any ﬁeld. Further discussion and reﬁnements of the Berlekamp-Massey algorithm are given by Blahut [144]. There are numerous other algorithms for computing the linear complexity of a se- quence. For example, Games and Chan [439] and Robshaw [1062] present efﬁcient algo- rithms for determining the linear complexity of binary sequences of period2 n; these algo- rithms have limited practical use since they require an entire cycle of the sequence. Jansen and Boekee [632] deﬁned themaximum order complexityof a sequence to be the length of the shortest (not necessarily linear) feedback shift register (FSR) that can gener- ate the sequence. The expected maximum order complexity of a random binary sequence of lengthnis approximately2l gn. An efﬁcient linear-time algorithm for computing this complexity measure was also presented; see also Jansen and Boekee [631]. Another complexity measure, theZiv-Lempel complexity measure, was proposed by Ziv and Lempel [1273]. This measure quantiﬁes the rate at which new patterns appear in a sequence. Mund [912] used a heuristic argument to derive the expected Ziv-Lempel complexity of a random binary sequence of a given length. For a detailed study of the relative strengths and weaknesses of the linear, maximum order, and Ziv-Lempel complexity measures, see Erdmann [372]. Kolmogorov [704] and Chaitin [236] introduced the notion of so-calledTuring-Kolmogorov -Chaitin complexity, which measures the minimum size of the input to a ﬁxed universal Turing machine which can generate a given sequence; see also Martin-L¨of [783]. While this complexity measure is of theoretical interest, there is no algorithm known for computing it and, hence, it has no apparent practical signiﬁcance. Beth and Dai [124] have shown that the Turing-Kolmogorov-Chaitin complexity is approximately twice the linear complexity for most sequences of sufﬁcient length. Fact 6.39 is due to Golomb and Welch, and appears in the book of Golomb [498, p.115]. Lai [725] showed that Fact 6.39 is only true for the binary case, and established necessary and sufﬁcient conditions for an FSR over a general ﬁnite ﬁeld to be nonsingular. Klapper and Goresky [677] introduced a new type of feedback register called afeedback with carry shift register(FCSR), which is equipped with auxiliary memory for storing the (integer) carry. An FCSR is similar to an LFSR (see Figure 6.4), except that the contents of the tapped stages of the shift register are addedas integersto the current content of the memory to form a sumS. The least signiﬁcant bit ofS(i.e.,Smod 2) is then fed back into the ﬁrst (leftmost) stage of the shift register, while the remaining higher order bits (i.e., ⌊S/2⌋) are retained as the new value of the memory. If the FCSR hasLstages, then the space required for the auxiliary memory is at mostlg Lbits. FCSRs can be conveniently analyzed using the algebra over the2-adic numbers just as the algebra over ﬁnite ﬁelds is used to analyze LFSRs. 218 Ch. 6 Stream Ciphers Any periodic binary sequence can be generated by a FCSR. The2-adic spanof a periodic sequence is the number of stages and memory bits in the smallest FCSR that generates the sequence. Letsbe a periodic sequence having a2-adic span ofT; note thatTis no more than the period ofs. Klapper and Goresky [678] presented an efﬁcient algorithm for ﬁnding an FCSR of lengthTwhich generatess, given2T+2 ⌈lg T⌉+4 of the initial bits ofs.A comprehensive treatment of FCSRs and the2-adic span is given by Klapper and Goresky [676]. §6.3 Notes 6.46 and 6.47 on the selection of connection polynomials were essentially ﬁrst point- ed out by Meier and Staffelbach [834] and Chepyzhov and Smeets [256] in relation to fast correlation attacks on regularly clocked LFSRs. Similar observations were made by Coppersmith, Krawczyk, and Mansour [279] in connection with the shrinking generator. More generally, to withstand sophisticated correlation attacks (e.g., see Meier and Staffel- bach [834]), the connection polynomials should not have low-weight polynomial multiples whose degrees are not sufﬁciently large. Klapper [675] provides examples of binary sequences having high linear complexity, but whose linear complexity is low when considered as sequences (whose elements happen to be only0 or1) over a larger ﬁnite ﬁeld. This demonstrates that high linear complexity (over Z 2) by itself is inadequate for security. Fact 6.49 was proven by Rueppel and Staffelbach [1085]. The Geffe generator (Example 6.50) was proposed by Geffe [446]. ThePless generator (Arrangement D of [978]) was another early proposal for a nonlinear combination genera- tor, and uses four J-K ﬂip-ﬂops to combine the output of eight LFSRs. This generator also succumbs to a divide-and-conquer attack, as was demonstrated by Rubin [1074]. The linear syndrome attackof Zeng, Yang, and Rao [1265] is a known-plaintext attack on keystream generators, and is based on earlier work of Zeng and Huang [1263]. It is effective when the known keystreamBcan be written in the formB= A⊕X, whereAis the output sequence of an LFSR with known connection polynomial, and the sequenceXis unknown but sparse in the sense that it contains more0’s than1’s. If the connection polynomials of the Geffe generator are all known to an adversary, and are primitive trinomials of degrees not exceedingn, then the initial states of the three component LFSRs (i.e., the secret key) can be efﬁciently recovered from a known keystream segment of length37nbits. The correlation attack (Note 6.51) on nonlinear combination generators was ﬁrst devel- oped by Siegenthaler [1133], and estimates were given for the length of the observed keystream required for the attack to succeed with high probability. The importance of correlation immunity to nonlinear combining functions was pointed out by Siegenthaler [1132], who showed the tradeoff between high correlation immunity and high nonlinear or- der (Fact 6.53). Meier and Staffelbach [834] presented two new so-calledfast correlation attackswhich are more efﬁcient than Siegenthaler’s attack in the case where the component LFSRs have sparse feedback polynomials, or if they have low-weight polynomial multiples (e.g., each having fewer than 10 non-zero terms) of not too large a degree. Further exten- sions and reﬁnements of correlation attacks can be found in the papers of Mihaljevi´c and Goli´c [874], Chepyzhov and Smeets [256], Goli´c and Mihaljevi´c [491], Mihaljevi´c and J. Goli´c [875], Mihaljevi´c [873], Clark, Goli´c, and Dawson [262], and Penzhorn and K¨uhn [967]. A comprehensive survey of correlation attacks on LFSR-based stream ciphers is the paper by Goli´c [486]; the cases where the combining function is memoryless or with mem- ory, as well as when the LFSRs are clocked regularly or irregularly, are all considered. The summation generator (Example 6.54) was proposed by Rueppel [1075, 1076]. Meier §6.5 Notes and further references 219 and Staffelbach [837] presented correlation attacks on combination generators having mem- ory, cracked the summation generator having only two component LFSRs, and as a result recommended using several LFSRs of moderate lengths rather than just a few long LFSRs in the summation generator. As an example, if a summation generator employs two LF- SRs each having length approximately200, and if50 000keystream bits are known, then Meier and Staffelbach’s attack is expected to take less than700 trials, where the dominant step in each trial involves solving a400 ×400 system of binary linear equations. Dawson [312] presented another known-plaintext attack on summation generators having two com- ponent LFSRs, which requires fewer known keystream bits than Meier and Staffelbach’s attack. Dawson’s attack is only faster than that of Meier and Staffelbach in the case where both LFSRs are relatively short. Recently, Klapper and Goresky [678] showed that the sum- mation generator has comparatively low2-adic span (see page 218). More precisely, ifa and bare two sequences of2-adic spanλ 2(a) and λ2(b), respectively, and ifsis the re- sult of combining them with the summation generator, then the2-adic span ofsis at most λ2(a)+ λ2(b)+2 ⌈lg(λ2(a))⌉+2 ⌈lg(λ2(b))⌉+6 . For example, ifm-sequences of period 2L−1 forL=7 ,11,13,15,16,17 are combined with the summation generator, then the resulting sequence has linear complexity nearly279, but the2-adic span is less than218. Hence, the summation generator is vulnerable to a known-plaintext attack when the com- ponent LFSRs are all relatively short. The probability distribution of the carry for addition ofnrandom integers was analyzed by Staffelbach and Meier [1167]. It was proven that the carry is balanced for evennand biased for oddn. Forn=3 the carry is strongly biased, however, the bias converges to0 asntends to∞. Goli´c [485] pointed out the importance of the correlation between linear functions of the output and input in general combiners with memory, and introduced the so-calledlinear sequential circuit approximation methodfor ﬁnding such functions that produce correlated sequences. Goli´c [488] used this as a basis for developing alinear cryptanalysistechnique for stream ciphers, and in the same paper proposed a stream cipher called GOAL, incorpo- rating principles of modiﬁed truncated linear congruential generators (see page 187), self- clock-control, and randomly generated combiners with memory. Fact 6.55(i) is due to Key [670], while Fact 6.55(ii) was proven by Rueppel [1075]. Massey and Serconek [794] gave an alternate proof of Key’s bound that is based on the Discrete Fourier Transform. Siegenthaler [1134] described a correlation attack on nonlinear ﬁlter generators. Forr´e [418] has applied fast correlation attacks to such generators. Anderson [29] demonstrated other correlations which may be useful in improving the success of cor- relation attacks. An attack called theinversion attack, proposed by Goli´c [490], may be more effective than Anderson’s attack. Goli´c also provides a list of design criteria for non- linear ﬁlter generators. Ding [349] introduced the notion of differential cryptanalysis for nonlinear ﬁlter generators where the LFSR is replaced by a simple counter having arbitrary period. The linear consistency attackof Zeng, Yang, and Rao [1264] is a known-plaintext attack on keystream generators which can discover key redundancies in various generators. It is effective in situations where it is possible to single out a certain portionk 1 of the secret key k, and form a linear system of equationsAx= bwhere the matrixAis determined byk1, and bis determined from the known keystream. The system of equations should have the property that it is consistent (and with high probability has a unique solution) ifk1 is the true value of the subkey, while it is inconsistent with high probability otherwise. In these circumstances, one can mount an exhaustive search fork 1, and subsequently mount a sepa- rate attack for the remaining bits ofk. If the bitlengths ofk1 and karel1 andl, respectively, the attack demonstrates that the security level of the generator is2l1 +2 l−l1, rather than2l. 220 Ch. 6 Stream Ciphers The multiplexer generatorwas proposed by Jennings [637]. Two maximum-length LFSRs having lengthsL1,L2 that are relatively prime are employed. Lethbe a positive integer satisfyingh≤min(L1,lg L2). After each clock cycle, the contents of a ﬁxed subset ofh stages of the ﬁrst LFSR are selected, and converted to an integertin the interval[0,L2 −1] using a1 −1 mapping θ. Finally, the content of stagetof the second LFSR is output as part of the keystream. Assuming that the connection polynomials of the LFSRs are known, the linear consistency attack provides a known-plaintext attack on the multiplexer gener- ator requiring a known keystream sequence of lengthN≥L1 + L22hand 2L1+hlinear consistency tests. This demonstrates that the choice of the mappingθand the second LFSR do not contribute signiﬁcantly to the security of the generator. The linear consistency attack has also been considered by Zeng, Yang, and Rao [1264] for themultispeed inner-product generatorof Massey and Rueppel [793]. In this generator, two LFSRs of lengthsL1 and L2 are clocked at different rates, and their contents combined at the lower clock rate by taking the inner-product of themin(L1,L2) stages of the two LFSRs. The paper by Zeng et al. [1266] is a readable survey describing the effectiveness of the linear consistency and linear syndrome attacks in cryptanalyzing stream ciphers. The knapsack generator (Example 6.56) was proposed by Rueppel and Massey [1084] and extensively analyzed by Rueppel [1075], however, no concrete suggestions on selecting ap- propriate parameters (the lengthLof the LFSR and the knapsack weights) for the generator were given. No weaknesses of the knapsack generator have been reported in the literature. The idea of using the output of a register to control the stepping of another register was used in several rotor machines during the second world war, for example, the German Lorenz SZ40 cipher. A description of this cipher, and also an extensive survey of clock-controlled shift registers, is provided by Gollmann and Chambers [496]. The alternating step generator (Algorithm 6.57) was proposed in 1987 by G¨unther [528], who also proved Fact 6.59 and described the divide-and-conquer attack mentioned in Note 6.60. The alternating step generator is based on thestop-and-gogenerator of Beth and Piper [126]. In the stop-and-go generator, a control registerR 1 is used to control the stepping of another registerR2 as follows. If the output ofR1 is1, thenR2 is clocked; if the output ofR1 is0, thenR2 is not clocked, however, its previous output is repeated. The output ofR2 is then XORed with the output sequence of a third registerR3 which is clocked at the same rate asR1. Beth and Piper showed how a judicious choice of registersR1,R2, and R3 can guarantee that the output sequence has high linear complexity and period, and good statistical properties. Unfortunately, the generator succumbs to the linear syndrome attack of Zeng, Yang, and Rao [1265] (see also page 218): if the connection polynomials of R 1 and R2 are primitive trinomials of degree not exceedingn, and known to the adversary, then the initial states of the three component LFSRs (i.e., the secret key) can be efﬁciently recovered from a known-plaintext segment of length37nbits. Another variant of the stop-and-go generator is thestep-1/step-2generator due to Gollmann and Chambers [496]. This generator uses two maximum-length registersR 1 and R2 of the same length. RegisterR1 is used to control the stepping ofR2 as follows. If the output of R1 is0, thenR2 is clocked once; if the output ofR1 is1, thenR2 is clocked twice before producing the next output bit.ˇZivkovi´c [1274] proposed anembedding correlation attack on R2 whose complexity ofO(2L2), whereL2 is the length ofR2. A cyclic registerof lengthLis an LFSR with feedback polynomialC(D)=1 + DL. Goll- mann [494] proposedcascadingncyclic registers of the same prime lengthpby arranging them serially in such a way that all except the ﬁrst register are clock-controlled by their pre- decessors; the Gollmannp-cycle cascadecan be viewed as an extension of the stop-and-go §6.5 Notes and further references 221 generator (page 220). The ﬁrst register is clocked regularly, and its output bit is the input bit to the second register. In general, if the input bit to theith register (fori≥2) at time tisat, then theith register is clocked ifat =1 ;i fat =0 , the register is not clocked but its previous output bit is repeated. The output bit of theith register is then XORed withat, and the result becomes the input bit to the(i+1) st register. The output of the last register is the output of thep-cycle cascade. The initial (secret) stage of a component cyclic register should not be the all-0’s vector or the all-1’s vector. Gollmann proved that the period of the output sequence ispn. Moreover, ifpis a prime such that2 is a generator ofZ∗ p, then the output sequence has linear complexitypn. This suggests very strongly using long cascades (i.e.,nlarge) of shorter registers rather than short cascades of longer registers. A variant of the Gollmann cascade, called anm-sequence cascade, has the cyclic registers replaced by maximum-length LFSRs of the same lengthL. Chambers [237] showed that the output se- quence of such anm-sequence cascade has period(2L−1)nand linear complexity at least L(2L−1)n−1. Park, Lee, and Goh [964] extended earlier work of Menicocci [845] and re- ported breaking 9-stagem-sequence cascades where each LFSR has length 100; they also suggested that 10-stagem-sequence cascades may be insecure. Chambers and Gollmann [239] studied an attack onp-cycle andm-sequence cascades calledlock-in, which results in a reduction in the effective key space of the cascades. The shrinking generator (Algorithm 6.61) was proposed in 1993 by Coppersmith, Krawczyk, and Mansour [279], who also proved Fact 6.63 and described the attacks men- tioned in Note 6.64. The irregular output rate of the shrinking generator can be overcome by using a short buffer for the output; the inﬂuence of such a buffer is analyzed by Kessler and Krawczyk [669]. Krawczyk [716] mentions some techniques for improving software im- plementations. A throughput of 2.5 Mbits/sec is reported for a C language implementation on a 33MHz IBM workstation, when the two shift registers each have lengths in the range 61–64 bits and secret connections are employed. The security of the shrinking generator is studied further by Goli´c [487]. A key generator related to the shrinking generator is theself-shrinking generator(SSG) of Meier and Staffelbach [838]. The self-shrinking generator uses only one maximum-length LFSR R. The output sequence ofRis partitioned into pairs of bits. The SSG outputs a 0 if a pair is10, and outputs a1 if a pair is11;01 and 00 pairs are discarded. Meier and Staffelbach proved that the self-shrinking generator can be implemented as a shrinking gen- erator. Moreover, the shrinking generator can be implemented as a self-shrinking genera- tor (whose component LFSR is not maximum-length). More precisely, if the component LFSRs of a shrinking generator have connection polynomialsC 1(D) and C2(D), its out- put sequence can be produced by a self-shrinking generator with connection polynomial C(D)= C 1(D)2 ·C2(D)2. Meier and Staffelbach also proved that if the length ofRisL, then the period and linear complexity of the output sequence of the SSG are at least2⌊L/2⌋ and 2⌊L/2⌋−1, respectively. Moreover, they provided strong evidence that this period and linear complexity is in fact about2L−1. Assuming a randomly chosen, but known, connec- tion polynomial, the best attack presented by Meier and Staffelbach on the SSG takes20.79L steps. More recently, Mihaljevi´c [871] presented a signiﬁcantly faster probabilistic attack on the SSG. For example, ifL= 100, then the new attack takes257 steps and requires a portion of the output sequence of length4.9 ×108. The attack does not have an impact on the security of the shrinking generator. A recent survey of techniques for attacking clock-controlled generators is given by Goll- mann [495]. For some newer attack techniques, see Mihaljevi´c [872], Goli´c and O’Connor [492], and Goli´c [489]. Chambers [238] proposed a clock-controlled cascade composed of LFSRs each of length32. Each32-bit portion of the output sequence of a component LFSR 222 Ch. 6 Stream Ciphers is passed through an invertible scrambler box (S-box), and the resulting32-bit sequence is used to control the clock of the next LFSR. Baum and Blackburn [77] generalized the notion of a clock-controlled shift register to that of a register based on a ﬁnite group. §6.4 SEAL (Algorithm 6.68) was designed and patented by Coppersmith and Rogaway [281]. Rogaway and Coppersmith [1066] report an encryption speed of 7.2 Mbytes/sec for an as- sembly language implementation on a 50 MHz 486 processor withL= 4096bits, assuming precomputed tables (cf. Note 6.66). Although the stream cipher RC4 remains proprietary, alleged descriptions have been pub- lished which are output compatible with certiﬁed implementations of RC4; for example, see Schneier [1094]. Bl¨ocher and Dichtl [156] proposed a fast software stream cipher called FISH (Fibonacci Shrinking generator), which is based on the shrinking generator principle applied to the lagged Fibonacci generator (also known as the additive generator) of Knuth [692, p.27]. Anderson [28] subsequently presented a known-plaintext attack on FISH which requires a few thousand 32-bit words of known plaintext and a work factor of about2 40 computations. Anderson also proposed a fast software stream cipher calledPIKE based on the Fibonacci generator and the stream cipher A5; a description of A5 is given by Anderson [28]. Wolfram [1251, 1252] proposed a stream cipher based on one-dimensional cellular automa- ta with nonlinear feedback. Meier and Staffelbach [835] presented a known-plaintext attack on this cipher which demonstrated that key lengths of 127 bits suggested by Wolfram [1252] are insecure; Meier and Staffelbach recommend key sizes of about 1000 bits. Klapper and Goresky [679] presented constructions for FCSRs (see page 217) whose output sequences have nearly maximal period, are balanced, and are nearly de Bruijn sequences in the sense that for any ﬁxed non-negative integert, the number of occurrences of any two t-bit sequences as subsequences of a period differs by at most2. Such FCSRs are good candidates for usage in the construction of secure stream ciphers, just as maximum-length LFSRs were used in§6.3. Goresky and Klapper [518] introduced a generalization of FCSRs calledd-FCSRs, based onramiﬁedextensions of the2-adic numbers (dis the ramiﬁcation). Chapter /7 Block Ciphers Contents in Brief 7.1 Introduction and overview ..................... 223 7.2 Background and general concepts ................. 224 7.3 Classical ciphers and historical development............ 237 7.4 DES ................................. 250 7.5 FEAL ................................ 259 7.6 IDEA ................................ 263 7.7 SAFER, RC5, and other block ciphers............... 266 7.8 Notes and further references.................... 271 7.1 Introduction and overview Symmetric-key block ciphers are the most prominent and important elements in many cryp- tographic systems. Individually, they provide conﬁdentiality. As a fundamental building block, their versatility allows construction of pseudorandom number generators, stream ci- phers, MACs, and hash functions. They may furthermore serve as a central component in message authentication techniques, data integrity mechanisms, entity authentication proto- cols, and (symmetric-key) digital signature schemes. This chapter examines symmetric-key block ciphers, including both general concepts and details of speciﬁc algorithms. Public- key block ciphers are discussed in Chapter 8. No block cipher is ideally suited for all applications, even one offering a high level of security. This is a result of inevitable tradeoffs required in practical applications, including those arising from, for example, speed requirements and memory limitations (e.g., code size, data size, cache memory), constraints imposed by implementation platforms (e.g., hardware, software, chipcards), and differing tolerances of applications to properties of var- ious modes of operation. In addition, efﬁciency must typically be traded off against security. Thus it is beneﬁcial to have a number of candidate ciphers from which to draw. Of the many block ciphers currently available, focus in this chapter is given to a sub- set of high proﬁle and/or well-studied algorithms. While not guaranteed to be more secure than other published candidate ciphers (indeed, this status changes as new attacks become known), emphasis is given to those of greatest practical interest. Among these, DES is paramount; FEAL has received both serious commercial backing and a large amount of in- dependent cryptographic analysis; and IDEA (originally proposed as a DES replacement) is widely known and highly regarded. Other recently proposed ciphers of both high promise and high proﬁle (in part due to the reputation of their designers) are SAFER and RC5. Ad- ditional ciphers are presented in less detail. 223 224 Ch. 7 Block Ciphers Chapter outline Basic background on block ciphers and algorithm-independent concepts are presented in §7.2, including modes of operation, multiple encryption, and exhaustive search techniques. Classical ciphers and cryptanalysis thereof are addressed in§7.3, including historical details on cipher machines. Modern block ciphers covered in chronological order are DES (§7.4), FEAL (§7.5), and IDEA (§7.6), followed by SAFER, RC5, and other ciphers in§7.7, col- lectively illustrating a wide range of modern block cipher design approaches. Further notes, including details on additional ciphers (e.g., Lucifer) and references for the chapter, may be found in§7.8. 7.2 Background and general concepts Introductory material on block ciphers is followed by subsections addressing modes of op- eration, and discussion of exhaustive key search attacks and multiple encryption. 7.2.1 Introduction to block ciphers Block ciphers can be either symmetric-key or public-key. The main focus of this chapter is symmetric-key block ciphers; public-key encryption is addressed in Chapter 8. (i) Block cipher deﬁnitions A block cipher is a function (see§1.3.1) which mapsn-bit plaintext blocks ton-bit cipher- text blocks;nis called theblocklength. It may be viewed as a simple substitution cipher with large character size. The function is parameterized by ak-bit keyK,1 taking values from a subsetK(thekey space) of the set of allk-bit vectorsVk. It is generally assumed that the key is chosen at random. Use of plaintext and ciphertext blocks of equal size avoids data expansion. To allow unique decryption, the encryption function must be one-to-one (i.e., invert- ible). Forn-bit plaintext and ciphertext blocks and a ﬁxed key, the encryption function is a bijection, deﬁning a permutation onn-bit vectors. Each key potentially deﬁnes a differ- ent bijection. The number of keys is\|K\|, and theeffective key sizeislg\|K\|; this equals the key length if allk-bit vectors are valid keys (K=Vk). If keys are equiprobable and each deﬁnes a different bijection, theentropyof the key space is alsolg\|K\|. 7.1 DeﬁnitionAn n-bitblock cipheris a functionE : Vn ×K→ Vn, such that for each key K ∈K,E(P,K)is an invertible mapping (theencryption functionforK) fromVn toVn, writtenEK(P). The inverse mapping is thedecryption function, denotedDK(C). C=EK(P)denotes that ciphertextCresults from encrypting plaintextPunderK. Whereas block ciphers generally process plaintext in relatively large blocks (e.g.,n≥ 64), stream ciphers typically process smaller units (see Note 6.1); the distinction, however, is not deﬁnitive (see Remark 7.25). For plaintext messages exceeding one block in length, various modes of operation for block ciphers are used (see§7.2.2). The most general block cipher implements every possible substitution, as per Deﬁni- tion 7.2. To represent the key of such ann-bit (true) random block cipher would require 1This use of symbolsk and K may differ from other chapters. §7.2 Background and general concepts 225 lg(2n!)≈(n−1.44)2nbits, or roughly2ntimes the number of bits in a message block. This excessive bitsize makes (true) random ciphers impractical. Nonetheless, it is an ac- cepted design principle that the encryption function corresponding to a randomly selected key shouldappearto be a randomly chosen invertible function. 7.2 DeﬁnitionA( true)random cipheris ann-bit block cipher implementing all2n!bijections on 2nelements. Each of the2n!keys speciﬁes one such permutation. A block cipher whose block sizenis too small may be vulnerable to attacks based on statistical analysis. One such attack involves simple frequency analysis of ciphertext blocks (see Note 7.74). This may be thwarted by appropriate use of modes of operation (e.g., Al- gorithm 7.13). Other such attacks are considered in Note 7.8. However, choosing too large a value for the blocksizenmay create difﬁculties as the complexity of implementation of many ciphers grows rapidly with block size. In practice, consequently, for largern, easily- implementable functions are necessary whichappearto be random (without knowledge of the key). An encryption function per Deﬁnition 7.1 is a deterministic mapping. Each pairing of plaintext blockPand keyKmaps to a unique ciphertext block. In contrast, in a randomized encryption technique (Deﬁnition 7.3; see also Remark 8.22), each(P,K)pair is associated with a setC (P,K) of eligible ciphertext blocks; each timePis encrypted underK, an out- putRfrom a random source non-deterministically selects one of these eligible blocks. To ensure invertibility, for every ﬁxed keyK, the subsetsC(P,K) over all plaintextsPmust be disjoint. Since the encryption function is essentially one-to-many involving an additional parameterR(cf. homophonic substitution,§7.3.2), the requirement for invertibility implies data expansion, which is a disadvantage of randomized encryption and is often unaccept- able. 7.3 DeﬁnitionA randomized encryptionmapping is a functionEfrom a plaintext spaceV n to a ciphertext spaceVm,m>n , drawing elements from a space of random numbersR =Vt.Eis deﬁned byE:Vn×K×R→ Vm, such that for each keyK∈K and R∈R, E(P,K,R), also writtenER K(P), mapsP ∈ Vn toVm; and an inverse (corresponding decryption) function exists, mappingVm×K→ Vn. (ii) Practical security and complexity of attacks The objective of a block cipher is to provide conﬁdentiality. The corresponding objective of an adversary is to recover plaintext from ciphertext. A block cipher istotally brokenif a key can be found, andpartially brokenif an adversary is able to recover part of the plaintext (but not the key) from ciphertext. 7.4 Note (standard assumptions) To evaluate block cipher security, it is customary to always assume that an adversary (i) has access to all data transmitted over the ciphertext channel; and (ii) (Kerckhoffs’ assumption) knows all details of the encryption function except the secret key (which security consequently rests entirely upon). Under the assumptions of Note 7.4, attacks are classiﬁed based on what information a cryptanalyst has access to in addition to intercepted ciphertext (cf.§1.13.1). The most prominent classes of attack for symmetric-key ciphers are (for a ﬁxed key): 1. ciphertext-only– no additional information is available. 2. known-plaintext– plaintext-ciphertext pairs are available. 226 Ch. 7 Block Ciphers 3. chosen-plaintext– ciphertexts are available corresponding to plaintexts of the adver- sary’s choice. A variation is anadaptive chosen-plaintextattack, where the choice of plaintexts may depend on previous plaintext-ciphertext pairs. Additional classes of attacks are given in Note 7.6; while somewhat more hypothetical, these are nonetheless of interest for the purposes of analysis and comparison of ciphers. 7.5 Remark (chosen-plaintext principle) It is customary to use ciphers resistant to chosen- plaintext attack even when mounting such an attack is not feasible. A cipher secure against chosen-plaintext attack is secure against known-plaintext and ciphertext-only attacks. 7.6 Note (chosen-ciphertext and related-key attacks)A chosen-ciphertextattack operates un- der the following model: an adversary is allowed access to plaintext-ciphertext pairs for some number of ciphertexts of his choice, and thereafter attempts to use this information to recover the key (or plaintext corresponding to some new ciphertext). In arelated-key at- tack, an adversary is assumed to have access to the encryption of plaintexts under both an unknown key and (unknown) keys chosen to have or known to have certain relationships with this key. With few exceptions (e.g., the one-time pad), the best available measure of security for practical ciphers is the complexity of the best (currently) known attack. Various aspects of such complexity may be distinguished as follows: 1. data complexity– expected number of input data units required (e.g., ciphertext). 2. storage complexity– expected number of storage units required. 3. processing complexity– expected number of operations required to process input data and/or ﬁll storage with data (at least one time unit per storage unit). The attack complexityis the dominant of these (e.g., for linear cryptanalysis on DES, essen- tially the data complexity). When parallelization is possible, processing complexity may be divided across many processors (but not reduced), reducing attack time. Given a data complexity of2 n, an attack is always possible; this many differentn- bit blocks completely characterize the encryption function for a ﬁxedk-bit key. Similarly, given a processing complexity of2k, an attack is possible by exhaustive key search (§7.2.3). Thus as a minimum, the effective key size should be sufﬁciently large to preclude exhaus- tive key search, and the block size sufﬁciently large to preclude exhaustive data analysis. A block cipher is consideredcomputationally secureif these conditions hold and no known attack has both data and processing complexity signiﬁcantly less than, respectively,2nand 2k. However, see Note 7.8 for additional concerns related to block size. 7.7 Remark (passive vs. active complexity) For symmetric-key block ciphers, data complex- ity is beyond the control of the adversary, and ispassive complexity(plaintext-ciphertext pairs cannot be generated by the adversary itself). Processing complexity isactive com- plexitywhich typically beneﬁts from increased resources (e.g., parallelization). 7.8 Note (attacks based on small block size) Security concerns which arise if the block size nis too small include the feasibility oftext dictionary attacksand matching ciphertext at- tacks. A text dictionary may be assembled if plaintext-ciphertext pairs become known for a ﬁxed key. The more pairs available, the larger the dictionary and the greater the chance of locating a random ciphertext block therein. A complete dictionary results if2 nplaintext- ciphertext pairs become known, and fewer sufﬁce if plaintexts contain redundancy and a non-chaining mode of encryption (such as ECB) is used. Moreover, if about2n/2 such pairs §7.2 Background and general concepts 227 are known, and about2n/2 ciphertexts are subsequently created, then by the birthday para- dox one expects to locate a ciphertext in the dictionary. Relatedly, from ciphertext blocks alone, as the number of available blocks approaches2n/2, one expects to ﬁnd matching ci- phertext blocks. These may reveal partial information about the corresponding plaintexts, depending on the mode of operation of the block cipher, and the amount of redundancy in the plaintext. Computational and unconditional security are discussed in§1.13.3. Unconditional se- curity is both unnecessary in many applications and impractical; for example, it requires as many bits of secret key as plaintext, and cannot be provided by a block cipher used to encrypt more than one block (due to Fact 7.9, since identical ciphertext implies matching plaintext). Nonetheless, results on unconditional security provide insight for the design of practical ciphers, and has motivated many of the principles of cryptographic practice cur- rently in use (see Remark 7.10). 7.9 FactA cipher providesperfect secrecy(unconditional security) if the ciphertext and plain- text blocks are statistically independent. 7.10 Remark (theoretically-motivated principles) The unconditional security of the one-time- pad motivates both additive stream ciphers (Chapter 6) and the frequent changing of cryp- tographic keys (§13.3.1). Theoretical results regarding the effect of redundancy on unicity distance (Fact 7.71) motivate the principle that for plaintext conﬁdentiality, the plaintext data should be as random as possible, e.g., via data-compression prior to encryption, use of random-bit ﬁelds in message blocks, or randomized encryption (Deﬁnition 7.3). The latter two techniques may, however, increase the data length or allow covert channels. (iii) Criteria for evaluating block ciphers and modes of operation Many criteria may be used for evaluating block ciphers in practice, including: 1. estimated security level. Conﬁdence in the (historical) security of a cipher grows if it has been subjected to and withstood expert cryptanalysis over a substantial time pe- riod, e.g., several years or more; such ciphers are certainly considered more secure than those which have not. This may include the performance of selected cipher com- ponents relative to various design criteria which have been proposed or gained favor in recent years. The amount of ciphertext required to mount practical attacks often vastly exceeds a cipher’s unicity distance (Deﬁnition 7.69), which provides a theo- retical estimate of the amount of ciphertext required to recover the unique encryption key. 2. key size. The effective bitlength of the key, or more speciﬁcally, the entropy of the key space, deﬁnes an upper bound on the security of a cipher (by considering exhaustive search). Longer keys typically impose additional costs (e.g., generation, transmis- sion, storage, difﬁculty to remember passwords). 3. throughput. Throughput is related to the complexity of the cryptographic mapping (see below), and the degree to which the mapping is tailored to a particular imple- mentation medium or platform. 4. block size. Block size impacts both security (larger is desirable) and complexity (larger is more costly to implement). Block size may also affect performance, for example, if padding is required. 5. complexity of cryptographic mapping. Algorithmic complexity affects the imple- mentation costs both in terms of development and ﬁxed resources (hardware gate 228 Ch. 7 Block Ciphers count or software code/data size), as well as real-time performance for ﬁxed resources (throughput). Some ciphers speciﬁcally favor hardware or software implementations. 6. data expansion. It is generally desirable, and often mandatory, that encryption does not increase the size of plaintext data. Homophonic substitution and randomized en- cryption techniques result in data expansion. 7. error propagation. Decryption of ciphertext containing bit errors may result in vari- ous effects on the recovered plaintext, including propagation of errors to subsequent plaintext blocks. Different error characteristics are acceptable in various applica- tions. Block size (above) typically affects error propagation. 7.2.2 Modes of operation A block cipher encrypts plaintext in ﬁxed-sizen-bit blocks (oftenn=64). For messages exceedingnbits, the simplest approach is to partition the message inton-bit blocks and encrypt each separately. This electronic-codebook (ECB) mode has disadvantages in most applications, motivating other methods of employing block ciphers (modes of operation) on larger messages. The four most common modes are ECB, CBC, CFB, and OFB. These are summarized in Figure 7.1 and discussed below. In what follows,EK denotes the encryption function of the block cipherEparame- terized by keyK, whileE−1 K denotes decryption (cf. Deﬁnition 7.1). A plaintext message x=x1 ...xt is assumed to consist ofn-bit blocks for ECB and CBC modes (see Algo- rithm 9.58 regarding padding), andr-bit blocks for CFB and OFB modes for appropriate ﬁxed r≤n. (i) ECB mode The electronic codebook(ECB) mode of operation is given in Algorithm 7.11 and illustrated in Figure 7.1(a). 7.11 AlgorithmECB mode of operation INPUT: k-bit keyK;n-bit plaintext blocksx1,...,x t. SUMMARY: produce ciphertext blocksc1,...,c t; decrypt to recover plaintext. 1. Encryption: for1≤j≤t, cj ←EK(xj). 2. Decryption: for1≤j≤t, xj ←E−1 K (cj). Properties of the ECB mode of operation: 1. Identical plaintext blocks (under the same key) result in identical ciphertext. 2. Chaining dependencies: blocks are enciphered independently of other blocks. Re- ordering ciphertext blocks results in correspondingly re-ordered plaintext blocks. 3. Error propagation: one or more bit errors in a single ciphertext block affect decipher- ment of that block only. For typical ciphersE, decryption of such a block is then ran- dom (with about 50% of the recovered plaintext bits in error). Regarding bits being deleted, see Remark 7.15. 7.12 Remark (use of ECB mode) Since ciphertext blocks are independent, malicious substi- tution of ECB blocks (e.g., insertion of a frequently occurring block) does not affect the decryption of adjacent blocks. Furthermore, block ciphers do not hide data patterns – iden- tical ciphertext blocks imply identical plaintext blocks. For this reason, the ECB mode is not recommended for messages longer than one block, or if keys are reused for more than §7.2 Background and general concepts 229 cj xj (i) encipherment (ii) decipherment x′ j = xj x′ j = xj cj (ii) decipherment(i) encipherment c0 = IV b) Cipher-block Chaining (CBC)a) Electronic Codebook (ECB) x′ j = xj n r c) Cipher feedback (CFB),r-bit characters/r-bit feedback I1 = IV rxj cj−1 cj−1 (i) encipherment cj (ii) decipherment key x′ j = xj Ij Ij E rxj (i) encipherment leftmost rbits cj (ii) decipherment d) Output feedback (OFB),r-bit characters/n-bit feedback r Oj−1Oj−1 I1 = IV EE −1 E E−1 cj−1 cj cj−1 r-bit shift r-bit shift Ij Ij Ekey rbits leftmost key key Oj Oj E E n n xj n n n r Oj Oj r n n key key key key Figure 7.1:Common modes of operation for ann-bit block cipher. 230 Ch. 7 Block Ciphers a single one-block message. Security may be improved somewhat by inclusion of random padding bits in each block. (ii) CBC mode The cipher-block chaining(CBC) mode of operation, speciﬁed in Algorithm 7.13 and il- lustrated in Figure 7.1(b), involves use of ann-bit initialization vector, denotedIV. 7.13 AlgorithmCBC mode of operation INPUT: k-bit keyK;n-bitIV;n-bit plaintext blocksx1,...,x t. SUMMARY: produce ciphertext blocksc1,...,c t; decrypt to recover plaintext. 1. Encryption:c0 ←IV. For1≤j≤t, cj ←EK(cj−1⊕xj). 2. Decryption:c0 ←IV. For1≤j≤t, xj ←cj−1⊕E−1 K (cj). Properties of the CBC mode of operation: 1. Identical plaintexts: identical ciphertext blocks result when the same plaintext is en- ciphered under the same key andIV. Changing theIV, key, or ﬁrst plaintext block (e.g., using a counter or random ﬁeld) results in different ciphertext. 2. Chaining dependencies: the chaining mechanism causes ciphertextcj to depend on xjand all preceding plaintext blocks (the entire dependency on preceding blocks is, however, contained in the value of the previous ciphertext block). Consequently, re- arranging the order of ciphertext blocks affects decryption. Proper decryption of a correct ciphertext block requires a correct preceding ciphertext block. 3. Error propagation: a single bit error in ciphertext blockcj affects decipherment of blockscj and cj+1 (sincexj depends oncj and cj−1). Blockx′ j recovered fromcj is typically totally random (50% in error), while the recovered plaintextx′ j+1 has bit errors precisely wherecj did. Thus an adversary may cause predictable bit changes inxj+1 by altering corresponding bits ofcj. See also Remark 7.14. 4. Error recovery: the CBC mode isself-synchronizingorciphertext autokey(see Re- mark 7.15) in the sense that if an error (including loss of one or more entire blocks) occurs in blockcjbut notcj+1,cj+2 is correctly decrypted toxj+2. 7.14 Remark (error propagation in encryption) Although CBC mode decryption recovers from errors in ciphertext blocks, modiﬁcations to a plaintext blockxjduring encryption alter all subsequent ciphertext blocks. This impacts the usability of chaining modes for applications requiring random read/write access to encrypted data. The ECB mode is an alternative (but see Remark 7.12). 7.15 Remark (self-synchronizing vs. framing errors) Although self-synchronizing in the sense of recovery from bit errors, recovery from “lost” bits causing errors in block boundaries (framing integrity errors) is not possible in the CBC or other modes. 7.16 Remark (integrity of IV in CBC) While theIV in the CBC mode need not be secret, its integrity should be protected, since malicious modiﬁcation thereof allows an adversary to make predictable bit changes to the ﬁrst plaintext block recovered. Using a secretIV is one method for preventing this. However, if message integrity is required, an appropriate mechanism should be used (see§9.6.5); encryption mechanisms typically guarantee conﬁ- dentiality only. §7.2 Background and general concepts 231 (iii) CFB mode While the CBC mode processes plaintextnbits at a time (using ann-bit block cipher), some applications require thatr-bit plaintext units be encrypted and transmitted without delay, for some ﬁxed r<n (oftenr=1 orr=8). In this case, thecipher feedback(CFB) mode may be used, as speciﬁed in Algorithm 7.17 and illustrated in Figure 7.1(c). 7.17 AlgorithmCFB mode of operation (CFB-r) INPUT: k-bit keyK;n-bitIV;r-bit plaintext blocksx1,...,x u(1≤r≤n). SUMMARY: produce r-bit ciphertext blocksc1,...,c u; decrypt to recover plaintext. 1. Encryption:I1 ←IV.(Ijis the input value in a shift register.) For1≤j≤u: (a)Oj ←EK(Ij). (Compute the block cipher output.) (b) tj ←therleftmost bits ofOj. (Assume the leftmost is identiﬁed as bit 1.) (c)cj ←xj⊕tj. (Transmit ther-bit ciphertext blockcj.) (d) Ij+1 ←2r·Ij+cj mod2n. (Shiftcjinto right end of shift register.) 2. Decryption:I1 ←IV. For1≤j≤u, upon receivingcj: xj ←cj⊕tj, wheretj,Ojand Ijare computed as above. Properties of the CFB mode of operation: 1. Identical plaintexts: as per CBC encryption, changing theIV results in the same plaintext input being enciphered to a different output. TheIV need not be secret (although an unpredictableIV may be desired in some applications). 2. Chaining dependencies: similar to CBC encryption, the chaining mechanism causes ciphertext blockcjto depend on bothxjand preceding plaintext blocks; consequent- ly, re-ordering ciphertext blocks affects decryption. Proper decryption of a correct ciphertext block requires the preceding⌈n/r⌉ciphertext blocks to be correct (so that the shift register contains the proper value). 3. Error propagation: one or more bit errors in any singler-bit ciphertext blockcj af- fects the decipherment of that and the next⌈n/r⌉ciphertext blocks (i.e., untilnbits of ciphertext are processed, after which the error blockcjhas shifted entirely out of the shift register). The recovered plaintextx′ jwill differ fromxjprecisely in the bit positionscj was in error; the other incorrectly recovered plaintext blocks will typi- cally be random vectors, i.e., have 50% of bits in error. Thus an adversary may cause predictable bit changes inxjby altering corresponding bits ofcj. 4. Error recovery: the CFB mode is self-synchronizing similar to CBC, but requires ⌈n/r⌉ciphertext blocks to recover. 5. Throughput: forr<n , throughput is decreased by a factor ofn/r(vs. CBC) in that each execution ofEyields onlyrbits of ciphertext output. 7.18 Remark (CFB use of encryption only) Since the encryption functionEis used for both CFB encryption and decryption, the CFB mode must not be used if the block cipherEis a public-key algorithm; instead, the CBC mode should be used. 7.19 Example (ISO variant of CFB) The CFB mode of Algorithm 7.17 may be modiﬁed as follows, to allow processing of plaintext blocks (characters) whose bitsizesis less than the bitsizerof the feedback variable (e.g., 7-bit characters using 8-bit feedback;s<r ). The leftmosts(rather thanr) bits ofOj are assigned totj; thes-bit ciphertext charactercj is computed; the feedback variable is computed fromcjby pre-prepending (on the left)r−s 1-bits; the resultingr-bit feedback variable is shifted into the least signiﬁcant (LS) end of the shift register as before. □ 232 Ch. 7 Block Ciphers (iv) OFB mode The output feedback(OFB) mode of operation may be used for applications in which all error propagation must be avoided. It is similar to CFB, and allows encryption of various block sizes (characters), but differs in that the output of the encryption block functionE (rather than the ciphertext) serves as the feedback. Two versions of OFB using ann-bit block cipher are common. The ISO version (Fig- ure 7.1(d) and Algorithm 7.20) requires ann-bit feedback, and is more secure (Note 7.24). The earlier FIPS version (Algorithm 7.21) allowsr<n bits of feedback. 7.20 AlgorithmOFB mode with full feedback (per ISO 10116) INPUT: k-bit keyK;n-bitIV;r-bit plaintext blocksx1,...,x u(1≤r≤n). SUMMARY: produce r-bit ciphertext blocksc1,...,c u; decrypt to recover plaintext. 1. Encryption:I1 ←IV. For1≤j≤u, given plaintext blockxj: (a)Oj ←EK(Ij). (Compute the block cipher output.) (b) tj ←therleftmost bits ofOj. (Assume the leftmost is identiﬁed as bit 1.) (c)cj ←xj⊕tj. (Transmit ther-bit ciphertext blockcj.) (d) Ij+1 ←Oj. (Update the block cipher input for the next block.) 2. Decryption:I1 ←IV. For1≤j≤u, upon receivingcj: xj ←cj⊕tj, wheretj,Oj, andIj are computed as above. 7.21 AlgorithmOFB mode withr-bit feedback (per FIPS 81) INPUT: k-bit keyK;n-bitIV;r-bit plaintext blocksx1,...,x u(1≤r≤n). SUMMARY: produce r-bit ciphertext blocksc1,...,c u; decrypt to recover plaintext. As per Algorithm 7.20, but with “Ij+1 ←Oj” replaced by: Ij+1 ←2r·Ij+tj mod2n. (Shift outputtjinto right end of shift register.) Properties of the OFB mode of operation: 1. Identical plaintexts: as per CBC and CFB modes, changing theIVresults in the same plaintext being enciphered to a different output. 2. Chaining dependencies: the keystream is plaintext-independent (see Remark 7.22). 3. Error propagation: one or more bit errors in any ciphertext charactercj affects the decipherment of only that character, in the precise bit position(s)cjis in error, causing the corresponding recovered plaintext bit(s) to be complemented. 4. Error recovery: the OFB mode recovers from ciphertext bit errors, but cannot self- synchronize after loss of ciphertext bits, which destroys alignment of the decrypting keystream (in which case explicit re-synchronization is required). 5. Throughput: forr<n , throughput is decreased as per the CFB mode. However, in all cases, since the keystream is independent of plaintext or ciphertext, it may be pre-computed (given the key andIV). 7.22 Remark (changing IV in OFB) TheIV, which need not be secret, must be changed if an OFB key Kis re-used. Otherwise an identical keystream results, and by XORing corre- sponding ciphertexts an adversary may reduce cryptanalysis to that of a running-key cipher with one plaintext as the running key (cf. Example 7.58 ff.). Remark 7.18 on public-key block ciphers applies to the OFB mode as well as CFB. §7.2 Background and general concepts 233 7.23 Example (counter mode) A simpliﬁcation of OFB involves updating the input block as a counter,Ij+1 =Ij+1, rather than using feedback. This both avoids the short-cycle prob- lem of Note 7.24, and allows recovery from errors in computingE. Moreover, it provides a random-access property: ciphertext blockineed not be decrypted in order to decrypt block i+1. □ 7.24 Note (OFB feedback size) In OFB with fulln-bit feedback (Algorithm 7.20), the keystre- am is generated by the iterated functionOj = EK(Oj−1). SinceEK is a permutation, and under the assumption that for randomK,EKis effectively a random choice among all (2n)!permutations onnelements, it can be shown that for a ﬁxed (random) key and starting value, the expected cycle length before repeating any valueOjis about2n−1. On the other hand, if the number of feedback bits isr<n as allowed in Algorithm 7.21, the keystream is generated by the iterationOj =f(Oj−1)for some non-permutationfwhich, assuming it behaves as a random function, has an expected cycle length of about2n/2. Consequently, it is strongly recommended to use the OFB mode with fulln-bit feedback. 7.25 Remark (modes as stream ciphers) It is clear that both the OFB mode with full feedback (Algorithm 7.20) and the counter mode (Example 7.23) employ a block cipher as a keystre- am generator for a stream cipher. Similarly the CFB mode encrypts a character stream using the block cipher as a (plaintext-dependent) keystream generator. The CBC mode may also be considered a stream cipher withn-bit blocks playing the role of very large characters. Thus modes of operation allow one to deﬁne stream ciphers from block ciphers. 7.2.3 Exhaustive key search and multiple encryption A ﬁxed-size key deﬁnes an upper bound on the security of a block cipher, due to exhaustive key search (Fact 7.26). While this requires either known-plaintext or plaintext containing redundancy, it has widespread applicability since cipher operations (including decryption) are generally designed to be computationally efﬁcient. A design technique which complicates exhaustive key search is to make the task of changing cipher keys computationally expensive, while allowing encryption with a ﬁxed key to remain relatively efﬁcient. Examples of ciphers with this property include the block cipher Khufu and the stream cipher SEAL. 7.26 Fact(exhaustive key search) For ann-bit block cipher withk-bit key, given a small num- ber (e.g.,⌈(k+4)/n⌉) of plaintext-ciphertext pairs encrypted under keyK,Kcan be re- covered by exhaustive key search in an expected time on the order of2 k−1 operations. Justiﬁcation: Progress through the entire key space, decrypting a ﬁxed ciphertextCwith each trial key, and discarding those keys which do not yield the known plaintextP. The target key is among the undiscarded keys. The number of false alarms expected (non-target keys which mapCtoP) depends on the relative size ofkand n, and follows from unicity distance arguments; additional(P ′,C′)pairs sufﬁce to discard false alarms. One expects to ﬁnd the correct key after searching half the key space. 7.27 Example (exhaustive DES key search) For DES,k=56,n=64, and the expected re- quirement by Fact 7.26 is255 decryptions and a single plaintext-ciphertext pair.□ If the underlying plaintext is known to contain redundancy as in Example 7.28, then ciphertext-only exhaustive key search is possible with a relatively small number of cipher- texts. 234 Ch. 7 Block Ciphers 7.28 Example (ciphertext-only DES key search) Suppose DES is used to encrypt 64-bit blocks of 8 ASCII characters each, with one bit per character serving as an even parity bit. Trial decryption with an incorrect keyKyields all 8 parity bits correct with probability2−8, and correct parity fortdifferent blocks (each encrypted byK) with probability2−8t. If this is used as a ﬁlter over all256 keys, the expected number of unﬁltered incorrect keys is256/28t. For most practical purposes,t=10 sufﬁces. □ (i) Cascades of ciphers and multiple encryption If a block cipher is susceptible to exhaustive key search (due to inadequate keylength), en- cipherment of the same message block more than once may increase security. Various such techniques for multiple encryption ofn-bit messages are considered here. Once deﬁned, they may be extended to messages exceeding one block by using standard modes of oper- ation (§7.2.2), withEdenoting multiple rather than single encryption. 7.29 DeﬁnitionA cascade cipheris the concatenation ofL≥2block ciphers (calledstages), each with independent keys. Plaintext is input to ﬁrst stage; the output of stageiis input to stagei+1; and the output of stageLis the cascade’s ciphertext output. In the simplest case, all stages in a cascade cipher havek-bit keys, and the stage in- puts and outputs are alln-bit quantities. The stage ciphers may differ (general cascade of ciphers), or all be identical (cascade of identical ciphers). 7.30 DeﬁnitionMultiple encryptionis similar to a cascade ofLidentical ciphers, but the stage keys need not be independent, and the stage ciphers may be either a block cipherEor its corresponding decryption functionD=E−1. Two important cases of multiple encryption are double and triple encryption, as illus- trated in Figure 7.2 and deﬁned below. EE M E(1) E(2) E(3) K1 K3 B (b) triple encryption (K1 = K3 for two-key variant) K1 K2 K2 (a) double encryption A plaintext P plaintext P ciphertext ciphertext C C Figure 7.2:Multiple encryption. 7.31 DeﬁnitionDouble encryptionis deﬁned asE(x)= EK2 (EK1 (x)), whereEKdenotes a block cipherEwith keyK. §7.2 Background and general concepts 235 7.32 DeﬁnitionTriple encryptionis deﬁned asE(x)= E(3) K3 (E(2) K2 (E(1) K1 (x))), whereE(j) K de- notes eitherEK orDK =E−1 K . The caseE(x)= EK3 (DK2 (EK1 (x)))is calledE-D-E triple-encryption; the subcaseK1 =K3 is often calledtwo-key triple-encryption. Independent stage keysK1 and K2 are typically used in double encryption. In triple encryption (Deﬁnition 7.32), to save on key management and storage costs, dependent stage keys are often used. E-D-E triple-encryption withK1 =K2 =K3 is backwards compati- ble with (i.e., equivalent to) single encryption. (ii) Meet-in-the-middle attacks on multiple encryption A naive exhaustive key search attack on double encryption tries all22kkey pairs. The attack of Fact 7.33 reduces time from22k, at the cost of substantial space. 7.33 FactFor a block cipher with ak-bit key, a known-plaintextmeet-in-the-middleattack de- feats double encryption using on the order of2koperations and2kstorage. Justiﬁcation(basic meet-in-the-middle): Noting Figure 7.2(a), given a(P,C)pair, com- puteMi =Ei(P)under all2kpossible key valuesK1 =i; store all pairs(Mi,i), sorted or indexed onMi(e.g., using conventional hashing). DecipherCunder all2kpossible val- ues K2 =j, and for each pair(Mj,j)where Mj =Dj(C), check forhitsMj =Mi against entriesMi in the ﬁrst table. (This can be done creating a second sorted table, or simply checking eachMjentry as generated.) Each hit identiﬁes a candidate solution key pair(i,j), sinceEi(P)= M=Dj(C). Using a second known-plaintext pair(P′,C′)(cf. Fact 7.35), discard candidate key pairs which do not mapP′toC′. A concept analogous to unicity distance for ciphertext-only attack (Deﬁnition 7.69) can be deﬁned for known-plaintext key search, based on the following strategy. Select a key; check if it is consistent with a given set (history) of plaintext-ciphertext pairs; if so, label the key ahit. A hit that is not the target key is afalse key hit. 7.34 DeﬁnitionThe number of plaintext-ciphertext pairs required to uniquely determine a key under a known-plaintext key search is theknown-plaintext unicity distance. This is the smallest integertsuch that a history of lengthtmakes false key hits improbable. Using Fact 7.35, the (known-plaintext) unicity distance of a cascade ofLrandom ci- phers can be estimated. Less than one false hit is expected whent>Lk/n . 7.35 FactFor anL-stage cascade of random block ciphers withn-bit blocks andk-bit keys, the expected number of false key hits for a history of lengthtis about2Lk−tn. Fact 7.35 holds with respect to random block ciphers deﬁned as follows (cf. Deﬁni- tions 7.2 and 7.70): givennand k, of the possible(2n)!permutations on2n elements, choose2krandomly and with equal probabilities, and associate these with the2kkeys. 7.36 Example (meet-in-the-middle – double-DES) Applying Fact 7.33 to DES (n=64,k= 56), the number of candidate key pairs expected for one(P,C)pair is248 =2k·2k/2n, and the likelihood of a false key pair satisfying a second(P′,C′)sample is2−16 =248/2n. Thus with high probability, two(P,C)pairs sufﬁce for key determination. This agrees with the unicity distance estimate of Fact 7.35: forL=2, a history of lengtht=2 yields2−16 expected false key hits. □ 236 Ch. 7 Block Ciphers A naive exhaustive attack on all key pairs in double-DES uses2112 time and negligi- ble space, while the meet-in-the-middle attack (Fact 7.33) requires256 time and256 space. Note 7.37 illustrates that the latter can be modiﬁed to yield a time-memory trade-off at any point between these two extremes, with the time-memory product essentially constant at 2112 (e.g.,272 time,240 space). 7.37 Note (time-memory tradeoff – double-encryption) In the attack of Example 7.36, memory may be reduced (from tables of256 entries) by independently guessingsbits of each ofK1, K2 (for any ﬁxeds,0≤s≤k). The tables then each have2k−sentries (ﬁxingskey bits eliminates2sentries), but the attack must be run over2s·2spairs of such tables to allow all possible key pairs. The memory requirement is2·2k−sentries (eachn+k−sbits, omitting sﬁxed key bits), while time is on the order of22s·2k−s=2k+s. The time-memory product is22k+1. 7.38 Note (generalized meet-in-the-middle trade-off) Variations of Note 7.37 allow time-space tradeoffs for meet-in-the-middle key search on any concatenation ofL≥2ciphers. ForL even, meeting between the ﬁrst and lastL/2stages results in requirements on the order of 2·2(kL/2)−s space and2(kL/2)+s time,0 ≤ s ≤ kL/2. ForLodd, meeting after the ﬁrst(L−1)/2and before the last(L+1)/2stages results in requirements on the order of 2·2k(L−1)/2 −sspace and2k(L+1)/2+ stime,1≤s≤k(L−1)/2. For a block cipher withk-bit key, a naive attack on two-key triple encryption (Deﬁni- tion 7.32) involves trying all22kkey pairs. Fact 7.39 notes a chosen-plaintext alternative. 7.39 FactFor ann-bit block cipher withk-bit key, two-key triple encryption may be defeated by a chosen-plaintext attack requiring on the order of2kof each of the following: cipher operations, words of(n+k)-bit storage, and plaintext-ciphertext pairs with plaintexts cho- sen. Justiﬁcation(chosen-plaintext attack on two-key triple-encryption): Using2kchosen plain- texts, two-key triple encryption may be reduced to double-encryption as follows. Noting Figure 7.2(b), focus on the case where the result after the ﬁrst encryption stage is the all- zero vectorA=0. For all2 kvaluesK1 =i, computePi=E−1 i (A). Submit each result- ingPias a chosen plaintext, obtaining the corresponding ciphertextCi. For each, compute Bi=E−1 i (Ci), representing an intermediate resultBafter the second of three encryption stages. Note that the valuesPialso represent candidate valuesB. Sort the valuesPjandBj in a table (using standard hashing for efﬁciency). Identify the keys corresponding to pairs P j =Bias candidate solution key pairsK1 =i,K2 =jto the given problem. Conﬁrm these by testing each key pair on a small number of additional known plaintext-ciphertext pairs as required. While generally impractical due to the storage requirement, the attack of Fact 7.39 is referred to as acertiﬁcational attackon two-key triple encryption, demonstrating it to be weaker than triple encryption. This motivates consideration of triple-encryption with three independent keys, although a penalty is a third key to manage. Fact 7.40, stated speciﬁcally for DES (n=64,k=56), indicates that for the price of additional computation, the memory requirement in Fact 7.39 may be reduced and the chosen-plaintext condition relaxed to known-plaintext. The attack, however, appears im- practical even with extreme parallelization; for example, forlgt=40, the number of op- erations is still2 80. §7.3 Classical ciphers and historical development 237 7.40 FactIftknown plaintext-ciphertext pairs are available, an attack on two-key triple-DES requiresO(t)space and2120−lg toperations. (iii) Multiple-encryption modes of operation In contrast to thesingle modesof operation in Figure 7.1,multiple modesare variants of multiple encryption constructed by concatenating selected single modes. For example, the combination of three single-mode CBC operations providestriple-inner-CBC; an alterna- tive istriple-outer-CBC, the composite operation of triple encryption (per Deﬁnition 7.32) with one outer ciphertext feedback after the sequential application of three single-ECB op- erations. With replicated hardware, multiple modes such as triple-inner-CBC may be pipe- lined allowing performance comparable to single encryption, offering an advantage over triple-outer-CBC. Unfortunately (Note 7.41), they are often less secure. 7.41 Note (security of triple-inner-CBC) Many multiple modes of operation are weaker than the corresponding multiple-ECB mode (i.e., multiple encryption operating as a black box with only outer feedbacks), and in some cases multiple modes (e.g., ECB-CBC-CBC) are not signiﬁcantly stronger than single encryption. In particular, under some attacks triple- inner-CBC is signiﬁcantly weaker than triple-outer-CBC; against other attacks based on the block size (e.g., Note 7.8), it appears stronger. (iv) Cascade ciphers Counter-intuitively, it is possible to devise examples whereby cascading of ciphers (Def- inition 7.29) actually reduces security. However, Fact 7.42 holds under a wide variety of attack models and meaningful deﬁnitions of “breaking”. 7.42 FactA cascade ofn(independently keyed) ciphers is at least as difﬁcult to break as the ﬁrst component cipher. Corollary: for stage ciphers which commute (e.g., additive stream ciphers), a cascade is at least as strong as the strongest component cipher. Fact 7.42 does not apply to product ciphers consisting of component ciphers which may have dependent keys (e.g., two-key triple-encryption); indeed, keying dependencies across stages may compromise security entirely, as illustrated by a two-stage cascade wherein the components are two binary additive stream ciphers using an identical keystream – in this case, the cascade output is the original plaintext. Fact 7.42 may suggest the following practical design strategy: cascade a set of key- stream generators each of which relies on one or more different design principles. It is not clear, however, if this is preferable to one large keystream generator which relies on a single principle. The cascade may turn out to be less secure for a ﬁxed set of parameters (number of key bits, block size), since ciphers built piecewise may often be attacked piecewise. 7.3 Classical ciphers and historical development The termclassical ciphersrefers to encryption techniques which have become well-known over time, and generally created prior to the second half of the twentieth century (in some cases, many hundreds of years earlier). Many classical techniques are variations of sim- ple substitution and simple transposition. Some techniques that are not technically block ciphers are also included here for convenience and context. 238 Ch. 7 Block Ciphers Classical ciphers and techniques are presented under§7.3 for historical and pedagogi- cal reasons only. They illustrate important basic principles and common pitfalls. However, since these techniques are neither sophisticated nor secure against current cryptanalytic ca- pabilities,they are not generally suitable for practical use. 7.3.1 Transposition ciphers (background) For asimple transpositioncipher with ﬁxed periodt, encryption involves grouping the plaintext into blocks oftcharacters, and applying to each block a single permutationeon the numbers 1 throught. More precisely, the ciphertext corresponding to plaintext block m=m1 ...mtisc=Ee(m)= me(1) ...me(t). The encryption key ise, which implic- itly deﬁnest; the key spaceKhas cardinalityt!for a given valuet. Decryption involves use of the permutationdwhich invertse. The above corresponds to Deﬁnition 1.32. The mathematical notation obscures the simplicity of the encryption procedure, as is evident from Example 7.43. 7.43 Example (simple transposition) Consider a simple transposition cipher witht=6 and e=(641352) . The messagem=CAESAR is encrypted toc=RSCEAA. Decryption uses the inverse permutationd=(364251) . The transposition may be represented by a two-row matrix with the second indicating the position to which the element indexed by the corresponding number of the ﬁrst is mapped to: ( 123456 364251 ) . Encryption may be done by writing a block of plaintext under headings “364251 ”, and then reading off the characters under the headings in numerical order. □ 7.44 Note (terminology: transposition vs. permutation) While the term “transposition” is tra- ditionally used to describe a transposition cipher, the mapping of Example 7.43 may alter- nately be called apermutationon the set{1,2,..., 6}. The latter terminology is used, for example, in substitution-permutation networks, and in DES (§7.4). A mnemonic keyword may be used in place of a key, although this may seriously de- crease the key space entropy. For example, forn=6, the keyword “CIPHER” could be used to specify the column ordering 1, 5, 4, 2, 3, 6 (by alphabetic priority). 7.45 DeﬁnitionSequential composition of two or more simple transpositions with respective periodst1,t2,...,t iis called acompound transposition. 7.46 FactThe compound transposition of Deﬁnition 7.45 is equivalent to a simple transposition of periodt=lcm(t1,...,t i). 7.47 Note (recognizing simple transposition) Although simple transposition ciphers alter de- pendencies between consecutive characters, they are easily recognized because they pre- serve the frequency distribution of each character. 7.3.2 Substitution ciphers (background) This section considers the following types of classical ciphers: simple (or mono-alphabetic) substitution, polygram substitution, and homophonic substitution. The difference between codes and ciphers is also noted. Polyalphabetic substitution ciphers are considered in§7.3.3. §7.3 Classical ciphers and historical development 239 (i) Mono-alphabetic substitution Suppose the ciphertext and plaintext character sets are the same. Letm=m1m2m3 ... be a plaintext message consisting of juxtaposed charactersmi∈A, whereAis some ﬁxed character alphabet such asA= {A,B,...,Z }.A simple substitution cipheror mono- alphabetic substitution cipheremploys a permutationeoverA, with encryption mapping Ee(m)= e(m1)e(m2)e(m3).... Here juxtaposition indicates concatenation (rather than multiplication), ande(mi)is the character to whichmiis mapped bye. This corresponds to Deﬁnition 1.27. 7.48 Example (trivial shift cipher/Caesar cipher)A shift cipheris a simple substitution cipher with the permutationeconstrained to an alphabetic shift throughkcharacters for some ﬁxed k. More precisely, if\|A\|=s, andmiis associated with the integer valuei,0≤i≤s−1, thenci =e(mi)= mi+kmods. The decryption mapping is deﬁned byd(ci)= ci− kmods. For English text,s=26, and characters A through Z are associated with integers 0through25. Fork=1, the messagem=HAL is encrypted toc=IBM. According to folklore, Julius Caesar used the keyk=3. □ The shift cipher can be trivially broken because there are onlys=\|A\|keys (e.g.,s= 26) to exhaustively search. A similar comment holds for afﬁne ciphers (Example 7.49). More generally, see Fact 7.68. 7.49 Example (afﬁne cipher – historical) The afﬁne cipher on a 26-letter alphabet is deﬁned by eK(x)= ax+bmod26, where0≤a,b≤25. The key is(a,b). Ciphertextc=eK(x)is decrypted usingdK(c)=( c−b)a−1 mod26, with the necessary and sufﬁcient condition for invertibility thatgcd(a,26)=1 . Shift ciphers are a subclass deﬁned bya=1. □ 7.50 Note (recognizing simple substitution) Mono-alphabetic substitution alters the frequency of individual plaintext characters, but does not alter the frequency distribution of the overall character set. Thus, comparing ciphertext character frequencies to a table of expected letter frequencies (unigram statistics) in the plaintext language allows associations between ci- phertext and plaintext characters. (E.g., if the most frequent plaintext character X occurred twelve times, then the ciphertext character that X maps to will occur twelve times). (ii) Polygram substitution A simple substitution cipher substitutes for single plaintext letters. In contrast,polygram substitution ciphersinvolve groups of characters being substituted by other groups of char- acters. For example, sequences of two plaintext characters (digrams) may be replaced by other digrams. The same may be done with sequences of three plaintext characters (tri- grams), or more generally usingn-grams. In full digram substitution over an alphabet of 26 characters, the key may be any of the 26 2 digrams, arranged in a table with row and column indices corresponding to the ﬁrst and second characters in the digram, and the table entries being the ciphertext digrams substi- tuted for the plaintext pairs. There are then(262)!keys. 7.51 Example (Playfair cipher – historical) A digram substitution may be deﬁned by arrang- ing the characters of a 25-letter alphabet (Iand Jare equated) in a5×5matrixM. Adja- cent plaintext characters are paired. The pair(p1,p2)is replaced by the digram(c3,c4)as follows. Ifp1 and p2 are in distinct rows and columns, they deﬁne the corners of a subma- trix (possiblyMitself), with the remaining cornersc3 and c4;c3 is deﬁned as the character in the same column asp1.I fp1 and p2 are in a common row,c3 is deﬁned as the charac- ter immediately to the right ofp1 and c4 that immediately right ofp2 (the ﬁrst column is 240 Ch. 7 Block Ciphers viewed as being to the right of the last). Ifp1 and p2 are in the same column, the charac- ters immediately (circularly) below them arec3 and c4.I fp1 =p2, an infrequent plaintext character (e.g.,X) is inserted between them and the plaintext is re-grouped. While crypt- analysis based on single character frequencies fails for the Playfair cipher (each letter may be replaced by any other), cryptanalysis employing digram frequencies succeeds.□ The key for a Playfair cipher is the5×5square. A mnemonic aid may be used to more easily remember the square. An example is the use of a meaningful keyphrase, with repeated letters deleted and the remaining alphabet characters included alphabetically at the end. The keyphrase “PLAYFAIR IS A DIGRAM CIPHER” would deﬁne a square with rows PLAYF, IRSDG, MCHEB, KNOQT, VWXYZ. To avoid the trailing characters always being from the end of the alphabet, a further shift cipher (Example 7.48) could be applied to the resulting 25-character string. Use of keyphrases may seriously reduce the key space entropy. This effect is reduced if the keyphrase is not directly written into the square. For example, the non-repeated key- phrase characters might be written into an 8-column rectangle (followed by the remaining alphabet letters), the trailing columns being incomplete. The 25-character string obtained by reading the columns vertically is then used to ﬁll the5×5square row by row. 7.52 Example (Hill cipher – historical)A n n-gram substitution may be deﬁned using an in- vertiblen×nmatrixA=a ij as the key to map ann-character plaintextm1 ...mnto a ciphertextn-gramci=∑ n j=1 aijmj,i=1,...,n . Decryption involves usingA−1. Here characters A–Z, for example, are associated with integers 0–25. This polygram substitution cipher is a linear transformation, and falls under known-plaintext attack.□ (iii) Homophonic substitution The idea of homophonic substitution, introduced in§1.5, is for each ﬁxed keykto asso- ciate with each plaintext unit (e.g., character)ma setS(k,m)of potential corresponding ciphertext units (generally all of common size). To encryptmunderk, randomly choose one element from this set as the ciphertext. To allow decryption, for each ﬁxed key this one-to-many encryption function must be injective on ciphertext space. Homophonic sub- stitution results in ciphertext data expansion. In homophonic substitution,\|S(k,m)\|should be proportional to the frequency ofmin the message space. The motivation is to smooth out obvious irregularities in the frequency distribution of ciphertext characters, which result from irregularities in the plaintext fre- quency distribution when simple substitution is used. While homophonic substitution complicates cryptanalysis based on simple frequency distribution statistics, sufﬁcient ciphertext may nonetheless allow frequency analysis, in conjunction with additional statistical properties of plaintext manifested in the ciphertext. For example, in long ciphertexts each element ofS(k,m)will occur roughly the same num- ber of times. Digram distributions may also provide information. (iv) Codes vs. ciphers A technical distinction is made betweenciphersand codes. Ciphers are encryption tech- niques which are applied to plaintext units (bits, characters, or blocks) independent of their semantic or linguistic meaning; the result is called ciphertext. In contrast, cryptographic codes operate on linguistic units such as words, groups of words, or phrases, and substitute (replace) these by designated words, letter groups, or number groups calledcodegroups. The key is a dictionary-likecodebooklisting plaintext units and their corresponding code- groups, indexed by the former; a corresponding codebook for decoding is reverse-indexed. §7.3 Classical ciphers and historical development 241 When there is potential ambiguity, codes in this context (vs. ciphers) may be qualiﬁed as cryptographic codebooks, to avoid confusion with error-correcting codes (EC-codes) used to detect and/or correct non-malicious errors and authentication codes (A-codes, or MACs as per Deﬁnition 9.7) which provide data origin authentication. Several factors suggest that codes may be more difﬁcult to break than ciphers: the key (codebook) is vastly larger than typical cipher keys; codes may result in data compression (cf. Fact 7.71); and statistical analysis is complicated by the large plaintext unit block size (cf. Note 7.74). Opposing this are several major disadvantages: the coding operation not being easily automated (relative to an algorithmic mapping); and identical encryption of re- peated occurrences of plaintext units implies susceptibility to known-plaintext attacks, and allows frequency analysis based on observed trafﬁc. This implies a need for frequent rekey- ing (changing the codebook), which is both more costly and inconvenient. Consequently, codes are not commonly used to secure modern telecommunications. 7.3.3 Polyalphabetic substitutions and Vigen`ere ciphers (historical) A simple substitution cipher involves a single mapping of the plaintext alphabet onto ci- phertext characters. A more complex alternative is to use different substitution mappings (calledmultiple alphabets) on various portions of the plaintext. This results in so-called polyalphabetic substitution(also introduced in Deﬁnition 1.30). In the simplest case, the different alphabets are used sequentially and then repeated, so the position of each plain- text character in the source string determines which mapping is applied to it. Under different alphabets, the same plaintext character is thus encrypted to different ciphertext characters, precluding simple frequency analysis as per mono-alphabetic substitution (§7.3.5). The simple Vigen`ere cipher is a polyalphabetic substitution cipher, introduced in Ex- ample 1.31. The deﬁnition is repeated here for convenience. 7.53 DeﬁnitionA simple Vigen`erecipher of periodt, over ans-character alphabet, involves a t-character keyk 1k2 ...kt. The mapping of plaintextm=m1m2m3 ... to ciphertext c=c1c2c3 ... is deﬁned on individual characters byci=mi+kimods, where subscript iinkiis taken modulot(the key is re-used). The simple Vigen`ere usestshift ciphers (see Example 7.48), deﬁned bytshift values ki, each specifying one ofs(mono-alphabetic) substitutions;kiis used on the characters in positioni,i+s,i+2s,.... In general, each of thetsubstitutions is different; this is referred to as usingtalphabets rather than a single substitution mapping. The shift cipher (Example 7.48) is a simple Vigen`ere with periodt=1. 7.54 Example (Beaufort variants of Vigen`ere) Compared to the simple Vigen`ere mappingci= mi+kimods, theBeaufort cipherhasci=ki−mimods, and is its own inverse. The variant Beauforthas encryption mappingci=mi−kimods. □ 7.55 Example (compound Vigen`ere) The compound Vigen`ere has encryption mappingci = mi+(k1 i +k2 i +··· +kr i)mod s, where in general the keyskj,1≤j≤r, have distinct periodstj, and the subscriptiinkj i, indicating theith character ofkj, is taken modulotj. This corresponds to the sequential application ofrsimple Vigen`eres, and is equivalent to a simple Vigen`ere of periodlcm(t1,...,t r). □ 242 Ch. 7 Block Ciphers 7.56 Example (single mixed alphabet Vigen`ere) A simple substitution mapping deﬁned by a general permutatione(not restricted to an alphabetic shift), followed by a simple Vigen`ere, is deﬁned by the mappingci=e(mi)+kimods, with inversemi=e−1(ci−ki)mod s. An alternative is a simple Vigen`ere followed by a simple substitution:ci=e(mi+kimod s), with inversemi=e−1(ci)−kimods. □ 7.57 Example (full Vigen`ere) In a simple Vigen`ere of periodt, replace the mapping deﬁned by the shift valueki(for shifting charactermi) by a general permutationeiof the alphabet. The result is the substitution mappingci=ei(mi), where the subscriptiineiis taken modulo t. The key consists oftpermutationse1,...,e t. □ 7.58 Example (running-key Vigen`ere) If the keystreamki of a simple Vigen`ere is as long as the plaintext, the cipher is called arunning-key cipher. For example, the key may be mean- ingful text from a book. □ While running-key ciphers prevent cryptanalysis by the Kasiski method (§7.3.5), if the key has redundancy, cryptanalysis exploiting statistical imbalances may nonetheless suc- ceed. For example, when encrypting plaintext English characters using a meaningful text as a running key, cryptanalysis is possible based on the observation that a signiﬁcant pro- portion of ciphertext characters results from the encryption of high-frequency running text characters with high-frequency plaintext characters. 7.59 FactA running-key cipher can be strengthened by successively enciphering plaintext un- der two or more distinct running keys. For typical English plaintext and running keys, it can be shown that iterating four such encipherments appears unbreakable. 7.60 DeﬁnitionAn auto-key cipheris a cipher wherein the plaintext itself serves as the key (typically subsequent to the use of an initial priming key). 7.61 Example (auto-key Vigen`ere) In a running-key Vigen`ere (Example 7.58) with ans-char- acter alphabet, deﬁne apriming keyk=k 1k2 ...kt. Plaintext charactersmiare encrypted asci =mi+kimodsfor1≤i≤t(simplest case:t=1). Fori>t ,ci =(mi+ mi−t)mod s. An alternative involving more keying material is to replace the simple shift by a full Vigen`ere with permutationsei,1≤i≤s, deﬁned by the keykior charactermi: for1≤i≤t,ci=eki (mi), and fori>t ,ci=emi−t (mi). □ An alternative to Example 7.61 is to auto-key a cipher using the resulting ciphertext as the key: for example, fori>t ,ci =(mi+ci−t)mod s. This, however, is far less desirable, as it provides an eavesdropping cryptanalyst the key itself. 7.62 Example (Vernam viewed as a Vigen`ere) Consider a simple Vigen`ere deﬁned byci = mi+kimods. If the keystream is truly random and independent – as long as the plain- text and never repeated (cf. Example 7.58) – this yields the unconditionally secure Vernam cipher (Deﬁnition 1.39;§6.1.1), generalized from a binary to an arbitrary alphabet.□ 7.3.4 Polyalphabetic cipher machines and rotors (historical) The Jefferson cylinderis a deceptively simple device which implements a polyalphabetic substitution cipher; conceived in the late 18th century, it had remarkable cryptographic §7.3 Classical ciphers and historical development 243 strength for its time. Polyalphabetic substitution ciphers implemented by a class of rotor- based machines were the dominant cryptographic tool in World War II. Such machines, in- cluding the Enigma machine and those of Hagelin, have an alphabet which changes con- tinuously for a very long period before repeating; this provides protection against Kasiski analysis and methods based on the index of coincidence (§7.3.5). (i) Jefferson cylinder The Jefferson cylinder(Figure 7.3) implements a polyalphabetic substitution cipher while avoiding complex machinery, extensive user computations, and Vigen`ere tableaus. A solid cylinder 6 inches long is sliced into 36 disks. A rod inserted through the cylinder axis allows the disks to rotate. The periphery of each disk is divided into 26 parts. On each disk, the letters A–Z are inscribed in a (different) random ordering. Plaintext messages are encrypted in 36-character blocks. A reference bar is placed along the cylinder’s length. Each of the 36 wheels is individually rotated to bring the appropriate character (matching the plaintext block) into position along the reference line. The 25 other parallel reference positions then each deﬁne a ciphertext, from which (in an early instance of randomized encryption) one is selected as the ciphertext to transmit. A S Q B N RCRL X S TF R F I KD L M O JE H Y P OW S Z Figure 7.3:The Jefferson cylinder. The second party possesses a cylinder with identically marked and ordered disks (1– 36). The ciphertext is decrypted by rotating each of the 36 disks to obtain characters along a ﬁxed reference line matching the ciphertext. The other 25 reference positions are exam- ined for a recognizable plaintext. If the original message is not recognizable (e.g., random data), both parties agree beforehand on an index 1 through 25 specifying the offset between plaintext and ciphertext lines. To accommodate plaintext digits 0–9 without extra disk sections, each digit is per- manently assigned to one of 10 letters (a,e,i,o,u,y and f,l,r,s) which is encrypted as above but annotated with an overhead dot, identifying that the procedure must be reversed. Re- ordering disks (1 through 36) alters the polyalphabetic substitution key. The number of pos- sible orderings is36!≈3.72×10 41. Changing the ordering of letters on each disk affords 25!further mappings (per disk), but is more difﬁcult in practice. (ii) Rotor-based machines – technical overview A simpliﬁed generic rotor machine (Figure 7.4) consists of a number ofrotors(wired code- wheels) each implementing a different ﬁxed mono-alphabetic substitution, mapping a char- acter at its input face to one on its output face. A plaintext character input to the ﬁrst rotor generates an output which is input to the second rotor, and so on, until the ﬁnal ciphertext character emerges from the last. For ﬁxed rotor positions, the bank of rotors collectively implements a mono-alphabetic substitution which is the composition of the substitutions deﬁned by the individual rotors. To provide polyalphabetic substitution, the encipherment of each plaintext character causes various rotors to move. The simplest case is an odometer-like movement, with a single rotor stepped until it completes a full revolution, at which time it steps the adjacent 244 Ch. 7 Block Ciphers A B C D E plaintext E A B C D ciphertext Figure 7.4:A rotor-based machine. rotor one position, and so on. Stepping a rotor changes the mono-alphabetic substitution it deﬁnes (theactivemapping). More precisely, each rotorRi effects a mono-alphabetic substitutionfi.Rican rotate intotipositions (e.g.,ti=26). When offsetjplaces from a reference setting,Rimaps inputatofi(a−j)+j, where both the input tofiand the ﬁnal output are reduced mod 26. The cipher key is deﬁned by the mono-alphabetic substitutions determined by the ﬁxed wheel wirings and initial rotor positions. Re-arranging the order of rotors provides addi- tional variability. Providing a machine with more rotors than necessary for operation at any one time allows further keying variation (by changing the active rotors). 7.63 FactTwo properties of rotor machines desirable for security-related reasons are: (1) long periods; and (2) state changes which are almost all “large”. The second property concerns the motion of rotors relative to each other, so that the sub-mappings between rotor faces change when the state changes. Rotor machines with odometer-like state changes fail to achieve this second property. 7.64 Note (rotor machine output methods) Rotor machines were categorized by their method of providing ciphertext output. Inindicating machines, ciphertext output characters are indi- cated by means such as lighted lamps or displayed characters in output apertures. Inprint- ing machines, ciphertext is printed or typewritten onto an output medium such as paper. With on-line machines, output characters are produced in electronic form suitable for di- rect transmission over telecommunications media. (iii) Rotor-based machines – historical notes A number of individuals are responsible for the development of early machines based on ro- tor principles. In 1918, the American E.H. Hebern built the ﬁrst rotor apparatus, based on an earlier typewriting machine modiﬁed with wired connections to generate a mono-alphabetic substitution. The output was originally by lighted indicators. The ﬁrst rotor patent was ﬁled in 1921, the yearHebern Electric Code, Inc.became the ﬁrst U.S. cipher machine company (and ﬁrst to bankrupt in 1926). The U.S. Navy (circa 1929-1930 and some years thereafter) used a number of Hebern’s ﬁve-rotor machines. In October 1919, H.A. Koch ﬁled Netherlands patent no.10,700 (“Geheimschrijfma- chine” – secret writing machine), demonstrating a deep understanding of rotor principles; no machine was built. In 1927, the patent rights were assigned to A. Scherbius. The German inventor Scherbius built a rotor machine called theEnigma. Model A was replaced by Model B with typewriter output, and a portable Model C with indicator lamps. §7.3 Classical ciphers and historical development 245 The company set up in 1923 dissolved in 1934, but thereafter the Germans used the portable battery-powered Enigma, including for critical World War II operations. In October 1919, three days after Koch, A.G. Damm ﬁled Swedish patent no.52,279 de- scribing a double-rotor device. His ﬁrm was joined by the Swede, B. Hagelin, whose 1925 modiﬁcation yielded the B-21 rotor machine (with indicating lamps) used by the Swedish army. The B-21 hadkeywheelswith varying number of teeth or gears, each of which was associated with a settable two-state pin. The period of the resulting polyalphabetic substi- tution was the product of the numbers of keywheel pins; the key was deﬁned by the state of each pin and the initial keywheel positions. Hagelin later produced other models: B-211 (a printing machine); a more compact (phone-sized) model C-36 for the French in 1934; and based on alterations suggested by Friedman and others, model C-48 (of which over140000 were produced) which was called M-209 when used by the U.S. Army as a World War II ﬁeld cipher. His 1948 Swiss factory later produced: model C-52, a strengthened version of M-209 (C-48) with period exceeding2.75×10 9 (with keywheels of 47, 43, 41, 37, 31, 29 pins); CD-55, a pocket-size version of the C-52; and T-55, an on-line version of the same, modiﬁable to use a one-time tape. A further model was CD-57. 7.65 Note (Enigma details) The Enigma initially had three rotorsR i, each with 26 positions. R1 steppedR2 which steppedR3 odometer-like, withR2 also stepping itself; the period was 26·25·26≈17000. The key consisted of the initial positions of these rotors (≈17000 choices), their order (3!=6 choices), and the state of aplugboard, which implemented a ﬁxed but easily changed (e.g., manually, every hour) mono-alphabetic substitution (26! choices), in addition to that carried out by rotor combinations. 7.66 Note (Hagelin M-209 details) The Hagelin M-209 rotor machine implements a polyalpha- betic substitution using 6 keywheels – more speciﬁcally, a self-decrypting Beaufort cipher (Example 7.54),E ki (mi)= ki−mimod26, of period101405850=26 ·25·23·21·19·17 letters. Thus for a ﬁxed ordered set of 6 keywheels, the cipher period exceeds108.kimay be viewed as theith character in the key stream, as determined by a particular ordering of keywheels, their pin settings, and starting positions. All keywheels rotate one position for- ward after each character is enciphered. The wheels simultaneously return to their initial position only after a period equal to the least-common-multiple of their gear-counts, which (since these are co-prime) is their product. A ciphertext-only attack is possible with 1000- 2000 characters, using knowledge of the machine’s internal mechanical details, and assum- ing natural language redundancy in the plaintext; a known-plaintext attack is possible with 50-100 characters. 7.3.5 Cryptanalysis of classical ciphers (historical) This section presents background material on redundancy and unicity distance, and tech- niques for cryptanalysis of classical ciphers, (i) Redundancy All natural languages are redundant. This redundancy results from linguistic structure. For example, in English the letter “E” appears far more frequently than “Z”, “Q” is almost al- ways followed by “U”, and “TH” is a common digram. An alphabet with 26 characters (e.g., Roman alphabet) can theoretically carry up to lg26=4 .7bits of information per character. Fact 7.67 indicates that, on average, far less information is actually conveyed by a natural language. 246 Ch. 7 Block Ciphers 7.67 FactThe estimated average amount of information carried per character (per-character en- tropy) in meaningful English alphabetic text is 1.5 bits. The per-character redundancy of English is thus about4.7−1.5=3 .2bits. 7.68 FactEmpirical evidence suggests that, for essentially any simple substitution cipher on a meaningful message (e.g., with redundancy comparable to English), as few as 25 ciphertext characters sufﬁces to allow a skilled cryptanalyst to recover the plaintext. (ii) Unicity distance and random cipher model 7.69 DeﬁnitionThe unicity distanceof a cipher is the minimum amount of ciphertext (number of characters) required to allow a computationally unlimited adversary to recover the unique encryption key. The unicity distance is primarily a theoretical measure, useful in relation to uncondi- tional security. A small unicity distance does not necessarily imply that a block cipher is insecure in practice. For example, consider a 64-bit block cipher with a unicity distance of two ciphertext blocks. It may still be computationally infeasible for a cryptanalyst (of reasonable but bounded computing power) to recover the key, although theoretically there is sufﬁcient information to allow this. The random cipher model (Deﬁnition 7.70) is a simpliﬁed model of a block cipher pro- viding a reasonable approximation for many purposes, facilitating results on block cipher properties not otherwise easily established (e.g., Fact 7.71). 7.70 DeﬁnitionLetCand Kbe random variables, respectively, denoting the ciphertext block and the key, and letDdenote the decryption function. Under therandom cipher model, D K(C)is a random variable uniformly distributed over all possible pre-images ofC(mean- ingful messages and otherwise, with and without redundancy). In an intuitive sense, a random cipher as per the model of Deﬁnition 7.70 is a random mapping. (A more precise approximation would be as a random permutation.) 7.71 FactUnder the random cipher model, the expected unicity distanceN0 of a cipher isN0 = H(K)/D, whereH(K)is the entropy of the key space (e.g., 64 bits for264 equiprobable keys), andDis the plaintext redundancy (in bits/character). For a one-time pad, the unbounded entropy of the key space implies, by Fact 7.71, that the unicity distance is likewise unbounded. This is consistent with the one-time pad being theoretically unbreakable. Data compression reduces redundancy. Fact 7.71 implies that data compression prior to encryption increases the unicity distance, thus increasing security. If the plaintext con- tains no redundancy whatsoever, then the unicity distance is inﬁnite; that is, the system is theoretically unbreakable under a ciphertext-only attack. 7.72 Example (unicity distance – transposition cipher) The unicity distance of a simple trans- position cipher of periodtcan be estimated under the random cipher model using Fact 7.71, and the assumption of plaintext redundancy ofD =3 .2bits/character. In this case, H(K)/D =l g (t!)/3.2and fort =1 2the estimated unicity distance is 9 characters, which is very crude, this being less than one 12-character block. Fort =2 7, the esti- mated unicity distance is a more plausible 29 characters; this can be computed using Stir- ling’s approximation of Fact 2.57(iii) (t! ≈ √ 2πt(t/e)t, for largetand e =2 .718)a s H(K)/D=lg(t!)/3.2≈(0.3t)·lg(t/e). □ §7.3 Classical ciphers and historical development 247 7.73 Example (unicity distance – simple substitution) The number of keys for a mono-alphab- etic substitution cipher over alphabetAis\|K\|=s!, wheres=\|A\|. For example,s=26 (Roman alphabet) yields26!≈4×1026 keys. Assuming equiprobable keys, an estimate of the entropy of the key space is then (cf. Example 7.72)H(K)=lg(26!) ≈88.4bits. As- suming English text withD=3.2bits of redundancy per character (Fact 7.67), a theoretical estimate of the unicity distance of a simple substitution cipher isH(K)/D=88.4/3.2≈ 28characters. This agrees closely with empirical evidence (Fact 7.68).□ (iii) Language statistics Cryptanalysis of classical ciphers typically relies on redundancy in the source language (plaintext). In many cases a divide-and-conquerapproach is possible, whereby the plaintext or key is recovered piece by piece, each facilitating further recovery. Mono-alphabetic substitution on short plaintext blocks (e.g., Roman alphabet char- acters) is easily defeated by associating ciphertext characters with plaintext characters (Note 7.50). The frequency distribution of individual ciphertext characters can be compared to that of single characters in the source language, as given by Figure 7.5 (estimated from 1964 English text). This is facilitated by grouping plaintext letters by frequency into high, medium, low, and rare classes; focussing on the high-frequency class, evidence support- ing trial letter assignments can be obtained by examining how closely hypothesized assign- ments match those of the plaintext language. Further evidence is available by examination of digram and trigram frequencies. Figure 7.6 gives the most common English digrams as a percentage of all digrams; note that of26 2 =676possible digrams, the top 15 account for 27% of all occurrences. Other examples of plaintext redundancy appearing in the cipher- text include associations of vowels with consonants, and repeated letters inpattern words (e.g., “that”, “soon”, “three”). 0 1 2 3 4 5 6 7 8 9 10 11 12 13 8.04 A 1.54 B 3.06 C 3.99 D 12.51 E 2.30 F 1.96 G 5.49 H 7.26 I 0.16 J 0.67 K 4.14 L 2.53 M 7.09 N 7.60 O 2.00 P 0.11 Q 6.12 R 6.54 S 9.25 T 2.71 U 0.99 V 1.92 W 0.19 X 1.73 Y 0.09 Z % Figure 7.5:Frequency of single characters in English text. 7.74 Note (large blocks preclude statistical analysis)A nn-bit block size implies2nplaintext units (“characters”). Compilation of frequency statistics on plaintext units thus becomes infeasible as the block size of the simple substitution increases; for example, this is clearly infeasible for DES (§7.4), wheren=64. 248 Ch. 7 Block Ciphers Cryptanalysis of simple transposition ciphers is similarly facilitated by source language statistics (see Note 7.47). Cryptanalyzing transposed blocks resembles solving an anagram. Attempts to reconstruct common digrams and trigrams are facilitated by frequency statis- tics. Solutions may be constructed piecewise, with the appearance of digrams and trigrams in trial decryptions conﬁrming (partial) success. 0 1 2 3 4 1.81 AN 1.51 AT 1.32 ED 1.53 EN 2.13 ER 1.36 ES 3.05 HE 2.30 IN 1.83 ON 1.28 OR 1.90 RE 1.22 ST 1.30 TE 3.21 TH 1.28 TI % Figure 7.6:Frequency of 15 common digrams in English text. Cryptanalysis of polyalphabetic ciphers is possible by various methods, including Ka- siski’s method and methods based on the index of coincidence, as discussed below. (iv) Method of Kasiski (vs. polyalphabetic substitution) Kasiski’s method provides a general technique for cryptanalyzing polyalphabetic ciphers with repeated keywords, such as the simple Vigen`ere cipher (Deﬁnition 7.53), based on the following observation: repeated portions of plaintext encrypted with the same portion of the keyword result in identical ciphertext segments. Consequently one expects the num- ber of characters between the beginning of repeated ciphertext segments to be a multiple of the keyword length. Ideally, it sufﬁces to compute the greatest common divisor of the var- ious distances between such repeated segments, but coincidental repeated ciphertext seg- ments may also occur. Nonetheless, an analysis (Kasiski examination) of the common fac- tors among all such distances is possible; the largest factor which occurs most commonly is the most likely keyword length. Repeated ciphertext segments of length 4 or longer are most useful, as coincidental repetitions are then less probable. The number of letters in the keyword indicates the number of alphabetstin the polyal- phabetic substitution. Ciphertext characters can then be partitioned intotsets, each of which is then the result of a mono-alphabetic substitution. Trial values fortare conﬁrmed if the frequency distribution of the (candidate) mono-alphabetic groups matches the fre- quency distribution of the plaintext language. For example, the proﬁle for plaintext English (Figure 7.5) exhibits a long trough characterizinguvwxyz, followed by a spike ata, and preceded by the triple-peak ofrst. The resulting mono-alphabetic portions can be solved in- dividually, with additional information available by combining their solution (based on di- grams, probable words, etc.). If the source language is unknown, comparing the frequency distribution of ciphertext characters to that of candidate languages may allow determination of the source language itself. (v) Index of coincidence (vs. polyalphabetic substitution) The index of coincidence(IC) is a measure of the relative frequency of letters in a cipher- text sample, which facilitates cryptanalysis of polyalphabetic ciphers by allowing determi- nation of the periodt(as an alternative to Kasiski’s method). For concreteness, consider a Vig `enere cipher and assume natural language English plaintext. §7.3 Classical ciphers and historical development 249 Let the ciphertext alphabet be{a0,a1,...,a n−1}, and letpibe the unknown probabil- ity that an arbitrarily chosen character in a random ciphertext isai. Themeasure of rough- nessmeasures the deviation of ciphertext characters from a ﬂat frequency distribution as follows: MR= n−1∑ i=0 ( pi−1 n )2 = n−1∑ i=0 pi 2 − 1 n (7.1) The minimum value isMRmin =0, corresponding to a ﬂat distribution (for equiprobable ai,pi=1/n). The maximum value occurs when the frequency distribution ofpihas great- est variability, corresponding to a mono-alphabetic substitution (the plaintext frequency dis- tribution is then manifested). Deﬁne this maximum valueMRmax =κp−1/n, whereκp corresponds to∑ pi2 when piare plaintext frequencies. For English as per Figure 7.5, the maximum value is MR=κp−1/n≈0.0658−0.0385=0 .0273. (This varies with letter frequency estimates;κp=0.0667, yieldingκp−1/n=0.0282is commonly cited, and is used in Table 7.1.) While MR cannot be computed directly from a ciphertext sample (since the periodtis unknown, the mono-alphabetic substitutions cannot be separated), it may be estimated from the frequency distribution of ciphertext characters as follows. Letf idenote the number of appearances ofaiin anL-character ciphertext sample (thus∑ fi=L). The number of pairs of letters among theseLisL(L−1)/2, of whichfi(fi− 1)/2are the pair(ai,ai)for any ﬁxed characterai. Deﬁne IC as the probability that two characters arbitrarily chosen from thegivenciphertext sample are equal: IC= ∑ n−1 i=0 (fi 2 ) (L 2 ) = ∑ n−1 i=0 fi(fi−1) L(L−1) (7.2) Independent of this given ciphertext sample, the probability that two randomly chosen ci- phertext characters are equal is∑ n−1 i=0 pi2. Thus (comparing word deﬁnitions) IC is an esti- mate of∑ pi2, and by equation (7.1), thereby an estimate of MR +1/n. Moreover, IC can be directly computed from a ciphertext sample, allowing estimation of MR itself. Since MR varies from 0 toκ p−1/n, one expects IC to range from1/n(for polyalphabetic sub- stitution with inﬁnite period) toκp(for mono-alphabetic substitution). More precisely, the following result may be established. 7.75 FactFor a polyalphabetic cipher of periodt,E(IC) as given below is the expected value of the index of coincidence for a ciphertext string of lengthL, wherenis the number of alphabet characters,κr=1/n, andκpis given in Table 7.1: E(IC)= 1 t·L−t L−1·κp + t−1 t · L L−1·κr (7.3) (pinκpis intended to denote a plaintext frequency distribution, while therinκrdenotes a distribution for random characters.) For Roman-alphabet languages,n=26 impliesκr= 0.03846; for the Russian Cyrillic alphabet,n=30. 7.76 Example (estimating polyalphabetic period using IC) Tabulating the expected values for IC for periodst=1,2,... using Equation (7.3) (which is essentially independent ofL for largeLand smallt), and comparing this to that obtained from a particular ciphertext using Equation (7.2) allows a crude estimate of the periodtof the cipher, e.g., whether it is mono-alphabetic or polyalphabetic with small period. Candidate valuestin the range thus determined may be tested for correctness by partitioning ciphertext characters into groups of letters separated bytciphertext positions, and in one or more such groups, comparing the character frequency distribution to that of plaintext. □ 250 Ch. 7 Block Ciphers Language κp French 0.0778 Spanish 0.0775 German 0.0762 Italian 0.0738 English 0.0667 Russian 0.0529 Table 7.1:Estimated roughness constantκp for various languages (see Fact 7.75). A polyalphabetic periodtmay be determined either by Example 7.76 or the alternative of Example 7.77, based on the same underlying ideas. Oncetis determined, the situation is as per after successful completion of the Kasiski method. 7.77 Example (determining period by ciphertext auto-correlation) Given a sample of polyal- phabetic ciphertext, the unknown periodtmay be determined by examining the number of coincidences when the ciphertext is auto-correlated. More speciﬁcally, given a ciphertext samplec1c2 ...cL, starting witht=1, count the total number of occurrencesci=ci+tfor 1≤i≤L−t. Repeat fort=2,3,... and tabulate the counts (or plot a bar graph). The actual periodt∗is revealed as follows: for valuestthat are a multiple oft∗, the counts will be noticeably higher (easily recognized as spikes on the bar graph). In fact, forLappro- priately large, one expects approximatelyL·κpcoincidences in this case, and signiﬁcantly fewer in other cases. □ In the auto-correlation method of coincidences of Example 7.77, the spikes on the bar graph reveal the period, independent of the source language. Once the period is determined, ciphertext characters from like alphabets can be grouped, and the proﬁle of single-character letter frequencies among these, which differs for each language, may be used to determine the plaintext language. 7.4 DES The Data Encryption Standard (DES) is the most well-known symmetric-key block cipher. Recognized world-wide, it set a precedent in the mid 1970s as the ﬁrst commercial-grade modern algorithm with openly and fully speciﬁed implementation details. It is deﬁned by the American standard FIPS 46–2. 7.4.1 Product ciphers and Feistel ciphers The design of DES is related to two general concepts: product ciphers and Feistel ciphers. Each involves iterating a common sequence or round of operations. The basic idea of a product cipher (see§1.5.3) is to build a complex encryption func- tion by composing several simple operations which offer complementary, but individually insufﬁcient, protection (note cascade ciphers per Deﬁnition 7.29 use independent keys). Ba- sic operations include transpositions, translations (e.g., XOR) and linear transformations, arithmetic operations, modular multiplication, and simple substitutions. §7.4 DES 251 7.78 DeﬁnitionA product ciphercombines two or more transformations in a manner intending that the resulting cipher is more secure than the individual components. 7.79 DeﬁnitionA substitution-permutation(SP)networkis a product cipher composed of a number of stages each involving substitutions and permutations (Figure 7.7). SSSS P ciphertext SSSS P plaintext Figure 7.7:Substitution-permutation (SP) network. Many SP networks are iterated ciphers as per Deﬁnition 7.80. 7.80 DeﬁnitionAn iterated block cipheris a block cipher involving the sequential repetition of an internal function called around function. Parameters include the number of roundsr, the block bitsizen, and the bitsizekof the input keyKfrom whichrsubkeysKi(round keys) are derived. For invertibility (allowing unique decryption), for each valueKi the round function is a bijection on the round input. 7.81 DeﬁnitionA Feistel cipheris an iterated cipher mapping a2t-bit plaintext(L0,R0), for t-bit blocksL0 and R0, to a ciphertext(Rr,Lr), through anr-round process wherer≥1. For 1≤i≤r, roundimaps (Li−1,Ri−1) Ki →(Li,Ri)as follows:Li =Ri−1,Ri = Li−1⊕f(Ri−1,Ki), where eachsubkeyKiis derived from the cipher keyK. Typically in a Feistel cipher,r≥3and often is even. The Feistel structure speciﬁcally orders the ciphertext output as(Rr,Lr)rather than(Lr,Rr); the blocks are exchanged from their usual order after the last round. Decryption is thereby achieved using the same r-round process but with subkeys used in reverse order,KrthroughK1; for example, the last round is undone by simply repeating it (see Note 7.84). Theffunction of the Feistel cipher may be a product cipher, thoughfitself need not be invertible to allow inversion of the Feistel cipher. Figure 7.9(b) illustrates that successive rounds of a Feistel cipher operate on alternat- ing halves of the ciphertext, while the other remains constant. Note the round function of Deﬁnition 7.81 may also be re-written to eliminateL i: Ri =Ri−2⊕f(Ri−1,Ki). In this case, the ﬁnal ciphertext output is(Rr,Rr−1), with input labeled(R−1,R0). 252 Ch. 7 Block Ciphers 7.4.2 DES algorithm DES is a Feistel cipher which processes plaintext blocks ofn=64 bits, producing 64-bit ciphertext blocks (Figure 7.8). The effective size of the secret keyKisk=56 bits; more precisely, the input keyKis speciﬁed as a 64-bit key, 8 bits of which (bits8,16,..., 64) may be used as parity bits. The256 keys implement (at most)256 of the264!possible bijec- tions on 64-bit blocks. A widely held belief is that the parity bits were introduced to reduce the effective key size from 64 to 56 bits, to intentionally reduce the cost of exhaustive key search by a factor of 256. 64 64 PC C 56 K key K ciphertextC plaintextP 56 K PDES DES −1 Figure 7.8:DES input-output. Full details of DES are given in Algorithm 7.82 and Figures 7.9 and 7.10. An overview follows. Encryption proceeds in 16 stages orrounds. From the input keyK, sixteen 48-bit subkeysKiare generated, one for each round. Within each round, 8 ﬁxed, carefully selected 6-to-4 bit substitution mappings (S-boxes)Si, collectively denotedS, are used. The 64-bit plaintext is divided into 32-bit halvesL0 and R0. Each round is functionally equivalent, taking 32-bit inputsLi−1 and Ri−1 from the previous round and producing 32-bit outputs Liand Rifor1≤i≤16, as follows: Li = Ri−1; (7.4) Ri = Li−1 ⊕f(Ri−1,Ki), wheref(Ri−1,Ki)= P(S(E(Ri−1)⊕Ki))(7.5) Here Eis a ﬁxed expansion permutation mappingRi−1 from 32 to 48 bits (all bits are used once; some are used twice).Pis another ﬁxed permutation on 32 bits. An initial bit per- mutation (IP) precedes the ﬁrst round; following the last round, the left and right halves are exchanged and, ﬁnally, the resulting string is bit-permuted by the inverse of IP. Decryption involves the same key and algorithm, but with subkeys applied to the internal rounds in the reverse order (Note 7.84). A simpliﬁed view is that the right half of each round (after expanding the 32-bit input to 8 characters of 6 bits each) carries out a key-dependent substitution on each of 8 charac- ters, then uses a ﬁxed bit transposition to redistribute the bits of the resulting characters to produce 32 output bits. Algorithm 7.83 speciﬁes how to compute the DES round keysK i, each of which con- tains 48 bits ofK. These operations make use of tables PC1 and PC2 of Table 7.4, which are calledpermuted choice 1and permuted choice 2. To begin, 8 bits (k8,k16,...,k 64)o f Kare discarded (by PC1). The remaining 56 bits are permuted and assigned to two 28-bit variablesCand D; and then for 16 iterations, bothCand Dare rotated either 1 or 2 bits, and 48 bits (Ki) are selected from the concatenated result. §7.4 DES 253 7.82 AlgorithmData Encryption Standard (DES) INPUT: plaintextm1 ...m64; 64-bit keyK=k1 ...k64 (includes 8 parity bits). OUTPUT: 64-bit ciphertext blockC=c1 ...c64. (For decryption, see Note 7.84.) 1. (key schedule) Compute sixteen 48-bit round keysKifromKusing Algorithm 7.83. 2. (L0,R0)←IP(m1m2 ...m64). (Use IP from Table 7.2 to permute bits; split the result into left and right 32-bit halvesL0 =m58m50 ...m8,R0 =m57m49 ...m7.) 3. (16 rounds) forifrom 1 to 16, computeLiand Riusing Equations (7.4) and (7.5) above, computingf(Ri−1,Ki)= P(S(E(Ri−1)⊕Ki))as follows: (a) ExpandRi−1 =r1r2 ...r32 from 32 to 48 bits usingEper Table 7.3: T←E(Ri−1). (ThusT=r32r1r2 ...r32r1.) (b) T′←T⊕Ki. RepresentT′as eight 6-bit character strings:(B1,...,B 8)= T′. (c)T′′←(S1(B1),S2(B2),...S 8(B8)). (HereSi(Bi)maps Bi =b1b2 ...b 6 to the 4-bit entry in rowrand columncof Si in Table 7.8, page 260 where r=2·b1 +b6, andb2b3b4b5 is the radix-2 representation of0≤c≤15. Thus S1(011011)yieldsr=1,c=13, and output 5, i.e., binary 0101.) (d) T′′′←P(T′′). (UsePper Table 7.3 to permute the 32 bits ofT′′=t1t2 ...t32, yieldingt16t7 ...t25.) 4. b1b2 ...b 64 ←(R16,L16). (Exchange ﬁnal blocksL16,R16.) 5. C←IP−1(b1b2 ...b 64). (Transpose using IP−1 from Table 7.2;C=b40b8 ...b 25.) IP 58 50 42 34 26 18 10 2 60 52 44 36 28 20 12 4 62 54 46 38 30 22 14 6 64 56 48 40 32 24 16 8 57 49 41 33 25 17 9 1 59 51 43 35 27 19 11 3 61 53 45 37 29 21 13 5 63 55 47 39 31 23 15 7 IP−1 40 8 48 16 56 24 64 32 39 7 47 15 55 23 63 31 38 6 46 14 54 22 62 30 37 5 45 13 53 21 61 29 36 4 44 12 52 20 60 28 35 3 43 11 51 19 59 27 34 2 42 10 50 18 58 26 33 1 41 9 49 17 57 25 Table 7.2:DES initial permutation and inverse (IP and IP−1). E 32 1 2 3 4 5 4 5 6 7 8 9 8 9 10 11 12 13 12 13 14 15 16 17 16 17 18 19 20 21 20 21 22 23 24 25 24 25 26 27 28 29 28 29 30 31 32 1 P 16 7 20 21 29 12 28 17 1 15 23 26 5 18 31 10 2 8 24 14 32 27 3 9 19 13 30 6 22 11 4 25 Table 7.3:DES per-round functions: expansionEand permutationP. 254 Ch. 7 Block Ciphers R0 L1 R1 L15 R15 IP−1 initial permutation output c1c2 ··· c64 64 64 irregular swap K16 R16 K2 48 32 32 inverse permutation (a) twisted ladder input 64 K1 m1m2 ··· m64 64 IP L0 32 f f L16 f K1 f K2 f K3 f K4 f K16 IP−1 input L0 L16 R16 output R2 f R16 L15 R0 L1 L3 L16 IP R0 L0 R1 L2 R3 R15 Li = Ri−1 Ri = Li−1 ⊕f(Ri−1,Ki) (b) untwisted ladder Figure 7.9:DES computation path. §7.4 DES 255 S1 S2 S3 S4 S5 S6 S7 S8 permutation f(Ri−1,Ki)= P(S(E(Ri−1) ⊕Ki)) 4 Ri−1 Ki 8 ×4 bits 8 ×6 bits substitution P 32 32 expansion 32 48 48 48 6 E Figure 7.10:DES inner functionf. 7.83 AlgorithmDES key schedule INPUT: 64-bit keyK=k1 ...k64 (including 8 odd-parity bits). OUTPUT: sixteen 48-bit keysKi,1≤i≤16. 1. Deﬁne vi,1≤i≤16as follows:vi =1 fori∈{1,2,9,16};vi =2 otherwise. (These are left-shift values for 28-bit circular rotations below.) 2. T←PC1(K); representTas 28-bit halves (C0,D0). (Use PC1 in Table 7.4 to select bits fromK:C0 =k57k49 ...k36,D0 =k63k55 ...k4.) 3. Forifrom 1 to 16, computeKias follows:Ci ←(Ci−1 ←↩vi),Di ←(Di−1 ←↩ vi),Ki←PC2(Ci,Di). (Use PC2 in Table 7.4 to select 48 bits from the concatena- tionb1b2 ...b 56 ofCiand Di:Ki=b14b17 ...b 32.‘←↩’ denotes left circular shift.) If decryption is designed as a simple variation of the encryption function, savings result in hardware or software code size. DES achieves this as outlined in Note 7.84. 7.84 Note (DES decryption) DES decryption consists of the encryption algorithm with the same key but reversed key schedule, using in orderK16,K15,...,K 1 (see Note 7.85). This works as follows (refer to Figure 7.9). The effect ofIP−1 is cancelled by IP in decryp- tion, leaving(R16,L16); consider applying round 1 to this input. The operation on the left half yields, rather thanL0⊕f(R0,K1), nowR16⊕f(L16,K16)which, sinceL16 =R15 and R16 =L15⊕f(R15,K16), is equal toL15⊕f(R15,K16)⊕f(R15,K16)= L15. Thus round 1 decryption yields(R15,L15), i.e., inverting round 16. Note that the cancellation 256 Ch. 7 Block Ciphers PC1 57 49 41 33 25 17 9 1 58 50 42 34 26 18 10 2 59 51 43 35 27 19 11 3 60 52 44 36 above forCi; below forDi 63 55 47 39 31 23 15 7 62 54 46 38 30 22 14 6 61 53 45 37 29 21 13 5 28 20 12 4 PC2 14 17 11 24 1 5 3 28 15 6 21 10 23 19 12 4 26 8 16 7 27 20 13 2 41 52 31 37 47 55 30 40 51 45 33 48 44 49 39 56 34 53 46 42 50 36 29 32 Table 7.4:DES key schedule bit selections (PC1 and PC2). of each round is independent of the deﬁnition offand the speciﬁc value ofKi; the swap- ping of halves combined with the XOR process is inverted by the second application. The remaining 15 rounds are likewise cancelled one by one in reverse order of application, due to the reversed key schedule. 7.85 Note (DES decryption key schedule) SubkeysK 1,...,K 16 may be generated by Algo- rithm 7.83 and used in reverse order, or generated in reverse order directly as follows. Note that afterK 16 is generated, the original values of the 28-bit registersCand Dare restored (each has rotated 28 bits). Consequently, and due to the choice of shift-values, modifying Algorithm 7.83 as follows generates subkeys in orderK16,...,K 1: replace the left-shifts by right-shift rotates; change the shift valuev1 to0. 7.86 Example (DES test vectors) The plaintext “Now is the time for all ”, represented as a string of 8-bit hex characters (7-bit ASCII characters plus leading 0-bit), and encrypted us- ing the DES key speciﬁed by the hex stringK =0123456789ABCDEFresults in the following plaintext/ciphertext: P= 4E6F772069732074 68652074696D6520 666F7220616C6C20 C= 3FA40E8A984D4815 6A271787AB8883F9 893D51EC4B563B53. □ 7.4.3 DES properties and strength There are many desirable characteristics for block ciphers. These include: each bit of the ciphertext should depend on all bits of the key and all bits of the plaintext; there should be no statistical relationship evident between plaintext and ciphertext; altering any single plain- text or key bit should alter each ciphertext bit with probability 1 2 ; and altering a ciphertext bit should result in an unpredictable change to the recovered plaintext block. Empirically, DES satisﬁes these basic objectives. Some known properties and anomalies of DES are given below. (i) Complementation property 7.87 FactLetEdenote DES, and xthe bitwise complement ofx. Theny=EK(x)implies y=E K( x). That is, bitwise complementing both the keyKand the plaintextxresults in complemented DES ciphertext. Justiﬁcation: Compare the ﬁrst round output (see Figure 7.10) to(L 0,R0)for the uncom- plemented case. The combined effect of the plaintext and key being complemented results §7.4 DES 257 in the inputs to the XOR preceding the S-boxes (the expandedRi−1 and subkeyKi) both being complemented; this double complementation cancels out in the XOR operation, re- sulting in S-box inputs, and thus an overall resultf(R0,K1), as before. This quantity is then XORed (Figure 7.9) to L0 (previouslyL0), resulting in L1 (rather thanL1). The same effect follows in the remaining rounds. The complementation property is normally of no help to a cryptanalyst in known-plain- text exhaustive key search. If an adversary has, for a ﬁxed unknown keyK, a chosen- plaintext set of(x,y)data(P1,C1),( P1,C2), thenC2 =EK( P1)implies C2 =E K(P1). Checking if the keyKwith plaintextP1 yields eitherC1 or C2 now rules out two keys with one encryption operation, thus reducing the expected number of keys required before success from255 to254. This is not a practical concern. (ii) Weak keys, semi-weak keys, and ﬁxed points If subkeysK1 toK16 are equal, then the reversed and original schedules create identical subkeys:K1 =K16,K2 =K15, and so on. Consequently, the encryption and decryption functions coincide. These are called weak keys (and also:palindromic keys). 7.88 DeﬁnitionA DES weak keyis a keyKsuch thatEK(EK(x))= xfor allx, i.e., deﬁning an involution. A pair of DESsemi-weakkeys is a pair(K1,K2)withEK1 (EK2 (x))= x. Encryption with one key of a semi-weak pair operates as does decryption with the other. 7.89 FactDES has four weak keys and six pairs of semi-weak keys. The four DES weak keys are listed in Table 7.5, along with corresponding 28-bit vari- ablesC0 and D0 of Algorithm 7.83; here{0}jrepresentsjrepetitions of bit0. SinceC0 and D0 are all-zero or all-one bit vectors, and rotation of these has no effect, it follows that all subkeysKiare equal and an involution results as noted above. The six pairs of DES semi-weak keys are listed in Table 7.6. Note their deﬁning prop- erty (Deﬁnition 7.88) occurs when subkeysK1 throughK16 of the ﬁrst key, respectively, equal subkeysK16 throughK1 of the second. This requires that a 1-bit circular left-shift of each ofC0 and D0 for the ﬁrst 56-bit key results in the(C0,D0)pair for the second 56-bit key (see Note 7.84), and thereafter left-rotatingCiand Dione or two bits for the ﬁrst re- sults in the same value as right-rotating those for the second the same number of positions. The values in Table 7.6 satisfy these conditions. Given any one 64-bit semi-weak key, its paired semi-weak key may be obtained by splitting it into two halves and rotating each half through 8 bits. 7.90 FactLetEdenote DES. For each of the four DES weak keysK, there exist2 32 ﬁxed points ofEK, i.e., plaintextsxsuch thatEK(x)= x. Similarly, four of the twelve semi-weak keys Keach have232 anti-ﬁxed points, i.e.,xsuch thatEK(x)= x. The four semi-weak keys of Fact 7.90 are in the upper portion of Table 7.6. These are calledanti-palindromic keys, since for theseK1 = K16,K2 = K15, and so on. (iii) DES is not a group For a ﬁxed DES keyK, DES deﬁnes a permutation from{0,1}64 to{0,1}64. The set of DES keys deﬁnes256 such (potentially different) permutations. If this set of permutations was closed under composition (i.e., given any two keysK1,K2, there exists a third keyK3 such thatEK3 (x)= EK2 (EK1 (x))for allx) then multiple encryption would be equivalent to single encryption. Fact 7.91 states that this is not the case for DES. 258 Ch. 7 Block Ciphers weak key (hexadecimal) C0 D0 0101 0101 0101 0101 {0}28 {0}28 FEFE FEFE FEFE FEFE {1}28 {1}28 1F1F 1F1F 0E0E 0E0E {0}28 {1}28 E0E0 E0E0 F1F1 F1F1 {1}28 {0}28 Table 7.5:Four DES weak keys. C0 D0 semi-weak key pair (hexadecimal) C0 D0 {01}14 {01}14 01FE 01FE 01FE 01FE, FE01 FE01 FE01 FE01 {10}14 {10}14 {01}14 {10}14 1FE0 1FE0 0EF1 0EF1, E01F E01F F10E F10E {10}14 {01}14 {01}14 {0}28 01E0 01E0 01F1 01F1, E001 E001 F101 F101 {10}14 {0}28 {01}14 {1}28 1FFE 1FFE 0EFE 0EFE, FE1F FE1F FE0E FE0E {10}14 {1}28 {0}28 {01}14 011F 011F 010E 010E, 1F01 1F01 0E01 0E01 {0}28 {10}14 {1}28 {01}14 E0FE E0FE F1FE F1FE, FEE0 FEE0 FEF1 FEF1 {1}28 {10}14 Table 7.6:Six pairs of DES semi-weak keys (one pair per line). 7.91 FactThe set of256 permutations deﬁned by the256 DES keys is not closed under func- tional composition. Moreover, a lower bound on the size of the group generated by com- posing this set of permutations is102499. The lower bound in Fact 7.91 is important with respect to using DES for multiple en- cryption. If the group generated by functional composition was too small, then multiple encryption would be less secure than otherwise believed. (iv) Linear and differential cryptanalysis of DES Assuming that obtaining enormous numbers of known-plaintext pairs is feasible, linear cryptanalysis provides the most powerful attack on DES to date; it is not, however, con- sidered a threat to DES in practical environments. Linear cryptanalysis is also possible in a ciphertext-only environment if some underlying plaintext redundancy is known (e.g., parity bits or high-order 0-bits in ASCII characters). Differential cryptanalysis is one of the most general cryptanalytic tools to date against modern iterated block ciphers, including DES, Lucifer, and FEAL among many others. It is, however, primarily a chosen-plaintext attack. Further information on linear and differential cryptanalysis is given in§7.8. 7.92 Note (strength of DES) The complexity (see§7.2.1) of the best attacks currently known against DES is given in Table 7.7; percentages indicate success rate for speciﬁed attack pa- rameters. The ‘processing complexity’ column provides only an estimate of the expected cost (operation costs differ across the various attacks); for exhaustive search, the cost is in DES operations. Regarding storage complexity, both linear and differential cryptanalysis require only negligible storage in the sense that known or chosen texts can be processed individually and discarded, but in a practical attack, storage for accumulated texts would be required if ciphertext was acquired prior to commencing the attack. §7.5 FEAL 259 attack method data complexity storage processing known chosen complexity complexity exhaustive precomputation — 1 256 1 (table lookup) exhaustive search 1 — negligible 255 linear cryptanalysis 243 (85%) — for texts 243 238 (10%) — for texts 250 differential cryptanalysis — 247 for texts 247 255 — for texts 255 Table 7.7:DES strength against various attacks. 7.93 Remark (practicality of attack models) To be meaningful, attack comparisons based on different models (e.g., Table 7.7) must appropriately weigh the feasibility of extracting (ac- quiring) enormous amounts of chosen (known) plaintexts, which is considerably more dif- ﬁcult to arrange than a comparable number of computing cycles on an adversary’s own ma- chine. Exhaustive search with one known plaintext-ciphertext pair (for ciphertext-only, see Example 7.28) and2 55 DES operations is signiﬁcantly more feasible in practice (e.g., using highly parallelized custom hardware) than linear cryptanalysis (LC) requiring243 known pairs. While exhaustive search, linear, and differential cryptanalysis allow recovery of a DES key and, therefore, the entire plaintext, the attacks of Note 7.8, which become feasible once about2 32 ciphertexts are available, may be more efﬁcient if the goal is to recover only part of the text. 7.5 FEAL The Fast Data Encipherment Algorithm (FEAL) is a family of algorithms which has played a critical role in the development and reﬁnement of various advanced cryptanalytic tech- niques, including linear and differential cryptanalysis. FEAL-N maps 64-bit plaintext to 64-bit ciphertext blocks under a 64-bit secret key. It is anN-round Feistel cipher similar to DES (cf. Equations (7.4), (7.5)), but with a far simplerf-function, and augmented by initial and ﬁnal stages which XOR the two data halves as well as XOR subkeys directly onto the data halves. FEAL was designed for speed and simplicity, especially for software on 8-bit micro- processors (e.g., chipcards). It uses byte-oriented operations (8-bit addition mod 256, 2-bit left rotation, and XOR), avoids bit-permutations and table look-ups, and offers small code size. The initial commercially proposed version with 4 rounds (FEAL-4), positioned as a fast alternative to DES, was found to be considerably less secure than expected (see Ta- ble 7.10). FEAL-8 was similarly found to offer less security than planned. FEAL-16 or FEAL-32 may yet offer security comparable to DES, but throughput decreases as the num- ber of rounds rises. Moreover, whereas the speed of DES implementations can be improved through very large lookup tables, this appears more difﬁcult for FEAL. Algorithm 7.94 speciﬁes FEAL-8. Thef-functionf(A,Y)maps an input pair of32× 16bits to a 32-bit output. Within theffunction, two byte-oriented data substitutions (S- boxes)S 0 and S1 are each used twice; each maps a pair of 8-bit inputs to an 8-bit output 260 Ch. 7 Block Ciphers row column number [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] S1 [0] 14 4 13 1 2 15 11 8 3 10 6 12 5 9 0 7 [1] 0 15 7 4 14 2 13 1 10 6 12 11 9 5 3 8 [2] 4 1 14 8 13 6 2 11 15 12 9 7 3 10 5 0 [3] 15 12 8 2 4 9 1 7 5 11 3 14 10 0 6 13 S2 [0] 15 1 8 14 6 11 3 4 9 7 2 13 12 0 5 10 [1] 3 13 4 7 15 2 8 14 12 0 1 10 6 9 11 5 [2] 0 14 7 11 10 4 13 1 5 8 12 6 9 3 2 15 [3] 13 8 10 1 3 15 4 2 11 6 7 12 0 5 14 9 S3 [0] 10 0 9 14 6 3 15 5 1 13 12 7 11 4 2 8 [1] 13 7 0 9 3 4 6 10 2 8 5 14 12 11 15 1 [2] 13 6 4 9 8 15 3 0 11 1 2 12 5 10 14 7 [3] 1 10 13 0 6 9 8 7 4 15 14 3 11 5 2 12 S4 [0] 7 13 14 3 0 6 9 10 1 2 8 5 11 12 4 15 [1] 13 8 11 5 6 15 0 3 4 7 2 12 1 10 14 9 [2] 10 6 9 0 12 11 7 13 15 1 3 14 5 2 8 4 [3] 3 15 0 6 10 1 13 8 9 4 5 11 12 7 2 14 S5 [0] 2 12 4 1 7 10 11 6 8 5 3 15 13 0 14 9 [1] 14 11 2 12 4 7 13 1 5 0 15 10 3 9 8 6 [2] 4 2 1 11 10 13 7 8 15 9 12 5 6 3 0 14 [3] 11 8 12 7 1 14 2 13 6 15 0 9 10 4 5 3 S6 [0] 12 1 10 15 9 2 6 8 0 13 3 4 14 7 5 11 [1] 10 15 4 2 7 12 9 5 6 1 13 14 0 11 3 8 [2] 9 14 15 5 2 8 12 3 7 0 4 10 1 13 11 6 [3] 4 3 2 12 9 5 15 10 11 14 1 7 6 0 8 13 S7 [0] 4 11 2 14 15 0 8 13 3 12 9 7 5 10 6 1 [1] 13 0 11 7 4 9 1 10 14 3 5 12 2 15 8 6 [2] 1 4 11 13 12 3 7 14 10 15 6 8 0 5 9 2 [3] 6 11 13 8 1 4 10 7 9 5 0 15 14 2 3 12 S8 [0] 13 2 8 4 6 15 11 1 10 9 3 14 5 0 12 7 [1] 1 15 13 8 10 3 7 4 12 5 6 11 0 14 9 2 [2] 7 11 4 1 9 12 14 2 0 6 10 13 15 3 5 8 [3] 2 1 14 7 4 10 8 13 15 12 9 0 3 5 6 11 Table 7.8:DES S-boxes. §7.5 FEAL 261 (see Table 7.9).S0 and S1 add a single bitd∈{0,1}to 8-bit argumentsxand y, ignore the carry out of the top bit, and left rotate the result 2 bits (ROT2): Sd(x,y)= ROT2(x+y+dmod256) (7.6) The key schedule uses a functionfK(A,B)similar to thef-function (see Table 7.9;Ai, Bi,Yi,ti, andUiare 8-bit variables), mapping two 32-bit inputs to a 32-bit output. U←f(A,Y) U←fK(A,B) t1 = (A0⊕A1)⊕Y0 A0⊕A1 t2 = (A2⊕A3)⊕Y1 A2⊕A3 U1 = S1(t1,t2) S1(t1,t2⊕B0) U2 = S0(t2,U1) S0(t2,U1⊕B1) U0 = S0(A0,U1) S0(A0,U1⊕B2) U3 = S1(A3,U2) S1(A3,U2⊕B3) Table 7.9:OutputU=( U0,U1,U2,U3) for FEAL functionsf,fK (Algorithm 7.94). As the operations of 2-bit rotation and XOR are both linear, the only nonlinear elemen- tary operation in FEAL is addition mod 256. 7.94 AlgorithmFast Data Encipherment Algorithm (FEAL-8) INPUT: 64-bit plaintextM=m1 ...m64; 64-bit keyK=k1 ...k64. OUTPUT: 64-bit ciphertext blockC=c1 ...c64. (For decryption, see Note 7.96.) 1. (key schedule) Compute sixteen 16-bit subkeysKifrom Kusing Algorithm 7.95. 2. Deﬁne ML=m1 ···m32,MR=m33 ···m64. 3. (L0,R0)←(ML,MR)⊕((K8,K9),(K10,K11)). (XOR initial subkeys.) 4. R0 ←R0⊕L0. 5. Forifrom 1 to 8 do:Li←Ri−1,Ri←Li−1⊕f(Ri−1,Ki−1). (Use Table 7.9 for f(A,Y)withA=Ri−1 =(A0,A1,A2,A3)and Y=Ki−1 =(Y0,Y1).) 6. L8 ←L8⊕R8. 7. (R8,L8)←(R8,L8)⊕((K12,K13),(K14,K15)). (XOR ﬁnal subkeys.) 8. C←(R8,L8). (Note the order of the ﬁnal blocks is exchanged.) 7.95 AlgorithmFEAL-8 key schedule INPUT: 64-bit keyK=k1 ...k64. OUTPUT: 256-bit extended key (16-bit subkeysKi,0≤i≤15). 1. (initialize)U(−2) ←0, U(−1) ←k1 ...k32, U(0) ←k33 ...k64. 2. U def =(U0,U1,U2,U3)for 8-bitUi. ComputeK0,...,K 15 asiruns from 1 to 8: (a)U←fK(U(i−2),U(i−1)⊕U(i−3)).(fK is deﬁned in Table 7.9, whereAand Bdenote 4-byte vectors(A0,A1,A2,A3),(B0,B1,B2,B3).) (b) K2i−2 =(U0,U1), K2i−1 =(U2,U3), U(i) ←U. 7.96 Note (FEAL decryption) Decryption may be achieved using Algorithm 7.94 with the same key Kand ciphertextC=(R8,L8)as the plaintext inputM, but with the key schedule reversed. More speciﬁcally, subkeys((K12,K13),(K14,K15))are used for the initial XOR (step 3),((K8,K9),(K10,K11))for the ﬁnal XOR (step 7), and the round keys are used from K7 back toK0 (step 5). This is directly analogous to decryption for DES (Note 7.84). 262 Ch. 7 Block Ciphers 7.97 Note (FEAL-N ) FEAL with 64-bit key can be generalized toN-rounds,Neven.N=2x is recommended;x=3yields FEAL-8 (Algorithm 7.94). FEAL-N usesN+8sixteen-bit subkeys:K0,...,K N−1, respectively, in roundi; KN,...,K N+3 for the initial XOR; and KN+4,...K N+7 for the ﬁnal XOR. The key schedule of Algorithm 7.95 is directly generalized to compute keysK0 throughKN+7 asiruns from1to(N/2)+4. 7.98 Note (FEAL-NX ) Extending FEAL-N to use a 128-bit key results in FEAL-NX, with al- tered key schedule as follows. The key is split into 64-bit halves(KL,KR). KRis parti- tioned into 32-bit halves(KR1,KR2). For1≤i≤(N/2)+4, deﬁneQi =KR1⊕KR2 fori ≡ 1mod3 ; Qi = KR1 fori ≡ 2mod3 ; andQi = KR2 fori ≡ 0mod3 . The second argument (U(i−1)⊕U(i−3))t ofK in step 2a of Algorithm 7.95 is replaced by U(i−1)⊕U(i−3)⊕Qi. ForKR =0, FEAL-NX matches FEAL-N with KL as the 64-bit FEAL-N key K. 7.99 Example (FEAL test vectors) For hex plaintextM=00000000 00000000and hex key K =01234567 89ABCDEF, Algorithm 7.95 generates subkeys(K0,...,K 7)= DF3BCA36 F17C1AEC 45A5B9C7 26EBAD25, (K8,...,K 15)= 8B2AECB7 AC509D4C22CD479BA8D50CB5. Algorithm 7.94 generates FEAL-8 ciphertextC= CEEF2C86F2490752. For FEAL-16, the corresponding ciphertext isC′=3ADE0D2A D84D0B6F; for FEAL-32,C′′=69B0FAE6 DDED6B0B. For 128-bit key(KL,KR) with KL = KR = Kas above,Mhas corresponding FEAL-8X ciphertextC′′′= 92BEB65D0E9382FB. □ 7.100 Note (strength of FEAL) Table 7.10 gives various published attacks on FEAL; LC and DC denote linear and differential cryptanalysis, and times are on common personal computers or workstations. attack data complexity storage processing method known chosen complexity complexity FEAL-4 – LC 5 — 30K bytes 6 minutes FEAL-6 – LC 100 — 100K bytes 40 minutes FEAL-8 – LC 224 10 minutes FEAL-8 – DC 27 pairs 280K bytes 2 minutes FEAL-16 – DC — 229 pairs 230 operations FEAL-24 – DC — 245 pairs 246 operations FEAL-32 – DC — 266 pairs 267 operations Table 7.10:FEAL strength against various attacks. §7.6 IDEA 263 7.6 IDEA The cipher named IDEA (International Data Encryption Algorithm) encrypts 64-bit plain- text to 64-bit ciphertext blocks, using a 128-bit input keyK. Based in part on a novel generalization of the Feistel structure, it consists of 8 computationally identical rounds fol- lowed by an output transformation (see Figure 7.11). Roundruses six 16-bit subkeysK(r) i , 1≤i≤6, to transform a 64-bit inputXinto an output of four 16-bit blocks, which are in- put to the next round. The round 8 output enters the output transformation, employing four additional subkeysK (9) i ,1≤i≤4to produce the ﬁnal ciphertextY =(Y1,Y2,Y3,Y4). All subkeys are derived fromK. A dominant design concept in IDEA is mixing operations from three different alge- braic groups of2nelements. The corresponding group operations on sub-blocksaand bof bitlengthn=16are bitwise XOR:a⊕b; addition mod2n:(a+b)AND 0xFFFF, denoted a⊞b; and (modiﬁed) multiplication mod2n+1, with0∈Z2n associated with2n∈Z2n+1: a⊙b(see Note 7.104). bitwise XOR addition mod216 multiplication mod216 +1 (with0 interpreted as216) 16 K(1) 5 X1 X2 K(1) 3K(1) 2 round 1 output transformation 16 16 16 K(1) 4 16 16 16 Y1 Y2 Y3 Y4 16 X3 X4 K(9) 2 K(9) 3 K(9) 4 (2 ≤r≤8) roundr K(1) 6 MA-box t2 t0 t1 16161616 K(1) 1 K(9) 1 ciphertext(Y1,Y2,Y3,Y4) subkeysK(r) i for roundr plaintext(X1,X2,X3,X4) Figure 7.11:IDEA computation path. 264 Ch. 7 Block Ciphers 7.101 AlgorithmIDEA encryption INPUT: 64-bit plaintextM=m1 ...m64; 128-bit keyK=k1 ...k128. OUTPUT: 64-bit ciphertext blockY=(Y1,Y2,Y3,Y4). (For decryption, see Note 7.103.) 1. (key schedule) Compute 16-bit subkeysK(r) 1 ,...,K (r) 6 for rounds1≤r≤8, and K(9) 1 ,...,K (9) 4 for the output transformation, using Algorithm 7.102. 2. (X1,X2,X3,X4)←(m1 ...m16,m17 ...m32,m33 ...m48,m49 ...m64), where Xiis a 16-bit data store. 3. For roundrfrom 1 to 8 do: (a)X1 ←X1⊙K(r) 1 ,X4 ←X4⊙K(r) 4 ,X2 ←X2 ⊞ K(r) 2 ,X3 ←X3 ⊞ K(r) 3 . (b) t0 ←K(r) 5 ⊙(X1⊕X3),t1 ←K(r) 6 ⊙(t0 ⊞ (X2⊕X4)),t2 ←t0 ⊞ t1. (c)X1 ←X1⊕t1,X4 ←X4⊕t2,a←X2⊕t2,X2 ←X3⊕t1,X3 ←a. 4. (output transformation)Y1 ←X1⊙K(9) 1 ,Y4 ←X4⊙K(9) 4 ,Y2 ←X3 ⊞K(9) 2 ,Y3 ← X2 ⊞ K(9) 3 . 7.102 AlgorithmIDEA key schedule (encryption) INPUT: 128-bit keyK=k1 ...k128. OUTPUT: 52 16-bit key sub-blocksK(r) i for 8 roundsrand the output transformation. 1. Order the subkeysK(1) 1 ...K (1) 6 ,K(2) 1 ...K (2) 6 ,...,K (8) 1 ...K (8) 6 ,K(9) 1 ...K (9) 4 . 2. PartitionKinto eight 16-bit blocks; assign these directly to the ﬁrst 8 subkeys. 3. Do the following until all 52 subkeys are assigned: cyclic shiftKleft 25 bits; parti- tion the result into 8 blocks; assign these blocks to the next 8 subkeys. The key schedule of Algorithm 7.102 may be converted into a table which lists, for each of the 52 keys blocks, which 16 (consecutive) bits of the input keyKform it. 7.103 Note (IDEA decryption) Decryption is achieved using Algorithm 7.101 with the cipher- textY provided as inputM, and the same encryption keyK, but the following change to the key schedule. First useKto derive all encryption subkeysK(r) i ; from these com- pute the decryption subkeysK′(r) i per Table 7.11; then useK′(r) i in place ofK(r) i in Algo- rithm 7.101. In Table 7.11,−Kidenotes the additive inverse (mod216)o fKi: the integer u=(216 −Ki)AND 0xFFFF,0≤u≤216 −1.K−1 i denotes the multiplicative inverse (mod 216 +1)o fKi, also in{0,1,..., 216 −1}, derivable by the Extended Euclidean al- gorithm (Algorithm 2.107), which on inputsa≥b≥0returns integersxand ysuch that ax+by=gcd(a,b). Usinga=216 +1 and b=Ki, the gcd is always1(except for Ki=0, addressed separately) and thusK−1 i =y,o r216 +1+ yify< 0. When Ki=0, this input is mapped to216 (since the inverse is deﬁned byKi⊙K−1 i =1; see Note 7.104) and (216)−1 =216 is then deﬁned to giveK−1 i =0. 7.104 Note (deﬁnition of⊙) In IDEA,a⊙bcorresponds to a (modiﬁed) multiplication, modulo 216+1, of unsigned 16-bit integersaandb, where0∈Z216 is associated with216 ∈Z∗ 216+1 as follows:2 ifa=0 orb=0, replace it by216 (which is≡−1mod2 16 +1) prior to modular multiplication; and if the result is216, replace this by0. Thus,⊙maps two 16- bit inputs to a 16-bit output. Pseudo-code for⊙is as follows (cf. Note 7.105, for ordinary 2Thus the operands of⊙ are from a set of cardinality216 (Z∗ 216+1) as are those of⊕ and ⊞. §7.6 IDEA 265 roundr K′(r) 1 K′(r) 2 K′(r) 3 K′(r) 4 K′(r) 5 K′(r) 6 r=1 (K(10−r) 1 )−1 −K(10−r) 2 −K(10−r) 3 (K(10−r) 4 )−1 K(9−r) 5 K(9−r) 6 2≤r≤8 (K(10−r) 1 )−1 −K(10−r) 3 −K(10−r) 2 (K(10−r) 4 )−1 K(9−r) 5 K(9−r) 6 r=9 (K(10−r) 1 )−1 −K(10−r) 2 −K(10−r) 3 (K(10−r) 4 )−1 — — Table 7.11:IDEA decryption subkeysK′(r) i derived from encryption subkeysK(r) i . multiplication mod216 +1), forca 32-bit unsigned integer: if(a=0) r←(0x10001 −b) (since216b≡−b), elseif(b=0) r←(0x10001−a) (by similar reasoning), else {c←ab;r←((cAND 0xFFFF) −(c>> 16)); if(r< 0)r←(0x10001+r)}, with return value(rAND 0xFFFF) in all 3 cases. 7.105 Note (implementingabmod2n+1) Multiplication mod216 +1may be efﬁciently imple- mented as follows, for0≤a,b≤216 (cf.§14.3.4). Letc=ab=c0 ·232 +cH·216 +cL, where c0 ∈{0,1}and 0≤cL,cH <216. To computec′=cmod(216 +1), ﬁrst obtain cLand cH by standard multiplication. Fora=b=216, note thatc0 =1,cL =cH =0, and c′=(−1)(−1)=1 , since216 ≡−1mod (216 +1); otherwise,c0 =0. Consequently, c′=cL−cH+c0 ifcL ≥cH, whilec′=cL−cH +(216 +1) ifcL <cH (since then −216 <cL−cH <0). 7.106 Example (IDEA test vectors) Sample data for IDEA encryption of 64-bit plaintextMus- ing 128-bit keyKis given in Table 7.12. All entries are 16-bit values displayed in hexadeci- mal. Table 7.13 details the corresponding decryption of the resulting 64-bit ciphertextC under the same keyK. □ 128-bit keyK=( 1,2,3,4,5,6,7,8) 64-bit plaintextM=( 0,1,2,3) r K(r) 1 K(r) 2 K(r) 3 K(r) 4 K(r) 5 K(r) 6 X1 X2 X3 X4 1 0001 0002 0003 0004 0005 0006 00f0 00f5 010a 0105 2 0007 0008 0400 0600 0800 0a00 222f 21b5 f45e e959 3 0c00 0e00 1000 0200 0010 0014 0f86 39be 8ee8 1173 4 0018 001c 0020 0004 0008 000c 57df ac58 c65b ba4d 5 2800 3000 3800 4000 0800 1000 8e81 ba9c f77f 3a4a 6 1800 2000 0070 0080 0010 0020 6942 9409 e21b 1c64 7 0030 0040 0050 0060 0000 2000 99d0 c7f6 5331 620e 8 4000 6000 8000 a000 c000 e001 0a24 0098 ec6b 4925 9 0080 00c0 0100 0140 — — 11fb ed2b 0198 6de5 Table 7.12:IDEA encryption sample: round subkeys and ciphertext(X1,X2,X3,X4). 7.107 Note (security of IDEA) For the full 8-round IDEA, other than attacks on weak keys (see page 279), no published attack is better than exhaustive search on the 128-bit key space. The security of IDEA currently appears bounded only by the weaknesses arising from the relatively small (compared to its keylength) blocklength of 64 bits. 266 Ch. 7 Block Ciphers K=( 1,2,3,4,5,6,7,8) C= (11fb,ed2b,0198,6de5) r K′(r) 1 K′(r) 2 K′(r) 3 K′(r) 4 K′(r) 5 K′(r) 6 X1 X2 X3 X4 1 fe01 ff40 ff00 659a c000 e001 d98d d331 27f6 82b8 2 fffd 8000 a000 cccc 0000 2000 bc4d e26b 9449 a576 3 a556 ffb0 ffc0 52ab 0010 0020 0aa4 f7ef da9c 24e3 4 554b ff90 e000 fe01 0800 1000 ca46 fe5b dc58 116d 5 332d c800 d000 fffd 0008 000c 748f 8f08 39da 45cc 6 4aab ffe0 ffe4 c001 0010 0014 3266 045e 2fb5 b02e 7 aa96 f000 f200 ff81 0800 0a00 0690 050a 00fd 1dfa 8 4925 fc00 fff8 552b 0005 0006 0000 0005 0003 000c 9 0001 fffe fffd c001 — — 0000 0001 0002 0003 Table 7.13:IDEA decryption sample: round subkeys and variables(X1,X2,X3,X4). 7.7 SAFER, RC5, and other block ciphers 7.7.1 SAFER SAFER K-64 (Secure And Fast Encryption Routine, with 64-bit key) is an iterated block cipher with 64-bit plaintext and ciphertext blocks. It consists ofridentical rounds followed by an output transformation. The original recommendation of 6 rounds was followed by a recommendation to adopt a slightly modiﬁed key schedule (yielding SAFER SK-64, which should be used rather than SAFER K-64 – see Note 7.110) and to use 8 rounds (maximum r=10). Both key schedules expand the 64-bit external key into2r+1subkeys each of 64- bits (two for each round plus one for the output transformation). SAFER consists entirely of simple byte operations, aside from byte-rotations in the key schedule; it is thus suitable for processors with small word size such as chipcards (cf. FEAL). Details of SAFER K-64 are given in Algorithm 7.108 and Figure 7.12 (see also page 280 regarding SAFER K-128 and SAFER SK-128). The XOR-addition stage beginning each round (identical to the output transformation) XORs bytes 1, 4, 5, and 8 of the (ﬁrst) round subkey with the respective round input bytes, and respectively adds (mod 256) the re- maining 4 subkey bytes to the others. The XOR and addition (mod 256) operations are inter- changed in the subsequent addition-XOR stage. The S-boxes are an invertible byte-to-byte substitution using one ﬁxed 8-bit bijection (see Note 7.111). A linear transformationf(the Pseudo-Hadamard Transform) used in the 3-level linear layer was specially constructed for rapid diffusion. The introduction of additive key biases in the key schedule eliminates weak keys (cf. DES, IDEA). In contrast to Feistel-like and many other ciphers, in SAFER the op- erations used for encryption differ from those for decryption (see Note 7.113). SAFER may be viewed as an SP network (Deﬁnition 7.79). Algorithm 7.108 uses the following deﬁnitions (L,Rdenote left, right 8-bit inputs): 1. f(L,R)=(2 L+R,L +R). Addition here is mod256(also denoted by⊞); 2. tablesSand S inv, and the constant table forkey biasesBi[j]as per Note 7.111. §7.7 SAFER, RC5, and other block ciphers 267 ffff ffff ffff Y1 Y2 Y3 Y4 transformation output (2 ≤i≤r) roundi Y5 Y6 Y8Y7 round1 X1 X2 X3 X4 X6 X7 X8 8 8 64-bit plaintext 64-bit ciphertext X5 SS −1 SS S −1 S−1 S K2i−1[1,...,8] K2i[1,...,8] K2r+1[1,...,8] 8 8 8 64 64 bitwise XOR addition mod28 f(x,y)=( 2x⊞ y,x⊞ y) K1[1,...,8] K2[1,...,8] S−1 Figure 7.12:SAFER K-64 computation path (rrounds). 268 Ch. 7 Block Ciphers 7.108 AlgorithmSAFER K-64 encryption (rrounds) INPUT: r, 6≤r≤10; 64-bit plaintextM=m1 ···m64 and keyK=k1 ···k64. OUTPUT: 64-bit ciphertext blockY=(Y1,...,Y 8). (For decryption, see Note 7.113.) 1. Compute 64-bit subkeysK1,...,K 2r+1 by Algorithm 7.109 with inputsKand r. 2. (X1,X2,...,X 8)←(m1 ···m8,m9 ···m16,...,m 57 ···m64). 3. Forifrom 1 tordo: (XOR-addition, S-box, addition-XOR, and 3 linear layers) (a) Forj=1,4,5,8:Xj ←Xj ⊕K2i−1[j]. For j=2,3,6,7:Xj ←Xj⊞ K2i−1[j]. (b) Forj=1,4,5,8:Xj ←S[Xj]. Forj=2,3,6,7:Xj ←Sinv[Xj]. (c) Forj=1,4,5,8:Xj ←Xj⊞K2i[j]. Forj=2,3,6,7:Xj ←Xj ⊕K2i[j]. (d) Forj=1,3,5,7:(Xj,Xj+1)←f(Xj,Xj+1). (e)(Y1,Y2)←f(X1,X3), (Y3,Y4)←f(X5,X7), (Y5,Y6)←f(X2,X4), (Y7,Y8)←f(X6,X8). For jfrom 1 to 8 do:Xj ←Yj. (f)(Y1,Y2)←f(X1,X3), (Y3,Y4)←f(X5,X7), (Y5,Y6)←f(X2,X4), (Y7,Y8)←f(X6,X8). For jfrom 1 to 8 do:Xj ←Yj. (This mimics the previous step.) 4. (output transformation): For j=1,4,5,8:Yj ←Xj ⊕K2r+1[j]. Forj=2,3,6,7:Yj ←Xj⊞ K2r+1[j]. 7.109 AlgorithmSAFER K-64 key schedule INPUT: 64-bit keyK=k1 ···k64; number of roundsr. OUTPUT: 64-bit subkeysK1,...,K 2r+1.Ki[j]is bytejofKi(numbered left to right). 1. LetR[i]denote an 8-bit data store and letBi[j]denote bytejofBi(Note 7.111). 2. (R[1],R[2],...,R [8])←(k1 ···k8,k9 ···k16,...,k 57 ···k64). 3. (K1[1],K1[2],...,K 1[8])←(R[1],R[2],...,R [8]). 4. Forifrom 2to2r+1do: (rotate key bytes left 3 bits, then add in the bias) (a) Forjfrom 1 to 8 do:R[j]←(R[j]←↩3). (b) Forjfrom 1 to 8 do:Ki[j]←R[j]⊞ Bi[j]. (See Note 7.110.) 7.110 Note (SAFER SK-64 – strengthened key schedule) An improved key schedule for Algo- rithm 7.108, resulting in SAFER SK-64, involves three changes as follows. (i) After ini- tializing theR[i]in step 1 of Algorithm 7.109, setR[9]←R[1]⊕R[2]⊕···⊕ R[8]. (ii) Change the upper bound on the loop index in step 4a from 8 to 9. (iii) Replace the iterated line in step 4b by:Ki[j]←R[((i+j−2)mod9)+1] ⊞Bi[j]. Thus, key bytes1,..., 8 ofR[·]are used forK1; bytes2,..., 9forK2; bytes3,... 9,1forK3, etc. Here and origi- nally,⊞denotes addition mod 256. No attack against SAFER SK-64 better than exhaustive key search is known. 7.111 Note (S-boxes and key biases in SAFER) The S-box, inverse S-box, and key biases for Al- gorithm 7.108 are constant tables as follows.g←45.S[0]←1,Sinv[1]←0. forifrom 1 to 255 do:t←g·S[i−1]mod257 ,S[i]←t,Sinv[t]←i. Finally,S[128]←0, Sinv[0]←128. (SinceggeneratesZ∗ 257,S[i]is a bijection on{0,1,..., 255}. (Note that g128 ≡256(mod257) , and associating 256 with 0 makesSa mapping with 8-bit input and output.) Theadditive key biasesare 8-bit constants used in the key schedule (Algo- rithm 7.109), intended to behave as random numbers, and deﬁnedBi[j]= S[S[9i+j]]fori from2to2r+1andjfrom 1 to 8. For example:B2 =(22,115,59,30,142,112,189,134) and B13 =(143,41,221,4,128,222,231,49). §7.7 SAFER, RC5, and other block ciphers 269 7.112 Remark (S-box mapping) The S-box of Note 7.111 is based on the functionS(x)= gx mod257using a primitive elementg=45 ∈Z257. This mapping is nonlinear with respect to bothZ257 arithmetic and the vector space of 8-tuples overF2 under the XOR operation. The inverse S-box is based on the base-glogarithm function. 7.113 Note (SAFER K-64 decryption) For decryption of Algorithm 7.108, the same keyKand subkeysKi are used as for encryption. Each encryption step is undone in reverse order, from last to ﬁrst. Begin with an input transformation (XOR-subtraction stage) with key K2r+1 to undo the output transformation, replacing modular addition with subtraction. Fol- low withrdecryption rounds using keysK2rthroughK1 (two-per-round), inverting each round in turn. Each starts with a 3-stage inverse linear layer usingfinv(L,R)=( L− R, 2R−L), with subtraction here mod256, in a 3-step sequence deﬁned as follows (to invert the byte-permutations between encryption stages): Level 1 (forj=1,3,5,7):(Xj,Xj+1)←finv(Xj,Xj+1). Levels 2 and 3 (each):(Y1,Y2)←finv(X1,X5),(Y3,Y4)←finv(X2,X6), (Y5,Y6)←finv(X3,X7),(Y7,Y8)←finv(X4,X8); forjfrom 1 to 8 do:Xj ←Yj. A subtraction-XOR stage follows (replace modular addition with subtraction), then an in- verse substitution stage (exchangeSand S−1), and an XOR-subtraction stage. 7.114 Example (SAFER test vectors) Using 6-round SAFER K-64 (Algorithm 7.108) on the 64- bit plaintextM =(1,2,3,4,5,6,7,8)with the keyK =(8,7,6,5,4,3,2,1)results in the ciphertextC=(200,242,156,221,135,120,62,217), written as 8 bytes in decimal. Using 6-round SAFER SK-64 (Note 7.110) on the plaintextMabove with the keyK= (1,2,3,4,5,6,7,8)results in the ciphertextC=(95,206,155,162,5,132,56,199). □ 7.7.2 RC5 The RC5 block cipher has a word-oriented architecture for variable word sizesw=16,32, or64bits. It has an extremely compact description, and is suitable for hardware or software. The number of roundsrand the key byte-lengthbare also variable. It is successively more completely identiﬁed as RC5–w, RC5–w/r, and RC5–w/r/b. RC5-32/12/16 is considered a common choice of parameters;r=12rounds are recommended for RC5–32, andr=16 for RC5–64. Algorithm 7.115 speciﬁes RC5. Plaintext and ciphertext are blocks of bitlength2w. Each ofrrounds updates bothw-bit data halves, using 2 subkeys in an input transformation and 2 more for each round. The only operations used, all onw-bit words, are addition mod 2w(⊞), XOR (⊕), and rotations (left←↩and right↪→). The XOR operation is linear, while the addition may be considered nonlinear depending on the metric for linearity. The data- dependent rotations featured in RC5 are the main nonlinear operation used:x←↩ydenotes cyclically shifting aw-bit word leftybits; the rotation-countymay be reduced modw(the low-orderlg(w)bits ofysufﬁce). The key schedule expands a key ofbbytes into2r+2 subkeysK iofwbits each. Regarding packing/unpacking bytes into words, the byte-order islittle-endian: forw =3 2, the ﬁrst plaintext byte goes in the low-order end ofA, the fourth inA’s high-order end, the ﬁfth inB’s low order end, and so on. 270 Ch. 7 Block Ciphers 7.115 AlgorithmRC5 encryption (w-bit wordsize,rrounds,b-byte key) INPUT: 2w-bit plaintextM=(A,B);r; keyK=K[0]...K [b−1]. OUTPUT: 2w-bit ciphertextC. (For decryption, see Note 7.117.) 1. Compute 2r+2subkeysK0,...,K 2r+1 by Algorithm 7.116 from inputsKand r. 2. A←A⊞ K0, B←B⊞ K1. (Use addition modulo2w.) 3. Forifrom 1 tordo: A←((A⊕B)←↩B)⊞K2i, B←((B⊕A)←↩A)⊞K2i+1. 4. The output isC←(A,B). 7.116 AlgorithmRC5 key schedule INPUT: word bitsizew; number of roundsr;b-byte keyK[0]...K [b−1]. OUTPUT: subkeys K0,...,K 2r+1 (whereKiiswbits). 1. Letu=w/8(number of bytes per word) andc=⌈b/u⌉(number of wordsKﬁlls). Pad Kon the right with zero-bytes if necessary to achieve a byte-count divisible by u(i.e.,K[j]←0forb≤j≤c·u−1). Forifrom 0 toc−1do:Li←∑ u−1 j=0 28j K[i·u+j](i.e., ﬁllLilow-order to high-order byte using each byte ofK[·]once). 2. K0 ←Pw; forifrom 1 to2r+1do:Ki←Ki−1 ⊞ Qw. (Use Table 7.14.) 3. i←0,j←0,A←0,B←0,t←max(c,2r+2). Forsfrom 1 to3tdo: (a)Ki←(Ki⊞ A⊞ B)←↩3, A←Ki, i←i+1mod(2 r+2). (b) Lj ←(Lj⊞ A⊞ B)←↩(A⊞ B), B←Lj, j←j+1mod c. 4. The output isK0,K1,...,K 2r+1. (TheLiare not used.) 7.117 Note (RC5 decryption) Decryption uses the Algorithm 7.115 subkeys, operating on ci- phertextC=(A,B)as follows (subtraction is mod2w, denoted⊟). Forifrom rdown to1do: B←((B⊟ K2i+1)↪→A)⊕A,A←((A⊟ K2i)↪→B)⊕B. FinallyM ← (A⊟ K0,B⊟ K1). w: 16 32 64 Pw : B7E1 B7E15163 B7E15162 8AED2A6B Qw : 9E37 9E3779B9 9E3779B9 7F4A7C15 Table 7.14:RC5 magic constants (given as hex strings). 7.118 Example (RC5–32/12/16 test vectors) For the hexadecimal plaintextM =65C178B2 84D197CCand keyK=5269F149 D41BA015 2497574D 7F153125, RC5 with w=32,r=12, andb=16 generates ciphertextC=EB44E415 DA319824. □ 7.7.3 Other block ciphers LOKI’91 (and earlier, LOKI’89) was proposed as a DES alternative with a larger 64-bit key, a matching 64-bit blocksize, and 16 rounds. It differs from DES mainly in key-scheduling and thef-function. Thef-function of each round uses four identical 12-to-8 bit S-boxes, §7.8 Notes and further references 271 4 input bits of which select one of 16 functions, each of which implements exponentia- tion with a ﬁxed exponent in a different representation of GF(28). While no signiﬁcant ex- ploitable weaknesses have been found in LOKI’91 when used for encryption, related-key attacks (see page 281) are viewed as a certiﬁcational weakness. Khufu and Khafre are DES-like ciphers which were proposed as fast software-oriented alternatives to DES. They have 64-bit blocks,8×32bit S-boxes, and a variable number of rounds (typically 16, 24, or 32). Khufu keys may be up to 512 bits. Khafre keys have bitlength that is a multiple of 64 (64 and 128-bit keys are typical); 64 key bits are XORed onto the data block before the ﬁrst and thereafter following every 8 rounds. Whereas a DES round involves eight 6-to-4 bit S-boxes, one round of Khufu involves a single 8-to-32 bit table look-up, with a different S-box for every 8 rounds. The S-boxes are generated pseu- dorandomly from the user key. Khafre uses ﬁxed S-boxes generated pseudorandomly from an initial S-box constructed from random numbers published by the RAND corporation in 1955. Under the best currently known attacks, 16-round Khufu and 24-round Khafre are each more difﬁcult to break than DES. 7.8 Notes and further references §7.1 The extensive and particularly readable survey by Difﬁe and Hellman [347], providing a broad introduction to cryptography especially noteworthy for its treatment of Hagelin and rotor machines and the valuable annotated bibliography circa 1979, is a source for much of the material in§7.2,§7.3, and§7.4 herein. Aside from the appearance of DES [396] in the mid 1970s and FEAL [884] later in the 1980s, prior to 1990 few fully-speciﬁed seri- ous symmetric block cipher proposals were widely available or discussed. (See Chapter 15 for Pohlig and Hellman’s 1978 discrete exponentiation cipher.) With the increasing feasi- bility of exhaustive search on 56-bit DES keys, the period 1990-1995 resulted in a large number of proposals, beginning with PES [728], the preliminary version of IDEA [730]. The Fast Software Encryptionworkshops (Cambridge, U.K., Dec. 1993; Leuven, Belgium, Dec. 1994; and again Cambridge, Feb. 1996) were a major stimulus and forum for new pro- posals. The most signiﬁcant cryptanalytic advances over the 1990-1995 period were Matsui’s linear cryptanalysis [796, 795], and the differential cryptanalysis of Biham and Shamir [138] (see also [134, 139]). Extensions of these included the differential-linear analysis by Langford and Hellman [741], and the truncated differential analysis of Knudsen [686]. For additional background on linear cryptanalysis, see Biham [132]; see also Matsui and Yamagishi [798] for a preliminary version of the method. Additional background on differential cryptanal- ysis is provided by many authors including Lai [726], Lai, Massey, and Murphy [730], and Coppersmith [271]; although more efﬁcient 6-round attacks are known, Stinson [1178] pro- vides detailed examples of attacks on 3-round and 6-round DES. Regarding both linear and differential cryptanalysis, see also Knudsen [684] and Kaliski and Yin [656]. §7.2 Lai [726, Chapter 2] provides an excellent concise introduction to block ciphers, including a lucid discussion of design principles (recommendedfor all block cipher designers). Regard- ing text dictionary and matching ciphertext attacks (Note 7.8), see Coppersmith, Johnson, and Matyas [278]. Rivest and Sherman [1061] provide a uniﬁed framework for random- ized encryption (Deﬁnition 7.3); a common example is the use of random “salt” appended 272 Ch. 7 Block Ciphers to passwords prior to password encryption in some operating systems (§10.2.3). Fact 7.9 is due to Shannon [1121], whose contributions are many (see below). The four basic modes of operation (includingk-bit OFB feedback) were originally deﬁned speciﬁcally for DES in 1980 by FIPS 81 [398] and in 1983 by ANSI X3.106 [34], while ISO 8732 [578] and ISO/IEC 10116 [604], respectively, deﬁned these modes for general 64-bit and generaln-bit block ciphers, mandatingn-bit OFB feedback (see also Chapter 15). Bras- sard [192] gives a concise summary of modes of operation; Davies and Price [308] provide a comprehensive discussion, including OFB cycling (Note 7.24; see also Jueneman [643] and Davies and Parkin [307]), and a method for encrypting incomplete CBC ﬁnal blocks with- out data expansion, which is important if plaintext must be encrypted and returned into its original store. See V oydock and Kent [1225] for additional requirements onIVs. Recom- mending r=sfor maximum strength, ISO/IEC 10116 [604] speciﬁes the CFB variation of Example 7.19, and provides extensive discussion of properties of the various modes. The counter mode (Example 7.23) was suggested by Difﬁe and Hellman [347]. The 1977 exhaustive DES key search machine (Example 7.27) proposed by Difﬁe and Hell- man [346] contained10 6 DES chips, with estimated cost US$20 million (1977 technology) and 12-hour expected search time; Difﬁe later revised the estimate upwards one order of magnitude in a BNR Inc. report (US$50 million machine, 2-day expected search time, 1980 technology). Difﬁe and Hellman noted the feasibility of a ciphertext-only attack (Exam- ple 7.28), and that attempting to preclude exhaustive search by changing DES keys more frequently, at best, doubles the expected search time before success. Subsequently Wiener [1241] provided a gate-level design for a US$1 million machine (1993 technology) using57600DES chips with expected success in 3.5 hours. Each chip con- tains 16 pipelined stages, each stage completing in one clock tick at 50 MHz; a chip with full pipeline completes a key test every 20 nanoseconds, providing a machine57600×50 times faster than the 1142 years noted in FIPS 74 [397] as the time required to check2 55 keys if one key can be tested each microsecond. Comparable key search machines of equiv- alent cost by Eberle [362] and Wayner [1231] are, respectively, 55 and 200 times slower, although the former does not require a chip design, and the latter uses a general-purpose machine. Wiener also noted adaptations of the ECB known-plaintext attack to other 64-bit modes (CBC, OFB, CFB) and 1-bit and 8-bit CFB. Even and Goldreich [376] discuss the unicity distance of cascade ciphers under known- plaintext attack (Fact 7.35), present a generalized time-memory meet-in-the-middle trade- off (Note 7.38), and give several other concise results on cascades, including that under reasonable assumptions, the number of permutations realizable by a cascade ofLrandom cipher stages is, with high probability,2 Lk. Difﬁe and Hellman [346] noted the meet-in-the-middle attack on double encryption (Fact 7.33), motivating their recommendation that multiple encipherment, if used, should be at least three-fold; Hoffman [558] credits them with suggesting E-E-E triple encryption with three independent keys. Merkle’s June 1979 thesis [850] explains the attack on two-key triple-encryption of Fact 7.39 (see also Merkle and Hellman [858]), and after noting Tuch- man’s proposal of two-key E-D-E triple encryption in a June 1978 conference talk (National Computer Conference, Anaheim, CA; see also [1199]), recommended that E-D-E be used with three independent keys:E K3(E−1 K2(EK1(x))). The two-key E-D-E idea, adopted in ANSI X9.17 [37] and ISO 8732 [578], was reportedly conceived circa April 1977 by Tuch- man’s colleagues, Matyas and Meyer. The attack of Fact 7.40 is due to van Oorschot and Wiener [1206]. See Coppersmith, Johnson, and Matyas [278] for a proposed construction for a triple-DES algorithm. Other techniques intended to extend the strength of DES in- §7.8 Notes and further references 273 clude theDESX proposal of Rivest as analyzed by Kilian and Rogaway [672], and the work of Biham and Biryukov [133]. Hellman [549] proposes a time-memory tradeoff for exhaustive key search on a cipher with N=2mciphertexts requiring a chosen-plaintext attack,O(N2/3)time andO(N2/3)space after anO(N)precomputation; search time can be reduced somewhat by use of Rivest’s suggestion of distinguished points (see Denning [326, p.100]). Kusuda and Matsumoto [722] recently extended this analysis. Fiat and Naor [393] pursue time-memory tradeoffs for more general functions. Amirazizi and Hellman [25] note that time-memory tradeoff with constant time-memory product offers no asymptotic cost advantage over exhaustive search; they examine tradeoffs between time, memory, and parallel processing, and using standard parallelization techniques, propose under a simpliﬁed model a search machine ar- chitecture for which doubling the machine budget (cost) increases the solution rate four- fold. This approach may be applied to exhaustive key search on double-encryption, as can the parallel collision search technique of van Oorschot and Wiener [1207, 1208]; see also Quisquater and Delescaille [1017, 1018]. Regarding Note 7.41, see Biham [131] (and earlier [130]) as well as Coppersmith, John- son, and Matyas [278]. Biham’s analysis on DES and FEAL shows that, in many cases, the use of intermediate data as feedback into an intermediate stage reduces security. 15 years earlier, reﬂecting on his chosen-plaintext attack on two-key triple-encryption, Merkle [850, p.149] noted “multiple encryption with any cryptographic system is liable to be much less secure than a system designed originally for the longer key”. Maurer and Massey [822] formalize Fact 7.42, where “break” means recovering plaintext from ciphertext (under a known-plaintext attack) or recovering the key; the results hold also for chosen-plaintext and chosen-ciphertext attack. They illustrate, however, that the ear- lier result and commonly-held belief proven by Even and Goldreich [376] – that a cascade is as strong as any of its component ciphers – requires the important qualifying (and non- practical) assumption that an adversary will not exploit statistics of the underlying plaintext; thus, the intuitive result is untrue for most practical ciphertext-only attacks. §7.3 Kahn [648] is the deﬁnitive historical reference for classical ciphers and machines up to 1967, including much of§7.3 and the notes below. The selection of classical ciphers pre- sented largely follows Shannon’s lucid 1949 paper [1121]. Standard references for classical cryptanalysis include Friedman [423], Gaines [436], and Sinkov [1152]. More recent books providing expository material on classical ciphers, machines, and cryptanalytic examples include Beker and Piper [84], Meyer and Matyas [859], Denning [326], and Davies and Price [308]. Polyalphabetic ciphers were invented circa 1467 by the Florentine architect Alberti, who devised a cipher disk with a larger outer and smaller inner wheel, respectively indexed by plaintext and ciphertext characters. Letter alignments deﬁned a simple substitution, modi- ﬁed by rotating the disk after enciphering a few words. The ﬁrst printed book on cryptogra- phy,Polygraphia, written in 1508 by the German monk Trithemius and published in 1518, contains the ﬁrsttableau– a square table on 24 characters listing all shift substitutions for a ﬁxed ordering of plaintext alphabet characters. Tableau rows were used sequentially to sub- stitute one plaintext character each for 24 letters, where-after the same tableau or one based on a different alphabet ordering was used. In 1553 Belaso (from Lombardy) suggested us- ing an easily changed key (and key-phrases as memory aids) to deﬁne the ﬁxed alphabetic (shift) substitutions in a polyalphabetic substitution. The 1563 book of Porta (from Naples) noted the ordering of tableau letters may deﬁne arbitrary substitutions (vs. simply shifted 274 Ch. 7 Block Ciphers alphabets). Various polyalphabetic auto-key ciphers, wherein the key changes with each message (the alteration depending on the message), were explored in the 16th century, most signiﬁcantly by the Frenchman B. de Vigen`ere. His 1586 bookTraict´e des Chiffresproposed the com- bined use of a mixed tableau (mixed alphabet on both the tableau top and side) and an auto- keying technique (cf. Example 7.61). A single character served as a priming key to select the tableau row for the ﬁrst character substitution, where-after theith plaintext character determined the alphabet (tableau row) for substituting the next. The far less secure simple Vigen`ere cipher (Deﬁnition 7.53) is incorrectly attributed to Vigen`ere. The Playfair cipher (Example 7.51), popularized by L. Playfair in England circa 1854 and invented by the British scientist C. Wheatstone, was used as a British ﬁeld cipher [648, p.6]. J. Mauborgne (see also the Vernam and PURPLE ciphers below) is credited in 1914 with the ﬁrst known solution of this digram cipher. The Jefferson cylinder was designed by American statesman T. Jefferson, circa 1790-1800. In 1817, fellow American D. Wadsworth introduced the principle of plaintext and cipher- text alphabets of different lengths. His disk (cf. Alberti above) implemented a cipher similar to Trithemius’ polyalphabetic substitution, but wherein the various alphabets were brought into play irregularly in a plaintext-dependent manner, foreshadowing both the polyalpha- betic ciphers of later 20th century rotor machines, and the concept of chaining. The inner disk had 26 letters while the outer had an additional 7 digits; one full revolution of the larger caused the smaller to advance 7 characters into its second revolution. The driving disk was always turned in the same clockwise sense; when the character revealed through an aperture in the plaintext disk matched the next plaintext character, that visible through a correspond- ing ciphertext aperture indicated the resulting ciphertext. In 1867, Wheatstone displayed an independently devised similar device thereafter called theWheatstone disc, receiving greater attention although less secure (having disks of respectively 26 and 27 characters, the extra character a plaintext space). Vernam [1222] recorded his idea for telegraph encryption in 1917; a patent ﬁled in Septem- ber 1918 was issued July 1919. Vernam’s device combined a stream of plaintext (5-bit Bau- dot coded) characters, via XOR, with a keystream of 5-bit (key) values, resulting in theVer- nam cipher(a term often used for related techniques). This, the ﬁrst polyalphabetic substi- tution automated using electrical impulses, had period equal to the length of the key stream; each 5-bit key value determined one of 32 ﬁxed mono-alphabetic substitutions. Credit for the actualone-time systemgoes to J. Mauborgne (U.S. Army) who, after seeing Vernam’s device with a repeated tape, realized that use of a random, non-repeated key improved se- curity. While Vernam’s device was a commercial failure, a related German system engi- neered by W. Kunze, R. Schaufﬂer, and E. Langlotz was put into practice circa 1921-1923 for German diplomatic communications; their encryption system, which involved manu- ally adding a key string to decimal-coded plaintext, was secured by using as the numerical key a random non-repeating decimal digit stream – the originalone-time pad. Pads of 50 numbered sheets were used, each with 48 ﬁve-digit groups; no pads were repeated aside for one identical pad for a communicating partner, and no sheet was to be used twice; sheets were destroyed once used. The Vernam cipher proper, when used as a one-time system, in- volves only 32 alphabets, but provides more security than rotor machines with a far greater number of alphabets because the latter eventually repeat, whereas there is total randomness (for each plaintext character) in selecting among the 32 Vernam alphabets. The matrix cipher of Example 7.52 was proposed in 1929 by Hill [557], providing a practi- cal method for polygraphic substitution, albeit a linear transformation susceptible to known- §7.8 Notes and further references 275 plaintext attack. Hill also recognized that using an involution as the encryption mapping al- lowed the same function to provide decryption. Recent contributions on homophonic sub- stitution include G¨unther [529] and Jendal, Kuhn, and Massey [636]. Among the unrivalled cryptanalytic contributions of the Russian-born American Friedman is his 1920 Riverbank Publication no.22 [426] on cryptanalysis using the index of coinci- dence. Friedman coined the termcryptanalysisin 1920, using it in his 1923 bookElements of Cryptanalysis[425], a 1944 expansion of which,Military Cryptanalysis[423], remains highly recommended. The method of Kasiski (from West Prussia) was originally published in 1863; see Kahn [648, pp.208-213] for a detailed example. The discussion on IC and MR follows that of Denning [326], itself based on Sinkov [1152]. Fact 7.75 follows from a stan- dard expectation computation weighted byκ porκrdepending on whether the second of a pair of randomly selected ciphertext characters is from the same ciphertext alphabet or one of thet−1remaining alphabets. The values in Table 7.1 are from Kahn [648], and vary somewhat over time as languages evolve. Friedman teaches how to cryptanalyze running-key ciphers in his (circa 1918) Riverbank Publication no.16,Methods for the Solution of Running-Key Ciphers; the two basic tech- niques are outlined by Difﬁe and Hellman [347]. The ﬁrst is aprobable wordattack wherein an attacker guesses an (e.g., 10 character) word hopefully present in underlying text, and subtracts that word (mod 26) from all possible starting locations in the ciphertext in hopes of ﬁnding a recognizable 10-character result, where-after the guessed word (as either par- tial running-key or plaintext) might be extended using context. Probable-word attacks also apply to polyalphabetic substitution. The second technique is based on the fact that each ciphertext lettercresults from a pair of plaintext/running-key letters(m i,m′ i), and is most likely to result from such pairs wherein bothmiand m′ iare high-frequency characters; one isolates the highest-probability pairs for each such ciphertext character valuec, makes trial assumptions, and attempts to extend apparently successful guesses by similarly decrypting adjacent ciphertext characters; see Denning [326, p.83] for a partial example. Difﬁe and Hellman [347] note Fact 7.59 as an obvious method that is little-used (modern ciphers be- ing more convenient); their suggestion that use of four iterative running keys is unbreakable follows from English being 75% redundant. They also brieﬂy summarize variousscram- blingtechniques (encryption via analog rather than digital methods), noting that analog scramblers are sometimes used in practice due to lower bandwidth and cost requirements, although such known techniques appear relatively insecure (possibly an inherent character- istic) and their use is waning as digital networks become prevalent. Denning [326] tabulates digrams into high, medium, low, and rare classes. Konheim [705, p.24] provides transition probabilitiesp(t\|s), the probability that the next letter istgiven that the current character issin English text, in a table also presented by H. van Tilborg [1210]. Single-letter distributions in plaintext languages other than English are given by Davies and Price [308]. The letter frequencies in Figure 7.5, which should be interpreted only as an estimate, were derived by Meyer and Matyas [859] using excerpts totaling 4 mil- lion characters from the 1964 publication: W. Francis,A Standard Sample of Present-Day Edited American English for Use with Digital Computers, Linguistics Dept., Brown Uni- versity, Providence, Rhode Island, USA. Figure 7.6 is based on data from Konheim [705, p.19] giving an estimated probability distribution of 2-grams in English, derived from a sample of size67320 digrams. See Shannon [1122] and Cover and King [285] regarding redundancy and Fact 7.67. While not proven in any concrete manner, Fact 7.68 is noted by Friedman [424] and generally accepted. Unicity distance was deﬁned by Shannon [1121]. Related issues are discussed in detail in various appendices of Meyer and Matyas [859]. Fact 7.71 and the random cipher 276 Ch. 7 Block Ciphers model are due to Shannon [1121]; see also Hellman [548]. Difﬁe and Hellman [347] give an instructive overview of rotor machines (see also Denning [326]), and note their use in World War II by the Americans in their highest level system, the British, and the Germans (Enigma); they also give Fact 7.63 and the number of characters required under ciphertext-only and known-plaintext attacks (Note 7.66). Beker and Piper [84] provide technical details of the Hagelin M-209, as does Kahn [648, pp.427-431] who notes its remarkable compactness and weight: 3.25 x 5.5 x 7 inches and 6 lb. (including case); see also Barker [74], Morris [906], and Rivest [1053]. Davies and Price [308] brieﬂy discuss the Enigma, noting it was cryptanalyzed during World War II in Poland, France, and then in the U.K. (Bletchley Park); see also Konheim [705]. The Japanese PURPLE cipher, used during World War II, was a polyalphabetic cipher crypt- analyzed August 1940 [648, p.18-23] by Friedman’s team in the U.S. Signal Intelligence Service, under (Chief Signal Ofﬁcer) Mauborgne. The earlier RED cipher used two rotor arrays; preceding it, the ORANGE system implemented a vowels-to-vowels, consonants- to-consonants cipher using sets of rotors. §7.4 The concept offractionation, related to product ciphers, is noted by Feistel [387], Shannon [1121], and Kahn [648, p.344] who identiﬁes this idea in an early product cipher, the WWI German ADFGVX ﬁeld cipher. As an example, an encryption function might operate on a block oft=8 plaintext characters in three stages as follows: the ﬁrst substitutes two symbols for each individual character; the second transposes (mixes) the substituted sym- bols among themselves; the third re-groups adjacent resulting symbols and maps them back to the plaintext alphabet. The action of the transposition on partial (rather than complete) characters contributes to the strength of the principle. Shannon [1121,§5 and§23-26] explored the idea of the product of two ciphers, noted the principles of confusion and diffusion (Remark 1.36), and introduced the idea of amixing transformationF(suggesting a preliminary transposition followed by a sequence of alter- nating substitution and simple linear operations), and combining ciphers in a product using an intervening transformationF. Transposition and substitution, respectively, rest on the principles of diffusion and confusion. Harpes, Kramer, and Massey [541] discuss a general model for iterated block ciphers (cf. Deﬁnition 7.80). The name Luciferis associated with two very different algorithms. The ﬁrst is an SP net- work described by Feistel [387], which employs (bitwise nonlinear)4×4invertible S- boxes; the second, closely related to DES (albeit signiﬁcantly weaker), is described by Smith [1160] (see also Sorkin [1165]). Principles related to both are discussed by Feis- tel, Notz, and Smith [388]; both are analyzed by Biham and Shamir [138], and the latter in greater detail by Ben-Aroya and Biham [108] whose extension of differential cryptanaly- sis allows, using2 36 chosen plaintexts and complexity, attack on 55% of the key space in Smith’s Lucifer – still infeasible in practice, but illustrating inferiority to DES despite the longer 128-bit key. Feistel’s product cipher Lucifer [387], instantiated by a blocksizen=128, consists of an unspeciﬁed number of alternating substitution and permutation (transposition) stages, using a ﬁxed (unpublished)n-bit permutationPand 32 parallel identical S-boxes each effecting a mappingS0 orS1 (ﬁxed but unpublished bijections on{0,1}4), depending on the value of one key bit; the unpublished key schedule requires 32-bits per S-box stage. Each stage operates on allnbits; decryption is by stage-wise inversion ofPand S i. The structure of so-called Feistel ciphers (Deﬁnition 7.81) was ﬁrst introduced in the Lu- cifer algorithm of Smith [1160], the direct predecessor of DES. This 16-round algorithm §7.8 Notes and further references 277 with 128-bit key operates on alternating half-blocks of a 128-bit message block with a sim- pliﬁedffunction based on two published invertible4×4bit S-boxesS0 andS1 (cf. above). Feistel, Notz, and Smith [388] discuss both the abstract Feistel cipher structure (suggesting its use with non-invertible S-boxes) and SP networks based on invertible (distinct) S-boxes. Suggestions for SP networks include the use of single key bits to select one of two map- pings (a ﬁxed bijection or its inverse) from both S-boxes and permutation boxes; decryption then uses a reversed key schedule with complemented key. They also noted the multi-round avalanche effectof changing a single input bit, subsequently pursued by Kam and Davida [659] in relation to SP networks and S-boxes having acompletenessproperty: for every pair of bit positionsi,j, there must exist at least two input blocksx,ywhich differ only in biti and whose outputs differ in at least bitj. More simply, a function iscompleteif each output bit depends on all input bits. Webster and Tavares [1233] proposed the more stringentstrict avalanche criterion: whenever one input bit is changed, every output bit must change with probability 1/2. DES resulted from IBM’s submission to the 1974 U.S. National Bureau of Standards (NBS) solicitation for encryption algorithms for the protection of computer data. The original speciﬁcation is the 1977 U.S. Federal Information Processing Standards Publication 46 [396], reprinted in its entirety as Appendix A in Meyer and Matyas [859]. DES is now spec- iﬁed in FIPS 46–2, which succeeded FIPS 46–1; the same cipher is deﬁned in the American standard ANSI X3.92 [33] and referred to as the Data Encryption Algorithm (DEA). Differ- ences between FIPS 46/46–1 and ANSI X3.92 included the following: these earlier FIPS required that DES be implemented in hardware and that the parity bits be used for parity; ANSI X3.92 speciﬁes that the parity bitsmay be used for parity. Although no purpose was stated by the DES designers for the permutations IP and IP −1, Preneel et al. [1008] provided some evidence of their cryptographic value in the CFB mode. FIPS 81 [398] speciﬁes the common modes of operation. Davies and Price [308] provide a comprehensive discussion of both DES and modes of operation; see also Difﬁe and Hellman [347], and the extensive treatment of Meyer and Matyas [859]. The survey of Smid and Branstad [1156] discusses DES, its history, and its use in the U.S. government. Test vectors for various modes of DES, including the ECB vectors of Example 7.86, may be found in ANSI X3.106 [34]. Regarding exhaustive cryptanalysis of DES and related issues, see also the notes under§7.2. The 1981 publication FIPS 74 [397] notes that DES is not (generally) commutative under two keys, and summarizes weak and semi-weak keys using the termdual keysto include both (weak keys being self-dual); see also Davies [303] and Davies and Price [308]. Cop- persmith [268] noted Fact 7.90; Moore and Simmons [900] pursue weak and semi-weak DES keys and related phenomena more rigorously. The 56-bit keylength of DES was criticized from the outset as being too small (e.g., see Difﬁe and Hellman [346], and p.272 above). Claims which have repeatedly arisen and been denied (e.g., see Tuchman [1199]) over the past 20 years regarding built-in weaknesses of DES (e.g., trap-door S-boxes) remain un-substantiated. Fact 7.91 is signiﬁcant in that if the permutation group were closed under composition, DES would fall to a known-plaintext attack requiring2 28 steps – see Kaliski, Rivest, and Sherman [654], whose cycling exper- iments provided strong evidence against this. Campbell and Wiener [229] prove the fact conclusively (and give the stated lower bound), through their own cycling experiments uti- lizing collision key search and an idea outlined earlier by Coppersmith [268] for establish- ing a lower bound on the group size; they attribute to Coppersmith the same result (in un- published work), which may also be deduced from the cycle lengths published by Moore and Simmons [901]. 278 Ch. 7 Block Ciphers Countless papers have analyzed various properties of DES; Davies and Price [308, pp.73- 75] provide a partial summary to 1987. Subsequent to the discovery of differential crypt- analysis (DC) by Biham and Shamir, Coppersmith [271] explains how DES was speciﬁcally designed 15 years earlier to counter DC, citing national security concerns regarding the de- sign team publishing neither the attack nor design criteria; then gives the (relevant) design criteria – some already noted by others, e.g., see Hellman et al. [552] – for DES S-boxes and the permutationP, explaining how these preclude DC. Coppersmith notes elements of DC were present in the work of den Boer [322], followed shortly by Murphy [913]. DES was not, however, speciﬁcally designed to preclude linear cryptanalysis (LC); Matsui [797] illustrates the order of the 8 DES S-boxes, while a strong (but not optimal) choice against DC, is relatively weak against LC, and that DES can be strengthened (vs. DC and LC) by carefully re-arranging these. Despite Remark 7.93, a DES key has actually been recovered by Matsui [795] using LC under experimental conditions (using2 43 known-plaintext pairs from randomly generated plaintexts, and243 complexity running twelve 99 MHz machines over 50 days); such a result remains to be published for exhaustive search or DC. Ben-Aroya and Biham [108] note that often suggestions to redesign DES, some based on de- sign criteria and attempts to speciﬁcally resist DC, have resulted in (sometimes far) weaker systems, including the RDES (randomized DES) proposal of Koyama and Terada [709], which fall to variant attacks. The lesson is that in isolation, individual design principles do not guarantee security. DES alternatives are sought not only due to the desire for a keylength exceeding 56 bits, but also because its bit-oriented operations are inconvenient in conventional software im- plementations, often resulting in poor performance; this makes triple-DES less attractive. Regarding fast software implementations of DES, see Shepherd [1124], Pﬁtzmann and Aß- mann [970], and Feldmeier and Karn [391]. §7.5 FEAL stimulated the development of a sequence of advanced cryptanalytic techniques of unparalleled richness and utility. While it appears to remain relatively secure when iterated a sufﬁcient number of rounds (e.g., 24 or more), this defeats its original objective of speed. FEAL-4 as presented at Eurocrypt’87 (Abstracts of Eurocrypt’87, April 1987) was found to have certain vulnerabilities by den Boer (unpublished Eurocrypt’87 rump session talk), re- sulting in Shimizu and Miyaguchi [1126] (or see Miyaguchi, Shiraishi, and Shimizu [887]) increasing FEAL to 8 rounds in the ﬁnal proceedings. In 1988 den Boer [322] showed FEAL-4 vulnerable to an adaptive chosen plaintext attack with100to10000plaintexts. In 1990, Gilbert and Chass´e [455] devised a chosen-plaintext attack (called a statistical meet- in-the-middle attack) on FEAL-8 requiring10000pairs of plaintexts, the bitwise XOR of each pair being selected to be an appropriate constant (thus another early variant of differ- ential cryptanalysis). FEAL-N with Nrounds, and its extension FEAL-NX with 128-bit key (Notes 7.97 and 7.98) were then published by Miyaguchi [884] (or see Miyaguchi et al. [885]), who nonethe- less opined that chosen-plaintext attacks on FEAL-8 were not practical threats. However, improved chosen-plaintext attacks were subsequently devised, as well as known-plaintext attacks. Employing den Boer’sGfunction expressing linearity in the FEALf-function, Murphy [913] defeated FEAL-4 with 20 chosen plaintexts in under 4 hours (under 1 hour for most keys) on a Sun 3/60 workstation. A statistical method of Tardy-Corfdir and Gilbert [1187] then allowed a known-plaintext attack on FEAL-4 (1000 texts; or 200 in an an- nounced improvement) and FEAL-6 (2×10000texts), involving linear approximation of FEAL S-boxes. Thereafter, the ﬁrst version of linear cryptanalysis (LC) introduced by Mat- sui and Yamagishi [798] allowed known-plaintext attack of FEAL-4 (5 texts, 6 minutes on §7.8 Notes and further references 279 a 25MHz 68040 processor), FEAL-6 (100 texts, 40 minutes), and FEAL-8 (228 texts, in time equivalent to exhaustive search on 50-bit keys); the latter betters the238 texts required for FEAL-8 by Biham and Shamir [136] in their known-plaintext conversion of differen- tial cryptanalysis (DC). Biham and Shamir [138, p.101] later implemented a DC chosen- plaintext attack recovering FEAL-8 keys in two minutes on a PC using 128 chosen pairs, the program requiring 280K bytes of storage. Biham [132] subsequently used LC to defeat FEAL-8 with2 24 known-plaintexts in 10 minutes on a personal computer. Ohta and Aoki [943] suggest that FEAL-32 is as secure as DES against DC, while FEAL-16 is as secure as DES against certain restricted forms of LC. Differential-linear cryptanalysiswas introduced by Langford and Hellman [741], combin- ing linear and differential cryptanalysis to allow a reduced 8-round version of DES to be attacked with fewer chosen-plaintexts than previous attacks. Aoki and Ohta [53] reﬁned these ideas for FEAL-8 yielding a differential-linear attack requiring only 12 chosen texts and 35 days of computer time (cf. Table 7.10). Test vectors for FEAL-N and FEAL-NX (Example 7.99) are given by Miyaguchi [884]. The DC attack of Biham and Shamir [137], which ﬁnds FEAL-N subkeys themselves, is equally as effective on FEAL-NX. Biham [132] notes that an LC attack on FEAL-N is pos- sible with less than2 64 known plaintexts (and complexity) for up toN=20. For additional discussion of properties of FEAL, see Biham and Shamir [138,§6.3]. §7.6 The primary reference for IDEA is Lai [726]. A preliminary version introduced by Lai and Massey [728] was named PES (Proposed Encryption Standard). Lai, Massey, and Murphy [730] showed that a generalization (see below) of differential cryptanalysis (DC) allowed recovery of PES keys, albeit requiring all264 possible ciphertexts (cf. exhaustive search of2128 operations). Minor modiﬁcations resulted in IPES (Improved PES): in stager,1≤ r≤9, the group operations keyed byK(r) 2 andK(r) 4 (⊞and⊙in Figure 7.11) were reversed from PES; the permutation on 16-bit blocks after stager,1≤r ≤9, was altered; and necessary changes were made in the decryption (but not encryption) key schedule. IPES was commercialized under the name IDEA, and is patented (see Chapter 15). The ingenious design of IDEA is supported by a careful analysis of the interaction and alge- braic incompatibilities of operations across the groups(F2 n,⊕),(Z2n ,⊞), and(Z∗ 2n+1,⊙). The design of the MA structure (see Figure 7.11) results in IDEA being “complete” after a single round; for other security properties, see Lai [726]. Regarding mixing operations from different algebraic systems, see also the 1974 examination by Grossman [522] of transfor- mations arising by alternating mod2nand mod 2addition (⊕), and the use of arithmetic modulo 232 −1and 232 −2in MAA (Algorithm 9.68). Daemen [292, 289] identiﬁes several classes of so-calledweak keys for IDEA, and notes a small modiﬁcation to the key schedule to eliminate them. The largest is a class of251 keys for which membership can be tested in two encryptions plus a small number of computa- tions, whereafter the key itself can be recovered using 16 chosen plaintext-difference en- cryptions, on the order of216 group operations, plus217 key search encryptions. The prob- ability of a randomly chosen key being in this class is251/2128 =2−77. A smaller number of weak key blocks were observed earlier by Lai [726], and dismissed as inconsequential. The analysis of Meier [832] revealed no attacks feasible against full 8-round IDEA, and supports the conclusion of Lai [726] that IDEA appears to be secure against DC after 4 of its 8 rounds (cf. Note 7.107). Daemen [289] also references attacks on reduced-round vari- ants of IDEA. While linear cryptanalysis (LC) can be applied to any iterated block cipher, 280 Ch. 7 Block Ciphers Harpes, Kramer, and Massey [541] provide a generalization thereof; IDEA and SAFER K- 64 are argued to be secure against this particular generalization. Lai, Massey, and Murphy [730] (see also Lai [726]) generalized DC to apply toMarkov ciphers(which they introduced for this purpose; DES, FEAL, and LOKI are all examples under the assumption of independent round keys) including IDEA; broadened the notion of a differencefrom that based on⊕to:∆X=X⊗(X∗)−1 where ⊗is a speciﬁed group operation and(X∗)−1 is the group inverse of an elementX∗; and deﬁned ani-round differ- ential(as opposed to ani-round characteristic used by Biham and Shamir [138] on DES) to be a pair(α,β)such that two distinct plaintexts with difference∆X=αresults in a pair of roundioutputs with differenceβ. Decimal values corresponding to Tables 7.12 and 7.13 may be found in Lai [726]. A table- based alternative for multiplication mod2 16 +1(cf. Note 7.104) is to look up the anti-log of logα(a)+log α(b)mod2 16, relative to a generatorαof Z∗ 216+1; the required tables, however, are quite large. §7.7 Massey [787] introduced SAFER K-64 with a 64-bit key and initially recommended 6 rounds, giving a reference implementation and test vectors (cf. Example 7.114). It is not patented. Massey [788] then published SAFER K-128 (with a reference implementation), differing only in its use of a non-proprietary (and backwards compatible) key schedule ac- commodating 128-bit keys, proposed by a Singapore group; 10 rounds were recommended (12 maximum). Massey [788] gave further justiﬁcation for design components of SAFER K-64. Vaudenay [1215] showed SAFER K-64 is weakened if the S-box mapping (Re- mark 7.112) is replaced by a random permutation. Knudsen [685] proposed the modiﬁed key schedule of Note 7.110 after ﬁnding a weakness in 6-round SAFER K-64 that, while not of practical concern for encryption (with2 45 chosen plaintexts, it ﬁnds 8 bits of the key), permitted collisions when using the cipher for hashing. This and a subsequent certiﬁcational attack on SAFER K-64 by S. Murphy (to be published) lead Massey (“Strengthened key schedule for the cipher SAFER”, posted to the USENET newsgroup sci.crypt, September 9 1995) to advise adoption of the new key schedule, with the resulting algorithm distinguished as SAFER SK-64 with 8 rounds recommended (min- imum 6, maximum 10); an analogous change to the 128-bit key schedule yields SAFER SK-128 for which 10 rounds remain recommended (maximum 12). A new variant of DC by Knudsen and Berson [687] usingtruncated differentials(building on Knudsen [686]) yields a certiﬁcational attack on 5-round SAFER K-64 with2 45 chosen plaintexts; the at- tack, which does not extend to 6 rounds, indicates that security is less than argued by Massey [788], who also notes that preliminary attempts at linear cryptanalysis of SAFER were un- successful. RC5 was designed by Rivest [1056], and published along with a reference implementation. The magic constants of Table 7.14 are based on the golden ratio and the base of natural log- arithms. The data-dependent rotations (which vary across rounds) distinguish RC5 from iterated ciphers which have identical operations each round; Madryga [779] proposed an earlier (less elegant) cipher involving data-dependent rotations. A preliminary examination by Kaliski and Yin [656] suggested that, while variations remain to be explored, standard linear and differential cryptanalysis appear impractical for RC5–32 (64-bit blocksize) for r =1 2: their differential attacks on 9 and 12 round RC5 require, respectively,2 45,262 chosen-plaintext pairs, while their linear attacks on 4, 5, and 6-round RC5–32 require, re- spectively,237,247,257 known plaintexts. Both attacks depend on the number of rounds and the blocksize, but not the byte-length of the input key (since subkeys are recovered di- §7.8 Notes and further references 281 rectly). Knudsen and Meier [689] subsequently presented differential attacks on RC5 which improved on those of Kaliski and Yin by a factor up to 512, and showed that RC5 has so- calledweak keys(independent of the key schedule) for which these differential attacks per- form even better. LOKI was introduced by Brown, Pieprzyk, and Seberry [215] and renamed LOKI’89 after the discovery of weaknesses lead to the introduction of LOKI’91 by Brown et al. [214]. Knudsen [682] noted each LOKI’89 key fell into a class of 16 equivalent keys, and the differential cryptanalysis of Biham and Shamir [137] was shown to be effective against reduced-round versions. LOKI’91 failed to succumb to differential analysis by Knudsen [683]; Tokita et al. [1193] later conﬁrmed the optimality of Knudsen’s characteristics, sug- gesting that LOKI’89 and LOKI’91 were resistant to both ordinary linear and differential cryptanalysis. However, neither should be used for hashing as originally proposed (see Knudsen [682]) or in other modes (see Preneel [1003]). Moreover, both are susceptible torelated-key attacks(Note 7.6), popularized by Biham [128, 129]; but see also the ear- lier ideas of Knudsen [683]. Distinct from these arekey clustering attacks(see Difﬁe and Hellman [347, p.410]), wherein a cryptanalyst ﬁrst ﬁnds a key “close” to the correct key, and then searches a cluster of “nearby” keys to ﬁnd the correct one. 8×32bit S-boxes ﬁrst appeared in the Snefru hash function of Merkle [854]; here such ﬁxed S-boxes created from random numbers were used in its internal encryption mapping. Regarding large S-boxes, see also Gordon and Retkin [517], Adams and Tavares [7], and Biham [132]. Merkle [856] again used8×32S-boxes in Khufu and Khafre (see also §15.2.3(viii)). In this 1990 paper, Merkle gives a chosen-plaintext differential attack de- feating 8 rounds of Khufu (with secret S-box). Regarding 16-round Khafre, a DC attack by Biham and Shamir [138, 137] requires somewhat over 1500 chosen plaintexts and one hour on a personal computer, and their known-plaintext differential attack requires2 37.5 plain- texts; for 24-round Khafre, they require253 chosen plaintexts or258.5 known plaintexts. Khufu with 16 rounds was examined by Gilbert and Chauvaud [456], who gave an attack using2 43 chosen plaintexts and about243 operations. CAST is a design procedure for a family of DES-like ciphers, featuring ﬁxedm×nbit S-boxes(m<n )based on bent functions. Adams and Tavares [7] examine the construc- tion of large S-boxes resistant to differential cryptanalysis, and give a partial example (with 64-bit blocklength and8×32bit S-boxes) of a CAST cipher. CAST ciphers have variable keysize and numbers of rounds. Rijmen and Preneel [1049] presented a cryptanalytic tech- nique applicable to Feistel ciphers with non-surjective round functions (e.g., LOKI’91 and an example CAST cipher), noting cases where 6 to 8 rounds is insufﬁcient. Blowﬁsh is a 16-round DES-like cipher due to Schneier [1093], with 64-bit blocks and keys of length up to 448 bits. The computationally intensive key expansion phase creates eigh- teen 32-bit subkeys plus four8×32bit S-boxes derived from the input key (cf. Khafre above), for a total of 4168 bytes. See Vaudenay [1216] for a preliminary analysis of Blow- ﬁsh. 3-WAY is a block cipher with 96-bit blocksize and keysize, due to Daemen [289] and intro- duced by Daemen, Govaerts, and Vandewalle [290] along with a reference C implementa- tion and test vectors. It was designed for speed in both hardware and software, and to resist differential and linear attacks. Its core is a 3-bit nonlinear S-box and a linear mapping rep- resentable as polynomial multiplication inZ 12 2 . SHARK is an SP-network block cipher due to Rijmen et al. [1048] (coordinates for a refer- ence implementation are given) which may be viewed as a generalization of SAFER, em- ploying highly nonlinear S-boxes and the idea of MDS codes (cf. Note 12.36) for diffusion 282 Ch. 7 Block Ciphers to allow a small number of rounds to sufﬁce. The block ciphers BEAR and LION of An- derson and Biham [30] are 3-round unbalanced Feistel networks, motivated by the earlier construction of Luby and Rackoff [776] (see also Maurer [816] and Lucks [777]) which provides a provably secure (under suitable assumptions) block cipher from pseudorandom functions using a 3-round Feistel structure. SHARK, BEAR, and LION all remain to be subjected to independent analysis in order to substantiate their conjectured security levels. SKIPJACK is a classiﬁed block cipher whose speciﬁcation is maintained by the U.S. Na- tional Security Agency (NSA). FIPS 185 [405] notes that its speciﬁcation is available to organizations entering into a Memorandum of Agreement with the NSA, and includes in- terface details (e.g., it has an 80-bit secret key). A public report contains results of a pre- liminary security evaluation of this 64-bit block cipher (“SKIPJACK Review, Interim Re- port, The SKIPJACK Algorithm”, 1993 July 28, by E.F. Brickell, D.E. Denning, S.T. Kent, D.P. Maher, and W. Tuchman). See also Roe [1064, p.312] regarding curious results on the cyclic closure tests on SKIPJACK, which give evidence related to the size of the cipher keyspace. GOST 28147-89 is a Soviet government encryption algorithm with a 32-round Feistel struc- ture and unspeciﬁed S-boxes; see Charnes et al. [241]. RC2 is a block cipher proprietary to RSA Data Security Inc. (as is the stream cipher RC4). WAKE is a block cipher due to Wheeler [1237] employing a key-dependent table, intended for fast encryption of bulk data on processors with 32-bit words. TEA (Tiny Encryption Algorithm) is a block cipher proposed by Wheeler and Needham [1238]. Chapter /8 Public-Key Encryption Contents in Brief 8.1 Introduction ............................. 283 8.2 RSA public-key encryption ..................... 285 8.3 Rabin public-key encryption.................... 292 8.4 ElGamal public-key encryption................... 294 8.5 McEliece public-key encryption.................. 298 8.6 Knapsack public-key encryption.................. 300 8.7 Probabilistic public-key encryption................. 306 8.8 Notes and further references.................... 312 8.1 Introduction This chapter considers various techniques for public-key encryption, also referred to as asymmetric encryption. As introduced previously (§1.8.1), in public-key encryption sys- tems each entityAhas apublic keyeand a correspondingprivate keyd. In secure systems, the task of computingdgiveneis computationally infeasible. The public key deﬁnes anen- cryption transformationEe, while the private key deﬁnes the associateddecryption trans- formationDd. Any entityBwishing to send a messagemtoAobtains an authentic copy ofA’s public keye, uses the encryption transformation to obtain the ciphertextc= Ee(m), and transmitsctoA. To decryptc,Aapplies the decryption transformation to obtain the original messagem= Dd(c). The public key need not be kept secret, and, in fact, may be widely available – only its authenticity is required to guarantee thatAis indeed the only party who knows the corre- sponding private key. A primary advantage of such systems is that providing authentic pub- lic keys is generally easier than distributing secret keys securely, as required in symmetric- key systems. The main objective of public-key encryption is to provideprivacyorconﬁdentiality. SinceA’s encryption transformation is public knowledge, public-key encryption alone does not providedata origin authentication(Deﬁnition 9.76) ordata integrity(Deﬁnition 9.75). Such assurances must be provided through use of additional techniques (see§9.6), including message authentication codes and digital signatures. Public-key encryption schemes are typically substantially slower than symmetric-key encryption algorithms such as DES (§7.4). For this reason, public-key encryption is most commonly used in practice for the transport of keys subsequently used for bulk data en- cryption by symmetric algorithms and other applications including data integrity and au- thentication, and for encrypting small data items such as credit card numbers and PINs. 283 284 Ch. 8 Public-Key Encryption Public-key decryption may also provide authentication guarantees in entity authentication and authenticated key establishment protocols. Chapter outline The remainder of the chapter is organized as follows.§8.1.1 provides introductory material. The RSA public-key encryption scheme is presented in§8.2; related security and implemen- tation issues are also discussed. Rabin’s public-key encryption scheme, which is provably as secure as factoring, is the topic of§8.3.§8.4 considers the ElGamal encryption scheme; related security and implementation issues are also discussed. The McEliece public-key encryption scheme, based on error-correcting codes, is examined in§8.5. Although known to be insecure, the Merkle-Hellman knapsack public-key encryption scheme is presented in §8.6 for historical reasons – it was the ﬁrst concrete realization of a public-key encryption scheme. Chor-Rivest encryption is also presented (§8.6.2) as an example of an as-yet un- broken public-key encryption scheme based on the subset sum (knapsack) problem.§8.7 introduces the notion of probabilistic public-key encryption, designed to meet especially stringent security requirements.§8.8 concludes with Chapter notes and references. The number-theoretic computational problems which form the security basis for the public-key encryption schemes discussed in this chapter are listed in Table 8.1. public-key encryption scheme computational problem RSA integer factorization problem (§3.2) RSA problem (§3.3) Rabin integer factorization problem (§3.2) square roots modulo compositen(§3.5.2) ElGamal discrete logarithm problem (§3.6) Difﬁe-Hellman problem (§3.7) generalized ElGamal generalized discrete logarithm problem (§3.6) generalized Difﬁe-Hellman problem (§3.7) McEliece linear code decoding problem Merkle-Hellman knapsack subset sum problem (§3.10) Chor-Rivest knapsack subset sum problem (§3.10) Goldwasser-Micali probabilistic quadratic residuosity problem (§3.4) Blum-Goldwasser probabilistic integer factorization problem (§3.2) Rabin problem (§3.9.3) Table 8.1:Public-key encryption schemes discussed in this chapter, and the related computational problems upon which their security is based. 8.1.1 Basic principles Objectives of adversary The primary objective of an adversary who wishes to “attack” a public-key encryption sch- eme is to systematically recover plaintext from ciphertext intended for some other entityA. If this is achieved, the encryption scheme is informally said to have beenbroken. A more ambitious objective iskey recovery– to recoverA’s private key. If this is achieved, the en- §8.2 RSA public-key encryption 285 cryption scheme is informally said to have beencompletely brokensince the adversary then has the ability to decryptallciphertext sent toA. Types of attacks Since the encryption transformations are public knowledge, a passive adversary can al- ways mount achosen-plaintext attackon a public-key encryption scheme (cf.§1.13.1). A stronger attack is achosen-ciphertext attackwhere an adversary selects ciphertext of its choice, and then obtains by some means (from the victimA) the corresponding plaintext (cf.§1.13.1). Two kinds of these attacks are usually distinguished. 1. In anindifferentchosen-ciphertext attack, the adversary is provided with decryptions of any ciphertexts of its choice, but these ciphertexts must be chosen prior to receiving the (target) ciphertextcit actually wishes to decrypt. 2. In anadaptivechosen-ciphertext attack, the adversary may use (or have access to)A’s decryption machine (but not the private key itself) even after seeing the target cipher- textc. The adversary may request decryptions of ciphertext which may be related to both the target ciphertext, and to the decryptions obtained from previous queries; a restriction is that it may not request the decryption of the targetcitself. Chosen-ciphertext attacks are of concern if the environment in which the public-key en- cryption scheme is to be used is subject to such an attack being mounted; if not, the exis- tence of a chosen-ciphertext attack is typically viewed as acertiﬁcationalweakness against a particular scheme, although apparently not directly exploitable. Distributing public keys The public-key encryption schemes described in this chapter assume that there is a means for the sender of a message to obtain anauthenticcopy of the intended receiver’s public key. In the absence of such a means, the encryption scheme is susceptible to animperson- ationattack, as outlined in§1.8.2. There are many techniques in practice by which authentic public keys can be distributed, including exchanging keys over a trusted channel, using a trusted public ﬁle, using an on-line trusted server, and using an off-line server and certiﬁ- cates. These and related methods are discussed in§13.4. Message blocking Some of the public-key encryption schemes described in this chapter assume that the mes- sage to be encrypted is, at most, some ﬁxed size (bitlength). Plaintext messages longer than this maximum must be broken intoblocks, each of the appropriate size. Speciﬁc tech- niques for breaking up a message into blocks are not discussed in this book. The compo- nent blocks can then be encrypted independently (cf. ECB mode in§7.2.2(i)). To provide protection against manipulation (e.g., re-ordering) of the blocks, theCipher Block Chaining (CBC) mode may be used (cf.§7.2.2(ii) and Example 9.84). Since the CFB and OFB modes (cf.§7.2.2(iii) and§7.2.2(iv)) employ only single-block encryption (and not decryption) for both message encryption and decryption, they cannot be used with public-key encryption schemes. 8.2 RSA public-key encryption The RSA cryptosystem, named after its inventors R. Rivest, A. Shamir, and L. Adleman, is the most widely used public-key cryptosystem. It may be used to provide both secrecy and digital signatures and its security is based on the intractability of the integer factorization 286 Ch. 8 Public-Key Encryption problem (§3.2). This section describes the RSA encryption scheme, its security, and some implementation issues; the RSA signature scheme is covered in§11.3.1. 8.2.1 Description 8.1 AlgorithmKey generation for RSA public-key encryption SUMMARY: each entity creates an RSA public key and a corresponding private key. Each entityAshould do the following: 1. Generate two large random (and distinct) primespand q, each roughly the same size. 2. Compute n= pqand φ=( p−1)(q−1). (See Note 8.5.) 3. Select a random integere,1 <e<φ , such thatgcd(e,φ)=1 . 4. Use the extended Euclidean algorithm (Algorithm 2.107) to compute the unique in- tegerd,1 <d<φ , such thated≡1( m o dφ). 5. A’s public key is(n,e);A’s private key isd. 8.2 DeﬁnitionThe integerseand din RSA key generation are called theencryption exponent and thedecryption exponent, respectively, whilenis called themodulus. 8.3 AlgorithmRSA public-key encryption SUMMARY: Bencrypts a messagemforA, whichAdecrypts. 1. Encryption.Bshould do the following: (a) ObtainA’s authentic public key(n,e). (b) Represent the message as an integermin the interval[0,n−1]. (c) Computec= memod n(e.g., using Algorithm 2.143). (d) Send the ciphertextctoA. 2. Decryption.To recover plaintextmfrom c,Ashould do the following: (a) Use the private keydto recoverm= cdmod n. Proof that decryption works.Sinceed≡1( m o dφ), there exists an integerksuch that ed=1+ kφ. Now, ifgcd(m,p)=1 then by Fermat’s theorem (Fact 2.127), mp−1 ≡1( m o d p). Raising both sides of this congruence to the powerk(q−1)and then multiplying both sides by myields m1+k(p−1)(q−1) ≡m (mod p). On the other hand, ifgcd(m,p)= p, then this last congruence is again valid since each side is congruent to0 modulo p. Hence, in all cases med≡m (mod p). By the same argument, med≡m (mod q). Finally, sincepand qare distinct primes, it follows that med≡m (mod n), §8.2 RSA public-key encryption 287 and, hence, cd≡(me)d≡m (mod n). 8.4 Example (RSA encryption with artiﬁcially small parameters) Key generation. EntityAchooses the primesp= 2357 ,q = 2551, and computesn= pq= 6012707 and φ=( p−1)(q−1) = 6007800.Achoosese= 3674911 and, using the extended Euclidean algorithm, ﬁndsd= 422191 such thated≡1( m o dφ). A’s public key is the pair(n= 6012707,e= 3674911), whileA’s private key isd= 422191. Encryption. To encrypt a messagem= 5234673,Buses an algorithm for modular expo- nentiation (e.g., Algorithm 2.143) to compute c = memod n = 5234673 3674911 mod 6012707 = 3650502 , and sends this toA. Decryption. To decryptc,Acomputes cdmod n = 3650502 422191 mod 6012707 = 5234673 . □ 8.5 Note (universal exponent) The numberλ=l c m (p−1,q−1), sometimes called theuni- versal exponentofn, may be used instead ofφ=( p−1)(q−1) in RSA key generation (Algorithm 8.1). Observe thatλis a proper divisor ofφ. Usingλcan result in a smaller decryption exponentd, which may result in faster decryption (cf. Note 8.9). However, ifp andqare chosen at random, thengcd(p−1,q−1)is expected to be small, and consequently φand λwill be roughly of the same size. 8.2.2 Security of RSA This subsection discusses various security issues related to RSA encryption. Various attacks which have been studied in the literature are presented, as well as appropriate measures to counteract these threats. (i) Relation to factoring The task faced by a passive adversary is that of recovering plaintextmfrom the correspond- ing ciphertextc, given the public information(n,e) of the intended receiverA. This is called theRSA problem(RSAP), which was introduced in§3.3. There is no efﬁcient algo- rithm known for this problem. One possible approach which an adversary could employ to solving the RSA problem is to ﬁrst factorn, and then computeφand djust asAdid in Algorithm 8.1. Oncedis obtained, the adversary can decrypt any ciphertext intended forA. On the other hand, if an adversary could somehow computed, then it could subse- quently factornefﬁciently as follows. First note that sinceed≡1( m o dφ), there is an integerksuch thated−1= kφ. Hence, by Fact 2.126(i),a ed−1 ≡1( m o dn) for all a∈Z∗ n. Leted−1=2 st, wheretis an odd integer. Then it can be shown that there exists ani∈[1,s]such thata2i−1t̸≡±1( m o dn)and a2it≡1( m o dn)for at least half of alla∈Z∗ n;i faand iare such integers thengcd(a2i−1t−1,n) is a non-trivial factor ofn. Thus the adversary simply needs to repeatedly select randoma∈Z∗ nand check if an i∈[1,s] satisfying the above property exists; the expected number of trials before a non-trivial factor ofnis obtained is2. This discussion establishes the following. 8.6 FactThe problem of computing the RSA decryption exponentdfrom the public key(n,e), and the problem of factoringn, are computationally equivalent. 288 Ch. 8 Public-Key Encryption When generating RSA keys, it is imperative that the primespand qbe selected in such a way that factoringn= pqis computationally infeasible; see Note 8.8 for more details. (ii) Small encryption exponente In order to improve the efﬁciency of encryption, it is desirable to select a small encryption exponente(see Note 8.9) such ase=3 . A group of entities may all have the same encryp- tion exponente, however, each entity in the group must have its own distinct modulus (cf. §8.2.2(vi)). If an entityAwishes to send the same messagemto three entities whose pub- lic moduli aren1,n2,n3, and whose encryption exponents aree=3 , thenAwould send ci = m3 mod ni, fori=1 ,2,3. Since these moduli are most likely pairwise relatively prime, an eavesdropper observingc1,c2,c3 can use Gauss’s algorithm (Algorithm 2.121) to ﬁnd a solutionx,0 ≤x<n 1n2n3, to the three congruences    x≡c1 (mod n1) x≡c2 (mod n2) x≡c3 (mod n3). Sincem3 <n1n2n3, by the Chinese remainder theorem (Fact 2.120), it must be the case thatx= m3. Hence, by computing the integer cube root ofx, the eavesdropper can recover the plaintextm. Thus a small encryption exponent such ase=3 should not be used if the same mes- sage, or even the same message with known variations, is sent to many entities. Alter- natively, to prevent against such an attack, a pseudorandomly generated bitstring of ap- propriate length (taking into account Coppersmith’s attacks mentioned on pages 313–314) should be appended to the plaintext message prior to encryption; the pseudorandom bit- string should be independently generated for each encryption. This process is sometimes referred to assaltingthe message. Small encryption exponents are also a problem for small messagesm, because ifm< n 1/e, thenmcan be recovered from the ciphertextc= memod nsimply by computing the integereth root ofc; salting plaintext messages also circumvents this problem. (iii) Forward search attack If the message space is small or predictable, an adversary can decrypt a ciphertextcby sim- ply encrypting all possible plaintext messages untilcis obtained. Salting the message as described above is one simple method of preventing such an attack. (iv) Small decryption exponentd As was the case with the encryption exponente, it may seem desirable to select a small de- cryption exponentdin order to improve the efﬁciency of decryption.1 However, ifgcd(p− 1,q−1) is small, as is typically the case, and ifdhas up to approximately one-quarter as many bits as the modulusn, then there is an efﬁcient algorithm (referenced on page 313) for computingdfrom the public information(n,e). This algorithm cannot be extended to the case wheredis approximately the same size asn. Hence, to avoid this attack, the de- cryption exponentdshould be roughly the same size asn. (v) Multiplicative properties Letm1 and m2 be two plaintext messages, and letc1 and c2 be their respective RSA en- cryptions. Observe that (m1m2)e ≡ me 1me 2 ≡ c1c2 (mod n). 1In this case, one would selectd ﬁrst and then computeein Algorithm 8.1, rather than vice-versa. §8.2 RSA public-key encryption 289 In other words, the ciphertext corresponding to the plaintextm= m1m2 mod nisc= c1c2 mod n; this is sometimes referred to as thehomomorphic propertyof RSA. This ob- servation leads to the followingadaptive chosen-ciphertext attackon RSA encryption. Suppose that an active adversary wishes to decrypt a particular ciphertextc= memod nintended forA. Suppose also thatAwill decrypt arbitrary ciphertext for the adversary, other thancitself. The adversary can concealcby selecting a random integerx ∈ Z∗ n and computing c= cxemod n. Upon presentation of c,Awill compute for the adversary m=( c)dmod n. Since m ≡ ( c)d ≡ cd(xe)d ≡ mx (mod n), the adversary can then computem= mx−1 mod n. This adaptive chosen-ciphertext attack should be circumvented in practice by imposing some structural constraints on plaintext messages. If a ciphertextcis decrypted to a message not possessing this structure, thencis rejected by the decryptor as being fraudulent. Now, if a plaintext messagemhas this (carefully chosen) structure, then with high probability mxmod nwill not forx∈Z∗ n. Thus the adaptive chosen-ciphertext attack described in the previous paragraph will fail becauseAwill not decrypt cfor the adversary. Note 8.63 provides a powerful technique for guarding against adaptive chosen-ciphertext and other kinds of attacks. (vi) Common modulus attack The following discussion demonstrates why it is imperative for each entity to choose its own RSA modulus n. It is sometimes suggested that a central trusted authority should select a single RSA modulus n, and then distribute a distinct encryption/decryption exponent pair (ei,di) to each entity in a network. However, as shown in (i) above, knowledge of any(ei,di)pair al- lows for the factorization of the modulusn, and hence any entity could subsequently deter- mine the decryption exponents of all other entities in the network. Also, if a single message were encrypted and sent to two or more entities in the network, then there is a technique by which an eavesdropper (any entity not in the network) could recover the message with high probability using only publicly available information. (vii) Cycling attacks Letc= m emod nbe a ciphertext. Letkbe a positive integer such thatcek ≡c(mod n); since encryption is a permutation on the message space{0,1,...,n −1}such an integer kmust exist. For the same reason it must be the case thatcek−1 ≡m(mod n). This ob- servation leads to the followingcycling attackon RSA encryption. An adversary computes cemod n,ce2 mod n,ce3 mod n,... untilcis obtained for the ﬁrst time. Ifcek mod n= c, then the previous number in the cycle, namelycek−1 mod n, is equal to the plaintextm. A generalized cycling attackis to ﬁnd the smallest positive integerusuch thatf = gcd(ceu −c,n) >1.I f ceu ≡c (mod p) and ceu ̸≡c (mod q) (8.1) thenf= p. Similarly, if ceu ̸≡c (mod p) and ceu ≡c (mod q) (8.2) thenf= q. In either case,nhas been factored, and the adversary can recoverdand then m. On the other hand, if both ceu ≡c (mod p) and ceu ≡c (mod q), (8.3) 290 Ch. 8 Public-Key Encryption thenf = nand ceu ≡ c(mod n). In fact,umust be the smallest positive integerk for whichcek ≡c(mod n). In this case, the basic cycling attack has succeeded and so m= ceu−1 mod ncan be computed efﬁciently. Since (8.3) is expected to occur much less frequently than (8.1) or (8.2), the generalized cycling attack usually terminates before the cycling attack does. For this reason, the generalized cycling attack can be viewed as being essentially an algorithm for factoringn. Since factoringnis assumed to be intractable, these cycling attacks do not pose a threat to the security of RSA encryption. (viii) Message concealing A plaintext messagem,0 ≤m≤n−1, in the RSA public-key encryption scheme is said to beunconcealedif it encrypts to itself; that is,me≡m(mod n). There are always some messages which are unconcealed (for examplem=0 ,m=1 , andm= n−1). In fact, the number of unconcealed messages is exactly [1 +gcd(e−1,p−1)]·[1 + gcd(e−1,q−1)]. Sincee−1,p−1and q−1are all even, the number of unconcealed messages is always at least9.I fpand qare random primes, and ifeis chosen at random (or ifeis chosen to be a small number such ase=3 ore=2 16 + 1 = 65537), then the proportion of messages which are unconcealed by RSA encryption will, in general, be negligibly small, and hence unconcealed messages do not pose a threat to the security of RSA encryption in practice. 8.2.3 RSA encryption in practice There are numerous ways of speeding up RSA encryption and decryption in software and hardware implementations. Some of these techniques are covered in Chapter 14, includ- ing fast modular multiplication (§14.3), fast modular exponentiation (§14.6), and the use of the Chinese remainder theorem for faster decryption (Note 14.75). Even with these im- provements, RSA encryption/decryption is substantially slower than the commonly used symmetric-key encryption algorithms such as DES (Chapter 7). In practice, RSA encryp- tion is most commonly used for the transport of symmetric-key encryption algorithm keys and for the encryption of small data items. The RSA cryptosystem has been patented in the U.S. and Canada. Several standards organizations have written, or are in the process of writing, standards that address the use of the RSA cryptosystem for encryption, digital signatures, and key establishment. For dis- cussion of patent and standards issues related to RSA, see Chapter 15. 8.7 Note (recommended size of modulus) Given the latest progress in algorithms for factoring integers (§3.2), a512-bit modulusnprovides only marginal security from concerted attack. As of 1996, in order to foil the powerful quadratic sieve (§3.2.6) and number ﬁeld sieve (§3.2.7) factoring algorithms, a modulusnof at least768 bits is recommended. For long- term security,1024-bit or larger moduli should be used. 8.8 Note (selecting primes) (i) As mentioned in§8.2.2(i), the primespand qshould be selected so that factoring n= pqis computationally infeasible. The major restriction onpand qin order to avoid the elliptic curve factoring algorithm (§3.2.4) is thatpand qshould be about the same bitlength, and sufﬁciently large. For example, if a1024-bit modulusnis to be used, then each ofpand qshould be about512 bits in length. §8.2 RSA public-key encryption 291 (ii) Another restriction on the primespand qis that the differencep−qshould not be too small. Ifp−qis small, thenp ≈ qand hencep ≈ √ n. Thus,ncould be factored efﬁciently simply by trial division by all odd integers close to√ n.I fpand qare chosen at random, thenp−qwill be appropriately large with overwhelming probability. (iii) In addition to these restrictions, many authors have recommended thatpand qbe strong primes. A primepis said to be astrong prime(cf. Deﬁnition 4.52) if the fol- lowing three conditions are satisﬁed: (a)p−1 has a large prime factor, denotedr; (b) p+1 has a large prime factor; and (c)r−1 has a large prime factor. An algorithm for generating strong primes is presented in§4.4.2. The reason for con- dition (a) is to foil Pollard’sp−1factoring algorithm (§3.2.3) which is efﬁcient only ifnhas a prime factorpsuch thatp−1 is smooth. Condition (b) foils thep+1 factoring algorithm mentioned on page 125 in§3.12, which is efﬁcient only ifnhas a prime factorpsuch thatp+1 is smooth. Finally, condition (c) ensures that the cycling attacks described in§8.2.2(vii) will fail. If the primepis randomly chosen and is sufﬁciently large, then bothp−1and p+1 can be expected to have large prime factors. In any case, while strong primes protect against thep−1and p+1 factoring algorithms, they do not protect against their gen- eralization, the elliptic curve factoring algorithm (§3.2.4). The latter is successful in factoringnif a randomly chosen number of the same size asp(more precisely, this number is the order of a randomly selected elliptic curve deﬁned overZp) has only small prime factors. Additionally, it has been shown that the chances of a cycling at- tack succeeding are negligible ifpand qare randomly chosen (cf.§8.2.2(vii)). Thus, strong primes offer little protection beyond that offered by random primes. Given the current state of knowledge of factoring algorithms, there is no compelling reason for requiring the use of strong primes in RSA key generation. On the other hand, they are no less secure than random primes, and require only minimal additional running time to compute; thus there is little real additional cost in using them. 8.9 Note (small encryption exponents) (i) If the encryption exponenteis chosen at random, then RSA encryption using the re- peated square-and-multiply algorithm (Algorithm 2.143) takeskmodular squarings and an expectedk/2 (less with optimizations) modular multiplications, wherekis the bitlength of the modulusn. Encryption can be sped up by selectingeto be small and/or by selectingewith a small number of 1’s in its binary representation. (ii) The encryption exponente=3 is commonly used in practice; in this case, it is nec- essary that neitherp−1norq−1be divisible by3. This results in a very fast encryp- tion operation since encryption only requires 1 modular multiplication and 1 modular squaring. Another encryption exponent used in practice ise=2 16 + 1 = 65537 . This number has only two 1’s in its binary representation, and so encryption using the repeated square-and-multiply algorithm requires only16 modular squarings and 1 modular multiplication. The encryption exponente=2 16 +1 has the advantage overe=3 in that it resists the kind of attack discussed in§8.2.2(ii), since it is un- likely the same message will be sent to216+1 recipients. But see also Coppersmith’s attacks mentioned on pages 313–314. 292 Ch. 8 Public-Key Encryption 8.3 Rabin public-key encryption A desirable property of any encryption scheme is a proof that breaking it is as difﬁcult as solving a computational problem that is widely believed to be difﬁcult, such as integer fac- torization or the discrete logarithm problem. While it is widely believed that breaking the RSA encryption scheme is as difﬁcult as factoring the modulusn, no such equivalence has been proven. The Rabin public-key encryption scheme was the ﬁrst example of aprovably securepublic-key encryption scheme – the problem faced by a passive adversary of recov- ering plaintext from some given ciphertext is computationally equivalent to factoring. 8.10 AlgorithmKey generation for Rabin public-key encryption SUMMARY: each entity creates a public key and a corresponding private key. Each entityAshould do the following: 1. Generate two large random (and distinct) primespand q, each roughly the same size. 2. Compute n= pq. 3. A’s public key isn;A’s private key is(p,q). 8.11 AlgorithmRabin public-key encryption SUMMARY: Bencrypts a messagemforA, whichAdecrypts. 1. Encryption.Bshould do the following: (a) ObtainA’s authentic public keyn. (b) Represent the message as an integermin the range{0,1,...,n −1}. (c) Computec= m2 mod n. (d) Send the ciphertextctoA. 2. Decryption.To recover plaintextmfrom c,Ashould do the following: (a) Use Algorithm 3.44 to ﬁnd the four square rootsm1,m2,m3, andm4 ofcmod- ulon.2 (See also Note 8.12.) (b) The message sent was eitherm1,m2,m3,o rm4.Asomehow (cf. Note 8.14) decides which of these ism. 8.12 Note (ﬁnding square roots ofcmodulo n= pqwhen p≡q≡3( m o d4 ))I fpand qare both chosen to be≡3( m o d4 ), then Algorithm 3.44 for computing the four square roots ofcmodulo nsimpliﬁes as follows: 1. Use the extended Euclidean algorithm (Algorithm 2.107) to ﬁnd integersaand bsat- isfyingap+bq=1 . Note thataand bcan be computed once and for all during the key generation stage (Algorithm 8.10). 2. Compute r= c(p+1)/4 mod p. 3. Compute s= c(q+1)/4 mod q. 4. Compute x=( aps+bqr)m o dn. 5. Compute y=( aps−bqr)m o dn. 6. The four square roots ofcmodulo narex,−xmod n,y, and−ymod n. 2In the very unlikely case thatgcd(m,n) ̸=1, the ciphertextcdoes not have four distinct square roots modulo n, but rather only one or two. §8.3 Rabin public-key encryption 293 8.13 Note (security of Rabin public-key encryption) (i) The task faced by a passive adversary is to recover plaintextmfrom the correspond- ing ciphertextc. This is precisely the SQROOT problem of§3.5.2. Recall (Fact 3.46) that the problems of factoringnand computing square roots modulonare computa- tionally equivalent. Hence, assuming that factoringnis computationally intractable, the Rabin public-key encryption scheme isprovably secureagainst a passive adver- sary. (ii) While provably secure against a passive adversary, the Rabin public-key encryption scheme succumbs to a chosen-ciphertext attack (but see Note 8.14(ii)). Such an at- tack can be mounted as follows. The adversary selects a random integerm∈Z∗ nand computesc= m2 mod n. The adversary then presentsctoA’s decryption machine, which decryptscand returns some plaintexty. SinceAdoes not knowm, andmis randomly chosen, the plaintextyis not necessarily the same asm. With probability 1 2,y̸≡±mmod n, in which casegcd(m−y,n)is one of the prime factors ofn.I f y≡±mmod n, then the attack is repeated with a newm.3 (iii) The Rabin public-key encryption scheme is susceptible to attacks similar to those on RSA described in§8.2.2(ii),§8.2.2(iii), and§8.2.2(v). As is the case with RSA, at- tacks (ii) and (iii) can be circumvented by salting the plaintext message, while attack (v) can be avoided by adding appropriate redundancy prior to encryption. 8.14 Note (use of redundancy) (i) A drawback of Rabin’s public-key scheme is that the receiver is faced with the task of selecting the correct plaintext from among four possibilities. This ambiguity in decryption can easily be overcome in practice by adding prespeciﬁed redundancy to the original plaintext prior to encryption. (For example, the last 64 bits of the message may be replicated.) Then, with high probability, exactly one of the four square roots m 1,m2,m3,m4 of a legitimate ciphertextcwill possess this redundancy, and the receiver will select this as the intended plaintext. If none of the square roots ofc possesses this redundancy, then the receiver should rejectcas fraudulent. (ii) If redundancy is used as above, Rabin’s scheme is no longer susceptible to the chosen- ciphertext attack of Note 8.13(ii). If an adversary selects a messagemhaving the re- quired redundancy and givesc= m2 mod ntoA’s decryption machine, with very high probability the machine will return the plaintextmitself to the adversary (since the other three square roots ofcwill most likely not contain the required redundancy), providing no new information. On the other hand, if the adversary selects a message mwhich does not contain the required redundancy, then with high probability none of the four square roots ofc= m 2 mod nwill possess the required redundancy. In this case, the decryption machine will fail to decryptcand thus will not provide a re- sponse to the adversary. Note that the proof of equivalence of breaking the modiﬁed scheme by a passive adversary to factoring is no longer valid. However, if the natu- ral assumption is made that Rabin decryption is composed of two processes, the ﬁrst which ﬁnds the four square roots ofcmod n, and the second which selects the distin- guished square root as the plaintext, then the proof of equivalence holds. Hence, Ra- bin public-key encryption, suitably modiﬁed by adding redundancy, is of great prac- tical interest. 3This chosen-ciphertext attack is an execution of the constructive proof of the equivalence of factoringn and the SQROOT problem (Fact 3.46), whereA’s decryption machine is used instead of the hypothetical polynomial- time algorithm for solving the SQROOT problem in the proof. 294 Ch. 8 Public-Key Encryption 8.15 Example (Rabin public-key encryption with artiﬁcially small parameters) Key generation. EntityAchooses the primesp= 277,q= 331, and computesn= pq= 91687.A’s public key isn= 91687, whileA’s private key is(p= 277,q= 331). Encryption. Suppose that the last six bits of original messages are required to be repli- cated prior to encryption (cf. Note 8.14(i)). In order to encrypt the 10-bit message m= 1001111001,Breplicates the last six bits of mto obtain the 16-bit message m= 1001111001111001, which in decimal notation ism= 40569.Bthen computes c = m2 mod n = 40569 2 mod 91687 = 62111 and sends this toA. Decryption. To decryptc,Auses Algorithm 3.44 and her knowledge of the factors ofnto compute the four square roots ofcmod n: m1 = 69654,m 2 = 22033,m 3 = 40569,m 4 = 51118, which in binary are m1 = 10001000000010110,m 2 = 101011000010001, m3 = 1001111001111001,m 4 = 1100011110101110. Since onlym3 has the required redundancy,Adecryptsctom3 and recovers the original message m= 1001111001. □ 8.16 Note (efﬁciency) Rabin encryption is an extremely fast operation as it only involves a sin- gle modular squaring. By comparison, RSA encryption withe=3 takes one modular mul- tiplication and one modular squaring. Rabin decryption is slower than encryption, but com- parable in speed to RSA decryption. 8.4 ElGamal public-key encryption The ElGamal public-key encryption scheme can be viewed as Difﬁe-Hellman key agree- ment (§12.6.1) in key transfer mode (cf. Note 8.23(i)). Its security is based on the intractabil- ity of the discrete logarithm problem (see§3.6) and the Difﬁe-Hellman problem (§3.7). The basic ElGamal and generalized ElGamal encryption schemes are described in this section. 8.4.1 Basic ElGamal encryption 8.17 AlgorithmKey generation for ElGamal public-key encryption SUMMARY: each entity creates a public key and a corresponding private key. Each entityAshould do the following: 1. Generate a large random primepand a generatorαof the multiplicative groupZ∗ pof the integers modulop(using Algorithm 4.84). 2. Select a random integera,1 ≤a≤p−2, and computeαamod p(using Algo- rithm 2.143). 3. A’s public key is(p,α,αa);A’s private key isa. §8.4 ElGamal public-key encryption 295 8.18 AlgorithmElGamal public-key encryption SUMMARY: Bencrypts a messagemforA, whichAdecrypts. 1. Encryption.Bshould do the following: (a) ObtainA’s authentic public key(p,α,αa). (b) Represent the message as an integermin the range{0,1,...,p −1}. (c) Select a random integerk,1 ≤k≤p−2. (d) Computeγ= αk mod pand δ= m·(αa)kmod p. (e) Send the ciphertextc=( γ,δ) toA. 2. Decryption.To recover plaintextmfrom c,Ashould do the following: (a) Use the private keyato computeγp−1−amod p(note:γp−1−a = γ−a = α−ak). (b) Recovermby computing(γ−a)·δmod p. Proof that decryption works.The decryption of Algorithm 8.18 allows recovery of original plaintext because γ−a·δ ≡ α−akmαak ≡ m (mod p). 8.19 Example (ElGamal encryption with artiﬁcially small parameters) Key generation. EntityAselects the primep= 2357 and a generatorα=2 ofZ∗ 2357. A chooses the private keya= 1751 and computes αamod p =2 1751 mod 2357 = 1185 . A’s public key is(p= 2357,α=2 ,αa= 1185). Encryption. To encrypt a messagem= 2035,Bselects a random integerk= 1520 and computes γ =2 1520 mod 2357 = 1430 and δ = 2035 ·11851520 mod 2357 = 697 . Bsendsγ= 1430 and δ= 697 toA. Decryption. To decrypt,Acomputes γp−1−a = 1430 605 mod 2357 = 872 , and recoversmby computing m = 872 ·697 mod 2357 = 2035 . □ 8.20 Note (common system-wide parameters) All entities may elect to use the same primep and generatorα, in which casepand αneed not be published as part of the public key. This results in public keys of smaller sizes. An additional advantage of having a ﬁxed base αis that exponentiation can then be expedited via precomputations using the techniques described in§14.6.3. A potential disadvantage of common system-wide parameters is that larger modulipmay be warranted (cf. Note 8.24). 296 Ch. 8 Public-Key Encryption 8.21 Note (efﬁciency of ElGamal encryption) (i) The encryption process requires two modular exponentiations, namelyαk mod pand (αa)kmod p. These exponentiations can be sped up by selecting random exponents khaving some additional structure, for example, having low Hamming weights. Care must be taken that the possible number of exponents is large enough to preclude a search via a baby-step giant-step algorithm (cf. Note 3.59). (ii) A disadvantage of ElGamal encryption is that there ismessage expansionby a factor of 2. That is, the ciphertext is twice as long as the corresponding plaintext. 8.22 Remark (randomized encryption) ElGamal encryption is one of many encryption schemes which utilizes randomization in the encryption process. Others include McEliece encryp- tion (§8.5), and Goldwasser-Micali (§8.7.1), and Blum-Goldwasser (§8.7.2) probabilistic encryption. Deterministic encryption schemes such as RSA may also employ randomiza- tion in order to circumvent some attacks (e.g., see§8.2.2(ii) and§8.2.2(iii)). The fundamen- tal idea behind randomized encryption (see Deﬁnition 7.3) techniques is to use randomiza- tion to increase the cryptographic security of an encryption process through one or more of the following methods: (i) increasing the effective size of the plaintext message space; (ii) precluding or decreasing the effectiveness of chosen-plaintext attacks by virtue of a one-to-many mapping of plaintext to ciphertext; and (iii) precluding or decreasing the effectiveness of statistical attacks by leveling the a priori probability distribution of inputs. 8.23 Note (security of ElGamal encryption) (i) The problem of breaking the ElGamal encryption scheme, i.e., recoveringmgiven p,α,α a,γ, andδ, is equivalent to solving the Difﬁe-Hellman problem (see§3.7). In fact, the ElGamal encryption scheme can be viewed as simply comprising a Difﬁe- Hellman key exchange to determine a session keyαak, and then encrypting the mes- sage by multiplication with that session key. For this reason, the security of the El- Gamal encryption scheme is said to bebased on the discrete logarithm problem in Z ∗ p, although such an equivalence has not been proven. (ii) It is critical that different random integerskbe used to encrypt different messages. Suppose the samekis used to encrypt two messagesm1 and m2 and the resulting ciphertext pairs are(γ1,δ1) and (γ2,δ2). Thenδ1/δ2 = m1/m2, andm2 could be easily computed ifm1 were known. 8.24 Note (recommended parameter sizes) Given the latest progress on the discrete logarithm problem inZ∗ p(§3.6), a512-bit moduluspprovides only marginal security from concerted attack. As of 1996, a moduluspof at least768 bits is recommended. For long-term secu- rity,1024-bit or larger moduli should be used. For common system-wide parameters (cf. Note 8.20) even larger key sizes may be warranted. This is because the dominant stage in the index-calculus algorithm (§3.6.5) for discrete logarithms inZ∗ p is the precomputa- tion of a database of factor base logarithms, following which individual logarithms can be computed relatively quickly. Thus computing the database of logarithms for one particular modulus pwill compromise the secrecy of all private keys derived usingp. §8.4 ElGamal public-key encryption 297 8.4.2 Generalized ElGamal encryption The ElGamal encryption scheme is typically described in the setting of the multiplicative groupZ∗ p, but can be easily generalized to work in any ﬁnite cyclic groupG. As with ElGamal encryption, the security of the generalized ElGamal encryption sch- eme isbased on the intractability of the discrete logarithm problem in the groupG. The groupGshould be carefully chosen to satisfy the following two conditions: 1. forefﬁciency, the group operation inGshould be relatively easy to apply; and 2. forsecurity, the discrete logarithm problem inGshould be computationally infeasi- ble. The following is a list of groups that appear to meet these two criteria, of which the ﬁrst three have received the most attention. 1. The multiplicative groupZ∗ pof the integers modulo a primep. 2. The multiplicative groupF∗ 2 m of the ﬁnite ﬁeldF2m of characteristic two. 3. The group of points on an elliptic curve over a ﬁnite ﬁeld. 4. The multiplicative groupF∗ qof the ﬁnite ﬁeldFq, whereq= pm,pa prime. 5. The group of unitsZ∗ n , wherenis a composite integer. 6. The jacobian of a hyperelliptic curve deﬁned over a ﬁnite ﬁeld. 7. The class group of an imaginary quadratic number ﬁeld. 8.25 AlgorithmKey generation for generalized ElGamal public-key encryption SUMMARY: each entity creates a public key and a corresponding private key. Each entityAshould do the following: 1. Select an appropriate cyclic groupGof ordern, with generatorα. (It is assumed here thatGis written multiplicatively.) 2. Select a random integera,1 ≤a≤n−1, and compute the group elementαa. 3. A’s public key is(α,αa), together with a description of how to multiply elements in G;A’s private key isa. 8.26 AlgorithmGeneralized ElGamal public-key encryption SUMMARY: Bencrypts a messagemforA, whichAdecrypts. 1. Encryption.Bshould do the following: (a) ObtainA’s authentic public key(α,αa). (b) Represent the message as an elementmof the groupG. (c) Select a random integerk,1 ≤k≤n−1. (d) Computeγ= αkand δ= m·(αa)k. (e) Send the ciphertextc=( γ,δ) toA. 2. Decryption.To recover plaintextmfrom c,Ashould do the following: (a) Use the private keyato computeγaand then computeγ−a. (b) Recovermby computing(γ−a)·δ. 8.27 Note (common system-wide parameters) All entities may elect to use the same cyclic groupGand generatorα, in which caseαand the description of multiplication inGneed not be published as part of the public key (cf. Note 8.20). 298 Ch. 8 Public-Key Encryption 8.28 Example (ElGamal encryption using the multiplicative group ofF2m, with artiﬁcially small parameters) Key generation. EntityAselects the groupGto be the multiplicative group of the ﬁnite ﬁeld F24, whose elements are represented by the polynomials overF2 of degree less than4, and where multiplication is performed modulo the irreducible polynomialf(x)= x4 +x+1 (cf. Example 2.231). For convenience, a ﬁeld elementa3x3 + a2x2 + a1x+ a0 is repre- sented by the binary string(a3a2a1a0). The groupGhas ordern=1 5 and a generator is α= (0010). Achooses the private keya=7 and computesαa = α7 = (1011) . A’s public key is αa = (1011) (together withα= (0010) and the polynomialf(x) which deﬁnes the mul- tiplication inG, if these parameters are not common to all entities). Encryption. To encrypt a messagem= (1100),Bselects a random integerk=1 1 and computesγ= α11 = (1110),(αa)11 = (0100), andδ= m·(αa)11 = (0101).Bsends γ= (1110) and δ= (0101) toA. Decryption. To decrypt,Acomputesγa= (0100),(γa)−1 = (1101) and ﬁnally recovers mby computingm=( γ−a)·δ= (1100). □ 8.5 McEliece public-key encryption The McEliece public-key encryption scheme is based on error-correcting codes. The idea behind this scheme is to ﬁrst select a particular code for which an efﬁcient decoding algo- rithm is known, and then to disguise the code as a general linear code (see Note 12.36). Since the problem of decoding an arbitrary linear code isNP -hard (Deﬁnition 2.73), a de- scription of the original code can serve as the private key, while a description of the trans- formed code serves as the public key. The McEliece encryption scheme (when used with Goppa codes) has resisted crypt- analysis to date. It is also notable as being the ﬁrst public-key encryption scheme to use randomization in the encryption process. Although very efﬁcient, the McEliece encryption scheme has received little attention in practice because of the very large public keys (see Remark 8.33). 8.29 AlgorithmKey generation for McEliece public-key encryption SUMMARY: each entity creates a public key and a corresponding private key. 1. Integersk,n, andtare ﬁxed as common system parameters. 2. Each entityAshould perform steps 3 – 7. 3. Choose ak×ngenerator matrixGfor a binary(n,k)-linear code which can correct terrors, and for which an efﬁcient decoding algorithm is known. (See Note 12.36.) 4. Select a randomk×kbinary non-singular matrixS. 5. Select a randomn×npermutation matrixP. 6. Compute thek×nmatrixˆG= SGP. 7. A’s public key is(ˆG,t);A’s private key is(S,G,P). §8.5 McEliece public-key encryption 299 8.30 AlgorithmMcEliece public-key encryption SUMMARY: Bencrypts a messagemforA, whichAdecrypts. 1. Encryption.Bshould do the following: (a) ObtainA’s authentic public key(ˆG,t). (b) Represent the message as a binary stringmof lengthk. (c) Choose a random binary error vectorzof lengthnhaving at mostt1’s. (d) Compute the binary vectorc= mˆG+z. (e) Send the ciphertextctoA. 2. Decryption.To recover plaintextmfrom c,Ashould do the following: (a) Computeˆc= cP−1, whereP−1 is the inverse of the matrixP. (b) Use the decoding algorithm for the code generated byGto decodeˆcto ˆm. (c) Computem= ˆmS−1. Proof that decryption works.Since ˆc = cP−1 =( mˆG+z)P−1 =( mSGP+z)P−1 =( mS)G+zP−1, and zP−1 is a vector with at mostt1’s, the decoding algorithm for the code generated by Gcorrectsˆcto ˆm= mS. Finally,ˆmS−1 = m, and, hence, decryption works. A special type of error-correcting code, called aGoppa code, may be used in step 3 of the key generation. For each irreducible polynomialg(x)of degreetoverF2m, there exists a binary Goppa code of lengthn=2 mand dimensionk≥n−mtcapable of correcting any pattern oftor fewer errors. Furthermore, efﬁcient decoding algorithms are known for such codes. 8.31 Note (security of McEliece encryption) There are two basic kinds of attacks known. (i) From the public information, an adversary may try to compute the keyGor a keyG′ for a Goppa code equivalent to the one with generator matrixG. There is no efﬁcient method known for accomplishing this. (ii) An adversary may try to recover the plaintextmdirectly given some ciphertextc. The adversary pickskcolumns at random fromˆG.I fˆGk,ckand zkdenote the restriction of ˆG,cand z, respectively, to thesekcolumns, then(ck+zk)= mˆGk.I fzk=0 and ifˆGk is non-singular, thenmcan be recovered by solving the system of equations ck = mˆGk. Since the probability thatzk =0 , i.e., the selectedkbits were not in error, is only (n−t k ) / (n k ) , the probability of this attack succeeding is negligibly small. 8.32 Note (recommended parameter sizes) The original parameters suggested by McEliece were n= 1024 ,t=5 0 , andk ≥524. Based on the security analysis (Note 8.31), an optimum choice of parameters for the Goppa code which maximizes the adversary’s work factor appears to ben= 1024,t=3 8, andk≥644. 8.33 Remark (McEliece encryption in practice) Although the encryption and decryption oper- ations are relatively fast, the McEliece scheme suffers from the drawback that the public key is very large. A (less signiﬁcant) drawback is that there is message expansion by a fac- tor ofn/k. For the recommended parametersn= 1024,t=3 8,k≥644, the public key is about2 19 bits in size, while the message expansion factor is about1.6. For these reasons, the scheme receives little attention in practice. 300 Ch. 8 Public-Key Encryption 8.6 Knapsack public-key encryption Knapsack public-key encryption schemes are based on the subset sum problem, which is NP -complete (see§2.3.3 and§3.10). The basic idea is to select an instance of the subset sum problem that is easy to solve, and then to disguise it as an instance of the general subset sum problem which is hopefully difﬁcult to solve. The original knapsack set can serve as the private key, while the transformed knapsack set serves as the public key. The Merkle-Hellman knapsack encryption scheme (§8.6.1) is important for historical reasons, as it was the ﬁrst concrete realization of a public-key encryption scheme. Many variations have subsequently been proposed but most, including the original, have been demonstrated to be insecure (see Note 8.40), a notable exception being the Chor-Rivest knapsack scheme (§8.6.2). 8.6.1 Merkle-Hellman knapsack encryption The Merkle-Hellman knapsack encryption scheme attempts to disguise an easily solved in- stance of the subset sum problem, called asuperincreasing subset sum problem, by modular multiplication and a permutation. It is however not recommended for use (see Note 8.40). 8.34 DeﬁnitionA superincreasing sequenceis a sequence(b 1,b2,...,b n)of positive integers with the property thatbi>∑i−1 j=1 bjfor eachi,2 ≤i≤n. Algorithm 8.35 efﬁciently solves the subset sum problem for superincreasing sequences. 8.35 AlgorithmSolving a superincreasing subset sum problem INPUT: a superincreasing sequence(b1,b2,...,b n)and an integerswhich is the sum of a subset of thebi. OUTPUT: (x1,x2,...,x n) where xi∈{0,1}, such that∑n i=1 xibi= s. 1. i←n. 2. Whilei≥1 do the following: 2.1 Ifs≥bithenxi←1 and s←s−bi. Otherwisexi←0. 2.2 i←i−1. 3. Return((x1,x2,...,x n)). 8.36 AlgorithmKey generation for basic Merkle-Hellman knapsack encryption SUMMARY: each entity creates a public key and a corresponding private key. 1. An integernis ﬁxed as a common system parameter. 2. Each entityAshould perform steps 3 – 7. 3. Choose a superincreasing sequence(b1,b2,...,b n) and modulusMsuch thatM> b1 +b2 +··· +bn. 4. Select a random integerW,1 ≤W≤M−1, such thatgcd(W,M)=1 . 5. Select a random permutationπof the integers{1,2,...,n }. 6. Compute ai= Wbπ(i) mod Mfori=1 ,2,...,n . 7. A’s public key is(a1,a2,...,a n);A’s private key is(π,M,W, (b1,b2,...,b n)). §8.6 Knapsack public-key encryption 301 8.37 AlgorithmBasic Merkle-Hellman knapsack public-key encryption SUMMARY: Bencrypts a messagemforA, whichAdecrypts. 1. Encryption.Bshould do the following: (a) ObtainA’s authentic public key(a1,a2,...,a n). (b) Represent the messagemas a binary string of lengthn,m= m1m2 ···mn. (c) Compute the integerc= m1a1 +m2a2 +··· +mnan. (d) Send the ciphertextctoA. 2. Decryption.To recover plaintextmfrom c,Ashould do the following: (a) Computed= W−1cmod M. (b) By solving a superincreasing subset sum problem (Algorithm 8.35), ﬁnd inte- gersr1,r2,...,r n,ri∈{0,1}, such thatd= r1b1 +r2b2 +··· +rnbn. (c) The message bits aremi= rπ(i),i=1 ,2,...,n . Proof that decryption works.The decryption of Algorithm 8.37 allows recovery of original plaintext because d ≡ W−1c ≡ W−1 n∑ i=1 miai ≡ n∑ i=1 mibπ(i) (mod M). Since0 ≤d<M ,d= ∑n i=1 mibπ(i) mod M, and hence the solution of the superincreas- ing subset sum problem in step (b) of the decryption gives the message bits, after application of the permutationπ. 8.38 Example (basic Merkle-Hellman knapsack encryption with artiﬁcially small parameters) Key generation. Letn=6 . EntityAchooses the superincreasing sequence(12,17,33,74, 157,316),M = 737 ,W = 635 , and the permutationπof {1,2,3,4,5,6}deﬁned by π(1) = 3,π(2) = 6,π(3) = 1,π(4) = 2,π(5) = 5, andπ(6) = 4.A’s public key is the knapsack set(319,196,250,477,200,559), whileA’s private key is(π,M,W, (12,17,33, 74,157,316)). Encryption. To encrypt the messagem= 101101,Bcomputes c = 319+250+477+559 = 1605 and sends this toA. Decryption. To decrypt,Acomputesd= W−1cmod M= 136, and solves the superin- creasing subset sum problem 136 = 12r1 +17r2 +33r3 +74r4 +157r5 +316r6 to get136 = 12+17+33+74 . Hence,r1 =1 ,r2 =1 ,r3 =1 ,r4 =1 ,r5 =0 ,r6 =0 , and application of the permutationπyields the message bitsm1 = r3 =1 ,m2 = r6 =0 , m3 = r1 =1 ,m4 = r2 =1 ,m5 = r5 =0 ,m6 = r4 =1 . □ Multiple-iterated Merkle-Hellman knapsack encryption One variation of the basic Merkle-Hellman scheme involves disguising the easy superin- creasing sequence by a series of modular multiplications. The key generation for this vari- ation is as follows. 302 Ch. 8 Public-Key Encryption 8.39 AlgorithmKey generation for multiple-iterated Merkle-Hellman knapsack encryption SUMMARY: each entity creates a public key and a corresponding private key. 1. Integersnand tare ﬁxed as common system parameters. 2. Each entityAshould perform steps 3 – 6. 3. Choose a superincreasing sequence(a(0) 1 ,a(0) 2 ,...,a (0) n ). 4. Forjfrom 1 totdo the following: 4.1 Choose a modulusMjwithMj >a(j−1) 1 +a(j−1) 2 +··· +a(j−1) n . 4.2 Select a random integerWj,1 ≤Wj ≤Mj−1, such thatgcd(Wj,Mj)=1 . 4.3 Compute a(j) i = a(j−1) i Wj mod Mjfori=1 ,2,...,n . 5. Select a random permutationπof the integers{1,2,...,n }. 6. A’s public key is(a1,a2,...,a n), whereai= a(t) π(i) fori=1 ,2,...,n ;A’s private key is(π,M1,...,M t,W1,...,W t,a(0) 1 ,a(0) 2 ,...,a (0) n ). Encryption is performed in the same way as in the basic Merkle-Hellman scheme (Al- gorithm 8.37). Decryption is performed by successively computingdj = W−1 j dj+1 mod Mjforj= t,t−1,..., 1, wheredt+1 = c. Finally, the superincreasing subset sum prob- lem d1 = r1a(0) 1 +r2a(0) 2 +···+rna(0) n is solved forri, and the message bits are recovered after application of the permutationπ. 8.40 Note (insecurity of Merkle-Hellman knapsack encryption) (i) A polynomial-time algorithm for breaking the basic Merkle-Hellman scheme is known. Given the public knapsack set, this algorithm ﬁnds a pair of integersU′,M′ such thatU′/M′is close toU/M(whereWand Mare part of the private key, and U= W−1 mod M) and such that the integersb′ i= U′aimod M,1 ≤i≤n, form a superincreasing sequence. This sequence can then be used by an adversary in place of(b1,b2,...,b n) to decrypt messages. (ii) The most powerful general attack known on knapsack encryption schemes is the tech- nique discussed in§3.10.2 which reduces the subset sum problem to the problem of ﬁnding a short vector in a lattice. It is typically successful if the density (see Deﬁ- nition 3.104) of the knapsack set is less than0.9408. This is signiﬁcant because the density of a Merkle-Hellman knapsack set must be less than 1, since otherwise there will in general be many subsets of the knapsack set with the same sum, in which case some ciphertexts will not be uniquely decipherable. Moreover, since each iteration in the multiple-iterated scheme lowers the density, this attack will succeed if the knap- sack set has been iterated a sufﬁcient number of times. Similar techniques have since been used to break most knapsacks schemes that have been proposed, including the multiple-iterated Merkle-Hellman scheme. The most promi- nent knapsack scheme that has resisted such attacks to date is the Chor-Rivest scheme (but see Note 8.44). 8.6.2 Chor-Rivest knapsack encryption The Chor-Rivest scheme is the only known knapsack public-key encryption scheme that does not use some form of modular multiplication to disguise an easy subset sum problem. §8.6 Knapsack public-key encryption 303 8.41 AlgorithmKey generation for Chor-Rivest public-key encryption SUMMARY: each entity creates a public key and a corresponding private key. Each entityAshould do the following: 1. Select a ﬁnite ﬁeldFq of characteristicp, whereq= ph,p≥h, and for which the discrete logarithm problem is feasible (see Note 8.45(ii)). 2. Select a random monic irreducible polynomialf(x) of degreehoverZp(using Al- gorithm 4.70). The elements ofFq will be represented as polynomials inZp[x] of degree less thanh, with multiplication performed modulof(x). 3. Select a random primitive elementg(x) of the ﬁeldFq(using Algorithm 4.80). 4. For each ground ﬁeld elementi∈Zp, ﬁnd the discrete logarithmai=l o gg(x)(x+i) of the ﬁeld element(x+i) to the baseg(x). 5. Select a random permutationπon the set of integers{0,1,2,...,p −1}. 6. Select a random integerd,0 ≤d≤ph−2. 7. Compute ci=( aπ(i) +d)m o d(ph−1) ,0 ≤i≤p−1. 8. A’s public key is((c0,c1,...,c p−1),p,h);A’s private key is(f(x),g(x),π,d). 8.42 AlgorithmChor-Rivest public-key encryption SUMMARY: Bencrypts a messagemforA, whichAdecrypts. 1. Encryption.Bshould do the following: (a) ObtainA’s authentic public key((c0,c1,...,c p−1),p,h). (b) Represent the messagemas a binary string of length⌊lg (p h ) ⌋, where (p h ) is a binomial coefﬁcient (Deﬁnition 2.17). (c) Considermas the binary representation of an integer. Transform this integer into a binary vectorM=( M0,M1,...,M p−1) of lengthphaving exactlyh 1’s as follows: i. Setl←h. ii. Forifrom 1 topdo the following: Ifm≥ (p−i l ) then setMi−1←1, m←m− (p−i l ) , l←l−1. Otherwise, setMi−1←0. (Note: (n 0 ) =1 forn≥0; (0 l ) =0 forl≥1.) (d) Computec= ∑p−1 i=0 Micimod (ph−1). (e) Send the ciphertextctoA. 2. Decryption.To recover plaintextmfrom c,Ashould do the following: (a) Computer=( c−hd)m o d(ph−1). (b) Computeu(x)= g(x)r mod f(x) (using Algorithm 2.227). (c) Computes(x)= u(x)+ f(x), a monic polynomial of degreehoverZp. (d) Factors(x) into linear factors overZp:s(x)= ∏h j=1(x+tj), wheretj ∈Zp (cf. Note 8.45(iv)). (e) Compute a binary vectorM =( M0,M1,...,M p−1) as follows. The com- ponents ofMthat are 1 have indicesπ−1(tj),1 ≤ j ≤ h. The remaining components are0. (f) The messagemis recovered fromMas follows: i. Setm←0,l←h. ii. Forifrom 1 topdo the following: IfMi−1 =1 then setm←m+ (p−i l ) and l←l−1. 304 Ch. 8 Public-Key Encryption Proof that decryption works.Observe that u(x)= g(x)rmod f(x) ≡ g(x)c−hd ≡ g(x)(∑p−1 i=0 Mici)−hd (mod f(x)) ≡ g(x)(∑p−1 i=0 Mi(aπ(i)+d))−hd ≡ g(x) ∑p−1 i=0 Miaπ(i) (mod f(x)) ≡ p−1∏ i=0 [g(x)aπ(i)]Mi ≡ p−1∏ i=0 (x+π(i))Mi (mod f(x)). Since∏p−1 i=0 (x+ π(i))Mi and s(x) are monic polynomials of degreehand are congruent modulo f(x), it must be the case that s(x)= u(x)+ f(x)= p−1∏ i=0 (x+π(i))Mi. Hence, thehroots ofs(x)all lie inZp, and applyingπ−1 to these roots gives the coordinates ofMthat are 1. 8.43 Example (Chor-Rivest public-key encryption with artiﬁcially small parameters) Key generation. EntityAdoes the following: 1. Selectsp=7 and h=4 . 2. Selects the irreducible polynomialf(x)= x4 +3 x3 +5 x2 +6 x+2 of degree4 overZ7. The elements of the ﬁnite ﬁeldF74 are represented as polynomials inZ7[x] of degree less than4, with multiplication performed modulof(x). 3. Selects the random primitive elementg(x)=3 x3 +3x2 +6. 4. Computes the following discrete logarithms: a0 =l o gg(x)(x) = 1028 a1 =l o gg(x)(x+1) = 1935 a2 =l o gg(x)(x+2) = 2054 a3 =l o gg(x)(x+3) = 1008 a4 =l o gg(x)(x+4) = 379 a5 =l o gg(x)(x+5) = 1780 a6 =l o gg(x)(x+6) = 223 . 5. Selects the random permutationπon {0,1,2,3,4,5,6}deﬁned byπ(0) = 6,π(1) = 4,π(2) = 0,π(3) = 2,π(4) = 1,π(5) = 5,π(6) = 3. 6. Selects the random integerd= 1702. 7. Computes c0 =( a6 +d) mod 2400 = 1925 c1 =( a4 +d) mod 2400 = 2081 c2 =( a0 +d) mod 2400 = 330 c3 =( a2 +d) mod 2400 = 1356 c4 =( a1 +d) mod 2400 = 1237 c5 =( a5 +d) mod 2400 = 1082 c6 =( a3 +d) mod 2400 = 310. §8.6 Knapsack public-key encryption 305 8. A’s public key is((c0,c1,c2,c3,c4,c5,c6),p =7 ,h=4 ), whileA’s private key is (f(x),g(x),π,d). Encryption. To encrypt a messagem=2 2 forA,Bdoes the following: (a) Obtains authenticA’s public key. (b) Representsmas a binary string of length5:m= 10110. (Note that⌊lg (7 4 ) ⌋=5 .) (c) Uses the method outlined in step 1(c) of Algorithm 8.42 to transformmto the binary vectorM=( 1,0,1,1,0,0,1)of length7. (d) Computesc=( c0 +c2 +c3 +c6) mod 2400 = 1521. (e) Sendsc= 1521 toA. Decryption. To decrypt the ciphertextc= 1521,Adoes the following: (a) Computesr=( c−hd) mod 2400 = 1913. (b) Computesu(x)= g(x)1913 mod f(x)= x3 +3x2 +2x+5. (c) Computess(x)= u(x)+ f(x)= x4 +4x3 +x2 +x. (d) Factorss(x)= x(x+2) (x+3) (x+6) (sot1 =0 ,t2 =2 ,t3 =3 ,t4 =6 ). (e) The components ofMthat are 1 have indicesπ−1(0) = 2,π−1(2) = 3,π−1(3) = 6, and π−1(6) = 0. Hence,M=( 1,0,1,1,0,0,1). (f) Uses the method outlined in step 2(f) of Algorithm 8.42 to transformMto the integer m=2 2, thus recovering the original plaintext. □ 8.44 Note (security of Chor-Rivest encryption) (i) When the parameters of the system are carefully chosen (see Note 8.45 and page 318), there is no feasible attack known on the Chor-Rivest encryption scheme. In partic- ular, the density of the knapsack set(c0,c1,...,c p−1) isp/lg(maxci), which is large enough to thwart the low-density attacks on the general subset sum problem (§3.10.2). (ii) It is known that the system is insecure if portions of the private key are revealed, for example, ifg(x) and din some representation ofFqare known, or iff(x) is known, or ifπis known. 8.45 Note (implementation) (i) Although the Chor-Rivest scheme has been described only for the casepa prime, it extends to the case where the base ﬁeldZpis replaced by a ﬁeld of prime power order. (ii) In order to make the discrete logarithm problem feasible in step 1 of Algorithm 8.41, the parameterspand hmay be chosen so thatq= ph−1 has only small factors. In this case, the Pohlig-Hellman algorithm (§3.6.4) can be used to efﬁciently compute discrete logarithms in the ﬁnite ﬁeldFq. (iii) In practice, the recommended size of the parameters arep≈200 and h≈25. One particular choice of parameters originally suggested isp= 197 and h=2 4; in this case, the largest prime factor of19724 −1 is10316017, and the density of the knap- sack set is about1.077. Other parameter sets originally suggested are{p= 211,h= 24},{p=3 5,h=2 4}(base ﬁeldF35), and{p=2 8,h=2 5}(base ﬁeldF28). (iv) Encryption is a very fast operation. Decryption is much slower, the bottleneck being the computation ofu(x)in step 2b. The roots ofs(x)in step 2d can be found simply by trying all possibilities inZp. (v) A major drawback of the Chor-Rivest scheme is that the public key is fairly large, namely, about(ph·lgp) bits. For the parametersp= 197 and h=2 4, this is about 36000 bits. 306 Ch. 8 Public-Key Encryption (vi) There is message expansion by a factor oflgph/lg (p h ) . Forp= 197 and h=2 4, this is1.797. 8.7 Probabilistic public-key encryption A minimal security requirement of an encryption scheme is that it must be difﬁcult, in es- sentially all cases, for a passive adversary to recover plaintext from the corresponding ci- phertext. However, in some situations, it may be desirable to impose more stringent security requirements. The RSA, Rabin, and knapsack encryption schemes aredeterministicin the sense that under a ﬁxed public key, a particular plaintextmis always encrypted to the same ciphertext c. A deterministic scheme has some or all of the following drawbacks. 1. The scheme is not secure for all probability distributions of the message space. For example, in RSA the messages0and1always get encrypted to themselves, and hence are easy to detect. 2. It is sometimes easy to compute partial information about the plaintext from the ci- phertext. For example, in RSA ifc= m emod nis the ciphertext corresponding to a plaintextm, then (c n ) = (me n ) = (m n )e = (m n ) sinceeis odd, and hence an adversary can easily gain one bit of information about m, namely the Jacobi symbol (m n ) . 3. It is easy to detect when the same message is sent twice. Of course, any deterministic encryption scheme can be converted into a randomized scheme by requiring that a portion of each plaintext consist of a randomly generated bit- string of a pre-speciﬁed lengthl. If the parameterlis chosen to be sufﬁciently large for the purpose at hand, then, in practice, the attacks listed above are thwarted. However, the re- sulting randomized encryption scheme is generally not provably secure against the different kinds of attacks that one could conceive. Probabilistic encryptionutilizes randomness to attain aprovableand very strong level of security. There are two strong notions of security that one can strive to achieve. 8.46 DeﬁnitionA public-key encryption scheme is said to bepolynomially secureif no passive adversary can, in expected polynomial time, select two plaintext messagesm 1 and m2 and then correctly distinguish between encryptions ofm1 and m2 with probability signiﬁcantly greater than1 2. 8.47 DeﬁnitionA public-key encryption scheme is said to besemantically secureif, for all probability distributions over the message space, whatever a passive adversary can compute in expected polynomial time about the plaintext given the ciphertext, it can also compute in expected polynomial time without the ciphertext. Intuitively, a public-key encryption scheme is semantically secure if the ciphertext does not leak any partial information whatsoever about the plaintext that can be computed in expected polynomial time. §8.7 Probabilistic public-key encryption 307 8.48 Remark (perfect secrecy vs. semantic security) In Shannon’s theory (see§1.13.3(i)), an encryption scheme hasperfect secrecyif a passive adversary, even with inﬁnite computa- tional resources, can learn nothing about the plaintext from the ciphertext, except possibly its length. The limitation of this notion is that perfect secrecy cannot be achieved unless the key is at least as long as the message. By contrast, the notion of semantic security can be viewed as a polynomially bounded version of perfect secrecy — a passive adversary with polynomially bounded computational resources can learn nothing about the plaintext from the ciphertext. It is then conceivable that there exist semantically secure encryption sch- emes where the keys are much shorter that the messages. Although Deﬁnition 8.47 appears to be stronger than Deﬁnition 8.46, the next result asserts that they are, in fact, equivalent. 8.49 FactA public-key encryption scheme is semantically secure if and only if it is polynomi- ally secure. 8.7.1 Goldwasser-Micali probabilistic encryption The Goldwasser-Micali scheme is a probabilistic public-key system which is semantically secure assuming the intractability of the quadratic residuosity problem (see§3.4). 8.50 AlgorithmKey generation for Goldwasser-Micali probabilistic encryption SUMMARY: each entity creates a public key and corresponding private key. Each entityAshould do the following: 1. Select two large random (and distinct) primespand q, each roughly the same size. 2. Compute n= pq. 3. Select ay∈Z nsuch thatyis a quadratic non-residue modulonand the Jacobi sym- bol (y n ) =1 (yis a pseudosquare modulon); see Remark 8.54. 4. A’s public key is(n,y);A’s private key is the pair(p,q). 8.51 AlgorithmGoldwasser-Micali probabilistic public-key encryption SUMMARY: Bencrypts a messagemforA, whichAdecrypts. 1. Encryption.Bshould do the following: (a) ObtainA’s authentic public key(n,y). (b) Represent the messagemas a binary stringm= m1m2 ···mtof lengtht. (c) Forifrom 1 totdo: i. Pick anx∈Z∗ nat random. ii. Ifmi=1 then setci←yx2 mod n; otherwise setci←x2 mod n. (d) Send thet-tuplec=( c1,c2,...,c t) toA. 2. Decryption.To recover plaintextmfrom c,Ashould do the following: (a) Forifrom 1 totdo: i. Compute the Legendre symbolei= (ci p ) (using Algorithm 2.149). ii. Ifei=1 then setmi←0; otherwise setmi←1. (b) The decrypted message ism= m1m2 ···mt. 308 Ch. 8 Public-Key Encryption Proof that decryption works.If a message bitmiis 0, thenci = x2 mod nis a quadratic residue modulon. If a message bitmi is 1, then sinceyis a pseudosquare modulon, ci= yx2 mod nis also a pseudosquare modulon. By Fact 2.137,ciis a quadratic residue modulo nif and only ifciis a quadratic residue modulop, or equivalently (ci p ) =1 . Since Aknows p, she can compute this Legendre symbol and hence recover the message bitmi. 8.52 Note (security of Goldwasser-Micali probabilistic encryption) Sincexis selected at ran- dom fromZ∗ n,x2 mod nis a random quadratic residue modulon, andyx2 mod nis a ran- dom pseudosquare modulon. Hence, an eavesdropper sees random quadratic residues and pseudosquares modulon. Assuming that the quadratic residuosity problem is difﬁcult, the eavesdropper can do no better that guess each message bit. More formally, if the quadratic residuosity problem is hard, then the Goldwasser-Micali probabilistic encryption scheme is semantically secure. 8.53 Note (message expansion) A major disadvantage of the Goldwasser-Micali scheme is the message expansion by a factor oflgnbits. Some message expansion is unavoidable in a probabilistic encryption scheme because there are many ciphertexts corresponding to each plaintext. Algorithm 8.56 is a major improvement of the Goldwasser-Micali scheme in that the plaintext is only expanded by a constant factor. 8.54 Remark (ﬁnding pseudosquares) A pseudosquareymodulo ncan be found as follows. First ﬁnd a quadratic non-residueamodulo pand a quadratic non-residuebmodulo q(see Remark 2.151). Then use Gauss’s algorithm (Algorithm 2.121) to compute the integery, 0 ≤y≤n−1, satisfying the simultaneous congruencesy≡a(mod p),y≡b(mod q). Sincey(≡ a(mod p)) is a quadratic non-residue modulop, it is also a quadratic non- residue modulon(Fact 2.137). Also, by the properties of the Legendre and Jacobi symbols (§2.4.5), ( y n ) = (y p )(y q ) =( −1)(−1) = 1. Hence,yis a pseudosquare modulon. 8.7.2 Blum-Goldwasser probabilistic encryption The Blum-Goldwasser probabilistic public-key encryption scheme is the most efﬁcient probabilistic encryption scheme known and is comparable to the RSA encryption scheme, both in terms of speed and message expansion. It is semantically secure (Deﬁnition 8.47) assuming the intractability of the integer factorization problem. It is, however, vulnerable to a chosen-ciphertext attack (see Note 8.58(iii)). The scheme uses the Blum-Blum-Shub generator (§5.5.2) to generate a pseudorandom bit sequence which is then XORed with the plaintext. The resulting bit sequence, together with an encryption of the random seed used, is transmitted to the receiver who uses his trapdoor information to recover the seed and sub- sequently reconstruct the pseudorandom bit sequence and the plaintext. 8.55 AlgorithmKey generation for Blum-Goldwasser probabilistic encryption SUMMARY: each entity creates a public key and a corresponding private key. Each entityAshould do the following: 1. Select two large random (and distinct) primesp,q, each congruent to3 modulo 4. 2. Compute n= pq. 3. Use the extended Euclidean algorithm (Algorithm 2.107) to compute integersaand bsuch thatap+bq=1 . 4. A’s public key isn;A’s private key is(p,q,a,b ). §8.7 Probabilistic public-key encryption 309 8.56 AlgorithmBlum-Goldwasser probabilistic public-key encryption SUMMARY: Bencrypts a messagemforA, whichAdecrypts. 1. Encryption.Bshould do the following: (a) ObtainA’s authentic public keyn. (b) Letk = ⌊lgn⌋and h= ⌊lgk⌋. Represent the messagemas a stringm= m1m2 ···mtof lengtht, where eachmiis a binary string of lengthh. (c) Select as a seedx0, a random quadratic residue modulon. (This can be done by selecting a random integerr∈Z∗ nand settingx0←r2 mod n.) (d) Forifrom 1 totdo the following: i. Computexi= x2 i−1 mod n. ii. Letpibe thehleast signiﬁcant bits ofxi. iii. Computeci= pi⊕mi. (e) Computext+1 = x2 t mod n. (f) Send the ciphertextc=( c1,c2,...,c t,xt+1) toA. 2. Decryption.To recover plaintextmfrom c,Ashould do the following: (a) Computed1 =( (p+1)/4)t+1 mod (p−1). (b) Computed2 =( (q+1)/4)t+1 mod (q−1). (c) Computeu= xd1 t+1 mod p. (d) Computev= xd2 t+1 mod q. (e) Computex0 = vap+ubqmod n. (f) Forifrom 1 totdo the following: i. Computexi= x2 i−1 mod n. ii. Letpibe thehleast signiﬁcant bits ofxi. iii. Computemi= pi⊕ci. Proof that decryption works.Sincextis a quadratic residue modulon, it is also a quadratic residue modulop; hence,x(p−1)/2 t ≡1( m o dp). Observe that x(p+1)/4 t+1 ≡ (x2 t)(p+1)/4 ≡ x(p+1)/2 t ≡ x(p−1)/2 t xt ≡ xt (mod p). Similarly,x(p+1)/4 t ≡xt−1 (mod p) and so x((p+1)/4)2 t+1 ≡ xt−1 (mod p). Repeating this argument yields u ≡ xd1 t+1 ≡ x((p+1)/4)t+1 t+1 ≡ x0 (mod p). Analogously, v ≡ xd2 t+1 ≡ x0 (mod q). Finally, sinceap+ bq=1 ,vap+ ubq≡x0 (mod p) and vap+ ubq≡x0 (mod q). Hence,x0 = vap+ubqmod n, andArecovers the same random seed thatBused in the encryption, and consequently also recovers the original plaintext. 8.57 Example (Blum-Goldwasser probabilistic encryption with artiﬁcially small parameters) Key generation. EntityAselects the primesp= 499,q= 547, each congruent to3modulo 4, and computesn= pq= 272953. Using the extended Euclidean algorithm,Acomputes 310 Ch. 8 Public-Key Encryption the integersa= −57,b=5 2 satisfyingap+bq=1 .A’s public key isn= 272953, while A’s private key is(p,q,a,b ). Encryption. The parameterskand hhave the values18 and 4, respectively.Brepresents the messagemas a stringm1m2m3m4m5 (t=5 ) wherem1 = 1001,m2 = 1100,m3 = 0001,m4 = 0000,m5 = 1100.Bthen selects a random quadratic residuex0 = 159201 (= 3992 mod n), and computes: i xi = x2 i−1 mod n pi ci = pi ⊕mi 1 180539 1011 0010 2 193932 1100 0000 3 245613 1101 1100 4 130286 1110 1110 5 40632 1000 0100 and x6 = x2 5 mod n= 139680.Bsends the ciphertext c = (0010 ,0000,1100,1110,0100,139680) toA. Decryption. To decryptc,Acomputes d1 =( (p+1)/4)6 mod (p−1) = 463 d2 =( (q+1)/4)6 mod (q−1) = 337 u= x463 6 mod p =2 0 v= x337 6 mod q =2 4 x0 = vap+ubqmod n = 159201. Finally,Ausesx0 to construct thexiand pijust asBdid for encryption, and recovers the plaintextmiby XORing thepiwith the ciphertext blocksci. □ 8.58 Note (security of Blum-Goldwasser probabilistic encryption) (i) Observe ﬁrst thatnis a Blum integer (Deﬁnition 2.156). An eavesdropper sees the quadratic residuext+1. Assuming that factoringnis difﬁcult, thehleast signiﬁcant bits of the principal square rootxtofxt+1 modulo nare simultaneously secure (see Deﬁnition 3.82 and Fact 3.89). Thus the eavesdropper can do no better than to guess the pseudorandom bitspi,1 ≤i≤t. More formally, if the integer factorization problem is hard, then the Blum-Goldwasser probabilistic encryption scheme is se- mantically secure. Note, however, that for a modulusnof a ﬁxed bitlength (e.g., 1024 bits), this statement is no longer true, and the scheme should only be consid- ered computationally secure. (ii) As of 1996, the modulusnshould be at least1024bits in length if long-term security is desired (cf. Note 8.7). Ifnis a1025-bit integer, thenk= 1024 and h=1 0. (iii) As with the Rabin encryption scheme (Algorithm 8.11), the Blum-Goldwasser sch- eme is also vulnerable to a chosen-ciphertext attack that recovers the private key from the public key. It is for this reason that the Blum-Goldwasser scheme has not received much attention in practice. 8.59 Note (efﬁciency of Blum-Goldwasser probabilistic encryption) (i) Unlike Goldwasser-Micali encryption, the ciphertext in Blum-Goldwasser encryp- tion is only longer than the plaintext by a constant number of bits, namelyk+1 (the size in bits of the integerx t+1). §8.7 Probabilistic public-key encryption 311 (ii) The encryption process is quite efﬁcient — it takes only 1 modular multiplication to encrypthbits of plaintext. By comparison, the RSA encryption process (Algo- rithm 8.3) requires 1 modular exponentiation (memod n) to encryptkbits of plain- text. Assuming that the parametereis randomly chosen and assuming that an (unop- timized) modular exponentiation takes3k/2modular multiplications, this translates to an encryption rate for RSA of2/3bits per modular multiplication. If one chooses a special value fore, such ase=3 (see Note 8.9), then RSA encryption is faster than Blum-Goldwasser encryption. (iii) Blum-Goldwasser decryption (step 2 of Algorithm 8.56) is also quite efﬁcient, requir- ing 1 exponentiation modulop−1(step 2a), 1 exponentiation moduloq−1(step 2b), 1 exponentiation modulop(step 2c), 1 exponentiation moduloq(step 2d), andtmul- tiplications modulon(step 2f) to decrypthtciphertext bits. (The time to perform step 2e is negligible.) By comparison, RSA decryption (step 2 of Algorithm 8.3) re- quires 1 exponentiation modulon(which can be accomplished by doing 1 exponen- tiation modulopand 1 exponentiation moduloq) to decryptkciphertext bits. Thus, for short messages (<k bits), Blum-Goldwasser decryption is slightly slower than RSA decryption, while for longer messages, Blum-Goldwasser is faster. 8.7.3 Plaintext-aware encryption While semantic security (Deﬁnition 8.47) is a strong security requirement for public-key encryption schemes, there are other measures of security. 8.60 DeﬁnitionA public-key encryption scheme is said to benon-malleableif given a cipher- text, it is computationally infeasible to generate a different ciphertext such that the respec- tive plaintexts are related in a known manner. 8.61 FactIf a public-key encryption scheme is non-malleable, it is also semantically secure. Another notion of security is that of being plaintext-aware. In Deﬁnition 8.62,valid ci- phertextmeans those ciphertext which are the encryptions of legitimate plaintext messages (e.g. messages containing pre-speciﬁed forms of redundancy). 8.62 DeﬁnitionA public-key encryption scheme is said to beplaintext-awareif it is computa- tionally infeasible for an adversary to produce a valid ciphertext without knowledge of the corresponding plaintext. In the “random oracle model”, the property of being plaintext-aware is a strong one — coupled with semantic security, it can be shown to imply that the encryption scheme is non-malleable and also secure against adaptive chosen-ciphertext attacks. Note 8.63 gives one method of transforming anyk-bit tok-bit trapdoor one-way permutation (such as RSA) into an encryption scheme that is plaintext-aware and semantically secure. 8.63 Note (Bellare-Rogaway plaintext-aware encryption) Letfbe ak-bit tok-bit trapdoor one- way permutation (such as RSA). Letk 0 and k1 be parameters such that2k0 and 2k1 steps each represent infeasible amounts of work (e.g.,k0 = k1 = 128). The length of the plain- textmis ﬁxed to ben= k−k0 −k1 (e.g., fork= 1024,n= 768). LetG: {0,1}k0 −→ {0,1}n+k1 and H: {0,1}n+k1 −→ {0,1}k0 be random functions. Then the encryption function, as depicted in Figure 8.1, is E(m)= f({m0k1 ⊕G(r)}∥{r⊕H(m0k1 ⊕G(r))}), 312 Ch. 8 Public-Key Encryption where m0k1 denotesmconcatenated with a string of0’s of bitlengthk1,ris a random bi- nary string of bitlengthk0, and∥denotes concatenation. G f H rm0k1 m0k1 ⊕G(r) r ⊕H(m0k1 ⊕G(r)) n+ k1 k0 n+ k0 + k1 E(m) m plaintext E(m) ciphertext r random bit string Figure 8.1:Bellare-Rogaway plaintext-aware encryption scheme. Under the assumption thatGand Hare random functions, the encryption schemeEof Note 8.63 can be proven to be plaintext-aware and semantically secure. In practice,Gand Hcan be derived from a cryptographic hash function such as the Secure Hash Algorithm (§9.4.2(iii)). In this case, the encryption scheme can no longer be proven to be plaintext- aware because the random function assumption is not true; however, such a scheme appears to provides greater security assurances than those designed using ad hoc techniques. 8.8 Notes and further references §8.1 For an introduction to public-key cryptography and public-key encryption in particular, see §1.8. A particularly readable introduction is the survey by Difﬁe [343]. Historical notes on public-key cryptography are given in the notes to§1.8 on page 47. A comparison of the features of public-key and symmetric-key encryption is given in§1.8.4; see also§13.2.5. Other recent proposals for public-key encryption schemes include those based on ﬁnite au- tomata (Renji [1032]); hidden ﬁeld equations (Patarin [965]); and isomorphism of polyno- mials (Patarin [965]). §8.2 The RSA cryptosystem was invented in 1977 by Rivest, Shamir, and Adleman [1060]. Kal- iski and Robshaw [655] provide an overview of the major attacks on RSA encryption and §8.8 Notes and further references 313 signatures, and the practical methods of counteracting these threats. The computational equivalence of computing the decryption exponentdand factoringn (§8.2.2(i)) was shown by Rivest, Shamir and Adleman [1060], based on earlier work by Miller [876]. The attack on RSA with small encryption exponent (§8.2.2(ii)) is discussed by H˚astad [544], who showed more generally that sending the encryptions of more thane(e+1)/2linearly related messages(messages of the form (aim+ bi), where theai and biare known) en- ables an eavesdropper to recover the messages provided that the modulini satisfyni > 2(e+1)(e+2)/4(e+1)(e+1).H ˚astad also showed that sending three linearly related messages using the Rabin public-key encryption scheme (Algorithm 8.11) is insecure. The attack on RSA with small decryption exponentd(§8.2.2(iv)) is due to Wiener [1240]. Wiener showed that his attack can be avoided if the encryption exponenteis chosen to be at least50%longer than the modulusn. In this case,dshould be at least160 bits in length to avoid the square-root discrete logarithm algorithms such as Pollard’s rho algorithm (Al- gorithm 3.60) and the parallelized variant of van Oorschot and Wiener [1207]. The adaptive chosen-ciphertext attack on RSA encryption (§8.2.2(v)) is due to Davida [302]. See also the related discussion in Denning [327]. Desmedt and Odlyzko [341] de- scribed an indifferent chosen-ciphertext attack in which the adversary has to obtain the plaintext corresponding to aboutLn[1 2,1 2]carefully chosen-ciphertext, subsequent to which it can decrypt all further ciphertext inLn[1 2,1 2] time without having to use the authorized user’s decryption machine. The common modulus attacks on RSA (§8.2.2(vi)) are due to DeLaurentis [320] and Sim- mons [1137]. The cycling attack (§8.2.2(vii)) was proposed by Simmons and Norris [1151]. Shortly after, Rivest [1052] showed that the cycling attack is extremely unlikely to succeed if the primes pand qare chosen so that: (i)p−1 and q−1 have large prime factorsp ′and q′, respec- tively; and (ii)p′−1 and q′−1 have large prime factorsp′′and q′′, respectively. Maurer [818] showed that condition (ii) is unnecessary. Williams and Schmid [1249] proposed the generalized cycling attack and showed that this attack is really a factoring algorithm. Rivest [1051] provided heuristic evidence that if the primespand qare selected at random, each having the same bitlength, then the expected time before the generalized cycling attack suc- ceeds is at leastp 1/3. The note on message concealing (§8.2.2(viii)) is due to Blakley and Borosh [150], who also extended this work to all composite integersnand determined the number ofderanging exponents for a ﬁxedn, i.e., exponentsefor which the number of unconcealed messages is the minimum possible. For further work see Smith and Palmer [1158]. Suppose that two or more plaintext messages which have a (known) polynomial relation- ship (e.g.m1 and m2 might belinearly related: m1 = am2 + b) are encrypted with the same small encryption exponent (e.g.e=3 or e=2 16 +1 ). Coppersmith et al. [277] presented a new class of attacks on RSA which enable a passive adversary to recover such plaintext from the corresponding ciphertext. This attack is of practical signiﬁcance because various cryptographic protocols have been proposed which require the encryption of poly- nomially related messages. Examples include the key distribution protocol of Tatebayashi, Matsuzaki, and Newman [1188], and the veriﬁable signature scheme of Franklin and Reiter [421]. Note that these attacks are different from those of§8.2.2(ii) and§8.2.2(vi) where the same plaintext is encrypted under different public keys. Coppersmith [274] presented an efﬁcient algorithm for ﬁnding a root of a polynomial of de- greekoverZ n, wherenis an RSA-like modulus, provided that there there is a root smaller 314 Ch. 8 Public-Key Encryption thann1/k. The algorithm yielded the following two attacks on RSA with small encryption exponents. Ife=3 and if an adversary knows a ciphertextcand more than2/3of the plain- textmcorresponding toc, then the adversary can efﬁciently recover the rest ofm. Suppose now that messages are padded with random bitstrings and encrypted with exponente=3 . If an adversary knows two ciphertextsc1 and c2 which correspond to two encryptions of the same messagem(with different padding), then the adversary can efﬁciently recovery m, provided that the padding is less than1/9of the length ofn. The latter attack suggests that caution must be exercised when using random padding in conjunction with a small en- cryption exponent. Letn= pqbe ak-bit RSA modulus, wherepand qarek/2-bit primes. Coppersmith [273] showed how ncan be factored in polynomial time if the high orderk/4bits ofpare known. This improves an algorithm of Rivest and Shamir [1058], which requires knowledge of the high orderk/3bits ofp. For related theoretical work, see Maurer [814]. One implication of Coppersmith’s result is that the method of Vanstone and Zuccherato [1214] for generating RSA moduli having a predetermined set of bits is insecure. A trapdoor in the RSA cryptosystem was proposed by Anderson [26] whereby a hardware device generates the RSA modulusn= pqin such a way that the hardware manufacturer can easily factorn, but factoringnremains difﬁcult for all other parties. However, Kaliski [652] subsequently showed how to efﬁciently detect such trapdoors and, in some cases, to actually factor the modulus. The arguments and recommendations about the use of strong primes in RSA key generation (Note 8.8) are taken from the detailed article by Rivest [1051]. Shamir [1117] proposed a variant of the RSA encryption scheme calledunbalanced RSA, which makes it possible to enhance security by increasing the modulus size (e.g. from 500 bits to 5000 bits) without any deterioration in performance. In this variant, the public mod- ulusnis the product of two primespand q, where one prime (sayq) is signiﬁcantly larger in size than the other; plaintext messagesmare in the interval[0,p−1]. For concrete- ness, consider the situation wherepis a 500-bit prime, andqis a 4500-bit prime. Fac- toring such a 5000-bit modulusnis well beyond the reach of the special-purpose elliptic curve factoring algorithm of§3.2.4 (whose running time depends on the size of the smallest prime factor ofn) and general-purpose factoring algorithms such as the number ﬁeld sieve of§3.2.7. Shamir recommends that the encryption exponentebe in the interval[20,100], which makes the encryption time with a 5000-bit modulus comparable to the decryption time with a 500-bit modulus. Decryption of the ciphertextc(= m dmod n) is accom- plished by computingm1 = cd1 mod p, whered1 = dmod (p−1). Since0 ≤m<p , m1 is in fact equal tom. Decryption in unbalanced RSA thus only involves one exponenti- ation modulo a 500-bit prime, and takes the same time as decryption in ordinary RSA with a 500-bit modulus. This optimization does not apply to the RSA signature scheme (§11.3.1), since the veriﬁer does not know the factorpof the public modulusn. A permutation polynomialofZnis a polynomialf(x) ∈Zn[x] which induces a permuta- tion ofZnupon substitution of the elements ofZn; that is,{f(a)\|a∈Zn}= Zn. In RSA encryption the permutation polynomialxeofZnis used, wheregcd(e,φ)=1 .M ¨uller and N¨obauer [910] suggested replacing the polynomialxeby the so-calledDickson polynomi- alsto create a modiﬁed RSA encryption scheme called theDickson scheme. The Dickson scheme was further studied by M¨uller and N¨obauer [909]. Other suitable classes of permu- tation polynomials were investigated by Lidl and M¨uller [763]. Smith and Lennon [1161] proposed an analogue of the RSA cryptosystem called LUC which is based on Lucas se- quences. Due to the relationships between Dickson polynomials and the Lucas sequences, §8.8 Notes and further references 315 the LUC cryptosystem is closely related to the Dickson scheme. Bleichenbacher, Bosma, and Lenstra [154] presented a chosen-message attack on the LUC signature scheme, under- mining the primary advantage claimed for LUC over RSA. Pinch [976, 977] extended the attacks on RSA with small encryption exponent (§8.2.2(ii)) and small decryption exponent (§8.2.2(iv)) to the LUC system. An analogue of the RSA cryptosystem which uses special kinds of elliptic curves overZn, where nis a composite integer, was proposed by Koyama et al. [708]. Demytko [321] pre- sented an analogue where there is very little restriction on the types of elliptic curves that can be used. A new cryptosystem based on elliptic curves overZnin which the message is held in the exponent instead of the group element was proposed by Vanstone and Zuccherato [1213]. The security of all these schemes is based on the difﬁculty of factoringn. Kuro- sawa, Okada, and Tsujii [721] showed that the encryption schemes of Koyama et al. and Demytko are vulnerable to low exponent attacks (cf.§8.2.2(ii)); Pinch [977] demonstrated that the attack on RSA with small decryption exponentd(§8.2.2(iv)) also extends to these schemes. Kaliski [649] presented a chosen-ciphertext attack on the Demytko encryption scheme (and also a chosen-message attack on the corresponding signature scheme), and concluded that the present beneﬁts of elliptic curve cryptosystems based on a composite modulus do not seem signiﬁcant. §8.3 The Rabin public-key encryption scheme (Algorithm 8.11) was proposed in 1979 by Ra- bin [1023]. In Rabin’s paper, the encryption function was deﬁned to beE(m)= m(m+ b)m o dn, whereband ncomprise the public key. The security of this scheme is equiv- alent to the security of the scheme described in Algorithm 8.11 with encryption function E(m)= m 2 mod n. A related digital signature scheme is described in§11.3.4. Schwenk and Eisfeld [1104] consider public-key encryption and signature schemes whose security relies on the intractability of factoring polynomials overZn. Williams [1246] presented a public-key encryption scheme similar in spirit to Rabin’s but using composite integersn = pqwith primesp ≡ 3( m o d8 )and q ≡ 7( m o d8 ). Williams’ scheme also has the property that breaking it (that is, recovering plaintext from some given ciphertext) is equivalent to factoringn, but has the advantage over Rabin’s sch- eme that there is an easy procedure for identifying the intended message from the four roots of a quadratic polynomial. The restrictions on the forms of the primespandqwere removed later by Williams [1248]. A simpler and more efﬁcient scheme also having the properties of provable security and unique decryption was presented by Kurosawa, Ito, and Takeuchi [720]. As with Rabin, all these schemes are vulnerable to a chosen-ciphertext attack (but see Note 8.14). It is not the case that all public-key encryption schemes for which the decryption problem is provably as difﬁcult as recovering the private key from the public key must succumb to a chosen-ciphertext attack. Goldwasser, Micali, and Rivest [484] were the ﬁrst to observe this, and presented a digital signature scheme provably secure against an adaptive chosen- ciphertext attack (see§11.6.4). Naor and Yung [921] proposed the ﬁrst concrete public-key encryption scheme that is semantically secure againstindifferentchosen-ciphertext attack. The Naor-Yung scheme uses two independent keys of a probabilistic public-encryption sch- eme that is secure against a passive adversary (for example, the Goldwasser-Micali scheme of Algorithm 8.51) to encrypt the plaintext, and then both encryptions are sent along with a non-interactive zero-knowledge proof that the same message was encrypted with both keys. Following this work, Rackoff and Simon [1029] gave the ﬁrst concrete construction for a public-key encryption scheme that is semantically secure against anadaptivechosen- 316 Ch. 8 Public-Key Encryption ciphertext attack. Unfortunately, these schemes are all impractical because of the degree of message expansion. Damg˚ard [297] proposed simple and efﬁcient methods for making public-key encryption schemes secure against indifferent chosen-ciphertext attacks. Zheng and Seberry [1269] noted that Damg˚ard’s schemes are insecure against an adaptive chosen-ciphertext attack, and proposed three practical schemes intended to resist such an attack. The Damg˚ard and Zheng-Seberry schemes were not proven to achieve their claimed levels of security. Bel- lare and Rogaway [93] later proved that one of the Zheng-Seberry schemes is provably se- cure against adaptive chosen-ciphertext attacks for theirrandom oracle model. Lim and Lee [766] proposed another method for making public-key schemes secure against adap- tive chosen-ciphertext attacks; this scheme was broken by Frankel and Yung [419]. §8.4 The ElGamal cryptosystem was invented by ElGamal [368]. Haber and Lenstra (see Ruep- pel et al. [1083]) raised the possibility of a trapdoor in discrete logarithm cryptosystems whereby a moduluspis generated (e.g., by a hardware manufacturer) that is intentionally “weak”; cf. Note 4.58. Here, a “weak” primepis one for which the discrete logarithm prob- lem inZ ∗ pis relatively easy. For example,p−1 may contain only small prime factors, in which case the Pohlig-Hellman algorithm (§3.6.4) would be especially effective. Another example is a primepfor which the number ﬁeld sieve for discrete logarithms (page 128) is especially well-suited. However, Gordon [509] subsequently showed how such trapdoors can be easily detected. Gordon also showed that the probability of a randomly chosen prime possessing such a trapdoor is negligibly small. Rivest and Sherman [1061] gave an overview and uniﬁed framework for randomized en- cryption, including comments on chosen-plaintext and chosen-ciphertext attacks. Elliptic curves were ﬁrst proposed for use in public-key cryptography by Koblitz [695] and Miller [878]. Recent work on the security and implementation of elliptic curve systems is reported by Menezes [840]. Menezes, Okamoto, and Vanstone [843] showed that if the elliptic curve belongs to a special family calledsupersingular curves, then the discrete log- arithm problem in the elliptic curve group can be reduced in expected polynomial time to the discrete logarithm problem in a small extension of the underlying ﬁnite ﬁeld. Hence, if a supersingular elliptic curve is desired in practice, then it should be carefully chosen. A modiﬁcation of ElGamal encryption employing the group of unitsZ ∗ n, wherenis a com- posite integer, was proposed by McCurley [825]; the scheme has the property that breaking it isprovablyat least as difﬁcult as factoring the modulusn(cf. Fact 3.80). If a cryptanalyst somehow learns the factors ofn, then in order to recover plaintext from ciphertext it is still left with the task of solving the Difﬁe-Hellman problem (§3.7) modulo the factors ofn. Hyperelliptic curve cryptosystems were proposed by Koblitz [696] but little research has since been done regarding their security and practicality. The possibility of using the class group of an imaginary quadratic number ﬁeld in public- key cryptography was suggested by Buchmann and Williams [218], however, the attrac- tiveness of this choice was greatly diminished after the invention of a subexponential-time algorithm for computing discrete logarithms in these groups by McCurley [826]. Smith and Skinner [1162] proposed analogues of the Difﬁe-Hellman key exchange (called LUCDIF) and ElGamal encryption and digital signature schemes (called LUCELG) which use Lucas sequences modulo a primepinstead of modular exponentiation. Shortly there- after, Laih, Tu, and Tai [733] and Bleichenbacher, Bosma, and Lenstra [154] showed that the analogue of the discrete logarithm problem for Lucas functions polytime reduces to the §8.8 Notes and further references 317 discrete logarithm problem in the multiplicative group of the ﬁnite ﬁeldFp2. Since there are subexponential-time algorithms known for the discrete logarithm problem in these ﬁelds (cf.§3.6), LUCDIF and LUCELG appear not to offer any advantages over the original sch- emes. §8.5 The McEliece encryption scheme (Algorithm 8.30) was introduced in 1978 by McEliece [828]. For information on Goppa codes and their decoding algorithms, see MacWilliams and Sloane [778]. The problem of decoding an arbitrary linear code was shown to beNP - hard by Berlekamp, McEliece, and van Tilborg [120]. The security of the McEliece scheme has been studied by Adams and Meijer [6], Lee and Brickell [742], van Tilburg [1212], Gib- son [451], and by Chabaud [235]. Gibson showed that there are, in fact, many trapdoors to a given McEliece encryption transformation, any of which may be used for decryption; this is contrary to the results of Adams and Meijer. However, Gibson notes that there are proba- bly sufﬁciently few trapdoors that ﬁnding one by brute force is computationally infeasible. The cryptanalytic attack reported by Korzhik and Turkin [707] has not been published in its entirety, and is not believed to be an effective attack. The strength of the McEliece encryption scheme can be severely weakened if the Goppa code is replaced with another type of error-correcting code. For example, Gabidulin, Para- monov, and Tretjakov [435] proposed a modiﬁcation which uses maximum-rank-distance (MRD) codes in place of Goppa codes. This scheme, and a modiﬁcation of it by Gabidulin [434], were subsequently shown to be insecure by Gibson [452, 453]. §8.6 The basic and multiple-iterated Merkle-Hellman knapsack encryption schemes (§8.6.1) we- re introduced by Merkle and Hellman [857]. An elementary overview of knapsack systems is given by Odlyzko [941]. The ﬁrst polynomial-time attack on the basic Merkle-Hellman scheme (cf. Note 8.40(i)) was devised by Shamir [1114] in 1982. The attack makes use of H. Lenstra’s algorithm for inte- ger programming which runs in polynomial time when the number of variables is ﬁxed, but is inefﬁcient in practice. Lagarias [723] improved the practicality of the attack by reducing the main portion of the procedure to a problem of ﬁnding an unusually good simultane- ous diophantine approximation; the latter can be solved by the more efﬁcientL 3-lattice ba- sis reduction algorithm (§3.10.1). The ﬁrst attack on the multiple-iterated Merkle-Hellman scheme was by Brickell [200]. For surveys of the cryptanalysis of knapsack schemes, see Brickell [201] and Brickell and Odlyzko [209]. Orton [960] proposed a modiﬁcation to the multiple-iterated Merkle-Hellman scheme that permits a knapsack density approaching 1, thus avoiding currently known attacks. The high density also allows for a fast digital sig- nature scheme. Shamir [1109] proposed a fast signature scheme based on the knapsack problem, later bro- ken by Odlyzko [939] using theL 3-lattice basis reduction algorithm. The Merkle-Hellman knapsack scheme illustrates the limitations of using anNP -complete problem to design a secure public-key encryption scheme. Firstly, Brassard [190] showed that under reasonable assumptions, the problem faced by the cryptanalyst cannot beNP - hard unlessNP =co-NP, which would be a very surprising result in computational complex- ity theory. Secondly, complexity theory is concerned primarily withasymptoticcomplex- ity of a problem. By contrast, in practice one works with a problem instance of aﬁxedsize. Thirdly,NP -completeness is a measure of theworst-casecomplexity of a problem. By con- trast, cryptographic security should depend on theaverage-casecomplexity of the problem (or even better, the problem should be intractable foressentially allinstances), since the 318 Ch. 8 Public-Key Encryption cryptanalyst’s task should be hard for virtually all instances and not merely in the worst case. There are manyNP -complete problems that are known to have polynomial-time average- case algorithms, for example, the graph coloring problem; see Wilf [1243]. Another inter- esting example is provided by Even and Yacobi [379] who describe a symmetric-key en- cryption scheme based on the subset sum problem for which breaking the scheme (under a chosen-plaintext attack) is anNP -hard problem, yet an algorithm exists which solves most instances in polynomial time. The Chor-Rivest knapsack scheme (Algorithm 8.42) was proposed by Chor and Rivest [261]. Recently, Schnorr and H¨orner [1100] introduced new algorithms for lattice ba- sis reduction that are improvements on theL 3-lattice basis reduction algorithm (Algo- rithm 3.101), and used these to break the Chor-Rivest scheme with parameters{p = 103,h =1 2}. Since the density of such knapsack sets is1.271, the attack demonstrated that subset sum problems with density greater than 1 can be solved via lattice basis re- duction. Schnorr and H¨orner also reported some success solving Chor-Rivest subset sum problems with parameters{p= 151 ,h =1 6 }. It remains to be seen whether the tech- niques of Schnorr and H¨orner can be successfully applied to the recommended parameter case{p= 197,h=2 4}. Depending on the choice of parameters, the computation of discrete logarithms in the Chor- Rivest key generation stage (step 4 of Algorithm 8.41) may be a formidable task. A mod- iﬁed version of the scheme which does not require the computation of discrete logarithms in a ﬁeld was proposed by H. Lenstra [758]. This modiﬁed scheme is called thepowerline systemand is not a knapsack system. It was proven to be at least as secure as the original Chor-Rivest scheme, and is comparable in terms of encryption and decryption speeds. Qu and Vanstone [1013] showed how the Merkle-Hellman knapsack schemes can be viewed as special cases of certain knapsack-like encryption schemes arising from subset factoriza- tions of ﬁnite groups. They also proposed an efﬁcient public-key encryption scheme based on subset factorizations of the additive groupZ nof integers modulon. Blackburn, Mur- phy, and Stern [143] showed that a simpliﬁed variant which uses subset factorizations of then-dimensional vector spaceZn 2 overZ2 is insecure. §8.7 The notion of probabilistic public-key encryption was conceived by Goldwasser and Micali [479], who also introduced the notions of polynomial and semantic security. The equiva- lence of these two notions (Fact 8.49) was proven by Goldwasser and Micali [479] and Mi- cali, Rackoff, and Sloan [865]. Polynomial security was also studied by Yao [1258], who referred to it aspolynomial-time indistinguishability. The Goldwasser-Micali scheme (Algorithm 8.51) can be described in a general setting by using the notion of a trapdoor predicate. Brieﬂy, atrapdoor predicateis a Boolean function B: {0,1} ∗ −→ {0,1}such that given a bitvit is easy to choose anxat random satisfy- ingB(x)= v. Moreover, given a bitstringx, computingB(x) correctly with probability signiﬁcantly greater than1 2 is difﬁcult; however, if certain trapdoor information is known, then it is easy to computeB(x). If entityA’s public key is a trapdoor predicateB, then any other entity encrypts a message bitmiby randomly selecting anxisuch thatB(xi)= mi, and then sendsxitoA. SinceAknows the trapdoor information, she can computeB(xi)to recovermi, but an adversary can do no better than guess the value ofmi. Goldwasser and Micali [479] proved that if trapdoor predicates exist, then this probabilistic encryption sch- eme is polynomially secure. Goldreich and Levin [471] simpliﬁed the work of Yao [1258], and showed how any trapdoor length-preserving permutationfcan be used to obtain a trap- door predicate, which in turn can be used to construct a probabilistic public-key encryption §8.8 Notes and further references 319 scheme. The Blum-Goldwasser scheme (Algorithm 8.56) was proposed by Blum and Goldwasser [164]. The version given here follows the presentation of Brassard [192]. Two probabilis- tic public-key encryption schemes, one whose breaking is equivalent to solving the RSA problem (§3.3), and the other whose breaking is equivalent to factoring integers, were pro- posed by Alexi et al. [23]. The scheme based on RSA is as follows. Leth= ⌊lglg n⌋, where (n,e) is entityA’s RSA public key. To encrypt anh-bit messagemforA, choose a randomy∈Z ∗ nsuch that thehleast signiﬁcant bits ofyequalm, and compute the ci- phertextc= yemod n.Acan recovermby computingy= cdmod n, and extracting the hleast signiﬁcant bits ofy. While both the schemes proposed by Alexi et al. are more ef- ﬁcient than the Goldwasser-Micali scheme, they suffer from large message expansion and are consequently not as efﬁcient as the Blum-Goldwasser scheme. The idea of non-malleable cryptography (Deﬁnition 8.60) was introduced by Dolev, Dwork, and Naor [357], who also observed Fact 8.61. The paper gives the example of two con- tract bidders who encrypt their bids. It should not be possible for one bidderAto see the encrypted bid of the other bidderBand somehow be able to offer a bid that was slightly lower, even ifAmay not know what the resulting bid actually is at that time. Bellare and Rogaway [95] introduced the notion of plaintext-aware encryption (Deﬁnition 8.62). They presented the scheme described in Note 8.63, building upon earlier work of Johnson et al. [639]. Rigorous deﬁnitions and security proofs were provided, as well as a concrete instan- tiation of the plaintext-aware encryption scheme using RSA as the trapdoor permutation, and constructing the random functionsGandHfrom the SHA-1 hash function (§9.4.2(iii)). Johnson and Matyas [640] presented some enhancements to the plaintext-aware encryption scheme. Bellare and Rogaway [93] presented various techniques for deriving appropriate random functions from standard cryptographic hash functions. Chapter /9 Hash Functions and Data Integrity Contents in Brief 9.1 Introduction ............................. 321 9.2 Classiﬁcation and framework.................... 322 9.3 Basic constructions and general results............... 332 9.4 Unkeyed hash functions (MDCs) .................. 338 9.5 Keyed hash functions (MACs) ................... 352 9.6 Data integrity and message authentication............. 359 9.7 Advanced attacks on hash functions................ 368 9.8 Notes and further references.................... 376 9.1 Introduction Cryptographic hash functionsplay a fundamental role in modern cryptography. While re- lated to conventional hash functions commonly used in non-cryptographic computer appli- cations – in both cases, larger domains are mapped to smaller ranges – they differ in several important aspects. Our focus is restricted to cryptographic hash functions (hereafter, simply hash functions), and in particular to their use for data integrity and message authentication. Hash functions take a message as input and produce an output referred to as ahash- code,hash-result,hash-value, or simplyhash. More precisely, a hash functionhmaps bit- strings of arbitrary ﬁnite length to strings of ﬁxed length, saynbits. For a domainDand rangeRwithh: D→Rand\|D\|>\|R\|, the function is many-to-one, implying that the exis- tence ofcollisions(pairs of inputs with identical output) is unavoidable. Indeed, restricting hto a domain oft-bit inputs (t>n ), ifhwere “random” in the sense that all outputs were essentially equiprobable, then about2 t−n inputs would map to each output, and two ran- domly chosen inputs would yield the same output with probability2−n (independent oft). The basic idea of cryptographic hash functions is that a hash-value serves as a compact rep- resentative image (sometimes called animprint,digital ﬁngerprint,o rmessage digest)o f an input string, and can be used as if it were uniquely identiﬁable with that string. Hash functions are used for data integrity in conjunction with digital signature sch- emes, where for several reasons a message is typically hashed ﬁrst, and then the hash-value, as a representative of the message, is signed in place of the original message (see Chap- ter 11). A distinct class of hash functions, called message authentication codes (MACs), allows message authentication by symmetric techniques. MAC algorithms may be viewed as hash functions which take two functionally distinct inputs, a message and a secret key, and produce a ﬁxed-size (sayn-bit) output, with the design intent that it be infeasible in 321 322 Ch. 9 Hash Functions and Data Integrity practice to produce the same output without knowledge of the key. MACs can be used to provide data integrity and symmetric data origin authentication, as well as identiﬁcation in symmetric-key schemes (see Chapter 10). A typical usage of (unkeyed) hash functions for data integrity is as follows. The hash- value corresponding to a particular messagexis computed at timeT1. The integrity of this hash-value (but not the message itself) is protected in some manner. At a subsequent time T2, the following test is carried out to determine whether the message has been altered, i.e., whether a messagex′is the same as the original message. The hash-value ofx′is computed and compared to the protected hash-value; if they are equal, one accepts that the inputs are also equal, and thus that the message has not been altered. The problem of preserving the integrity of a potentially large message is thus reduced to that of a small ﬁxed-size hash- value. Since the existence of collisions is guaranteed in many-to-one mappings, the unique association between inputs and hash-values can, at best, be in the computational sense. A hash-value should be uniquely identiﬁable with a single inputin practice, and collisions should becomputationallydifﬁcult to ﬁnd (essentially never occurring in practice). Chapter outline The remainder of this chapter is organized as follows.§9.2 provides a framework including standard deﬁnitions, a discussion of the desirable properties of hash functions and MACs, and consideration of one-way functions.§9.3 presents a general model for iterated hash functions, some general construction techniques, and a discussion of security objectives and basic attacks (i.e., strategies an adversary may pursue to defeat the objectives of a hash function).§9.4 considers hash functions based on block ciphers, and a family of functions based on the MD4 algorithm.§9.5 considers MACs, including those based on block ciphers and customized MACs.§9.6 examines various methods of using hash functions to provide data integrity.§9.7 presents advanced attack methods.§9.8 provides chapter notes with references. 9.2 Classiﬁcation and framework 9.2.1 General classiﬁcation At the highest level, hash functions may be split into two classes:unkeyed hash functions, whose speciﬁcation dictates a single input parameter (a message); andkeyed hash functions, whose speciﬁcation dictates two distinct inputs, a message and a secret key. To facilitate discussion, a hash function is informally deﬁned as follows. 9.1 DeﬁnitionA hash function(in the unrestricted sense) is a functionhwhich has, as a min- imum, the following two properties: 1. compression— hmaps an inputxof arbitrary ﬁnite bitlength, to an outputh(x) of ﬁxed bitlengthn. 2. ease of computation— given hand an inputx,h(x) is easy to compute. §9.2 Classiﬁcation and framework 323 As deﬁned here,hash functionimplies an unkeyed hash function. On occasion when discussion is at a generic level, this term is abused somewhat to mean both unkeyed and keyed hash functions; hopefully ambiguity is limited by context. For actual use, a more goal-oriented classiﬁcation of hash functions (beyondkeyedvs. unkeyed) is necessary, based on further properties they provide and reﬂecting requirements of speciﬁc applications. Of the numerous categories in such afunctional classiﬁcation, two types of hash functions are considered in detail in this chapter: 1. modiﬁcation detection codes(MDCs) Also known asmanipulation detection codes, and less commonly asmessage integri- ty codes(MICs), the purpose of an MDC is (informally) to provide a representative image orhash of a message, satisfying additional properties as reﬁned below. The end goal is to facilitate, in conjunction with additional mechanisms (see§9.6.4), data integrity assurances as required by speciﬁc applications. MDCs are a subclass ofun- keyedhash functions, and themselves may be further classiﬁed; the speciﬁc classes of MDCs of primary focus in this chapter are (cf. Deﬁnitions 9.3 and 9.4): (i)one-way hash functions(OWHFs): for these, ﬁnding an input which hashes to a pre-speciﬁed hash-value is difﬁcult; (ii)collision resistant hash functions(CRHFs): for these, ﬁnding any two inputs having the same hash-value is difﬁcult. 2. message authentication codes(MACs) The purpose of a MAC is (informally) to facilitate, without the use of any additional mechanisms, assurances regarding both the source of a message and its integrity (see §9.6.3). MACs have two functionally distinct parameters, a message input and a se- cret key; they are a subclass ofkeyedhash functions (cf. Deﬁnition 9.7). Figure 9.1 illustrates this simpliﬁed classiﬁcation. Additional applications of unkeyed hash functions are noted in§9.2.6. Additional applications of keyed hash functions in- clude use in challenge-response identiﬁcation protocols for computing responses which are a function of both a secret key and a challenge message; and for key conﬁrmation (Deﬁni- tion 12.7). Distinction should be made between a MAC algorithm, and the use of an MDC with a secret key included as part of its message input (see§9.5.2). It is generally assumed that the algorithmic speciﬁcation of a hash function is public knowledge. Thus in the case of MDCs, given a message as input, anyone may compute the hash-result; and in the case of MACs, given a message as input, anyone with knowledge of the key may compute the hash-result. 9.2.2 Basic properties and deﬁnitions To facilitate further deﬁnitions, three potential properties are listed (in addition toease of computationand compressionas per Deﬁnition 9.1), for an unkeyed hash functionhwith inputsx,x′and outputsy,y′. 1. preimage resistance— for essentially all pre-speciﬁed outputs, it is computationally infeasible to ﬁnd any input which hashes to that output, i.e., to ﬁnd any preimagex′ such thath(x′)= ywhen given anyyfor which a corresponding input is not known.1 2. 2nd-preimage resistance— it is computationally infeasible to ﬁnd any second input which has the same output as any speciﬁed input, i.e., givenx, to ﬁnd a 2nd-preimage x′̸=xsuch thath(x)= h(x′). 1This acknowledges that an adversary may easily precompute outputs for any small set of inputs, and thereby invert the hash function trivially for such outputs (cf. Remark 9.35). 324 Ch. 9 Hash Functions and Data Integrity authentication message (MACs) other applications other applications modiﬁcation detection (MDCs) keyed OWHF CRHF unkeyed hash functions preimage resistant collision resistant preimage resistant 2nd Figure 9.1:Simpliﬁed classiﬁcation of cryptographic hash functions and applications. 3. collision resistance— it is computationally infeasible to ﬁnd any two distinct inputs x,x′which hash to the same output, i.e., such thath(x)= h(x′). (Note that here there is free choice of both inputs.) Here and elsewhere, the terms “easy” and “computationally infeasible” (or “hard”) are intentionally left without formal deﬁnition; it is intended they be interpreted relative to an understood frame of reference. “Easy” might mean polynomial time and space; or more practically, within a certain number of machine operations or time units – perhaps seconds or milliseconds. A more speciﬁc deﬁnition of “computationally infeasible” might involve super-polynomial effort; require effort far exceeding understood resources; specify a lower bound on the number of operations or memory required in terms of a speciﬁed security pa- rameter; or specify the probability that a property is violated be exponentially small. The properties as deﬁned above, however, sufﬁce to allow practical deﬁnitions such as Deﬁni- tions 9.3 and 9.4 below. 9.2 Note (alternate terminology) Alternate terms used in the literature are as follows: preim- age resistant≡one-way(cf. Deﬁnition 9.9); 2nd-preimage resistance≡weak collision re- sistance; collision resistance≡strong collision resistance. For context, one motivation for each of the three major properties above is now given. Consider a digital signature scheme wherein the signature is applied to the hash-valueh(x) rather than the messagex. Herehshould be an MDC with 2nd-preimage resistance, oth- erwise, an adversaryCmay observe the signature of some partyAon h(x), then ﬁnd an x ′such thath(x)= h(x′), and claim thatAhas signedx′.I fCis able to actually choose the message whichAsigns, thenCneed only ﬁnd a collision pair(x,x′) rather than the harder task of ﬁnding a second preimage ofx; in this case, collision resistance is also nec- essary (cf. Remark 9.93). Less obvious is the requirement of preimage resistance for some public-key signature schemes; consider RSA (Chapter 11), where partyAhas public key §9.2 Classiﬁcation and framework 325 (e,n). Cmay choose a random valuey, computez= ye mod n, and (depending on the particular RSA signature veriﬁcation process used) claim thatyisA’s signature onz. This (existential) forgery may be of concern ifCcan ﬁnd a preimagexsuch thath(x)= z, and for whichxis of practical use. 9.3 DeﬁnitionA one-way hash function(OWHF) is a hash functionhas per Deﬁnition 9.1 (i.e., offering ease of computation and compression) with the following additional proper- ties, as deﬁned above: preimage resistance, 2nd-preimage resistance. 9.4 DeﬁnitionA collision resistant hash function(CRHF) is a hash functionhas per Deﬁni- tion 9.1 (i.e., offering ease of computation and compression) with the following additional properties, as deﬁned above: 2nd-preimage resistance, collision resistance (cf. Fact 9.18). Although in practice a CRHF almost always has the additional property of preimage re- sistance, for technical reasons (cf. Note 9.20) this property is not mandated in Deﬁnition 9.4. 9.5 Note (alternate terminology for OWHF , CRHF) Alternate terms used in the literature are as follows: OWHF≡weak one-way hash function(but here preimage resistance is often not explicitly considered); CRHF≡strong one-way hash function. 9.6 Example (hash function properties) (i) A simple modulo-32 checksum (32-bit sum of all 32-bit words of a data string) is an easily computed function which offers compression, but is not preimage resistant. (ii) The functiong(x) of Example 9.11 is preimage resistant but provides neither com- pression nor 2nd-preimage resistance. (iii) Example 9.13 presents a function with preimage resistance and 2nd-preimage resis- tance (but not compression). □ 9.7 DeﬁnitionA message authentication code (MAC)algorithm is a family of functionshk parameterized by a secret keyk, with the following properties: 1. ease of computation— for a known functionhk, given a valuekand an inputx, hk(x) is easy to compute. This result is called theMAC-value orMAC . 2. compression— hk maps an inputxof arbitrary ﬁnite bitlength to an outputhk(x) of ﬁxed bitlengthn. Furthermore, given a description of the function familyh, for every ﬁxed allowable value ofk(unknown to an adversary), the following property holds: 3. computation-resistance— given zero or more text-MAC pairs(xi,hk(xi)), it is com- putationally infeasible to compute any text-MAC pair(x,hk(x)) for any new input x̸=xi (including possibly forhk(x)= hk(xi) for somei). If computation-resistance does not hold, a MAC algorithm is subject toMAC forgery. While computation-resistance implies the property ofkey non-recovery(it must be computation- ally infeasible to recoverk, given one or more text-MAC pairs(xi,hk(xi)) for thatk), key non-recovery does not imply computation-resistance (a key need not always actually be re- covered to forge new MACs). 9.8 Remark (MAC resistance when key known) Deﬁnition 9.7 does not dictate whether MACs need be preimage- and collision resistant for parties knowing the keyk(as Fact 9.21 implies for parties withoutk). 326 Ch. 9 Hash Functions and Data Integrity (i) Objectives of adversaries vs. MDCs The objective of an adversary who wishes to “attack” an MDC is as follows: (a) to attack a OWHF: given a hash-valuey, ﬁnd a preimagexsuch thaty= h(x);o r given one such pair(x,h(x)), ﬁnd a second preimagex′such thath(x′)= h(x). (b) to attack a CRHF: ﬁnd any two inputsx,x′, such thath(x′)= h(x). A CRHF must be designed to withstand standard birthday attacks (see Fact 9.33). (ii) Objectives of adversaries vs. MACs The corresponding objective of an adversary for a MAC algorithm is as follows: (c) to attack a MAC: without prior knowledge of a keyk, compute a new text-MAC pair (x,hk(x)) for some textx̸=xi, given one or more pairs(xi,hk(xi)). Computation-resistance here should hold whether the textsxi for which matching MACs are available are given to the adversary, or may be freely chosen by the adversary. Similar to the situation for signature schemes, the following attack scenarios thus exist for MACs, for adversaries with increasing advantages: 1. known-text attack. One or more text-MAC pairs(xi,hk(xi)) are available. 2. chosen-text attack. One or more text-MAC pairs(xi,hk(xi)) are available forxi chosen by the adversary. 3. adaptive chosen-text attack. Thexi may be chosen by the adversary as above, now allowing successive choices to be based on the results of prior queries. As a certiﬁcational checkpoint, MACs should withstand adaptive chosen-text attack regard- less of whether such an attack may actually be mounted in a particular environment. Some practical applications may limit the number of interactions allowed over a ﬁxed period of time, or may be designed so as to compute MACs only for inputs created within the appli- cation itself; others may allow access to an unlimited number of text-MAC pairs, or allow MAC veriﬁcation of an unlimited number of messages and accept any with a correct MAC for further processing. (iii) Types of forgery (selective, existential) When MAC forgery is possible (implying the MAC algorithm has been technically de- feated), the severity of the practical consequences may differ depending on the degree of control an adversary has over the valuexfor which a MAC may be forged. This degree is differentiated by the following classiﬁcation of forgeries: 1. selective forgery– attacks whereby an adversary is able to produce a new text-MAC pair for a text of his choice (or perhaps partially under his control). Note that here the selected value is the text for which a MAC is forged, whereas in a chosen-text attack the chosen value is the text of a text-MAC pair used for analytical purposes (e.g., to forge a MAC on a distinct text). 2. existential forgery– attacks whereby an adversary is able to produce a new text-MAC pair, but with no control over the value of that text. Key recovery of the MAC key itself is the most damaging attack, and trivially allows se- lective forgery. MAC forgery allows an adversary to have a forged text accepted as authen- tic. The consequences may be severe even in the existential case. A classic example is the replacement of a monetary amount known to be small by a number randomly distributed between0 and 2 32 −1. For this reason, messages whose integrity or authenticity is to be veriﬁed are often constrained to have pre-determined structure or a high degree of veriﬁable redundancy, in an attempt to preclude meaningful attacks. §9.2 Classiﬁcation and framework 327 Analogously to MACs, attacks on MDC schemes (primarily 2nd-preimage and colli- sion attacks) may be classiﬁed as selective or existential. If the message can be partially controlled, then the attack may be classiﬁed as partially selective (e.g., see§9.7.1(iii)). 9.2.3 Hash properties required for speciﬁc applications Because there may be costs associated with speciﬁc properties – e.g., CRHFs are in gen- eral harder to construct than OWHFs and have hash-values roughly twice the bitlength – it should be understood which properties are actually required for particular applications, and why. Selected techniques whereby hash functions are used for data integrity, and the cor- responding properties required thereof by these applications, are summarized in Table 9.1. In general, an MDC should be a CRHF if an untrusted party has control over the exact content of hash function inputs (see Remark 9.93); a OWHF sufﬁces otherwise, including the case where there is only a single party involved (e.g., a store-and-retrieve application). Control over precise format of inputs may be eliminated by introducing into the message randomization that is uncontrollable by one or both parties. Note, however, that data in- tegrity techniques based on a shared secret key typically involve mutual trust and do not address non-repudiation; in this case, collision resistance may or may not be a requirement. Hash properties required→ Preimage 2nd- Collision Details Integrity application↓ resistant preimage resistant MDC + asymmetric signature yes yes yes† page 324 MDC + authentic channel yes yes† page 364 MDC + symmetric encryption page 365 hash for one-way password ﬁle yes page 389 MAC (key unknown to attacker) yes yes yes† page 326 MAC (key known to attacker) yes‡ page 325 Table 9.1:Resistance properties required for speciﬁed data integrity applications. †Resistance required if attacker is able to mount a chosen message attack. ‡Resistance required in rare case of multi-cast authentication (see page 378). 9.2.4 One-way functions and compression functions Related to Deﬁnition 9.3 of a OWHF is the following, which is unrestrictive with respect to a compression property. 9.9 DeﬁnitionA one-way function(OWF) is a functionfsuch that for eachxin the domain of f, it is easy to computef(x); but for essentially allyin the range off, it is computationally infeasible to ﬁnd anyxsuch thaty= f(x). 9.10 Remark (OWF vs. domain-restricted OWHF) A OWF as deﬁned here differs from a OWHF with domain restricted to ﬁxed-size inputs in that Deﬁnition 9.9 does not require 2nd-preimage resistance. Many one-way functions are, in fact, non-compressing, in which case most image elements have unique preimages, and for these 2nd-preimage resistance holds vacuously – making the difference minor (but see Example 9.11). 328 Ch. 9 Hash Functions and Data Integrity 9.11 Example (one-way functions and modular squaring) The squaring of integers modulo a primep, e.g.,f(x)= x2 −1m o dp, behaves in many ways like a random mapping. How- ever,f(x) is not a OWF because ﬁnding square roots modulo primes is easy (§3.5.1). On the other hand,g(x)= x2 mod nis a OWF (Deﬁnition 9.9) for appropriate randomly chosen primespand qwhere n= pqand the factorization ofnis unknown, as ﬁnding a preimage (i.e., computing a square root modn) is computationally equivalent to factoring (Fact 3.46) and thus intractable. Nonetheless, ﬁnding a 2nd-preimage, and, therefore, collisions, is triv- ial (givenx,−xyields a collision), and thusgﬁts neither the deﬁnition of a OWHF nor a CRHF with domain restricted to ﬁxed-size inputs. □ 9.12 Remark (candidate one-way functions) There are, in fact, no known instances of functions which are provably one-way (with no assumptions); indeed, despite known hash function constructions which are provably as secure asNP -complete problems, there is no assur- ance the latter are difﬁcult. All instances of “one-way functions” to date should thus more properly be qualiﬁed as “conjectured” or “candidate” one-way functions. (It thus remains possible, although widely believed most unlikely, that one-way functions do not exist.) A proof of existence would establishP ̸=NP , while non-existence would have devastating cryptographic consequences (see page 377), although not directly implyingP = NP . Hash functions are often used in applications (cf.§9.2.6) which require the one-way property, but not compression. It is, therefore, useful to distinguish three classes of func- tions (based on the relative size of inputs and outputs): 1. (general) hash functions. These are functions as per Deﬁnition 9.1, typically with ad- ditional one-way properties, which compress arbitrary-length inputs ton-bit outputs. 2. compression functions(ﬁxed-size hash functions). These are functions as per Deﬁ- nition 9.1, typically with additional one-way properties, but with domain restricted to ﬁxed-size inputs – i.e., compressingm-bit inputs ton-bit outputs,m>n . 3. non-compressing one-way functions. These are ﬁxed-size hash functions as above, except thatn= m. These includeone-way permutations, and can be more explicitly described as computationally non-invertible functions. 9.13 Example (DES-based OWF ) A one-way function can be constructed from DES or any block cipherEwhich behaves essentially as a random function (see Remark 9.14), as fol- lows:f(x)= E k(x)⊕x, for any ﬁxed known keyk. The one-way nature of this construc- tion can be proven under the assumption thatEis a random permutation. An intuitive ar- gument follows. For any choice ofy, ﬁnding anyx(and keyk) such thatEk(x)⊕x= yis difﬁcult because for any chosenx,Ek(x) will be essentially random (for any keyk) and thus so willEk(x)⊕x; hence, this will equalywith no better than random chance. By similar reasoning, if one attempts to use decryption and chooses anx, the probability that E−1 k (x⊕y)= xis no better than random chance. Thusf(x) appears to be a OWF. While f(x) is not a OWHF (it handles only ﬁxed-length inputs), it can be extended to yield one (see Algorithm 9.41). □ 9.14 Remark (block ciphers and random functions) Regarding random functions and their properties, see§2.1.6. If a block cipher behaved as a random function, then encryption and decryption would be equivalent to looking up values in a large table of random numbers; for a ﬁxed input, the mapping from a key to an output would behave as a random mapping. However, block ciphers such as DES are bijections, and thus at best exhibit behavior more like random permutations than random functions. §9.2 Classiﬁcation and framework 329 9.15 Example (one-wayness w.r.t. two inputs) Considerf(x,k)= Ek(x), whereErepre- sents DES. This is not a one-way function of the joint input(x,k), because given any func- tion valuey= f(x,k), one can choose any keyk′and computex′= E−1 k′(y) yielding a preimage(x′,k′). Similarly,f(x,k) is not a one-way function ofxifkis known, as giveny= f(x,k) and k, decryption ofyusingkyieldsx. (However, a “black-box” which computesf(x,k) for ﬁxed, externally-unknownkis a one-way function ofx.) In contrast, f(x,k) is a one-way function ofk; giveny= f(x,k) and x, it is not known how to ﬁnd a preimagekin less than about255 operations. (This latter concept is utilized in one-time digital signature schemes – see§11.6.2.) □ 9.16 Example (OWF - multiplication of large primes) For appropriate choices of primespand q,f(p,q)= pqis a one-way function: givenpand q, computingn= pqis easy, but given n, ﬁndingpandq, i.e.,integer factorization, is difﬁcult. RSA and many other cryptographic systems rely on this property (see Chapter 3, Chapter 8). Note that contrary to many one- way functions, this functionfdoes not have properties resembling a “random” function.□ 9.17 Example (OWF - exponentiation in ﬁnite ﬁelds) For most choices of appropriately large primespand any elementα∈Z∗ p of sufﬁciently large multiplicative order (e.g., a gen- erator),f(x)= αx mod pis a one-way function. (For example,pmust not be such that all the prime divisors ofp−1 are small, otherwise the discrete log problem is feasible by the Pohlig-Hellman algorithm of§3.6.4.)f(x) is easily computed givenα,x, andpusing the square-and-multiply technique (Algorithm 2.143), but for most choicespit is difﬁcult, given(y,p,α), to ﬁnd anxin the range0 ≤x≤p−2 such thatαx mod p= y, due to the apparent intractability of the discrete logarithm problem (§3.6). Of course, for speciﬁc values off(x) the function can be inverted trivially. For example, the respective preimages of1 and −1 are known to be0 and (p−1)/2, and by computingf(x) for any small set of values forx(e.g.,x=1 ,2,..., 10), these are also known. However, foressentiallyally in the range, the preimage ofyis difﬁcult to ﬁnd. □ 9.2.5 Relationships between properties In this section several relationships between the hash function properties stated in the pre- ceding section are examined. 9.18 FactCollision resistance implies 2nd-preimage resistance of hash functions. Justiﬁcation.Suppose hhas collision resistance. Fix an inputxj.I fhdoes not have 2nd- preimage resistance, then it is feasible to ﬁnd a distinct inputxi such thath(xi)= h(xj), in which case(xi,xj) is a pair of distinct inputs hashing to the same output, contradicting collision resistance. 9.19 Remark (one-way vs. preimage and 2nd-preimage resistant) While the term “one-way” is generally taken to mean preimage resistant, in the hash function literature it is some- times also used to imply that a function is 2nd-preimage resistant or computationally non- invertible. (Computationally non-invertibleis a more explicit term for preimage resistance when preimages are unique, e.g., for one-way permutations. In the case that two or more preimages exist, a function fails to be computationally non-invertible if any one can be found.) This causes ambiguity as 2nd-preimage resistance does not guarantee preimage- resistance (Note 9.20), nor does preimage resistance guarantee 2nd-preimage resistance (Example 9.11); see also Remark 9.10. An attempt is thus made to avoid unqualiﬁed use of the term “one-way”. 330 Ch. 9 Hash Functions and Data Integrity 9.20 Note (collision resistance does not guarantee preimage resistance) Letgbe a hash func- tion which is collision resistant and maps arbitrary-length inputs ton-bit outputs. Consider the functionhdeﬁned as (here and elsewhere,\|\|denotes concatenation): h(x)= { 1 \|\| x, if xhas bitlengthn 0 \|\| g(x), otherwise. Then his an(n+1 )-bit hash function which is collision resistant but not preimage resis- tant. As a simpler example, the identity function on ﬁxed-length inputs is collision and 2nd- preimage resistant (preimages are unique) but not preimage resistant. While such patholog- ical examples illustrate that collision resistance does not guarantee the difﬁculty of ﬁnding preimages of speciﬁc (or even most) hash outputs, for most CRHFs arising in practice it nonetheless appears reasonable to assume that collision resistance does indeed imply preim- age resistance. 9.21 Fact(implications of MAC properties) Leth k be a keyed hash function which is a MAC algorithm per Deﬁnition 9.7 (and thus has the property of computation-resistance). Then h k is, against chosen-text attack by an adversary without knowledge of the keyk, (i) both 2nd-preimage resistant and collision resistant; and (ii) preimage resistant (with respect to the hash-input). Justiﬁcation.For (i), note that computation-resistance implies hash-results should not even be computable by those without secret keyk. For (ii), by way of contradiction, assume hwere not preimage resistant. Then recovery of the preimagexfor a randomly selected hash-outputyviolates computation-resistance. 9.2.6 Other hash function properties and applications Most unkeyed hash functions commonly found in practice were originally designed for the purpose of providing data integrity (see§9.6), including digital ﬁngerprinting of messages in conjunction with digital signatures (§9.6.4). The majority of these are, in fact, MDCs designed to have preimage, 2nd-preimage, or collision resistance properties. Because one- way functions are a fundamental cryptographic primitive, many of these MDCs, which typ- ically exhibit behavior informally equated with one-wayness and randomness, have been proposed for use in various applications distinct from data integrity, including, as discussed below: 1. conﬁrmation of knowledge 2. key derivation 3. pseudorandom number generation Hash functions used for conﬁrmation of knowledge facilitate commitment to data values, or demonstrate possession of data, without revealing such data itself (until possibly a later point in time); veriﬁcation is possible by parties in possession of the data. This resembles the use of MACs where one also essentially demonstrates knowledge of a secret (but with the demonstration bound to a speciﬁc message). The property of hash functions required is preimage resistance (see also partial-preimage resistance below). Speciﬁc examples in- clude use in password veriﬁcation using unencrypted password-image ﬁles (Chapter 10); symmetric-key digital signatures (Chapter 11); key conﬁrmation in authenticated key es- tablishment protocols (Chapter 12); and document-dating or timestamping by hash-code registration (Chapter 13). In general, use of hash functions for purposes other than which they were originally de- signed requires caution, as such applications may require additional properties (see below) §9.2 Classiﬁcation and framework 331 these functions were not designed to provide; see Remark 9.22. Unkeyed hash functions having properties associated with one-way functions have nonetheless been proposed for a wide range of applications, including as noted above: •key derivation– to compute sequences of new keys from prior keys (Chapter 13). A primary example is key derivation in point-of-sale (POS) terminals; here an impor- tant requirement is that the compromise of currently active keys must not compromise the security of previous transaction keys. A second example is in the generation of one-time password sequences based on one-way functions (Chapter 10). •pseudorandom number generation– to generate sequences of numbers which have various properties of randomness. (A pseudorandom number generator can be used to construct a symmetric-key block cipher, among other things.) Due to the difﬁculty of producing cryptographically strong pseudorandom numbers (see Chapter 5), MDCs should not be used for this purpose unless the randomness requirements are clearly understood, and the MDC is veriﬁed to satisfy these. For the applications immediately above, rather than hash functions, the cryptographic prim- itive which is needed may be apseudorandom function(or keyed pseudorandom function). 9.22 Remark (use of MDCs) Many MDCs used in practice may appear to satisfy additional requirements beyond those for which they were originally designed. Nonetheless, the use of arbitrary hash functions cannot be recommended for any applications without careful analysis precisely identifying both the critical properties required by the application and those provided by the function in question (cf.§9.5.2). Additional properties of one-way hash functions Additional properties of one-way hash functions called for by the above-mentioned appli- cations include the following. 1. non-correlation. Input bits and output bits should not be correlated. Related to this, an avalanche property similar to that of good block ciphers is desirable whereby every input bit affects every output bit. (This rules out hash functions for which preimage resistance fails to imply 2nd-preimage resistance simply due to the function effec- tively ignoring a subset of input bits.) 2. near-collision resistance. It should be hard to ﬁnd any two inputsx,x ′such thath(x) and h(x′) differ in only a small number of bits. 3. partial-preimage resistanceorlocal one-wayness. It should be as difﬁcult to recover any substring as to recover the entire input. Moreover, even if part of the input is known, it should be difﬁcult to ﬁnd the remainder (e.g., iftinput bits remain un- known, it should take on average2t−1 hash operations to ﬁnd these bits.) Partial preimage resistance is an implicit requirement in some of the proposed applications of§9.5.2. One example where near-collision resistance is necessary is when only half of the output bits of a hash function are used. Many of these properties can be summarized as requirements that there be neither lo- cal nor global statistical weaknesses; the hash function should not be weaker with respect to some parts of its input or output than others, and all bits should be equally hard. Some of these may be calledcertiﬁcational properties– properties which intuitively appear de- sirable, although they cannot be shown to be directly necessary. 332 Ch. 9 Hash Functions and Data Integrity 9.3 Basic constructions and general results 9.3.1 General model for iterated hash functions Most unkeyed hash functionshare designed as iterative processes which hash arbitrary- length inputs by processing successive ﬁxed-size blocks of the input, as illustrated in Fig- ure 9.2. output ﬁxed length preprocessing Hi original inputx inputx = x1x2 ··· xt formatted compression xi Hi−1 iterated compression (a) high-level view (b) detailed view transformation optional output output append padding bits append length block arbitrary length input function iterated processing functionf g outputh(x)= g(Ht) f H0 = IV hash functionh Ht Figure 9.2:General model for an iterated hash function. A hash inputxof arbitrary ﬁnite length is divided into ﬁxed-lengthr-bit blocksxi. This preprocessing typically involves appending extra bits (padding) as necessary to attain an overall bitlength which is a multiple of the blocklengthr, and often includes (for security reasons – e.g., see Algorithm 9.26) a block or partial block indicating the bitlength of the unpadded input. Each blockxi then serves as input to an internal ﬁxed-size hash function f, thecompression functionofh, which computes a new intermediate result of bitlengthn for some ﬁxedn, as a function of the previousn-bit intermediate result and the next input blockxi. LettingHi denote the partial result after stagei, the general process for an iterated §9.3 Basic constructions and general results 333 hash function with inputx= x1x2 ...xt can be modeled as follows: H0 = IV; Hi = f(Hi−1,xi), 1 ≤i≤t; h(x)= g(Ht). (9.1) Hi−1 serves as then-bitchaining variablebetween stagei−1 and stagei, andH0 is a pre-deﬁned starting value orinitializing value(IV). An optional output transformationg (see Figure 9.2) is used in a ﬁnal step to map then-bit chaining variable to anm-bit result g(Ht);gis often the identity mappingg(Ht)= Ht. Particular hash functions are distinguished by the nature of the preprocessing, com- pression function, and output transformation. 9.3.2 General constructions and extensions To begin, an example demonstrating an insecure construction is given. Several secure gen- eral constructions are then discussed. 9.23 Example (insecure trivial extension of OWHF to CRHF) In the case that an iterated OWHF hyieldingn-bit hash-values is not collision resistant (e.g., when a2n/2 birthday collision attack is feasible – see§9.7.1) one might propose constructing fromha CRHF using as output the concatenation of the last twon-bit chaining variables, so that at-block message has hash-valueHt−1\|\|Ht rather thanHt. This is insecure as the ﬁnal message blockxt can be held ﬁxed along withHt, reducing the problem to ﬁnding a collision on Ht−1 forh. □ Extending compression functions to hash functions Fact 9.24 states an important relationship between collision resistant compression functions and collision resistant hash functions. Not only can the former be extended to the latter, but this can be done efﬁciently using Merkle’s meta-method of Algorithm 9.25 (also called the Merkle-Damg˚ard construction). This reduces the problem of ﬁnding such a hash function to that of ﬁnding such a compression function. 9.24 Fact(extending compression functions) Any compression functionfwhich is collision resistant can be extended to a collision resistant hash functionh(taking arbitrary length inputs). 9.25 AlgorithmMerkle’s meta-method for hashing INPUT: compression functionfwhich is collision resistant. OUTPUT: unkeyed hash functionhwhich is collision resistant. 1. Supposefmaps (n+ r)-bit inputs ton-bit outputs (for concreteness, considern= 128 and r= 512). Construct a hash functionhfrom f, yieldingn-bit hash-values, as follows. 2. Break an inputxof bitlengthbinto blocksx1x2 ...xt each of bitlengthr, padding out the last blockxt with 0-bits if necessary. 3. Deﬁne an extra ﬁnal blockxt+1, the length-block, to hold the right-justiﬁed binary representation ofb(presume thatb< 2r). 4. Letting0j represent the bitstring ofj0’s, deﬁne then-bit hash-value ofxto be h(x)= Ht+1 = f(Ht \|\|xt+1) computed from: H0 =0 n; Hi = f(Hi−1 \|\|xi), 1 ≤i≤t+1 . 334 Ch. 9 Hash Functions and Data Integrity The proof that the resulting functionhis collision resistant follows by a simple argu- ment that a collision forhwould imply a collision forffor some stagei. The inclusion of the length-block, which effectively encodes all messages such that no encoded input is the tail end of any other encoded input, is necessary for this reasoning. Adding such a length- block is sometimes called Merkle-Damg˚ard strengthening (MD-strengthening), which is now stated separately for future reference. 9.26 AlgorithmMD-strengthening Before hashing a messagex= x1x2 ...xt (wherexi is a block of bitlengthrappropriate for the relevant compression function) of bitlengthb, append a ﬁnal length-block,xt+1, containing the (say) right-justiﬁed binary representation ofb. (This presumesb< 2r.) Cascading hash functions 9.27 Fact(cascading hash functions)I feitherh1 or h2 is a collision resistant hash function, thenh(x)= h1(x) \|\|h2(x) is a collision resistant hash function. If bothh1 and h2 in Fact 9.27 aren-bit hash functions, thenhproduces2n-bit out- puts; mapping this back down to ann-bit output by ann-bit collision-resistant hash func- tion (h1 and h2 are candidates) would leave the overall mapping collision-resistant. Ifh1 and h2 are independent, then ﬁnding a collision forhrequires ﬁnding a collision for both simultaneously (i.e., on the same input), which one could hope would require the product of the efforts to attack them individually. This provides a simple yet powerful way to (almost surely) increase strength using only available components. 9.3.3 Formatting and initialization details 9.28 Note (data representation) As hash-values depend on exact bitstrings, different data rep- resentations (e.g., ASCII vs. EBCDIC) must be converted to a common format before com- puting hash-values. (i) Padding and length-blocks For block-by-block hashing methods, extra bits are usually appended to a hash input string before hashing, to pad it out to a number of bits which make it a multiple of the relevant block size. The padding bits need not be transmitted/stored themselves, provided the sender and recipient agree on a convention. 9.29 AlgorithmPadding Method 1 INPUT: datax; bitlengthngiving blocksize of data input to processing stage. OUTPUT: padded datax′, with bitlength a multiple ofn. 1. Append toxas few (possibly zero) 0-bits as necessary to obtain a stringx′whose bitlength is a multiple ofn. 9.30 AlgorithmPadding Method 2 INPUT: datax; bitlengthngiving blocksize of data input to processing stage. OUTPUT: padded datax′, with bitlength a multiple ofn. 1. Append toxa single 1-bit. §9.3 Basic constructions and general results 335 2. Then append as few (possibly zero) 0-bits as necessary to obtain a stringx′whose bitlength is a multiple ofn. 9.31 Remark (ambiguous padding) Padding Method 1 isambiguous – trailing 0-bits of the original data cannot be distinguished from those added during padding. Such methods are acceptable if the length of the data (before padding) is known by the recipient by other means. Padding Method 2 is not ambiguous – each padded stringx ′corresponds to a unique unpadded stringx. When the bitlength of the original dataxis already a multiple ofn, Padding Method 2 results in the creation of an extra block. 9.32 Remark (appended length blocks) Appending a logical length-block prior to hashing prevents collision and pseudo-collision attacks which ﬁnd second messages of different length, including trivial collisions for random IVs (Example 9.96), long-message attacks (Fact 9.37), and ﬁxed-point attacks (page 374). This further justiﬁes the use of MD- strengthening (Algorithm 9.26). Trailing length-blocks and padding are often combined. For Padding Method 2, a len- gth ﬁeld of pre-speciﬁed bitlengthwmay replace the ﬁnalw0-bits padded if padding would otherwise causewor more redundant such bits. By pre-agreed convention, the length ﬁeld typically speciﬁes the bitlength of the original message. (If used instead to specify the num- ber of padding bits appended, deletion of leading blocks cannot be detected.) (ii) IVs Whether the IV is ﬁxed, is randomly chosen per hash function computation, or is a function of the data input, the same IV must be used to generate and verify a hash-value. If not known a prioriby the veriﬁer, it must be transferred along with the message. In the latter case, this generally should be done with guaranteed integrity (to cut down on the degree of freedom afforded to adversaries, in line with the principle that hash functions should be deﬁned with a ﬁxed or a small set of allowable IVs). 9.3.4 Security objectives and basic attacks As a framework for evaluating the computational security of hash functions, the objectives of both the hash function designer and an adversary should be understood. Based on Deﬁ- nitions 9.3, 9.4, and 9.7, these are summarized in Table 9.2, and discussed below. Hash type Design goal Ideal strength Adversary’s goal OWHF preimage resistance; 2n produce preimage; 2nd-preimage resistance 2n ﬁnd 2nd input, same image CRHF collision resistance 2n/2 produce any collision MAC key non-recovery; 2t deduce MAC key; computation resistance Pf =m a x ( 2−t, 2−n) produce new (msg, MAC) Table 9.2:Design objectives forn-bit hash functions (t-bit MAC key).Pf denotes the probability of forgery by correctly guessing a MAC. Given a speciﬁc hash function, it is desirable to be able to prove a lower bound on the com- plexity of attacking it under speciﬁed scenarios, with as few or weak a set of assumptions as possible. However, such results are scarce. Typically the best guidance available regarding 336 Ch. 9 Hash Functions and Data Integrity the security of a particular hash function is the complexity of the (most efﬁcient) applicable known attack, which gives anupperbound on security. An attack ofcomplexity2t is one which requires approximately2t operations, each being an appropriate unit of work (e.g., one execution of the compression function or one encryption of an underlying cipher). The storage complexity of an attack (i.e., storage required) should also be considered. (i) Attacks on the bitsize of an MDC Given a ﬁxed messagexwithn-bit hashh(x), a naive method for ﬁnding an input colliding withxis to pick a random bitstringx′(of bounded bitlength) and check ifh(x′)= h(x). The cost may be as little as one compression function evaluation, and memory is negligi- ble. Assuming the hash-code approximates a uniform random variable, the probability of a match is2 −n. The implication of this is Fact 9.33, which also indicates the effort required to ﬁnd collisions ifxmay itself be chosen freely. Deﬁnition 9.34 is motivated by the de- sign goal that the best possible attack should require no less than such levels of effort, i.e., essentially brute force. 9.33 Fact(basic hash attacks) For ann-bit hash functionh, one may expect a guessing attack to ﬁnd a preimage or second preimage within2n hashing operations. For an adversary able to choose messages, a birthday attack (see§9.7.1) allows colliding pairs of messagesx,x′ withh(x)= h(x′) to be found in about2n/2 operations, and negligible memory. 9.34 DeﬁnitionAn n-bit unkeyed hash function hasideal securityif both: (1) given a hash output, producing each of a preimage and a 2nd-preimage requires approximately2n oper- ations; and (2) producing a collision requires approximately2n/2 operations. (ii) Attacks on the MAC key space An attempt may be made to determine a MAC key using exhaustive search. With a sin- gle known text-MAC pair, an attacker may compute then-bit MAC on that text under all possible keys, and then check which of the computed MAC-values agrees with that of the known pair. For at-bit key space this requires2t MAC operations, after which one expects 1+2 t−n candidate keys remain. Assuming the MAC behaves as a random mapping, it can be shown that one can expect to reduce this to a unique key by testing the candidate keys us- ing just overt/ntext-MAC pairs. Ideally, a MAC key (or information of cryptographically equivalent value) would not be recoverable in fewer than2t operations. As a probabilistic attack on the MAC key space distinct from key recovery, note that for at-bit key and a ﬁxed input, a randomly guessed key will yield a correct (n-bit) MAC with probability≈2−t fort<n . (iii) Attacks on the bitsize of a MAC MAC forgery involves producing any inputxand the corresponding correct MAC without having obtained the latter from anyone with knowledge of the key. For ann-bit MAC al- gorithm, either guessing a MAC for a given input, or guessing a preimage for a given MAC output, has probability of success about2−n, as for an MDC. A difference here, however, is that guessed MAC-values cannot be veriﬁed off-line without known text-MAC pairs – either knowledge of the key, or a “black-box” which provides MACs for given inputs (i.e., a chosen-text scenario) is required. Since recovering the MAC key trivially allows forgery, an attack on thet-bit key space (see above) must be also be considered here. Ideally, an ad- versary would be unable to produce new (correct) text-MAC pairs(x,y) with probability signiﬁcantly better thanmax(2−t,2−n), i.e., the better of guessing a key or a MAC-value. §9.3 Basic constructions and general results 337 (iv) Attacks using precomputations, multiple targets, and long messages 9.35 Remark (precomputation of hash values) For both preimage and second preimage attacks, an opponent who precomputes a large number of hash function input-output pairs may trade off precomputation plus storage for subsequent attack time. For example, for a 64-bit hash value, if one randomly selects240 inputs, then computes their hash values and stores (hash value, input) pairs indexed by hash value, this precomputation ofO(240) time and space allows an adversary to increase the probability of ﬁnding a preimage (per one subsequent hash function computation) from2 −64 to2−24. Similarly, the probability of ﬁnding a sec- ond preimage increases tortimes its original value (when no stored pairs are known) ifr input-output pairs of a OWHF are precomputed and tabulated. 9.36 Remark (effect of parallel targets for OWHFs) In a basic attack, an adversary seeks a sec- ond preimage for one ﬁxed target (the image computed from a ﬁrst preimage). If there arer targets and the goal is to ﬁnd a second preimage for any one of theser, then the probability of success increases tortimes the original probability. One implication is that when using hash functions in conjunction with keyed primitives such as digital signatures, repeated use of the keyed primitive may weaken the security of the combined mechanism in the follow- ing sense. Ifrsigned messages are available, the probability of a hash collision increases r-fold (cf. Remark 9.35), and colliding messages yield equivalent signatures, which an op- ponent could not itself compute off-line. Fact 9.37 reﬂects a related attack strategy of potential concern when using iterated hash functions on long messages. 9.37 Fact(long-message attack for 2nd-preimage) Lethbe an iteratedn-bit hash function with compression functionf(as in equation (9.1), without MD-strengthening). Letxbe a mes- sage consisting oftblocks. Then a 2nd-preimage forh(x) can be found in time(2 n/s)+ s operations off, and in spacen(s+lg(s)) bits, for anysin the range1 ≤s≤min(t,2n/2). Justiﬁcation.The idea is to use a birthday attack on the intermediate hash-results; a sketch for the choices= tfollows. Computeh(x), storing(Hi,i) for each of thetintermediate hash-resultsHi corresponding to thetinput blocksxi in a table such that they may be later indexed by value. Computeh(z) for random choicesz, checking for a collision involving h(z) in the table, until one is found; approximately2n/svalueszwill be required, by the birthday paradox. Identify the indexjfrom the table responsible for the collision; the input zxj+1xj+2 ...xt then collides withx. 9.38 Note (implication of long messages) Fact 9.37 implies that for “long” messages, a 2nd- preimage is generally easier to ﬁnd than a preimage (the latter takes at most2n operations), becoming moreso with the length ofx. Fort≥2n/2, computation is minimized by choos- ings=2 n/2 in which case a 2nd-preimage costs about2n/2 executions off(comparable to the difﬁculty of ﬁnding a collision). 9.3.5 Bitsizes required for practical security Suppose that a hash function producesn-bit hash-values, and as a representative benchmark assume that280 (but not fewer) operations is acceptably beyond computational feasibility.2 Then the following statements may be made regardingn. 2Circa 1996,240 simple operations is quite feasible, and256 is considered quite reachable by those with suf- ﬁcient motivation (possibly using parallelization or customized machines). 338 Ch. 9 Hash Functions and Data Integrity 1. For a OWHF, n ≥ 80 is required. Exhaustive off-line attacks require at most2n operations; this may be reduced with precomputation (Remark 9.35). 2. For a CRHF,n≥160 is required. Birthday attacks are applicable (Fact 9.33). 3. For a MAC,n≥64 along with a MAC key of 64-80 bits is sufﬁcient for most ap- plications and environments (cf. Table 9.1). If a single MAC key remains in use, off-line attacks may be possible given one or more text-MAC pairs; but for a proper MAC algorithm, preimage and 2nd-preimage resistance (as well as collision resis- tance) should follow directly from lack of knowledge of the key, and thus security with respect to such attacks should depend on the keysize rather thann. For attacks requiring on-line queries, additional controls may be used to limit the number of such queries, constrain the format of MAC inputs, or prevent disclosure of MAC outputs for random (chosen-text) inputs. Given special controls, values as small asn=3 2or 40 may be acceptable; but caution is advised, since even with one-time MAC keys, the chance any randomly guessed MAC being correct is2 −n, and the relevant factors are the total number of trials a system is subject to over its lifetime, and the conse- quences of a single successful forgery. These guidelines may be relaxed somewhat if a lower threshold of computational infeasi- bility is assumed (e.g.,2 64 instead of280). However, an additional consideration to be taken into account is that for both a CRHF and a OWHF, not only can off-line attacks be carried out, but these can typically be parallelized. Key search attacks against MACs may also be parallelized. 9.4 Unkeyed hash functions (MDCs) A move from general properties and constructions to speciﬁc hash functions is now made, and in this section the subclass of unkeyed hash functions known as modiﬁcation detection codes (MDCs) is considered. From a structural viewpoint, these may be categorized based on the nature of the operations comprising their internal compression functions. From this viewpoint, the three broadest categories of iterated hash functions studied to date are hash functionsbased on block ciphers,customized hash functions, and hash functionsbased on modular arithmetic. Customized hash functions are those designed speciﬁcally for hashing, with speed in mind and independent of other system subcomponents (e.g., block cipher or modular multiplication subcomponents which may already be present for non-hashing pur- poses). Table 9.3 summarizes the conjectured security of a subset of the MDCs subsequently discussed in this section. Similar to the case of block ciphers for encryption (e.g. 8- or 12- round DES vs. 16-round DES), security of MDCs often comes at the expense of speed, and tradeoffs are typically made. In the particular case of block-cipher-based MDCs, a provably secure scheme of Merkle (see page 378) with rate0.276 (see Deﬁnition 9.40) is known but little-used, while MDC-2 is widely believed to be (but not provably) secure, has rate=0 .5, and receives much greater attention in practice. 9.4.1 Hash functions based on block ciphers A practical motivation for constructing hash functions from block ciphers is that if an efﬁ- cient implementation of a block cipher is already available within a system (either in hard- ware or software), then using it as the central component for a hash function may provide §9.4 Unkeyedhash functions (MDCs) 339 ↓Hash function n m Preimage Collision Comments Matyas-Meyer-Oseasa n n 2n 2n/2 for keylength= n MDC-2 (with DES)b 64 128 2 ·282 2 ·254 rate 0.5 MDC-4 (with DES) 64 128 2109 4 ·254 rate 0.25 Merkle (with DES) 106 128 2112 256 rate 0.276 MD4 512 128 2128 220 Remark 9.50 MD5 512 128 2128 264 Remark 9.52 RIPEMD-128 512 128 2128 264 – SHA-1, RIPEMD-160 512 160 2160 280 – aThe same strength is conjectured for Davies-Meyer and Miyaguchi-Preneel hash functions. bStrength could be increased using a cipher with keylength equal to cipher blocklength. Table 9.3:Upper bounds on strength of selected hash functions.n-bit message blocks are processed to producem-bit hash-values. Number of cipher or compression function operations currently be- lieved necessary to ﬁnd preimages and collisions are speciﬁed, assuming no underlying weaknesses for block ciphers (ﬁgures for MDC-2 and MDC-4 account for DES complementation and weak key properties). Regarding rate, see Deﬁnition 9.40. the latter functionality at little additional cost. The (not always well-founded) hope is that a good block cipher may serve as a building block for the creation of a hash function with properties suitable for various applications. Constructions for hash functions have been given which are “provably secure” assum- ing certain ideal properties of the underlying block cipher. However, block ciphers do not possess the properties of random functions (for example, they are invertible – see Re- mark 9.14). Moreover, in practice block ciphers typically exhibit additional regularities or weaknesses (see§9.7.4). For example, for a block cipherE, double encryption using an encrypt-decrypt (E-D) cascade with keysK 1,K2 results in the identity mapping when K1 = K2. In summary, while various necessary conditions are known, it is unclear ex- actly what requirements of a block cipher are sufﬁcient to construct a secure hash function, and properties adequate for a block cipher (e.g., resistance to chosen-text attack) may not guarantee a good hash function. In the constructions which follow, Deﬁnition 9.39 is used. 9.39 DeﬁnitionAn (n,r) block cipheris a block cipher deﬁning an invertible function from n-bit plaintexts ton-bit ciphertexts using anr-bit key. IfEis such a cipher, thenEk(x) denotes the encryption ofxunder keyk. Discussion of hash functions constructed fromn-bit block ciphers is divided between those producingsingle-length(n-bit) anddouble-length(2n-bit) hash-values, where single and double are relative to the size of the block cipher output. Under the assumption that computations of2 64 operations are infeasible,3 the objective of single-length hash functions is to provide a OWHF for ciphers of blocklength nearn=6 4, or to provide CRHFs for cipher blocklengths nearn= 128. The motivation for double-length hash functions is that many n-bit block ciphers exist of size approximatelyn=6 4, and single-length hash-codes of this size are not collision resistant. For such ciphers, the goal is to obtain hash-codes of bitlength2nwhich are CRHFs. In the simplest case, the size of the key used in such hash functions is approximately the same as the blocklength of the cipher (i.e.,nbits). In other cases, hash functions use 3The discussion here is easily altered for a more conservative bound, e.g.,280 operations as used in§9.3.5. Here 264 is more convenient for discussion, due to the omnipresence of 64-bit block ciphers. 340 Ch. 9 Hash Functions and Data Integrity larger (e.g., double-length) keys. Another characteristic to be noted in such hash functions is the number of block cipher operations required to produce a hash output of blocklength equal to that of the cipher, motivating the following deﬁnition. 9.40 DeﬁnitionLethbe an iterated hash function constructed from a block cipher, with com- pression functionfwhich performssblock encryptions to process each successiven-bit message block. Then therateofhis1/s. The hash functions discussed in this section are summarized in Table 9.4. The Matyas- Meyer-Oseas and MDC-2 algorithms are the basis, respectively, of the two generic hash functions in ISO standard 10118-2, each allowing use of anyn-bit block cipherEand pro- viding hash-codes of bitlengthm≤nand m≤2n, respectively. Hash function (n,k,m) Rate Matyas-Meyer-Oseas (n,k,n) 1 Davies-Meyer (n,k,n) k/n Miyaguchi-Preneel (n,k,n) 1 MDC-2 (with DES) (64,56,128) 1/2 MDC-4 (with DES) (64,56,128) 1/4 Table 9.4:Summary of selected hash functions based onn-bit block ciphers.k = key bitsize (ap- proximate); function yieldsm-bit hash-values. (i) Single-length MDCs of rate 1 The ﬁrst three schemes described below, and illustrated in Figure 9.3, are closely related single-length hash functions based on block ciphers. These make use of the following pre- deﬁned components: 1. a genericn-bit block cipherEK parametrized by a symmetric keyK; 2. a functiongwhich mapsn-bit inputs to keysKsuitable forE(if keys forEare also of lengthn,gmight be the identity function); and 3. a ﬁxed (usuallyn-bit) initial valueIV, suitable for use withE. xi Hi xi E Hi Matyas-Meyer-Oseas Miyaguchi-Preneel Hi−1 gHi−1g E Hi−1 E Hi xi Davies-Meyer Figure 9.3:Three single-length, rate-one MDCs based on block ciphers. §9.4 Unkeyedhash functions (MDCs) 341 9.41 AlgorithmMatyas-Meyer-Oseas hash INPUT: bitstringx. OUTPUT: n-bit hash-code ofx. 1. Inputxis divided inton-bit blocks and padded, if necessary, to complete last block. Denote the padded message consisting oftn-bit blocks:x1x2 ...xt. A constantn- bit initial valueIV must be pre-speciﬁed. 2. The output isHt deﬁned by:H0 = IV;Hi = Eg(Hi−1)(xi)⊕xi,1 ≤i≤t. 9.42 AlgorithmDavies-Meyer hash INPUT: bitstringx. OUTPUT: n-bit hash-code ofx. 1. Inputxis divided intok-bit blocks wherekis the keysize, and padded, if necessary, to complete last block. Denote the padded message consisting oftk-bit blocks:x1x2 ...x t. A constantn-bit initial valueIV must be pre-speciﬁed. 2. The output isHt deﬁned by:H0 = IV;Hi = Exi (Hi−1)⊕Hi−1,1 ≤i≤t. 9.43 AlgorithmMiyaguchi-Preneel hash This scheme is identical to that of Algorithm 9.41, except the outputHi−1 from the previous stage is also XORed to that of the current stage. More precisely,Hi is redeﬁned as:H0 = IV; Hi = Eg(Hi−1)(xi)⊕xi⊕Hi−1,1 ≤i≤t. 9.44 Remark (dual schemes) The Davies-Meyer hash may be viewed as the ‘dual’ of the Mat- yas-Meyer-Oseas hash, in the sense thatxi and Hi−1 play reversed roles. When DES is used as the block cipher in Davies-Meyer, the input is processed in 56-bit blocks (yield- ing rate56/64 <1), whereas Matyas-Meyer-Oseas and Miyaguchi-Preneel process 64-bit blocks. 9.45 Remark (black-box security) Aside from heuristic arguments as given in Example 9.13, it appears that all three of Algorithms 9.41, 9.42, and 9.43 yield hash functions which are provably secure under an appropriate “black-box” model (e.g., assumingEhas the required randomness properties, and that attacks may not make use of any special properties or in- ternal details ofE). “Secure” here means that ﬁnding preimages and collisions (in fact, pseudo-preimages and pseudo-collisions – see§9.7.2) require on the order of2 n and 2n/2 n-bit block cipher operations, respectively. Due to their single-length nature, none of these three is collision resistant for underlying ciphers of relatively small blocklength (e.g., DES, which yields 64-bit hash-codes). Several double-length hash functions based on block ciphers are considered next. (ii) Double-length MDCs: MDC-2 and MDC-4 MDC-2 and MDC-4 are manipulation detection codes requiring 2 and 4, respectively, block cipher operations per block of hash input. They employ a combination of either 2 or 4 itera- tions of the Matyas-Meyer-Oseas (single-length) scheme to produce a double-length hash. When used as originally speciﬁed, using DES as the underlying block cipher, they produce 128-bit hash-codes. The general construction, however, can be used with other block ci- phers. MDC-2 and MDC-4 make use of the following pre-speciﬁed components: 342 Ch. 9 Hash Functions and Data Integrity 1. DES as the block cipherEK of bitlengthn=6 4parameterized by a56-bit keyK; 2. two functionsgand ˜gwhich map64-bit valuesUto suitable56-bit DES keys as fol- lows. ForU= u1u2 ...u64, delete every eighth bit starting withu8, and set the 2nd and 3rd bits to ‘10’ forg, and ‘01’ for˜g: g(U)= u1 10 u4u5u6u7u9u10 ...u 63. ˜g(U)= u1 01 u4u5u6u7u9u10 ...u 63. (The resulting values are guaranteed not to be weak or semi-weak DES keys, as all such keys have bit 2 = bit 3; see page 375. Also, this guarantees the security require- ment thatg(IV) ̸=˜g( ˜IV).) MDC-2 is speciﬁed in Algorithm 9.46 and illustrated in Figure 9.4. CD CBA A Eg Xi in2 in4 Hi out1 out2 Hi−1 ˜Hi−1 in3 in1 E ˜g B D ˜Hi Figure 9.4:Compression function of MDC-2 hash function.E = DES. 9.46 AlgorithmMDC-2 hash function (DES-based) INPUT: stringxof bitlengthr=6 4tfort≥2. OUTPUT: 128-bit hash-code ofx. 1. Partitionxinto64-bit blocksxi:x= x1x2 ...xt. 2. Choose the 64-bit non-secret constantsIV, ˜IV (the same constants must be used for MDC veriﬁcation) from a set of recommended prescribed values. A default set of prescribed values is (in hexadecimal):IV = 0x5252525252525252, ˜IV = 0x2525252525252525. §9.4 Unkeyedhash functions (MDCs) 343 3. Let\|\|denote concatenation, andCL i ,CR i the left and right32-bit halves ofCi. The output ish(x)= Ht \|\|˜Ht deﬁned as follows (for1 ≤i≤t): H0 = IV; ki = g(Hi−1); Ci = Eki (xi)⊕xi; Hi = CL i \|\| ˜Ci R ˜H0 = ˜IV; ˜ki = ˜g( ˜Hi−1); ˜Ci = E˜ki (xi)⊕xi; ˜Hi = ˜Ci L \|\| Ci R . In Algorithm 9.46, padding may be necessary to meet the bitlength constraint on the inputx. In this case, an unambiguous padding method may be used (see Remark 9.31), possibly including MD-strengthening (see Remark 9.32). MDC-4 (see Algorithm 9.47 and Figure 9.5) is constructed using the MDC-2 compres- sion function. One iteration of the MDC-4 compression function consists of two sequential executions of the MDC-2 compression function, where: 1. the two 64-bit data inputs to the ﬁrst MDC-2 compression are both the same next 64-bit message block; 2. the keys for the ﬁrst MDC-2 compression are derived from the outputs (chaining vari- ables) of the previous MDC-4 compression; 3. the keys for the second MDC-2 compression are derived from the outputs (chaining variables) of the ﬁrst MDC-2 compression; and 4. the two 64-bit data inputs for the second MDC-2 compression are the outputs (chain- ing variables) from the opposite sides of the previous MDC-4 compression. 9.47 AlgorithmMDC-4 hash function (DES-based) INPUT: stringxof bitlengthr=6 4tfort≥2. (See MDC-2 above regarding padding.) OUTPUT: 128-bit hash-code ofx. 1. As in step 1 of MDC-2 above. 2. As in step 2 of MDC-2 above. 3. With notation as in MDC-2, the output ish(x)= Gt \|\|˜Gt deﬁned as follows (for 1 ≤i≤t): G0 = IV; ˜G0 = ˜IV; ki = g(Gi−1); Ci = Eki (xi)⊕xi; Hi = CL i \|\| ˜Ci R ˜ki = ˜g(˜Gi−1); ˜Ci = E˜ki (xi)⊕xi; ˜Hi = ˜Ci L \|\| Ci R ji = g(Hi); Di = Eji (˜Gi−1)⊕˜Gi−1; Gi = DL i \|\| ˜Di R ˜ji = ˜g(˜Hi); ˜Di = E˜ji (Gi−1)⊕Gi−1; ˜Gi = ˜Di L \|\| Di R . 9.4.2 Customized hash functions based on MD4 Customized hash functionsare those which are speciﬁcally designed “from scratch” for the explicit purpose of hashing, with optimized performance in mind, and without being con- strained to reusing existing system components such as block ciphers or modular arithmetic. Those having received the greatest attention in practice are based on the MD4 hash function. Number 4 in a series of hash functions (Message Digestalgorithms), MD4 was de- signed speciﬁcally for software implementation on 32-bit machines. Security concerns mo- tivated the design of MD5 shortly thereafter, as a more conservative variation of MD4. 344 Ch. 9 Hash Functions and Data Integrity MDC-2 compression function MDC-2 compression function Xi Gi Hi Gi−1 out1 in3 in4 out2 in1 in2 in3 in4 ˜Gi−1 ˜Gi ˜Hiout1 out2 ˜Gi−1 Gi−1 in1 in2 Figure 9.5:Compression function of MDC-4 hash function Other important subsequent variants include the Secure Hash Algorithm (SHA-1), the hash function RIPEMD, and its strengthened variants RIPEMD-128 and RIPEMD-160. Param- eters for these hash functions are summarized in Table 9.5. “Rounds×Steps per round” refers to operations performed on input blocks within the corresponding compression func- tion. Table 9.6 speciﬁes test vectors for a subset of these hash functions. Notation for description of MD4-family algorithms Table 9.7 deﬁnes the notation for the description of MD4-family algorithms described be- low. Note 9.48 addresses the implementation issue of converting strings of bytes to words in an unambiguous manner. 9.48 Note (little-endian vs. big-endian) For interoperable implementations involving byte-to- word conversions on different processors (e.g., converting between 32-bit words and groups of four 8-bit bytes), an unambiguous convention must be speciﬁed. Consider a stream of bytesB i with increasing memory addressesi, to be interpreted as a 32-bit word with nu- merical valueW.I nlittle-endianarchitectures, the byte with the lowest memory address (B1) is the least signiﬁcant byte:W =2 24B4 +2 16B3 +2 8B2 + B1.I nbig-endian architectures, the byte with the lowest address (B1) is the most signiﬁcant byte:W = 224B1 +2 16B2 +2 8B3 + B4. (i) MD4 MD4 (Algorithm 9.49) is a 128-bit hash function. The original MD4 design goals were that breaking it should require roughly brute-force effort: ﬁnding distinct messages with the same hash-value should take about264 operations, and ﬁnding a message yielding a §9.4 Unkeyedhash functions (MDCs) 345 Name Bitlength Rounds ×Steps per round Relative speed MD4 128 3 ×16 1.00 MD5 128 4 ×16 0.68 RIPEMD-128 128 4 ×16 twice (in parallel) 0.39 SHA-1 160 4 ×20 0.28 RIPEMD-160 160 5 ×16 twice (in parallel) 0.24 Table 9.5:Summary of selected hash functions based on MD4. Name String Hash value (as a hex byte string) MD4 “” 31d6cfe0d16ae931b73c59d7e0c089c0 “a” bde52cb31de33e46245e05fbdbd6fb24 “abc” a448017aaf21d8525fc10ae87aa6729d “abcdefghijklmnopqrstuvwxyz” d79e1c308aa5bbcdeea8ed63df412da9 MD5 “” d41d8cd98f00b204e9800998ecf8427e “a” 0cc175b9c0f1b6a831c399e269772661 “abc” 900150983cd24fb0d6963f7d28e17f72 “abcdefghijklmnopqrstuvwxyz” c3fcd3d76192e4007dfb496cca67e13b SHA-1 “” da39a3ee5e6b4b0d3255bfef95601890afd80709 “a” 86f7e437faa5a7fce15d1ddcb9eaeaea377667b8 “abc” a9993e364706816aba3e25717850c26c9cd0d89d “abcdefghijklmnopqrstuvwxyz” 32d10c7b8cf96570ca04ce37f2a19d84240d3a89 RIPEMD-160 “” 9c1185a5c5e9fc54612808977ee8f548b2258d31 “a” 0bdc9d2d256b3ee9daae347be6f4dc835a467ffe “abc” 8eb208f7e05d987a9b044a8e98c6b087f15a0bfc “abcdefghijklmnopqrstuvwxyz” f71c27109c692c1b56bbdceb5b9d2865b3708dbc Table 9.6:Test vectors for selected hash functions. Notation Meaning u,v,w variables representing 32-bit quantities 0x67452301 hexadecimal 32-bit integer (least signiﬁcant byte: 01) + addition modulo232 u bitwise complement u←↩s result of rotatinguleft throughspositions uv bitwise AND u∨v bitwise inclusive-OR u⊕v bitwise exclusive-OR f(u,v,w) uv∨ uw g(u,v,w) uv∨uw∨vw h(u,v,w) u⊕v⊕w (X1,...,X j) ← simultaneous assignments(Xi ←Yi), (Y1,...,Y j) where (Y1,...,Y j) is evaluated prior to any assignments Table 9.7:Notation for MD4-family algorithms. 346 Ch. 9 Hash Functions and Data Integrity pre-speciﬁed hash-value about2128 operations. It is now known that MD4 fails to meet this goal (Remark 9.50). Nonetheless, a full description of MD4 is included as Algorithm 9.49 for historical and cryptanalytic reference. It also serves as a convenient reference for de- scribing, and allowing comparisons between, other hash functions in this family. 9.49 AlgorithmMD4 hash function INPUT: bitstringxof arbitrary bitlengthb≥0. (For notation see Table 9.7.) OUTPUT: 128-bit hash-code ofx. (See Table 9.6 for test vectors.) 1. Deﬁnition of constants.Deﬁne four 32-bit initial chaining values (IVs): h1 = 0x67452301,h2 = 0xefcdab89,h3 = 0x98badcfe,h4 = 0x10325476. Deﬁne additive 32-bit constants: y[j]= 0,0 ≤j≤15; y[j]= 0x5a827999,16 ≤j≤31; (constant = square-root of 2) y[j]= 0x6ed9eba1,32 ≤j≤47; (constant = square-root of 3) Deﬁne order for accessing source words (each list contains 0 through 15): z[0..15] = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], z[16..31] = [0,4,8,12,1,5,9,13,2,6,10,14,3,7,11,15], z[32..47] = [0,8,4,12,2,10,6,14,1,9,5,13,3,11,7,15]. Finally deﬁne the number of bit positions for left shifts (rotates): s[0..15] = [3,7,11,19,3,7,11,19,3,7,11,19,3,7,11,19], s[16..31] = [3,5,9,13,3,5,9,13,3,5,9,13,3,5,9,13], s[32..47] = [3,9,11,15,3,9,11,15,3,9,11,15,3,9,11,15]. 2. Preprocessing.Pad xsuch that its bitlength is a multiple of 512, as follows. Append a single 1-bit, then appendr−1 (≥0)0-bits for the smallestrresulting in a bitlength 64 less than a multiple of 512. Finally append the 64-bit representation ofbmod 2 64, as two 32-bit words with least signiﬁcant word ﬁrst. (Regarding converting between streams of bytes and 32-bit words, the convention is little-endian; see Note 9.48.) Let mbe the number of 512-bit blocks in the resulting string (b+ r+ 64 = 512m= 32 ·16m). The formatted input consists of16m32-bit words:x 0x1 ...x16m−1. Ini- tialize:(H1,H2,H3,H4) ←(h1,h2,h3,h4). 3. Processing.For eachifrom 0 tom−1, copy theith block of 16 32-bit words into temporary storage:X[j] ←x16i+j, 0 ≤j≤15, then process these as below in three 16-step rounds before updating the chaining variables: (initialize working variables)(A,B,C,D ) ←(H1,H2,H3,H4). (Round 1) Forjfrom 0 to15 do the following: t ←(A+ f(B,C,D)+ X[z[j]] +y[j]),(A,B,C,D ) ←(D,t←↩s[j],B,C). (Round 2) Forjfrom 16 to31 do the following: t ←(A+ g(B,C,D)+ X[z[j]] +y[j]),(A,B,C,D ) ←(D,t←↩s[j]),B,C). (Round 3) Forjfrom 32 to47 do the following: t ←(A+ h(B,C,D)+ X[z[j]] +y[j]),(A,B,C,D ) ←(D,t←↩s[j]),B,C). (update chaining values)(H1,H2,H3,H4) ←(H1 +A,H2 +B,H3 +C,H4 +D). 4. Completion.The ﬁnal hash-value is the concatenation:H1\|\|H2\|\|H3\|\|H4 (with ﬁrst and last bytes the low- and high-order bytes ofH1,H4, respectively). 9.50 Remark (MD4 collisions) Collisions have been found for MD4 in220 compression func- tion computations (cf. Table 9.3). For this reason, MD4 is no longer recommended for use as a collision-resistant hash function. While its utility as a one-way function has not been studied in light of this result, it is prudent to expect a preimage attack on MD4 requiring fewer than2 128 operations will be found. §9.4 Unkeyedhash functions (MDCs) 347 (ii) MD5 MD5 (Algorithm 9.51) was designed as a strengthened version of MD4, prior to actual MD4 collisions being found. It has enjoyed widespread use in practice. It has also now been found to have weaknesses (Remark 9.52). The changes made to obtain MD5 from MD4 are as follows: 1. addition of a fourth round of 16 steps, and a Round 4 function 2. replacement of the Round 2 function by a new function 3. modiﬁcation of the access order for message words in Rounds 2 and 3 4. modiﬁcation of the shift amounts (such that shifts differ in distinct rounds) 5. use of unique additive constants in each of the4 ×16 steps, based on the integer part of2 32 ·sin(j) for stepj(requiring overall, 256 bytes of storage) 6. addition of output from the previous step into each of the 64 steps. 9.51 AlgorithmMD5 hash function INPUT: bitstringxof arbitrary bitlengthb≥0. (For notation, see Table 9.7.) OUTPUT: 128-bit hash-code ofx. (See Table 9.6 for test vectors.) MD5 is obtained from MD4 by making the following changes. 1. Notation.Replace the Round 2 function by:g(u,v,w) def = uw ∨v w. Deﬁne a Round 4 function:k(u,v,w) def = v⊕(u∨ w). 2. Deﬁnition of constants.Redeﬁne unique additive constants: y[j]= ﬁrst 32 bits of binary valueabs(sin(j+1) ),0 ≤j≤63, wherejis in radians and “abs” denotes absolute value. Redeﬁne access order for words in Rounds 2 and 3, and deﬁne for Round 4: z[16..31] = [1,6,11,0,5,10,15,4,9,14,3,8,13,2,7,12], z[32..47] = [5,8,11,14,1,4,7,10,13,0,3,6,9,12,15,2], z[48..63] = [0,7,14,5,12,3,10,1,8,15,6,13,4,11,2,9]. Redeﬁne number of bit positions for left shifts (rotates): s[0..15] = [7,12,17,22,7,12,17,22,7,12,17,22,7,12,17,22], s[16..31] = [5,9,14,20,5,9,14,20,5,9,14,20,5,9,14,20], s[32..47] = [4,11,16,23,4,11,16,23,4,11,16,23,4,11,16,23], s[48..63] = [6,10,15,21,6,10,15,21,6,10,15,21,6,10,15,21]. 3. Preprocessing.As in MD4. 4. Processing.In each of Rounds 1, 2, and 3, replace “B←(t←↩s[j])”b y“B← B+( t←↩s[j])”. Also, immediately following Round 3 add: (Round 4) Forjfrom 48 to63 do the following: t ←(A+k(B,C,D)+X[z[j]]+y[j]),(A,B,C,D ) ←(D,B+(t←↩s[j]),B,C). 5. Completion.As in MD4. 9.52 Remark (MD5 compression function collisions) While no collisions for MD5 have yet been found (cf. Table 9.3), collisions have been found for the MD5 compression function. More speciﬁcally, these are called collisions for random IV . (See§9.7.2, and in particular Deﬁnition 9.97 and Note 9.98.) 348 Ch. 9 Hash Functions and Data Integrity (iii) SHA-1 The Secure Hash Algorithm (SHA-1), based on MD4, was proposed by the U.S. National Institute for Standards and Technology (NIST) for certain U.S. federal government appli- cations. The main differences of SHA-1 from MD4 are as follows: 1. The hash-value is 160 bits, and ﬁve (vs. four) 32-bit chaining variables are used. 2. The compression function has four rounds instead of three, using the MD4 step func- tionsf,g, andhas follows:fin the ﬁrst,gin the third, andhin both the second and fourth rounds. Each round has 20 steps instead of 16. 3. Within the compression function, each 16-word message block is expanded to an 80- word block, by a process whereby each of the last 64 of the 80 words is the XOR of 4 words from earlier positions in the expanded block. These 80 words are then input one-word-per-step to the 80 steps. 4. The core step is modiﬁed as follows: the only rotate used is a constant 5-bit rotate; the ﬁfth working variable is added into each step result; message words from the ex- panded message block are accessed sequentially; andCis updated asBrotated left 30 bits, rather than simplyB. 5. SHA-1 uses four non-zero additive constants, whereas MD4 used three constants only two of which were non-zero. The byte ordering used for converting between streams of bytes and 32-bit words in the ofﬁcial SHA-1 speciﬁcation is big-endian (see Note 9.48); this differs from MD4 which is little-endian. 9.53 AlgorithmSecure Hash Algorithm – revised (SHA-1) INPUT: bitstringxof bitlengthb≥0. (For notation, see Table 9.7.) OUTPUT: 160-bit hash-code ofx. (See Table 9.6 for test vectors.) SHA-1 is deﬁned (with reference to MD4) by making the following changes. 1. Notation. As in MD4. 2. Deﬁnition of constants.Deﬁne a ﬁfth IV to match those in MD4:h5 = 0xc3d2e1f0. Deﬁne per-round integer additive constants:y1 = 0x5a827999,y2 = 0x6ed9eba1, y3 = 0x8f1bbcdc,y4 = 0xca62c1d6. (No order for accessing source words, or spec- iﬁcation of bit positions for left shifts is required.) 3. Overall preprocessing.Pad as in MD4, except the ﬁnal two 32-bit words specifying the bitlengthbis appended with most signiﬁcant word preceding least signiﬁcant. As in MD4, the formatted input is16m32-bit words:x0x1 ...x16m−1. Initialize chaining variables:(H1,H2,H3,H4,H5) ←(h1,h2,h3,h4,h5). 4. Processing.For eachifrom 0 tom−1, copy theith block of sixteen 32-bit words into temporary storage:X[j] ←x16i+j,0 ≤j≤15, and process these as below in four 20-step rounds before updating the chaining variables: (expand 16-word block into 80-word block; letXj denoteX[j]) forjfrom 16 to79,Xj ←(( Xj−3⊕Xj−8⊕Xj−14⊕Xj−16 ) ←↩1). (initialize working variables)(A,B,C,D,E ) ←(H1,H2,H3,H4,H5). (Round 1) Forjfrom 0 to19 do the following: t ←((A←↩5) +f(B,C,D)+ E+ Xj + y1), (A,B,C,D,E ) ←(t,A,B ←↩30,C,D). (Round 2) Forjfrom 20 to39 do the following: t ←((A←↩5) +h(B,C,D)+ E+ Xj + y2), (A,B,C,D,E ) ←(t,A,B ←↩30,C,D). §9.4 Unkeyedhash functions (MDCs) 349 (Round 3) Forjfrom 40 to59 do the following: t ←((A←↩5) +g(B,C,D)+ E+ Xj + y3), (A,B,C,D,E ) ←(t,A,B ←↩30,C,D). (Round 4) Forjfrom 60 to79 do the following: t ←((A←↩5) +h(B,C,D)+ E+ Xj + y4), (A,B,C,D,E ) ←(t,A,B ←↩30,C,D). (update chaining values) (H1,H2,H3,H4,H5) ←(H1 + A,H2 + B,H3 + C,H4 + D,H5 + E). 5. Completion.The hash-value is:H1\|\|H2\|\|H3\|\|H4\|\|H5 (with ﬁrst and last bytes the high- and low-order bytes ofH1,H5, respectively). 9.54 Remark (security of SHA-1) Compared to 128-bit hash functions, the 160-bit hash-value of SHA-1 provides increased security against brute-force attacks. SHA-1 and RIPEMD- 160 (see§9.4.2(iv)) presently appear to be of comparable strength; both are considered stronger than MD5 (Remark 9.52). In SHA-1, a signiﬁcant effect of the expansion of 16- word message blocks to 80 words in the compression function is that any two distinct 16- word blocks yield 80-word values which differ in a larger number of bit positions, signif- icantly expanding the number of bit differences among message words input to the com- pression function. The redundancy added by this preprocessing evidently adds strength. (iv) RIPEMD-160 RIPEMD-160 (Algorithm 9.55) is a hash function based on MD4, taking into account knowledge gained in the analysis of MD4, MD5, and RIPEMD. The overall RIPEMD-160 compression function maps 21-word inputs (5-word chaining variable plus 16-word mes- sage block, with 32-bit words) to 5-word outputs. Each input block is processed in parallel by distinct versions (theleft lineand right line) of the compression function. The 160-bit outputs of the separate lines are combined to give a single 160-bit output. Notation Deﬁnition f(u,v,w) u⊕v⊕w g(u,v,w) uv∨ uw h(u,v,w) (u∨ v)⊕w k(u,v,w) uw∨v w l(u,v,w) u⊕(v∨ w) Table 9.8:RIPEMD-160 round function deﬁnitions. The RIPEMD-160 compression function differs from MD4 in the number of words of chaining variable, the number of rounds, the round functions themselves (Table 9.8), the order in which the input words are accessed, and the amounts by which results are rotated. The left and and right computation lines differ from each other in these last two items, in their additive constants, and in the order in which the round functions are applied. This de- sign is intended to improve resistance against known attack strategies. Each of the parallel lines uses the same IV as SHA-1. When writing the IV as a bitstring, little-endian ordering is used for RIPEMD-160 as in MD4 (vs. big-endian in SHA-1; see Note 9.48). 350 Ch. 9 Hash Functions and Data Integrity 9.55 AlgorithmRIPEMD-160 hash function INPUT: bitstringxof bitlengthb≥0. OUTPUT: 160-bit hash-code ofx. (See Table 9.6 for test vectors.) RIPEMD-160 is deﬁned (with reference to MD4) by making the following changes. 1. Notation. See Table 9.7, with MD4 round functionsf,g,hredeﬁned per Table 9.8 (which also deﬁnes the new round functionsk,l). 2. Deﬁnition of constants.Deﬁne a ﬁfth IV:h5 = 0xc3d2e1f0. In addition: (a) Use the MD4 additive constants for the left line, renamed:yL[j]= 0,0 ≤j≤ 15;yL[j]= 0x5a827999,16 ≤j≤31;yL[j]= 0x6ed9eba1,32 ≤j≤47. Deﬁne two further constants (square roots of 5,7):yL[j]= 0x8f1bbcdc,48 ≤ j≤63;yL[j]= 0xa953fd4e,64 ≤j≤79. (b) Deﬁne ﬁve new additive constants for the right line (cube roots of 2,3,5,7): yR[j]= 0x50a28be6,0 ≤j≤15; yR[j]= 0x5c4dd124,16 ≤j≤31; yR[j]= 0x6d703ef3,32 ≤j ≤47; yR[j]= 0x7a6d76e9,48 ≤j ≤63; yR[j]= 0,64 ≤j≤79. (c) See Table 9.9 for constants for stepjof the compression function:zL[j],zR[j] specify the access order for source words in the left and right lines;sL[j],sR[j] the number of bit positions for rotates (see below). 3. Preprocessing.As in MD4, with addition of a ﬁfth chaining variable:H5 ←h5. 4. Processing.For eachifrom 0 tom−1, copy theith block of sixteen 32-bit words into temporary storage:X[j] ←x16i+j,0 ≤j≤15. Then: (a) Execute ﬁve 16-step rounds of the left line as follows: (AL,BL,CL,DL,EL) ←(H1,H2,H3,H4,H5). (left Round 1) Forjfrom 0 to15 do the following: t ←(AL + f(BL,CL,DL)+ X[zL[j]] +yL[j]), (AL,BL,CL,DL,EL) ←(EL,EL +( t←↩sL[j]),BL,CL ←↩10,DL). (left Round 2) Forjfrom 16 to31 do the following: t ←(AL + g(BL,CL,DL)+ X[zL[j]] +yL[j]), (AL,BL,CL,DL,EL) ←(EL,EL +( t←↩sL[j]),BL,CL ←↩10,DL). (left Round 3) Forjfrom 32 to47 do the following: t ←(AL + h(BL,CL,DL)+ X[zL[j]] +yL[j]), (AL,BL,CL,DL,EL) ←(EL,EL +( t←↩sL[j]),BL,CL ←↩10,DL). (left Round 4) Forjfrom 48 to63 do the following: t ←(AL + k(BL,CL,DL)+ X[zL[j]] +yL[j]), (AL,BL,CL,DL,EL) ←(EL,EL +( t←↩sL[j]),BL,CL ←↩10,DL). (left Round 5) Forjfrom 64 to79 do the following: t ←(AL + l(BL,CL,DL)+ X[zL[j]] +yL[j]), (AL,BL,CL,DL,EL) ←(EL,EL +( t←↩sL[j]),BL,CL ←↩10,DL). (b) Execute in parallel with the above ﬁve rounds an analogous right line with (AR,BR,CR,DR,ER),yR[j],zR[j],sR[j] replacing the corresponding quan- tities with subscriptL; and the order of the round functions reversed so that their order is:l,k,h,g, andf. Start by initializing the right line working variables: (AR,BR,CR,DR,ER) ←(H1,H2,H3,H4,H5). (c) After executing both the left and right lines above, update the chaining values as follows:t←H1,H1 ←H2 + CL + DR,H2 ←H3 + DL + ER,H3 ← H4 + EL + AR,H4 ←H5 + AL + BR,H5 ←t+ BL + CR. 5. Completion.The ﬁnal hash-value is the concatenation:H1\|\|H2\|\|H3\|\|H4\|\|H5 (with ﬁrst and last bytes the low- and high-order bytes ofH1,H5, respectively). §9.4 Unkeyedhash functions (MDCs) 351 Variable Value zL[ 0..15] [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15] zL[16..31] [ 7, 4,13, 1,10, 6,15, 3,12, 0, 9, 5, 2,14,11, 8] zL[32..47] [ 3,10,14, 4, 9,15, 8, 1, 2, 7, 0, 6,13,11, 5,12] zL[48..63] [ 1, 9,11,10, 0, 8,12, 4,13, 3, 7,15,14, 5, 6, 2] zL[64..79] [ 4, 0, 5, 9, 7,12, 2,10,14, 1, 3, 8,11, 6,15,13] zR[ 0..15] [ 5,14, 7, 0, 9, 2,11, 4,13, 6,15, 8, 1,10, 3,12] zR[16..31] [ 6,11, 3, 7, 0,13, 5,10,14,15, 8,12, 4, 9, 1, 2] zR[32..47] [15, 5, 1, 3, 7,14, 6, 9,11, 8,12, 2,10, 0, 4,13] zR[48..63] [ 8, 6, 4, 1, 3,11,15, 0, 5,12, 2,13, 9, 7,10,14] zR[64..79] [12,15,10, 4, 1, 5, 8, 7, 6, 2,13,14, 0, 3, 9,11] sL[ 0..15] [11,14,15,12, 5, 8, 7, 9,11,13,14,15, 6, 7, 9, 8] sL[16..31] [ 7, 6, 8,13,11, 9, 7,15, 7,12,15, 9,11, 7,13,12] sL[32..47] [11,13, 6, 7,14, 9,13,15,14, 8,13, 6, 5,12, 7, 5] sL[48..63] [11,12,14,15,14,15, 9, 8, 9,14, 5, 6, 8, 6, 5,12] sL[64..79] [ 9,15, 5,11, 6, 8,13,12, 5,12,13,14,11, 8, 5, 6] sR[ 0..15] [ 8, 9, 9,11,13,15,15, 5, 7, 7, 8,11,14,14,12, 6] sR[16..31] [ 9,13,15, 7,12, 8, 9,11, 7, 7,12, 7, 6,15,13,11] sR[32..47] [ 9, 7,15,11, 8, 6, 6,14,12,13, 5,14,13,13, 7, 5] sR[48..63] [15, 5, 8,11,14,14, 6,14, 6, 9,12, 9,12, 5,15, 8] sR[64..79] [ 8, 5,12, 9,12, 5,14, 6, 8,13, 6, 5,15,13,11,11] Table 9.9:RIPEMD-160 word-access orders and rotate counts (cf. Algorithm 9.55). 9.4.3 Hash functions based on modular arithmetic The basic idea of hash functions based on modular arithmetic is to construct an iterated hash function using modMarithmetic as the basis of a compression function. Two moti- vating factors are re-use of existing software or hardware (in public-key systems) for mod- ular arithmetic, and scalability to match required security levels. Signiﬁcant disadvantages, however, include speed (e.g., relative to the customized hash functions of§9.4.2), and an embarrassing history of insecure proposals. MASH MASH-1 ( Modular Arithmetic Secure Hash, algorithm 1) is a hash function based on mod- ular arithmetic. It has been proposed for inclusion in a draft ISO/IEC standard. MASH-1 involves use of an RSA-like modulusM, whose bitlength affects the security.Mshould be difﬁcult to factor, and forMof unknown factorization, the security is based in part on the difﬁculty of extracting modular roots (§3.5.2). The bitlength ofMalso determines the blocksize for processing messages, and the size of the hash-result (e.g., a 1025-bit modulus yields a 1024-bit hash-result). As a recent proposal, its security remains open to question (page 381). Techniques for reducing the size of the ﬁnal hash-result have also been pro- posed, but their security is again undetermined as yet. 352 Ch. 9 Hash Functions and Data Integrity 9.56 AlgorithmMASH-1 (version of Nov. 1995) INPUT: dataxof bitlength0 ≤b< 2n/2. OUTPUT: n-bit hash ofx(nis approximately the bitlength of the modulusM). 1. System setup and constant deﬁnitions.Fix an RSA-like modulusM= pqof bitlength m, wherepand qare randomly chosen secret primes such that the factorization of Mis intractable. Deﬁne the bitlengthnof the hash-result to be the largest multiple of 16 less thanm(i.e.,n=1 6n′<m).H0 =0 is deﬁned as an IV , and ann- bit integer constantA= 0xf0...0. “∨” denotes bitwise inclusive-OR; “⊕” denotes bitwise exclusive-OR. 2. Padding, blocking, and MD-strengthening.Pad xwith 0-bits, if necessary, to obtain a string of bitlengtht·n/2 for the smallest possiblet≥1. Divide the padded text into (n/2)-bit blocksx1,...,x t, and append a ﬁnal blockxt+1 containing the (n/2)-bit representation ofb. 3. Expansion.Expand eachxi to ann-bit blockyi by partitioning it into (4-bit) nibbles and inserting four 1-bits preceding each, except foryt+1 wherein the inserted nibble is 1010 (not1111). 4. Compression function processing.For1 ≤i≤t+1, map twon-bit inputs (Hi−1,yi) to onen-bit output as follows:Hi ←((((Hi−1⊕yi) ∨A)2 mod M) ⊣n)⊕Hi−1. Here ⊣ndenotes keeping the rightmostnbits of them-bit result to its left. 5. Completion.The hash is then-bit blockHt+1. MASH-2 is deﬁned as per MASH-1 with the exponente=2 used for squaring in the compression function processing stage (step 4) replaced withe=2 8 +1 . 9.5 Keyed hash functions (MACs) Keyed hash functions whose speciﬁc purpose is message authentication are called message authentication code (MAC) algorithms. Compared to the large number of MDC algorithms, prior to 1995 relatively few MAC algorithms had been proposed, presumably because the original proposals, which were widely adopted in practice, were adequate. Many of these are for historical reasons block-cipher based. Those with relatively short MAC bitlengths (e.g., 32-bits for MAA) or short keys (e.g., 56 bits for MACs based on DES-CBC) may still offer adequate security, depending on the computational resources available to adversaries, and the particular environment of application. Many iterated MACs can be described as iterated hash functions (see Figure 9.2, and equation (9.1) on page 333). In this case, the MAC key is generally part of the output trans- formationg; it may also be an input to the compression function in the ﬁrst iteration, and be involved in the compression functionfat every stage. Fact 9.57 is a general result giving an upper bound on the security of MACs. 9.57 Fact(birthday attack on MACs) Lethbe a MAC algorithm based on an iterated com- pression function, which hasnbits of internal chaining variable, and is deterministic (i.e., them-bit result is fully determined by the message). Then MAC forgery is possible using O(2 n/2) known text-MAC pairs plus a numbervof chosen text-MAC pairs which (depend- ing onh) is between1 and about2n−m. §9.5 Keyed hash functions (MACs) 353 9.5.1 MACs based on block ciphers CBC-based MACs The most commonly used MAC algorithm based on a block cipher makes use of cipher- block-chaining (§7.2.2(ii)). When DES is used as the block cipherE,n=6 4in what fol- lows, and the MAC key is a 56-bit DES key. 9.58 AlgorithmCBC-MAC INPUT: datax; speciﬁcation of block cipherE; secret MAC keykforE. OUTPUT: n-bit MAC onx(nis the blocklength ofE). 1. Padding and blocking.Pad xif necessary (e.g., using Algorithm 9.30). Divide the padded text inton-bit blocks denotedx1,...,x t. 2. CBC processing.LettingEk denote encryption usingEwith keyk, compute the blockHt as follows:H1 ←Ek(x1);Hi ←Ek(Hi−1⊕xi),2 ≤i≤t. (This is standard cipher-block-chaining,IV =0 , discarding ciphertext blocksCi = Hi.) 3. Optional process to increase strength of MAC.Using a second secret keyk′̸=k, optionally compute:H′ t ←E−1 k′(Ht),Ht ←Ek(H′ t). (This amounts to using two- key triple-encryption on the last block; see Remark 9.59.) 4. Completion.The MAC is then-bit blockHt. E-1 Ek x1 Ek x2 xt optional H1 Ek x3 H2 H3 Ht−1 E E H 0 k k Ht k′ Figure 9.6:CBC-based MAC algorithm. For CBC-MAC with n=6 4= m, Fact 9.57 applies withv=1 . 9.59 Remark (CBC-MAC strengthening) The optional process reduces the threat of exhaus- tive key search, and prevents chosen-text existential forgery (Example 9.62), without im- pacting the efﬁciency of the intermediate stages as would using two-key triple-encryption 354 Ch. 9 Hash Functions and Data Integrity throughout. Alternatives to combat such forgery include prepending the input with a length block before the MAC computation; or using keyKto encrypt the lengthmyieldingK′= EK(m), before usingK′as the key to MAC the message. 9.60 Remark (truncated MAC outputs) Exhaustive attack may, depending on the unicity dis- tance of the MAC, be precluded (information-theoretically) by using less thannbits of the ﬁnal output as them-bit MAC. (This must be traded off against an increase in the proba- bility of randomly guessing the MAC:2−m.) Form=3 2 and E= DES, an exhaustive attack reduces the key space to about224 possibilities. However, even form<n , a second text-MAC pair almost certainly determines a unique MAC key. 9.61 Remark (CBC-MAC IV ) While a random IV in CBC encryption serves to prevent a code- book attack on the ﬁrst ciphertext block, this is not a concern in a MAC algorithm. 9.62 Example (existential forgery of CBC-MAC) While CBC-MAC is secure for messages of a ﬁxed numbertof blocks, additional measures (beyond simply adding a trailing length- block) are required if variable length messages are allowed, otherwise (adaptive chosen- text) existential forgery is possible as follows. Assumex i is ann-bit block, and let⊥b denote then-bit binary representation ofb. Let(x1,M1) be a known text-MAC pair, and request the MACM2 for the one-block messagex2 = M1; thenM2 = Ek(Ek(x1)) is also the MAC for the 2-block message(x1\|\|⊥0). As a less trivial example, given two known text-MAC pairs(x1,H1),(x2,H2) for one-block messagesx1,x2, and request- ing the MAC Mon a chosen 2-block third message(x1\|\|z) for a third text-MAC pair ((x1\|\|z),M), thenHi = Ek(xi),M = Ek(H1⊕z), and the MAC for the new 2-block message X = x2\|\|(H1⊕z⊕H2) is known – it isMalso. Moreover, MD-strengthening (Algorithm 9.26) does not address the problem: assume padding by Algorithm 9.29, re- place the third message above by the 3-block message(x1\|\|⊥64\|\|z), note H′ i = Ek(Ek(xi)⊕⊥64),M 3 = Ek(Ek(Ek(Ek(x1)⊕⊥64)⊕z)⊕⊥192), and M3 is also the MAC for the new 3-block messageX=( x2\|\|⊥64\|\|H′ 1⊕H′ 2⊕z). □ 9.63 Example (RIPE-MAC ) RIPE-MAC is a variant of CBC-MAC. Two versions RIPE- MAC1 and RIPE-MAC3, both producing 64-bit MACs, differ in their internal encryption function E being either single DES or two-key triple-DES, respectively, requiring a 56- or 112-bit keyk(cf. Remark 9.59). Differences from Algorithm 9.58 are as follows: the compression function uses a non-invertible chaining best described as CBC with data feed- forward:Hi ← Ek(Hi−1⊕xi)⊕xi; after padding using Algorithm 9.30, a ﬁnal 64-bit length-block (giving bitlength of original input) is appended; the optional process of Al- gorithm 9.58 is mandatory with ﬁnal output block encrypted using keyk ′derived by com- plementing alternating nibbles ofk: fork= k0 ...k63 a 56-bit DES key with parity bits k7k15 ...k63,k′= k⊕0xf0f0f0f0f0f0f0f0. □ 9.5.2 Constructing MACs from MDCs A common suggestion is to construct a MAC algorithm from an MDC algorithm, by simply including a secret keykas part of the MDC input. A concern with this approach is that implicit but unveriﬁed assumptions are often made about the properties that MDCs have; in particular, while most MDCs are designed to provide one-wayness or collision resistance, §9.5 Keyed hash functions (MACs) 355 the requirements of a MAC algorithm differ (Deﬁnition 9.7). Even in the case that a one- way hash function precludes recovery of a secret key used as a partial message input (cf. partial-preimage resistance, page 331), this does not guarantee the infeasibility of producing MACs for new inputs. The following examples suggest that construction of a MAC from a hash function requires careful analysis. 9.64 Example (secret preﬁx method) Consider a messagex= x1x2 ...xt and an iterated MDC hwith compression functionf, with deﬁnition:H0 = IV,Hi = f(Hi−1,xi); h(x)= Ht. (1) Suppose one attempts to usehas a MAC algorithm by prepending a secret keyk, so that the proposed MAC onxisM = h(k\|\|x). Then, extending the messagexby an arbitrary single blocky, one may deduceM′= h(k\|\|x\|\|y) as f(M,y) without knowing the secret keyk(the original MACMserves as chaining variable). This is true even for hash functions whose preprocessing pads inputs with length indicators (e.g., MD5); in this case, the padding/length-blockzfor the original messagexwould appear as part of the extended message,x\|\|z\|\|y, but a forged MAC on the latter may nonetheless be deduced. (2) For similar reasons, it is insecure to use an MDC to construct a MAC algorithm by using the secret MAC keykas IV . Ifkcomprises the entire ﬁrst block, then for efﬁciencyf(IV,k) may be precomputed, illustrating that an adversary need only ﬁnd ak ′(not necessarilyk) such thatf(IV,k)= f(IV,k′); this is equivalent to using a secret IV . □ 9.65 Example (secret sufﬁx method) An alternative proposal is to use a secret key as a sufﬁx, i.e., then-bit MAC onxisM= h(x\|\|k). In this case, a birthday attack applies (§9.7.1). An adversary free to choose the messagex(or a preﬁx thereof) may, in O(2n/2) operations, ﬁnd a pair of messagesx,x′for whichh(x)= h(x′). (This can be done off-line, and does not require knowledge ofk; the assumption here is thatnis the size of both the chaining variable and the ﬁnal output.) Obtaining a MACMon xby legitimate means then allows an adversary to produce a correct text-MAC pair(x′,M) for a new messagex′. Note that this method essentially hashes and then encrypts the hash-value in the ﬁnal iteration; in this weak form of MAC, the MAC-value depends only on the last chaining value, and the key is used in only one step. □ The above examples suggest that a MAC key should be involved at both the start and the end of MAC computations, leading to Example 9.66. 9.66 Example (envelope method with padding) For a keykand MDC h, compute the MAC on a messagexas:h k(x)= h(k\|\|p\|\|x\|\|k). Herepis a string used to padkto the length of one block, to ensure that the internal computation involves at least two iterations. For example, ifhis MD5 andkis 128 bits,pis a 384-bit pad string. □ Due to both a certiﬁcational attack against the MAC construction of Example 9.66 and theoretical support for that of Example 9.67 (see page 382), the latter construction is fa- vored. 9.67 Example (hash-based MAC) For a keykand MDC h, compute the MAC on a message xas HMAC (x)= h(k\|\|p1 \|\|h(k\|\|p2 \|\|x)), wherep1,p2 are distinct strings of sufﬁcient length to padkout to a full block for the compression function. The overall construction is quite efﬁcient despite two calls toh, since the outer execution processes only (e.g., ifhis MD5) a two-block input, independent of the length ofx. □ Additional suggestions for achieving MAC-like functionality by combining MDCs and encryption are discussed in§9.6.5. 356 Ch. 9 Hash Functions and Data Integrity 9.5.3 Customized MACs Two algorithms designed for the speciﬁc purpose of message authentication are discussed in this section: MAA and MD5-MAC. Message Authenticator Algorithm (MAA) The Message Authenticator Algorithm (MAA), dating from 1983, is a customized MAC algorithm for 32-bit machines, involving 32-bit operations throughout. It is speciﬁed as Algorithm 9.68 and illustrated in Figure 9.7. The main loop consists of two parallel inter- dependent streams of computation. Messages are processed in 4-byte blocks using 8 bytes of chaining variable. The execution time (excluding key expansion) is proportional to mes- sage length; as a rough guideline, MAA is twice as slow as MD4. 9.68 AlgorithmMessage Authenticator Algorithm (MAA) INPUT: dataxof bitlength32j,1 ≤j≤106; secret 64-bit MAC keyZ= Z[1]..Z[8]. OUTPUT: 32-bit MAC on x. 1. Message-independent key expansion.Expand keyZto six 32-bit quantitiesX,Y,V, W,S,T(X,Y are initial values;V,W are main loop variables;S,T are appended to the message) as follows. 1.1 First replace any bytes 0x00 or 0xff inZas follows.P←0; forifrom 1 to 8 (P←2P;i fZ[i]= 0x00 or 0xff then (P←P+1 ;Z[i] ←Z[i] OR P)). 1.2 LetJand Kbe the ﬁrst 4 bytes and last 4 bytes ofZ, and compute:4 X←J4 (mod 232 −1)⊕J4 (mod 232 −2) Y←[K5 (mod 232 −1)⊕K5 (mod 232 −2)](1 +P)2 (mod 232 −2) V←J6 (mod 232 −1)⊕J6 (mod 232 −2) W←K7 (mod 232 −1)⊕K7 (mod 232 −2) S←J8 (mod 232 −1)⊕J8 (mod 232 −2) T←K9 (mod 232 −1)⊕K9 (mod 232 −2) 1.3 Process the 3 resulting pairs(X,Y),(V,W),(S,T) to remove any bytes 0x00, 0xff as forZearlier. Deﬁne the AND-OR constants:A= 0x02040801,B= 0x00804021,C= 0xbfef7fdf,D= 0x7dfefbff. 2. Initialization and preprocessing.Initialize the rotating vector:v←V, and the chain- ing variables:H1 ←X,H2 ←Y. Append the key-derived blocksS,Ttox, and letx1 ...xt denote the resulting augmented segment of 32-bit blocks. (The ﬁnal 2 blocks of the segment thus involve key-derived secrets.) 3. Block processing.Process each 32-bit blockxi (forifrom 1 tot) as follows. v←(v←↩1),U ←(v⊕W) t1 ←(H1⊕xi) ×1 (((H2⊕xi)+ U)O RA) ANDC) t2 ←(H2⊕xi) ×2 (((H1⊕xi)+ U)O RB) ANDD) H1 ←t1,H2 ←t2 where ×i denotes special multiplication mod232 −ias noted above (i=1 or2); “+” is addition mod232; and “←↩1” denotes rotation left one bit. (Each combined AND-OR operation on a 32-bit quantity sets 4 bits to 1, and 4 to 0, precluding 0- multipliers.) 4. Completion.The resulting MAC is:H= H 1⊕H2. 4In ISO 8731-2, a well-deﬁned but unconventional deﬁnition of multiplication mod232 − 2 is speciﬁed, pro- ducing 32-bit results which in some cases are232 − 1 or232 − 2; for this reason, specifying e.g.,J6 here may be ambiguous; the standard should be consulted for exact details. §9.5 Keyed hash functions (MACs) 357 xt−2 xt−1x1 ···x2 x3 xt delay ∗(mod 232 −1) ∗(mod 232 −2) V W S H1 H2 B D A C i> 1 i=1 VT Y i=1 i> 1 x i X H1 U key (J,K) xi i←i+1 64 message x key expansion AND AND OR delayi←i+1 OR ++ H2 [from messagex] Figure 9.7:The Message Authenticator Algorithm (MAA). Since the relatively complex key expansion stage is independent of the message, a one- time computation sufﬁces for a ﬁxed key. The mixing of various operations (arithmetic mod 232 −i, fori=0 ,1 and 2; XOR; and nonlinear AND-OR computations) is intended to strengthen the algorithm against arithmetic cryptanalytic attacks. MD5-MAC A more conservative approach (cf. Example 9.66) to building a MAC from an MDC is to arrange that the MAC compression function itself depend onk, implying the secret key be involved in all intervening iterations; this provides additional protection in the case that weaknesses of the underlying hash function become known. Algorithm 9.69 is such a tech- nique, constructed using MD5. It provides performance close to that of MD5 (5-20% slower in software). 358 Ch. 9 Hash Functions and Data Integrity 9.69 AlgorithmMD5-MAC INPUT: bitstringxof arbitrary bitlengthb≥0; keykof bitlength≤128. OUTPUT: 64-bit MAC-value ofx. MD5-MAC is obtained from MD5 (Algorithm 9.51) by the following changes. 1. Constants.The constantsUi and Ti are as deﬁned in Example 9.70. 2. Key expansion. (a) Ifkis shorter than 128 bits, concatenatekto itself a sufﬁcient number of times, and redeﬁnekto be the leftmost 128 bits. (b) Let MD5 denote MD5 with both padding and appended length omitted. Expand kinto three 16-byte subkeysK0,K1, andK2 as follows: forifrom 0 to 2, Ki ← MD5 (k∥Ui ∥k). (c) Partition each ofK0 and K1 into four 32-bit substringsKj[i],0 ≤i≤3. 3. K0 replaces the four 32-bitIV’s of MD5 (i.e.,hi = K0[i]). 4. K1[i] is addedmod 232 to each constanty[j] used in Roundiof MD5. 5. K2 is used to construct the following 512-bit block, which is appended to the padded inputxsubsequent to the regular padding and length block as deﬁned by MD5: K2 ∥K2 ⊕T0 ∥K2 ⊕T1 ∥K2 ⊕T2. 6. The MAC-value is the leftmost 64 bits of the 128-bit output from hashing this padded and extended input string using MD5 with the above modiﬁcations. 9.70 Example (MD5-MAC constants/test vectors) The 16-byte constantsTi and three test vec- tors (x, MD5-MAC( x)) for keyk = 00112233445566778899aabbccddeeffare given below. (TheTi themselves are derived using MD5 on pre-deﬁned constants.) With subscripts inTi taken mod 3, the 96-byte constantsU0,U1,U2 are deﬁned: Ui = Ti ∥Ti+1 ∥Ti+2 ∥Ti ∥Ti+1 ∥Ti+2. T0: 97 ef 45 ac 29 0f 43 cd 45 7e 1b 55 1c 80 11 34 T1: b1 77 ce 96 2e 72 8e 7c 5f 5a ab 0a 36 43 be 18 T2: 9d 21 b4 21 bc 87 b9 4d a2 9d 27 bd c7 5b d7 c3 ("", 1f1ef2375cc0e0844f98e7e811a34da8) ("abc", e8013c11f7209d1328c0caa04fd012a6) ("abcdefghijklmnopqrstuvwxyz", 9172867eb60017884c6fa8cc88ebe7c9) □ 9.5.4 MACs for stream ciphers Providing data origin authentication and data integrity guarantees for stream ciphers is par- ticularly important due to the fact that bit manipulations in additive stream-ciphers may di- rectly result in predictable modiﬁcations of the underlying plaintext (e.g., Example 9.83). While iterated hash functions process message data a block at a time (§9.3.1), MACs de- signed for use with stream ciphers process messages either one bit or one symbol (block) at a time, and those which may be implemented using linear feedback shift registers (LFSRs) are desirable for reasons of efﬁciency. One such MAC technique, Algorithm 9.72 below, is based on cyclic redundancy codes (cf. Example 9.80). In this case, the polynomial division may be implemented using an LFSR. The following deﬁnition is of use in what follows. §9.6 Data integrity and message authentication 359 9.71 DeﬁnitionA (b,m) hash-familyHis a collection of hash functions mappingb-bit mes- sages tom-bit hash-values. A(b,m) hash-family isε-balancedif for all messagesB̸=0 and allm-bit hash-valuesc,probh(h(B)= c)) ≤ε, where the probability is over all ran- domly selected functionsh∈H. 9.72 AlgorithmCRC-based MAC INPUT: b-bit messageB; shared key (see below) between MAC source and veriﬁer. OUTPUT: m-bit MAC-value onB(e.g.,m=6 4). 1. Notation. AssociateB= Bb−1 ...B1B0 with the polynomialB(x)= ∑b−1 i=0 Bixi. 2. Selection of MAC key. (a) Select a random binary irreducible polynomialp(x) of degreem. (This repre- sents randomly drawing a functionhfrom a(b,m) hash-family.) (b) Select a randomm-bit one-time keyk(to be used as a one-time pad). The secret MAC key consists ofp(x) and k, both of which must be shared a priori between the MAC originator and veriﬁer. 3. Compute h(B)= coef(B(x) ·xm mod p(x)), them-bit string of coefﬁcients from the degreem−1 remainder polynomial after dividingB(x) ·xm by p(x). 4. Them-bit MAC-value forBis:h(B)⊕k. 9.73 Fact(security of CRC-based MAC) For any valuesbandm> 1, the hash-family resulting from Algorithm 9.72 isε-balanced forε=( b+ m)/(2m−1), and the probability of MAC forgery is at mostε. 9.74 Remark (polynomial reuse) The hash functionhin Algorithm 9.72 is determined by the irreducible polynomialp(x). In practice,p(x) may be re-used for different messages (e.g., within a session), but for each message a new random keykshould be used. 9.6 Data integrity and message authentication This section considers the use of hash functions for data integrity and message authenti- cation. Following preliminary subsections, respectively, providing background deﬁnitions and distinguishing non-malicious from malicious threats to data integrity, three subsequent subsections consider three basic approaches to providing data integrity using hash func- tions, as summarized in Figure 9.8. 9.6.1 Background and deﬁnitions This subsection discusses data integrity, data origin authentication (message authentica- tion), and transaction authentication. Assurances are typically required both that data actually came from its reputed source (data origin authentication), and that its state is unaltered (data integrity). These issues can- not be separated – data which has been altered effectively has a new source; and if a source cannot be determined, then the question of alteration cannot be settled (without reference to a source). Integrity mechanisms thus implicitly provide data origin authentication, and vice versa. 360 Ch. 9 Hash Functions and Data Integrity encryption algorithm algorithm MDC unsecured channel authentic channel (a) MAC only (c) MDC & authentic encrypted (b) MDC & encipherment channel secret key message algorithm MAC message MAC unsecured channel secret key message message algorithm MDC unsecured channelmessage MDC message MDC MDC Figure 9.8:Three methods for providing data integrity using hash functions. The second method provides encipherment simultaneously. §9.6 Data integrity and message authentication 361 (i) Data integrity 9.75 DeﬁnitionData integrityis the property whereby data has not been altered in an unautho- rized manner since the time it was created, transmitted, or stored by an authorized source. Veriﬁcation of data integrity requires that only a subset of all candidate data items sat- isﬁes particular criteria distinguishing the acceptable from the unacceptable. Criteria al- lowing recognizability of data integrity include appropriate redundancy or expectation with respect to format. Cryptographic techniques for data integrity rely on either secret informa- tion or authentic channels (§9.6.4). The speciﬁc focus of data integrity is on the bitwise composition of data (cf. transac- tion authentication below). Operations which invalidate integrity include: insertion of bits, including entirely new data items from fraudulent sources; deletion of bits (short of deleting entire data items); re-ordering of bits or groups of bits; inversion or substitution of bits; and any combination of these, such as message splicing (re-use of proper substrings to construct new or altered data items). Data integrity includes the notion that data items are complete. For items split into multiple blocks, the above alterations apply analogously with blocks envisioned as substrings of a contiguous data string. (ii) Data origin authentication (message authentication) 9.76 DeﬁnitionData origin authenticationis a type of authentication whereby a party is cor- roborated as the (original) source of speciﬁed data created at some (typically unspeciﬁed) time in the past. By deﬁnition, data origin authentication includes data integrity. 9.77 DeﬁnitionMessage authenticationis a term used analogously with data origin authenti- cation. It provides data origin authentication with respect to the original message source (and data integrity, but no uniqueness and timeliness guarantees). Methods for providing data origin authentication include the following: 1. message authentication codes (MACs) 2. digital signature schemes 3. appending (prior to encryption) a secret authenticator value to encrypted text. 5 Data origin authentication mechanisms based on shared secret keys (e.g., MACs) do not allow a distinction to be made between the parties sharing the key, and thus (as opposed to digital signatures) do not provide non-repudiation of data origin – either party can equally originate a message using the shared key. If resolution of subsequent disputes is a potential requirement, either an on-line trusted third party in a notary role, or asymmetric techniques (see Chapter 11) may be used. While MACs and digital signatures may be used to establish that data was generated by a speciﬁed party at some time in the past, they provide no inherent uniqueness or timeliness guarantees. These techniques alone thus cannot detect message re-use or replay, which is necessary in environments where messages may have renewed effect on second or subse- quent use. Such message authentication techniques may, however, be augmented to provide these guarantees, as next discussed. 5Such asealed authenticator(cf. a MAC, sometimes called anappended authenticator) is used along with an encryption method which provides error extension. While this resembles the technique of using encryption and an MDC (§9.6.5), whereas the MDC is a (known) function of the plaintext, a sealed authenticator is itself secret. 362 Ch. 9 Hash Functions and Data Integrity (iii) Transaction authentication 9.78 DeﬁnitionTransaction authenticationdenotes message authentication augmented to ad- ditionally provide uniqueness and timeliness guarantees on data (thus preventing unde- tectable message replay). The uniqueness and timeliness guarantees of Deﬁnition 9.78 are typically provided by appropriate use of time-variant parameters (TVPs). These include random numbers in challenge-response protocols, sequence numbers, and timestamps as discussed in§10.3.1. This may be viewed as a combination of message authentication and entity authentication (Deﬁnition 10.1). Loosely speaking, message authentication + TVP = transaction authentication. As a simple example, sequence numbers included within the data of messages authen- ticated by a MAC or digital signature algorithm allow replay detection (see Remark 9.79), and thus provide transaction authentication. As a second example, for exchanges between two parties involving two or more mes- sages, transaction authentication on each of the second and subsequent messages may be provided by including in the message data covered by a MAC a random number sent by the other party in the previous message. This chaining of messages through random numbers prevents message replay, since any MAC values in replayed messages would be incorrect (due to disagreement between the random number in the replayed message, and the most recent random number of the veriﬁer). Table 9.10 summarizes the properties of these and other types of authentication. Au- thentication in the broadest sense encompasses not only data integrity and data origin au- thentication, but also protection from all active attacks including fraudulent representation and message replay. In contrast, encryption provides protection only from passive attacks. →Property identiﬁcation data timeliness or deﬁned ↓Type of authentication of source integrity uniqueness in message authentication yes yes — §9.6.1 transaction authentication yes yes yes §9.6.1 entity authentication yes — yes §10.1.1 key authentication yes yes desirable §12.2.1 Table 9.10:Properties of various types of authentication. 9.79 Remark (sequence numbers and authentication) Sequence numbers may provide unique- ness, but not (real-time) timeliness, and thus are more appropriate to detect message replay than for entity authentication. Sequence numbers may also be used to detect the deletion of entire messages; they thus allow data integrity to be checked over an ongoing sequence of messages, in addition to individual messages. 9.6.2 Non-malicious vs. malicious threats to data integrity The techniques required to provide data integrity on noisy channels differ substantially from those required on channels subject to manipulation by adversaries. Checksums provide protection against accidental or non-malicious errors on channels which are subject to transmission errors. The protection is non-cryptographic, in the sense §9.6 Data integrity and message authentication 363 that neither secret keys nor secured channels are used. Checksums generalize the idea of a parity bit by appending a (small) constant amount of message-speciﬁc redundancy. Both the data and the checksum are transmitted to a receiver, at which point the same redundancy computation is carried out on the received data and compared to the received checksum. Checksums can be used either for error detection or in association with higher-level error- recovery strategies (e.g., protocols involving acknowledgements and retransmission upon failure). Trivial examples include an arithmetic checksum (compute the running 32-bit sum of all 32-bit data words, discarding high-order carries), and a simple XOR (XOR all 32- bit words in a data string).Error-correcting codesgo one step further than error-detecting codes, offering the capability to actually correct a limited number of errors without retrans- mission; this is sometimes calledforward error correction. 9.80 Example (CRCs )Cyclic redundancy codesor CRCs are commonly used checksums. A k-bit CRC algorithm maps arbitrary length inputs intok-bit imprints, and provides signif- icantly better error-detection capability thank-bit arithmetic checksums. The algorithm is based on a carefully chosen(k+1 )-bit vector represented as a binary polynomial; for k=1 6, a commonly used polynomial (CRC-16) isg(x)=1 + x 2 +x15 +x16.A t-bit data input is represented as a binary polynomiald(x) of degreet−1, and the CRC-value cor- responding tod(x) is the 16-bit string represented by the polynomial remainderc(x) when x16 ·d(x) is divided byg(x);6 polynomial remaindering is analogous to computing integer remainders by long division. For all messagesd(x) witht< 32 768, CRC-16 can detect all errors that consist of only a single bit, two bits, three bits, or any odd number of bits, all burst errors of bitlength 16 or less, 99.997% (1 −2−15) of 17-bit burst errors, and 99.998% (1 −2−16) of all bursts 18 bits or longer. (Aburst errorof bitlengthbis any bitstring of ex- actlybbits beginning and ending with a1.) Analogous to the integer case, other data strings d′(x) yielding the same remainder asd(x) can be trivially found by adding multiples of the divisorg(x) tod(x), or inserting extra blocks representing a multiple ofg(x). CRCs thus do not provide one-wayness as required for MDCs; in fact, CRCs are a class of linear (error correcting) codes, with one-wayness comparable to an XOR-sum. □ While of use for detection of random errors,k-bit checksums are not of cryptographic use, because typically a data string checksumming to any target value can be easily created. One method is to simply insert or append to any data string of choice ak-bit correcting- blockcwhich has the effect of correcting the overall checksum to the desired value. For example, for the trivial XOR checksum, if the target checksum isc ′, insert as blockcthe XOR of c′and the XOR of all other blocks. In contrast to checksums, data integrity mechanisms based on (cryptographic) hash functions are speciﬁcally designed to preclude undetectable intentional modiﬁcation. The hash-values resulting are sometimes calledintegrity check values (ICV),o rcryptographic check valuesin the case of keyed hash functions. Semantically, it should not be possible for an adversary to take advantage of the willingness of users to associate a given hash output with a single, speciﬁc input, despite the fact that each such output typically corresponds to a large set of inputs. Hash functions should exhibit no predictable relationships or correla- tions between inputs and outputs, as these may allow adversaries to orchestrate unintended associations. 6A modiﬁcation is typically used in practice (e.g., complementingc(x)) to address the combination of an input d(x)=0 and a “stuck-at-zero” communications fault yielding a successful CRC check. 364 Ch. 9 Hash Functions and Data Integrity 9.6.3 Data integrity using a MAC alone Message Authentication Codes (MACs) as discussed earlier are designed speciﬁcally for applications where data integrity (but not necessarily privacy) is required. The originator of a messagexcomputes a MAC hk(x) over the message using a secret MAC keykshared with the intended recipient, and sends both (effectivelyx\|\|hk(x)). The recipient deter- mines by some means (e.g., a plaintext identiﬁer ﬁeld) the claimed source identity, sepa- rates the received MAC from the received data, independently computes a MAC over this data using the shared MAC key, and compares the computed MAC to the received MAC. The recipient interprets the agreement of these values to mean the data is authentic and has integrity – that is, it originated from the other party which knows the shared key, and has not been altered in transit. This corresponds to Figure 9.8(a). 9.6.4 Data integrity using an MDC and an authentic channel The use of a secret key is not essential in order to provide data integrity. It may be eliminated by hashing a message and protecting the authenticity of the hash via an authentic (but not necessarily private) channel. The originator computes a hash-code using an MDC over the message data, transmits the data to a recipient over an unsecured channel, and transmits the hash-code over an independent channel known to provide data origin authentication. Such authentic channels may include telephone (authenticity through voice recognition), any data medium (e.g., ﬂoppy disk, piece of paper) stored in a trusted place (e.g., locked safe), or publication over any difﬁcult-to-forgepublic medium (e.g., daily newspaper). The recipient independently hashes the received data, and compares the hash-code to that received. If these values agree, the recipient accepts the data as having integrity. This corresponds to Figure 9.8(c). Example applications include virus protection of software, and distribution of software or public keys via untrusted networks. For virus checking of computer source or object code, this technique is preferable to one resulting in encrypted text. A common example of combining an MDC with an authentic channel to provide data integrity is digital signa- ture schemes such as RSA, which typically involve the use of MDCs, with the asymmetric signature providing the authentic channel. 9.6.5 Data integrity combined with encryption Whereas digital signatures provide assurances regarding both integrity and authentication, in general, encryption alone provides neither. This issue is ﬁrst examined, and then the question of how hash functions may be employed in conjunction with encryption to pro- vide data integrity. (i) Encryption alone does not guarantee data integrity A common misconception is that encryption provides data origin authentication and data integrity, under the argument that if a message is decrypted with a key shared only with partyA, and if the decrypted message is meaningful, then it must have originated fromA. Here “meaningful” means the message contains sufﬁcient redundancy or meets some other a priori expectation. While the intuition is that an attacker must know the secret key in order to manipulate messages, this is not always true. In some cases he may be able to choose the plaintext message, while in other cases he may be able to effectively manipulate §9.6 Data integrity and message authentication 365 plaintext despite not being able to control its speciﬁc content. The extent to which encrypted messages can be manipulated undetectably depends on many factors, as illustrated by the following examples. 9.81 Example (re-ordering ECB blocks) The ciphertext blocks of any block cipher used only in ECB mode are subject to re-ordering. □ 9.82 Example (encryption of random data) If the plaintext corresponding to a given cipher- text contains no redundancy (e.g., a random key), then all attempted decryptions thereof are meaningful, and data integrity cannot be veriﬁed. Thus, some form of redundancy is always required to allow veriﬁcation of integrity; moreover, to facilitate veriﬁcation in practice, ex- plicit redundancy veriﬁable by automated means is required. □ 9.83 Example (bit manipulations in additive stream ciphers) Despite the fact that the one-time pad offers unconditional secrecy, an attacker can change any single bit of plaintext by mod- ifying the corresponding bit of ciphertext. For known-plaintext attacks, this allows an at- tacker to substitute selected segments of plaintext by plaintext of his own choosing. An example target bit is the high-order bit in a numeric ﬁeld known to represent a dollar value. Similar comments apply to any additive stream cipher, including the OFB mode of any block cipher. □ 9.84 Example (bit manipulation in DES ciphertext blocks) Several standard modes of opera- tion for any block cipher are subject to selective bit manipulation. Modifying the last cipher- text block in a CFB chain is undetectable. A ciphertext block in CFB mode which yields random noise upon decryption is an indication of possible selective bit-manipulation of the preceding ciphertext block. A ciphertext block in CBC mode which yields random noise upon decryption is an indication of possible selective bit-manipulation of the following ci- phertext block. For further discussion regarding error extension in standard modes of op- eration, see§7.2.2. □ (ii) Data integrity using encryption and an MDC If both conﬁdentiality and integrity are required, then the following data integrity technique employing anm-bit MDC hmay be used. The originator of a messagexcomputes a hash valueH = h(x) over the message, appends it to the data, and encrypts the augmented message using a symmetric encryption algorithmEwith shared keyk, producing ciphertext C= E k(x\|\|h(x)) (9.2) (Note that this differs subtlely from enciphering the message and the hash separately as (Ek(x),Ek(h(x))), which e.g. using CBC requires two IVs.) This is transmitted to a recip- ient, who determines (e.g., by a plaintext identiﬁer ﬁeld) which key to use for decryption, and separates the recovered datax′from the recovered hashH′. The recipient then indepen- dently computes the hashh(x′) of the received datax′, and compares this to the recovered hashH′. If these agree, the recovered data is accepted as both being authentic and having integrity. This corresponds to Figure 9.8(b). The intention is that the encryption protects the appended hash, and that it be infeasi- ble for an attacker without the encryption key to alter the message without disrupting the correspondence between the decrypted plaintext and the recovered MDC. The properties required of the MDC here may be notably weaker, in general, than for an MDC used in con- junction with, say, digital signatures. Here the requirement, effectively a joint condition on the MDC and encryption algorithm, is that it not be feasible for an adversary to manipulate 366 Ch. 9 Hash Functions and Data Integrity or create new ciphertext blocks so as to produce a new ciphertextC′which upon decryp- tion will yield plaintext blocks having the same MDC as that recovered, with probability signiﬁcantly greater than 1 in2m. 9.85 Remark (separation of integrity and privacy) While this approach appears to separate pri- vacy and data integrity from a functional viewpoint, the two are not independent with re- spect to security. The security of the integrity mechanism is, at most, that of the encryption algorithm regardless of the strength of the MDC (consider exhaustive search of the encryp- tion key). Thought should, therefore, be given to the relative strengths of the components. 9.86 Remark (vulnerability to known-plaintext attack) In environments where known-plain- text attacks are possible, the technique of equation (9.2) should not be used in conjunction with additive stream ciphers unless additional integrity techniques are used. In this sce- nario, an attacker can recover the key stream, then make plaintext changes, recompute a new MDC, and re-encrypt the modiﬁed message. Note this attack compromises the man- ner in which the MDC is used, rather than the MDC or encryption algorithm directly. If conﬁdentiality is not essential other than to support the requirement of integrity, an apparent option is to encrypt only either the messagexor the MDCh(x). Neither approach is common, for reasons including Remark 9.85, and the general undesirability to utilize en- cryption primitives in systems requiring only integrity or authentication services. The fol- lowing further comments apply: 1. encrypting the hash-code only:(x,E k(h(x))) Applying the key to the hash-value only (cf. Example 9.65) results in a property (typi- cal for public-key signatures but) atypical for MACs: pairs of inputsx,x′with collid- ing outputs (MAC-values here) can be veriﬁably pre-determined without knowledge ofk. Thushmust be collision-resistant. Other issues include: pairs of inputs having the same MAC-value under one key also do under other keys; if the blocklength of the cipherE k is less than the bitlengthmof the hash-value, splitting the latter across encryption blocks may weaken security;kmust be reserved exclusively for this in- tegrity function (otherwise chosen-text attacks on encryption allow selective MAC forgery); andE k must not be an additive stream cipher (see Remark 9.86). 2. encrypting the plaintext only:(Ek(x),h(x)) This offers little computational savings over encrypting both message and hash (ex- cept for very short messages) and, as above,h(x) must be collision-resistant and thus twice the typical MAC bitlength. Correct guesses of the plaintextxmay be conﬁrmed (candidate valuesx ′forxcan be checked by comparingh(x′) toh(x)). (iii) Data integrity using encryption and a MAC It is sometimes suggested to use a MAC rather than the MDC in the mechanism of equa- tion (9.2) on page 365. In this case, a MAC algorithmhk′replaces the MDCh, and rather thanC= Ek(x\|\|h(x)), the message sent is C′= Ek(x\|\|hk′(x)) (9.3) The use of a MAC here offers the advantage (over an MDC) that should the encryption al- gorithm be defeated, the MAC still provides integrity. A drawback is the requirement of managing both an encryption key and a MAC key. Care must be exercised to ensure that dependencies between the MAC and encryption algorithms do not lead to security weak- nesses, and as a general recommendation these algorithms should be independent (see Ex- ample 9.88). §9.6 Data integrity and message authentication 367 9.87 Remark (precluding exhaustive MAC search) Encryption of the MAC-value in equation (9.3) precludes an exhaustive key search attack on the MAC key. Two alternatives here include encrypting the plaintext ﬁrst and then computing a MAC over the ciphertext, and encrypting the message and MAC separately. These are discussed in turn. 1. computing a MAC over the ciphertext:(Ek(x),hk′(Ek(x))). This allows message authentication without knowledge of the plaintextx(or cipher- text key). However, as the message authentication is on the ciphertext rather than the plaintext directly, there are no guarantees that the party creating the MAC knew the plaintextx. The recipient, therefore, must be careful about conclusions drawn – for example, ifE k is public-key encryption, the originator ofxmay be independent of the party sharing the keyk′with the recipient. 2. separate encryption and MAC:(Ek(x),hk′(x)). This alternative requires that neither the encryption nor the MAC algorithm compro- mises the objectives of the other. In particular, in this case an additional requirement on the algorithm is that the MAC onxmust not compromise the conﬁdentiality of x(cf. Deﬁnition 9.7). Keys(k,k′) should also be independent here, e.g., to pre- clude exhaustive search on the weaker algorithm compromising the other (cf. Ex- ample 9.88). Ifkand k ′are not independent, exhaustive key search is theoretically possible even without known plaintext. (iv) Data integrity using encryption – examples 9.88 Example (improper combination of CBC-MAC and CBC encryption) Consider using the data integrity mechanism of equation (9.3) withEk being CBC-encryption with keykand initialization vectorIV,hk′(x) being CBC-MAC with k′and IV′, andk= k′,IV = IV′. The datax= x1x2 ...xt can then be processed in a single CBC pass, since the CBC-MAC is equal to the last ciphertext blockct = Ek(ct−1⊕xt), and the last data block isxt+1 = ct, yielding ﬁnal ciphertext blockct+1 = Ek(ct⊕xt+1)= Ek(0). The encrypted MAC is thus independent of both plaintext and ciphertext, rendering the integrity mechanism completely insecure. Care should thus be taken in combining a MAC with an encryption scheme. In general, it is recommended that distinct (and ideally, independent) keys be used. In some cases, one key may be derived from the other by a simple technique; a common sugges- tion for DES keys is complementation of every other nibble. However, arguments favoring independent keys include the danger of encryption algorithm weaknesses compromising authentication (or vice-versa), and differences between authentication and encryption keys with respect to key management life cycle. See also Remark 13.32. □ An efﬁciency drawback in using distinct keys for secrecy and integrity is the cost of two separate passes over the data. Example 9.89 illustrates a proposed data integrity mechanism (which appeared in a preliminary draft of U.S. Federal Standard 1026) which attempts this by using an essentially zero-cost linear checksum; it is, however, insecure. 9.89 Example (CBC encryption with XOR checksum – CBCC) Consider using the data integ- rity mechanism of equation (9.2) withE k being CBC-encryption with keyk,x= x1x2 ... xt a message oftblocks, and as MDC function the simple XOR of all plaintext blocks, h(x) = ⨁i=t i=1 xi. The quantityM = h(x) which serves as MDC then becomes plain- text blockxt+1. The resulting ciphertext blocks using CBC encryption withc0 = IV are ci = Ek(xi⊕ci−1),1 ≤i≤t+1 . In the absence of manipulation, the recovered plain- text isxi = ci−1⊕Dk(ci). To see that this scheme is insecure as an integrity mechanism, letc′ i denote the actual ciphertext blocks received by a recipient, resulting from possibly 368 Ch. 9 Hash Functions and Data Integrity manipulated blocksci, and letx′ i denote the plaintext recovered by the recipient by CBC decryption with the proper IV . The MDC computed over the recovered plaintext blocks is then M′= h(x′)= i=t⨁ i=1 x′ i = i=t⨁ i=1 (c′ i−1 ⊕Dk(c′ i )) =IV⊕( i=t−1⨁ i=1 c′ i )⊕( i=t⨁ i=1 Dk(c′ i )) M′is compared for equality withx′ t+1(= c′ t ⊕Dk(c′ t+1 )) as a check for data integrity, or equivalently, thatS= M′⊕x′ t+1 =0 . By construction,S=0 if there is no manipula- tion (i.e., ifc′ i = ci, which impliesx′ i = xi). Moreover, the sumSis invariant under any permutation of the valuesc′ i,1 ≤i≤t(sinceDk(ct+1) appears as a term inS, butct+1 does not,ct+1 must be excluded from the permutable set). Thus, any of the ﬁrsttciphertext blocks can be permuted without affecting the successful veriﬁcation of the MDC. Further- more, insertion into the ciphertext stream of any random blockc∗ j twice, or any set of such pairs, will cancel itself out in the sumS, and thus also cannot be detected. □ 9.90 Example (CBC encryption with mod2n −1 checksum) Consider as an alternative to Ex- ample 9.89 the simple MDC functionh(x)= ∑t i=1 xi, the sum of plaintext blocks asn-bit integers with wrap-around carry (add overﬂow bits back into units bit), i.e., the sum modulo 2n −1; considern=6 4for ciphers of blocklength 64. The sumSfrom Example 9.89 in this case involves both XOR and addition modulo2n −1; both permutations of ciphertext blocks and insertions of pairs of identical random blocks are now detected. (This technique should not, however, be used in environments subject to chosen-plaintext attack.)□ 9.91 Example (PCBC encryption with mod2 n checksum) A non-standard, non-self-synchron- izing mode of DES known asplaintext-ciphertext block chaining(PCBC) is deﬁned as fol- lows, fori≥0 and plaintextx= x1x2 ...xt: ci+1 = Ek(xi+1⊕Gi) where G0 = IV, Gi = g(xi,ci) fori ≥ 1, andga simple function such asg(xi,ci)= ( xi + ci) mod 264. A one-pass technique providing both encryption and integrity, which exploits the error- propagation property of this mode, is as follows. Append an additional plaintext block to provide redundancy, e.g.,x t+1 = IV (alternatively: a ﬁxed constant orx1). Encrypt all blocks of the augmented plaintext using PCBC encryption as deﬁned above. The quantity ct+1 =Ek(xt+1⊕g(xt,ct)) serves as MAC. Upon decipherment ofct+1, the receiver ac- cepts the message as having integrity if the expected redundancy is evident in the recovered blockx t+1. (To avoid a known-plaintext attack, the functiongin PCBC should not be a simple XOR for this integrity application.) □ 9.7 Advanced attacks on hash functions A deeper understanding of hash function security can be obtained through consideration of various general attack strategies. The resistance of a particular hash function to known gen- eral attacks provides a (partial) measure of security. A selection of prominent attack strate- gies is presented in this section, with the intention of providing an introduction sufﬁcient to establish that designing (good) cryptographic hash functions is not an easily mastered art. Many other attack methods and variations exist; some are general methods, while others rely on peculiar properties of the internal workings of speciﬁc hash functions. §9.7 Advanced attacks on hash functions 369 9.7.1 Birthday attacks Algorithm-independent attacksare those which can be applied to any hash function, treat- ing it as a black-box whose only signiﬁcant characteristics are the output bitlengthn(and MAC key bitlength for MACs), and the running time for one hash operation. It is typi- cally assumed the hash output approximates a uniform random variable. Attacks falling under this category include those based on hash-result bitsize (page 336); exhaustive MAC key search (page 336); and birthday attacks on hash functions (including memoryless vari- ations) as discussed below. (i) Yuval’s birthday attack on hash functions Yuval’s birthday attack was one of the ﬁrst (and perhaps the most well-known) of many cryptographic applications of the birthday paradox arising from the classical occupancy distribution (§2.1.5): when drawing elements randomly, with replacement, from a set of Nelements, with high probability a repeated element will be encountered after O( √ N) selections. Such attacks are among those calledsquare-root attacks. The relevance to hash functions is that it is easier to ﬁnd collisions for a one-way hash function than to ﬁnd pre-images or second preimages of speciﬁc hash-values. As a result, signature schemes which employ one-way hash functions may be vulnerable to Yuval’s at- tack outlined below. The attack is applicable to all unkeyed hash functions (cf. Fact 9.33), with running timeO(2 m/2) varying with the bitlengthmof the hash-value. 9.92 AlgorithmYuval’s birthday attack INPUT: legitimate messagex1; fraudulent messagex2;m-bit one-way hash functionh. OUTPUT: x′ 1,x′ 2 resulting from minor modiﬁcations ofx1,x2 withh(x′ 1 ) = h(x′ 2 ) (thus a signature onx′ 1 serves as a valid signature onx′ 2 ). 1. Generatet=2 m/2 minor modiﬁcationsx′ 1 ofx1. 2. Hash each such modiﬁed message, and store the hash-values (grouped with corre- sponding message) such that they can be subsequently searched on hash-value. (This can done in O(t) total time using conventional hashing.) 3. Generate minor modiﬁcationsx′ 2 ofx2, computingh(x′ 2 ) for each and checking for matches with anyx′ 1 above; continue until a match is found. (Each table lookup will require constant time; a match can be expected after abouttcandidatesx′ 2.) 9.93 Remark (application of birthday attack) The idea of this attack can be used by a dishon- est signer who provides to an unsuspecting party his signature onx′ 1 and later repudiates signing that message, claiming instead that the message signed wasx′ 2 ; or by a dishonest veriﬁer, who is able to convince an unsuspecting party to sign a prepared messagex′ 1, and later claim that party’s signature onx′ 2. This remark generalizes to other schemes in which the hash of a message is taken to represent the message itself. Regarding practicality, the collisions produced by the birthday attack are “real” (vs. pseudo-collisions or compression function collisions), and moreover of direct practical con- sequence when messages are constructed to be meaningful. The latter may often be done as follows: alter inputs via individual minor modiﬁcations which create semantically equiva- lent messages (e.g., substituting tab characters in text ﬁles for spaces, unprintable characters for each other, etc.). For 128-bit hash functions, 64 such potential modiﬁcation points are 370 Ch. 9 Hash Functions and Data Integrity required to allow264 variations. The attack then requires O(264) time (feasible with ex- treme parallelization); and while it requires space for O(264) messages (which is impracti- cal), the memory requirement can be addressed as discussed below. (ii) Memoryless variation of birthday attack To remove the memory requirement of Algorithm 9.92, a deterministic mapping may be used which approximates a random walk through the hash-value space. By the birthday paradox, in a random walk through a space of2m points, one expects to encounter some point a second time (i.e., obtain a collision) afterO(2m/2) steps, after which the walk will repeat its previous path (and begin to cycle). General memoryless cycle-ﬁnding techniques may then be used to ﬁnd this collision. (Herememorylessmeans requiring negligible mem- ory, rather than in the stochastic sense.) These include Floyd’s cycle-ﬁnding algorithm (§3.2.2) and improvements to it. Following Algorithm 9.92, letgbe a function such thatg(x 1,H)= x′ 1 is a minor modiﬁcation, determined by the hash-valueH, of messagex1 (each bit ofHmight deﬁne whether or not to modifyx1 at a pre-determined modiﬁcation point). Ifx1 is ﬁxed, then gessentially maps a hash-result to a message and it is convenient to writegx1(H)= x′ 1 . Moreover, letgbe injective so that distinct hashesHresult in distinctx′ 1. Then, with ﬁxed messagesx1,x2, and using some easily distinguishable property (e.g., parity) which splits the space of hash-values into two roughly equal-sized subsets, deﬁne a functionrmapping hash-results to hash-results by: r(H)= {h(gx1 (H)) ifHis even h(gx2 (H)) ifHis odd (9.4) The memoryless collision search technique (see above) is then used to ﬁnd two inputs tor which map to the same output (i.e., collide). Ifhbehaves statistically as a random mapping then, with probability0.5, the parity will differ inHand H′for the colliding inputs, in which case without loss of generalityh(gx1 (H)) = h(gx2 (H′)). This yields a colliding pair of variationsx′ 1 = gx1 (H), x′ 2 = gx2 (H′) of distinct messagesx1,x2, respectively, such thath(x′ 1 )= h(x′ 2 ), as per the output of Algorithm 9.92. (iii) Illustrative application to MD5 Actual application of the above generic attack to a speciﬁc hash function raises additional technicalities. To illustrate how these may be addressed, such application is now examined, with assumptions and choices made for exposition only. Lethbe an iterated hash function processing messages in 512-bit blocks and producing 128-bit hashes (e.g., MD5, RIPEMD- 128). To minimize computational expense, restrictr(effectivelygand h) in equation (9.4) to single 512-bit blocks ofxi, such that each iteration ofrinvolves only the compression functionfon inputs one message block and the current chaining variable. Let the legitimate message inputx1 consist ofs512-bit blocks (s≥1, prior to MD- strengthening). Create a fraudulent messagex2 of equal bitlength. Allowx2 to differ from x1 up to and including thejth block, for any ﬁxedj≤s−1. Use the(j+1 )st block ofxi, denotedBi (i=1 ,2), as a matching/replacement block, to be replaced by the 512-bit blocks resulting from the collision search. Set all blocks inx2 subsequent toBi identically equal to those inx1;x′ i will then differ fromxi only in the single block(j+1 ). For maximum freedom in the construction ofx2, choosej= s−1. Letc1,c2 be the respective 128-bit intermediate results (chaining variables) after the iterated hash operates on the ﬁrstjblocks ofx1,x2. Compression functionfmaps (128 + 512 =)640-bit inputs to128-bit outputs. Since the chaining variables depend onxi,gxi (= g) may be deﬁned independent ofxi here (cf. equation (9.4)); assume both entire blocksBi may be replaced without practical §9.7 Advanced attacks on hash functions 371 implication. Letg(H)= Bdenote an injective mapping from the space of 128-bit hash- values to the space of 512-bit potential replacement blocks, deﬁned as follows: map each two-bit segment ofHto one of four 8-bit values in the replacement blockB. (A practical motivation for this is that ifxi is an ASCII message to be printed, and the four 8-bit values are selected to represent non-printable characters, then upon printing, the resulting blocks Bare all indistinguishable, leaving no evidence of adversarial manipulation.) The collision-ﬁnding functionrfor this speciﬁc example (corresponding to the generic equation (9.4)) is then: r(H)= { f(c1,g(H)) ifHis even f(c2,g(H)) ifHis odd Collisions for MD5 (and similar hash functions) can thus be found inO(264) operations and without signiﬁcant storage requirements. 9.7.2 Pseudo-collisions and compression function attacks The exhaustive or brute force methods discussed in§9.3.4, producing preimages, 2nd-pre- images, and collisions for hash functions, are always theoretically possible. They are not considered true “attacks” unless the number of operations required is signiﬁcantly less than both the strength conjectured by the hash function designer and that of hash functions of similar parameters with ideal strength. An attack requiring such a reduced number of oper- ations is informally said tobreakthe hash function, whether or not this computational effort is feasible in practice. Any attack method which demonstrates that conjectured properties do not hold must be taken seriously; when this occurs, one must admit the possibility of additional weaknesses. In addition to considering the complexity of ﬁnding (ordinary) preimages and colli- sions, it is common to examine the feasibility of attacks on slightly modiﬁed versions of the hash function in question, for reasons explained below. The most common case is ex- amination of the difﬁculty of ﬁnding preimages or collisions if one allows free choice of IVs. Attacks on hash functions with unconstrained IVs dictate upper bounds on the security of the actual algorithms. Vulnerabilities found, while not direct weaknesses in the overall hash function, are nonetheless considered certiﬁcational weaknesses and cast suspicion on overall security. In some cases, restricted attacks can be extended to full attacks by standard techniques. Table 9.11 lists the most commonly examined variations, includingpseudo-collisions – collisions allowing different IVs for the different message inputs. In contrast to preim- ages and collisions, pseudo-preimages and pseudo-collisions are of limited direct practical signiﬁcance. 9.94 Note (alternate names for collision and preimage attacks) Alternate names for those in Table 9.11 are as follows: preimage or 2nd-preimage≡target attack; pseudo-preimage ≡free-start target attack; collision (ﬁxed IV)≡collision attack; collision (random IV)≡ semi-free-start collision attack; pseudo-collision≡free-start collision attack. 9.95 Note (relative difﬁculty of attacks) Finding a collision can be no harder than ﬁnding a 2nd- preimage. Similarly, ﬁnding a pseudo-collision can be no harder than ﬁnding (two distinct) pseudo-preimages. 372 Ch. 9 Hash Functions and Data Integrity ↓Type of attack V V′ x x′ y Find... preimage V0 — * — y0 x: h(V0,x)= y0 pseudo-preimage * — * — y0 x,V: h(V,x)= y0 2nd-preimage V0 V0 x0 * h(V0,x0) x′: h(V0,x0)= h(V0,x′) collision (ﬁxed IV) V0 V0 * * — x,x′: h(V0,x)= h(V0,x′) collision (random IV) * V * * — x,x′,V: h(V,x)= h(V,x′) pseudo-collision * * * * — x,x′,V,V ′: h(V,x)= h(V′,x′) Table 9.11:Deﬁnition of preimage and collision attacks.V and V′denote (potentially different) IVs used for MDCh applied to inputsx and x′, respectively;V0 denotes the IV pre-speciﬁed in the deﬁnition ofh,x0 a pre-speciﬁed target input, andy = y0 a pre-speciﬁed target output. * Denotes IVs or inputs which may be freely chosen by an attacker;h(V0,x 0) denotes the hash-code resulting from applyingh with ﬁxed IVV = V0 to inputx = x0. — Means not applicable. 9.96 Example (trivial collisions for random IVs) If free choice of IV is allowed, then trivial pseudo-collisions can be found by deleting leading blocks from a target message. For exam- ple, for an iterated hash (cf. equation (9.1) on page 333),h(IV,x1x2)= f(f(IV,x1),x2). Thus, forIV′= f(IV,x1),h(IV′,x2)= h(IV,x1x2) yields a pseudo-collision ofh, in- dependent of the strength off. (MD-strengthening as per Algorithm 9.26 precludes this.) □ Another common analysis technique is to consider the strength of weakened variants of an algorithm, or attack speciﬁc subcomponents, akin to cryptanalyzing an 8-round version of DES in place of the full 16 rounds. 9.97 DeﬁnitionAn attack on the compression functionof an iterated hash function is any attack as per Table 9.11 withf(H i−1,xi) replacingh(V0,x) – the compression functionfin place of hash functionh, chaining variableHi−1 in place of initializing valueV, and a single input blockxi in place of the arbitrary-length messagex. An attack on a compression function focuses on one ﬁxed stepiof the iterative func- tion of equation (9.1). The entire message consists of a single blockxi = x(without MD-strengthening), and the hash output is taken to be the compression function output so h(x)= H i. The importance of such attacks arises from the following. 9.98 Note (compression function vs. hash function attacks) Any of the six attacks of Table 9.11 which is found for the compression function of an iterated hash can be extended to a similar attack of roughly equal complexity on the overall hash. An iterated hash function is thus in this regard at most as strong as its compression function. (However note, for example, an overall pseudo-collision is not always of practical concern, since most hash functions specify a ﬁxed IV .) For example, consider a messagex= x 1x2 ...xt. Suppose a successful 2nd-preimage attack on compression functionfyields a 2nd-preimagex′ 1 ̸=x1 such thatf(IV,x′ 1 )= f(IV,x1). Then,x′= x′ 1x2 ...xt is a preimage ofh(x). More positively, if MD-strengthening is used, the strength of an iterated hash with respect to the attacks of Table 9.11 is the same as that of its compression function (cf. §9.7 Advanced attacks on hash functions 373 Fact 9.24). However, an iterated hash may certainly be weaker than its compression func- tion (e.g., Example 9.96; Fact 9.37). In summary, a compression function secure against preimage, 2nd-preimage, and col- lision (ﬁxed IV) attacks is necessary and sometimes, but not always, sufﬁcient for a secure iterated hash; and security against the other (i.e., free-start) attacks of Table 9.11 is desir- able, but not always necessary for a secure hash function in practice. For this reason, com- pression functions are analyzed in isolation, and attacks on compression functions as per Deﬁnition 9.97 are considered. A further result motivating the study of pseudo-preimages is the following. 9.99 Fact(pseudo-preimages yielding preimages) If the compression functionfof ann-bit iterated hash functionhdoes not have ideal computational security (2 n) against pseudo- preimage attacks, then preimages forhcan be found in fewer than2n operations (cf.§9.3.4, Table 9.2). This result is true even ifhhas MD-strengthening. Justiﬁcation.The attack requires messages of 3 or more blocks, with 2 or more uncon- strained to allow a meet-in-the-middle attack (page 374). If pseudo-preimages can be found in2 s operations, then2(n+s)/2 forward points and2(n−s)/2 backward points are employed (fewer backward points are used since they are more costly). Preimages can thus be found in2 ·2 (n+s)/2 operations. 9.7.3 Chaining attacks Chaining attacks are those which are based on the iterative nature of hash functions and, in particular, the use of chaining variables. These focus on the compression functionfrather than the overall hash functionh, and may be further classiﬁed as below. An example for context is ﬁrst given. 9.100 Example (chaining attack) Consider a (candidate) collision resistant iterative hash func- tionhproducing a 128-bit hash-result, with a compression functionftaking as inputs a 512-bit message blockxi and 128-bit chaining variableHi (H0 = IV) and producing out- putHi+1 = f(Hi,xi). For a ﬁxed 10-block messagex(640 bytes), considerH= h(x). Suppose one picks any one of the 10 blocks, and wishes to replace it with another block without affecting the hashH.I fhbehaves like a random mapping, the number of such 512-bit blocks is approximately2 512/2128 =2 384. Any efﬁcient method for ﬁnding any one of these2384 blocks distinct from the original constitutes an attack onh. The challenge is that such blocks are a sparse subset of all possible blocks, about 1 in2128. □ (i) Correcting-block chaining attacks Using the example above for context, one could attempt to (totally) replace a messagex with a new messagex′, such thath(x)= h(x′), by using a single unconstrained “correct- ing” block inx′, designated ahead of time, to be determined later such that it produces a chaining value which results in the overall hash being equal to target valueh(x). Such acor- recting block attackcan be used to ﬁnd both preimages and collisions. If the unconstrained block is the ﬁrst (last) block in the message, it is called acorrecting ﬁrst (last) block at- tack. These attacks may be precluded by requiring per-block redundancy, but this results in an undesirable bandwidth penalty. Example 9.101 illustrates a correcting ﬁrst block attack. The extension of Yuval’s birthday attack (page 369), with message alterations restricted to the last block of candidate messages, resembles a correcting last block attack applied simul- taneously to two messages, seeking a (birthday) collision rather than a ﬁxed overall target hash-value. 374 Ch. 9 Hash Functions and Data Integrity 9.101 Example (correcting block attack on CBC cipher mode) The CBC mode of encryption with non-secret key (H0 = IV;Hi = Ek(Hi−1⊕xi)) is unsuitable as an MDC algorithm, because it fails to be one-way – the compression function is reversible when the encryption key is known. A messagex′, of unconstrained length (saytblocks) can be constructed to have any speciﬁed target hash-valueHas follows. Letx′ 2,...x ′ t be t−1 blocks chosen freely. SetH′ t ←H, then forifromtto1 computeH′ i−1 ←Dk(H′ i)⊕x′ i. Finally, compute x∗ 1 ←Dk(H′ 1)⊕IV. Then, forx′= x∗ 1 x′ 2 ...x′ t ,h(x′)= Hand all but blockx∗ 1 (which will appear random) can be freely chosen by an adversary; even this minor drawback can be partially addressed by a meet-in-the-middle strategy (see below). Analogous remarks apply to the CFB mode. □ (ii) Meet-in-the-middle chaining attacks These are birthday attacks similar to Yuval’s (and which can be made essentially memory- less) but which seek collisions on intermediate results (i.e., chaining variables) rather than the overall hash-result. When applicable, they allow (unlike Yuval’s attack) one to ﬁnd a message with a pre-speciﬁed hash-result, for either a 2nd-preimage or a collision. An at- tack point is identiﬁed between blocks of a candidate (fraudulent) message. Variations of the blocks preceding and succeeding this point are generated. The variations are hashed forward from the algorithm-speciﬁed IV (computingH i = f(Hi−1,xi) as usual) and back- ward from the target ﬁnal hash-result (computingHi = f−1(Hi+1,xi+1) for someHi+1, xi+1, ideally forxi+1 chosen by the adversary), seeking a collision in the chaining vari- ableHi at the attack point. For the attack to work, the attacker must be able to efﬁciently go backwards through the chain (certainly moreso than by brute force – e.g., see Exam- ple 9.102), i.e., invert the compression function in the following manner: given a value H i+1, ﬁnd a pair(Hi,xi+1) such thatf(Hi,xi+1)= Hi+1. 9.102 Example (meet-in-the-middle attack on invertible key chaining modes) Chaining modes which allow easily derived stage keys result in reversible compression functions unsuitable for use in MDCs due to lack of one-wayness (cf. Example 9.101). An example of such invertible key chainingmethods is Bitzer’s scheme:H0 = IV,Hi = f(Hi−1,xi)= Eki (Hi−1) whereki = xi⊕s(Hi−1) ands(Hi−1) is a function mapping chaining variables to the key space. For exposition, letsbe the identity function. This compression function is unsuitable because it falls to a meet-in-the-middle attack as outlined above. The ability to move backwards through chaining variables, as required by such an attack, is possible here with the chaining variableHi computed fromHi+1 as follows. Choose a ﬁxed value ki+1 ←k, computeHi ←Dk(Hi+1), then choose as message blockxi+1 ←k⊕Hi. □ (iii) Fixed-point chaining attacks A ﬁxed pointof a compression function is a pair(Hi−1,xi) such thatf(Hi−1,xi)= Hi−1. For such a pair of message block and chaining value, the overall hash on a message is un- changed upon insertion of an arbitrary number of identical blocksxi at the chain point at which that chaining value arises. Such attacks are thus of concern if it can be arranged that the chaining variable has a value for which a ﬁxed point is known. This includes the fol- lowing cases: if ﬁxed points can be found and it can be easily arranged that the chaining variable take on a speciﬁc value; or if for arbitrary chaining valuesHi−1, blocksxi can be found which result in ﬁxed-points. Fixed points allow 2nd-preimages and collisions to be produced; their effect can be countered by inclusion of a trailing length-block (Algo- rithm 9.26). §9.7 Advanced attacks on hash functions 375 (iv) Differential chaining attacks Differential cryptanalysis has proven to be a powerful tool for the cryptanalysis of not only block ciphers but also of hash functions (including MACs). For multi-round block ciphers this attack method examines input differences (XORs) to round functions and the corre- sponding output differences, searching for statistical anomalies. For hash functions, the examination is of input differences to compression functions and the corresponding output differences; a collision corresponds to an output difference of zero. 9.7.4 Attacks based on properties of underlying cipher The implications of certain properties of block ciphers, which may be of no practical con- cern when used for encryption, must be carefully examined when such ciphers are used to construct iterated hash functions. The general danger is that such properties may facil- itate adversarial manipulation of compression function inputs so as to allow prediction or greater control of outputs or relations between outputs of successive iterations. Included among block cipher properties of possible concern are the following (cf. Chapter 7): 1. complementation property: y= E k(x) ⇐⇒ y= E k( x), where xdenotes bitwise complement. This makes it trivial to ﬁnd key-message pairs of block cipher inputs whose outputs differ in a pre-determined manner. For example, for such a block ci- pherE, the compression functionf(H i−1,xi)= EHi−1⊕xi (xi)⊕xi (a linear trans- formation of the Matyas-Meyer-Oseas function) produces the same output forxi and its bitwise complement xi. 2. weak keys: Ek(Ek(x)) = x(for allx). This property of involution of the block cipher may allow an adversary to easily create a two-step ﬁxed point of the compres- sion functionfin the case that message blocksxi have direct inﬂuence on the block cipher key input (e.g., iff= Exi (Hi−1), insert 2 blocksxi containing a weak key). The threat is similar forsemi-weak keys, whereEk′(Ek(x)) =x. 3. ﬁxed points:Ek(x)= x. Block cipher ﬁxed points may facilitate ﬁxed-point attacks if an adversary can control the block cipher key input. For example, for the Davies- Meyer compression functionf(Hi−1,xi)= Exi (Hi−1)⊕Hi−1,i fHi−1 is a ﬁxed point of the block cipher for keyxi (i.e.,Exi (Hi−1)= Hi−1), then this yields a predictable compression function outputf(Hi−1,xi)=0 . 4. key collisions:Ek(x)= Ek′(x). These may allow compression function collisions. Although they may serve as distinguishing metrics, attacks which appear purely certi- ﬁcational in nature should be noted separately from others; for example, ﬁxed point attacks appear to be of limited practical consequence. 9.103 Example (DES-based hash functions) Consider DES as the block cipher in question (see §7.4). DES has the complementation property; has 4 weak keys and 6 pairs of semi-weak keys (each with bit 2 equal to bit 3); each weak key has2 32 ﬁxed points (thus a random plaintext is a ﬁxed point of some weak key with probability2−30), as do 4 of the semi- weak keys; and key collisions can be found in232 operations. The security implications of these properties must be taken into account in the design of any DES-based hash function. Concerns regarding both weak keys and the complementation property can be eliminated by forcing key bits 2 and 3 to be 10 or 01 within the compression function.□ 376 Ch. 9 Hash Functions and Data Integrity 9.8 Notes and further references §9.1 The deﬁnitive reference for cryptographic hash functions, and an invaluable source for the material in this chapter (including many otherwise unattributed results), is the comprehen- sive treatment of Preneel [1003, 1004]; see also the surveys of Preneel [1002] and Preneel, Govaerts, and Vandewalle [1006]. Davies and Price [308] also provide a solid treatment of message authentication and data integrity. An extensive treatment of conventional hash- ing, including historical discussion tracing origins back to IBM in 1953, is given by Knuth [693, p.506-549]. Independent of cryptographic application,universal classes of hash func- tionswere introduced by Carter and Wegman [234] in the late 1970s, the idea being to ﬁnd a class of hash functions such that for every pair of inputs, the probability was low that a randomly chosen function from the class resulted in that pair colliding. Shortly thereafter, Wegman and Carter [1234] noted the cryptographic utility of these hash functions, when combined with secret keys, for (unconditionally secure)message authentication tag sys- tems; they formalized this concept, earlier considered by Gilbert, MacWilliams, and Sloane [454] (predating the concept of digital signatures) who attribute the problem to Simmons. Simmons ([1138],[1144]; see also Chapter 10 of Stinson [1178]) independently developed a general theory of unconditionally secure message authentication schemes and the subject ofauthentication codes(see also§9.5 below). Rabin [1022, 1023] ﬁrst suggested employing a one-way hash function (constructed by us- ing successive message blocks to key an iterated block encryption) in conjunction with a one-time signature scheme and later in a public-key signature scheme; Rabin essentially noted the requirements of 2nd-preimage resistance and collision resistance. Merkle [850] explored further uses of one-way hash functions for authentication, including the idea of tree authentication[852] for both one-time signatures and authentication of public ﬁles. §9.2 Merkle [850] (partially published as [853]) was the ﬁrst to give a substantial (informal) def- inition of one-way hash functions in 1979, specifying the properties of preimage and 2nd- preimage resistance. Foreshadowing UOWHFs (see below), he suggested countering the effect of Remark 9.36 by using slightly different hash functionshover time; Merkle [850, p.16-18] also proposed a public key distribution method based on a one-way hash function (effectively used as a one-way pseudo-permutation) and the birthday paradox, in a precur- sor to his “puzzle system” (see page 537). The ﬁrst formal deﬁnition of a CRHF was given in 1988 by Damg˚ard [295] (an informal deﬁnition was later given by Merkle [855, 854]; see also [853]), who was ﬁrst to explore collision resistant hash functions in a complexity- theoretic setting. Working from the idea ofclaw-resistant pairs of trapdoor permutations due to Goldwasser, Micali, and Rivest [484], Damg˚ard deﬁnedclaw-resistant families of permutations(without the trapdoor property). The termclaw-resistant(originally:claw- free) originates from the pictorial representation of a functional mapping showing two dis- tinct domain elements being mapped to the same range element under distinct functionsf (i) and f(j) (colliding atz= f(i)(x)= f(j)(y)), thereby tracing out a claw. Goldwasser et al. [484] established that the intractability of factoring sufﬁces for the exis- tence of claw-resistant pairs of permutations. Damg˚ard showed that the intractability of the discrete logarithm problem likewise sufﬁces. Using several reasonably efﬁcient number- theoretic constructions for families of claw-resistant permutations, he gave the ﬁrst prov- ably collision resistant hash functions, under such intractability assumptions (for discrete §9.8 Notes and further references 377 logarithms, the assumption required is that takingspeciﬁcdiscrete logarithms be difﬁcult). Russell [1088] subsequently established that a collection of collision resistant hash func- tions exists if and only if there exists a collection ofclaw-resistant pairs of pseudo-permu- tations; a pseudo-permutation on a set is a function computationally indistinguishable from a permutation (pairs of elements demonstrating non-injectivity are hard to ﬁnd). It remains open whether the existence of one-way functions sufﬁces for the existence of collision re- sistant hash functions. The deﬁnition of a one-way function (Deﬁnition 9.9) was given in the seminal paper of Difﬁe and Hellman [345], along with the use of the discrete exponential function modulo a prime as a candidate OWF, which they credit to Gill. The idea of providing the hash- value of some data, to indicate prior commitment to (or knowledge of) that data, was uti- lized in Lamport’s one-time signature scheme (circa 1976); see page 485. The OWF of Example 9.13 was known to Matyas and Meyer circa 1979. As noted by Massey [786], the idea of one-wayness was published in 1873 by J.S. Jevons, who noted (preceding RSA by a century) that multiplying two primes is easy whereas factoring the result is not. Published work dated 1968 records the use of ciphers essentially as one-way functions (decryption was not required) in a technique to avoid storing cleartext computer account passwords in time-shared systems. These were referred to asone-way ciphersby Wilkes [1244] (p.91- 93 in 1968 or 1972 editions; p.147 in 1975 edition), who credits Needham with the idea and an implementation thereof. The ﬁrst proposal of a non-invertible function for the same purpose appears to be that of Evans, Kantrowitz, and Weiss [375], while Purdy [1012] pro- posed extremely high-degree, sparse polynomials over a prime ﬁeld as a class of functions which were computationally difﬁcult to invert. Foreshadowing later research into collision resistance, Purdy also deﬁned thedegeneracyof such a function to be the maximum number of preimages than any image could have, noting that “if the degeneracy is catastrophically large there may be no security at all”. Naor and Yung [920] introduced the cryptographic primitive known as auniversal one-way hash function (UOWHF)family, and give a provably secure construction for a one-way hash function from a one-way hash function which compresses by a single bit (t+1 totbits); the main property of a UOWHF family is 2nd-preimage resistance as for a OWHF, but here an instance of the function is picked at random from a family of hash functions after ﬁxing an input, as might be modeled in practice by using a random IV with a OWHF. Naor and Yung [920] also prove by construction that UOWHFs exist if and only if one-way permu- tations do, and show how to use UOWHFs to construct provably secure digital signature schemes assuming the existence of any one-way permutation. Building on this, Rompel [1068] showed how to construct a UOWHF family from any one-way function, and based signature schemes on such hash functions; combining this with the fact that a one-way func- tion can be constructed from any secure signature scheme, the result is that the existence of one-way functions is necessary and sufﬁcient for the existence of secure digital signature schemes. De Santis and Yung [318] proceed with more efﬁcient reductions from one-way functions to UOWHFs, and show the equivalence of a number of complexity-theoretic def- initions regarding collision resistance. Impagliazzo and Naor [569] give an efﬁcient con- struction for a UOWHF and prove security equivalent to the subset-sum problem (anNP - hard problem whose corresponding decision problem isNP -complete); for parameters for which a random instance of subset-sum is hard, they argue that this UOWHF is secure (cf. Remark 9.12). Impagliazzo, Levin, and Luby [568] prove the existence of one-way func- tions is necessary and sufﬁcient for that of secure pseudorandom generators. Application-speciﬁc (often unprovable) hash function properties beyond collision resist- ance (but short of preimage resistance) may often be identiﬁed as necessary, e.g., for or- 378 Ch. 9 Hash Functions and Data Integrity dinary RSA signatures computed directly after hashing, the multiplicative RSA property dictates that for the hash functionhused it be infeasible to ﬁnd messagesx,x1,x2 such thath(x)= h(x1) ·h(x2). Anderson [27] discusses such additional requirements on hash functions. For a summary of requirements on a MAC in the special case of multi-cast au- thentication, see Preneel [1003]. Bellare and Rogaway [93] include discussion of issues related to the random nature of practical hash functions, and cryptographic uses thereof. Damg˚ard [295] showed that the security of a digital signature scheme which is not existen- tially forgeable under an adaptive chosen-message attack will not be decreased if used in conjunction with a collision-resistant hash function. Bellare, Goldreich, and Goldwasser [88] (see also [89]) introduce the idea ofincremental hashing, involving computing a hash value over data and then updating the hash-value after changing the data; the objective is that the computation required for the update be propor- tional to the amount of change. §9.3 Merkle’s meta-method [854] (Algorithm 9.25) was based on ideas from his 1979 Ph.D. the- sis [850]. An equivalent construction was given by Damg˚ard [296], which Gibson [450] remarks on again yielding Merkle’s method. Naor and Yung [920] give a related construc- tion for a UOWHF. See Preneel [1003] for fundamental results (cf. Remarks 9.35 and 9.36, and Fact 9.27 on cascading hash functions which follow similar results on stream ciphers by Maurer and Massey [822]). The padding method of Algorithms 9.29 and 9.30 originated from ISO/IEC 10118-4 [608]. The basic idea of the long-message attack (Fact 9.37) is from Winternitz [1250]. §9.4 The hash function of Algorithm 9.42 and referred to as Davies-Meyer (as cited per Quis- quater and Girault [1019]) has been attributed by Davies to Meyer; apparently known to Meyer and Matyas circa 1979, it was published along with Algorithm 9.41 by Matyas, Meyer, and Oseas [805]. The Miyaguchi-Preneel scheme (Algorithm 9.43) was proposed circa 1989 by Preneel [1003], and independently by Miyaguchi, Ohta, and Iwata [886]. The three single-length rate-one schemes discussed (Remark 9.44) are among 12 compression functions employingnon-invertible chainingfound through systematic analysis by Preneel et al. [1007] to be provably secure under black-box analysis, 8 being certiﬁcationally vul- nerable to ﬁxed-point attack nonetheless. These 12 are linear transformations on the mes- sage block and chaining variable (i.e.,[x ′,H′]= A[x,H] for any of the 6 invertible2 ×2 binary matricesA) of the Matyas-Meyer-Oseas (Algorithm 9.41) and Miyaguchi-Preneel schemes; these latter two themselves are among the 4 recommended when the underlying cipher is resistant to differential cryptanalysis (e.g., DES), while Davies-Meyer is among the remaining 8 recommended otherwise (e.g., for FEAL). MDC-2 and MDC-4 are of IBM origin, proposed by Brachtl et al. [184], and reported by Meyer and Schilling [860]; details of MDC-2 are also reported by Matyas [803]. For a description of MDC-4, see Bosselaers and Preneel [178]. The DES-based hash function of Merkle [855] which is mentioned uses the meta-method and employs a compression functionfmapping 119-bit input to 112-bit output in 2 DES operations, allowing 7-bit message blocks to be processed (with rate 0.055). An optimized version maps 234 bits to 128 bits in 6 DES operations, processing 106-bit message blocks (with rate 0.276); unfortunately, overheads related to “bit chopping” and the inconvenient block size are substantial in practice. This construction is provably as secure as the under- lying block cipher assuming an unﬂawed cipher (cf. Table 9.3; Preneel [1003] shows that accounting for DES weak keys and complementation drops the rate slightly to 0.266). Win- §9.8 Notes and further references 379 ternitz [1250] considers the security of the Davies-Meyer hash under a black-box model (cf. Remark 9.45). The search for secure double-length hash functions of rate 1 is ongoing, the goal being security better than single-length Matyas-Meyer-Oseas and approaching that of MDC-2. Quisquater and Girault [1019] proposed two functions, one (QG-original) appearing in the Abstracts of Eurocrypt’89 and a second (QG-revised) in the ﬁnal proceedings altered to counter an attack of Coppersmith [276] on the ﬁrst. The attack, restricted to the case of DES as underlying block cipher, uses ﬁxed points resulting from weak keys to ﬁnd colli- sions in2 36 DES operations. A general attack of Knudsen and Lai [688], which (unfortu- nately) applies to a large class of double-length (i.e.,2n-bit) rate-one block-cipher-based hashes including QG-original, ﬁnds preimages in about2n operations plus2n storage. The systematic method used to establish this result was earlier used by Hohl et al. [560] to prove that pseudo-preimage and pseudo-collision attacks on a large class of double-length hash functions of rate 1/2 and 1, including MDC-2, are no more difﬁcult than on the single-length rate-one Davies-Meyer hash; related results are summarized by Lai and Knudsen [727]. A second attack due to Coppersmith [276], not restricted to DES, employs 88 correcting blocks to ﬁnd collisions for QG-revised in2 40 steps. Another modiﬁcation of QG-original, the LOKI Double Hash Function (LOKI-DBH) of Brown, Pieprzyk, and Seberry [215], ap- pears as a general construction to offer the same security as QG-revised (provided the un- derlying block cipher is not LOKI). The speeds in Table 9.5 are normalized from the timings reported by Dobbertin, Bosse- laers, and Preneel [355], relative to an assembly code MD4 implementation optimized for the Pentium processor, with a throughput (90 MHz clock) of 165.7 Mbit/s (optimized C code was roughly a factor of 2 slower). See Bosselaers, Govaerts, and Vandewalle [177] for a detailed MD5 implementation discussion. MD4 and MD5 (Algorithms 9.49, 9.51) were designed by Rivest [1055, 1035]. An Aus- tralian extension of MD5 known as HA V AL has also been proposed by Zheng, Pieprzyk, and Seberry [1268]. The ﬁrst published partial attack on MD4 was by den Boer and Bosse- laers [324], who demonstrated collisions could be found when Round 1 (of the three) was omitted from the compression function, and conﬁrmed unpublished work of Merkle show- ing that collisions could be found (for input pairs differing in only 3 bits) in under a mil- lisecond on a personal computer if Round 3 was omitted. More devastating was the partial attack by Vaudenay [1215] on the full MD4, which provided only near-collisions, but al- lowed sets of inputs to be found for which, of the corresponding four 32-bit output words, three are constant while the remaining word takes on all possible 32-bit values. This re- vealed the word access-order in MD4 to be an unfortunate choice. Finally, late in 1995, using techniques related to those which earlier allowed a partial attack on RIPEMD (see below), Dobbertin [354] broke MD4 as a CRHF by ﬁnding not only collisions as stated in Remark 9.50 (taking a few seconds on a personal computer), but collisions for meaningful messages (in under one hour, requiring 20 free bytes at the start of the messages). A ﬁrst partial attack on MD5 was published by den Boer and Bosselaers [325], who found pseudo-collisions for its compression functionf, which maps a 128-bit chaining variable and sixteen 32-bit words down to 128-bits; using2 16 operations, they found a 16-word message Xand chaining variablesS1 ̸=S2 (these differing only in 4 bits, the most sig- niﬁcant of each word), such thatf(S1,X)= f(S2,X). Because this specialized internal pseudo-collision could not be extended to an external collision due to the ﬁxed initial chain- ing values (and due to the special relation between the inputs), this attack was considered by many to have little practical signiﬁcance, although exhibiting a violation of the design goal to build a CRHF from a collision resistant compression function. But in May of 1996, us- 380 Ch. 9 Hash Functions and Data Integrity ing techniques related to his attack on MD4 above, Dobbertin (rump session, Eurocrypt’96) found MD5 compression function collisions (Remark 9.52) in 10 hours on a personal com- puter (about234 compress function computations). Anticipating the feasibility of264 operations, Rivest [1055] proposed a method to extend MD4 to 256 bits by running two copies of MD4 in parallel over the input, with different initial chaining values and constants for the second, swapping the values of the variableA with the ﬁrst after processing each 16-word block and, upon completion, concatenating the 128-bit hash-values from each copy. However, in October of 1995 Dobbertin [352] found collisions for the compression function of extended MD4 in2 26 compress function opera- tions, and conjectured that a more sophisticated attack could ﬁnd a collision for extended MD4 itself in O(2 40) operations. MD2, an earlier and slower hash function, was designed in 1988 by Rivest; see Kaliski [1033] for a description. Rogier and Chauvaud [1067] demonstrated that collisions can be efﬁciently found for the compression function of MD2, and that the MD2 checksum block is necessary to preclude overall MD2 collisions. RIPEMD [178] was designed in 1992 by den Boer and others under the European RACE Integrity Primitives Evaluation (RIPE) project. A version of MD4 strengthened to counter known attacks, its compression function has two parallel computation lines of three 16- step rounds. Nonetheless, early in 1995, Dobbertin [353] demonstrated that if the ﬁrst or last (parallel) round of the 3-round RIPEMD compress function is omitted, collisions can be found in2 31 compress function computations (one day on a 66 MHz personal com- puter). This result coupled with concern about inherent limitations of 128-bit hash results motivated RIPEMD-160 (Algorithm 9.55) by Dobbertin, Bosselaers, and Preneel [355]; but for corrections, see the directory/pub/COSIC/bosselae/ripemd/at ftp site ftp.esat.kuleuven.ac.be. Increased security is provided by ﬁve rounds (each with two lines) and greater independence between the parallel lines, at a performance penalty of a factor of 2. RIPEMD-128 (with 128-bit result and chaining variable) was si- multaneously proposed as a drop-in upgrade for RIPEMD; it scales RIPEMD-160 back to four rounds (each with two lines). SHA-1 (Algorithm 9.53) is a U.S. government standard [404]. It differs from the original standard SHA [403], which it supersedes, only in the inclusion of the 1-bit rotation in the block expansion from 16 to 80 words. For discussion of how this expansion in SHA is re- lated to linear error correcting codes, see Preneel [1004]. Lai and Massey [729] proposed two hash functions of rate 1/2 with2m-bit hash values, Tandem Davies-MeyerandAbreast Davies-Meyer, based on anm-bit block cipher with2m- bit key (e.g., IDEA), and a thirdm-bit hash function using a similar block cipher. Merkle’s public-domain hash function Snefru [854] and the FEAL-based N-Hash proposed by Miya- guchi, Ohta, and Iwata [886] are other hash functions which have attracted considerable at- tention. Snefru, one of the earliest proposals, is based on the idea of Algorithm 9.41, (typi- cally) using asEthe ﬁrst 128 bits of output of a custom-designed symmetric 512-bit block cipher with ﬁxed keyk=0 . Differential cryptanalysis has been used by Biham and Shamir [137] to ﬁnd collisions for Snefru with 2 passes, and is feasible for Snefru with 4 passes; Merkle currently recommends 8 passes (impacting performance). Cryptanalysis of the 128- bit hash N-Hash has been carried out by Biham and Shamir [136], with attacks on 3, 6, 9, and 12 rounds being of respective complexity2 8,224,240, and256 for the more secure of the two proposed variations. Despite many proposals, few hash functions based on modular arithmetic have withstood attack, and most that have (including those which are provably secure) tend to be relatively §9.8 Notes and further references 381 inefﬁcient. MASH-1 (Algorithm 9.56), from Committee Draft ISO/IEC 10118-4 [608], evolved from a long line of related proposals successively broken and repaired, includ- ing contributions by Jueneman; Davies and Price; A. Jung; Girault [457] (which includes a summary); and members of ISO SC27/WG2 circa 1994-95 (e.g., in response to the crypt- analysis of the 1994 draft proposal, by Coppersmith and Preneel, in ISO/IEC JTC1/SC27 N1055, Attachment 12, “Comments on MASH-1 and MASH-2 (Feb.21 1995)”). Most prominent among prior proposals was thesqmodn algorithm (due to Jung) in informative Annex D of CCITT Recommendation X.509 (1988 version), which despite suffering ig- nominy at the hands of Coppersmith [275], was resurrected with modiﬁcations as the basis for MASH-1. §9.5 Simmons [1146] notes that techniques for message authentication without secrecy (today called MACs) were known to Simmons, Stewart, and Stokes already in the early 1970s. In the open literature, the idea of using DES to provide a MAC was presented already in Feb. 1977 by Campbell [230], who wrote “... Each group of 64 message bits is passed through the algorithm after being combined with the output of the previous pass. The ﬁnal DES output is thus a residue which is a cryptographic function of the entire message”, and noted that to detect message replay or deletion each message could be made unique by using per-message keys or cryptographically protected sequence numbers. Page 121 of this same publication describes the use of encryption in conjunction with an appended redundancy check code for manipulation detection (cf. Figure 9.8(b)). The termMAC itself evolved in the period 1979-1982 during development of ANSI X9.9 [36], where it is deﬁned as “an eight-digit number in hexadecimal format which is the result of passing a ﬁnancial message through the authentication algorithm using a speciﬁc key.” FIPS 81 [398] standardizes MACs based on CBC and CFB modes (CFB-based MACs are little-used, having some disadvantages over CBC-MAC and apparently no advantages); see also FIPS 113 [400]. Algorithm 9.58 is generalized by ISO/IEC 9797 [597] to a CBC-based MAC for ann-bit block cipher providing anm-bit MAC,m≤n, including an alternative to the optional strengthening process of Algorithm 9.58: a second keyk ′(possibly dependent on k) is used to encrypt the ﬁnal output block. As discussed in Chapter 15, using ISO/IEC 9797 with DES to produce a 32-bit MAC and Algorithm 9.29 for padding is equivalent to the MAC speciﬁed in ISO 8731-1, ANSI X9.9 and required by ANSI X9.17. Regard- ing RIPE-MAC (Example 9.63) [178], other than the2−64 probability of guessing a 64-bit MAC, and MAC forgery as applicable to all iterated MACs (see below), the best known at- tacks providing key recovery are linear cryptanalysis using2 42 known plaintexts for RIPE- MAC1, and a2112 exhaustive search for RIPE-MAC3. Bellare, Kilian, and Rogaway [91] formally examine the security of CBC-based MACs and provide justiﬁcation, establishing (via exact rather than asymptotic arguments) that pseudorandom functions are preserved under cipher block chaining; they also propose solutions to the problem of Example 9.62 (cf. Remark 9.59). The MAA (Algorithm 9.68) was developed in response to a request by the Bankers Auto- mated Clearing Services (U.K.), and ﬁrst appeared as a U.K. National Physical Laboratory Report (NPL Report DITC 17/83 February 1983). It has been part of an ISO banking stan- dard [577] since 1987, and is due to Davies and Clayden [306]; comments on its security (see also below) are offered by Preneel [1003], Davies [304], and Davies and Price [308], who note that its design follows the general principles of the Decimal Shift and Add (DSA) algorithm proposed by Sievi in 1980. As a consequence of the conjecture that MAA may show weaknesses in the case of very long messages, ISO 8731-2 speciﬁes a special mode of operation for messages over 1024 bytes. For more recent results on MAA including ex- 382 Ch. 9 Hash Functions and Data Integrity ploration of a key recovery attack, see Preneel and van Oorschot [1010]. Methods for constructing a MAC algorithm from an MDC, including the secret preﬁx, suf- ﬁx, and envelope methods, are discussed by Tsudik [1196]; Galvin, McCloghrie, and Davin [438] suggest addressing the message extension problem (Example 9.65) in the secret suf- ﬁx method by using a prepended length ﬁeld (this requires two passes over the message if the length is not knowna priori). Preneel and van Oorschot [1009] compare the secu- rity of these methods; propose MD5-MAC (Algorithm 9.69) and similar constructions for customized MAC functions based on RIPEMD and SHA; and provide Fact 9.57, which ap- plies to MAA (n =6 4=2 m) withu =2 32.5 and v =2 32.3, while for MD5-MAC (n= 128 = 2m) bothuand vare on the order of264. Remark 9.60 notwithstanding, the use of ann-bit internal chaining variable with a MAC-value of bitlengthm= n/2 is supported by these results. The envelope method with padding (Example 9.66) is discussed by Kaliski and Robshaw (CryptoBytesvol.1 no.1, Spring 1995). Preneel and van Oorschot [1010] proposed a key recovery attack on this method, which although clearly impractical by requiring over264 known text-MAC pairs (for MD5 with 128-bit key), reveals an architectural ﬂaw. Bellare, Canetti, and Krawczyk [86] rigorously examined the security of a nested MAC construction (NMAC), and the practical variation HMAC thereof (Example 9.67), proving HMAC to be secure provided the hash function used exhibits certain appropriate characteristics. Prior to this, the related constructionh(k 1\|\|h(k2\|\|x)) was considered in the note of Kaliski and Robshaw (see above). Other recent proposals for practical MACs include the bucket hashing construction of Rog- away [1065], and the XOR MAC scheme of Bellare, Gu´erin, and Rogaway [90]. The latter is a provably secure construction for MACs under the assumption of the availability of a ﬁnite pseudorandom function, which in practice is instantiated by a block cipher or hash function; advantages include that it is parallelizable and incremental. MACs intended to provide unconditional security are often calledauthentication codes(cf. §9.1 above), with anauthentication tag(cf. MAC value) accompanying data to provide origin authentication (including data integrity). More formally, an authentication code in- volves ﬁnite setsSof source states (plaintext),Aof authentication tags, andKof secret keys, and a set of rules such that eachk∈K deﬁnes a mappingeK : S→A . An (authen- ticated) message, consisting of a source state and a tag, can be veriﬁed only by the intended recipient (as for MACs) possessing a pre-shared key. Wegman and Carter [1234] ﬁrst com- bined one-time pads with hash functions for message authentication; this approach was pur- sued by Brassard [191] trading unconditional security for short keys. This approach was further reﬁned by Krawczyk [714] (see also [717]), whose CRC-based scheme (Algorithm 9.72) is a minor modiﬁcation of a construction by Rabin [1026]. A sec- ond LFSR-based scheme proposed by Krawczyk for producingm-bit hashes (again com- bined with one-time pads as per Algorithm 9.72) improves on a technique of Wegman and Carter, and involves matrix-vector multiplication by anm×bbinaryToeplitz matrixA(each left-to-right diagonal is ﬁxed:A i,j = Ak,l fork−i= l−j), itself generated from a ran- dom binary irreducible polynomial of degreem(deﬁning the LFSR), andmbits of initial state. Krawczyk proves that the probability of successful MAC forgery here for ab-bit mes- sage is at mostb/2m−1, e.g., less than2−30 even form=6 4and a 1 Gbyte message (cf. Fact 9.73). Earlier, Bierbrauer et al. [127] explored the relations between coding theory, universal hashing, and practical authentication codes with relatively short keys (see also Johansson, Kabatianskii, and Smeets [638]; and the survey of van Tilborg [1211]). These and other MAC constructions suitable for use with stream ciphers are very fast, scalable, §9.8 Notes and further references 383 and information-theoretically secure when the short keys they require are used as one-time pads; when used with key streams generated by pseudorandom generators, their security is dependent on the stream and (at best) computationally secure. Desmedt [335] investigated authenticity in stream ciphers, and proposed both uncondition- ally secure authentication systems and stream ciphers providing authenticity. Lai, Rueppel, and Woollven [731] deﬁne an efﬁcient MAC for use with stream ciphers (but see Preneel [1003] regarding a modiﬁcation to address tampering with ends of messages). Part of an initial secret key is used to seed a key stream generator, each bit of which selectively routes message bits to one of two feedback shift registers (FSRs), the initial states of which are part of the secret key and the ﬁnal states of which comprise the MAC. The number of pseudoran- dom bits required equals the number of message bits. Taylor [1189] proposes an alternate MAC technique for use with stream ciphers. §9.6 Simmons [1144] notes the use of sealed authenticators by the U.S. military. An early pre- sentation of MACs and authentication is given by Meyer and Matyas [859]; the third or later printings are recommended, and include the one-pass PCBC encryption-integrity method of Example 9.91. Example 9.89 was initially proposed by the U.S. National Bureau of Stan- dards, and was subsequently found by Jueneman to have deﬁciencies; this is included in the extensive discussion by Jueneman, Matyas, and Meyer [645] of using MDCs for integrity, along with the idea of Example 9.90, which Davies and Price [308, p.124] also consider for n=1 6. Later work by Jueneman [644] considers both MDCs and MACs; see also Meyer and Schilling [860]. Davies and Price also provide an excellent discussion of transaction au- thentication, noting additional techniques (cf.§9.6.1) addressing message replay including use of MAC values themselves from immediately preceding messages as chaining values in place of random number chaining. Subtle ﬂaws in various ﬁelded data integrity techniques are discussed by Stubblebine and Gligor [1179]. §9.7 The taxonomy of preimages and collisions is from Preneel [1003]. The alternate terminol- ogy of Note 9.94 is from Lai and Massey [729], who published the ﬁrst systematic treatment of attacks on iterated hash functions, including relationships between ﬁxed-start and free- start attacks, consideredideal security, and re-examined MD-strengthening. The idea of Algorithm 9.92 was published by Yuval [1262], but the implications of the birthday para- dox were known to others at the time, e.g., see Merkle [850, p.12-13]. The details of the memoryless version are from van Oorschot and Wiener [1207], who also show the process can be perfectly parallelized (i.e., attaining a factorrspeedup withrprocessors) using par- allel collision search methods; related independent work (unpublished) has been reported by Quisquater. Meet-in-the-middle chaining attacks can be extended to handle additional constraints and otherwise generalized. A “triple birthday” chaining attack, applicable when the compres- sion function is invertible, is given by Coppersmith [267] and generalized by Girault, Co- hen, Campana [460]; see also Jueneman [644]. For additional discussion of differential cryptanalysis of hash functions based on block ciphers, see Biham and Shamir [138], Pre- neel, Govaerts, and Vandewalle [1005], and Rijmen and Preneel [1050]. Chapter /1/0 Identiﬁcation and Entity Authentication Contents in Brief 10.1 Introduction............................. 385 10.2 Passwords (weak authentication).................. 388 10.3 Challenge-response identiﬁcation (strong authentication)..... 397 10.4 Customized and zero-knowledge identiﬁcation protocols..... 405 10.5 Attacks on identiﬁcation protocols................. 417 10.6 Notes and further references.................... 420 10.1 Introduction This chapter considers techniques designed to allow one party (theveriﬁer) to gain assur- ances that the identity of another (theclaimant) is as declared, thereby preventing imper- sonation. The most common technique is by the veriﬁer checking the correctness of a mes- sage (possibly in response to an earlier message) which demonstrates that the claimant is in possession of a secret associated by design with the genuine party. Names for such tech- niques includeidentiﬁcation,entity authentication, and (less frequently)identity veriﬁca- tion. Related topics addressed elsewhere include message authentication (data origin au- thentication) by symmetric techniques (Chapter 9) and digital signatures (Chapter 11), and authenticated key establishment (Chapter 12). A major difference between entity authentication and message authentication (as pro- vided by digital signatures or MACs) is that message authentication itself provides no time- liness guarantees with respect to when a message was created, whereas entity authentica- tion involves corroboration of a claimant’s identity through actual communications with an associated veriﬁer during execution of the protocol itself (i.e., inreal-time, while the ver- ifying entity awaits). Conversely, entity authentication typically involves no meaningful message other than the claim of being a particular entity, whereas message authentication does. Techniques which provide both entity authentication and key establishment are de- ferred to Chapter 12; in some cases, key establishment is essentially message authentication where the message is the key. 385 386 Ch. 10 Identiﬁcation and Entity Authentication Chapter outline The remainder of§10.1 provides introductory material.§10.2 discusses identiﬁcation sch- emes involving ﬁxed passwords including Personal Identiﬁcation Numbers (PINs), and providing so-called weak authentication; one-time password schemes are also considered. §10.3 considers techniques providing so-called strong authentication, including challenge- response protocols based on both symmetric and public-key techniques. It includes discus- sion of time-variant parameters (TVPs), which may be used in entity authentication proto- cols and to provide uniqueness or timeliness guarantees in message authentication.§10.4 examines customized identiﬁcation protocols based on or motivated by zero-knowledge techniques.§10.5 considers attacks on identiﬁcation protocols.§10.6 provides references and further chapter notes. 10.1.1 Identiﬁcation objectives and applications The general setting for an identiﬁcation protocol involves aproverorclaimantAand averi- ﬁerB. The veriﬁer is presented with, or presumes beforehand, the purported identity of the claimant. The goal is to corroborate that the identity of the claimant is indeedA, i.e., to provide entity authentication. 10.1 DeﬁnitionEntity authenticationis the process whereby one party is assured (through ac- quisition of corroborative evidence) of the identity of a second party involved in a protocol, and that the second has actually participated (i.e., is active at, or immediately prior to, the time the evidence is acquired). 10.2 Remark (identiﬁcation terminology) The termsidentiﬁcationandentity authenticationare used synonymously throughout this book. Distinction is made between weak, strong, and zero-knowledge based authentication. Elsewhere in the literature, sometimes identiﬁcation implies only a claimed or stated identity whereas entity authentication suggests a corrobo- rated identity. (i) Objectives of identiﬁcation protocols From the point of view of the veriﬁer, the outcome of an entity authentication protocol is eitheracceptanceof the claimant’s identity as authentic (completion with acceptance), or termination without acceptance(rejection). More speciﬁcally, the objectives of an identi- ﬁcation protocol include the following. 1. In the case of honest partiesAand B,Ais able to successfully authenticate itself to B, i.e.,Bwill complete the protocol having acceptedA’s identity. 2. (transferability)Bcannot reuse an identiﬁcation exchange withAso as to success- fully impersonateAto a third partyC. 3. (impersonation) The probability is negligible that any partyCdistinct fromA, car- rying out the protocol and playing the role ofA, can causeBto complete and accept A’s identity. Herenegligibletypically means “is so small that it is not of practical signiﬁcance”; the precise deﬁnition depends on the application. 4. The previous points remain true even if: a (polynomially) large number of previous authentications betweenAand Bhave been observed; the adversaryChas partici- pated in previous protocol executions with either or bothAand B; and multiple in- stances of the protocol, possibly initiated byC, may be run simultaneously. §10.1 Introduction 387 The idea of zero-knowledge-based protocols is that protocol executions do not even reveal any partial information which makesC’s task any easier whatsoever. An identiﬁcation (or entity authentication) protocol is a “real-time” process in the sense that it provides an assurance that the party being authenticated is operational at the time of protocol execution – that party is taking part, having carried out some action since the start of the protocol execution. Identiﬁcation protocols provide assurances only at the particu- lar instant in time of successful protocol completion. If ongoing assurances are required, additional measures may be necessary; see§10.5. (ii) Basis of identiﬁcation Entity authentication techniques may be divided into three main categories, depending on which of the following the security is based: 1. something known. Examples include standard passwords (sometimes used to derive a symmetric key), Personal Identiﬁcation Numbers (PINs), and the secret or private keys whose knowledge is demonstrated in challenge-response protocols. 2. something possessed. This is typically a physical accessory, resembling a passport in function. Examples include magnetic-striped cards,chipcards(plastic cards the size of credit cards, containing an embedded microprocessor or integrated circuit; also calledsmart cardsorIC cards), and hand-held customized calculators (password generators) which provide time-variant passwords. 3. something inherent(to a human individual). This category includes methods which make use of human physical characteristics and involuntary actions (biometrics), such as handwritten signatures, ﬁngerprints, voice, retinal patterns, hand geome- tries, and dynamic keyboarding characteristics. These techniques are typically non- cryptographic and are not discussed further here. (iii) Applications of identiﬁcation protocols One of the primary purposes of identiﬁcation is to facilitate access control to a resource, when an access privilege is linked to a particular identity (e.g., local or remote access to computer accounts; withdrawals from automated cash dispensers; communications permis- sions through a communications port; access to software applications; physical entry to re- stricted areas or border crossings). A password scheme used to allow access to a user’s computer account may be viewed as the simplest instance of anaccess control matrix: each resource has a list of identities associated with it (e.g., a computer account which authorized entities may access), and successful corroboration of an identity allows access to the autho- rized resources as listed for that entity. In many applications (e.g., cellular telephony) the motivation for identiﬁcation is to allow resource usage to be tracked to identiﬁed entities, to facilitate appropriate billing. Identiﬁcation is also typically an inherent requirement in authenticated key establishment protocols (see Chapter 12). 10.1.2 Properties of identiﬁcation protocols Identiﬁcation protocols may have many properties. Properties of interest to users include: 1. reciprocity of identiﬁcation. Either one or both parties may corroborate their iden- tities to the other, providing, respectively,unilateralormutualidentiﬁcation. Some techniques, such as ﬁxed-password schemes, may be susceptible to an entity posing as a veriﬁer simply in order to capture a claimant’s password. 2. computational efﬁciency. The number of operations required to execute a protocol. 388 Ch. 10 Identiﬁcation and Entity Authentication 3. communication efﬁciency. This includes the number of passes (message exchanges) and the bandwidth required (total number of bits transmitted). More subtle properties include: 4. real-time involvement of a third party (if any). Examples of third parties include an on-linetrustedthird party to distribute common symmetric keys to communicating entities for authentication purposes; and an on-line (untrusted) directory service for distributing public-key certiﬁcates, supported by an off-line certiﬁcation authority (see Chapter 13). 5. nature of trust required in a third party (if any). Examples include trusting a third party to correctly authenticate and bind an entity’s name to a public key; and trusting a third party with knowledge of an entity’s private key. 6. nature of security guarantees. Examples include provable security and zero-know- ledge properties (see§10.4.1). 7. storage of secrets. This includes the location and method used (e.g., software only, local disks, hardware tokens, etc.) to store critical keying material. Relation between identiﬁcation and signature schemes Identiﬁcation schemes are closely related to, but simpler than, digital signature schemes, which involve a variable message and typically provide a non-repudiation feature allowing disputes to be resolved by judges after the fact. For identiﬁcation schemes, the semantics of the message are essentially ﬁxed – a claimed identity at the current instant in time. The claim is either corroborated or rejected immediately, with associated privileges or access either granted or denied in real time. Identiﬁcations do not have “lifetimes” as signatures do 1 – disputes need not typically be resolved afterwards regarding a prior identiﬁcation, and attacks which may become feasible in the future do not affect the validity of a prior identiﬁcation. In some cases, identiﬁcation schemes may also be converted to signature schemes using a standard technique (see Note 10.30). 10.2 Passwords (weak authentication) Conventional password schemes involve time-invariant passwords, which provide so-call- ed weak authentication. The basic idea is as follows. Apassword, associated with each user (entity), is typically a string of 6 to 10 or more characters the user is capable of com- mitting to memory. This serves as a shared secret between the user and system. (Conven- tional password schemes thus fall under the category of symmetric-key techniques provid- ing unilateral authentication.) To gain access to a system resource (e.g., computer account, printer, or software application), the user enters a (userid, password) pair, and explicitly or implicitly speciﬁes a resource; hereuseridis a claim of identity, andpasswordis the evi- dence supporting the claim. The system checks that the password matches corresponding data it holds for that userid, and that the stated identity is authorized to access the resource. Demonstration of knowledge of this secret (by revealing the password itself) is accepted by the system as corroboration of the entity’s identity. Various password schemes are distinguished by the means by which information al- lowing password veriﬁcation is stored within the system, and the method of veriﬁcation. The collection of ideas presented in the following sections motivate the design decisions 1Some identiﬁcation techniques involve, as a by-product, the granting ofticketswhich provide time-limited access to speciﬁed resources (see Chapter 13). §10.2 Passwords (weak authentication) 389 made in typical password schemes. A subsequent section summarizes the standard attacks these designs counteract. Threats which must be guarded against include: password dis- closure (outside of the system) and line eavesdropping (within the system), both of which allow subsequent replay; and password guessing, including dictionary attacks. 10.2.1 Fixed password schemes: techniques (i) Stored password ﬁles The most obvious approach is for the system to store user passwords cleartext in a system password ﬁle, which is both read- and write-protected (e.g., via operating system access control privileges). Upon password entry by a user, the system compares the entered pass- word to the password ﬁle entry for the corresponding userid; employing no secret keys or cryptographic primitives such as encryption, this is classiﬁed as a non-cryptographic tech- nique. A drawback of this method is that it provides no protection against privileged in- siders orsuperusers(special userids which have full access privileges to system ﬁles and resources). Storage of the password ﬁle on backup media is also a security concern, since the ﬁle contains cleartext passwords. (ii) “Encrypted” password ﬁles Rather than storing a cleartext user password in a (read- and write-protected) password ﬁle, a one-way function of each user password is stored in place of the password itself (see Fig- ure 10.1). To verify a user-entered password, the system computes the one-way function of the entered password, and compares this to the stored entry for the stated userid. To pre- clude attacks suggested in the preceding paragraph, the password ﬁle need now only be write-protected. 10.3 Remark (one-way function vs. encryption) For the purpose of protecting password ﬁles, the use of a one-way function is generally preferable to reversible encryption; reasons in- clude those related to export restrictions, and the need for keying material. However, in both cases, for historical reasons, the resulting values are typically referred to as “encrypted” passwords. Protecting passwords by either method before transmission over public com- munications lines addresses the threat of compromise of the password itself, but alone does not preclude disclosure or replay of the transmission (cf. Protocol 10.6). (iii) Password rules Since dictionary attacks (see§10.2.2(iii)) are successful against predictable passwords, some systems impose “password rules” to discourage or prevent users from using “weak” passwords. Typical password rules include a lower bound on the password length (e.g., 8 or 12 characters); a requirement for each password to contain at least one character from each of a set of categories (e.g., uppercase, numeric, non-alphanumeric); or checks that candi- date passwords are not found in on-line or available dictionaries, and are not composed of account-related information such as userids or substrings thereof. Knowing which rules are in effect, an adversary may use a modiﬁed dictionary attack strategy taking into account the rules, and targeting the weakest form of passwords which nonetheless satisfy the rules. The objective of password rules is to increase the entropy (rather than just the length) of user passwords beyond the reach of dictionary and exhaus- tive search attacks.Entropyhere refers to the uncertainty in a password (cf.§2.2.1); if all passwords are equally probable, then the entropy is maximal and equals the base-2 loga- rithm of the number of possible passwords. 390 Ch. 10 Identiﬁcation and Entity Authentication Veriﬁer (system)B Password table A h(passwordA) h(passwordA) password,A h(password) A password = REJECT ACCEPTyes no ClaimantA h Figure 10.1:Use of one-way function for password-checking. Another procedural technique intended to improve password security ispassword ag- ing. A time period is deﬁned limiting the lifetime of each particular password (e.g., 30 or 90 days). This requires that passwords be changed periodically. (iv) Slowing down the password mapping To slow down attacks which involve testing a large number of trial passwords (see§10.2.2), the password veriﬁcation function (e.g., one-way function) may be made more computa- tionally intensive, for example, by iterating a simpler functiont> 1times, with the output of iterationiused as the input for iterationi+1. The total number of iterations must be restricted so as not to impose a noticeable or unreasonable delay for legitimate users. Also, the iterated function should be such that the iterated mapping does not result in a ﬁnal range space whose entropy is signiﬁcantly decimated. (v) Salting passwords To make dictionary attacks less effective, each password, upon initial entry, may be aug- mented with at-bit random string called asalt(it alters the “ﬂavor” of the password; cf. §10.2.3) before applying the one-way function. Both the hashed password and the salt are recorded in the password ﬁle. When the user subsequently enters a password, the system looks up the salt, and applies the one-way function to the entered password, as altered or augmented by the salt. The difﬁculty of exhaustive search on any particular user’s pass- word is unchanged by salting (since the salt is given in cleartext in the password ﬁle); how- ever, salting increases the complexity of a dictionary attack against a large set of passwords simultaneously, by requiring the dictionary to contain2 tvariations of each trial password, implying a larger memory requirement for storing an encrypted dictionary, and correspond- ingly more time for its preparation. Note that with salting, two users who choose the same password have different entries in the system password ﬁle. In some systems, it may be appropriate to use an entity’s userid itself as salt. (vi) Passphrases To allow greater entropy without stepping beyond the memory capacity of human users, passwords may be extended topassphrases; in this case, the user types in a phrase or sen- tence rather than a short “word”. The passphrase is hashed down to a ﬁxed-size value, which plays the same role as a password; here, it is important that the passphrase is not simply trun- §10.2 Passwords (weak authentication) 391 cated by the system, as passwords are in some systems. The idea is that users can remember phrases easier than random character sequences. If passwords resemble English text, then since each character contains only about 1.5 bits of entropy (Fact 7.67), a passphrase pro- vides greater security through increased entropy than a short password. One drawback is the additional typing requirement. 10.2.2 Fixed password schemes: attacks (i) Replay of ﬁxed passwords A weakness of schemes using ﬁxed, reusable passwords (i.e., the basic scheme of§10.2), is the possibility that an adversary learns a user’s password by observing it as it is typed in (or from where it may be written down). A second security concern is that user-entered passwords (or one-way hashes thereof) are transmitted in cleartext over the communications line between the user and the system, and are also available in cleartext temporarily during system veriﬁcation. An eavesdropping adversary may record this data, allowing subsequent impersonation. Fixed password schemes are thus of use when the password is transmitted over trusted communications lines safe from monitoring, but are not suitable in the case that passwords are transmitted over open communications networks. For example, in Figure 10.1, the claimantAmay be a user logging in from home over a telephone modem, to a remote ofﬁce siteBtwo (or two thousand) miles away; the cleartext password might then travel over an unsecured telephone network (including possibly a wireless link), subject to eavesdropping. In the case that remote identity veriﬁcation is used for access to a local resource, e.g., an automated cash dispenser with on-line identity veriﬁcation, the system response (ac- cept/reject) must be protected in addition to the submitted password, and must include vari- ability to prevent trivial replay of a time-invariant accept response. (ii) Exhaustive password search A very naive attack involves an adversary simply (randomly or systematically) trying pass- words, one at a time, on the actual veriﬁer, in hope that the correct password is found. This may be countered by ensuring passwords are chosen from a sufﬁciently large space, limit- ing the number of invalid (on-line) attempts allowed within ﬁxed time periods, and slowing down the password mapping or login-process itself as in§10.2.1(iv).Off-line attacks, in- volving a (typically large) computation which does not require interacting with the actual veriﬁer until a ﬁnal stage, are of greater concern; these are now considered. Given a password ﬁle containing one-way hashes of user passwords, an adversary may attempt to defeat the system by testing passwords one at a time, and comparing the one-way hash of each to passwords in the encrypted password ﬁle (see§10.2.1(ii)). This is theoreti- cally possible since both the one-way mapping and the (guessed) plaintext are known. (This could be precluded by keeping any or all of the details of the one-way mapping or the pass- word ﬁle itself secret, but it is not considered prudent to base the security of the system on the assumption that such details remain secret forever.) The feasibility of the attack depends on the number of passwords that need be checked before a match is expected (which itself depends on the number of possible passwords), and the time required to test each (see Ex- ample 10.4, Table 10.1, and Table 10.2). The latter depends on the password mapping used, its implementation, the instruction execution time of the host processor, and the number of processors available (note exhaustive search is parallelizable). The time required to actu- ally compare the image of each trial password to all passwords in a password ﬁle is typically negligible. 392 Ch. 10 Identiﬁcation and Entity Authentication 10.4 Example (password entropy) Suppose passwords consist of strings of 7-bit ASCII char- acters. Each has a numeric value in the range 0-127. (When 8-bit characters are used, val- ues 128-255 compose theextended character set, generally inaccessible from standard key- boards.) ASCII codes 0-31 are reserved for control characters; 32 is a space character; 33- 126 are keyboard-accessible printable characters; and 127 is a special character. Table 10.1 gives the number of distinctn-character passwords composed of typical combinations of characters, indicating an upper bound on the security of such password spaces.□ →c 26 36 (lowercase 62 (mixed case 95 (keyboard ↓n (lowercase) alphanumeric) alphanumeric) characters) 5 23.5 25.9 29.8 32.9 6 28.2 31.0 35.7 39.4 7 32.9 36.2 41.7 46.0 8 37.6 41.4 47.6 52.6 9 42.3 46.5 53.6 59.1 10 47.0 51.7 59.5 65.7 Table 10.1:Bitsize of password space for various character combinations. The number ofn- character passwords, givenc choices per character, iscn. The table gives the base-2 logarithm of this number of possible passwords. →c 26 36 (lowercase 62 (mixed case 95 (keyboard ↓n (lowercase) alphanumeric) alphanumeric) characters) 5 0.67 hr 3.4 hr 51 hr 430 hr 6 17 hr 120 hr 130 dy 4.7 yr 7 19 dy 180 dy 22 yr 440 yr 8 1.3 yr 18 yr 1400 yr 42000 yr 9 34 yr 640 yr 86000 yr 4.0 ×106 yr 10 890 yr 23000 yr 5.3 ×106 yr 3.8 ×108 yr Table 10.2:Time required to search entire password space. The table gives the timeT (in hours, days, or years) required to search or pre-compute over the entire speciﬁed spaces using a single processor (cf. Table 10.1).T = cn ·t ·y, wheret is the number of times the password mapping is iterated, andy the time per iteration, fort =2 5,y =1 /(125 000)sec. (This approximates theUNIX crypt command on a high-end PC performing DES at 1.0 Mbytes/s – see§10.2.3.) (iii) Password-guessing and dictionary attacks To improve upon the expected probability of success of an exhaustive search, rather than searching through the space of all possible passwords, an adversary may search the space in order of decreasing (expected) probability. While ideally arbitrary strings ofncharacters would be equiprobable as user-selected passwords, most (unrestricted) users select pass- words from a small subset of the full password space (e.g., short passwords; dictionary words; proper names; lowercase strings). Such weak passwords with low entropy are easily guessed; indeed, studies indicate that a large fraction of user-selected passwords are found in typical (intermediate) dictionaries of only150000 words, while even a large dictionary of250000 words represents only a tiny fraction of all possiblen-character passwords (see Table 10.1). Passwords found in any on-line or available list of words may be uncovered by an ad- versary who tries all words in this list, using a so-calleddictionary attack. Aside from tradi- tional dictionaries as noted above, on-line dictionaries of words from foreign languages, or §10.2 Passwords (weak authentication) 393 on specialized topics such as music, ﬁlm, etc. are available. For efﬁciency in repeated use by an adversary, an “encrypted” (hashed) list of dictionary or high-probability passwords may be created and stored on disk or tape; password images from system password ﬁles may then be collected, ordered (using a sorting algorithm or conventional hashing), and then compared to entries in the encrypted dictionary. Dictionary-style attacks are not gen- erally successful at ﬁnding a particular user’s password, but ﬁnd many passwords in most systems. 10.2.3 Case study – UNIX passwords The UNIX 2 operating system provides a widely known, historically important example of a ﬁxed password system, implementing many of the ideas of§10.2.1. AUNIX password ﬁle contains a one-way function of user passwords computed as follows: each user password serves as the key to encrypt a known plaintext (64 zero-bits). This yields a one-way function of the key, since only the user (aside from the system, temporarily during password veri- ﬁcation) knows the password. For the encryption algorithm, a minor modiﬁcation of DES (§7.4) is used, as described below; variations may appear in products outside of the USA. The technique described relies on the conjectured property that DES is resistant to known- plaintext attacks – given cleartext and the corresponding ciphertext, it remains difﬁcult to ﬁnd the key. The speciﬁc technique makes repeated use of DES, iterating the enciphermentt=25 times (see Figure 10.2). In detail, a user password is truncated to its ﬁrst 8 ASCII char- acters. Each of these provides7bits for a 56-bit DES key (padded with 0-bits if less than 8 characters). The key is used to DES-encrypt the 64-bit constant 0, with the output fed back as inputttimes iteratively. The 64-bit result is repacked into 11 printable characters (a 64-bit output and 12 salt bits yields 76 bits; 11 ASCII characters allow 77). In addition, a non-standard method of password salting is used, intended to simultaneously complicate dictionary attacks and preclude use of off-the-shelf DES hardware for attacks: 1. password salting. UNIX password salting associates a 12-bit “random” salt (12 bits taken from the system clock at time of password creation) with each user-selected password. The 12 bits are used to alter the standard expansion functionEof the DES mapping (see§7.4), providing one of 4096 variations. (The expansionEcreates a 48-bit block; immediately thereafter, the salt bits collectively determine one of 4096 permutations. Each bit is associated with a pre-determined pair from the 48-bit block, e.g., bit 1 with block bits 1 and 25, bit 2 with block bits 2 and 26, etc. If the salt bit is 1, the block bits are swapped, and otherwise they are not.) Both the hashed password and salt are recorded in the system password ﬁle. Security of any particular user’s password is unchanged by salting, but a dictionary attack now requires2 12 =4096 variations of each trial password. 2. preventing use of off-the-shelf DES chips. Because the DES expansion permutation Eis dependent on the salt, standard DES chips can no longer be used to implement theUNIX password algorithm. An adversary wishing to use hardware to speed up an attack must build customized hardware rather than use commercially available chips. This may deter adversaries with modest resources. The value stored for a given userid in the write-protected password ﬁle/etc/passwd is thus the iterated encryption of 0 under that user’s password, using the salted modiﬁcation of DES. The constant 0 here could be replaced by other values, but typically is not. The overall algorithm is called the UNIX cryptpassword algorithm. 2UNIX is a trademark of Bell Laboratories. 394 Ch. 10 Identiﬁcation and Entity Authentication 64 56 64 12 12 user password key K I1 =0 ··· 0 data Ii user salt next inputIi, O25 /etc/passwd into eleven 7-bit characters ASCII chars; 0-pad if necessary truncate to 8 output Oi repack 76 bits “encrypted” password DES ∗ 2 ≤i ≤25 Figure 10.2:UNIX cryptpassword mapping. DES* indicates DES with the expansion mappingE modiﬁed by a 12-bit salt. 10.5 Remark (performance advances) While theUNIX cryptmapping witht=25 iterations provided a reasonable measure of protection against exhaustive search when introduced in the 1970s, for equivalent security in a system designed today a more computationally in- tensive mapping would be provided, due to performance advances in both hardware and software. 10.2.4 PINs and passkeys (i) PINs Personal identiﬁcation numbers(PINs) fall under the category of ﬁxed (time-invariant) passwords. They are most often used in conjunction with “something possessed”, typically a physicaltokensuch as a plastic banking card with a magnetic stripe, or a chipcard. To prove one’s identity as the authorized user of the token, and gain access to the privileges associated therewith, entry of the correct PIN is required when the token is used. This pro- vides a second level of security if the token is lost or stolen. PINs may also serve as the second level of security for entry to buildings which have an independent ﬁrst level of se- curity (e.g., a security guard or video camera). For user convenience and historical reasons, PINs are typically short (relative to ﬁxed password schemes) and numeric, e.g., 4 to 8 digits. To prevent exhaustive search through such a small key space (e.g.,10000values for a 4-digit numeric PIN), additional procedural constraints are necessary. For example, some automated cash dispenser machines accessed §10.2 Passwords (weak authentication) 395 by banking cards conﬁscate a card if three incorrect PINs are entered successively; for oth- ers, incorrect entry of a number of successive PINs may cause the card to be “locked” or deactivated, thereafter requiring a longer PIN (e.g., 8 digits) for reactivation following such suspicious circumstances. In an on-line system using PINs or reusable passwords, a claimed identity accompanied by a user-entered PIN may be veriﬁed by comparison to the PIN stored for that identity in a system database. An alternative is to use the PIN as a key for a MAC (see Chapter 9). In an off-line system without access to a central database, information facilitating PIN veriﬁcation must be stored on the token itself. If the PIN need not be user-selected, this may be done by deﬁning the PIN to be a function of a secret key and the identity associated with the token; the PIN is then veriﬁable by any remote system knowing this master key. In an off-line system, it may also be desirable to allow the PIN to be user-selectable, to facilitate PIN memorization by users. In this case, the PIN may be encrypted under a master key and stored on the token, with the master key known to all off-line terminals that need to be capable of verifying the token. A preferable design is to store a one-way function of the PIN, user identity, and master key on the token. (ii) Two-stage authentication and password-derived keys Human users have difﬁculty remembering secret keys which have sufﬁcient entropy to pro- vide adequate security. Two techniques which address this issue are now described. When tokens are used with off-line PIN veriﬁcation, a common technique is for the PIN to serve to verify the user to the token, while the token contains additional independent information allowing the token to authenticate itself to the system (as a valid token repre- senting a legitimate user). The user is thereby indirectly authenticated to the system by a two-stage process. This requires the user have possession of the token but need remember only a short PIN, while a longer key (containing adequate entropy) provides cryptographic security for authentication over an unsecured link. A second technique is for a user password to be mapped by a one-way hash function into a cryptographic key (e.g., a 56-bit DES key). Such password-derived keys are called passkeys. The passkey is then used to secure a communications link between the user and a system which also knows the user password. It should be ensured that the entropy of the user’s password is sufﬁciently large that exhaustive search of the password space is not more efﬁcient than exhaustive search of the passkey space (i.e., guessing passwords is not easier than guessing 56-bit DES keys); see Table 10.1 for guidance. An alternative to having passkeys remain ﬁxed until the password is changed is to keep a running sequence number on the system side along with each user’s password, for use as a time-variant salt communicated to the user in the clear and incremented after each use. A ﬁxed per-user salt could also be used in addition to a running sequence number. Passkeys should be viewed as long-term keys, with use restricted to authentication and key management (e.g., rather than also for bulk encryption of user data). A disadvantage of using password-derived keys is that storing each user’s password within the system requires some mechanism to protect the conﬁdentiality of the stored passwords. 10.2.5 One-time passwords (towards strong authentication) A natural progression from ﬁxed password schemes to challenge-response identiﬁcation protocols may be observed by considering one-time password schemes. As was noted in §10.2.2, a major security concern of ﬁxed password schemes is eavesdropping and subse- quent replay of the password. A partial solution isone-time passwords: each password is 396 Ch. 10 Identiﬁcation and Entity Authentication used only once. Such schemes are safe from passive adversaries who eavesdrop and later attempt impersonation. Variations include: 1. shared lists of one-time passwords. The user and the system use a sequence or set oft secret passwords, (each valid for a single authentication), distributed as a pre-shared list. A drawback is maintenance of the shared list. If the list is not used sequen- tially, the system may check the entered password against all remaining unused pass- words. A variation involves use of achallenge-response table, whereby the user and the system share a table of matching challenge-response pairs, ideally with each pair valid at most once; this non-cryptographic technique differs from the cryptographic challenge-response of§10.3. 2. sequentially updated one-time passwords. Initially only a single secret password is shared. During authentication using passwordi, the user creates and transmits to the system a new password (passwordi+1) encrypted under a key derived from pass- word i. This method becomes difﬁcult if communication failures occur. 3. one-time password sequences based on a one-way function. Lamport’s one-time password scheme is described below. This method is more efﬁcient (with respect to bandwidth) than sequentially updated one-time passwords, and may be viewed as a challenge-response protocol where the challenge is implicitly deﬁned by the current position within the password sequence. One-time passwords based on one-way functions (Lamport’s scheme) In Lamport’s one-time password scheme, the user begins with a secretw. A one-way func- tion (OWF)His used to deﬁne the password sequence:w,H(w),H(H(w)),...,H t(w). The password for theith identiﬁcation session,1≤i≤t, is deﬁned to bewi=Ht−i(w). 10.6 ProtocolLamport’s OWF-based one-time passwords SUMMARY: Aidentiﬁes itself toBusing one-time passwords from a sequence. 1. One-time setup. (a) UserAbegins with a secretw. LetHbe a one-way function. (b) A constanttis ﬁxed (e.g.,t=100 or1000), deﬁning the number of identiﬁca- tions to be allowed. (The system is thereafter restarted with a neww, to avoid replay attacks.) (c)Atransfers (theinitial shared secret)w0 =Ht(w), in a manner guaranteeing its authenticity, to the systemB.Binitializes its counter forAtoiA=1. 2. Protocol messages. Theith identiﬁcation,1≤i≤t, proceeds as follows: A→B: A,i,w i(=Ht−i(w)) (1) Here A→B:XdenotesAsending the messageXtoB. 3. Protocol actions. To identify itself for sessioni,Adoes the following. (a)A’s equipment computeswi =Ht−i(w)(easily done either fromwitself, or from an appropriate intermediate value saved during the computation ofHt(w) initially), and transmits (1) toB. (b) Bchecks thati=iA, and that the received passwordwisatisﬁes:H(wi)= wi−1. If both checks succeed,Baccepts the password, setsiA←iA+1, and saveswifor the next session veriﬁcation. §10.3 Challenge-response identiﬁcation (strong authentication) 397 10.7 Note (pre-play attack) Protocol 10.6 and similar one-time password schemes including that of Note 10.8 remain vulnerable to an active adversary who intercepts and traps (or im- personates the system in order to extract) an as-yet unused one-time password, for the pur- pose of subsequent impersonation. To prevent this, a password should be revealed only to a party which itself is known to be authentic. Challenge-response techniques (see§10.3) address this threat. 10.8 Note (alternative one-time password scheme) The following one-time-password alterna- tive to Protocol 10.6 is suitable if storing actual passwords on the system side is acceptable (cf. Figure 10.1; compare also to§10.3.2(iii)). The claimantAhas a shared passwordP with the system veriﬁerB, to which it sends the data pair:(r,H(r,P)). The veriﬁer com- putes the hash of the received valuerand its local copy ofP, and declares acceptance if this matches the received hash value. To avoid replay,rshould be a sequence number, time- stamp, or other parameter which can be easily guaranteed to be accepted only once. 10.3 Challenge-response identiﬁcation (strong authentication) The idea of cryptographic challenge-response protocols is that one entity (the claimant) “proves” its identity to another entity (the veriﬁer) by demonstrating knowledge of a secret known to be associated with that entity, without revealing the secret itself to the veriﬁer dur- ing the protocol. 3 This is done by providing a response to a time-variant challenge, where the response depends on both the entity’s secret and the challenge. Thechallengeis typi- cally a number chosen by one entity (randomly and secretly) at the outset of the protocol. If the communications line is monitored, the response from one execution of the identiﬁ- cation protocol should not provide an adversary with useful information for a subsequent identiﬁcation, as subsequent challenges will differ. Before considering challenge-response identiﬁcation protocols based on symmetric- key techniques (§10.3.2), public-key techniques (§10.3.3), and zero-knowledge concepts (§10.4), background on time-variant parameters is ﬁrst provided. 10.3.1 Background on time-variant parameters Time-variant parameters may be used in identiﬁcation protocols to counteract replay and interleaving attacks (see§10.5), to provide uniqueness or timeliness guarantees, and to pre- vent certain chosen-text attacks. They may similarly be used in authenticated key estab- lishment protocols (Chapter 12), and to provide uniqueness guarantees in conjunction with message authentication (Chapter 9). Time-variant parameters which serve to distinguish one protocol instance from another are sometimes callednonces,unique numbers,o rnon-repeating values; deﬁnitions of these terms have traditionally been loose, as the speciﬁc properties required depend on the actual usage and protocol. 10.9 DeﬁnitionA nonce is a value used no more than once for the same purpose. It typically serves to prevent (undetectable) replay. 3In some mechanisms, the secret is known to the veriﬁer, and is used to verify the response; in others, the secret need not actually be known by the veriﬁer. 398 Ch. 10 Identiﬁcation and Entity Authentication The termnonce is most often used to refer to a “random” number in a challenge-response protocol, but the required randomness properties vary. Three main classes of time-variant parameters are discussed in turn below: random numbers, sequence numbers, and time- stamps. Often, to ensure protocol security, the integrity of such parameters must be guar- anteed (e.g., by cryptographically binding them with other data in a challenge-response sequence). This is particularly true of protocols in which the only requirement of a time- variant parameter is uniqueness, e.g., as provided by a never-repeated sequential counter. 4 Following are some miscellaneous points about time-variant parameters. 1. Veriﬁable timeliness may be provided through use of random numbers in challenge- response mechanisms, timestamps in conjunction with distributed timeclocks, or se- quence numbers in conjunction with the maintenance of pairwise (claimant, veriﬁer) state information. 2. To provide timeliness or uniqueness guarantees, the veriﬁer in the protocol controls the time-variant parameter, either directly (through choice of a random number) or indirectly (through information maintained regarding a shared sequence, or logically through a common time clock). 3. To uniquely identify a message or sequence of messages (protocol instance), nonces drawn from a monotonically increasing sequence may be used (e.g., sequence or se- rial numbers, and timestamps, if guaranteed to be increasing and unique), or random numbers of sufﬁcient size. Uniqueness is often required only within a given key life- time or time window. 4. Combinations of time-variant parameters may be used, e.g., random numbers con- catenated to timestamps or sequence numbers. This may guarantee that a pseudoran- dom number is not duplicated. (i) Random numbers Random numbers may be used in challenge-response mechanisms, to provide uniqueness and timeliness assurances, and to preclude certain replay and interleaving attacks (see§10.5, including Remark 10.42). Random numbers may also serve to provide unpredictability, for example, to preclude chosen-text attacks. The termrandom numbers, when used in the context of identiﬁcation and authentica- tion protocols, includes pseudorandom numbers which are unpredictable to an adversary (see Remark 10.11); this differs from randomness in the traditional statistical sense. In pro- tocol descriptions, “choose a random number” is usually intended to mean “pick a number with uniform distribution from a speciﬁed sample space” or “select from a uniform distri- bution”. Random numbers are used in challenge-response protocols as follows. One entity in- cludes a (new) random number in an outgoing message. An incoming message subsequen- tly received (e.g., the next protocol message of the same protocol instance), whose construc- tion required knowledge of this nonce and to which this nonce is inseparably bound, is then deemed to befresh(Remark 10.10) based on the reasoning that the random number links the two messages. The non-tamperable binding is required to prevent appending a nonce to an old message. Random numbers used in this manner serve to ﬁx a relative point in time for the parties involved, analogous to a shared timeclock. The maximum allowable time between protocol messages is typically constrained by atimeout period, enforced using local, independent countdown timers. 4Such predictable parameters differ from sequence numbers in that they might not be bound to any stored state. Without appropriate cryptographic binding, a potential concern then is a pre-play attack wherein an adversary obtains the response before the time-variant parameter is legitimately sent (see Note 10.7). §10.3 Challenge-response identiﬁcation (strong authentication) 399 10.10 Remark (freshness) In the context of challenge-response protocols,freshtypically means recent, in the sense of having originated subsequent to the beginning of the current protocol instance. Note that such freshness alone does not rule out interleaving attacks using parallel sessions (see§10.5). 10.11 Remark (birthday repetitions in random numbers) In generating pseudorandom numbers for use as time-variant parameters, it sufﬁces if the probability of a repeated number is ac- ceptably low and if numbers are not intentionally reused. This may be achieved by selecting the random value from a sufﬁciently large sample space, taking into account coincidences arising from the birthday paradox. The latter may be addressed by either using a larger sam- ple space, or by using a generation process guaranteed to avoid repetition (e.g., a bijection), such as using the counter or OFB mode of a block cipher (§7.2.2). 10.12 Remark (disadvantages of random numbers) Many protocols involving random numbers require the generation of cryptographically secure (i.e., unpredictable) random numbers. If pseudorandom number generators are used, an initial seed with sufﬁcient entropy is re- quired. When random numbers are used in challenge-response mechanisms in place of timestamps, typically the protocol involves one additional message, and the challenger must temporarily maintain state information, but only until the response is veriﬁed. (ii) Sequence numbers A sequence number (serial number, or counter value) serves as a unique number identify- ing a message, and is typically used to detect message replay. For stored ﬁles, sequence numbers may serve asversion numbersfor the ﬁle in question. Sequence numbers are spe- ciﬁc to a particular pair of entities, and must explicitly or implicitly be associated with both the originator and recipient of a message; distinct sequences are customarily necessary for messages fromAtoBand fromBtoA. Parties follow a pre-deﬁned policy for message numbering. A message is accepted only if the sequence number therein has not been used previously (or not used previously within a speciﬁed time period), and satisﬁes the agreed policy. The simplest policy is that a se- quence number starts at zero, is incremented sequentially, and each successive message has a number one greater than the previous one received. A less restrictive policy is that sequence numbers need (only) be monotonically increasing; this allows for lost messages due to non-malicious communications errors, but precludes detection of messages lost due to adversarial intervention. 10.13 Remark (disadvantages of sequence numbers) Use of sequence numbers requires an over- head as follows: each claimant must record and maintain long-term pairwise state infor- mation for each possible veriﬁer, sufﬁcient to determine previously used and/or still valid sequence numbers. Special procedures (e.g., for resetting sequence numbers) may be neces- sary following circumstances disrupting normal sequencing (e.g., system failures). Forced delays are not detectable in general. As a consequence of the overhead and synchronization necessary, sequence numbers are most appropriate for smaller, closed groups. (iii) Timestamps Timestamps may be used to provide timeliness and uniqueness guarantees, to detect mes- sage replay. They may also be used to implement time-limited access privileges, and to detect forced delays. 400 Ch. 10 Identiﬁcation and Entity Authentication Timestamps function as follows. The party originating a message obtains a timestamp from its local (host) clock, and cryptographically binds it to a message. Upon receiving a time-stamped message, the second party obtains the current time from its own (host) clock, and subtracts the timestamp received. The received message is valid provided: 1. the timestamp difference is within theacceptance window(a ﬁxed-size time interval, e.g., 10 milliseconds or 20 seconds, selected to account for the maximum message transit and processing time, plus clock skew); and 2. (optionally) no message with an identical timestamp has been previously received from the same originator. This check may be made by the veriﬁer maintaining a list of all timestamps received from each source entity within the current acceptance win- dow. Another method is to record the latest (valid) timestamp used by each source (in this case the veriﬁer accepts only strictly increasing time values). The security of timestamp-based veriﬁcation relies on use of a common time reference. This requires that host clocks be available and both “loosely synchronized” and secured from modiﬁcation. Synchronization is necessary to counter clock drift, and must be appro- priate to accommodate the acceptance window used. The degree of clock skew allowed, and the acceptance window, must be appropriately small to preclude message replay if the above optional check is omitted. The timeclock must be secure to prevent adversarial re- setting of a clock backwards so as to restore the validity of old messages, or setting a clock forward to prepare a message for some future point in time (cf. Note 10.7). 10.14 Remark (disadvantages of timestamps) Timestamp-based protocols require that time- clocks be both synchronized and secured. The preclusion of adversarial modiﬁcation of local timeclocks is difﬁcult to guarantee in many distributed environments; in this case, the security provided must be carefully re-evaluated. Maintaining lists of used timestamps within the current window has the drawback of a potentially large storage requirement, and corresponding veriﬁcation overhead. While technical solutions exist for synchronizing dis- tributed clocks, if synchronization is accomplished via network protocols, such protocols themselves must be secure, which typically requires authentication; this leads to a circular security argument if such authentication is itself timestamp-based. 10.15 Remark (comparison of time-variant parameters) Timestamps in protocols offer the ad- vantage of fewer messages (typically by one), and no requirement to maintain pairwise long-term state information (cf. sequence numbers) or per-connection short-term state in- formation (cf. random numbers). Minimizing state information is particularly important for servers in client-server applications. The main drawback of timestamps is the requirement of maintaining secure, synchronized distributed timeclocks. Timestamps in protocols may typically be replaced by a random number challenge plus a return message. 10.3.2 Challenge-response by symmetric-key techniques Challenge-response mechanisms based on symmetric-key techniques require the claimant and the veriﬁer to share a symmetric key. For closed systems with a small number of users, each pair of users may share a key a priori; in larger systems employing symmetric-key techniques, identiﬁcation protocols often involve the use of a trusted on-line server with which each party shares a key. The on-line server effectively acts like the hub of a spoked wheel, providing a common session key to two parties each time one requests authentication with the other. §10.3 Challenge-response identiﬁcation (strong authentication) 401 The apparent simplicity of the techniques presented below and in§10.3.3 is misleading. The design of such techniques is intricate and the security is brittle; those presented have been carefully selected. (i) Challenge-response based on symmetric-key encryption Both the Kerberos protocol (Protocol 12.24) and the Needham-Schroeder shared-key pro- tocol (Protocol 12.26) provide entity authentication based on symmetric encryption and in- volve use of an on-line trusted third party. These are discussed in Chapter 12, as they addi- tionally provide key establishment. Below, three simple techniques based on ISO/IEC 9798-2 are described. They assume the prior existence of a shared secret key (and no further requirement for an on-line server). In this case, two parties may carry out unilateral entity authentication in one pass using timestamps or sequence numbers, or two passes using random numbers; mutual authen- tication requires, respectively, two and three passes. The claimant corroborates its identity by demonstrating knowledge of the shared key by encrypting a challenge (and possibly ad- ditional data) using the key. These techniques are similar to those given in§12.3.1. 10.16 Remark (data integrity) When encipherment is used in entity authentication protocols, data integrity must typically also be guaranteed to ensure security. For example, for mes- sages spanning more than one block, the rearrangement of ciphertext blocks cannot be de- tected in the ECB mode of block encryption, and even CBC encryption may provide only a partial solution. Such data integrity should be provided through use of an accepted data integrity mechanism (see§9.6; cf. Remark 12.19). 9798-2 mechanisms: Regarding notation:r Aand tA, respectively, denote a random num- ber and a timestamp, generated byA. (In these mechanisms, the timestamptAmay be re- placed by a sequence numbernA, providing slightly different guarantees.)EK denotes a symmetric encryption algorithm, with a keyKshared byAand B; alternatively, distinct keysKABandKBAmay be used for unidirectional communication. It is assumed that both parties are aware of the claimed identity of the other, either by context or by additional (un- secured) cleartext data ﬁelds. Optional message ﬁelds are denoted by an asterisk (), while a comma (,) within the scope ofEKdenotes concatenation. 1. unilateral authentication, timestamp-based: A→B:EK(tA,B∗)( 1 ) Upon reception and decryption,Bveriﬁes that the timestamp is acceptable, and op- tionally veriﬁes the received identiﬁer as its own. The identiﬁerBhere prevents an adversary from re-using the message immediately onA, in the case that a single bi- directional keyKis used. 2. unilateral authentication, using random numbers: To avoid reliance on timestamps, the timestamp may be replaced by a random num- ber, at the cost of an additional message: A←B:rB (1) A→B:EK(rB,B∗)( 2 ) Bdecrypts the received message and checks that the random number matches that sent in (1). Optionally,Bchecks that the identiﬁer in (2) is its own; this prevents a reﬂection attack in the case of a bi-directional keyK. To prevent chosen-text at- tacks on the encryption schemeEK,Amay (as below) embed an additional random number in (2) or, alternately, the form of the challenges can be restricted; the critical requirement is that they be non-repeating. 402 Ch. 10 Identiﬁcation and Entity Authentication 3. mutual authentication, using random numbers: A←B:rB (1) A→B:EK(rA,rB,B∗)( 2 ) A←B:EK(rB,rA)( 3 ) Upon reception of (2),Bcarries out the checks as above and, in addition, recovers the decryptedrAfor inclusion in (3). Upon decrypting (3),Achecks that both random numbers match those used earlier. The second random numberrAin (2) serves both as a challenge and to prevent chosen-text attacks. 10.17 Remark (doubling unilateral authentication) While mutual authentication may be obtain- ed by running any of the above unilateral authentication mechanisms twice (once in each direction), such an ad-hoc combination suffers the drawback that the two unilateral authen- tications, not being linked, cannot logically be associated with a single protocol run. (ii) Challenge-response based on (keyed) one-way functions The encryption algorithm in the above mechanisms may be replaced by a one-way or non- reversible function of the shared key and challenge, e.g., having properties similar to a MAC (Deﬁnition 9.7). This may be preferable in situations where encryption algorithms are oth- erwise unavailable or undesirable (e.g., due to export restrictions or computational costs). The modiﬁcations required to the 9798-2 mechanisms above (yielding the analogous mech- anisms of ISO/IEC 9798-4) are the following: 1. the encryption functionE Kis replaced by a MAC algorithmhK; 2. rather than decrypting and verifying that ﬁelds match, the recipient now indepen- dently computes the MAC value from known quantities, and accepts if the computed MAC matches the received MAC value; and 3. to enable independent MAC computation by the recipient, the additional cleartext ﬁeldt Amust be sent in message (1) of the one-pass mechanism.rAmust be sent as an additional cleartext ﬁeld in message (2) of the three-pass mechanism. The revised three-pass challenge-response mechanism based on a MAChK, with ac- tions as noted above, provides mutual identiﬁcation. Essentially the same protocol, called SKID3 , has messages as follows: A←B: r B (1) A→B: rA,hK(rA,rB,B)( 2 ) A←B: hK(rB,rA,A)( 3 ) Note that the additional ﬁeldAis included in message (3). The protocolSKID2 , obtained by omitting the third message, provides unilateral entity authentication. (iii) Implementation using hand-held passcode generators Answering a challenge in challenge-response protocols requires some type of computing device and secure storage for long-term keying material (e.g., a ﬁle on a trusted local disk, perhaps secured under a local password-derived key). For additional security, a device such as a chipcard (and corresponding card reader) may be used for both the key storage and response computation. In some cases, a less expensive option is a passcode generator. Passcode generatorsare hand-held devices, resembling thin calculators in both size and display, and which provide time-variant passwords orpasscodes(see Figure 10.3). The generator contains a device-speciﬁc secret key. When a user is presented with a challenge (e.g., by a system displaying it on a computer terminal), the challenge is keyed into the gen- erator. The generator displays a passcode, computed as a function of the secret key and the §10.3 Challenge-response identiﬁcation (strong authentication) 403 challenge; this may be either an asymmetric function, or a symmetric function (e.g., encryp- tion or MAC as discussed above). The user returns the response (e.g., keys the passcode in at his terminal), which the system veriﬁes by comparison to an independently computed response, using the same information stored on the system side. For further protection against misplaced generators, the response may also depend on a user-entered PIN. Simpler passcode generators omit the user keypad, and use as an implicit challenge a time value (with a typical granularity of one minute) deﬁned by a timeclock loosely synchronized automatically between the system and the passcode generator. A more sophisticated device combines implicit synchronization with explicit challenges, presenting an explicit challenge only when synchronization is lost. A drawback of systems using passcode generators is, as per§10.2.1(i), the requirement to provide conﬁdentiality for user passwords stored on the system side. passcode generator PIN (optional) sA f display (response) y (challenge) A A (user) B (system) A (optional) REJECT secret database sA PINA ee no (login request) sA PINA user-entered yes f = challenge generator ACCEPT Figure 10.3:Functional diagram of a hand-held passcode generator.sA isA’ s user-speciﬁc secret. f is a one-way function. The (optional) PIN could alternatively be locally veriﬁed in the passcode generator only, makingy independent of it. 10.3.3 Challenge-response by public-key techniques Public-key techniques may be used for challenge-response based identiﬁcation, with a claimant demonstrating knowledge of its private key in one of two ways (cf.§12.5): 1. the claimant decrypts a challenge encrypted under its public key; 2. the claimant digitally signs a challenge. Ideally, the public-key pair used in such mechanisms should not be used for other pur- poses, since combined usage may compromise security (Remark 10.40). A second caution is that the public-key system used should not be susceptible to chosen-ciphertext attacks,5 5Both chosen-ciphertext and chosen-plaintext attacks are of concern for challenge-response techniques based on symmetric-key encryption. 404 Ch. 10 Identiﬁcation and Entity Authentication as an adversary may attempt to extract information by impersonating a veriﬁer and choos- ing strategic rather than random challenges. (See Notes 8.13 and 8.58 regarding the Ra- bin/Williams and Blum-Goldwasser schemes.) Incorporating a self-generated random number orconfounder(§10.5) into the data over which the response is computed may address both of these concerns. Such data may be made available to the veriﬁer in cleartext to allow veriﬁcation. (i) Challenge-response based on public-key decryption Identiﬁcation based on PK decryption and witness.Consider the following protocol: A←B: h(r),B,PA(r,B)( 1 ) A→B: r (2) Bchooses a randomr, computes thewitnessx=h(r)(xdemonstrates knowledge ofr without disclosing it – cf.§10.4.1), and computes the challengee=PA(r,B). HerePA denotes the public-key encryption (e.g., RSA) algorithm ofA, andhdenotes a one-way hash function.Bsends (1) toA.Adecryptseto recoverr′and B′, computesx′=h(r′), and quits ifx′̸=x(implyingr′̸=r)o ri fB′is not equal to its own identiﬁerB. Otherwise, Asendsr=r′toB.Bsucceeds with (unilateral) entity authentication ofAupon verify- ing the receivedragrees with that sent earlier. The use of the witness precludes chosen-text attacks. Modiﬁed Needham-Schroeder PK protocol for identiﬁcation.The modiﬁed Needham-Schr- oeder public-key protocol of Note 12.39 provides key transport of distinct keysk1,k2 from AtoBand BtoA, respectively, as well as mutual authentication. If the key establishment feature is not required,k1 andk2 may be omitted. WithPBdenoting the public-key encryp- tion algorithm forB(e.g., RSA), the messages in the modiﬁed protocol for identiﬁcation are then as follows: A→B: PB(r1,A)( 1 ) A←B: PA(r1,r2)( 2 ) A→B: r2 (3) Veriﬁcation actions are analogous to those of Note 12.39. (ii) Challenge-response based on digital signatures X.509 mechanisms based on digital signatures.The ITU-T (formerly CCITT) X.509 two- and three-way strong authentication protocols specify identiﬁcation techniques based on digital signatures and, respectively, timestamps and random number challenges. These are described in§12.5.2, and optionally provide key establishment in addition to entity authen- tication. 9798-3 mechanisms.Three challenge-response identiﬁcation mechanisms based on signa- tures are given below, analogous to those in§10.3.2(i) based on symmetric-key encryption, but, in this case, corresponding to techniques in ISO/IEC 9798-3. Regarding notation (cf. 9798-2 above):rAand tA, respectively, denote a random number and timestamp generated by A.SAdenotesA’s signature mechanism; if this mechanism provides message recovery, some of the cleartext ﬁelds listed below are redundant and may be omitted.certAdenotes the public-key certiﬁcate containingA’s signature public key. (In these mechanisms, if the veriﬁer has the authentic public key of the claimant a priori, certiﬁcates may be omitted; otherwise, it is assumed that the veriﬁer has appropriate information to verify the validity of the public key contained in a received certiﬁcate – see Chapter 13.) Remark 10.17 also applies here. §10.4 Customized and zero-knowledge identiﬁcation protocols 405 1. unilateral authentication with timestamps: A→B:certA,tA,B,SA(tA,B)( 1 ) Upon reception,Bveriﬁes that the timestamp is acceptable, the received identiﬁerB is its own, and (usingA’s public key extracted fromcertAafter verifying the latter) checks that the signature over these two ﬁelds is correct. 2. unilateral authentication with random numbers: Reliance on timestamps may be re- placed by a random number, at the cost of an additional message: A←B:rB (1) A→B:certA,rA,B,SA(rA,rB,B)( 2 ) Bveriﬁes that the cleartext identiﬁer is its own, and using a valid signature public key forA(e.g., fromcertA), veriﬁes thatA’s signature is valid over the cleartext random number rA, the same numberrB as sent in (1), and this identiﬁer. The signedrA explicitly prevents chosen-text attacks. 3. mutual authentication with random numbers: A←B:rB (1) A→B:certA,rA,B,SA(rA,rB,B)( 2 ) A←B:certB,A,SB(rB,rA,A)( 3 ) Processing of (1) and (2) is as above; (3) is processed analogously to (2). 10.4 Customized and zero-knowledge identiﬁcation protocols This section considers protocols speciﬁcally designed to achieve identiﬁcation, which use asymmetric techniques but do not rely on digital signatures or public-key encryption, and which avoid use of block ciphers, sequence numbers, and timestamps. They are similar in some regards to the challenge-response protocols of§10.3, but are based on the ideas of interactive proof systems and zero-knowledge proofs (see§10.4.1), employing random numbers not only as challenges, but also ascommitmentsto prevent cheating. 10.4.1 Overview of zero-knowledge concepts A disadvantage of simple password protocols is that when a claimantA(called aproverin the context of zero-knowledge protocols) gives the veriﬁerBher password,Bcan there- after impersonateA. Challenge-response protocols improve on this:Aresponds toB’s challenge to demonstrate knowledge ofA’s secret in a time-variant manner, providing in- formation not directly reusable byB. This might nonetheless reveal some partial informa- tion about the claimant’s secret; an adversarial veriﬁer might also be able to strategically select challenges to obtain responses providing such information (see chosen-text attacks, §10.5). Zero-knowledge (ZK) protocols are designed to address these concerns, by allowing a prover to demonstrate knowledge of a secret while revealing no information whatsoever (beyond what the veriﬁer was able to deduce prior to the protocol run) of use to the veriﬁer 406 Ch. 10 Identiﬁcation and Entity Authentication in conveying this demonstration of knowledge to others. The point is that only a single bit of information need be conveyed – namely, that the prover actually does know the secret. More generally, a zero-knowledge protocol allows a proof of the truth of an assertion, while conveying no information whatsoever (this notion can be quantiﬁed in a rigorous sense) about the assertion itself other than its actual truth. In this sense, a zero-knowledge proof is similar to an answer obtained from a (trusted)oracle. (i) Interactive proof systems and zero-knowledge protocols The ZK protocols to be discussed are instances ofinteractive proof systems, wherein a prov- er and veriﬁer exchange multiple messages (challenges and responses), typically dependent on random numbers (ideally: the outcomes of fair coin tosses) which they may keep secret. The prover’s objective is to convince (proveto) the veriﬁer the truth of an assertion, e.g., claimed knowledge of a secret. The veriﬁer either accepts or rejects theproof. The tradi- tional mathematical notion of a proof, however, is altered to an interactive game wherein proofs areprobabilisticrather than absolute; a proof in this context need be correct only with bounded probability, albeit possibly arbitrarily close to 1. For this reason, an interac- tive proof is sometimes called aproof by protocol. Interactive proofs used for identiﬁcation may be formulated as proofs of knowledge. Apossesses some secrets, and attempts to convinceBit hasknowledgeofsby correctly responding to queries (involving publicly known inputs and agreed upon functions) which require knowledge ofsto answer. Note that proving knowledge ofsdiffers from proving that suchsexists – for example, proving knowledge of the prime factors ofndiffers from proving thatnis composite. An interactive proof is said to be aproof of knowledgeif it has both the properties of completeness and soundness. Completeness may be viewed as the customary requirement that a protocol functions properly given honest participants. 10.18 Deﬁnition(completeness property) An interactive proof (protocol) iscompleteif, given an honest prover and an honest veriﬁer, the protocol succeeds with overwhelming probabil- ity (i.e., the veriﬁer accepts the prover’s claim). The deﬁnition ofoverwhelmingdepends on the application, but generally implies that the probability of failure is not of practical signiﬁcance. 10.19 Deﬁnition(soundness property) An interactive proof (protocol) issoundif there exists an expected polynomial-time algorithmMwith the following property: if a dishonest prover (impersonatingA) can with non-negligible probability successfully execute the protocol withB, thenMcan be used to extract from this prover knowledge (essentially equivalent toA’s secret) which with overwhelming probability allows successful subsequent protocol executions. An alternate explanation of the condition in Deﬁnition 10.19 is as follows: the prover’s se- cretstogether with public data satisﬁes some polynomial-time predicate, and another so- lution of this predicate (possibly the same) can be extracted, allowing successful execution of subsequent protocol instances. Since any party capable of impersonatingAmust know the equivalent ofA’s secret knowledge (Mcan be used to extract it from this party in polynomial time), soundness guar- antees that the protocol does indeed provide a proof of knowledge – knowledge equivalent to that being queried is required to succeed. Soundness thus prevents a dishonest prover from convincing an honest veriﬁer (but does does not by itself guarantee that acquiring the §10.4 Customized and zero-knowledge identiﬁcation protocols 407 prover’s secret is difﬁcult; see Remark 10.23). A standard method to establish the sound- ness of a particular protocol is to assume the existence of a dishonest prover capable of suc- cessfully executing the protocol, and show how this allows one to compute the real prover’s secret. While an interactive proof of knowledge (or protocol based thereon) must be sound to be of cryptographic use, the main property of zero-knowledge protocols is the zero- knowledge aspect itself. For what follows, deﬁne atranscript(or view) to be the collection of messages resulting from protocol execution. 10.20 Deﬁnition(zero-knowledge property) A protocol which is a proof of knowledge has the zero-knowledge propertyif it is simulatable in the following sense: there exists an expected polynomial-time algorithm (simulator) which can produce, upon input of the assertion(s) to be proven but without interacting with the real prover, transcripts indistinguishable from those resulting from interaction with the real prover. The zero-knowledge property implies that a prover executing the protocol (even when in- teracting with a malicious veriﬁer) does not release any information (about its secret knowl- edge, other than that the particular assertion itself is true) not otherwise computable in polynomial time from public information alone. Thus, participation does not increase the chances of subsequent impersonation. 10.21 Remark (simulated ZK protocols and protocol observers) Consider an observerCwho witnesses a zero-knowledge interactive proof (ZKIP) involving a proverAconvincing a veriﬁerB(B ̸=C) of some knowledgeAhas. The “proof” toBdoes not provide any guarantees toC. (Indeed,Aand Bmight have a prior agreement, conspiring againstC, on the challenges to be issued.) Similarly, a recorded ZKIP conveys no guarantees upon playback. This is fundamental to the idea of the zero-knowledge property and the condition that proofs be simulatable by a veriﬁer alone. Interactive proofs convey knowledge only to (interactive) veriﬁers able to select their own random challenges. 10.22 Deﬁnition(computational vs. perfect zero-knowledge) A protocol iscomputationally zero-knowledge if an observer restricted to probabilistic polynomial-time tests cannot dis- tinguish real from simulated transcripts. Forperfectzero-knowledge, the probability dis- tributions of the transcripts must be identical. By convention, when not further qualiﬁed, zero-knowledgemeans computational zero-knowledge. In the case of computational zero-knowledge, real and simulated transcripts are said to bepolynomially indistinguishable(indistinguishable using polynomial-time algorithms). Any information extracted by a veriﬁer through interaction with a prover provides no ad- vantage to the veriﬁer within polynomial time. 10.23 Remark (ZK property and soundness vs. security) The zero-knowledge property (Deﬁni- tion 10.20) does not guarantee that a protocol is secure (i.e., that the probability of it being easily defeated is negligible). Similarly, the soundness property (Deﬁnition 10.19) does not guarantee that a protocol is secure. Neither property has much value unless the underlying problem faced by an adversary is computationally hard. (ii) Comments on zero-knowledge vs. other asymmetric protocols The following observations may be made regarding zero-knowledge (ZK) techniques, as compared with other public-key (PK) techniques. 408 Ch. 10 Identiﬁcation and Entity Authentication 1. no degradation with usage: protocols proven to have the ZK property do not suffer degradation of security with repeated use, and resist chosen-text attacks. This is per- haps the most appealing practical feature of ZK techniques. A ZK technique which is not provably secure may or may not be viewed as more desirable than a PK technique which is provably secure (e.g., as difﬁcult as factoring). 2. encryption avoided: many ZK techniques avoid use of explicit encryption algo- rithms. This may offer political advantages (e.g., with respect to export controls). 3. efﬁciency: while some ZK-based techniques are extremely efﬁcient (see§10.4.5), protocols which formally have the zero-knowledge property typically have higher communications and/or computational overheads than PK protocols which do not. The computational efﬁciency of the more practical ZK-based schemes arises from their nature as interactive proofs, rather than their zero-knowledge aspect. 4. unproven assumptions: many ZK protocols (“proofs of knowledge”) themselves rely on the same unproven assumptions as PK techniques (e.g., the intractability of fac- toring or quadratic residuosity). 5. ZK-based vs. ZK: although supported by prudent underlying principles, many tech- niques based on zero-knowledge concepts fall short of formally being zero-knowled- ge and/or formally sound in practice, due to parameter selection for reasons of ef- ﬁciency, or for other technical reasons (cf. Notes 10.33 and 10.38). In fact, many such concepts are asymptotic, and do not apply directly to practical protocols (Re- mark 10.34). (iii) Example of zero-knowledge proof: Fiat-Shamir identiﬁcation protocol The general idea of a zero-knowledge (ZK) proof is illustrated by the basic version of the Fiat-Shamir protocol. The basic version is presented here for historical and illustrative pur- poses (Protocol 10.24). In practice, one would use a more efﬁcient variation, such as Pro- tocol 10.26, with multiple “questions” per iteration rather than as here, whereBposes only a single one-bit challenge per iteration. The objective is forAto identify itself by proving knowledge of a secrets(associated withAthrough authentic public data) to any veriﬁerB, without revealing any information aboutsnot known or computable byBprior to execution of the protocol (see Note 10.25). The security relies on the difﬁculty of extracting square roots modulo large composite in- tegersnof unknown factorization, which is equivalent to that of factoringn(Fact 3.46). 10.24 ProtocolFiat-Shamir identiﬁcation protocol (basic version) SUMMARY: Aproves knowledge ofstoBintexecutions of a 3-pass protocol. 1. One-time setup. (a) A trusted centerTselects and publishes an RSA-like modulusn=pqbut keeps primespand qsecret. (b) Each claimantAselects a secretscoprime ton,1≤s≤n−1, computes v=s2 modn, and registersvwithTas its public key.6 2. Protocol messages. Each oftrounds has three messages with form as follows. A→B: x=r2 modn (1) A←B: e∈{0,1} (2) A→B: y=r·semodn (3) 6Technically,T should verify the conditiongcd(s, n)=1 or equivalentlygcd(v, n)=1 , for this to be a sound proof of knowledge; andB should stop with failure ifgcd(y, n) ̸=1, wherey isA’s response in the third message. But either condition failing would allow the factorization ofn, violating the assumption thatn cannot be factored. §10.4 Customized and zero-knowledge identiﬁcation protocols 409 3. Protocol actions. The following steps are iteratedttimes (sequentially and indepen- dently).Baccepts the proof if alltrounds succeed. (a)Achooses a random (commitment)r,1≤r≤n−1, and sends (thewitness) x=r2 modntoB. (b) Brandomly selects a (challenge) bite=0 ore=1, and sendsetoA. (c)Acomputes and sends toB(theresponse)y, eithery=r(ife=0)o ry= rsmodn(ife=1). (d) Brejects the proof ify=0, and otherwise accepts upon verifyingy2 ≡x·ve (modn). (Depending one,y2 =xory2 =xvmodn, sincev=s2 modn. Note that checking fory=0 precludes the caser=0.) Protocol 10.24 may be explained and informally justiﬁed as follows. The challenge (or exam)erequires thatAbe capable of answering two questions, one of which demonstrates her knowledge of the secrets, and the other an easy question (for honest provers) to prevent cheating. An adversary impersonatingAmight try to cheat by selecting anyrand setting x=r2/v, then answering the challengee=1 with a “correct” answery=r; but would be unable to answer the exame=0 which requires knowing a square root ofxmod n. A proverAknowing scan answer both questions, but otherwise can at best answer one of the two questions, and so has probability only1/2of escaping detection. To decrease the probability of cheating arbitrarily to an acceptably small value of2−t(e.g.,t=20 or t=40), the protocol is iteratedttimes, withBacceptingA’s identity only if alltquestions (overtrounds) are successfully answered. 10.25 Note (secret information revealed byA) The responsey=ris independent ofA’s secrets, while the responsey=rsmodnalso provides no information aboutsbecause the random ris unknown toB. Information pairs(x,y)extracted fromAcould equally well be simu- lated by a veriﬁerBalone by choosingyrandomly, then deﬁningx=y2 ory2/vmodn. While this is not the method by whichAwould construct such pairs, such pairs(x,y)have a probability distribution which is indistinguishable from thoseAwould produce; this es- tablishes the zero-knowledge property. Despite the ability to simulate proofs,Bis unable to impersonateAbecauseBcannot predict the real-time challenges. As a minor technical point, however, the protocol does reveal a bit of information: the answery=rsprovides supporting evidence thatvis indeed a square modulon, and the soundness of the protocol allows one to conclude, aftertsuccessful iterations, that this is indeed the case. (iv) General structure of zero-knowledge protocols Protocol 10.24 illustrates the general structure of a large class of three-move zero-knowl- edge protocols: A→B: witness A←B: challenge A→B: response The prover claiming to beAselects a random element from a pre-deﬁned set as its secret commitment (providing hidden randomization or “private coin tosses”), and from this com- putes an associated (public) witness. This provides initial randomness for variation from other protocol runs, and essentially deﬁnes a set of questions all of which the prover claims to be able to answer, thereby a priori constraining her forthcoming response. By protocol design, only the legitimate partyA, with knowledge ofA’s secret, is truly capable of an- swering all the questions, and the answer to any one of these provides no information about 410 Ch. 10 Identiﬁcation and Entity Authentication A’s long-term secret.B’s subsequent challenge selects one of these questions.Aprovides its response, whichBchecks for correctness. The protocol is iterated, if necessary, to im- prove the bound limiting the probability of successful cheating. Zero-knowledge interactive protocols thus combine the ideas ofcut-and-choosepro- tocols (this terminology results from the standard method by which two children share a piece of cake: one cuts, the other chooses) and challenge-response protocols.Aresponds to at most one challenge (question) for a given witness, and should not reuse any witness; in many protocols, security (possibly of long-term keying material) may be compromised if either of these conditions is violated. 10.4.2 Feige-Fiat-Shamir identiﬁcation protocol The basic version of the Fiat-Shamir protocol is presented as Protocol 10.24. This can be generalized, and the Feige-Fiat-Shamir (FSS) identiﬁcation protocol (Protocol 10.26) is a minor variation of such a generalization. The FFS protocol involves an entity identifying itself by proving knowledge of a secret using a zero-knowledge proof; the protocol reveals no partial information whatsoever regarding the secret identiﬁcation value(s) ofA(cf. Def- inition 10.20). It requires limited computation (a small fraction of that required by RSA – see§10.4.5), and is thus well-suited for applications with low-power processors (e.g., 8-bit chipcard microprocessors). 10.26 ProtocolFeige-Fiat-Shamir identiﬁcation protocol SUMMARY: Aproves its identity toBintexecutions of a 3-pass protocol. 1. Selection of system parameters. A trusted centerTpublishes the common modulus n = pqfor all users, after selecting two secret primespand qeach congruent to 3mod4 , and such thatnis computationally infeasible to factor. (Consequently,n is a Blum integer per§2.4.6, and−1is a quadratic non-residue modnwith Jacobi symbol +1.) Integerskand tare deﬁned as security parameters (see Note 10.28). 2. Selection of per-entity secrets. Each entityAdoes the following. (a) Selectkrandom integerss1,s2,...,s k in the range1≤si ≤n−1, andk random bitsb1,...,b k. (For technical reasons,gcd(si,n)=1 is required, but is almost surely guaranteed as its failure allows factorization ofn.) (b) Computevi=(−1)bi ·(s2 i)−1 modnfor1≤i≤k. (This allowsvito range over all integers coprime tonwith Jacobi symbol+1, a technical condition re- quired to prove that no secret information is “leaked”; by choice ofn, precisely one signed choice forvihas a square root.) (c)Aidentiﬁes itself by non-cryptographic means (e.g., photo id) toT, which thereafter registersA’s public key(v1,...,v k;n), while onlyAknows its pri- vate key(s1,...,s k)and n. (To guarantee the bounded probability of attack speciﬁed per Note 10.28,Tmay conﬁrm that eachviindeed does have Jacobi symbol +1 relative ton.) This completes the one-time set-up phase. 3. Protocol messages. Each oftrounds has three messages with form as follows. A→B: x(=±r2 modn)( 1 ) A←B:( e1,...,e k),ei∈{0,1} (2) A→B: y(=r·∏ ej =1 sj modn)( 3 ) 4. Protocol actions. The following steps are executedttimes;BacceptsA’s identity if alltrounds succeed. AssumeBhasA’s authentic public key(v1,...,v k;n); other- wise, a certiﬁcate may be sent in message (1), and used as in Protocol 10.36. §10.4 Customized and zero-knowledge identiﬁcation protocols 411 (a)Achooses a random integerr,1≤r≤n−1, and a random bitb; computes x=(−1)b·r2 modn; and sendsx(thewitness)t oB. (b) Bsends toA(thechallenge,) a randomk-bit vector(e1,...,e k). (c)Acomputes and sends toB(theresponse):y=r·∏k j=1 sej j modn(the prod- uct ofrand thosesjspeciﬁed by the challenge). (d) Bcomputesz=y2 ·∏k j=1 vej j modn, and veriﬁes thatz=±xand z̸=0. (The latter precludes an adversary succeeding by choosingr=0.) 10.27 Example (Feige-Fiat-Shamir protocol with artiﬁcially small parameters) 1. The trusted centerTselects the primesp=683,q=811, and publishesn=pq= 553913. Integersk=3 and t=1 are deﬁned as security parameters. 2. EntityAdoes the following. (a) Selects 3 random integerss1 =157,s2 =43215,s3 =4646, and 3 bitsb1 =1, b2 =0,b3 =1. (b) Computesv1 =441845,v2 =338402, andv3 =124423. (c)A’s public key is(441845,338402,124423;553913)and private key is(157, 43215,4646). 3. See Protocol 10.26 for a summary of the messages exchanged. 4. (a) Achoosesr=1279,b=1, computesx=25898, and sends this toB. (b) Bsends toAthe 3-bit vector(0,0,1). (c)Acomputes and sends toBy =r·s3 modn=403104. (d) Bcomputesz=y2·v3 modn=25898and acceptsA’s identity sincez=+x and z̸=0. □ 10.28 Note (security of Feige-Fiat-Shamir identiﬁcation protocol) (i)probability of forgery.Protocol 10.26 is provably secure against chosen message at- tack in the following sense: provided that factoringnis difﬁcult, the best attack has a probability2−ktof successful impersonation. (ii)security assumption required.The security relies on the difﬁculty of extracting square roots modulo large composite integersnof unknown factorization. This is equivalent to that of factoringn(see Fact 3.46). (iii)zero-knowledge and soundness.The protocol is, relative to a trusted server, a (sound) zero-knowledge proof of knowledge providedk=O(loglogn)and t=Θ(log n). See Remark 10.34 regarding the practical signiﬁcance of such constraints. A simplis- tic view for ﬁxedkis that the veriﬁer, interested in soundness, favors largert(more iterations) for a decreased probability of fraud; while the prover, interested in zero- knowledge, favors smallert. (iv)parameter selection.Choosingkand tsuch thatkt =2 0allowsa1i na million chance of impersonation, which sufﬁces in the case that an identiﬁcation attempt re- quires a personal appearance by a would-be impersonator (see§10.5). Computation, memory, and communication can be traded off;1≤k≤18was originally suggested as appropriate. Speciﬁc parameter choices might be, for security2 −20:k=5,t=4; for2−30:k=6,t=5. (v) security trade-off.Both computation and communication may be reduced by trading off security parameters to yield a single iteration (t =1 ), holding the productkt constant and increasingkwhile decreasingt; however, in this case the protocol is no longer a zero-knowledge proof of knowledge. 412 Ch. 10 Identiﬁcation and Entity Authentication 10.29 Note (modiﬁcations to Feige-Fiat-Shamir) (i) As an alternative to step 1 of Protocol 10.26, each user may pick its own such modulus n.Tis still needed to associate each user with its modulus. (ii) The communication complexity can be reduced ifAsendsB(e.g., 128 bits of) a hash valueh(x)instead ofxin message (1), withB’s veriﬁcation modiﬁed accordingly. (iii) The scheme can be madeidentity-basedas follows (cf.§13.4.3).Tassigns a disting- uished identifying stringIAto each partyA(e.g.,A’s name, address, or other infor- mation which a veriﬁer may wish to corroborate).A’s public valuesvi,1≤i≤k are then derived by bothTand other partiesBasvi =f(IA,i)using an appropri- ate functionf. Then the trusted center, knowing the factorization ofn, computes a square rootsiof eachviand gives these toA. As an example off, consider, for a randomly chosen but known valuec,f(IA,i)= IA +i+cmodn. Since a square root offi =f(IA,i)is required, anyfi with Jacobi symbol−1mod nmay be multiplied by a ﬁxed number with Jacobi symbol −1. A non-residuefiwith Jacobi+1may be either discarded (Amust then indicate toB, e.g., in message (3), which valuesiallow computation of thevj); or mapped to a residue via multiplication by−1, again with an indication toBof this to allow computation ofvj. Note that both cases for dealing with a non-residuefiwith Jacobi +1reveal some (non-useful) information. (iv) Theparallel versionof the protocol, in which each of three messages contains the respective data for alltrounds simultaneously, can be shown to be secure (it releases no “transferable information”), but for technical reasons loses the zero-knowledge property. Such parallel execution (as opposed tosequential iteration) in interactive proofs allows the probability of error (forgery) to be decreased without increasing the number of rounds. 10.30 Note (converting identiﬁcation to signature scheme) The following general technique may be used to convert an identiﬁcation scheme involving a witness-challenge-response sequen- ce to a signature scheme: replace the random challengeeof the veriﬁer by the one-way hashe=h(x\|\|m), of the concatenation of the witnessxand the messagemto be signed (h essentially plays the role of veriﬁer). As this converts an interactive identiﬁcation scheme to a non-interactive signature scheme, the bitsize of the challengeemust typically be increased to preclude off-line attacks on the hash function. 10.4.3 GQ identiﬁcation protocol The Guillou-Quisquater (GQ) identiﬁcation scheme (Protocol 10.31) is an extension of the Fiat-Shamir protocol. It allows a reduction in both the number of messages exchanged and memory requirements for user secrets and, like Fiat-Shamir, is suitable for applications in which the claimant has limited power and memory. It involves three messages between a claimantAwhose identity is to be corroborated, and a veriﬁerB. 10.31 ProtocolGQ identiﬁcation protocol SUMMARY: Aproves its identity (via knowledge ofsA)t oBin a 3-pass protocol. 1. Selection of system parameters. (a) An authorityT, trusted by all parties with respect to binding identities to public keys, selects secret random RSA-like primespand qyielding a modulusn= pq. (As for RSA, it must be computationally infeasible to factorn.) §10.4 Customized and zero-knowledge identiﬁcation protocols 413 (b) Tdeﬁnes a public exponentv≥3withgcd(v,φ)=1 whereφ=(p−1)(q−1), and computes its private exponents=v−1 modφ. (See Note 10.33.) (c) System parameters(v,n)are made available (with guaranteed authenticity) for all users. 2. Selection of per-user parameters. (a) Each entityAis given a unique identityIA, from which (theredundant iden- tity) JA =f(IA), satisfying1<JA <n, is derived using a known redun- dancy functionf. (See Note 10.35. Assuming that factoringnis difﬁcult im- pliesgcd(JA,φ)=1 .) (b) Tgives toAthe secret (accreditation data)sA=(JA)−smodn. 3. Protocol messages. Each oftrounds has three messages as follows (oftent=1). A→B: IA,x =rv modn (1) A←B: e(where1 ≤e≤v)( 2 ) A→B: y=r·sAemodn (3) 4. Protocol actions.Aproves its identity toBby texecutions of the following;Bac- cepts the identity only if alltexecutions are successful. (a)Aselects a random secret integerr(thecommitment),1≤r≤n−1, and computes (thewitness)x=rv modn. (b) Asends toBthe pair of integers(IA,x). (c)Bselects and sends toAa random integere(thechallenge),1≤e≤v. (d) Acomputes and sends toB(theresponse)y=r·sAemodn. (e)Breceivesy, constructsJAfrom IAusingf(see above), computesz=JA e· yvmodn, and acceptsA’s proof of identity if bothz=xand z̸=0. (The latter precludes an adversary succeeding by choosingr=0.) 10.32 Example (GQ identiﬁcation protocol with artiﬁcially small parameters andt=1) 1. (a) The authorityTselects primesp=569,q=739, and computesn=pq= 420491. (b) Tcomputesφ=(p−1)(q−1)=419184 , selectsv=54955, and computes s=v−1 modφ=233875. (c) System parameters(54955,420491)are made available for all users. 2. (a) Suppose thatA’s redundant identity isJA=34579. (b) Tgives toAthe accreditation datasA=(JA)−smodn=403154. 3. See Protocol 10.31 for a summary of the messages exchanged. 4. (a) Aselectsr=65446 and computesx=rv modn=89525. (b) Asends toBthe pair(IA,89525). (c)Bsends toAthe random challengee=38980. (d) Asendsy=r·sAemodn=83551 toB. (e)Bcomputesz=JA e·yvmodn=89525and acceptsA’s identity sincez=x and z̸=0. □ 10.33 Note (security of GQ identiﬁcation protocol) (i)probability of forgery.In Protocol 10.31,vdetermines the security level (cf. Fiat- Shamir wherev=2but there are many rounds); some values such asv=216+1may offer computational advantages. A fraudulent claimant can defeat the protocol with a1i nvchance by guessingecorrectly a priori (and then formingx=JA e·yvas the veriﬁer would). The recommended bitlength ofvthus depends on the environment under which attacks could be mounted (see§10.5). 414 Ch. 10 Identiﬁcation and Entity Authentication (ii)security assumption required.Extractingvth roots modulo the composite integern (i.e., solving the RSA problem –§3.3) appears necessary to defeat the protocol; this is no harder than factoringn(Fact 3.30), and appears computationally intractable with- out knowing the factors ofn. (iii)soundness.In practice, GQ witht=1 and ak-bit primevis often suggested. For generalized parameters(n,v,t), the probability of forgery isv−t.I fvis constant, then technically for soundness,tmust grow asymptotically faster thanloglogn. (For soundness,v−t =O(e−kt)must be smaller than inverse-polynomial inlogn; only polynomial security is provided if for a constantc,vt =O((logn)c). See also Re- mark 10.34.) (iv)zero-knowledge property.In opposition to the soundness requirement, for GQ to be zero-knowledge apparently requirestv=O((logn)c)for constantc, imposing an upper bound ontasymptotically: forvconstant,tmust be no larger than polynomial inlogn. 10.34 Remark (asymptotic concepts vs. practical protocols) The asymptotic conditions for soundness speciﬁed in Note 10.33 have little meaning in practice, e.g., because big-O nota- tion is not applicable once ﬁxed values are assigned to parameters. Indeed, zero-knowledge is a theoretical concept; while complexity-theoretic deﬁnitions offer guidance in selecting practical security parameters, their signiﬁcance diminishes when parameters are ﬁxed. Re- garding Note 10.33, ift =1 is viewed as the instantiation of a non-constant parameter (e.g., the iterated logarithm ofn), thent=1 will sufﬁce for all practical purposes; con- sidern=1024,t=⌈lg 4 n⌉=1. 10.35 Note (redundancy function for identity-based GQ) (i) The protocol as given is an identity-based version (cf. Note 10.29), whereA’s public key is reconstructed from identiﬁerIAsent in message (1). Alternatively, a certiﬁed public key may be used, distributed in a certiﬁcate as per Protocol 10.36. (ii) One example of the redundancy functionfis the redundancy mapping of the prepro- cessing stage of ISO/IEC 9796 (see§11.3.5). A second example is a single function value offas in Note 10.29, for an appropriate valuei. (iii) The purpose of the redundancy is to preclude an adversary computing false accredi- tation data corresponding to a plausible identity; this would be equivalent to forging a certiﬁcate in certiﬁcate-based schemes. 10.4.4 Schnorr identiﬁcation protocol The Schnorr identiﬁcation protocol is an alternative to the Fiat-Shamir and GQ protocols. Its security is based on the intractability of the discrete logarithm problem. The design al- lows pre-computation, reducing the real-time computation for the claimant to one multi- plication modulo a primeq; it is thus particularly suitable for claimants of limited com- putational ability. A further important computational efﬁciency results from the use of a subgroup of orderqof the multiplicative group of integers modulop, whereq\|(p−1); this also reduces the required number of transmitted bits. Finally, the protocol was designed to require only three passes, and a low communications bandwidth (e.g., compared to Fiat- Shamir). The basic idea is thatAproves knowledge of a secreta(without revealing it) in a time- variant manner (depending on a challengee), identifyingAthrough the association ofa with the public keyvviaA’s authenticated certiﬁcate. §10.4 Customized and zero-knowledge identiﬁcation protocols 415 10.36 ProtocolSchnorr identiﬁcation protocol SUMMARY: Aproves its identity toBin a 3-pass protocol. 1. Selection of system parameters. (a) A suitable primepis selected such thatp−1is divisible by another primeq. (Discrete logarithms modulopmust be computationally infeasible – see§3.6; e.g.,p≈21024,q≥2160.) (b) An elementβis chosen,1≤β≤p−1, having multiplicative orderq. (For example, forαa generator modp,β=α(p−1)/q modp; see Note 4.81.) (c) Each party obtains an authentic copy of the system parameters(p,q,β)and the veriﬁcation function (public key) of the trusted partyT, allowing veriﬁcation ofT’s signaturesST(m)on messagesm.(ST involves a suitable known hash function prior to signing, and may be any signature mechanism.) (d) A parametert(e.g.,t≥40),2t<q, is chosen (deﬁning a security level2t). 2. Selection of per-user parameters. (a) Each claimantAis given a unique identityIA. (b) Achooses a private keya,0≤a≤q−1, and computesv=β−amodp. (c)Aidentiﬁes itself by conventional means (e.g., passport) toT, transfersvtoT with integrity, and obtains a certiﬁcatecertA =(IA,v ,ST(IA,v))from T bindingIAwithv. 3. Protocol messages. The protocol involves three messages. A→B: certA,x =βrmodp (1) A←B: e(where1 ≤e≤2t<q)( 2 ) A→B: y=ae+rmodq (3) 4. Protocol actions.Aidentiﬁes itself to veriﬁerBas follows. (a)Achooses a randomr(thecommitment),1≤r≤q−1, computes (thewitness) x=βrmodp, and sends (1) toB. (b) BauthenticatesA’s public keyvby verifyingT’s signature oncertA, then sends toAa (never previously used) randome(thechallenge),1≤e≤2t. (c)Achecks1≤e≤2tand sendsB(theresponse)y=ae+rmodq. (d) Bcomputesz=βyvemodp, and acceptsA’s identity providedz=x. 10.37 Example (Schnorr identiﬁcation protocol with artiﬁcially small parameters) 1. (a) The primep=48731is selected, wherep−1is divisible by the primeq=443. (b) A generator mod 48731 isα=6;βis computed asα(p−1)/q modp=11444. (c) The system parameters are(48731,443,11444). (d) The parametert=8 is chosen. 2. (b) Achooses a private keya=357 and computesv=β−amodp=7355. 3. See Protocol 10.36 for a summary of the messages exchanged. 4. (a) Achoosesr=274 and sendsx=βr modp=37123 toB. (b) Bsends toAthe random challengee=129. (c)AsendsBthe numbery=ae+rmodq=255. (d) Bcomputesz=βyvemodp=37123 and accept’sA’s identity sincez=x. □ 416 Ch. 10 Identiﬁcation and Entity Authentication 10.38 Note (security of Schnorr identiﬁcation protocol) (i)probability of forgery. In Protocol 10.36,tmust be sufﬁciently large to make the probability2−tof correctly guessing the challengeenegligible.t=40,q≥22t = 280 was originally suggested in the case that a response is required within seconds (see§10.5); largerqmay be necessary to preclude time-memory trade-offs, andq≥ 2160 is recommended to preclude other off-line discrete log attacks. Correctly guess- ingeallows an adversary to impersonateAby choosing anyy, sendingx=βyve modptoBin (1), then sendingyin (3). (ii)soundness.It can be shown that the protocol is a proof of knowledge ofa, i.e., any party completing the protocol asAmust be capable of computinga. Informally, the protocol reveals “no useful information” aboutabecausexis a random number, andy is perturbed by the random numberr. (However, this does not prove that adversarial discovery ofais difﬁcult.) (iii)zero-knowledge property.The protocol is not zero-knowledge for largee, because through interaction,Bobtains the solution(x,y,e)to the equationx=βyvemodp, which Bitself might not be able to compute (e.g., ifewere chosen to depend onx). 10.39 Note (reducing transmission bandwidth) The number of bits transmitted in the protocol can be reduced by replacingxin message (1) bytpre-speciﬁed bits ofx(e.g., the least signiﬁcanttbits), and havingBcompare this totcorresponding bits ofz. 10.4.5 Comparison: Fiat-Shamir, GQ, and Schnorr The protocols of Feige-Fiat-Shamir, Guillou-Quisquater, and Schnorr all provide solutions to the identiﬁcation problem. Each has relative advantages and disadvantages with respect to various performance criteria and for speciﬁc applications. To compare the protocols, a typical set of selected parameters must be chosen for each providing comparable estimated security levels. The protocols may then be compared based on the following criteria: 1. communications: number of messages exchanged, and total bits transferred; 2. computations: number of modular multiplications for each of prover and veriﬁer (noting on-line and off-line computations); 3. memory : storage requirements for secret keys (and signature size, in the case of sig- nature schemes); 4. security guarantees: comparisons should consider security against forgery by guess- ing (soundness), possible disclosure of secret information (zero-knowledge prop- erty), and status regarding provable security; and 5. trust required in third party: variations of the protocols may require different trust assumptions in the trusted party involved. The number of criteria and potential parameter choices precludes a comparison which is both deﬁnitive and concise. The following general comments may, however, be made. 1. computational efﬁciency.Fiat-Shamir requires between one and two orders of mag- nitude fewer full modular multiplications (steps) by the prover than an RSA private- key operation (cf.§10.3.3). Whenkt=20 and nis 512 bits, Fiat-Shamir uses from about 11 to about 30 steps (k=20,t=1; andk=1,t=20); GQ requires about 60 steps (fort=1,m=20=log 2(v)), or somewhat fewer ifvhas low Hamming weight; and full exponentiation in unoptimized RSA takes 768 steps. §10.5 Attacks on identiﬁcation protocols 417 2. off-line computations.Schnorr identiﬁcation has the advantage of requiring only a single on-line modular multiplication by the claimant, provided exponentiation may be done as a precomputation. (Such a trade-off of on-line for off-line computation is possible in some applications; in others, the total computation must be considered.) However, signiﬁcant computation is required by the veriﬁer compared to Fiat-Shamir and GQ. 3. bandwidth and memory for secrets.GQ allows the simultaneous reduction of both memory (parameterk) and transmission bandwidth (parametert) withk=t=1, by introducing the public exponentv> 2with the intention that the probability of successful cheating becomesv−kt; this simultaneous reduction is not possible in Fiat- Shamir, which requireskuser secrets andtiterations for an estimated security (prob- ability of cheating) of2−kt. Regarding other tradeoffs, see Note 10.28. 4. security assumptions.The protocols require the assumptions that the following un- derlying problems are intractable, for a composite (RSA) integern: Fiat-Shamir – extracting square roots modn; GQ – extractingvth roots modn(i.e., the RSA prob- lem); Schnorr identiﬁcation – computing discrete logs modulo a primep. 10.5 Attacks on identiﬁcation protocols The methods an adversary may employ in an attempt to defeat identiﬁcation protocols are a subset of those discussed in Chapter 12 for authenticated key establishment, and the types of adversaries may be similarly classiﬁed (e.g., passive vs. active, insider vs. outsider); for a discussion of attacks on simple password schemes, see§10.2.2. Identiﬁcation is, how- ever, less complex than authenticated key establishment, as there is no issue of an adver- sary learning a previous session key, or forcing an old key to be reused. For conciseness, the following deﬁnitions are made: 1. impersonation:a deception whereby one entity purports to be another. 2. replay attack:an impersonation or other deception involving use of information from a single previous protocol execution, on the same or a different veriﬁer. For stored ﬁles, the analogue of a replay attack is arestoreattack, whereby a ﬁle is replaced by an earlier version. 3. interleaving attack:an impersonation or other deception involving selective combi- nation of information from one or more previous or simultaneously ongoing protocol executions (parallel sessions), including possible origination of one or more protocol executions by an adversary itself. 4. reﬂection attack:an interleaving attack involving sending information from an on- going protocol execution back to the originator of such information. 5. forced delay:a forced delay occurs when an adversary intercepts a message (typically containing a sequence number), and relays it at some later point in time. Note the delayed message is not a replay. 6. chosen-text attack:an attack on a challenge-response protocol wherein an adver- sary strategically chooses challenges in an attempt to extract information about the claimant’s long-term key. Chosen-text attacks are sometimes referred to as using the claimant as anoracle, i.e., to obtain information not computable from knowledge of a claimant’s public key alone. The attack may involve chosen-plaintext if the claimant is required to sign, 418 Ch. 10 Identiﬁcation and Entity Authentication encrypt, or MAC the challenge, or chosen-ciphertext if the requirement is to decrypt a challenge. Potential threats to identiﬁcation protocols include impersonation by any of the follow- ing attacks: replay, interleaving, reﬂection, or forced delay. Impersonation is also trivial if an adversary is able to discover an entity’s long-term (secret or private) keying material, for example, using a chosen-text attack. This may be possible in protocols which are not zero- knowledge, because the claimant uses its private key to compute its response, and thus a response may reveal partial information. In the case of an active adversary, attacks may in- volve the adversary itself initiating one or more new protocol runs, and creating, injecting, or otherwise altering new or previous messages. Table 10.3 summarizes counter-measures for these attacks. Type of attack Principles to avoid attack replay use of challenge-response techniques; use of nonces; embed tar- get identity in response interleaving linking together all messages from a protocol run (e.g., using chained nonces) reﬂection embed identiﬁer of target party in challenge responses; construct protocols with each message of different form (avoid message symmetries); use of uni-directional keys chosen-text use of zero-knowledge techniques; embed in each challenge re- sponse a self-chosen random number (confounder) forced delay combined use of random numbers with short response time-outs; timestamps plus appropriate additional techniques Table 10.3:Identiﬁcation protocol attacks and counter-measures. 10.40 Remark (use of keys for multiple purposes) Caution is advised if any cryptographic key is used for more than one purpose. For example, using an RSA key for both entity authenti- cation and signatures may compromise security by allowing a chosen-text attack. Suppose authentication here consists ofBchallengingAwith a random numberrBRSA-encrypted underA’s public key, andAis required to respond with the decrypted random number. If BchallengesAwithrB =h(x),A’s response to this authentication request may (unwit- tingly) provide toBits RSA signature on the hash value of the (unknown toA) messagex. See also Example 9.88, where a DES key used for both CBC encryption and CBC-MAC leads to a security ﬂaw; and Remark 13.32. 10.41 Remark (adversary acting “as a wire”) In any identiﬁcation protocol betweenAand B, an adversaryCmay step into the communications path and simply relay (without changing) the messages between legitimates partiesAand B, itself acting as a part of the communi- cations link. Typically in practice, this is not considered a true “attack”, in the sense that it does not alter the aliveness assurance delivered by the protocol; however, in some special applications, this may be a concern (see Remark 10.42). 10.42 Remark (grandmaster postal-chess problem) Identiﬁcation protocols do not provide as- surances about the physical location of the authenticated party. Therefore, Remark 10.41 notwithstanding, a concern may arise in the special case that the following is possible: an adversaryCattempts to impersonateB, is challenged (to prove it isB)b yA, and is able to §10.5 Attacks on identiﬁcation protocols 419 relay (in real time, without detection or noticeable delay, and pretending to beA) the chal- lenge on to the realB, get a proper response fromB, and pass this response along back to A. In this case, additional measures are necessary to prevent a challenged entity from elic- iting aid in computing responses. This is related to the so-calledgrandmaster postal-chess problem, whereby an amateur’s chess rating may unfairly be improved by engaging in two simultaneous chess games with distinct grandmasters, playing black in one game and white in the second, and using the grandmaster’s moves from each game in the other. Either two draws, or a win and a loss, are guaranteed, both of which will improve the amateur’s rating. For further discussion of protocol attacks including speciﬁc examples of ﬂawed entity authentication protocols, see§12.9. (i) Maintaining authenticity Identiﬁcation protocols provide assurances corroborating the identity of an entity only at a given instant in time. If the continuity of such an assurance is required, additional tech- niques are necessary to counteract active adversaries. For example, if identiﬁcation is car- ried out at the beginning of a communications session to grant communications permis- sions, a potential threat is an adversary who “cuts in” on the communications line immedi- ately after the successful identiﬁcation of the legitimate party. Approaches to prevent this include: 1. performing re-authentication periodically, or for each discrete resource requested (e.g., each ﬁle access). A remaining threat here is an adversary who “steps out” ev- ery time re-authentication is performed, allowing the legitimate party to perform this task, before re-entering. 2. tying the identiﬁcation process to an ongoing integrity service. In this case, the iden- tiﬁcation process should be integrated with a key establishment mechanism, such that a by-product of successful identiﬁcation is a session key appropriate for use in a sub- sequent ongoing integrity mechanism. (ii) Security level required for on-line vs. off-line attacks The security level required for identiﬁcation protocols depends on the environment and the speciﬁc application at hand. The probability of success of “guessing attacks” should be considered, and distinguished from the amount of computation required to mount on-line or off-line attacks (using the best techniques known). Some illustrative notes follow (see also Note 10.28). 1. Local attacks. Selecting security parameters which limit the probability of successful impersonation of a guessing attack (an adversary simply guesses a legitimate party’s secret) to a1in2 20 chance (20bits of security) may sufﬁce if, for each attempted impersonation, a local appearance is required by the would-be impersonator and there is a penalty for failed attempts. Depending on the potential loss resulting relative to the penalty,10to30bits or more of security may be required. 2. Remote attacks. A higher level of security is required in environments where unlim- ited identiﬁcation attempts, each involving minimal computational effort, are pos- sible by remote electronic communications, by an anonymous claimant interacting with an on-line system, with no penalties for failed attempts.20to40bits of security or more may be called for here, unless the number of interactions may be somehow limited. 3. Off-line or non-interactive attacks. Selecting security parameters such that an attack requires2 40 computations in real-time (during a protocol execution) may be accept- able, but a bound of260 to280 computations (the latter should be adequate in all 420 Ch. 10 Identiﬁcation and Entity Authentication cases) may be called for if the computations can be carried out off-line, and the at- tack isveriﬁable(i.e., the adversary can conﬁrm, before interacting with the on-line system, that his probability of successful impersonation is near 1; or can recover a long-term secret by off-line computations subsequent to an interaction). 10.6 Notes and further references §10.1 Davies and Price [308] and Ford [414] provide extensive discussion of authentication and identiﬁcation; see also the former for biometric techniques, as well as Everett [380]. The comprehensive survey on login protocols by de Waleffe and Quisquater [319] is highly rec- ommended. Cr´epeau and Goutier provide a lucid concise summary of user identiﬁcation techniques with Brassard [192]. For standardized entity authentication mechanisms, see ISO/IEC 9798 [598, 599, 600, 601, 602]. §10.2 See the§9.2 notes on page 377 for historical discussion of using a one-way function (one- way cipher) for “encrypted” password ﬁles. Morris and Thompson [907] introduce the no- tion of password salting in their 1979 report on UNIX passwords; in one study of 3289 user passwords unconstrained by password rules, 86% fell within an easily-searched subset of passwords. Feldmeier and Karn [391] give an update 10 years later, indicating 30% of pass- words they encountered fell to their attack using a precomputed encrypted dictionary, sorted on tapes by salt values. See also Klein [680] and Lomas et al. [771]. Password salting is related to randomized encryption; the idea of padding plaintext with random bits before en- cryption may also be used to preventforward searchattacks on public-key encryption with small plaintext spaces. Password rules and procedures have been published by the U.S. De- partments of Commerce [399] and Defense [334]. Methods for computing password-derived keys (§10.2.4) are speciﬁed in the Kerberos Au- thentication Service [1041] and PKCS #5 [1072]. A concern related to password-derived keys is that known plaintext allows password-guessing attacks; protocols speciﬁcally de- signed to prevent such attacks are mentioned in Chapter 12 notes on§12.6. The idea of chaining one-time passwords by a one-way function (Protocol 10.6) is due to Lam- port [739]; for related practical applications, see RFC 1938 [1047]. Davies and Price [308, p.176] note a questionnaire-based identiﬁcation technique related to ﬁxed challenge- response tables, wherein the user is challenged by a random subset of previously answered questions. §10.3 Needham and Schroeder [923] stimulated much early work in the area of authentication pro- tocols in the late 1970s, and Needham was again involved with Burrows and Abadi [227] in the BAN logic work which stimulated considerable interest in protocol analysis beginning in the late 1980s; see Chapter 12 notes for further discussion. Gong [501] provides an overview of both time variant parameters and message replay; see also Neuman and Stubblebine [925], and the annexes of parts of ISO/IEC 9798 (e.g., [600]). For security arguments against the use of timestamps and a discussion of implemen- tation difﬁculties, see Bellovin and Merritt [103]; Gaarder and Snekkenes [433]; Difﬁe, van Oorschot, and Wiener [348]; and Gong [500], who considers postdated timestamps. See also§12.3 notes. Lam and Beth [734] note that timestamp-based protocols are appropriate §10.6 Notes and further references 421 for connectionless interactions whereas challenge-response suits connection-oriented com- munications, and suggest challenge-response techniques be used to securely synchronize timeclocks with applications themselves using timestamp-based authentication. ISO/IEC 9798 [598] parts 2 through 5 specify entity authentication protocols respectively based on symmetric encryption [599], digital signatures [600], keyed one-way functions [601], and zero-knowledge techniques [602]; a subset of these are presented in this chapter. FIPS 196 [407] is a subset of 9798-3 containing the unilateral and mutual authentication protocols involving challenge-response with random numbers. Several parts of 9798 were inﬂuenced by the SKID2 and SKID3 (Secret Key IDentiﬁcation) protocols from the RACE/RIPE project [178], which leave the keyed hash function unspec- iﬁed but recommend RIPE-MAC with 64-bit random-number challenges. Difﬁe [342, 345] notes that two-pass challenge-response identiﬁcation based on encryption and random chal- lenges has been used since the 1950s in militaryIdentiﬁcation Friend or Foe(IFF) systems to distinguish friendly from hostile aircraft. Mao and Boyd [781] discuss the danger of im- properly using encryption in authentication protocols, speciﬁcally the CBC mode without an integrity mechanism (cf. Remark 10.16). Stubblebine and Gligor [1179] discuss attacks involving this same mode; see also the much earlier paper by Akl [20]. Davies and Price [308] give a concise discussion of password generators. The identiﬁcation technique in§10.3.3(i) based on public-key decryption and witness is derived from a Dan- ish contribution to the 4th Working Draft of ISO/IEC 9798-5, specifying a protocol called COMSET and motivated in part by Brandt et al. [188], and related to ideas noted earlier by Blum et al. [163]. §10.4 A refreshingly non-mathematical introduction to zero-knowledge proofs is provided by Quisquater, Guillou, and Berson [1020], who document the secret of Ali Baba’s legendary cave, and its rediscovery by Mick Ali. Mitropoulos and Meijer [883] give an exception- ally readable and comprehensive survey (circa 1990) of interactive proofs and zero knowl- edge, with a focus on identiﬁcation. Other overviews include Johnson [641]; Stinson [1178, Ch.13]; and Brassard, Chaum, and Cr´epeau [193] (or [192]) for a discussion ofminimum disclosureproofs, based onbit commitmentand the primitive of ablob. Brassard and Cr´epeau [195] provide a user-friendly discussion of various deﬁnitions of zero-knowledge, while Goldreich and Oren [475] examine properties and relationships between various def- initions of ZK proof systems. Rabin [1022] employed the idea ofcut-and-choose protocolsfor cryptographic applications as early as 1978. While Babai (with Moran) [60, 61] independently developed a theory of randomized interactive proofs known asArthur-Merlin gamesin an attempt to “formalize the notion of efﬁcient provability by overwhelming statistical evidence”, interactive proof systems and the notion of zero-knowledge (ZK) proofs were formalized in 1985 by Gold- wasser, Micali, and Rackoff [481] in the context of an interactiveproof of membershipof a stringxin a languageL; they showed that the languages of quadratic-residues and of quadratic non-residues each have ZK interactive proof (ZKIP) systems revealing only a single bit of knowledge, namely, thatx∈L. Goldreich, Micali, and Wigderson [473, 474] prove likewise for graph non-isomorphism(known not to be inNP ) and graph isomorphism, and that assuming the existence of secure encryption schemes, every language inNP has a ZKIP; see also Chaum [244], and Brassard and Cr´epeau [194]. Motivated by cryptographic applications and identiﬁcation in particular, Feige, Fiat, and Shamir [383] adapted the concepts of interactive proofs of membership to interactiveproofs 422 Ch. 10 Identiﬁcation and Entity Authentication of knowledge, including reformulated deﬁnitions for completeness, soundness, and zero- knowledge; while proofs of membership reveal one bit of set membership information, proofs of knowledge reveal only one bit about the prover’sstateof knowledge. The deﬁni- tions given in§10.4.1 are based on these. These authors reﬁne the original scheme of Fiat and Shamir [395] to yield that of Protocol 10.26; both may be converted to identity-based schemes (Note 10.29) in the sense of Shamir [1115]. The Fiat-Shamir scheme is related to (but more efﬁcient than) an earlier protocol for proving quadratic residuosity (presented at Eurocrypt’84, but unpublished) by Fischer, Micali, and Rackoff [412]. The Fiat-Shamir protocol as per Protocol 10.24 includes an improvement noted by Desmedt et al. [340] to avoid inverses in the derivation of user secrets; this optimization may also be made to Pro- tocol 10.26. Related to deﬁnitions in§10.4.1, Bellare and Goldreich [87] noted that Goldwasser, Mi- cali, and Rackoff [481] did not formally propose a deﬁnition for a proof of knowledge, and suggested that the formal deﬁnitions of Feige, Fiat, and Shamir [383] and Tompa and Woll [1194] were unsatisfactory for some applications. To address these issues they proposed a new deﬁnition, having some common aspects with that of Feige and Shamir [384], but offering additional advantages. Micali and Shamir [868] provide preliminary notes on reducing computation in the Fiat- Shamir protocol by choosing the public keysv i,1≤i≤kto be the ﬁrstkprime numbers; each user then has an independent modulusn. A modiﬁcation of Fiat-Shamir identiﬁca- tion by Ong and Schnorr [957] decreases computational complexity, signature size, and the number of communications required, condensingtFiat-Shamir iterations into one iteration while leaving each user withkprivate keys (cf. thek=1 extension below); for computa- tional efﬁciency, they suggest using as secret keys (not too) small integers. The idea of generalizing Fiat-Shamir identiﬁcation in other ways, including “replacing square roots by cubic or higher roots”, was suggested in the original paper; using higher roots allows users to reduce their number of private keysk, including to the limiting case k=1. Guillou and Quisquater [524] proposed a speciﬁc formulation of this idea of “using deep coin tosses” as the GQ scheme (Protocol 10.31); apparently independently, Ohta and Okamoto [945, 944] proposed a similar formulation, including security analysis. The Ohta-Okamoto (OO) version of thisextended Fiat-Shamirscheme differs from the GQ version (Protocol 10.31) as follows: (1) in OO, rather thanTcomputings Afrom identity IA,Achooses its own secretsA∈Znand publishesIA=sAvmodn; and (2) the veriﬁ- cation relationx≡JA e·yv (modn)becomes yv ≡x·IA e. OO is more general in that, as originally proposed, it avoids the GQ (RSA) constraint thatgcd(v,φ(n))=1 . Subsequent analysis by Burmester and Desmedt [221] suggests that additional care may be required when vis not prime. While the OO version precludes an identity-based variation, a further subsequent version of extended Fiat-Shamir (GQ variation) by Okamoto [949] (“Scheme 3” of 5 protocols therein) is provably as secure as factoring, only slightly less efﬁcient, and is amenable to an identity-based variation. The zero-knowledge interactive protocols of Chaum et al. [248, 249] for proving possession of discrete logarithms, provided a basis for Protocol 10.36 which is due to Schnorr [1097, 1098]. Schnorr also proposed a preprocessing scheme to reduce real-time computation, but see de Rooij [314] regarding its security. The Schnorr identiﬁcation and signature schemes must not both be used with the same parametersβ,p[1098] (cf. Remark 10.40). Schnorr’s protocol is related to the log-based identiﬁcation scheme of Beth [123] also proven to be zero-knowledge. Burmester et al. [223] analyze (cf. Note 10.33) a generalized identiﬁcation protocol encompassing all the well-known variations related to Fiat-Shamir and including §10.6 Notes and further references 423 those of both Chaum et al. and Beth noted above. Van de Graaf and Peralta [1200] give a ZK interactive protocol for proving that a Blum integer is a Blum integer. Brickell and McCurley [207] propose a modiﬁcation of Schnorr’s identiﬁcation scheme, in which qis kept secret and exponent computations are reduced modulop−1rather thanq; it has provable security if factoringp−1is difﬁcult, and moreover security equivalent to that of Schnorr’s scheme otherwise; a drawback is that almost 4 times as much computa- tion is required by the claimant. Another variant of Schnorr’s scheme by Girault [458, 461] was the ﬁrst identity-based identiﬁcation scheme based on discrete logs; it uses a composite modulus, and features the user choosing its own secret key, which remains unknown to the trusted party (cf. implicitly-certiﬁed public keys,§12.6.2). A further variation of Schnorr’s identiﬁcation protocol by Okamoto [949] (“Scheme 1”) uses two elementsβ 1 andβ2,o fo r - derq, and is provably secure, assuming the computational infeasibility of computing theZp discrete logarithmlogβ1 β2 ofβ2 relative toβ1; it does, however, involve some additional computation. Aside from the above protocols based on the computational intractability of the standard number-theoretic problems (factoring and discrete logarithms), a number of very efﬁcient identiﬁcation protocols have more recently been proposed based onNP -hard problems. Shamir [1116] proposed a zero-knowledge identiﬁcation protocol based on theNP -hard permuted kernel problem: given anm×nmatrixAover Zp, pprime (and relatively small, e.g.,p= 251), and ann-vectorV, ﬁnd a permutationπon {1,...,n }such that Vπ ∈ker(A), whereker(A)is the kernel ofAconsisting of alln-vectorsWsuch that AW=[0...0]mod p. Patarin and Chauvaud [966] discuss attacks on the permuted ker- nel problem which are feasible for the smallest of parameter choices originally suggested, while earlier less efﬁcient attacks are presented by Baritaud et al. [73] and Georgiades [447]. Stern [1176] proposed a practical zero-knowledge identiﬁcation scheme based on theNP - hardsyndrome decodingproblem, following an earlier less practical scheme of Stern [1174] based on intractable problems in coding theory. Stern [1175] proposed another practi- cal identiﬁcation scheme based on anNP -hard combinatorialconstrained linear equations problem, offering a very short key length, which is of particular interest in speciﬁc applica- tions. Pointcheval [983] proposed another such scheme based on theNP -hardperceptrons problem: given anm×nmatrixMwith entries±1, ﬁnd ann-vectorywith entries±1 such thatMy≥0. Goldreich and Krawczyk [469] pursue the fact that the original deﬁnition of ZK of Gold- wasser, Micali, and Rackoff is not closed under sequential composition (this was noted ear- lier by D. Simon), establishing the importance of the stronger deﬁnitions of ZK formulated subsequently (e.g.,auxiliary-inputzero-knowledge – see Goldreich and Oren [475]), for which closure under sequential composition has been proven. They prove that even these strong formulations of ZK are not, however, closed under parallel composition (thus moti- vating the deﬁnition of weaker notions of zero-knowledge), and that 3-pass interactive ZK proofs of membership that areblack-box simulation ZKexist only for languages inBPP (Deﬁnition 2.77); while the deﬁnition of “black-box simulation ZK” is more restrictive than the original deﬁnition of ZK, all known ZK protocols are ZK by this deﬁnition also. Conse- quently, protocols that are (formally) ZK are less practical than their corresponding 3-pass parallel versions. As a replacement for the security requirement of zero knowledge in many protocols, Feige and Shamir [384] proposedwitness indistinguishabilityand the related notion ofwitness hiding protocols. Unlike zero knowledge, witness indistinguishability is preserved under arbitrary composition of protocols. 424 Ch. 10 Identiﬁcation and Entity Authentication Methods have been proposed to reduce the communication complexity of essentially all customized identiﬁcation protocols, including the use of hash values in the ﬁrst message (cf. Note 10.29; Note 10.39). Girault and Stern [462] examine the security implications of the length of such hash values, note that collision-resistance of the hash function sufﬁces for the typically claimed security levels, and examine further optimizations of the communication complexity of such protocols, including use ofr-collision resistanthash functions. Blum, Feldman, and Micali [163] introduced the idea of non-interactive (or more clearly: mono-directional) ZK proofs, separating the notions of interactive proof systems and zero- knowledge protocols; here the prover and veriﬁer share a random string, and communica- tion is restricted to one-way (or the prover may simply publish a proof, for veriﬁcation at some future time). De Santis, Micali, and Persiano [317] improve these results employing a weaker complexity assumption; Blum et al. [162] provide a summary and further improve- ments. While the technique of Remark 10.30, due to Fiat and Shamir [395], allows a zero- knowledge identiﬁcation scheme to be converted to a signature scheme, the latter cannot be a sound zero-knowledge signature scheme because the very simulatability of the identiﬁca- tion which establishes the ZK property would allow signature forgery (e.g., see Okamoto [949]). A further ﬂavor of zero-knowledge (cf. Deﬁnition 10.22) isstatistical(oralmost perfect) zero-knowledge; here the probability distributions of the transcripts must bestatistically indistinguishable(indistinguishable by an examiner with unlimited computing power but given only polynomially many samples). Pursuing other characterizations, interactive pro- tocols in which the assurance a veriﬁer obtains is based on some unproven assumption may be distinguished asarguments(see Brassard and Cr´epeau [195]), withproofsthen required to be free of any unproven assumptions, although possibly probabilistic. For performance comparisons and tradeoffs for the Fiat-Shamir, Guillou-Quisquater, and Schnorr schemes, see Fiat and Shamir [395], Schnorr [1098], Okamoto [949], and Lim and Lee [768], among others. For an overview of chipcard technology and the use thereof for identiﬁcation, see Guillou, Ugon, and Quisquater [527]; an earlier paper on chipcards is by Guillou and Ugon [526]. Knobloch [681] describes a preliminary chipcard implementation of the Fiat-Shamir protocol. §10.5 Bauspiess and Knobloch [78] discuss issues related to Remark 10.41, including taking over a communications line after entity authentication has completed. Bengio et al. [113] discuss implementation issues related to identiﬁcation schemes such as the Fiat-Shamir protocol, including Remark 10.42. Classes of replay attacks are discussed in several papers, e.g., see Syverson [1182] and the ISO/IEC 10181-2 authentication framework [610]. For further references on the analysis of entity authentication protocols and attacks, see the§12.9 notes. Chapter /1/1 Digital Signatures Contents in Brief 11.1 Introduction............................. 425 11.2 A framework for digital signature mechanisms.......... 426 11.3 RSA and related signature schemes................. 433 11.4 Fiat-Shamir signature schemes................... 447 11.5 The DSA and related signature schemes.............. 451 11.6 One-time digital signatures..................... 462 11.7 Other signature schemes...................... 471 11.8 Signatures with additional functionality.............. 474 11.9 Notes and further references.................... 481 11.1 Introduction This chapter considers techniques designed to provide the digital counterpart to a handwrit- ten signature. Adigital signatureof a message is a number dependent on some secret known only to the signer, and, additionally, on the content of the message being signed. Signatures must be veriﬁable; if a dispute arises as to whether a party signed a document (caused by ei- ther a lying signer trying torepudiatea signature it did create, or a fraudulent claimant), an unbiased third party should be able to resolve the matter equitably, without requiring access to the signer’s secret information (private key). Digital signatures have many applications in information security, including authenti- cation, data integrity, and non-repudiation. One of the most signiﬁcant applications of dig- ital signatures is the certiﬁcation of public keys in large networks. Certiﬁcation is a means for a trusted third party (TTP) to bind the identity of a user to a public key, so that at some later time, other entities can authenticate a public key without assistance from a trusted third party. The concept and utility of a digital signature was recognized several years before any practical realization was available. The ﬁrst method discovered was the RSA signature sch- eme, which remains today one of the most practical and versatile techniques available. Sub- sequent research has resulted in many alternative digital signature techniques. Some offer signiﬁcant advantages in terms of functionality and implementation. This chapter is an ac- count of many of the results obtained to date, with emphasis placed on those developments which are practical. 425 426 Ch. 11 Digital Signatures Chapter outline §11.2 provides terminology used throughout the chapter, and describes a framework for dig- ital signatures that permits a useful classiﬁcation of the various schemes. It is more abstract than succeeding sections.§11.3 provides an indepth discussion of the RSA signature sch- eme, as well as closely related techniques. Standards which have been adopted to imple- ment RSA and related signature schemes are also considered here.§11.4 looks at meth- ods which arise from identiﬁcation protocols described in Chapter 10. Techniques based on the intractability of the discrete logarithm problem, such as the Digital Signature Algo- rithm (DSA) and ElGamal schemes, are the topic of§11.5. One-time signature schemes, many of which arise from symmetric-key cryptography, are considered in§11.6.§11.7 de- scribes arbitrated digital signatures and the ESIGN signature scheme. Variations on the ba- sic concept of digital signatures, including blind, undeniable, and fail-stop signatures, are discussed in§11.8. Further notes, including subtle points on schemes documented in the chapter and variants (e.g., designated conﬁrmer signatures, convertible undeniable signa- tures, group signatures, and electronic cash) may be found in§11.9. 11.2 A framework for digital signature mechanisms §1.6 provides a brief introduction to the basic ideas behind digital signatures, and§1.8.3 shows how these signatures can be realized through reversible public-key encryption tech- niques. This section describes two general models for digital signature schemes. A com- plete understanding of the material in this section is not necessary in order to follow sub- sequent sections; the reader unfamiliar with some of the more concrete methods such as RSA (§11.3) and ElGamal (§11.5) is well advised not to spend an undue amount of time. The idea of a redundancy function is necessary in order to understand the algorithms which give digital signatures with message recovery. The notation provided in Table 11.1 will be used throughout the chapter. 11.2.1 Basic deﬁnitions 1. A digital signatureis a data string which associates a message (in digital form) with some originating entity. 2. A digital signature generation algorithm(orsignature generation algorithm)i sa method for producing a digital signature. 3. A digital signature veriﬁcation algorithm(orveriﬁcation algorithm) is a method for verifying that a digital signature is authentic (i.e., was indeed created by the speciﬁed entity). 4. A digital signature scheme(ormechanism) consists of a signature generation algo- rithm and an associated veriﬁcation algorithm. 5. A digital signature signing process(orprocedure) consists of a (mathematical) digi- tal signature generation algorithm, along with a method for formatting data into mes- sages which can be signed. 6. A digital signature veriﬁcation process(orprocedure) consists of a veriﬁcation algo- rithm, along with a method for recovering data from the message. 1 1Often little distinction is made between the terms scheme and process, and they are used interchangeably. §11.2 A framework for digital signature mechanisms 427 This chapter is, for the most part, concerned simply with digital signature schemes. In order to use a digital signature scheme in practice, it is necessary to have a digital signature process. Several processes related to various schemes have emerged as commercially rele- vant standards; two such processes, namely ISO/IEC 9796 and PKCS #1, are described in §11.3.5 and§11.3.6, respectively. Notation used in the remainder of this chapter is provided in Table 11.1. The sets and functions listed in Table 11.1 are all publicly known. Notation Meaning M a set of elements called themessage space. MS a set of elements called thesigning space. S a set of elements called thesignature space. R a 1 −1 mapping fromMtoMScalled theredundancy function. MR the image ofR(i.e.,MR=I m (R)). R−1 the inverse ofR(i.e.,R−1 : MR−→M). R a set of elements called theindexing set for signing. h a one-way function with domainM. Mh the image ofh(i.e.,h: M−→Mh);Mh⊆MScalled the hash value space. Table 11.1:Notation for digital signature mechanisms. 11.1 Note (comments on Table 11.1) (i) (messages)Mis the set of elements to which a signer can afﬁx a digital signature. (ii) (signing space)MSis the set of elements to which the signature transformations (to be described in§11.2.2 and§11.2.3) are applied. The signature transformations are not applied directly to the setM. (iii) (signature space)Sis the set of elements associated to messages inM. These ele- ments are used to bind the signer to the message. (iv) (indexing set)Ris used to identify speciﬁc signing transformations. A classiﬁcation of digital signature schemes §11.2.2 and§11.2.3 describe two general classes of digital signature schemes, which can be brieﬂy summarized as follows: 1. Digital signature schemes with appendix require the original message as input to the veriﬁcation algorithm. (See Deﬁnition 11.3.) 2. Digital signature schemes with message recovery do not require the original message as input to the veriﬁcation algorithm. In this case, the original message is recovered from the signature itself. (See Deﬁnition 11.7.) These classes can be further subdivided according to whether or not\|R\|=1 , as noted in Deﬁnition 11.2. 11.2 DeﬁnitionA digital signature scheme (with either message recovery or appendix) is said to be arandomized digital signature schemeif\|R\|>1; otherwise, the digital signature scheme is said to bedeterministic. Figure 11.1 illustrates this classiﬁcation. Deterministic digital signature mechanisms can be further subdivided intoone-time signature schemes(§11.6) andmultiple-use schemes. 428 Ch. 11 Digital Signatures Digital signature schemes message recovery appendix Randomized Deterministic Randomized Deterministic Figure 11.1:A taxonomy of digital signature schemes. 11.2.2 Digital signature schemes with appendix Digital signature schemes with appendix, as discussed in this section, are the most com- monly used in practice. They rely on cryptographic hash functions rather than customized redundancy functions, and are less prone to existential forgery attacks (§11.2.4). 11.3 DeﬁnitionDigital signature schemes which require the message as input to the veriﬁca- tion algorithm are calleddigital signature schemes with appendix. Examples of mechanisms providing digital signatures with appendix are the DSA (§11.5.1), ElGamal (§11.5.2), and Schnorr (§11.5.3) signature schemes. Notation for the following discussion is given in Table 11.1. 11.4 AlgorithmKey generation for digital signature schemes with appendix SUMMARY: each entity creates a private key for signing messages, and a corresponding public key to be used by other entities for verifying signatures. 1. Each entityAshould select a private key which deﬁnes a setSA= {SA,k: k∈R} of transformations. EachSA,kis a 1-1 mapping fromMhtoSand is called asigning transformation. 2. SAdeﬁnes a corresponding mappingVAfrom Mh×S to{true, false}such that VA( ˜m,s∗)= { true, ifSA,k( ˜m)= s∗, false, otherwise, for all˜m∈Mh,s∗ ∈S; here,˜m= h(m) form∈M.VAis called averiﬁcation transformationand is constructed such that it may be computed without knowledge of the signer’s private key. 3. A’s public key isVA;A’s private key is the setSA. §11.2 A framework for digital signature mechanisms 429 11.5 AlgorithmSignature generation and veriﬁcation (digital signature schemes with appendix) SUMMARY: entity Aproduces a signatures∈S for a messagem∈M, which can later be veriﬁed by any entityB. 1. Signature generation.EntityAshould do the following: (a) Select an elementk∈R. (b) Compute ˜m= h(m) and s∗= SA,k( ˜m). (c)A’s signature formiss∗. Bothmand s∗are made available to entities which may wish to verify the signature. 2. Veriﬁcation.EntityBshould do the following: (a) ObtainA’s authentic public keyVA. (b) Compute ˜m= h(m) and u= VA( ˜m,s∗). (c) Accept the signature if and only ifu= true. Figure 11.2 provides a schematic overview of a digital signature scheme with appendix. The following properties are required of the signing and veriﬁcation transformations: (i) for eachk∈R,SA,kshould be efﬁcient to compute; (ii)VAshould be efﬁcient to compute; and (iii) it should be computationally infeasible for an entity other thanAto ﬁnd anm∈M and ans∗∈S such thatVA( ˜m,s∗)= true, where˜m= h(m). VA true false Mh ×S m ˜m hS A,k MM h S s∗ = SA,k( ˜m) (a) The signing process (b) The veriﬁcation process Figure 11.2:Overview of a digital signature scheme with appendix. 11.6 Note (use of hash functions) Most digital signature schemes with message recovery (§11.2.3) are applied to messages of a ﬁxed length, while digital signatures with appendix are applied to messages of arbitrary length. The one-way functionhin Algorithm 11.5 is 430 Ch. 11 Digital Signatures typically selected to be a collision-free hash function (see Deﬁnition 9.3). An alternative to hashing is to break the message into blocks of a ﬁxed length which can be individually signed using a signature scheme with message recovery. Since signature generation is rel- atively slow for many schemes, and since reordering of multiple signed blocks presents a security risk, the preferred method is to hash. 11.2.3 Digital signature schemes with message recovery The digital signature schemes described in this section have the feature that the message signed can be recovered from the signature itself. In practice, this feature is of use for short messages (see§11.3.3(viii)). 11.7 DeﬁnitionA digital signature scheme with message recoveryis a digital signature scheme for which a priori knowledge of the message is not required for the veriﬁcation algorithm. Examples of mechanisms providing digital signatures with message recovery are RSA (§11.3.1), Rabin (§11.3.4), and Nyberg-Rueppel (§11.5.4) public-key signature schemes. 11.8 AlgorithmKey generation for digital signature schemes with message recovery SUMMARY: each entity creates a private key to be used for signing messages, and a cor- responding public key to be used by other entities for verifying signatures. 1. Each entityAshould select a setSA = {SA,k: k∈R} of transformations. Each SA,kis a 1-1 mapping fromMStoSand is called asigning transformation. 2. SAdeﬁnes a corresponding mappingVAwith the property thatVA◦SA,kis the iden- tity map onMS for allk ∈R . VA is called averiﬁcation transformationand is constructed such that it may be computed without knowledge of the signer’s private key. 3. A’s public key isV A;A’s private key is the setSA. 11.9 AlgorithmSignature generation and veriﬁcation for schemes with message recovery SUMMARY: entity Aproduces a signatures∈S for a messagem∈M, which can later be veriﬁed by any entityB. The messagemis recovered froms. 1. Signature generation.EntityAshould do the following: (a) Select an elementk∈R. (b) Compute ˜m= R(m) and s∗ = SA,k( ˜m).(Ris a redundancy function; see Table 11.1 and Note 11.10.) (c)A’s signature iss∗; this is made available to entities which may wish to verify the signature and recovermfrom it. 2. Veriﬁcation.EntityBshould do the following: (a) ObtainA’s authentic public keyVA. (b) Compute ˜m= VA(s∗). (c) Verify that˜m∈MR. (If˜m̸∈MR, then reject the signature.) (d) Recovermfrom ˜mby computingR−1( ˜m). §11.2 A framework for digital signature mechanisms 431 RM m MR MS SA,k ˜m s∗ = SA,k( ˜m) S Figure 11.3:Overview of a digital signature scheme with message recovery. Figure 11.3 provides a schematic overview of a digital signature scheme with message recovery. The following properties are required of the signing and veriﬁcation transforma- tions: (i) for eachk∈R,SA,kshould be efﬁcient to compute; (ii)VAshould be efﬁcient to compute; and (iii) it should be computationally infeasible for an entity other thanAto ﬁnd anys∗∈S such thatVA(s∗) ∈MR. 11.10 Note (redundancy function) The redundancy functionRand its inverseR−1 are publicly known. Selecting an appropriateRis critical to the security of the system. To illustrate this point, suppose thatMR = MS. SupposeRand SA,kare bijections fromMtoMR and MStoS, respectively. This implies thatMand Shave the same number of elements. Then for anys∗∈S,VA(s∗) ∈MR, and it is trivial to ﬁnd messagesmand corresponding signaturess∗which will be accepted by the veriﬁcation algorithm (step 2 of Algorithm 11.9) as follows. 1. Select randomk∈R and randoms∗∈S. 2. Compute ˜m= VA(s∗). 3. Compute m= R−1( ˜m). The elements∗is a valid signature for the messagemand was created without knowledge of the set of signing transformationsSA. 11.11 Example (redundancy function) SupposeM= {m: m∈{0,1}n}for some ﬁxed posi- tive integernand MS = {t: t∈{0,1}2n}. DeﬁneR: M− →MS by R(m)= m∥m, where ∥denotes concatenation; that is,MR = {m∥m: m∈M}⊆M S. For large val- ues ofn, the quantity\|MR\|/\|MS\|=( 1 2 )nis a negligibly small fraction. This redundancy function is suitable provided that no judicious choice ofs∗on the part of an adversary will have a non-negligible probability of yieldingVA(s∗) ∈MR. □ 11.12 Remark (selecting a redundancy function) Even though the redundancy functionRis pub- lic knowledge andR−1 is easy to compute, selection ofRis critical and should not be made independently of the choice of the signing transformations inSA. Example 11.21 provides a speciﬁc example of a redundancy function which compromises the security of the signa- ture scheme. An example of a redundancy function which has been accepted as an inter- national standard is given in§11.3.5. This redundancy function is not appropriate for all digital signature schemes with message recovery, but does apply to the RSA (§11.3.1) and Rabin (§11.3.4) digital signature schemes. 432 Ch. 11 Digital Signatures 11.13 Remark (a particular class of message recovery schemes)§1.8.3 describes a class of dig- ital signature schemes with message recovery which arise from reversible public-key en- cryption methods. Examples include the RSA (§8.2) and Rabin (§8.3) encryption schemes. The corresponding signature mechanisms are discussed in§11.3.1 and§11.3.4, respectively. 11.14 Note (signatures with appendix from schemes providing message recovery) Any digital signature scheme with message recovery can be turned into a digital signature scheme with appendix by simply hashing the message and then signing the hash value. The message is now required as input to the veriﬁcation algorithm. A schematic for this situation can be derived from Figure 11.3 and is illustrated in Figure 11.4. The redundancy functionRis no longer critical to the security of the signature scheme, and can be any1 −1 function from M htoMS. R MR MS SA,k ˜m s∗ = SA,k( ˜m) MhM m h h(m) S Figure 11.4:Signature scheme with appendix obtained from one providing message recovery. 11.2.4 Types of attacks on signature schemes The goal of an adversary is toforgesignatures; that is, produce signatures which will be accepted as those of some other entity. The following provides a set of criteria for what it means to break a signature scheme. 1. total break. An adversary is either able to compute the private key information of the signer, or ﬁnds an efﬁcient signing algorithm functionally equivalent to the valid signing algorithm. (For example, see§11.3.2(i).) 2. selective forgery. An adversary is able to create a valid signature for a particular mes- sage or class of messages chosen a priori. Creating the signature does not directly involve the legitimate signer. (See Example 11.21.) 3. existential forgery. An adversary is able to forge a signature for at least one mes- sage. The adversary has little or no control over the message whose signature is ob- tained, and the legitimate signer may be involved in the deception (for example, see Note 11.66(iii)). There are two basic attacks against public-key digital signature schemes. 1. key-only attacks. In these attacks, an adversary knows only the signer’s public key. 2. message attacks. Here an adversary is able to examine signatures corresponding ei- ther to known or chosen messages. Message attacks can be further subdivided into three classes: (a)known-message attack. An adversary has signatures for a set of messages which are known to the adversary but not chosen by him. §11.3 RSA and related signature schemes 433 (b) chosen-message attack. An adversary obtains valid signatures from a chosen list of messages before attempting to break the signature scheme. This attack isnon-adaptivein the sense that messages are chosen before any signatures are seen. Chosen-message attacks against signature schemes are analogous to chosen-ciphertext attacks against public-key encryption schemes (see§1.13.1). (c)adaptive chosen-message attack. An adversary is allowed to use the signer as an oracle; the adversary may request signatures of messages which depend on the signer’s public key and he may request signatures of messages which depend on previously obtained signatures or messages. 11.15 Note (adaptive chosen-message attack) In principle, an adaptive chosen-message attack is the most difﬁcult type of attack to prevent. It is conceivable that given enough messages and corresponding signatures, an adversary could deduce a pattern and then forge a signature of its choice. While an adaptive chosen-message attack may be infeasible to mount in prac- tice, a well-designed signature scheme should nonetheless be designed to protect against the possibility. 11.16 Note (security considerations) The level of security required in a digital signature scheme may vary according to the application. For example, in situations where an adversary is only capable of mounting a key-only attack, it may sufﬁce to design the scheme to prevent the adversary from being successful at selective forgery. In situations where the adversary is capable of a message attack, it is likely necessary to guard against the possibility of exis- tential forgery. 11.17 Note (hash functions and digital signature processes) When a hash functionhis used in a digital signature scheme (as is often the case),hshould be a ﬁxed part of the signature process so that an adversary is unable to take a valid signature, replacehwith a weak hash function, and then mount a selective forgery attack. 11.3 RSA and related signature schemes This section describes the RSA signature scheme and other closely related methods. The security of the schemes presented here relies to a large degree on the intractability of the integer factorization problem (see§3.2). The schemes presented include both digital signa- tures with message recovery and appendix (see Note 11.14). 11.3.1 The RSA signature scheme The message space and ciphertext space for the RSA public-key encryption scheme (§8.2) are bothZn = {0,1,2,...,n −1}where n= pqis the product of two randomly chosen distinct prime numbers. Since the encryption transformation is a bijection, digital signa- tures can be created by reversing the roles of encryption and decryption. The RSA signature scheme is a deterministic digital signature scheme which provides message recovery (see Deﬁnition 11.7). The signing spaceM Sand signature spaceSare bothZn(see Table 11.1 for notation). A redundancy functionR: M−→Znis chosen and is public knowledge. 434 Ch. 11 Digital Signatures 11.18 AlgorithmKey generation for the RSA signature scheme SUMMARY: each entity creates an RSA public key and a corresponding private key. Each entity A should do the following: 1. Generate two large distinct random primespand q, each roughly the same size (see §11.3.2). 2. Compute n= pqand φ=( p−1)(q−1). 3. Select a random integere,1 <e<φ , such thatgcd(e,φ)=1 . 4. Use the extended Euclidean algorithm (Algorithm 2.107) to compute the unique in- tegerd,1 <d<φ , such thated≡1( m o dφ). 5. A’s public key is(n,e);A’s private key isd. 11.19 AlgorithmRSA signature generation and veriﬁcation SUMMARY: entity Asigns a messagem∈M. Any entityBcan verifyA’s signature and recover the messagemfrom the signature. 1. Signature generation.EntityAshould do the following: (a) Compute ˜m= R(m), an integer in the range[0,n−1]. (b) Computes= ˜mdmod n. (c)A’s signature formiss. 2. Veriﬁcation.To verifyA’s signaturesand recover the messagem,Bshould: (a) ObtainA’s authentic public key(n,e). (b) Compute ˜m= semod n. (c) Verify that˜m∈MR; if not, reject the signature. (d) Recoverm= R−1( ˜m). Proof that signature veriﬁcation works.Ifsis a signature for a messagem, thens ≡ ˜mdmod nwhere ˜m= R(m). Sinceed≡1( m o dφ),se ≡ ˜med ≡ ˜m(mod n). Fi- nally,R−1( ˜m)= R−1(R(m)) =m. 11.20 Example (RSA signature generation with artiﬁcially small parameters) Key generation. EntityAselects primesp= 7927,q= 6997, and computesn= pq= 55465219 and φ= 7926×6996 = 55450296.Achoosese=5 and solvesed=5 d≡1 (mod 55450296), yieldingd= 44360237. A’s public key is(n= 55465219,e =5 ); A’s private key isd= 44360237. Signature generation. For the sake of simplicity (but see§11.3.3(ii)), assume thatM= Zn and that the redundancy functionR: M−→Znis the identity mapR(m)= mfor allm∈ M. To sign a messagem= 31229978,Acomputes ˜m= R(m) = 31229978, and com- putes the signatures= ˜mdmod n= 3122997844360237 mod 55465219 = 30729435. Signature veriﬁcation. Bcomputes ˜m = semod n = 307294355 mod 55465219 = 31229978. Finally,Baccepts the signature since˜mhas the required redundancy (i.e.,˜m∈ MR), and recoversm= R−1( ˜m) = 31229978. □ 11.3.2 Possible attacks on RSA signatures (i) Integer factorization If an adversary is able to factor the public modulusnof some entityA, then the adversary can computeφand then, using the extended Euclidean algorithm (Algorithm 2.107), deduce §11.3 RSA and related signature schemes 435 the private keydfrom φand the public exponenteby solvinged ≡ 1( m o dφ). This constitutes a total break of the system. To guard against this,Amust selectpand qso that factoringnis a computationally infeasible task. For further information, see§8.2.2(i) and Note 8.8. (ii) Multiplicative property of RSA The RSA signature scheme (as well as the encryption method, cf.§8.2.2(v)) has the follow- ing multiplicative property, sometimes referred to as thehomomorphic property.I fs1 = md 1 mod nand s2 = md 2 mod nare signatures on messagesm1 and m2, respectively (or more properly on messages with redundancy added), thens= s1s2 mod nhas the prop- erty thats=( m1m2)dmod n.I fm= m1m2 has the proper redundancy (i.e.,m∈MR), thenswill be a valid signature for it. Hence, it is important that the redundancy function Ris not multiplicative, i.e., for essentially all pairsa,b ∈M,R(a·b) ̸=R(a)R(b).A s Example 11.21 shows, this condition onRis necessary but not sufﬁcient for security. 11.21 Example (insecure redundancy function) Letnbe an RSA modulus anddthe private key. Letk= ⌈lg n⌉be the bitlength ofn, and lettbe a ﬁxed positive integer such thatt<k/ 2. Letw=2 tand let messages be integersmin the interval[1,n2−t−1]. The redundancy functionRis taken to beR(m)= m2t(the least signiﬁcanttbits of the binary representa- tion ofR(m) are 0’s). For most choices ofn,Rwill not have the multiplicative property. The general existential forgery attack described in Note 11.10 would have a probability of success of(1 2 )t. But for this redundancy function, a selective forgery attack (which is more serious) is possible, as is now explained. Suppose that an adversary wishes to forge a signature on a messagem. The adversary knows nbut notd. The adversary can mount the following chosen-message attack to obtain the signature onm. Apply the extended Euclidean algorithm (Algorithm 2.107) tonand ˜m= R(m)= m2t = mw. At each stage of the extended Euclidean algorithm, integers x,y, andrare computed such thatxn+ y˜m= r. It can be shown that at some stage there exists ayand rsuch that\|y\|<n/w and r<n/w , providedw≤√ n.I fy> 0, form integersm2 = rwand m3 = yw.I fy< 0, form integersm2 = rwand m3 = −yw.I n either case,m2 and m3 have the required redundancy. If signaturess2 = md 2 mod nand s3 = md 3 mod nare obtained from the legitimate signer, then the adversary can compute a signature formas follows: •ify> 0, computes2 s3 = md 2 md 3 =( rw yw)d=( r y)d= ˜mdmod n; •ify< 0, computes2 −s3 = md 2 (−m3)d =( rw yw)d=( r y)d= ˜mdmod n. In either case, the adversary has a signed message of its choice with the required redun- dancy. This attack is an example of a chosen-message attack providing selective forgery. It emphasizes the requirement for judicious choice of the redundancy functionR. □ 11.3.3 RSA signatures in practice (i) Reblocking problem One suggested use of RSA is to sign a message and then encrypt the resulting signature. One must be concerned about the relative sizes of the moduli involved when implementing this procedure. Suppose thatAwishes to sign and then encrypt a message forB. Suppose that (nA,eA) and (nB,eB) areA’s andB’s public keys, respectively. IfnA >nB, then there is a chance that the message cannot be recovered byB, as illustrated in Example 11.22. 436 Ch. 11 Digital Signatures 11.22 Example (reblocking problem) LetnA= 8387×7499 = 62894113,eA=5 , anddA= 37726937; andnB = 55465219,eB =5 ,dB = 44360237. Notice thatnA>nB. Suppose m= 1368797is a message with redundancy to be signed underA’s private key and then encrypted usingB’s public key.Acomputes the following: 1. s= mdA mod nA= 136879737726937 mod 62894113 = 59847900. 2. c= seB mod nB = 598479005 mod 55465219 = 38842235. To recover the message and verify the signature,Bcomputes the following: 1. ˆs= cdB mod nB = 3884223544360237 mod 55465219 = 4382681. 2. ˆm= ˆseA mod nA= 43826815 mod 62894113 = 54383568. Observe thatm̸=ˆm. The reason for this is thatsis larger than the modulusnB. Here, the probability of this problem occurring is(nA−nB)/nA≈0.12. □ There are various ways to overcome the reblocking problem. 1. reordering. The problem of incorrect decryption will never occur if the operation us- ing the smaller modulus is performed ﬁrst. That is, ifnA>nB, then entityAshould ﬁrst encrypt the message usingB’s public key, and then sign the resulting cipher- text usingA’s private key. The preferred order of operations, however, is always to sign the message ﬁrst and then encrypt the signature; for ifAencrypts ﬁrst and then signs, an adversary could remove the signature and replace it with its own signature. Even though the adversary will not know what is being signed, there may be situa- tions where this is advantageous to the adversary. Thus, reordering is not a prudent solution. 2. two moduli per entity. Have each entity generate separate moduli for encrypting and for signing. If each user’s signing modulus is smaller than all of the possible encrypt- ing moduli, then incorrect decryption never occurs. This can be guaranteed by requir- ing encrypting moduli to be(t+1 )-bit numbers and signing modulit-bit numbers. 3. prescribing the form of the modulus. In this method, one selects the primespandqso that the modulusnhas a special form: the highest-order bit is a1 and thekfollowing bits are all0’s. At-bit modulusnof this form can be found as follows. Fornto have the required form,2t−1 ≤n< 2t−1 +2 t−k−1. Select a random⌈t/2⌉-bit primep, and search for a primeqin the interval between⌈2t−1/p⌉and ⌊(2t−1 +2 t−k−1)/p⌋; thenn= pqis a modulus of the required type (see Example 11.23). This choice for the modulusndoes not completely prevent the incorrect decryption problem, but it can reduce the probability of its occurrence to a negligibly small number. Suppose thatn Ais such a modulus ands= mdA mod nAis a signature onm. Suppose fur- ther thatshas a1 in one of the high-orderk+1 bit positions, other than the highest. Then s, since it is smaller thannA, must have a0 in the highest-order bit position and so is necessarily smaller than any other modulus of a similar form. The proba- bility thatsdoes not have any1’s in the high-orderk+1 bit positions, other than the highest, is less than(1 2 )k, which is negligibly small ifkis selected to be around100. 11.23 Example (prescribing the form of the modulus) Suppose one wants to construct a 12-bit modulus nsuch that the high order bit is a1 and the nextk =3 bits are0’s. Begin by selecting a 6-bit primep=3 7. Select a primeqin the interval between⌈211/p⌉=5 6and ⌊(211 +2 8)/p⌋=6 2. The possibilities forqare59 and 61.I fq=5 9 is selected, then n=3 7×59 = 2183, having binary representation100010000111.I fq=6 1is selected, thenn=3 7×61 = 2257, having binary representation100011010001. □ §11.3 RSA and related signature schemes 437 (ii) Redundancy functions In order to avoid an existential forgery attack (see§11.2.4) on the RSA signature scheme, a suitable redundancy functionRis required.§11.3.5 describes one such function which has been accepted as an international standard. Judicious choice of a redundancy function is crucial to the security of the system (see§11.3.2(ii)). (iii) The RSA digital signature scheme with appendix Note 11.14 describes how any digital signature scheme with message recovery can be modiﬁed to give a digital signature scheme with appendix. For example, if MD5 (Algo- rithm 9.51) is used to hash messages of arbitrary bitlengths to bitstrings of length 128, then Algorithm 11.9 could be used to sign these hash values. Ifnis ak-bit RSA modulus, then a suitable redundancy functionRis required to assign 128-bit integers tok-bit integers. §11.3.6 describes a method for doing this which is often used in practice. (iv) Performance characteristics of signature generation and veriﬁcation Letn= pqbe a2k-bit RSA modulus wherepandqare eachk-bit primes. Computing a sig- natures= mdmod nfor a messagemrequiresO(k3) bit operations (regarding modular multiplication, see§14.3; and for modular exponentiation,§14.6). Since the signer typi- cally knowspand q, she can computes1 = mdmod p,s2 = mdmod q, and determines by using the Chinese remainder theorem (see Note 14.75). Although the complexity of this procedure remainsO(k3), it is considerably more efﬁcient in some situations. Veriﬁcation of signatures is signiﬁcantly faster than signing if the public exponent is chosen to be a small number. If this is done, veriﬁcation requiresO(k2) bit operations. Suggested values forein practice are3 or216 +1 ;2 of course,pand qmust be chosen so thatgcd(e,(p−1)(q−1)) = 1. The RSA signature scheme is thus ideally suited to situations where signature veriﬁca- tion is the predominant operation being performed. For example, when a trusted third party creates a public-key certiﬁcate for an entityA, this requires only one signature generation, and this signature may be veriﬁed many times by various other entities (see§13.4.2). (v) Parameter selection As of 1996, a minimum of 768 bits is recommended for RSA signature moduli. A modulus of at least 1024 bits is recommended for signatures which require much longer lifetimes or which are critical to the overall security of a large network. It is prudent to remain aware of progress in integer factorization, and to be prepared to adjust parameters accordingly. No weaknesses in the RSA signature scheme have been reported when the public expo- nenteis chosen to be a small number such as 3 or2 16 +1 . It is not recommended to restrict the size of the private exponentdin order to improve the efﬁciency of signature generation (cf.§8.2.2(iv)). (vi) Bandwidth efﬁciency Bandwidth efﬁciencyfor digital signatures with message recovery refers to the ratio of the logarithm (base 2) of the size of the signing spaceMSto the logarithm (base 2) of the size of MR, the image space of the redundancy function. Hence, the bandwidth efﬁciency is deter- mined by the redundancyR. For RSA (and the Rabin digital signature scheme,§11.3.4), the redundancy function speciﬁed by ISO/IEC 9796 (§11.3.5) takesk-bit messages and encodes them to2k-bit elements inMS from which a2k-bit signature is formed. The bandwidth 2The choice ofe=2 16 +1 is based on the fact thateis a prime number, and˜me mod ncan be computed with only16 modular squarings and one modular multiplication (see§14.6.1). 438 Ch. 11 Digital Signatures efﬁciency in this case is1 2 . For example, with a modulus of size1024 bits, the maximum size of a message which can be signed is512 bits. (vii) System-wide parameters Each entity must have a distinct RSA modulus; it is insecure to use a system-wide modulus (see§8.2.2(vi)). The public exponentecan be a system-wide parameter, and is in many applications (see Note 8.9(ii)). (viii) Short vs. long messages Suppose nis a2k-bit RSA modulus which is used in Algorithm 11.19 to signk-bit mes- sages (i.e., the bandwidth efﬁciency is1 2 ). Suppose entityAwishes to sign akt-bit message m. One approach is to partitionmintok-bit blocks such thatm= m1\|\|m2\|\|···\|\| mtand sign each block individually (but see Note 11.6 regarding why this is not recommended). The bandwidth requirement for this is2ktbits. Alternatively,Acould hash messagemto a bitstring of lengthl≤kand sign the hash value. The bandwidth requirement for this signa- ture iskt+2k, where the termktcomes from sending the messagem. Sincekt+2k≤2kt whenevert≥2, it follows that the most bandwidth efﬁcient method is to use RSA digital signatures with appendix. For a message of size at mostk-bits, RSA with message recovery is preferred. 11.3.4 The Rabin public-key signature scheme The Rabin public-key signature scheme is similar to RSA (Algorithm 11.19), but it uses an even public exponente. 3 For the sake of simplicity, it will be assumed thate=2 . The signing spaceMSisQn(the set of quadratic residues modulon— see Deﬁnition 2.134) and signatures are square roots of these. A redundancy functionRfrom the message space MtoMSis selected and is public knowledge. Algorithm 11.25 describes the basic version of the Rabin public-key signature scheme. A more detailed version (and one more useful in practice) is presented in Algorithm 11.30. 11.24 AlgorithmKey generation for the Rabin public-key signature scheme SUMMARY: each entity creates a public key and corresponding private key. Each entityAshould do the following: 1. Generate two large distinct random primespand q, each roughly the same size. 2. Compute n= pq. 3. A’s public key isn;A’s private key is(p,q). 11.25 AlgorithmRabin signature generation and veriﬁcation SUMMARY: entity Asigns a messagem∈M. Any entityBcan verifyA’s signature and recover the messagemfrom the signature. 1. Signature generation.EntityAshould do the following: (a) Compute ˜m= R(m). (b) Compute a square rootsof ˜mmod n(using Algorithm 3.44). (c)A’s signature formiss. 3Sincepand qare distinct primes in an RSA modulus,φ =( p−1)(q−1) is even. In RSA, the public exponentemust satisfygcd(e,φ)=1 and so must be odd. §11.3 RSA and related signature schemes 439 2. Veriﬁcation.To verifyA’s signaturesand recover the messagem,Bshould: (a) ObtainA’s authentic public keyn. (b) Compute ˜m= s2 mod n. (c) Verify that˜m∈MR; if not, reject the signature. (d) Recoverm= R−1( ˜m). 11.26 Example (Rabin signature generation with artiﬁcially small parameters) Key generation. EntityAselects primesp =7 ,q =1 1, and computesn =7 7. A’s public key isn=7 7;A’s private key is(p=7 ,q = 11). The signing space isMS = Q77 = {1,4,9,15,16,23,25,36,37,53,58,60,64,67,71}. For the sake of simplicity (but see Note 11.27), takeM= MSand the redundancy functionRto be the identity map (i.e., ˜m= R(m)= m). Signature generation. To sign a messagem=2 3,AcomputesR(m)= ˜m=2 3, and then ﬁnds a square root of˜mmodulo 77.I fsdenotes such a square root, thens≡±3( m o d7 ) and s≡±1 (mod 11), implyings=1 0,32,45,o r67. The signature formis chosen to be s=4 5. (The signature could be any one of the four square roots.) Signature veriﬁcation. Bcomputes ˜m= s2 mod 77 = 23. Since˜m=2 3 ∈MR,B accepts the signature and recoversm= R−1( ˜m)=2 3. □ 11.27 Note (redundancy) (i) As with the RSA signature scheme (Example 11.21), an appropriate choice of a re- dundancy functionRis crucial to the security of the Rabin signature scheme. For example, suppose thatM= MS = Qnand R(m)= mfor allm∈M.I f a n adversary selects any integers∈Z∗ nand squares it to get˜m= s2 mod n, thensis a valid signature for˜mand is obtained without knowledge of the private key. (Here, the adversary has little control over what the message will be.) In this situation, ex- istential forgery is trivial. (ii) In most practical applications of digital signature schemes with message recovery, the message spaceMconsists of bitstrings of some ﬁxed length. For the Rabin scheme, determining a redundancy functionRis a challenging task. For example, if a message mis a bitstring,Rmight assign it to the integer whose binary representation is the message. There is, however, no guarantee that the resulting integer is a quadratic residue modulon, and so computing a square root might be impossible. One might try to append a small number of random bits tomand applyRagain in the hope thatR(m) ∈Q n. On average, two such attempts would sufﬁce, but a deterministic method would be preferable. Modiﬁed-Rabin signature scheme To overcome the problem discussed in Note 11.27(ii), a modiﬁed version of the basic Rabin signature scheme is provided. The technique presented is similar to that used in the ISO/IEC 9796 digital signature standard (§11.3.5). It provides a deterministic method for associating messages with elements in the signing spaceMS, such that computing a square root (or something close to it) is always possible. An understanding of this method will facilitate the reading of§11.3.5. 11.28 FactLetpand qbe distinct primes each congruent to3 modulo 4, and letn= pq. (i) Ifgcd(x,n)=1 , thenx(p−1)(q−1)/2 ≡1( m o dn). (ii) Ifx∈Qn, thenx(n−p−q+5)/8 mod nis a square root ofxmodulo n. 440 Ch. 11 Digital Signatures (iii) Letxbe an integer having Jacobi symbol (x n ) =1 , and letd=( n−p−q+5 )/8. Then x2dmod n= { x, ifx∈Qn, n−x, ifx̸∈Qn. (iv) Ifp̸≡q(mod 8), then (2 n ) = −1. Hence, multiplication of any integerxby 2 or 2−1 mod nreverses the Jacobi symbol ofx. (Integers of the formn= pqwhere p≡q≡3( m o d4 )and p̸≡q(mod 8)are sometimes calledWilliams integers.) Algorithm 11.30 is a modiﬁed version of the Rabin digital signature scheme. Mes- sages to be signed are fromMS = {m∈Zn: m≡6 (mod 16)}. Notation is given in Table 11.2. In practice, the redundancy functionRshould be more complex to prevent existential forgery (see§11.3.5 for an example). Symbol Term Description M message space {m∈Zn: m≤⌊(n−6)/16⌋} MS signing space {m∈Zn: m≡6 (mod 16)} S signature space {s∈Zn:( s2 mod n) ∈MS} R redundancy function R(m)=1 6m+6 for allm∈M MR image ofR {m∈Zn: m≡6 (mod 16)} Table 11.2:Deﬁnition of sets and functions for Algorithm 11.30. 11.29 AlgorithmKey generation for the modiﬁed-Rabin signature scheme SUMMARY: each entity creates a public key and corresponding private key. Each entityAshould do the following: 1. Select random primesp≡3( m o d8 ),q≡7( m o d8 )and computen= pq. 2. A’s public key isn;A’s private key isd=( n−p−q+5 )/8. 11.30 AlgorithmModiﬁed-Rabin public-key signature generation and veriﬁcation SUMMARY: entity Asigns a messagem∈M. Any entityBcan verifyA’s signature and recover the messagemfrom the signature. 1. Signature generation.EntityAshould do the following: (a) Compute ˜m= R(m)=1 6m+6 . (b) Compute the Jacobi symbolJ= (˜m n ) (using Algorithm 2.149). (c) IfJ=1 then computes= ˜mdmod n. (d) IfJ= −1 then computes=( ˜m/2)dmod n.4 (e)A’s signature formiss. 2. Veriﬁcation.To verifyA’s signaturesand recover the messagem,Bshould: (a) ObtainA’s authentic public keyn. (b) Computem′= s2 mod n. (Note the original messagemitself is not required.) (c) Ifm′≡6( m o d8 ), take˜m= m′. (d) Ifm′≡3( m o d8 ), take˜m=2 m′. 4IfJ̸=1or−1 thenJ=0 , implyinggcd( ˜m,n) ̸=1. This leads to a factorization ofn. In practice, the probability that this will ever occur is negligible. §11.3 RSA and related signature schemes 441 (e) Ifm′≡7( m o d8 ), take˜m= n−m′. (f) Ifm′≡2( m o d8 ), take˜m=2 (n−m′). (g) Verify that˜m∈MR(see Table 11.2); if not, reject the signature. (h) Recoverm= R−1( ˜m)=( ˜m−6)/16. Proof that signature veriﬁcation works.The signature generation phase signs eitherv= ˜m orv= ˜m/2 depending upon which has Jacobi symbol1. By Fact 11.28(iv), exactly one of ˜m, ˜m/2 has Jacobi symbol 1. The valuevthat is signed is such thatv≡3 or6( m o d8 ). By Fact 11.28(iii),s2 mod n= vorn−vdepending on whether or notv∈Qn. Since n≡5( m o d8 ), these cases can be uniquely distinguished. 11.31 Example (modiﬁed-Rabin signature scheme with artiﬁcially small parameters) Key generation. Achoosesp =1 9,q =3 1, and computesn = pq = 589 and d = (n−p−q+5 )/8=6 8 . A’s public key isn= 589, whileA’s private key isd=6 8. The signing spaceMSis given in the following table, along with the Jacobi symbol of each element. m 6 22 54 70 86 102 118 134 150 166 ( m 589 ) −11 −1 −11111 −11 m 182 198 214 230 246 262 278 294 326 358 ( m 589 ) −11111 −11 −1 −1 −1 m 374 390 406 422 438 454 470 486 502 518 ( m 589 ) −1 −1 −1111 −1 −11 −1 m 534 550 566 582 ( m 589 ) −11 −11 Signature generation. To sign a messagem=1 2,Acomputes ˜m= R(12) = 198, (˜m n ) =(198 589 ) =1 , ands= 19868 mod 589 = 102.A’s signature form=1 2iss= 102. Signature veriﬁcation. Bcomputes m′= s2 mod n = 1022 mod 589 = 391. Since m′≡ 7( m o d8 ),Btakes˜m = n−m′= 589 −391 = 198. Finally,Bcomputes m= R−1( ˜m) = (198−6)/16 = 12, and accepts the signature. □ 11.32 Note (security of modiﬁed-Rabin signature scheme) (i) When using Algorithm 11.30, one should never sign a valuevhaving Jacobi symbol −1, since this leads to a factorization ofn. To see this, observe thaty= v2d = s2 must have Jacobi symbol 1; buty2 ≡ (v2)2d ≡ v2 (mod n) by Fact 11.28(iii). Therefore,(v−y)(v+y) ≡0( m o dn). Sincevandyhave opposite Jacobi symbols, v̸≡y(mod n) and thusgcd(v−y,n)= porq. (ii) Existential forgery is easily accomplished for the modiﬁed-Rabin scheme as it was for the original Rabin scheme (see Note 11.27(i)). One only needs to ﬁnd ans,1 ≤ s≤n−1, such that eithers2 orn−s2 or2s2 or2(n−s2)m o dnis congruent to 6 modulo 16. In any of these cases,sis a valid signature form′= s2 mod n. 11.33 Note (performance characteristics of the Rabin signature scheme) Algorithm 11.25 re- quires a redundancy function fromMtoMS = Qnwhich typically involves computing a Jacobi symbol (Algorithm 2.149). Signature generation then involves computing at least one Jacobi symbol (see Note 11.27) and a square root modulon. The square root compu- tation is comparable to an exponentiation modulon(see Algorithm 3.44). Since comput- ing the Jacobi symbol is equivalent to a small number of modular multiplications, Rabin 442 Ch. 11 Digital Signatures signature generation is not signiﬁcantly more computationally intensive than an RSA sig- nature generation with the same modulus size. Signature veriﬁcation is very fast ife=2 ; it requires only one modular multiplication. Squaring can be performed slightly more ef- ﬁciently than a general modular multiplication (see Note 14.18). This, too, compares fa- vorably with RSA signature veriﬁcation even when the RSA public exponent ise =3 . The modiﬁed Rabin scheme (Algorithm 11.30) speciﬁes the message space and redundancy function. Signature generation requires the evaluation of a Jacobi symbol and one modular exponentiation. 11.34 Note (bandwidth efﬁciency) The Rabin digital signature scheme is similar to the RSA sch- eme with respect to bandwidth efﬁciency (see§11.3.3(vi)). 11.3.5 ISO/IEC 9796 formatting ISO/IEC 9796 was published in 1991 by the International Standards Organization as the ﬁrst international standard for digital signatures. It speciﬁes a digital signature process which uses a digital signature mechanism providing message recovery. The main features of ISO/IEC 9796 are: (i) it is based on public-key cryptography; (ii) the particular signature algorithm is not speciﬁed but it must mapkbits tokbits; (iii) it is used to sign messages of limited length and does not require a cryptographic hash func- tion; (iv) it provides message recovery (see Note 11.14); and (v) it speciﬁes the message padding, where required. Examples of mechanisms suitable for the standard are RSA (Al- gorithm 11.19) and modiﬁed-Rabin (Algorithm 11.30). The speciﬁc methods used for padding, redundancy, and truncation in ISO/IEC 9796 prevent various means to forge sig- natures. Table 11.3 provides notation for this subsection. Symbol Meaning k the bitlength of the signature. d the bitlength of the messagemto be signed; it is required thatd≤8 ⌊(k+3 )/16⌋. z the number of bytes in the padded message;z= ⌈d/8⌉. r one more than the number of padding bits;r=8 z−d+1 . t the least integer such that a string of2tbytes includes at least k−1 bits;t= ⌈(k−1)/16⌉. Table 11.3:ISO/IEC 9796 notation. 11.35 Example (sample parameter values for ISO/IEC 9796) The following table lists sample values of parameters in the signing process for a150-bit message and a1024-bit signature. Parameter k(bits) d(bits) z(bytes) r(bits) t(bytes) Value 1024 150 19 3 64 □ §11.3 RSA and related signature schemes 443 (i) Signature process for ISO/IEC 9796 The signature process consists of 5 steps as per Figure 11.5(a). Padding Extension Redundancy Truncating and forcing Signature production Padding Extension Redundancy Truncating and forcing Signature production Message recovery Signature accepted Signature Reject Reject Reject YES YES NO NO Signature opening Redundancy checking Message YES Message NO (a) ISO/IEC 9796 signature process (b) ISO/IEC 9796 veriﬁcation process Figure 11.5:Signature and veriﬁcation processes for ISO/IEC 9796. 1. padding.I fmis the message, form the padded messageMP =0 r−1∥mwhere 1 ≤ r≤8, such that the number of bits inMP is a multiple of 8. The number of bytes in MP isz:MP = mz∥mz−1∥···∥m2∥m1 where eachmiis a byte. 2. message extension. The extended message, denotedME, is obtained fromMP by repeated concatenation on the left ofMP with itself untiltbytes are in the string: ME = MEt∥MEt−1∥···∥ME2∥ME1 (eachMEiis a byte). Iftis not a multiple ofz, then the last bytes to be concatenated are a partial set of bytes fromMP, where these bytes are consecutive bytes ofMP from the right. More precisely,MEi+1 = m(imodz)+1 for0 ≤i≤t−1. 3. message redundancy. Redundancy is added toME to get the byte stringMR = MR2t∥MR2t−1∥···∥MR2∥MR1 as follows.MR is obtained by interleaving thet bytes ofME withtredundant bytes and then adjusting byteMR2z of the resulting string. More precisely,MR2i−1 = MEiand MR2i= S(MEi) for1 ≤i≤t, where S(u) is called theshadow functionof the byteu, and is deﬁned as follows. Ifu= u2∥u1 whereu1 andu2 are nibbles (strings of bitlength 4), thenS(u)= π(u2)∥π(u1) where πis the permutation π= ( 01 2 3 4 5 6789 ABCDEF E 358942 F 0 DB 67 AC 1 ) . (For brevity,πis written with nibbles represented by hexadecimal characters.) Fi- nally,MR is obtained by replacingMR2zwithr⊕MR2z.5 4. truncation and forcing. Form thek-bit intermediate integerIR from MR as follows: (a) to the least signiﬁcantk−1 bits ofMR, append on the left a single bit1; (b) modify the least signiﬁcant byteu2∥u1 of the result, replacing it byu1∥0110. (This is done to ensure thatIR ≡6 (mod 16).) 5The purpose ofMR2z is to permit the veriﬁer of a signature to recover the lengthdof the message. Since d=8 z−r+1 , it sufﬁces to knowzand r. These values can be deduced fromMR. 444 Ch. 11 Digital Signatures 5. signature production. A signature mechanism is used which mapsk-bit integers to k-bit integers (and allows message recovery).IR is signed using this mechanism; let sdenote the resulting signature. 11.36 Note (RSA, Rabin) ISO/IEC 9796 was intended for use with the RSA (Algorithm 11.19)6 and Rabin (Algorithm 11.25)7 digital signature mechanisms. For these particular schemes, signature production is stated more explicitly. Letebe the public exponent for the RSA or Rabin algorithms,nthe modulus, anddthe private exponent. First form the representative elementRR which is: (i)IR ifeis odd, or ifeis even and the Jacobi symbol ofIR (treated as an integer) with respect to the modulusnis 1; (ii)IR/2 ifeis even and the Jacobi symbol ofIR with respect tonis−1. The signature formiss=( RR)dmod n. ISO/IEC 9796 speciﬁes that the signaturesshould be the lesser of(RR)dmod nandn−((RR)dmod n). (ii) Veriﬁcation process for ISO/IEC 9796 The veriﬁcation process for an ISO/IEC 9796 digital signature can be separated into three stages, as per Figure 11.5(b). 1. signature opening. Letsbe the signature. Then the following steps are performed. (a) Apply the public veriﬁcation transformation tosto recover an integerIR′. (b) Reject the signature ifIR′is not a string ofkbits with the most signiﬁcant bit being a 1, or if the least signiﬁcant nibble does not have value0110. 2. message recovery. A stringMR′of2tbytes is constructed fromIR′by performing the following steps. (a) LetXbe the least signiﬁcantk−1 bits ofIR′. (b) Ifu4∥u3∥u2∥0110 are the four least signiﬁcant nibbles ofX, replace the least signiﬁcant byte ofXby π−1(u4)∥u2. (c)MR′is obtained by paddingXwith between0 and 15 zero bits so that the re- sulting string has2tbytes. The valueszand rare computed as follows. (a) From the2tbytes ofMR′, compute thetsums MR′ 2i⊕S(MR′ 2i−1 ),1 ≤i≤t. If all sums are0, reject the signature. (b) Letzbe the smallest value ofifor whichMR′ 2i⊕S(MR′ 2i−1 ) ̸=0. (c) Letrbe the least signiﬁcant nibble of the sum found in step (b). Reject the signature if the hexadecimal value ofris not between1 and 8. From MR′, thez-byte stringMP′is constructed as follows. (a)MP′ i= MR′ 2i−1 for1 ≤i≤z. (b) Reject the signature if ther−1 most signiﬁcant bits ofMP′are not all0’s. (c) LetM′be the8z−r+1 least signiﬁcant bits ofMP′. 3. redundancy checking. The signaturesis veriﬁed as follows. (a) FromM′construct a stringMR′′by applying the message padding, message extension, and message redundancy steps of the signing process. (b) Accept the signature if and only if thek−1 least signiﬁcant bits ofMR′′are equal to thek−1 least signiﬁcant bits ofMR′. 6Since steps 1 through 4 of the signature process describe the redundancy functionR, ˜min step 1a of Algo- rithm 11.19 is taken to beIR. 7 ˜mis taken to beIR in step 1 of Algorithm 11.25. §11.3 RSA and related signature schemes 445 11.3.6 PKCS #1 formatting Public-key cryptography standards (PKCS) are a suite of speciﬁcations which include tech- niques for RSA encryption and signatures (see§15.3.6). This subsection describes the dig- ital signature process speciﬁed in PKCS #1 (“RSA Encryption Standard”). The digital signature mechanism in PKCS #1 does not use the message recovery feature of the RSA signature scheme. It requires a hashing function (either MD2, or MD5 — see Algorithm 9.51) and, therefore, is a digital signature scheme with appendix. Table 11.4 lists notation used in this subsection. Capital letters refer to octet strings. IfXis an octet string, thenXiis octeticounting from the left. Symbol Meaning Symbol Meaning k the length ofnin octets (k≥11) EB encryption block n the modulus,28(k−1) ≤n< 28k ED encrypted data p,q the prime factors ofn octet a bitstring of length 8 e the public exponent ab hexadecimal octet value d the private exponent BT block type M message PS padding string MD message digest S signature MD ′ comparative message digest ∥X∥ length ofXin octets Table 11.4:PKCS #1 notation. (i) PKCS #1 data formatting The data is an octet string D, where∥D ∥≤k−11. BT is a single octet whose hexadecimal representation is either00 or01. PS is an octet string with∥PS∥= k−3−∥D ∥.I f B T=0 0, then all octets in PS are00;i fB T=0 1, then all octets in PS are ff. The formatted data block (called theencryption block)i sE B=0 0∥BT ∥PS∥00∥D. 11.37 Note (data formatting rationale) (i) The leading00 block ensures that the octet string EB, when interpreted as an integer, is less than the modulusn. (ii) If the block type is BT=0 0, then either D must begin with a non-zero octet or its length must be known, in order to permit unambiguous parsing of EB. (iii) If BT=0 1, then unambiguous parsing is always possible. (iv) For the reason given in (iii), and to thwart certain potential attacks on the signature mechanism, BT=0 1is recommended. 11.38 Example (PKCS #1 data formatting for particular values) Suppose thatnis a 1024-bit modulus (sok= 128). If∥D ∥=2 0 octets, then∥PS∥= 105 octets, and∥EB ∥= 128 octets. □ (ii) Signature process for PKCS #1 The signature process involves the steps as per Figure 11.6(a). The input to the signature process is the message M, and the signer’s private exponentd and modulusn. 1. message hashing.Hash the message M using the selected message-digest algorithm to get the octet string MD. 446 Ch. 11 Digital Signatures encoding Message digest Message Data block RSA computation Integer-to-octet -string conversion Signature formatting REJECT Octet-string-to-integer Integer-to-octet-string RSA computation Parsing Data decoding andcomparison Message digesting YES YES Signature accepted YES YES REJECT REJECT REJECT NO NO NO NO conversion conversion Message hashing Signature and Message (a) PKCS #1 signature process (b) PKCS #1 veriﬁcation process Octet-string-to- integer conversion Figure 11.6:Signature and veriﬁcation processes for PKCS #1. 2. message digest encoding. MD and the hash algorithm identiﬁer are combined into an ASN.1 (abstract syntax notation) value and then BER-encoded (basic encoding rules) to give an octet data string D. 3. data block formatting. With data string input D, use the data formatting from §11.3.6(i) to form octet string EB. 4. octet-string-to-integer conversion. Let the octets of EB be EB1∥EB 2∥···∥EB k. De- ﬁne ˜EB ito be the integer whose binary representation is the octet EBi(least signiﬁ- cant bit is on the right). The integer representing EB ism= ∑ k i=1 28(k−i) ˜EB i.8 5. RSA computation. Computes= mdmod n. 6. integer-to-octet-string conversion. Convertsto an octet string ED= ED 1∥ED 2∥··· ∥ED k, where the octets EDisatisfys= ∑ k i=1 28(k−i) ˜ED i. The signature is S= ED. (iii) Veriﬁcation process for PKCS #1 The veriﬁcation process involves the steps as per Figure 11.6(b). The input to the veriﬁca- tion process is the message M, the signature S, the public exponente, and modulusn. 1. octet-string-to-integer conversion. (a) Reject S if the bitlength of S is not a multiple of 8. 8Since EB1 =0 0and n≥28(k−1), then0 ≤m<n . §11.4 Fiat-Shamir signature schemes 447 (b) Convert S to an integersas in step 4 of the signature process. (c) Reject the signature ifs>n . 2. RSA computation. Computem= semod n. 3. integer-to-octet-string conversion.Convertmto an octet string EB of lengthkoctets as in step 6 of the signature process. 4. parsing. Parse EB into a block type BT, a padding string PS, and the data D. (a) Reject the signature if EB cannot be parsed unambiguously. (b) Reject the signature if BT is not one of00 or01. (c) Reject the signature if PS consists of<8 octets or is inconsistent with BT. 5. data decoding. (a) BER-decode D to get a message digest MD and a hash algorithm identiﬁer. (b) Reject the signature if the hashing algorithm identiﬁer does not identify one of MD2 or MD5. 6. message digesting and comparison. (a) Hash the message M with the selected message-digest algorithm to get MD′. (b) Accept the signature S on M if and only if MD′= MD. 11.4 Fiat-Shamir signature schemes As described in Note 10.30, any identiﬁcation scheme involving a witness-challenge resp- onse sequence can be converted to a signature scheme by replacing the random challenge of the veriﬁer with a one-way hash function. This section describes two signature mechanisms which arise in this way. The basis for this methodology is the Fiat-Shamir identiﬁcation protocol (Protocol 10.24). 11.4.1 Feige-Fiat-Shamir signature scheme The Feige-Fiat-Shamir signature scheme is a modiﬁcation of an earlier signature scheme of Fiat and Shamir, and requires a one-way hash functionh: {0,1}∗−→{0,1}kfor some ﬁxed positive integerk. Here{0,1}kdenotes the set of bitstrings of bitlengthk, and{0,1}∗ denotes the set of all bitstrings (of arbitrary bitlengths). The method provides a digital sig- nature with appendix, and is a randomized mechanism. 11.39 AlgorithmKey generation for the Feige-Fiat-Shamir signature scheme SUMMARY: each entity creates a public key and corresponding private key. Each entityAshould do the following: 1. Generate random distinct secret primesp,qand formn= pq. 2. Select a positive integerkand distinct random integerss 1,s2,...,s k∈Z∗ n. 3. Compute vj = s−2 j mod n,1 ≤j≤k. 4. A’s public key is thek-tuple(v1,v2,...,v k) and the modulusn;A’s private key is thek-tuple(s1,s2,...,s k). 448 Ch. 11 Digital Signatures 11.40 AlgorithmFeige-Fiat-Shamir signature generation and veriﬁcation SUMMARY: entity Asigns a binary messagemof arbitrary length. Any entityBcan verify this signature by usingA’s public key. 1. Signature generation.EntityAshould do the following: (a) Select a random integerr,1 ≤r≤n−1. (b) Computeu= r2 mod n. (c) Computee=( e1,e2,...,e k)= h(m∥u); eachei∈{0,1}. (d) Computes= r·∏ k j=1 sej j mod n. (e)A’s signature formis(e,s). 2. Veriﬁcation.To verifyA’s signature(e,s) on m,Bshould do the following: (a) ObtainA’s authentic public key(v1,v2,...,v k) and n. (b) Computew= s2 ·∏ k j=1 vej j mod n. (c) Computee′= h(m∥w). (d) Accept the signature if and only ife= e′. Proof that signature veriﬁcation works. w≡s2 · k∏ j=1 vej j ≡r2 · k∏ j=1 s2ej j k∏ j=1 vej j ≡r2 · k∏ j=1 (s2 jvj)ej ≡r2 ≡u (mod n). Hence,w= uand thereforee= e′. 11.41 Example (Feige-Fiat-Shamir signature generation with artiﬁcially small parameters) Key generation. EntityAgenerates primesp= 3571,q= 4523, and computesn= pq= 16151633. The following table displays the selection ofsj (A’s private key) and integers vj(A’s public key) along with intermediate valuess−1 j . j 1 2 3 4 5 sj 42 73 85 101 150 s−1 j mod n 4999315 885021 6270634 13113207 11090788 vj = s−2 j mod n 503594 4879739 7104483 1409171 6965302 Signature generation. Supposeh: {0,1}∗−→{0,1}5 is a hash function.Aselects a ran- dom integerr= 23181and computesu= r2 mod n= 4354872. To sign messagem,A evaluatese= h(m∥u) = 10110(the hash value has been contrived for this example).A formss= rs1s3s4 mod n= (23181)(42)(85)(101) modn= 7978909; the signature for mis(e= 10110,s= 7978909). Signature veriﬁcation.Bcomputess2 mod n= 2926875and v1v3v4 mod n= (503594) (7104483)(1409171) modn= 15668174. Bthen computesw = s2v1v3v4 mod n= 4354872. Sincew= u, it follows thate′= h(m∥w)= h(m∥u)= eand, hence,Bac- cepts the signature. □ 11.42 Note (security of Feige-Fiat-Shamir signature scheme) (i) Unlike the RSA signature scheme (Algorithm 11.19), all entities may use the same modulus n(cf.§8.2.2(vi)). In this scenario, a trusted third party (TTP) would need to generate the primespand qand also public and private keys for each entity. §11.4 Fiat-Shamir signature schemes 449 (ii) The security of the Feige-Fiat-Shamir scheme is based on the intractability of com- puting square roots modulon(see§3.5.2). It has been proven to be secure against an adaptive chosen-message attack, provided that factoring is intractable,his a random function, and thesi’s are distinct. 11.43 Note (parameter selection and key storage requirements)I fnis at-bit integer, the private key constructed in Algorithm 11.39 isktbits in size. This may be reduced by selecting the random valuessj,1 ≤j≤k, as numbers of bitlengtht′<t;t′, however, should not be chosen so small that guessing thesjis feasible. The public key is(k+1 )tbits in size. For example, ift= 768and k= 128, then the private key requires98304 bits and the public key requires99072 bits. 11.44 Note (identity-based Feige-Fiat-Shamir signatures) Suppose a TTP constructs primesp and qand modulusn; the modulus is common to all entities in the system. Algorithm 11.39 can be modiﬁed so that the scheme is identity-based. EntityA’s bitstringIAcontains in- formation which identiﬁesA. The TTP computesvj = f(IA∥j),1 ≤j≤k, wherefis a one-way hash function from{0,1}∗toQnand jis represented in binary, and computes a square rootsjofv−1 j modulo n,1 ≤j≤k.A’s public key is simply the identity infor- mationIA, whileA’s private key (transported securely and secretly by the TTP toA) is the k-tuple(s1,s2,...,s k). The functionsh,f, and the modulusnare system-wide quantities. This procedure has the advantage that the public key generated in Algorithm 11.39 might be generated from a smaller quantityIA, potentially reducing the storage and trans- mission cost. It has the disadvantages that the private keys of entities are known to the TTP, and the modulusnis system-wide, making it a more attractive target. 11.45 Note (small prime variation of Feige-Fiat-Shamir signatures) This improvement aims to reduce the size of the public key and increase the efﬁciency of signature veriﬁcation. Unlike the modiﬁcation described in Note 11.44, each entityAgenerates its own modulusnAand a set ofksmall primesv1,v2,...,v k ∈Qn (each prime will require around 2 bytes to represent). EntityAselects one of the square rootssjofv−1 j modulo nfor eachj,1 ≤j≤ k; these form the private key. The public key consists ofnAand the valuesv1,v2,...,v k. Veriﬁcation of signatures proceeds more efﬁciently since computations are done with much smaller numbers. 11.46 Note (performance characteristics of Feige-Fiat-Shamir signatures) With the RSA sch- eme and a modulus of lengtht = 768, signature generation using naive techniques re- quires, on average,1152 modular multiplications (more precisely,768 squarings and384 multiplications). Signature generation for the Feige-Fiat-Shamir scheme (Algorithm 11.40) requires, on average,k/2 modular multiplications. To sign a message with this scheme, a modulus of lengtht= 768and k= 128requires, on average,64 modular multiplications, or less than6% of the work required by a naive implementation of RSA. Signature veriﬁ- cation requires only one modular multiplication for RSA if the public exponent ise=3 , and 64 modular multiplications, on average, for Feige-Fiat-Shamir. For applications where signature generation must be performed quickly and key space storage is not limited, the Feige-Fiat-Shamir scheme (or DSA-like schemes — see§11.5) may be preferable to RSA. 450 Ch. 11 Digital Signatures 11.4.2 GQ signature scheme The Guillou-Quisquater (GQ) identiﬁcation protocol (§10.4.3) can be turned into a digital signature mechanism (Algorithm 11.48) if the challenge is replaced with a one-way hash function. Leth: {0,1}∗−→Znbe a hash function wherenis a positive integer. 11.47 AlgorithmKey generation for the GQ signature scheme SUMMARY: each entity creates a public key(n,e,JA) and corresponding private keya. EntityAshould do the following: 1. Select random distinct secret primesp,qand formn= pq. 2. Select an integere∈{1,2,...,n −1}such thatgcd(e,(p−1)(q−1)) = 1. (See Note 11.50 for guidance on selectinge.) 3. Select an integerJA,1 <JA<n, which serves as an identiﬁer forAand such that gcd(JA,n)=1 . (The binary representation ofJAcould be used to convey informa- tion aboutAsuch as name, address, driver’s license number, etc.) 4. Determine an integera∈Znsuch thatJAae≡1( m o dn) as follows: 4.1 Compute J−1 A mod n. 4.2 Compute d1 = e−1 mod (p−1) and d2 = e−1 mod (q−1). 4.3 Compute a1 =( J−1 A )d1 mod pand a2 =( J−1 A )d2 mod q. 4.4 Find a solutionato the simultaneous congruencesa≡a1 (mod p),a≡a2 (mod q). 5. A’s public key is(n,e,JA);A’s private key isa. 11.48 AlgorithmGQ signature generation and veriﬁcation SUMMARY: entity Asigns a binary messagemof arbitrary length. Any entityBcan verify this signature by usingA’s public key. 1. Signature generation.EntityAshould do the following: (a) Select a random integerkand computer= kemod n. (b) Computel= h(m∥r). (c) Computes= kalmod n. (d) A’s signature formis the pair(s,l). 2. Veriﬁcation.To verifyA’s signature(s,l) on m,Bshould do the following: (a) ObtainA’s authentic public key(n,e,JA). (b) Computeu= seJA lmod nand l′= h(m∥u). (c) Accept the signature if and only ifl= l′. Proof that signature veriﬁcation works.Note thatu≡seJA l ≡(kal)eJA l ≡ke(aeJA)l ≡ke≡r(mod n). Hence,u= rand thereforel= l′. 11.49 Example (GQ signature generation with artiﬁcially small parameters) Key generation. EntityAchooses primesp= 20849,q= 27457, and computesn= pq= 572450993.Aselects an integere=4 7, an identiﬁerJA= 1091522, and solves the con- gruenceJAae ≡1( m o dn) to geta= 214611724.A’s public key is (n= 572450993, e=4 7,JA= 1091522), whileA’s private key isa= 214611724. Signature generation. To sign the messagem= 1101110001,Aselects a random integer §11.5 The DSA and related signature schemes 451 k= 42134and computesr= kemod n= 297543350.Athen computesl= h(m∥r)= 2713833 (the hash value has been contrived for this example) ands = kalmod n = (42134)2146117242713833 mod n = 252000854. A’s signature formis the pair (s = 252000854,l= 2713833). Signature veriﬁcation. Bcomputes semod n = 252000854 47 mod n = 398641962, JA lmod n= 10915222713833 mod n= 110523867, and ﬁnallyu= seJA lmod n= 297543350. Sinceu= r,l′= h(m∥u)= h(m∥r)= l, and soBaccepts the signature.□ 11.50 Note (security of GQ signature scheme) In Algorithm 11.47,emust be sufﬁciently large to exclude the possibility of forgery based on the birthday paradox (see§2.1.5). The potential attack proceeds along the following lines. The adversary selects a messagemand computes l= h(m∥JA t) for sufﬁciently many values oftuntill≡t(mod e); this is expected to occur withinO(√ e) trials. Having determined such a pair(l,t), the adversary determines an integerxsuch thatt= xe+ land computess= JA xmod n. Observe thatseJA l ≡ (JA x)eJA l≡JA xe+l≡JA t (mod n), and, hence,h(m∥JA t)= l. Thus,(s,l) is a valid (forged) signature for messagem. 11.51 Note (parameter selection) Current methods (as of 1996) for integer factorization suggest that a modulusnof size at least 768 bits is prudent. Note 11.50 suggests thateshould be at least 128 bits in size. Typical values for the outputs of secure hash functions are 128 or 160 bits. With a 768-bit modulus and a 128-bite, the public key for the GQ scheme is896 +u bits in size, whereuis the number of bits needed to representJA. The private keyais 768 bits in size. 11.52 Note (performance characteristics of GQ signatures) Signature generation for GQ (Algo- rithm 11.48) requires two modular exponentiations and one modular multiplication. Using a 768-bit modulusn, a 128-bit valuee, and a hash function with a 128-bit outputl, signature generation (using naive techniques for exponentiation) requires on average 384 modular multiplications (128 squarings and 64 multiplications for each ofeand l). Signature veri- ﬁcation requires a similar amount of work. Compare this with RSA (naively 1152 modular multiplications) and Feige-Fiat-Shamir (64 modular multiplications) for signature genera- tion (see Note 11.46). GQ is computationally more intensive than Feige-Fiat-Shamir but requires signiﬁcantly smaller key storage space (see Note 11.51). 11.53 Note (message recovery variant of GQ signatures) Algorithm 11.48 can be modiﬁed as follows to provide message recovery. Let the signing space beM S = Zn, and letm∈ MS. In signature generation, select a randomksuch thatgcd(k,n)=1 and compute r= kemod nand l= mrmod n. The signature iss= kalmod n. Veriﬁcation gives seJA l ≡keaelJA l ≡ke ≡r(mod n). Messagemis recovered fromlr−1 mod n.A s for all digital signature schemes with message recovery, a suitable redundancy functionR is required to guard against existential forgery. 11.5 The DSA and related signature schemes This section presents the Digital Signature Algorithm (DSA) and several related signature schemes. Most of these are presented overZ ∗ pfor some large primep, but all of these mech- anisms can be generalized to any ﬁnite cyclic group; this is illustrated explicitly for the El- 452 Ch. 11 Digital Signatures Gamal signature scheme in§11.5.2. All of the methods discussed in this section are ran- domized digital signature schemes (see Deﬁnition 11.2). All give digital signatures with appendix and can be modiﬁed to provide digital signatures with message recovery (see Note 11.14). A necessary condition for the security of all of the signature schemes described in this section is that computing logarithms inZ∗ pbe computationally infeasible. This con- dition, however, is not necessarily sufﬁcient for the security of these schemes; analogously, it remains unproven that RSA signatures are secure even if factoring integers is hard. 11.5.1 The Digital Signature Algorithm (DSA) In August of 1991, the U.S. National Institute of Standards and Technology (NIST) pro- posed a digital signature algorithm (DSA). The DSA has become a U.S. Federal Informa- tion Processing Standard (FIPS 186) called theDigital Signature Standard(DSS), and is the ﬁrst digital signature scheme recognized by any government. The algorithm is a variant of the ElGamal scheme (§11.5.2), and is a digital signature scheme with appendix. The signature mechanism requires a hash functionh: {0,1}∗ −→Zq for some inte- gerq. The DSS explicitly requires use of the Secure Hash Algorithm (SHA-1), given by Algorithm 9.53. 11.54 AlgorithmKey generation for the DSA SUMMARY: each entity creates a public key and corresponding private key. Each entityAshould do the following: 1. Select a prime numberqsuch that2 159 <q< 2160. 2. Choosetso that0 ≤t≤8, and select a prime numberpwhere 2511+64t <p< 2512+64t, with the property thatqdivides(p−1). 3. (Select a generatorαof the unique cyclic group of orderqinZ∗ p.) 3.1 Select an elementg∈Z∗ p and computeα= g(p−1)/q mod p. 3.2 Ifα=1 then go to step 3.1. 4. Select a random integerasuch that1 ≤a≤q−1. 5. Compute y= αamod p. 6. A’s public key is(p,q,α,y );A’s private key isa. 11.55 Note (generation of DSA primespand q) In Algorithm 11.54 one must select the primeq ﬁrst and then try to ﬁnd a primepsuch thatqdivides(p−1). The algorithm recommended by the DSS for accomplishing this is Algorithm 4.56. 11.56 AlgorithmDSA signature generation and veriﬁcation SUMMARY: entity Asigns a binary messagemof arbitrary length. Any entityBcan verify this signature by usingA’s public key. 1. Signature generation.EntityAshould do the following: (a) Select a random secret integerk,0 <k<q . (b) Computer=( αkmod p)m o dq(e.g., using Algorithm 2.143). (c) Computek−1 mod q(e.g., using Algorithm 2.142). (d) Computes= k−1{h(m)+ ar}mod q. (e)A’s signature formis the pair(r,s). §11.5 The DSA and related signature schemes 453 2. Veriﬁcation.To verifyA’s signature(r,s) on m,Bshould do the following: (a) ObtainA’s authentic public key(p,q,α,y ). (b) Verify that0 <r<q and 0 <s<q ; if not, then reject the signature. (c) Computew= s−1 mod qand h(m). (d) Computeu1 = w·h(m)m o dqand u2 = rwmod q. (e) Computev=( αu1 yu2 mod p)m o dq. (f) Accept the signature if and only ifv= r. Proof that signature veriﬁcation works.If(r,s) is a legitimate signature of entityAon message m, thenh(m) ≡−ar+ ks(mod q) must hold. Multiplying both sides of this congruence bywand rearranging givesw·h(m)+ arw≡k(mod q). But this is simply u1 + au2 ≡ k(mod q). Raisingαto both sides of this equation yields(αu1 yu2 mod p)m o dq=( αkmod p)m o dq. Hence,v= r, as required. 11.57 Example (DSA signature generation with artiﬁcially small parameters) Key generation.Aselects primesp= 124540019and q= 17389such thatqdivides(p− 1); here,(p−1)/q= 7162.Aselects a random elementg= 110217528∈Z∗ pand com- putesα= g7162 mod p= 10083255. Sinceα̸=1,αis a generator for the unique cyclic subgroup of orderqinZ∗ p .Anext selects a random integera= 12496satisfying1 ≤a≤ q−1, and computesy= αamod p= 1008325512496 mod 124540019 = 119946265. A’s public key is (p= 124540019,q= 17389,α= 10083255,y= 119946265), while A’s private key isa= 12496. Signature generation. To signm,Aselects a random integerk= 9557, and computesr= (αkmod p)m o dq = (100832559557 mod 124540019) mod 17389 = 27039929 mod 17389 = 34.Athen computesk−1 mod q= 7631,h(m) = 5246(the hash value has been contrived for this example), and ﬁnallys= (7631){5246+(12496)(34)}mod q= 13049. The signature formis the pair(r=3 4,s= 13049). Signature veriﬁcation. Bcomputes w = s−1 mod q = 1799,u1 = w·h(m)m o d q = (5246)(1799) mod 17389 = 12716, andu2 = rwmod q = (34)(1799) mod 17389 = 8999 . Bthen computesv =( αu1 yu2 mod p)m o dq = (1008325512716 · 1199462658999 mod 124540019) mod 17389 = 27039929 mod 17389 = 34. Sincev= r,Baccepts the signature. □ 11.58 Note (security of DSA) The security of the DSA relies on two distinct but related discrete logarithm problems. One is the logarithm problem inZ∗ p where the powerful index-calculus methods apply; the other is the logarithm problem in the cyclic subgroup of orderq, where the best current methods run in “square-root” time. For further discussion, see§3.6.6. Since the DSA is a special case of ElGamal signatures (§11.5.2) with respect to the equation for s, security considerations for the latter are pertinent here (see Note 11.66). 11.59 Note (recommended parameter sizes) The size ofqis ﬁxed by Algorithm 11.54 (as per FIPS 186) at 160 bits, while the size ofpcan be any multiple of 64 between 512 and 1024 bits inclusive. A 512-bit primepprovides marginal security against a concerted attack. As of 1996, a modulus of at least 768 bits is recommended. FIPS 186 does not permit primes plarger than 1024 bits. 11.60 Note (performance characteristics of the DSA) For concreteness, supposepis a 768-bit integer. Signature generation requires one modular exponentiation, taking on average (us- ing naive techniques for exponentiation) 240 modular multiplications, one modular inverse 454 Ch. 11 Digital Signatures with a 160-bit modulus, two 160-bit modular multiplications, and one addition. The 160-bit operations are relatively minor compared to the exponentiation. The DSA has the advantage that the exponentiation can be precomputed and need not be done at the time of signature generation. By comparison, no precomputation is possible with the RSA signature scheme. The major portion of the work for signature veriﬁcation is two exponentiations modulop, each to 160-bit exponents. On average, these each require 240 modular multiplications or 480 in total. Some savings can be realized by doing the two exponentiations simultaneously (cf. Note 14.91); the cost, on average, is then 280 modular multiplications. 11.61 Note (system-wide parameters) It is not necessary for each entity to select its own primes pand q. The DSS permitsp,q, andαto be system-wide parameters. This does, however, present a more attractive target for an adversary. 11.62 Note (probability of failure) Veriﬁcation requires the computation ofs−1 mod q.I fs=0 , thens−1 does not exist. To avoid this situation, the signer may check thats̸=0; but ifsis assumed to be a random element inZq, then the probability thats=0 is( 1 2 )160. In practice, this is extremely unlikely ever to occur. The signer may also check thatr̸=0. If the signer detects that eitherr=0 ors=0 , a new value ofkshould be generated. 11.5.2 The ElGamal signature scheme The ElGamal signature scheme is a randomized signature mechanism. It generates digital signatures with appendix on binary messages of arbitrary length, and requires a hash func- tionh: {0,1}∗−→Zpwhere pis a large prime number. The DSA (§11.5.1) is a variant of the ElGamal signature mechanism. 11.63 AlgorithmKey generation for the ElGamal signature scheme SUMMARY: each entity creates a public key and corresponding private key. Each entityAshould do the following: 1. Generate a large random primepand a generatorαof the multiplicative groupZ∗ p (using Algorithm 4.84). 2. Select a random integera,1 ≤a≤p−2. 3. Compute y= αamod p(e.g., using Algorithm 2.143). 4. A’s public key is(p,α,y);A’s private key isa. 11.64 AlgorithmElGamal signature generation and veriﬁcation SUMMARY: entity Asigns a binary messagemof arbitrary length. Any entityBcan verify this signature by usingA’s public key. 1. Signature generation.EntityAshould do the following: (a) Select a random secret integerk,1 ≤k≤p−2, withgcd(k,p−1) = 1. (b) Computer= αkmod p(e.g., using Algorithm 2.143). (c) Computek−1 mod (p−1) (e.g., using Algorithm 2.142). (d) Computes= k−1{h(m) −ar}mod (p−1). (e)A’s signature formis the pair(r,s). 2. Veriﬁcation.To verifyA’s signature(r,s) on m,Bshould do the following: §11.5 The DSA and related signature schemes 455 (a) ObtainA’s authentic public key(p,α,y). (b) Verify that1 ≤r≤p−1; if not, then reject the signature. (c) Computev1 = yrrsmod p. (d) Computeh(m) and v2 = αh(m) mod p. (e) Accept the signature if and only ifv1 = v2. Proof that signature veriﬁcation works.If the signature was generated byA, thens≡k−1 {h(m)−ar}(mod p−1). Multiplying both sides bykgivesks≡h(m)−ar(mod p−1), and rearranging yieldsh(m) ≡ar+ ks(mod p−1). This impliesαh(m) ≡αar+ks ≡ (αa)rrs (mod p). Thus,v1 = v2, as required. 11.65 Example (ElGamal signature generation with artiﬁcially small parameters) Key generation.Aselects the primep= 2357and a generatorα=2 ofZ∗ 2357.Achooses the private keya= 1751and computesy= αamod p=2 1751 mod 2357 = 1185.A’s public key is (p= 2357,α=2 ,y= 1185). Signature generation. For simplicity, messages will be integers fromZpand h(m)= m (i.e., for this example only, takehto be the identity function). To sign the messagem= 1463,Aselects a random integerk = 1529, computesr = αkmod p =2 1529 mod 2357 = 1490, andk−1 mod (p−1) = 245. Finally,Acomputes s = 245{1463 − 1751(1490)}mod 2356 = 1777.A’s signature form= 1463is the pair(r= 1490,s= 1777). Signature veriﬁcation.Bcomputesv1 = 11851490 ·14901777 mod 2357 = 1072,h(m)= 1463, andv2 =2 1463 mod 2357 = 1072.Baccepts the signature sincev1 = v2. □ 11.66 Note (security of ElGamal signatures) (i) An adversary might attempt to forgeA’s signature (per Algorithm 11.64) onmby selecting a random integerkand computingr = αk mod p. The adversary must then determines= k−1{h(m)−ar}mod (p−1). If the discrete logarithm problem is computationally infeasible, the adversary can do no better than to choose ansat random; the success probability is only1 p, which is negligible for largep. (ii) A differentkmust be selected for each message signed; otherwise, the private key can be determined with high probability as follows. Supposes1 = k−1{h(m1) − ar}mod (p−1) and s2 = k−1{h(m2) −ar}mod (p−1). Then(s1 −s2)k≡ (h(m1) −h(m2)) (mod p−1).I fs1 −s2 ̸≡0( m o dp−1), thenk=( s1 − s2)−1(h(m1) −h(m2)) mod (p−1). Oncekis known,ais easily found. (iii) If no hash functionhis used, the signing equation iss= k−1{m−ar}mod (p−1). It is then easy for an adversary to mount an existential forgery attack as follows. Se- lect any pair of integers(u,v) withgcd(v,p−1) = 1. Computer= αuyvmod p= αu+av mod pand s= −rv−1 mod (p−1). The pair(r,s) is a valid signature for the messagem= sumod (p−1), since(αmα−ar)s−1 = αuyv= r. (iv) Step 2b in Algorithm 11.64 requires the veriﬁer to check that0 <r<p . If this check is not done, then an adversary can sign messages of its choice provided it has one valid signature created by entityA, as follows. Suppose that(r,s) is a signature for mes- sagemproduced byA. The adversary selects a messagem ′of its choice and com- putesh(m′) andu= h(m′)·[h(m)]−1 mod (p−1) (assuming[h(m)]−1 mod (p−1) exists). It then computess′= sumod (p−1) and r′such thatr′≡ru(mod p−1) and r′≡r(mod p). The latter is always possible by the Chinese Remainder The- orem (Fact 2.120). The pair(r′,s′) is a signature for messagem′which would be accepted by the veriﬁcation algorithm (Algorithm 11.64) if step 2b were ignored. 456 Ch. 11 Digital Signatures 11.67 Note (security based on parameter selection) (i) (index-calculus attack) The primepshould be sufﬁciently large to prevent efﬁcient use of the index-calculus methods (§3.6.5). (ii) (Pohlig-Hellman attack)p−1 should be divisible by a prime numberqsufﬁciently large to prevent a Pohlig-Hellman discrete logarithm attack (§3.6.4). (iii) (weak generators) Suppose thatp ≡ 1( m o d4 )and the generatorαsatisﬁes the following conditions: (a)αdivides(p−1); and (b) computing logarithms in the subgroupSof orderαinZ∗ pcan be efﬁciently done (for example, if a Pohlig-Hellman attack (§3.6.4) can be mounted inS). It is then possible for an adversary to construct signatures (without knowledge ofA’s private key) which will be accepted by the veriﬁcation algorithm (step 2 of Algo- rithm 11.64). To see this, suppose thatp−1= αq. To sign a messagemthe adversary does the following: (a) Computet=( p−3)/2 and setr= q. (b) Determinezsuch thatαqz ≡yq (mod p) where yisA’s public key. (This is possible sinceαqand yqare elements ofSand αqis a generator ofS.) (c) Computes= t·{h(m) −qz}mod (p−1). (d) (r,s) is a signature onmwhich will be accepted by step 2 of Algorithm 11.64. This attack works because the veriﬁcation equationrsyr ≡ αh(m) (mod p) is satisﬁed. To see this, ﬁrst observe thatαq≡−1( m o dp),α≡−q−1 (mod p), and thatq(p−1)/2 ≡−1( m o dp).(The latter congruence follows from the fact that αis a generator ofZ∗ pand q≡−α−1 (mod p).) From these, one deduces thatqt= q(p−1)/2q−1 ≡−q−1 ≡α(mod p).Now rsyr=( qt)[h(m)−qz]yq≡αh(m)α−qzyq ≡αh(m)y−qyq = αh(m) (mod p).Notice in the case whereα=2 is a generator that the conditions speciﬁed in (iii) above are trivially satisﬁed. The attack can be avoided ifαis selected as a generator for a subgroup ofZ∗ pof prime order rather than a generator forZ∗ pitself. 11.68 Note (performance characteristics of ElGamal signatures) (i) Signature generation by Algorithm 11.64 is relatively fast, requiring one modu- lar exponentiation(αk mod p), the extended Euclidean algorithm (for computing k−1 mod (p−1)), and two modular multiplications. (Modular subtraction is neg- ligible when compared with modular multiplication.) The exponentiation and appli- cation of the extended Euclidean algorithm can be done off-line, in which case sig- nature generation (in instances where precomputation is possible) requires only two (on-line) modular multiplications. (ii) Signature veriﬁcation is more costly, requiring three exponentiations. Each exponen- tiation (using naive techniques) requires 3 2 ⌈lg p⌉modular multiplications, on aver- age, for a total cost of9 2 ⌈lg p⌉multiplications. The computing costs can be reduced by modifying the veriﬁcation slightly. Computev1 = α−h(m)yrrsmod p, and ac- cept the signature as valid if and only ifv1 =1 . Now, v1 can be computed more efﬁciently by doing the three exponentiations simultaneously (see Note 14.91); the total cost is now about15 8 ⌈lg p⌉modular multiplications, almost2.5 times as cost ef- ﬁcient as before. (iii) Signature veriﬁcation calculations are all performed modulop, while signature gen- eration calculations are done modulopand modulo(p−1). §11.5 The DSA and related signature schemes 457 11.69 Note (recommended parameter sizes) Given the latest progress on the discrete logarithm problem inZ∗ p(§3.6), a 512-bit moduluspprovides only marginal security from concerted attack. As of 1996, a moduluspof at least 768 bits is recommended. For long-term security, 1024-bit or larger moduli should be used. 11.70 Note (system-wide parameters) All entities may elect to use the same prime numberp and generatorα, in which casepand αare not required to be part of the public key (cf. Note 11.61). (i) Variations of the ElGamal scheme Many variations of the basic ElGamal signature scheme (Algorithm 11.64) have been pro- posed. Most of these alter what is commonly referred to as thesigning equation(given in step 1d of Algorithm 11.64). After suitable rearrangement, this signing equation can be written asu = av+ kwmod (p−1) where u = h(m),v = r, andw = s(i.e., h(m)= ar+ ksmod (p−1)). Other signing equations can be obtained by permitting u,v, andwto take on the valuess,r, andh(m) in different orders. Table 11.5 lists the 6 possibilities. u v w Signing equation Veriﬁcation 1 h(m) r s h(m)= ar+ ks αh(m) =( αa)rrs 2 h(m) s r h(m)= as+ kr αh(m) =( αa)srr 3 s r h(m) s= ar+ kh(m) αs=( αa)rrh(m) 4 s h(m) r s= ah(m)+ kr αs=( αa)h(m)rr 5 r s h(m) r= as+ kh(m) αr=( αa)srh(m) 6 r h(m) s r= ah(m)+ ks αr=( αa)h(m)rs Table 11.5:Variations of the ElGamal signing equation. Signing equations are computed modulo (p−1); veriﬁcation is done modulop. 11.71 Note (comparing variants of the ElGamal signature scheme) (i) Some of the signing equations listed in Table 11.5 are more efﬁcient to compute than the original ElGamal equation in Algorithm 11.64. For example, equations (3) and (4) of Table 11.5 do not require the computation of an inverse to determine the sig- natures. Equations (2) and (5) require the signer to computea −1 mod (p−1), but this ﬁxed quantity need only be computed once. (ii) Veriﬁcation equations (2) and (4) involve the expressionrr. Part of the security of signature schemes based on these signing equations is the intractability of ﬁnding so- lutions to an expression of the formxx ≡c(mod p) for ﬁxedc. This problem ap- pears to be intractable for large values ofp, but has not received the same attention as the discrete logarithm problem. (ii) The generalized ElGamal signature scheme The ElGamal digital signature scheme, originally described in the setting of the multiplica- tive groupZ∗ p, can be generalized in a straightforward manner to work in any ﬁnite abelian groupG. The introductory remarks for§8.4.2 are pertinent to the algorithm presented in this section. Algorithm 11.73 requires a cryptographic hash functionh: {0,1}∗ −→Zn 458 Ch. 11 Digital Signatures where nis the number of elements inG. It is assumed that each elementr∈Gcan be represented in binary so thath(r) is deﬁned.9 11.72 AlgorithmKey generation for the generalized ElGamal signature scheme SUMMARY: each entity selects a ﬁnite groupG; generator ofG; public and private keys. Each entityAshould do the following: 1. Select an appropriate cyclic groupGof ordern, with generatorα. (Assume thatG is written multiplicatively.) 2. Select a random secret integera,1 ≤a≤n−1. Compute the group elementy= αa. 3. A’s public key is(α,y), together with a description of how to multiply elements in G;A’s private key isa. 11.73 AlgorithmGeneralized ElGamal signature generation and veriﬁcation SUMMARY: entity Asigns a binary messagemof arbitrary length. Any entityBcan verify this signature by usingA’s public key. 1. Signature generation.EntityAshould do the following: (a) Select a random secret integerk,1 ≤k≤n−1, withgcd(k,n)=1 . (b) Compute the group elementr= αk. (c) Computek−1 mod n. (d) Computeh(m) and h(r). (e) Computes= k−1{h(m) −ah(r)}mod n. (f)A’s signature formis the pair(r,s). 2. Veriﬁcation.To verifyA’s signature(r,s) on m,Bshould do the following: (a) ObtainA’s authentic public key(α,y). (b) Computeh(m) and h(r). (c) Computev1 = yh(r) ·rs. (d) Computev2 = αh(m). (e) Accept the signature if and only ifv1 = v2. 11.74 Example (generalized ElGamal signatures with artiﬁcially small parameters) Key generation. Consider the ﬁnite ﬁeldF25 constructed from the irreducible polynomial f(x)= x5 + x2 +1 overF2. (See Example 2.231 for examples of arithmetic in the ﬁeld F24 .) The elements of this ﬁeld are the 31 binary 5-tuples displayed in Table 11.6, along with00000. The elementα= (00010)is a generator forG= F∗ 25 , the multiplicative cyclic group of the ﬁeld. The order of this groupGisn=3 1. Leth: {0,1}∗−→Z31 be a hash function. EntityAselects the private keya=1 9and computesy= αa = (00010)19 = (00110).A’s public key is (α= (00010),y= (00110)). Signature generation. To sign the messagem= 10110101,Aselects a random integer k=2 4, and computesr= α24 = (11110) and k−1 mod 31 = 22. Athen computes h(m)=1 6 and h(r)=7 (the hash values have been contrived for this example) ands= 22 ·{16 −(19)(7)}mod 31 = 30.A’s signature for messagemis(r= (11110),s= 30). Signature veriﬁcation. Bcomputesh(m)=1 6 ,h(r)=7 ,v1 = yh(r)rs = (00110)7· (11110)30 = (11011), andv2 = αh(m) = α16 = (11011).Baccepts the signature since v1 = v2. □ 9More precisely, one would deﬁne a functionf: G−→ {0,1}∗and writeh(f(r)) instead ofh(r). §11.5 The DSA and related signature schemes 459 i αi 0 00001 1 00010 2 00100 3 01000 4 10000 5 00101 6 01010 7 10100 i αi 8 01101 9 11010 10 10001 11 00111 12 01110 13 11100 14 11101 15 11111 i αi 16 11011 17 10011 18 00011 19 00110 20 01100 21 11000 22 10101 23 01111 i αi 24 11110 25 11001 26 10111 27 01011 28 10110 29 01001 30 10010 Table 11.6:The elements ofF25 as powers of a generatorα. 11.75 Note (security of generalized ElGamal) Much of the security of Algorithm 11.73 relies on the intractability of the discrete logarithm problem in the groupG(see§3.6). Most of the security comments in Note 11.66 apply to the generalized ElGamal scheme. 11.76 Note (signing and veriﬁcation operations) Signature generation requires computations in the groupG(i.e.,r= αk) and computations inZn. Signature veriﬁcation only requires computations in the groupG. 11.77 Note (generalized ElGamal using elliptic curves) One of the most promising implemen- tations of Algorithm 11.73 is the case where the ﬁnite abelian groupGis constructed from the set of points on an elliptic curve over a ﬁnite ﬁeldFq. The discrete logarithm problem in groups of this type appears to be more difﬁcult than the discrete logarithm problem in the multiplicative group of a ﬁnite ﬁeldFq. This implies thatqcan be chosen smaller than for corresponding implementations in groups such asG= F∗ q. 11.5.3 The Schnorr signature scheme Another well-known variant of the ElGamal scheme (Algorithm 11.64) is the Schnorr sig- nature scheme. As with the DSA (Algorithm 11.56), this technique employs a subgroup of orderqinZ ∗ p, wherepis some large prime number. The method also requires a hash func- tionh: {0,1}∗ −→Zq. Key generation for the Schnorr signature scheme is the same as DSA key generation (Algorithm 11.54), except that there are no constraints on the sizes of pand q. 11.78 AlgorithmSchnorr signature generation and veriﬁcation SUMMARY: entity Asigns a binary messagemof arbitrary length. Any entityBcan verify this signature by usingA’s public key. 1. Signature generation.EntityAshould do the following: (a) Select a random secret integerk,1 ≤k≤q−1. (b) Computer= αkmod p,e= h(m∥r), ands= ae+ kmod q. (c)A’s signature formis the pair(s,e). 460 Ch. 11 Digital Signatures 2. Veriﬁcation.To verifyA’s signature(s,e) on m,Bshould do the following: (a) ObtainA’s authentic public key(p,q,α,y ). (b) Computev= αsy−emod pand e′= h(m∥v). (c) Accept the signature if and only ife′= e. Proof that signature veriﬁcation works.If the signature was created byA, thenv≡αsy−e ≡αsα−ae≡αk≡r(mod p). Hence,h(m∥v)= h(m∥r) and e′= e. 11.79 Example (Schnorr’ s signature scheme with artiﬁcially small parameters) Key generation. Aselects primesp= 129841 and q= 541; here,(p−1)/q= 240. A then selects a random integerg= 26346∈Z∗ pand computesα= 26346240 mod p=2 6. Sinceα̸=1,αgenerates the unique cyclic subgroup of order541 inZ∗ p . Athen selects the private keya= 423 and computesy=2 6423 mod p= 115917. A’s public key is (p= 129841,q= 541,α=2 6,y= 115917). Signature generation. To sign the messagem= 11101101,Aselects a random number k= 327 such that1 ≤k≤540, and computesr =2 6327 mod p= 49375 and e= h(m∥r) = 155(the hash value has been contrived for this example). Finally,Acomputes s= 423·155 + 327 mod 541 = 431. The signature formis(s= 431,e= 155). Signature veriﬁcation. Bcomputesv=2 6431 ·115917−155 mod p= 49375 and e′= h(m∥v) = 155.Baccepts the signature sincee= e′. □ 11.80 Note (performance characteristics of the Schnorr scheme) Signature generation in Algo- rithm 11.78 requires one exponentiation modulopand one multiplication moduloq. The exponentiation modulopcould be done off-line. Depending on the hash algorithm used, the time to computeh(m∥r) should be relatively small. Veriﬁcation requires two exponen- tiations modulop. These two exponentiations can be computed by Algorithm 14.88 at a cost of about 1.17 exponentiations. Using the subgroup of orderqdoes not signiﬁcantly enhance computational efﬁciency over the ElGamal scheme of Algorithm 11.64, but does provide smaller signatures (for the same level of security) than those generated by the El- Gamal method. 11.5.4 The ElGamal signature scheme with message recovery The ElGamal scheme and its variants (§11.5.2) discussed so far are all randomized digital signature schemes with appendix (i.e., the message is required as input to the veriﬁcation algorithm). In contrast, the signature mechanism of Algorithm 11.81 has the feature that the message can be recovered from the signature itself. Hence, this ElGamal variant provides a randomized digital signature with message recovery. For this scheme, the signing space isM S = Z∗ p,pa prime, and the signature space is S= Zp×Zq,qa prime, whereqdivides(p−1). LetRbe a redundancy function from the set of messagesMtoMS(see Table 11.1). Key generation for Algorithm 11.81 is the same as DSA key generation (Algorithm 11.54), except that there are no constraints on the sizes ofpand q. §11.5 The DSA and related signature schemes 461 11.81 AlgorithmNyberg-Rueppel signature generation and veriﬁcation SUMMARY: entity Asigns a messagem∈M. Any entityBcan verifyA’s signature and recover the messagemfrom the signature. 1. Signature generation.EntityAshould do the following: (a) Compute ˜m= R(m). (b) Select a random secret integerk,1 ≤k≤q−1, and computer= α−kmod p. (c) Computee= ˜mrmod p. (d) Computes= ae+ kmod q. (e)A’s signature formis the pair(e,s). 2. Veriﬁcation.To verifyA’s signature(e,s) on m,Bshould do the following: (a) ObtainA’s authentic public key(p,q,α,y ). (b) Verify that0 <e<p ; if not, reject the signature. (c) Verify that0 ≤s<q ; if not, reject the signature. (d) Computev= αsy−emod pand ˜m= vemod p. (e) Verify that˜m∈MR;i f˜m̸∈MRthen reject the signature. (f) Recoverm= R−1( ˜m). Proof that signature veriﬁcation works.IfAcreated the signature, thenv ≡ αsy−e ≡ αs−ae≡αk (mod p). Thusve≡αk˜mα−k≡˜m(mod p), as required. 11.82 Example (Nyberg-Rueppel signature generation with artiﬁcially small parameters) Key generation. EntityAselects primesp= 1256993 and q = 3571, whereqdivides (p−1); here,(p−1)/q= 352. Athen selects a random numberg= 42077 ∈Z∗ pand computesα= 42077352 mod p= 441238. Sinceα̸=1,αgenerates the unique cyclic subgroup of order3571 inZ∗ p. Finally,Aselects a random integera= 2774and computes y= αamod p= 1013657.A’s public key is(p= 1256993,q= 3571,α= 441238,y= 1013657), whileA’s private key isa= 2774. Signature generation. To sign a messagem,Acomputes ˜m= R(m) = 1147892(the value R(m) has been contrived for this example).Athen randomly selectsk= 1001, computes r= α−kmod p= 441238−1001 mod p= 1188935,e= ˜mrmod p= 138207, ands= (2774)(138207) + 1001 modq= 1088. The signature formis(e= 138207,s= 1088). Signature veriﬁcation. Bcomputesv= 4412381088 ·1013657−138207 mod 1256993 = 504308, and˜m= v·138207 mod 1256993 = 1147892. Bveriﬁes that˜m∈MRand recoversm= R−1( ˜m). □ 11.83 Note (security of the Nyberg-Rueppel signature scheme) (i) Since Algorithm 11.81 is a variant of the basic ElGamal scheme (Algorithm 11.64), the security considerations of Note 11.66 apply. Like DSA (Algorithm 11.56), this ElGamal mechanism with message recovery relies on the difﬁculty of two related but distinct discrete logarithm problems (see Note 11.58). (ii) Since Algorithm 11.81 provides message recovery, a suitable redundancy functionR is required (see Note 11.10) to guard against existential forgery. As is the case with RSA, the multiplicative nature of this signature scheme must be carefully consid- ered when choosing a redundancy functionR. The following possible attack should be kept in mind. Supposem ∈M, ˜m= R(m), and(e,s) is a signature form. Then e = ˜mα−kmod pfor some integerkand s = ae+ kmod q. Let˜m∗ = ˜mαlmod pfor some integerl.I fs∗ = s+ lmod qand ˜m∗ ∈MR, then(e,s∗) 462 Ch. 11 Digital Signatures is a valid signature form∗ = R−1( ˜m∗). To see this, consider the veriﬁcation algo- rithm (step 2 of Algorithm 11.81).v≡αs∗ y−e≡αs+lα−ae≡αk+l (mod p), and ve≡αk+l˜mα−k ≡ ˜mαl ≡ ˜m∗ (mod p). Since˜m∗ ∈MR, the forged signature (e,s∗) will be accepted as a valid signature form∗. (iii) The veriﬁcation that0 <e<p given in step 2b of Algorithm 11.81 is crucial. Suppose (e,s) isA’s signature for the messagem. Thene= ˜mrmod pand s= ae+ kmod q. An adversary can use this signature to compute a signature on a mes- sagem∗of its choice. It determines ane∗such thate∗≡˜m∗r(mod p) and e∗≡e (mod q). (This is possible by the Chinese Remainder Theorem (Fact 2.120).) The pair(e∗,s) will pass the veriﬁcation algorithm provided that0 <e ∗ <p is not checked. 11.84 Note (a generalization of ElGamal signatures with message recovery) The expressione= ˜mrmod pin step 1c of Algorithm 11.81 provides a relatively simple way to encrypt˜mwith key rand could be generalized to any symmetric-key algorithm. LetE= {Er : r∈Zp} be a set of encryption transformations where eachEr is indexed by an elementr ∈Z∗ p and is a bijection fromMS = Z∗ p toZ∗ p . For anym∈M, select a random integerk, 1 ≤k≤q−1, computer= αkmod p,e= Er( ˜m), ands= ae+ kmod q. The pair (e,s) is a signature form. The fundamental signature equations= ae+ kmod qis a means to bind entityA’s private key and the messagemto a symmetric key which can then be used to recover the message by any other entity at some later time. 11.6 One-time digital signatures One-time digital signature schemes are digital signature mechanisms which can be used to sign, at most, one message; otherwise, signatures can be forged. A new public key is required for each message that is signed. The public information necessary to verify one- time signatures is often referred to asvalidation parameters. When one-time signatures are combined with techniques for authenticating validation parameters, multiple signatures are possible (see§11.6.3 for a description of authentication trees). Most, but not all, one-time digital signature schemes have the advantage that signature generation and veriﬁcation are very efﬁcient. One-time digital signature schemes are useful in applications such as chipcards, where low computational complexity is required. 11.6.1 The Rabin one-time signature scheme Rabin’s one-time signature scheme was one of the ﬁrst proposals for a digital signature of any kind. It permits the signing of a single message. The veriﬁcation of a signature requires interaction between the signer and veriﬁer. Unlike other digital signature schemes, veriﬁ- cation can be done only once. While not practical, it is presented here for historical reasons. Notation used in this section is given in Table 11.7. §11.6 One-time digital signatures 463 Symbol Meaning M0 0l= the all0’s string of bitlengthl. M0(i) 0l−e∥be−1 ···b1b0 where be−1 ···b1b0 is the binary representation ofi. K a set ofl-bit strings. E a set of encryption transformations indexed by a key spaceK. Et an encryption transformation belonging toEwitht∈K. EachEt maps l-bit strings tol-bit strings. h a publicly-known one-way hash function from{0,1}∗to{0,1}l. n a ﬁxed positive integer which serves as a security parameter. Table 11.7:Notation for the Rabin one-time signature scheme. 11.85 AlgorithmKey generation for the Rabin one-time signature scheme SUMMARY: each entity Aselects a symmetric-key encryption schemeE, generates2n random bitstrings, and creates a set of validation parameters. Each entityAshould do the following: 1. Select a symmetric-key encryption schemeE(e.g., DES). 2. Generate2nrandom secret stringsk1,k2,...,k 2n∈K, each of bitlengthl. 3. Compute yi= Eki (M0(i)),1 ≤i≤2n. 4. A’s public key is(y1,y2,...,y 2n);A’s private key is(k1,k2,...,k 2n). 11.86 AlgorithmRabin one-time signature generation and veriﬁcation SUMMARY: entity Asigns a binary messagemof arbitrary length. Signature veriﬁcation is interactive withA. 1. Signature generation.EntityAshould do the following: (a) Computeh(m). (b) Computesi= Eki (h(m)),1 ≤i≤2n. (c)A’s signature formis(s1,s2,...,s 2n). 2. Veriﬁcation.To verifyA’s signature(s1,s2,...,s 2n) on m,Bshould: (a) ObtainA’s authentic public key(y1,y2,...,y 2n). (b) Computeh(m). (c) Selectndistinct random numbersrj,1 ≤rj ≤2n, for1 ≤j≤n. (d) Request fromAthe keyskrj ,1 ≤j≤n. (e) Verify the authenticity of the received keys by computingzj = Ekrj (M0(rj)) and checking thatzj = yrj , for each1 ≤j≤n. (f) Verify thatsrj = Ekrj (h(m)),1 ≤j≤n. 11.87 Note (key sizes for Rabin’ s one-time signatures) SinceEtoutputslbits (see Table 11.7), the public and private keys in Algorithm 11.86 each consist of2nlbits. Forn=8 0 and l=6 4, the keys are each 1280 bytes long. 11.88 Note (resolution of disputes) To resolve potential disputes between the signerAand the veriﬁerBusing Algorithm 11.86, the following procedure is followed: 1. Bprovides a trusted third party (TTP) withmand the signature(s1,s2,...,s 2n). 464 Ch. 11 Digital Signatures 2. The TTP obtainsk1,k2,...,k 2nfrom A. 3. The TTP veriﬁes the authenticity of the private key by computingzi= Eki (M0(i)) and checking thatyi= zi,1 ≤i≤2n. If this fails, the TTP rules in favor ofB(i.e., the signature is deemed to be valid). 4. The TTP computesui = Eki (h(m)),1 ≤i≤2n.I fui = sifor at mostnvalues ofi,1 ≤i≤2n, the signature is declared a forgery and the TTP rules in favor ofA (who denies having created the signature). Ifn+1 or more values ofigiveui= si, the signature is deemed valid and the TTP rules in favor ofB. 11.89 Note (rationale for dispute resolution protocol) The rationale for adjudicating disputes in Rabin’s one-time signature scheme, as outlined in Note 11.88, is as follows. IfBhas at- tempted to forgeA’s signature on a new messagem′,Beither needs to determine at least one more keyk′so that at leastn+1 values ofigiveui = si, or determinem′such that h(m)= h(m′). This should be infeasible if the symmetric-key algorithm and hash function are chosen appropriately. IfAattempts to create a signature which it can later disavow,A must ensure thatui= sifor preciselynvalues ofiand hope thatBchooses thesenvalues in step 2c of the veriﬁcation procedure, the probability of which is only1/ (2n n ) . 11.90 Note (one-timeness of Algorithm 11.86)Acan sign at most one message with a given pri- vate key in Rabin’s one-time scheme; otherwise,Awill (with high probability) revealn+1 or more of the private key values and enableB(and perhaps collaborators) to forge signa- tures on new messages (see Note 11.89). A signature can only be veriﬁed once without revealing (with high probability) more thannof the2nprivate values. 11.6.2 The Merkle one-time signature scheme Merkle’s one-time digital signature scheme (Algorithm 11.92) differs substantially from that of Rabin (Algorithm 11.86) in that signature veriﬁcation is not interactive with the signer. A TTP or some other trusted means is required to authenticate the validation pa- rameters constructed in Algorithm 11.91. 11.91 AlgorithmKey generation for the Merkle one-time signature scheme SUMMARY: to sign n-bit messages,Ageneratest= n+⌊lg n⌋+1 validation parameters. Each entityAshould do the following: 1. Selectt= n+ ⌊lg n⌋+1 random secret stringsk1,k2,...,k teach of bitlengthl. 2. Compute vi = h(ki),1 ≤ i ≤ t. Here,his a preimage-resistant hash function h: {0,1}∗−→{0,1}l(see§9.2.2). 3. A’s public key is(v1,v2,...,v t);A’s private key is(k1,k2,...,k t). To sign ann-bit messagem, a bitstringw = m∥cis formed wherecis the binary representation for the number of0’s inm.cis assumed to be a bitstring of bitlength⌊lg n⌋+ 1 with high-order bits padded with0’s, if necessary. Hence,wis a bitstring of bitlength t= n+ ⌊lg n⌋+1 . §11.6 One-time digital signatures 465 11.92 AlgorithmMerkle one-time signature generation and veriﬁcation SUMMARY: entity Asigns a binary messagemof bitlengthn. Any entityBcan verify this signature by usingA’s public key. 1. Signature generation.EntityAshould do the following: (a) Computec, the binary representation for the number of0’s inm. (b) Formw= m∥c=( a1a2 ···at). (c) Determine the coordinate positionsi1 <i2 <··· <iuinwsuch thataij =1 , 1 ≤j≤u. (d) Letsj = kij ,1 ≤j≤u. (e)A’s signature formis(s1,s2,...,s u). 2. Veriﬁcation.To verifyA’s signature(s1,s2,...,s u) on m,Bshould: (a) ObtainA’s authentic public key(v1,v2,...,v t). (b) Computec, the binary representation for the number of0’s inm. (c) Formw= m∥c=( a1a2 ···at). (d) Determine the coordinate positionsi1 <i2 <··· <iuinwsuch thataij =1 , 1 ≤j≤u. (e) Accept the signature if and only ifvij = h(sj) for all1 ≤j≤u. 11.93 Note (security of Merkle’ s one-time signature scheme) Letmbe a message,w = m∥c the bitstring formed in step 1b of Algorithm 11.92, and(s1,s2,...,s u) a signature form. Ifhis a preimage-resistant hash function, the following argument shows that no signature for a messagem′̸=mcan be forged. Letw′= m′∥c′where c′is the(⌊lg n⌋+1 )-bit string which is the binary representation for the number of0’s inm′. Since an adversary has access to only that portion of the signer’s private key which consists of(s1,s2,...,s u), the set of coordinate positions inm′having a1 must be a subset of the coordinate positions inmhaving a1 (otherwise,m′will have a1 in some position wheremhas a0 and the adversary will require an element of the private key not revealed by the signer). But this means thatm′has more0’s thanmand thatc′>c (when considered as integers). In this case,c′will have a1 in some position wherechas a0. The adversary would require a private key element, corresponding to this position, which was not revealed by the signer. 11.94 Note (storage and computational requirements of Algorithm 11.92) (i) To sign ann-bit messagemwhich haskones requiresl·(n+ ⌊lg n⌋+1 )bits of storage for the validation parameters (public key), andl·(n+ ⌊lg n⌋+1 )bits for the private key. The signature requiresl·(k+ k′) bits of storage, wherek′is the number of1’s in the binary representation ofn−k. For example, ifn= 128,l=6 4, and k=7 2, then the public and private keys each require 8704 bits (1088 bytes). The signature requires 4800 bits (600 bytes). (ii) The private key can be made smaller by forming theki’s from a singleseedvalue. For example, ifk∗is a bitstring of bitlength at leastl, then formki = h(k∗∥i),1 ≤ i≤t. Since only the seedk∗need be stored, the size of the private key is drastically reduced. (iii) Signature generation is very fast, requiring no computation. Signature veriﬁcation requires the evaluation of the hash function for fewer thann+ ⌊lg n⌋+1 values. 466 Ch. 11 Digital Signatures 11.95 Note (improving efﬁciency of Merkle’ s one-time scheme) Algorithm 11.92 requiresl·(n+ ⌊lg n⌋+1 )bits for each of the public and private keys. The public key must necessarily be this large because the signing algorithm considers individual bits of the message. The scheme can be made more efﬁcient if the signing algorithm considers more than one bit at a time. Suppose entityAwishes to sign akt-bit messagem. Writem= m1∥m2∥···∥mt where eachmihas bitlengthkand each represents an integer between0 and2k−1 inclusive. Deﬁne U= ∑ t i=1(2k−mi) ≤t2k.Ucan be represented bylg U≤⌊lg t⌋+1+ kbits. Ifr = ⌈(⌊lg t⌋+1+ k)/k⌉, thenUcan be written in binary asU = u1∥u2∥···∥ur, where eachuihas bitlengthk. Form the bitstringw= m1∥m2∥···mt∥u1∥u2∥···∥ur. Generatet+ rrandom bitstringsk1,k2,...,k t+rand computevi = h2k−1(ki),1 ≤i≤ t+ r. The private key for the modiﬁed scheme is(k1,k2,...,k t+r) and the public key is (v1,v2,...,v t+r). The signature formis(s1,s2,...,s t+r) wheresi= hmi(ki),1 ≤i≤ t, andsi= hui(kt+i),1 ≤i≤r. Here,hcdenotes thec-fold composition ofhwith itself. As with the original scheme (Algorithm 11.92), the bits appended to the message act as a check-sum (see Note 11.93) as follows. Given an elementsi = ha(kj), an adversary can easily computeha+δ(kj) for0 ≤δ≤2k−a, but is unable to computeha−δfor anyδ> 0 ifhis a one-way hash function. To forge a signature on a new message, an adversary can only reduce the value of the check-sum, which will make it impossible for him to compute the required hash values on the appendedkrbits. 11.96 Example (signing more than one bit at a time) This example illustrates the modiﬁcation of the Merkle scheme described in Note 11.95. Letm= m1∥m2∥m3∥m4 where m1 = 1011,m2 = 0111,m3 = 1010, andm4 = 1101. m1,m2,m3, andm4 are the binary representations of11,7,10, and13, respectively.U=( 1 6−m1)+( 1 6−m2)+( 1 6− m3)+( 1 6−m4)= 5+9+6+3=2 3 . In binary,U= 10111. Formw= m∥0001 0111. The signature is(s1,s2,s3,s4,s5,s6) where s1 = h11(k1),s2 = h7(k2),s3 = h10(k3), s4 = h13(k4),s5 = h1(x5), ands6 = h7(x6). If an adversary tries to alter the message, he can only apply the functionhto somesi. This causes the sum of the exponents used (i.e.,∑ mi) to increase and, hence,t2d −∑ mi to decrease. An adversary would be unable to modify the last two blocks sinceh−1 is required to decrease the sum. But, sincehis preimage-resistant,h−1 cannot be computed by the adversary. □ 11.6.3 Authentication trees and one-time signatures §13.4.1 describes the basic structure of an authentication tree and relates how such a tree could be used, among other things, to authenticate a large number of public validation pa- rameters for a one-time signature scheme. This section describes how an authentication tree can be used in conjunction with a one-time signature scheme to provide a scheme which al- lows multiple signatures. A small example will serve to illustrate how this is done. 11.97 Example (an authentication tree for Merkle’ s one-time scheme) Consider the one-time signature scheme of Algorithm 11.92 for signingn-bit messages. Leth: {0,1} ∗ −→ {0,1}l be a preimage-resistant hash function andt = n+ ⌊lg n⌋+1 . Figure 11.7 il- lustrates a 5-vertex binary tree created by an entityAin the course of signing ﬁve mes- sagesm0,m1,m2,m3,m4. Each vertex in the tree is associated with one of the ﬁve mes- sages. For the vertex associated with messagemi,Ahas selectedXi=( x1i,x2i,...,x ti), Ui =( u1i,u2i,...,u ti) and Wi =( w1i,w2i,...,w ti),0 ≤ i ≤ 4, the elements of which are random bitstrings. From these lists,Ahas computedYi =( h(xji): 1 ≤j≤ t),Vi =( h(uji): 1 ≤ j ≤ t), andZi =( h(wji): 1 ≤ j ≤ t). Deﬁne h(Yi)= §11.6 One-time digital signatures 467 R2 R3 R4 R1 R0 Figure 11.7:An authentication tree for the Merkle one-time signature scheme (cf. Example 11.97). h(h(x1i)∥h(x2i)∥···∥h(xti)) for0 ≤i≤4, and deﬁneh(Vi) and h(Zi) analogously. Denote the Merkle one-time signature ofmi using private keyXi by SA(mi,Xi),0 ≤ i ≤ 4. Yi is the set of validation parameters for the signatureSA(mi,Xi). Finally, let Ri = h(h(Yi)∥h(Vi)∥h(Zi)),0 ≤i≤4. Table 11.8 summarizes the parameters asso- ciated with the vertexRi. The setsUiand Wi are used to sign the labels of the children message mi private parameters Xi,Ui,Wi public parameters Yi,Vi,Zi hash values h(Yi),h(Vi),h(Zi) Ri h(h(Yi)∥h(Vi)∥h(Zi)) signature SA(mi,Xi) validation parameters Yi Table 11.8:Parameters and signature associated with vertexRi,0 ≤i≤4 (cf. Figure 11.7). of vertexRi. The signature on vertexR0 is that of a trusted third party (TTP). Table 11.9 summarizes the parameters and signatures associated with each vertex label of the binary tree. To describe how the tree is used to verify signatures, consider messagem4 and signa- Message Vertex Signature on Authentication Label Vertex Label Parameters m0 R0 Signature of TTP — m1 R1 SA(R1,U0) V0,h(Y0),h(Z0) m2 R2 SA(R2,W0) Z0,h(Y0),h(V0) m3 R3 SA(R3,U1) V1,h(Y1),h(Z1) m4 R4 SA(R4,W1) Z1,h(Y1),h(V1) Table 11.9:Parameters and signatures associated with vertices of the binary tree (cf. Figure 11.7). tureSA(m4,X4). The signerAﬁrst provides the veriﬁerBwith the validation parameters Y4. The veriﬁer checks the Merkle one-time signature using step 2 of Algorithm 11.92.B must then be convinced thatY4 is an authentic set of validation parameters created byA. To accomplish this,AprovidesBwith a sequence of values enumerated in the steps below: 1. h(V4),h(Z4);Bcomputesh(Y4) and thenR4 = h(h(Y4)∥h(V4)∥h(Z4)). 2. SA(R4,W1) and Z1;Bveriﬁes the signature onR4 using Algorithm 11.92. 3. h(Y1),h(V1);Bcomputesh(Z1) and thenR1 = h(h(Y1)∥h(V1)∥h(Z1)). 4. SA(R1,U0) and V0;Bveriﬁes the signature using Algorithm 11.92. 468 Ch. 11 Digital Signatures 5. h(Y0),h(Z0);Bcomputesh(V0) and thenR0 = h(h(Y0)∥h(V0)∥h(Z0)). 6. the signature of the TTP forR0;Bveriﬁes the TTP’s signature using an algorithm appropriate to the signature mechanism for the TTP. The binary tree on 5 vertices (Figure 11.7) could be extended indeﬁnitely from any leaf as more signatures are created byA. The length of a longest authentication path (or equiva- lently, the depth of the tree) determines the maximum amount of information whichAmust provideBin order forBto verify the signature of a message associated with a vertex.□ 11.6.4 The GMR one-time signature scheme The Goldwasser, Micali, and Rivest (GMR) scheme (Algorithm 11.102) is a one-time sig- nature scheme which requires a pair of claw-free permutations (see Deﬁnition 11.98). When combined with a tree authentication procedure, it provides a mechanism for signing more than one message. The GMR scheme is noteworthy as it was the ﬁrst digital signature mech- anism proven to be secure against an adaptive chosen-message attack. Although the GMR scheme is not practical, variations of it have been proposed which suggest that the concept is not purely of theoretical importance. 11.98 DeﬁnitionLet g i: X −→X,i=0 ,1, be two permutations deﬁned on a ﬁnite setX. g0 and g1 are said to be aclaw-free pairof permutations if it is computationally infeasible to ﬁndx,y ∈ Xsuch thatg0(x)= g1(y). A triple(x,y,z) of elements fromXwith g0(x)= g1(y)= zis called aclaw. If bothgi,i=0 ,1, have the property that given additional information it is computationally feasible to determineg−1 0 ,g−1 1 , respectively, the permutations are called atrapdoor claw-free pairof permutations. In order forg0,g1 to be a claw-free pair, computingg−1 i (x), for bothi=0 and 1, must be computationally infeasible for essentially allx∈X. For, ifg−1 1 (and similarly for g−1 0 ) could be efﬁciently computed, one could select anx∈X, computeg0(x)= zand g−1 1 (z)= y, to obtain a claw(x,y,z). 11.99 Example (trapdoor claw-free permutation pair) Letn= pqwhere p≡3( m o d4 )and q≡7( m o d8 ). For this choice ofpand q, (−1 n ) =1 but−1 ̸∈Qn, and (2 n ) = −1. Here,(· n ) denotes the Jacobi symbol (Deﬁnition 2.147). DeﬁneDn= {x: (x n ) =1 and 0 <x< n 2 }. Deﬁneg0 : Dn−→Dnand g1 : Dn−→Dnby g0(x)= { x2 mod n, ifx2 mod n< n 2 , −x2 mod n, ifx2 mod n> n 2 , g1(x)= { 4x2 mod n, if4x2 mod n< n 2 , −4x2 mod n, if4x2 mod n> n 2 . If factoringnis intractable, theng0,g1 form a trapdoor claw-free pair of permutations; this can be seen as follows. (i) (g0 and g1 are permutations onDn)I fg0(x)= g0(y), thenx2 ≡y2 (mod n) (x2 ≡ −y2 (mod n) is not possible since−1 ̸∈Qn), whencex≡±y(mod n). Since 0 <x ,y<n /2, thenx = y, and henceg0 is a permutation onDn. A similar argument shows thatg1 is a permutation onDn. (ii) (g0 and g1 are claw-free) Suppose that there is an efﬁcient method for ﬁndingx,y∈ Dnsuch thatg0(x)= g1(y). Thenx2 ≡4y2 (mod n) (x2 ≡−4y2 (mod n) is §11.6 One-time digital signatures 469 impossible since−1 ̸∈Qn), whence(x−2y)(x+2y) ≡0( m o dn). Since (x n ) =1 and (±2y n ) = −1,x̸≡±2y(mod n) and, hence,gcd(x−2y,n) yields a non-trivial factor ofn. This contradicts the assumption that factoringnis intractable. (iii) (g0,g1 is a trapdoor claw-free pair) Knowing the factorization ofnpermits one to compute g−1 0 and g−1 1 . Hence,g0,g1 is a trapdoor claw-free permutation pair.□ The following example illustrates the general construction given in Example 11.99. 11.100 Example (pair of claw-free permutations for artiﬁcially small parameters) Letp=1 1, q=7 , andn= pq=7 7.D77 = {x:( x n)=1 and 0 <x< 38}= {1,4,6,9,10,13,15, 16,17,19,23,24,25,36,37}. The following table describesg0 and g1. x 1 4 6 9 10 13 15 16 17 19 23 24 25 36 37 g0(x) 11 63 6 4 2 31 5 6 2 51 92 41 03 7 9 1 31 7 g1(x) 41 31 01 61 51 72 42 3 1 1 93 7 6 3 62 5 9 Notice thatg0 and g1 are permutations onD77. □ 11.101 AlgorithmKey generation for the GMR one-time signature scheme SUMMARY: each entity selects a pair of trapdoor claw-free permutations and a validation parameter. Each entityAshould do the following: 1. Select a pairg0,g1 of trapdoor claw-free permutations on some setX. (It is “trap- door” in thatAitself can computeg−1 0 and g−1 1 .) 2. Select a random elementr∈X.(ris called avalidation parameter.) 3. A’s public key is(g0,g1,r);A’s private key is(g−1 0 ,g−1 1 ). In the following, the notation for the composition of functionsg0,g1 usually denotedg0 ◦g1 (see Deﬁnition 1.33) is simpliﬁed tog0g1. Also,(g0g1)(r) will be written asg0g1(r). The signing spaceMSconsists of binary strings which are preﬁx-free (see Note 11.103). 11.102 AlgorithmGMR one-time signature generation and veriﬁcation SUMMARY: Asigns a binary stringm= m1m2 ···mt.Bveriﬁes usingA’s public key. 1. Signature generation.EntityAshould do the following: (a) ComputeSr(m)= ∏ t−1 i=0 g−1 mt−i (r). (b) A’s signature formisSr(m). 2. Veriﬁcation.To verifyA’s signatureSr(m) on m,Bshould do the following: (a) ObtainA’s authentic public key(g0,g1,r). (b) Computer′= ∏ t i=1 gmi(Sr(m)). (c) Accept the signature if and only ifr′= r. Proof that signature veriﬁcation works. r′= t∏ i=1 gmi(Sr(m)) = t∏ i=1 gmi t−1∏ j=0 g−1 mt−j (r) = gm1 ◦gm2 ◦···◦ gmt ◦g−1 mt ◦g−1 mt−1 ◦···◦ g−1 m1 (r)= r. Thus r′= r, as required. 470 Ch. 11 Digital Signatures 11.103 Note (message encoding and security) The set of messages which can be signed using Al- gorithm 11.102 must come from a set of binary strings which arepreﬁx-free. (For example, 101 and 10111 cannot be in the same space since101 is a preﬁx of10111.) One method to accomplish this is to encode a binary stringb1b2 ···blasb1b1b2b2 ···blbl01. To see why the preﬁx-free requirement is necessary, supposem= m1m2 ···mtis a message whose signature isSr(m)= ∏ t−1 i=0 g−1 mt−i(r).I fm′= m1m2 ···mu,u<t , then an adversary can easily ﬁnd a valid signature form′from Sr(m) by computing Sr(m′)= t∏ j=u+1 gmj (Sr(m)) = u−1∏ i=0 g−1 mu−i(r). 11.104 Note (one-timeness of Algorithm 11.102) To see that the GMR signature scheme is a one- time scheme, suppose that two preﬁx-free messagesm= m1m2 ···mtandm′= n1n2 ··· nuare both signed with the same validation parameterr. ThenSr(m)= ∏ t−1 i=0 g−1 mt−i(r) and Sr(m′)= ∏ u−1 i=0 g−1 nu−i(r). Therefore,∏ t i=1 gmi(Sr(m)) =r= ∏ u i=1 gni(Sr(m′)). Since the message space is preﬁx-free, there is a smallest indexh≥1 for whichmh̸=nh. Since eachgjis a bijection, it follows that t∏ i=h gmi(Sr(m)) = u∏ i=h gni(Sr(m′)) or gmh t∏ i=h+1 gmi(Sr(m)) =gnh u∏ i=h+1 gni(Sr(m′)). Takingx = ∏ t i=h+1 gmi(Sr(m)), andy = ∏ u i=h+1 gni(Sr(m′)), the adversary has a claw (x,y,gmh (x)). This violates the basic premise that it is computationally infeasible to ﬁnd a claw. It should be noted that this does not necessarily mean that a signature for a new message can be forged. In the particular case given in Example 11.99, ﬁnding a claw factors the modulusnand permits anyone to sign an unlimited number of new messages (i.e., a total break of the system is possible). 11.105 Example (GMR with artiﬁcially small parameters.) Key generation. Letn,p,q,g0,g1 be those given in Example 11.100.Aselects the valida- tion parameterr=1 5∈D77. Signature generation. Letm= 1011000011be the message to be signed. Then Sr(m)= g−1 1 ◦g−1 1 ◦g−1 0 ◦g−1 0 ◦g−1 0 ◦g−1 0 ◦g−1 1 ◦g−1 1 ◦g−1 0 ◦g−1 1 (15) = 23. A’s signature for messagemis23. Signature veriﬁcation. To verify the signature,Bcomputes r′= g1 ◦g0 ◦g1 ◦g1 ◦g0 ◦g0 ◦g0 ◦g0 ◦g1 ◦g1(23) = 15. Sincer= r′,Baccepts the signature. □ GMR scheme with authentication trees In order to sign multiple messages using the GMR one-time signature scheme (Algorithm 11.102), authentication trees (see§13.4.1) are required. Although conceptually similar to the method described in§11.6.3, only the leaves are used to produce the signature. Before giving details, an overview and some additional notation are necessary. §11.7 Other signature schemes 471 11.106 DeﬁnitionA full binary treewithklevelsis a binary tree which has2k+1 −1 vertices and 2kleaves. The leaves are said to be at levelkof the tree. LetTbe a full binary tree withklevels. Select public parametersY1,Y2,...,Y nwhere n=2 k. Form an authentication treeT∗fromTwith root labelR(see below).Ris certiﬁed by a TTP and placed in a publicly available ﬁle.T∗can now be used to authenticate any of theYiby providing the authentication path values associated with the authentication path forYi. EachYican now be used as the public parameterrfor the GMR scheme. The details for constructing the authentication treeT∗now follow. The treeT∗is constructed recursively. For the root vertex, select a valuerand twot- bit binary stringsrLand rR. Sign the stringrL∥rRwith the GMR scheme using the public valuer. The label for the root consists of the valuesr,rL,rR, andSr(rL∥rR). To authen- ticate the children of the root vertex, selectt-bit binary stringsb0L,b1L,b0R, andb1R. The label for the left child of the root is the set of valuesrL,b0L,b1L,SrL(b0L∥b1L) and the label for the right child isrR,b0R,b1R,SrR(b0R∥b1R). Using the stringsb0L,b1L,b0R, and b1Ras public values for the signing mechanism, one can construct labels for the children of the children of the root. Continuing in this manner, each vertex ofT∗can be labeled. The method is illustrated in Figure 11.8. r,rL,rR,Sr(rL∥rR) rL,b0L,b1L,SrL(b0L∥b1L) b0L,c0L,c1L,Sb0L (c0L∥c1L) rR,b0R,b1R,SrR(b0R∥b1R) b1R,d0R,d1R,Sb1R (d0R∥d1R) b0R,d0L,d1L,Sb0R (d0L∥d1L)b1L,c0R,c1R,Sb1L (c0R∥c1R) Figure 11.8:A full binary authentication tree of level 2 for the GMR scheme. Each leaf of the authentication treeT∗can be used to sign a different binary message m. The signing procedure uses a pair of claw-free permutationsg0,g1.I fmis the binary message to be signed, andxis the public parameter in the label of a leaf which has not been used to sign any other message, then the signature formconsists of bothSx(m) and the authentication path labels. 11.7 Other signature schemes The signature schemes described in this section do not fall naturally into the general set- tings of§11.3 (RSA and related signature schemes),§11.4 (Fiat-Shamir signature schemes), §11.5 (DSA and related signature schemes), or§11.6 (one-time digital signatures). 472 Ch. 11 Digital Signatures 11.7.1 Arbitrated digital signatures 11.107 DeﬁnitionAn arbitrated digital signature schemeis a digital signature mechanism re- quiring an unconditionally trusted third party (TTP) as part of the signature generation and veriﬁcation. Algorithm 11.109 requires a symmetric-key encryption algorithmE= {Ek: k∈K} where Kis the key space. Assume that the inputs and outputs of eachEkarel-bit strings, and leth: {0,1}∗−→{0,1}lbe a one-way hash function. The TTP selects a keykT ∈K which it keeps secret. In order to verify a signature, an entity must share a symmetric key with the TTP. 11.108 AlgorithmKey generation for arbitrated signatures SUMMARY: each entity selects a key and transports it secretly with authenticity to the TTP. Each entityAshould do the following: 1. Select a random secret keykA∈K. 2. Secretly and by some authentic means, makekAavailable to the TTP. 11.109 AlgorithmSignature generation and veriﬁcation for arbitrated signatures SUMMARY: entity Agenerates signatures usingEkA . Any entityBcan verifyA’s signa- ture with the cooperation of the TTP. 1. Signature generation.To sign a messagem, entityAshould do the following: (a)AcomputesH= h(m). (b) AencryptsHwithEto getu= EkA (H). (c)Asendsualong with some identiﬁcation stringIAto the TTP. (d) The TTP computesE−1 kA (u) to getH. (e) The TTP computess= EkT (H\|\|IA) and sendsstoA. (f)A’s signature formiss. 2. Veriﬁcation.Any entityBcan verifyA’s signatureson mby doing the following: (a)Bcomputesv= EkB (s). (b) Bsendsvand some identiﬁcation stringIBto the TTP. (c) The TTP computesE−1 kB (v) to gets. (d) The TTP computesE−1 kT (s) to getH∥IA. (e) The TTP computesw= EkB (H∥IA) and sendswtoB. (f)BcomputesE−1 kB (w) to getH∥IA. (g) BcomputesH′= h(m) from m. (h) Baccepts the signature if and only ifH′= H. 11.110 Note (security of arbitrated signature scheme) The security of Algorithm 11.109 is based on the symmetric-key encryption scheme chosen and the ability to distribute keys to par- ticipants in an authentic manner.§13.3 discusses techniques for distributing conﬁdential keys. §11.7 Other signature schemes 473 11.111 Note (performance characteristics of arbitrated signatures) Since symmetric-key algo- rithms are typically much faster than public-key techniques, signature generation and veri- ﬁcation by Algorithm 11.109 are (relatively) very efﬁcient. A drawback is that interaction with the TTP is required, which places a much higher burden on the TTP and requires ad- ditional message exchanges between entities and the TTP. 11.7.2 ESIGN ESIGN (an abbreviation for Efﬁcient digital SIGNature) is another digital signature scheme whose security relies on the difﬁculty of factoring integers. It is a signature scheme with appendix and requires a one-way hash functionh: {0,1} ∗−→Zn. 11.112 AlgorithmKey generation for ESIGN SUMMARY: each entity creates a public key and corresponding private key. Each entityAshould do the following: 1. Select random primespand qsuch thatp ≥ qand p,qare roughly of the same bitlength. 2. Compute n= p 2q. 3. Select a positive integerk≥4. 4. A’s public key is(n,k);A’s private key is(p,q). 11.113 AlgorithmESIGN signature generation and veriﬁcation SUMMARY: the signing algorithm computes an integerssuch thatskmod nlies in a cer- tain interval determined by the message. Veriﬁcation demonstrates thatskmod ndoes in- deed lie in the speciﬁed interval. 1. Signature generation.To sign a messagemwhich is a bitstring of arbitrary length, entityAshould do the following: (a) Computev= h(m). (b) Select a random secret integerx,0 ≤x<pq . (c) Computew= ⌈((v−xk)m o dn)/(pq)⌉and y= w·(kxk−1)−1 mod p. (d) Computes= x+ ypqmod n. (e)A’s signature formiss. 2. Veriﬁcation.To verifyA’s signatureson m,Bshould do the following: (a) ObtainA’s authentic public key(n,k). (b) Computeu= skmod nand z= h(m). (c) Ifz≤u≤z+2 ⌈2 3 lg n⌉, accept the signature; else reject it. Proof that signature veriﬁcation works.Note thatsk≡(x+ypq)k≡∑ k i=0 (k i ) xk−i(ypq)i ≡xk+ kypqxk−1 (mod n). Butkxk−1y≡w(mod p) and, thus,kxk−1y= w+ lpfor some l∈Z. Hence,sk ≡xk+ pq(w+ lp) ≡xk+ pqw≡xk+ pq ⌈ (h(m)−xk)modn pq ⌉ ≡ xk+ pq ( h(m)−xk+jn+ϵ pq ) (mod n), whereϵ=( xk−h(m)) modpq. Therefore,sk ≡ xk+ h(m) −xk+ ϵ≡h(m)+ ϵ(mod n). Since0 ≤ϵ<pq, it follows thath(m) ≤ skmod n≤h(m)+ pq≤h(m)+2 ⌈2 3 lg n⌉, as required. 474 Ch. 11 Digital Signatures 11.114 Example (ESIGN for artiﬁcially small parameters) In Algorithm 11.113, take messages to be integersm,0 ≤m<n , and the hash functionhto beh(m)= m. Key generation. Aselects primesp = 17389 and q = 15401,k =4 , and computes n= p2q= 4656913120721.A’s public key is(n= 4656913120721,k=4 );A’s private key is(p= 17389,q= 15401). Signature generation. To sign the messagem= 3111527988477,Acomputesv= h(m) = 3111527988477, and selectsx= 14222such that0 ≤x<pq .Athen computesw=⌈ ((v−xk)m o dn)/(pq) ⌉ = ⌈2848181921806/267807989⌉= ⌈10635.16414⌉= 10636 and y= w(kxk−1)−1 mod p= 10636(4 ×142223)−1 mod 17389 = 9567. Finally,A computes the signatures= x+ ypqmod n= 2562119044985. Signature veriﬁcation.BobtainsA’s public key(n= 4656913120721,k=4 ), and com- putesu= skmod n= 3111751837675. Since3111527988477 ≤3111751837675 ≤ 3111527988477 + 229,Baccepts the signature (here,⌈2 3 lg n⌉=2 9). □ 11.115 Note (security of ESIGN) (i) The modulusn= p2qin Algorithm 11.113 differs from an RSA modulus by having a repeated factor ofp. It is unknown whether or not moduli of this form are easier to factor than integers which are simply the product of two distinct primes. (ii) Given a valid signaturesfor a messagem, an adversary could forge a signature for a messagem′ifh(m′) is such thath(m′) ≤u≤h(m′)+2 ⌈2 3 lg n⌉(whereu= skmod n). If anm′with this property is found, thenswill be a signature for it. This will occur ifh(m) and h(m′) agree in the high-order(lg n)/3 bits. Assuming thath behaves like a random function, one would expect to try2(lg n)/3 different values of m′before observing this. (iii) Another possible approach to forging is to ﬁnd a pair of messagesmand m′such thath(m) and h(m′) agree in the high-order(lg n)/3 bits. By the birthday paradox (Fact 2.27(ii)), one can expect to ﬁnd such a pair inO(2(lg n)/6) trials. If an adversary is able to get the legitimate signer to signm, the same signature will be a signature form′. (iv) For the size of the integernnecessary to make the factorization ofninfeasible, (ii) and (iii) above are extremely unlikely possibilities. 11.116 Note (performance characteristics of ESIGN signatures) Signature generation in Algo- rithm 11.113 is very efﬁcient. For small values ofk(e.g.,k=4 ), the most computationally intensive part is the modular inverse required in step 1c. Depending on the implementation, this corresponds to a small number of modular multiplications with modulusp. Fork=4 and a768-bit modulusn, ESIGN signature generation may be between one and two orders of magnitude (10 to100 times) faster than RSA signature generation with an equivalent modulus size. Signature veriﬁcation is also very efﬁcient and is comparable to RSA with a small public exponent. 11.8 Signatures with additional functionality The mechanisms described in this section provide functionality beyond authentication and non-repudiation. In most instances, they combine a basic digital signature scheme (e.g., RSA) with a speciﬁc protocol to achieve additional features which the basic method does not provide. §11.8 Signatures with additional functionality 475 11.8.1 Blind signature schemes Rather than signature schemes as described in§11.2,blind signature schemesare two-party protocols between asenderAand asignerB. The basic idea is the following.Asends a piece of information toBwhich Bsigns and returns toA. From this signature,Acan compute B’s signature on an a priori messagemofA’s choice. At the completion of the protocol,Bknows neither the messagemnor the signature associated with it. The purpose of a blind signature is to prevent the signerBfrom observing the message it signs and the signature; hence, it is later unable to associate the signed message with the senderA. 11.117 Example (applications of blind signatures) Blind signature schemes have applications where the senderA(the customer) does not want the signerB(the bank) to be capable of associating a postiori a messagemand a signatureSB(m) to a speciﬁc instance of the protocol. This may be important in electronic cash applications where a messagemmight represent a monetary value whichAcan spend. Whenmand SB(m) are presented toB for payment,Bis unable to deduce which party was originally given the signed value. This allowsAto remain anonymous so that spending patterns cannot be monitored.□ A blind signature protocol requires the following components: 1. A digital signature mechanism for signerB.SB(x) denotes the signature ofBon x. 2. Functionsfand g(known only to the sender) such thatg(SB(f(m))) =SB(m).f is called ablinding function,gan unblinding function, andf(m) a blinded message. Property 2 places many restrictions on the choice ofSBand g. 11.118 Example (blinding function based on RSA) Letn= pqbe the product of two large ran- dom primes. The signing algorithmSB for entityBis the RSA signature scheme (Algo- rithm 11.19) with public key(n,e) and private keyd. Letkbe some ﬁxed integer with gcd(n,k)=1 . The blinding functionf: Zn−→Znis deﬁned byf(m)= m·kemod n and the unblinding functiong: Zn −→Znby g(m)= k−1mmod n. For this choice of f,g, andSB,g(SB(f(m))) = g(SB(mkemod n)) = g(mdkmod n)= mdmod n= SB(m), as required by property 2. □ Protocol 11.119 presents a blind signature scheme which uses the digital signature mechanism and functionsfand gdescribed in Example 11.118. 11.119 ProtocolChaum’s blind signature protocol SUMMARY: sender Areceives a signature ofBon a blinded message. From this,Acom- putesB’s signature on a messagemchosen a priori byA,0 ≤m≤n−1. Bhas no knowledge ofmnor the signature associated withm. 1. Notation.B’s RSA public and private keys are(n,e) and d, respectively.kis a ran- dom secret integer chosen byAsatisfying0 ≤k≤n−1 and gcd(n,k)=1 . 2. Protocol actions. (a) (blinding)Acomputesm∗= mkemod nand sends this toB. (b) (signing)Bcomputess∗=( m∗)dmod nwhich it sends toA. (c) (unblinding)Acomputess= k−1s∗mod n, which isB’s signature onm. 476 Ch. 11 Digital Signatures 11.8.2 Undeniable signature schemes Undeniable signature schemesare distinct from digital signatures in the sense of§11.2 in that the signature veriﬁcation protocol requires the cooperation of the signer. The following example describes two scenarios where an undeniable signature could be applied. 11.120 Example (scenarios for undeniable signatures) (i) EntityA(the customer) wishes to gain access to a secured area controlled by entity B(the bank). The secured area might, for example, be a safety-deposit box room.B requiresAto sign a time and date document before access is granted. IfAuses an undeniable signature, thenBis unable to prove (at some later date) to anyone thatA used the facility withoutA’s direct involvement in the signature veriﬁcation process. (ii) Suppose some large corporationAcreates a software package.Asigns the package and sells it to entityB, who decides to make copies of this package and resell it to a third partyC.Cis unable to verify the authenticity of the software without the coop- eration ofA. Of course, this scenario does not preventBfrom re-signing the package with its own signature but the marketing advantage associated with corporationA’s name is lost toB. It will also be easier to trace the fraudulent activity ofB. □ 11.121 AlgorithmKey generation for Algorithm 11.122 SUMMARY: each entity selects a private key and corresponding public key. Each entityAshould do the following: 1. Select a random primep=2 q+1 where qis also a prime. 2. (Select a generatorαfor the subgroup of orderqinZ∗ p.) 2.1 Select a random elementβ∈Z∗ p and computeα= β(p−1)/q mod p. 2.2 Ifα=1 then go to step 2.1. 3. Select a random integera∈{1,2,...,q −1}and computey= αamod p. 4. A’s public key is(p,α,y);A’s private key isa. 11.122 AlgorithmChaum-van Antwerpen undeniable signature scheme SUMMARY: Asigns a messagembelonging to the subgroup of orderqinZ∗ p . Any entity Bcan verify this signature with the cooperation ofA. 1. Signature generation.EntityAshould do the following: (a) Computes= mamod p. (b) A’s signature on messagemiss. 2. Veriﬁcation.The protocol forBto verifyA’s signatureson mis the following: (a)BobtainsA’s authentic public key(p,α,y). (b) Bselects random secret integersx1,x2 ∈{1,2,...,q −1}. (c)Bcomputesz= sx1 yx2 mod pand sendsztoA. (d) Acomputesw=(z)a−1 mod p(whereaa−1 ≡1( m o dq)) and sendswtoB. (e)Bcomputesw′= mx1 αx2 mod pand accepts the signature if and only ifw= w′. §11.8 Signatures with additional functionality 477 Proof that signature veriﬁcation works. w≡(z)a−1 ≡(sx1 yx2 )a−1 ≡(max1 αax2 )a−1 ≡mx1 αx2 ≡w′mod p, as required. Fact 11.123 states that, with high probability, an adversary is unable to causeBto ac- cept a fraudulent signature. 11.123 Fact(detecting forgeries of undeniable signatures) Suppose thatsis a forgery ofA’s sig- nature for a messagem, i.e.,s̸=mamod p. Then the probability ofBaccepting the sig- nature in Algorithm 11.122 is only1/q; this probability is independent of the adversary’s computational resources. 11.124 Note (disavowing signatures) The signerAcould attempt to disavow a (valid) signature constructed by Algorithm 11.122 in one of three ways: (i) refuse to participate in the veriﬁcation protocol of Algorithm 11.122; (ii) perform the veriﬁcation protocol incorrectly; or (iii) claim a signature a forgery even though the veriﬁcation protocol is successful. Disavowing a signature by following (i) would be considered as an obvious attempt at (wrongful) repudiation. (ii) and (iii) are more difﬁcult to guard against, and require a dis- avowal protocol (Protocol 11.125). Protocol 11.125 essentially applies the veriﬁcation protocol of Algorithm 11.122 twice and then performs a check to verify thatAhas performed the protocol correctly. 11.125 ProtocolDisavowal protocol for Chaum-van Antwerpen undeniable signature scheme SUMMARY: this protocol determines whether the signerAis attempting to disavow a valid signaturesusing Algorithm 11.122, or whether the signature is a forgery. 1. BobtainsA’s authentic public key(p,α,y). 2. Bselects random secret integersx1,x2 ∈{ 1,2,...,q −1}, and computesz = sx1 yx2 mod p, and sendsztoA. 3. Acomputesw=( z)a−1 mod p(whereaa−1 ≡1( m o dq)) and sendswtoB. 4. Ifw= mx1 αx2 mod p,Baccepts the signaturesand the protocol halts. 5. Bselects random secret integersx′ 1,x′ 2 ∈{1,2,...,q −1}, and computesz′= sx′ 1 yx′ 2 mod p, and sendsz′toA. 6. Acomputesw′=( z′)a−1 mod pand sendsw′toB. 7. Ifw′= mx′ 1 αx′ 2 mod p,Baccepts the signaturesand the protocol halts. 8. Bcomputesc=( wα−x2 )x′ 1 mod pand c′=( w′α−x′ 2 )x1 mod p.I fc= c′, thenB concludes thatsis a forgery; otherwise,Bconcludes that the signature is valid and Ais attempting to disavow the signatures. Fact 11.126 states that Protocol 11.125 achieves its desired objectives. 11.126 FactLetmbe a message and suppose thatsisA’s (purported) signature onm. (i) Ifsis a forgery, i.e.,s̸=mamod p, and ifAandBfollow Protocol 11.125 correctly, thenw= w′(and hence,B’s conclusion thatsis a forgery is correct). (ii) Suppose thatsis indeedA’s signature form, i.e.,s = mamod p. Suppose that Bfollows Protocol 11.125 correctly, but thatAdoes not. Then the probability that w= w′(and henceAsucceeds in disavowing the signature) is only1/q. 478 Ch. 11 Digital Signatures 11.127 Note (security of undeniable signatures) (i) The security of Algorithm 11.122 is dependent on the intractability of the discrete logarithm problem in the cyclic subgroup of orderqinZ∗ p(see§3.6.6). (ii) Suppose veriﬁerBrecords the messages exchanged in step 2 of Algorithm 11.122, and also the random valuesx1,x2 used in the protocol. A third partyCshould never accept this transcript fromBas a veriﬁcation of signatures. To see why this is the case, it sufﬁces to show howBcould contrive a successful transcript of step 2 of Al- gorithm 11.122 without the signerA’s participation.Bchooses a messagem, inte- gersx1,x2 and lin the interval[1,q−1], and computess=( (mx1 αx2 )l−1 y−x2 )x−1 1 mod p. The protocol message fromBtoAwould bez= sx1 yx2 mod p, and from AtoBwould bew= zlmod p. Algorithm 11.122 will acceptsas a valid signature ofAfor messagem. This argument demonstrates that signatures can only be veriﬁed by interacting directly with the signer. 11.8.3 Fail-stop signature schemes Fail-stop digital signaturesare digital signatures which permit an entityAto prove that a signature purportedly (but not actually) signed byAis a forgery. This is done by showing that the underlying assumption on which the signature mechanism is based has been com- promised. The ability to prove a forgery does not rely on any cryptographic assumption, but may fail with some small probability; this failure probability is independent of the comput- ing power of the forger. Fail-stop signature schemes have the advantage that even if a very powerful adversary can forge a single signature, the forgery can be detected and the signing mechanism no longer used. Hence, the termfail-then-stopis also appropriate. A fail-stop signature scheme should have the following properties: 1. If a signer signs a message according to the mechanism, then a veriﬁer upon checking the signature should accept it. 2. A forger cannot construct signatures that pass the veriﬁcation algorithm without do- ing an exponential amount of work. 3. If a forger succeeds in constructing a signature which passes the veriﬁcation test then, with high probability, the true signer can produce a proof of forgery. 4. A signer cannot construct signatures which are at some later time claimed to be for- geries. Algorithm 11.130 is an example of a fail-stop mechanism. As described, it is a one-time sig- nature scheme, but there are ways to generalize it to allow multiple signings; using authen- tication trees is one possibility (see§11.6.3). The proof-of-forgery algorithm is presented in Algorithm 11.134. 11.128 AlgorithmKey generation for Algorithm 11.130 SUMMARY: key generation is divided between entityAand a trusted third party (TTP). 1. The TTP should do the following: (a) Select primespand qsuch thatqdivides(p−1) and the discrete logarithm problem inZ∗ qis intractable. (b) (Select a generatorαfor the cyclic subgroupGofZ∗ p having orderq.) (i) Select a random elementg∈Z∗ p and computeα= g(p−1)/q mod p. (ii) Ifα=1 then go to step (i). §11.8 Signatures with additional functionality 479 (c) Select a random integera,1 ≤a≤q−1, and computeβ= αamod p. The integerais kept secret by the TTP. (d) Send(p,q,α,β ) in the clear to entityA. 2. EntityAshould do the following: (a) Select random secret integersx1,x2,y1,y2 in the interval[0,q−1]. (b) Computeβ1 = αx1 βx2 and β2 = αy1 βy2 mod p. (c)A’s public key is(β1,β2,p,q,α,β ); A’s private key is the quadruple x=( x1,x2,y1,y2). 11.129 Note (TTP’ s secret information) Assuming that the discrete logarithm problem in the sub- group of orderqinZ∗ pis intractable in Algorithm 11.128, the only entity which knowsa, the discrete logarithm ofβto the baseα, is the TTP. 11.130 AlgorithmFail-stop signature scheme (van Heijst-Pedersen) SUMMARY: this is a one-time digital signature scheme whose security is based on the dis- crete logarithm problem in the subgroup of orderqinZ∗ p. 1. Signature generation.To sign a messagem∈[0,q−1],Ashould do the following: (a) Computes1,m= x1 + my1 mod qand s2,m= x2 + my2 mod q. (b) A’s signature formis(s1,m,s2,m). 2. Veriﬁcation.To verifyA’s signature(s1,m,s2,m) on m,Bshould do the following: (a) ObtainA’s authentic public key(β1,β2,p,q,α,β ). (b) Computev1 = β1βm 2 mod pand v2 = αs1,m βs2,m mod p. (c) Accept the signature if and only ifv1 = v2. Proof that signature veriﬁcation works. v1 ≡ β1βm 2 ≡(αx1 βx2 )(αy1 βy2 )m≡αx1+my1 βx2+my2 ≡ αs1,m βs2,m ≡v2 (mod p). Algorithm 11.130 is a one-time signature scheme sinceA’s private key xcan be com- puted if two messages are signed using x. Before describing the algorithm for proof of forgery (Algorithm 11.134), a number of facts are needed. These are given in Fact 11.131 and illustrated in Example 11.132. 11.131 Fact(number of distinct quadruples representing a public key and a signature) Suppose thatA’s public key in Algorithm 11.130 is(β1,β2,p,q,α,β ) and private key is the quadru- ple x=( x1,x2,y1,y2). (i) There are exactlyq2 quadruples x′=( x′ 1,x′ 2 ,y′ 1,y′ 2) withx′ 1 ,x′ 2 ,y′ 1,y′ 2 ∈Zqwhich yield the same portion(β1,β2) of the public key. (ii) LetTbe the set ofq2 quadruples which yield the same portion of the public key (β1,β2). For eachm∈Zq, there are exactlyqquadruples inTwhich give the same signature(s1,m,s2,m) form(where a signature is as described in Algorithm 11.130). Hence, theq2 quadruples inTgive exactlyqdifferent signatures form. (iii) Letm′∈Zqbe a message different fromm. Then theqquadruples inTwhich yield A’s signature(s1,m,s2,m) form, yieldqdifferent signatures form′. 480 Ch. 11 Digital Signatures 11.132 Example (illustration of Fact 11.131) Letp=2 9 and q=7 . α=1 6 is a generator of the subgroup of orderqinZ∗ p. Takeβ= α5 mod 29 = 23. SupposeA’s private key is x=( 2,3,5,2);A’s public key isβ1 = α2β3 mod 29 = 7,β2 = α5β2 mod 29 = 16. The following table lists theq2 =4 9quadruples which give the same public key. 1603 2303 3003 4403 5103 6503 0203 1610 2310 3010 4410 5110 6510 0210 1624 2324 3024 4424 5124 6524 0224 1631 2331 3031 4431 5131 6531 0231 1645 2345 3045 4445 5145 6545 0245 1652 2352 3052 4452 5152 6552 0252 1666 2366 3066 4466 5166 6566 0266 If the49 quadruples of this table are used to sign the messagem=1 , exactlyq=7 sig- nature pairs(s1,m,s2,m) arise. The next table lists the possibilities and those quadruples which generate each signature. signature pair (2,6) (3,3) (4,0) (5,4) (6,1) (0,5) (1,2) quadruples 1610 1624 1631 1645 1652 1666 1603 2303 2310 2324 2331 2345 2352 2366 3066 3003 3010 3024 3031 3045 3052 4452 4466 4403 4410 4424 4431 4445 5145 5152 5166 5103 5110 5124 5131 6531 6545 6552 6566 6503 6510 6524 0224 0231 0245 0252 0266 0203 0210 The next table lists, for each messagem′∈Z7, all signature pairs for the7 quadruples which yieldA’s signature(0,5) form=1 . m′ quadruple 0 1 2 3 4 5 6 1666 16 05 64 53 42 31 20 2352 23 05 50 32 14 66 41 3045 30 05 43 11 56 24 62 4431 44 05 36 60 21 52 13 5124 51 05 22 46 63 10 34 6510 65 05 15 25 35 45 55 0203 02 05 01 04 00 03 06 □ 11.133 Note (probability of successful forgery in Algorithm 11.130) Suppose that an adversary (the forger) wishes to deriveA’s signature on some messagem′. There are two possibilities to consider. (i) The forger has access only to the signer’s public key (i.e., the forger is not in pos- session of a message and valid signature). By Fact 11.131(ii), the probability that the signature created by the adversary is the same asA’s signature form′is only q/q2 =1 /q; this probability is independent of the adversary’s computational re- sources. (ii) The forger has access to a messagemand a signature(s1,m,s2,m) created by the signer. By Fact 11.131(iii), the probability that the signature created by the adversary is the same asA’s signature form ′is only1/q; again, this probability is independent of the adversary’s computational resources. §11.9 Notes and further references 481 Suppose now that an adversary has forgedA’s signature on a message, and the signa- ture passed the veriﬁcation stage in Algorithm 11.130. The objective is thatAshould be able to prove that this signature is a forgery. The following algorithm shows howAcan, with high probability, use the forged signature to derive the secreta. Sinceawas supposed to have been known only to the TTP (Note 11.129), it serves as proof of forgery. 11.134 AlgorithmProof-of-forgery algorithm for Algorithm 11.130 SUMMARY: to prove that a signatures′=( s′ 1,m,s′ 2,m ) on a messagemis a forgery, the signer derives the integera=l o gαβwhich serves as proof of forgery. The signer (entityA) should do the following: 1. Compute a signature pairs=( s1,m,s2,m) for messagemusing its private key x (see Algorithm 11.128). 2. Ifs= s′return to step 1. 3. Compute a=( s1,m−s′ 1,m) ·(s2,m−s′ 2,m )−1 mod q. Proof that Algorithm 11.134 works.By Fact 11.131(iii), the probability thats = s′in step 1 of Algorithm 11.134 is1/q. From the veriﬁcation algorithm (Algorithm 11.130), αs1,m βs2,m ≡αs′ 1,m βs′ 2,m (mod p) orαs1,m−s′ 1,m ≡αa(s′ 2,m −s2,m) (mod p) ors1,m− s′ 1,m≡a(s′ 2,m −s2,m)( m o dq). Hence,a=( s1,m−s′ 1,m ) ·(s2,m−s′ 2,m )−1 mod q. 11.135 Remark (disavowing signatures) In order for a signer to disavow a signature that it created with Algorithm 11.134, an efﬁcient method for computing logarithms is required. 11.9 Notes and further references §11.1 The concept of a digital signature was introduced in 1976 by Difﬁe and Hellman [344, 345]. Although the idea of a digital signature was clearly articulated, no practical realization emerged until the 1978 paper by Rivest, Shamir, and Adleman [1060]. Digital signatures appear to have been independently discovered by Merkle [849, 850] but not published until 1978. One of Merkle’s contributions is discussed in§11.6.2. Other early research was due to Lamport [738], Rabin [1022, 1023], and Matyas [801]. A detailed survey on digital signatures is given by Mitchell, Piper, and Wild [882]. A thor- ough discussion of a selected subset of topics in the area is provided by Stinson [1178]. Other sources which provide a good overview are Meyer and Matyas [859], Goldwasser, Micali, and Rivest [484], Rivest [1054], and Schneier [1094]. §11.2 The original proposal for a digital signature scheme by Difﬁe and Hellman [344] consid- ered only digital signatures with message recovery. The ﬁrst discussion of digital signature schemes with appendix (although the term was not used per se) appears to be in the patent by Merkle and Hellman [553]. Davies and Price [308] and Denning [326] give brief intro- ductions to digital signatures but restrict the discussion to digital signature schemes with message recovery and one-time digital signature schemes. Mitchell, Piper, and Wild [882] and Stinson [1178] give abstract deﬁnitions of digital signature schemes somewhat less gen- eral than those given in§11.2. 482 Ch. 11 Digital Signatures Excellent discussions on attacks against signature schemes are provided by Goldwasser, Micali, and Rivest [484] and Rivest [1054]. The former refers to the discovery of a func- tionally equivalent signing algorithm asuniversal forgery, and separates chosen-message attacks intogeneric chosen-message attacksand directed chosen-message attacks. Many proposed digital signature schemes have been shown to be insecure. Among the most prominent of these are the Merkle-Hellman knapsack scheme proposed by Merkle and Hell- man [857], shown to be totally breakable by Shamir [1114]; the Shamir fast signature sch- eme [1109], shown to be totally breakable by Odlyzko [939]; and the Ong-Schnorr-Shamir (OSS) scheme [958], shown to be totally breakable by Pollard (see Pollard and Schnorr [988]). Naccache [914] proposed a modiﬁcation of the Ong-Schnorr-Shamir scheme to avoid the earlier attacks. §11.3 The RSA signature scheme (Algorithm 11.19), discovered by Rivest, Shamir, and Adleman [1060], was the ﬁrst practical signature scheme based on public-key techniques. The multiplicative property of RSA (§11.3.2(ii)) was ﬁrst exploited by Davida [302]. Den- ning [327] reports and expands on Davida’s attack and credits Moore with a simpliﬁcation. Gordon [515] uses the multiplicative property of RSA to show how to create public-key pa- rameters and associated (forged) certiﬁcates if the signing authority does not take adequate precautions. The existential attack on RSA signatures having certain types of redundancy (Example 11.21) is due to de Jonge and Chaum [313]. Evertse and van Heijst [381] consider other types of attacks on RSA signatures which also rely on the multiplicative property. The reblocking problem (§11.3.3(i)) is discussed by Davies and Price [308], who attribute the method of prescribing the form of the modulus to Guillou. An alternate way of con- structing an (even)t-bit modulusn= pqhaving a 1 in the high-order position followed by k0’s is the following. Construct an integeru=2 t+ w2t/2 for some randomly selected (t/2 −k)-bit integerw. Select a(t/2)-bit primep, and dividepintouto get a quotient qand a remainderr(i.e.,u= pq+ r). Ifqis a prime number, thenn= pqis an RSA modulus of the required type. For example, ift=1 4and k=3 , letu=2 14 + w27 where w=1 1.I fp=8 9, thenq= 199and n= pq= 17711. The binary representation ofnis 100010100101111. The Rabin public-key signature scheme (Algorithm 11.25) is due to Rabin [1023]. Veriﬁca- tion of signatures using the Rabin scheme is efﬁcient since only one modular multiplication is required (cf. Note 11.33). Beller and Yacobi [101] take advantage of this aspect in their authenticated key transport protocol (see§12.5.3). The modiﬁed-Rabin signature scheme (Algorithm 11.30) is derived from the RSA variant proposed by Williams [1246] (see also page 315). The purpose of the modiﬁcation is to provide a deterministic procedure for signing. A similar methodology is incorporated in ISO/IEC 9796 (§11.3.5). The modiﬁed scheme can be generalized to other even public ex- ponents besidese=2 .I fgcd(e,(p−1)(q−1)/4) = 1, then exponentiation byeis a permutation ofQ n. ISO/IEC 9796 [596] became an international standard in October of 1991. This standard provides examples based on both the RSA and Rabin digital signature mechanisms. Al- though the standard permits the use of any digital signature scheme with message recovery which provides at-bit signature for a⌊ t 2 ⌋-bit message, the design was speciﬁcally tailored for the RSA and Rabin mechanisms. For design motivation, see Guillou et al. [525]. At the time of publication of ISO/IEC 9796, no other digital signature schemes providing message recovery were known, but since then several have been found; see Koyama et al. [708]. §11.9 Notes and further references 483 ISO/IEC 9796 is effective for signing messages which do not exceed a length determined by the signature process. Quisquater [1015] proposed a method for extending the utility of ISO/IEC 9796 to longer messages. Brieﬂy, the modiﬁed scheme is as follows. Select a one- way hash functionhwhich maps bitstrings of arbitrary length tok-bitstrings. If the signing capability of ISO/IEC 9796 istbits andmis ann-bit message wheren>t , thenmis partitioned into two bitstringsmcand ms, wheremcis(n−t+ k) bits long. Computed= h(m) and formm′= ms∥d;m′is a string of bitlengtht. Signm′using ISO/IEC 9796 to getJ. The signature on messagemismc∥J. This provides a randomized digital signature mechanism with message recovery, where the hash function provides the randomization. §11.3.6 is from PKCS #1 [1072]. This document describes formatting for both encryption and digital signatures but only those details pertinent to digital signatures are mentioned here. The speciﬁcation does not include message recovery as ISO/IEC 9796 does. It also does not specify the size of the primes, how they should be generated, nor the size of public and private keys. It is suggested thate=3 or e=2 16 +1 are widely used. The only attacks mentioned in PKCS #1 (which the formatting attempts to prevent) are those by den Boer and Bosselaers [324], and Desmedt and Odlyzko [341]. §11.4 The Feige-Fiat-Shamir digital signature scheme (Algorithm 11.40), proposed by Feige, Fiat, and Shamir [383], is a minor improvement of the Fiat-Shamir signature scheme [395], requiring less computation and providing a smaller signature. Fiat and Shamir [395] prove that their scheme is secure against existential forgery provided that factoring is intractable and thathis a truly random function. Feige, Fiat, and Shamir [383] prove that their modi- ﬁcation has the same property. Note 11.44 was suggested by Fiat and Shamir [395]. Note 11.45 is due to Micali and Shamir [868], who suggest that only the modulusn Aof entityAneeds to be public ifv1,v2,...,v k are system-wide parameters. Since all entities have distinct moduli, it is unlikely thatvj ∈ Qn,1 ≤j ≤k, for many different values ofn. To overcome this problem, Micali and Shamir claim that some perturbation ofkpublic values is possible to ensure that the result- ing values are quadratic residues with respect to a particular modulus, but do not specify any method which provides the necessary perturbation. The GQ signature scheme (Algorithm 11.48) is due to Guillou and Quisquater [524]. §11.5 The DSA (Algorithm 11.56) is due to Kravitz [711] and was proposed as a Federal Informa- tion Processing Standard in August of 1991 by the U.S. National Institute for Science and Technology. It became the Digital Signature Standard (DSS) in May 1994, as speciﬁed in FIPS 186 [406]. Smid and Branstad [1157] comment that the DSA was selected based on a number of important factors: the level of security provided, the applicability of patents, the ease of export from the U.S., the impact on national security and law enforcement, and the efﬁciency in a number of government and commercial applications. They provide a comparison of the computational efﬁciencies of DSA and RSA and address a number of negative responses received during the FIPS public comment period. Naccache et al. [916] describe a number of techniques for improving the efﬁciency of the DSA. For example, the computation ofk −1 mod qin step 1c of Algorithm 11.56 can be re- placed by the random generation of an integerb, the computation ofu= bkmod qands= b·{h(m)+ ar}mod q. The signature is(r,s,u). The veriﬁer computesu−1 mod qand u−1smod q= ˜s. Veriﬁcation of the signature(r,˜s) now proceeds as in Algorithm 11.56. This variant might be useful for signature generation in chipcard applications where com- puting power is limited. Naccache et al. also propose the idea ofuse and throw coupons 484 Ch. 11 Digital Signatures which eliminate the need to computer=( αk mod p)m o dq. Since this exponentiation is the most computationally intensive portion of DSA signature generation, use and throw coupons greatly improve efﬁciency. Coupons require storage, and only one signature can be created for each coupon. If storage is limited (as is often the case), only a ﬁxed number of DSA signatures can be created with this method. B´eguin and Quisquater [82] show how to use an insecure server to aid in computations asso- ciated with DSA signature generation and veriﬁcation. The method accelerates the compu- tation of modular multiplication and exponentiation by using an untrusted auxiliary device to provide the majority of the computing. As such, it also applies to schemes other than DSA. Arazi [54] shows how to integrate a Difﬁe-Hellman key exchange into the DSA. The ElGamal digital signature scheme (Algorithm 11.64) was proposed in 1984 by ElGamal [368]. ElGamal [368], Mitchell, Piper, and Wild [882], and Stinson [1178] comment further on its security. Note 11.66(iv) is due to Bleichenbacher [153], as is Note 11.67(iii), which is a special case of the following more general result. Supposepis a prime,αis a generator ofZ∗ p, andy is the public key of entityAfor an instance of the ElGamal signature scheme. Suppose p−1= bqand logarithms in the subgroup of orderbinZ∗ p can be efﬁciently computed. Finally, suppose that a generatorβ= cqfor somec,0 <c<b , and an integertare known such thatβt≡α(mod p). For messagem, the pair(r,s) withr= βand s= t·{h(m) − cqz}mod (p−1) where zsatisﬁesαqz ≡yq (mod p) is a signature for messagemwhich will be accepted by Algorithm 11.64. Bleichenbacher also describes how a trapdoor could be constructed for the ElGamal signature scheme when system-wide parameterspand α are selected by a fraudulent trusted third party. Variations of the ElGamal signing equation described in§11.5.2 were proposed by ElGamal [366], Agnew, Mullin, and Vanstone [19], Kravitz [711], Schnorr [1098], and Yen and Laih [1259]. Nyberg and Rueppel [938] and, independently, Horster and Petersen [564], placed these variations in a much more general framework and compared their various properties. ElGamal signatures based on elliptic curves over ﬁnite ﬁelds were ﬁrst proposed by Koblitz [695] and independently by Miller [878] in 1985. A variation of the DSA based on elliptic curves and referred to as the ECDSA is currently being drafted for an IEEE standard. The Schnorr signature scheme (Algorithm 11.78), due to Schnorr [1098], is derived from an identiﬁcation protocol given in the same paper (see§10.4.4). Schnorr proposed a prepro- cessing method to improve the efﬁciency of the signature generation in Algorithm 11.78. Instead of generating a random integerkand computingα k mod pfor each signature, a small number of integerskiand αki mod p,1 ≤i≤t, are precomputed and stored, and subsequently combined and refreshed for each signature. De Rooij [315] showed that this preprocessing is insecure iftis small. Brickell and McCurley [207] proposed a variant of the Schnorr scheme. Their method uses a primepsuch thatp−1 is hard to factor, a prime divisorqofp−1, and an elementαof order qinZ ∗ p. The signing equation iss= ae+kmod (p−1) as opposed to the Schnorr equation s= ae+kmod q. While computationally less efﬁcient than Schnorr’s, this variant has the advantage that its security is based on the difﬁculty of two hard problems: (i) computing logarithms in the cyclic subgroup of orderqinZ∗ p; and (ii) factoringp−1. If either of these problems is hard, then the problem of computing logarithms inZ∗ p is also hard. Okamoto [949] describes a variant of the Schnorr scheme which he proves to be secure, provided that the discrete logarithm problem inZ∗ pis intractable and that correlation-free hash functions exist (no instance of a correlation-free hash function is yet known). Signa- §11.9 Notes and further references 485 ture generation and veriﬁcation are not signiﬁcantly more computationally intensive than in the Schnorr scheme; however, the public key is larger. The Nyberg-Rueppel scheme (Algorithm 11.81) is due to Nyberg and Rueppel [936]. For an extensive treatment including variants, see Nyberg and Rueppel [938]. They note that unlike RSA, this signature scheme cannot be used for encryption since the signing trans- formationShas a left inverse, namely, the veriﬁcation transformationV, butSis not the left inverse ofV; in other words,V(S(m)) = mfor allm∈Z p, butS(V(m)) ̸=mfor most m∈Zp. The second paper also deﬁnes the notion of strong equivalence between signature schemes (two signature schemes are calledstrongly equivalentif the signature on a messagemin one scheme can be transformed into the corresponding signature in the other scheme, without knowledge of the private key), and discusses how to modify DSA to provide message recovery. Some digital signature schemes make it easy to conceal information in the signature which can only be recovered by entities privy to the concealment method. Information communi- cated this way is said to besubliminaland the conveying mechanism is called asubliminal channel. Among the papers on this subject are those of Simmons [1139, 1140, 1147, 1149]. Simmons [1139] shows that if a signature requiresl1 bits to convey and providesl2 bits of security, thenl1 −l2 bits are available for the subliminal channel. This does not imply that alll1 −l2 bits can, in fact, be used by the channel; this depends on the signature mechanism. If a large proportion of these bits are available, the subliminal channel is said to bebroad- band; otherwise, it isnarrowband. Simmons [1149] points out that ElGamal-like signature schemes provide a broadband subliminal channel. For example, if the signing equation is s= k −1 ·{h(m) −ar}mod (p−1) where ais the private key known to both the signer and the recipient of the signature, thenkcan be used to carry the subliminal message. This has the disadvantage that the signer must provide the recipient with the private key, allow- ing the recipient to sign messages that will be accepted as having originated with the signer. Simmons [1147] describes narrowband channels for the DSA. §11.6 Rabin [1022] proposed the ﬁrst one-time signature scheme (Algorithm 11.86) in 1978. Lamport [738] proposed a similar mechanism, popularized by Difﬁe and Hellman [347], which does not require interaction with the signer for veriﬁcation. Difﬁe suggested the use of a one-way hash function to improve the efﬁciency of the method. For this reason, the mechanism is often referred to as theDifﬁe-Lamport scheme. Lamport [738] also describes a more efﬁcient method for one-time digital signatures, which was rediscovered by Bos and Chaum [172]. Bos and Chaum provide more substantial modiﬁcations which lead to a scheme that can be proven to be existentially unforgeable under adaptive chosen-message attack, provided RSA is secure. Merkle’s one-time signature scheme (Algorithm 11.92) is due to Merkle [853]; see also §15.2.3(vi). The modiﬁcation described in Note 11.95 is attributed by Merkle [853] to Win- ternitz. Bleichenbacher and Maurer [155] generalize the methods of Lamport, Merkle, and Winternitz through directed acyclic graphs and one-way functions. Authentication trees were introduced by Merkle [850, 852, 853] at the time when public- key cryptography was in its infancy. Since public-key cryptography and, in particular, dig- ital signatures had not yet been carefully scrutinized, it seemed prudent to devise alternate methods for providing authentication over insecure channels. Merkle [853] suggests that authentication trees provide as much versatility as public-key techniques and can be quite practical. An authentication tree, constructed by a single user to authenticate a large num- ber of public values, requires the user to either regenerate the authentication path values 486 Ch. 11 Digital Signatures at the time of use or to store all authentication paths and values in advance. Merkle [853] describes a method to minimize the storage requirements if public values are used in a pre- scribed order. The GMR scheme (Algorithm 11.102) is due to Goldwasser, Micali, and Rivest [484], who introduced the notion of a claw-free pair of permutations, and described the construction of a claw-free pair of permutations (Example 11.99) based on the integer factorization prob- lem. Combining the one-time signature scheme with tree authentication gives a digital sig- nature mechanism which Goldwasser, Micali and Rivest prove existentially unforgeable un- der an adaptive chosen-message attack. In order to make their scheme more practical, the tree authentication structure is constructed in such a way that the system must retain some information about preceding signatures (i.e.,memory historyis required). Goldreich [465] suggested modiﬁcations to both the general scheme and the example based on integer fac- torization (Example 11.99), removing the memory constraint and, in the latter, improving the efﬁciency of the signing procedure. Bellare and Micali [92] generalized the GMR sch- eme by replacing the claw-free pair of permutations by any trapdoor one-way permutation (the latter requiring a weaker cryptographic assumption). Naor and Yung [920] further gen- eralized the scheme by requiring only the existence of a one-way permutation. The most general result is due to Rompel [1068], who proved that digital signature schemes which are secure against an adaptive chosen-message attack exist if and only if one-way functions exist. Although attractive in theory (due to the fact that secure digital signatures can be re- duced to the study of a single structure), none of these methods seem to provide techniques as efﬁcient as RSA and other methods which, although their security has yet to be proven rigorously, have withstood all attacks to date. On-line/off-line digital signatures(see also§15.2.3(ix)) were introduced by Even, Goldre- ich, and Micali [377, 378] as a means to speed up the signing process in applications where computing resources are limited and time to sign is critical (e.g., chipcard applications). The method uses both one-time digital signatures and digital signatures arising from public-key techniques (e.g., RSA, Rabin, DSA). The off-line portion of the signature generation is to create a set of validation parameters for a one-time signature scheme such as the Merkle sch- eme (Algorithm 11.92), and to hash this set and sign the resulting hash value using a public- key signature scheme. Since the public-key signature scheme is computationally more in- tensive, it is done off-line. The off-line computations are independent of the message to be signed. The on-line portion is to sign the message using the one-time signature scheme and the validation parameters which were constructed off-line; this part of the signature process is very efﬁcient. Signatures are much longer than would be the case if only the public-key signature mechanism were used to sign the message directly and, consequently, bandwidth requirements are a disadvantage of this procedure. §11.7 The arbitrated digital signature scheme of Algorithm 11.109 is from Davies and Price [308], based on work by Needham and Schroeder [923]. ESIGN (Algorithm 11.113; see also§15.2.2(i)), proposed by Okamoto and Shiraishi [953], was motivated by the signature mechanism OSS devised earlier by Ong, Schnorr, and Sha- mir [958]. The OSS scheme was shown to be insecure by Pollard in a private communi- cation. Ong, Schnorr, and Shamir [958] modiﬁed their original scheme but this too was shown insecure by Estes et al. [374]. ESIGN bases its security on the integer factorization problem and the problem of solving polynomial inequalities. The original version [953] proposedk=2 as the appropriate value for the public key. Brickell and DeLaurentis [202] demonstrated that this choice was insecure. Their attack also extends to the casek=3 ; see Brickell and Odlyzko [209, p.516]. Okamoto [948] revised the method by requiring §11.9 Notes and further references 487 k≥4. No weaknesses for these values ofkhave been reported in the literature. Fujioka, Okamoto, and Miyaguchi [428] describe an implementation of ESIGN which suggests that it is twenty times faster than RSA signatures with comparable key and signature lengths. §11.8 Blind signatures (§11.8.1) were introduced by Chaum [242], who described the concept, desired properties, and a protocol for untraceable payments. The ﬁrst concrete realization of the protocol (Protocol 11.119) was by Chaum [243]. Chaum and Pedersen [251] provide a digital signature scheme which is a variant of the ElGamal signature mechanism (§11.5.2), using a signing equation similar to the Schnorr scheme (§11.5.3), but computationally more intensive for both signing and veriﬁcation. This signature technique is then used to provide a blind signature scheme. The concept of a blind signature was extended by Chaum [245] to blinding for unantici- pated signatures. Camenisch, Piveteau, and Stadler [228] describe a blind signature pro- tocol based on the DSA (Algorithm 11.56) and one based on the Nyberg-Rueppel scheme (Algorithm 11.81). Horster, Petersen, and Michels [563] consider a number of variants of these protocols. Stadler, Piveteau, and Camenisch [1166] extend the idea of a blind signa- ture to afair blind signaturewhere the signer in cooperation with a trusted third party can link the message and signature, and trace the sender. Chaum, Fiat, and Naor [250] propose a scheme foruntraceable electronic cash, which al- lows a participantAto receive an electronic cash token from a bank.Acan subsequently spend the token at a shopB, which need not be on-line with the bank to accept and verify the authenticity of the token. When the token is cashed at the bank byB, the bank is unable to associate it withA. If, however,Aattempts to spend the token twice (double-spending), A’s identity is revealed. Okamoto [951] proposes a divisible electronic cash scheme. Adi- visible electronic coinis an element which has some monetary value associated with it, and which can be used to make electronic purchases many times, provided the total value of all transactions does not exceed the value of the coin. Undeniable signatures (§11.8.2) were ﬁrst introduced by Chaum and van Antwerpen [252], along with a disavowal protocol (Protocol 11.125). Chaum [246] shows how to modify the veriﬁcation protocol for undeniable signatures (step 2 of Algorithm 11.122) to obtain a zero-knowledge veriﬁcation. One shortcoming of undeniable signature schemes is the possibility that the signer is un- available or refuses to co-operate so that the signature cannot be veriﬁed by a recipient. Chaum [247] proposed the idea of adesignated conﬁrmer signaturewhere the signer des- ignates some entity as a conﬁrmer of its signature. If the signer is unavailable or refuses to co-operate, the conﬁrmer has the ability to interact with a recipient of a signature in order to verify it. The conﬁrmer is unable to create signatures for the signer. Chaum [247] describes an example of designated conﬁrmer signatures based on RSA encryption. Okamoto [950] provides a more indepth analysis of this technique and gives other realizations. A convertible undeniable digital signature, introduced by Boyar et al. [181], is an unde- niable signature (§11.8.2) with the property that the signerAcan reveal a secret piece of information, causing all undeniable signatures signed byAto become ordinary digital sig- natures. These ordinary digital signatures can be veriﬁed by anyone using only the public key ofAand requiring no interaction withAin the veriﬁcation process; i.e., the signatures become self-authenticating. This secret information which is made available should not permit anyone to create new signatures which will be accepted as originating fromA.A s an application of this type of signature, consider the following scenario. EntityAsigns all documents during her lifetime with convertible undeniable signatures. The secret piece of 488 Ch. 11 Digital Signatures information needed to convert these signatures to self-authenticating signatures is placed in trust with her lawyerB. After the death ofA, the lawyer can make the secret information public knowledge and all signatures can be veriﬁed.Bdoes not have the ability to alter or create new signatures on behalf ofA. Boyar et al. [181] give a realization of the con- cept of convertible undeniable signatures using ElGamal signatures (§11.5.2) and describe how one can reveal information selectively to convert some, but not all, previously created signatures to self-authenticating ones. Chaum, van Heijst, and Pﬁtzmann [254] provide a method for constructing undeniable sig- natures which are unconditionally secure for the signer. Fail-stop signatures were introduced by Waidner and Pﬁtzmann [1227] and formally de- ﬁned by Pﬁtzmann and Waidner [971]. The ﬁrst constructions for fail-stop signatures used claw-free pairs of permutations (Deﬁnition 11.98) and one-time signature methods (see Pﬁtzmann and Waidner [972]). More efﬁcient techniques were provided by van Heijst and Pedersen [1201], whose construction is the basis for Algorithm 11.130; they describe three methods for extending the one-time nature of the scheme to multiple signings. Van Heijst, Pedersen, and Pﬁtzmann [1202] extended the idea of van Heijst and Pedersen to fail-stop signatures based on the integer factorization problem. Damg˚ard [298] proposed a signature scheme in which the signer can gradually and veriﬁ- ably release the signature to a veriﬁer. Chaum and van Heijst [253] introduced the concept of agroup signature. A group signature has the following properties: (i) only members of a predeﬁned group can sign messages; (ii) anyone can verify the validity of a signature but no one is able to identify which member of the group signed; and (iii) in case of disputes, the signature can be opened (with or without the help of group members) to reveal the identity of the group member who signed it. Chen and Pedersen [255] extended this idea to provide group signatures with additional function- ality. Chapter /1/2 Key Establishment Protocols Contents in Brief 12.1 Introduction............................. 489 12.2 Classiﬁcation and framework.................... 490 12.3 Key transport based on symmetric encryption........... 497 12.4 Key agreement based on symmetric techniques.......... 505 12.5 Key transport based on public-key encryption........... 506 12.6 Key agreement based on asymmetric techniques.......... 515 12.7 Secret sharing............................ 524 12.8 Conference keying ......................... 528 12.9 Analysis of key establishment protocols.............. 530 12.10 Notes and further references.................... 534 12.1 Introduction This chapter considers key establishment protocols and related cryptographic techniques which provide shared secrets between two or more parties, typically for subsequent use as symmetric keys for a variety of cryptographic purposes including encryption, message authentication, and entity authentication. The main focus is two-party key establishment, with the aid of a trusted third party in some cases. While many concepts extend naturally to multi-party key establishment including conference keying protocols, such protocols rapid- ly become more complex, and are considered here only brieﬂy, as is the related area of secret sharing. Broader aspects of key management, including distribution of public keys, certiﬁ- cates, and key life cycle issues, are deferred to Chapter 13. Relationships to other cryptographic techniques.Key establishment techniques known as key transport mechanisms directly employ symmetric encryption (Chapter 7) or public- key encryption (Chapter 8). Authenticated key transport may be considered a special case of message authentication (Chapter 9) with privacy, where the message includes a cryp- tographic key. Many key establishment protocols based on public-key techniques employ digital signatures (Chapter 11) for authentication. Others are closely related to techniques for identiﬁcation (Chapter 10). Chapter outline The remainder of this chapter is organized as follows.§12.2 provides background mate- rial including a general classiﬁcation, basic deﬁnitions and concepts, and a discussion of 489 490 Ch. 12 Key Establishment Protocols objectives.§12.3 and§12.4 discuss key transport and agreement protocols, respectively, based on symmetric techniques; the former includes several protocols involving an on-line trusted third party.§12.5 and§12.6 discuss key transport and agreement protocols, respec- tively, based on asymmetric techniques; the former includes protocols based on public-key encryption, some of which also employ digital signatures, while the latter includes selected variations of Difﬁe-Hellman key agreement.§12.7 and§12.8 consider secret sharing and conference keying, respectively.§12.9 addresses the analysis of key establishment proto- cols and standard attacks which must be countered.§12.10 contains chapter notes with ref- erences. The particular protocols discussed provide a representative subset of the large number of practical key establishment protocols proposed to date, selected according to a number of criteria including historical signiﬁcance, distinguishing merits, and practical utility, with particular emphasis on the latter. 12.2 Classiﬁcation and framework 12.2.1 General classiﬁcation and fundamental concepts 12.1 DeﬁnitionA protocolis a multi-party algorithm, deﬁned by a sequence of steps precisely specifying the actions required of two or more parties in order to achieve a speciﬁed objec- tive. 12.2 DeﬁnitionKey establishmentis a process or protocol whereby a shared secret becomes available to two or more parties, for subsequent cryptographic use. Key establishment may be broadly subdivided intokey transportand key agreement, as deﬁned below and illustrated in Figure 12.1. 12.3 DeﬁnitionA key transportprotocol or mechanism is a key establishment technique where one party creates or otherwise obtains a secret value, and securely transfers it to the other(s). 12.4 DeﬁnitionA key agreementprotocol or mechanism is a key establishment technique in which a shared secret is derived by two (or more) parties as a function of information con- tributed by, or associated with, each of these, (ideally) such that no party can predetermine the resulting value. Additional variations beyond key transport and key agreement exist, including various forms ofkey update, such askey derivationin§12.3.1. Key establishment protocols involving authentication typically require a set-up phase whereby authentic and possibly secret initial keying material is distributed. Most protocols have as an objective the creation of distinct keys on each protocol execution. In some cases, the initial keying material pre-deﬁnes a ﬁxed key which will result every time the protocol is executed by a given pair or group of users. Systems involving such static keys are insecure under known-key attacks (Deﬁnition 12.17). 12.5 DeﬁnitionKey pre-distributionschemes are key establishment protocols whereby the re- sulting established keys are completely determineda prioriby initial keying material. In §12.2 Classiﬁcation and framework 491 contrast,dynamic key establishmentschemes are those whereby the key established by a ﬁxed pair (or group) of users varies on subsequent executions. Dynamic key establishment is also referred to assession key establishment. In this case the session keys are dynamic, and it is usually intended that the protocols are immune to known-key attacks. key establishment key transport key agreement asymmetric techniques techniques symmetric key pre-distributiondynamic key establishment Figure 12.1:Simpliﬁed classiﬁcation of key establishment techniques. Use of trusted servers Many key establishment protocols involve a centralized or trusted party, for either or both initial system setup and on-line actions (i.e., involving real-time participation). This party is referred to by a variety of names depending on the role played, including:trusted third party,trusted server,authentication server,key distribution center(KDC), key translation center(KTC), andcertiﬁcation authority (CA). The various roles and functions of such trusted parties are discussed in greater detail in Chapter 13. In the present chapter, discus- sion is limited to the actions required of such parties in speciﬁc key establishment protocols. Entity authentication, key authentication, and key conﬁrmation It is generally desired that each party in a key establishment protocol be able to determine the true identity of the other(s) which could possibly gain access to the resulting key, imply- ing preclusion of any unauthorized additional parties from deducing the same key. In this case, the technique is said (informally) to providesecure key establishment. This requires both secrecy of the key, and identiﬁcation of those parties with access to it. Furthermore, the identiﬁcation requirement differs subtly, but in a very important manner, from that of entity authentication – here the requirement is knowledge of the identity of parties which may gain access to the key, rather than corroboration that actual communication has been established with such parties. Table 12.1 distinguishes various such related concepts, which are highlighted by the deﬁnitions which follow. While authenticationmay be informally deﬁned as the process of verifying that an identity is as claimed, there are many aspects to consider, including who, what, and when. Entity authenticationis deﬁned in Chapter 10 (Deﬁnition 10.1), which presents protocols providing entity authentication alone.Data origin authenticationis deﬁned in Chapter 9 (Deﬁnition 9.76), and is quite distinct. 492 Ch. 12 Key Establishment Protocols Authentication term Central focus authentication depends on context of usage entity authentication identity of a party, and aliveness at a given instant data origin authentication identity of the source of data (implicit) key authentication identity of party which may possibly share a key key conﬁrmation evidence that a key is possessed by some party explicit key authentication evidence an identiﬁed party possesses a given key Table 12.1:Authentication summary – various terms and related concepts. 12.6 DeﬁnitionKey authenticationis the property whereby one party is assured that no other party aside from a speciﬁcally identiﬁed second party (and possibly additional identiﬁed trusted parties) may gain access to a particular secret key. Key authentication is independent of the actual possession of such key by the second party, or knowledge of such actual possession by the ﬁrst party; in fact, it need not involve any action whatsoever by the second party. For this reason, it is sometimes referred to more precisely as(implicit) key authentication. 12.7 DeﬁnitionKey conﬁrmationis the property whereby one party is assured that a second (possibly unidentiﬁed) party actually has possession of a particular secret key. 12.8 DeﬁnitionExplicit key authenticationis the property obtained when both (implicit) key authentication and key conﬁrmation hold. In the case of explicit key authentication, an identiﬁed party is known to actually pos- sess a speciﬁed key, a conclusion which cannot otherwise be drawn. Encryption applica- tions utilizing key establishment protocols which offer only implicit key authentication of- ten begin encryption with an initial known data unit serving as an integrity check-word, thus moving the burden of key conﬁrmation from the establishment mechanism to the applica- tion. The focus in key authentication is the identity of the second party rather than the value of the key, whereas in key conﬁrmation the opposite is true. Key conﬁrmation typically involves one party receiving a message from a second containing evidence demonstrating the latter’s possession of the key. In practice, possession of a key may be demonstrated by various means, including producing a one-way hash of the key itself, use of the key in a (keyed) hash function, and encryption of a known quantity using the key. These techniques may reveal some information (albeit possibly of no practical consequence) about the value of the key itself; in contrast, methods using zero-knowledge techniques (cf.§10.4.1) allow demonstration of possession of a key while providing no additional information (beyond that previously known) regarding its value. Entity authentication is not a requirement in all protocols. Some key establishment protocols (such as unauthenticated Difﬁe-Hellman key agreement) providenone of entity authentication, key authentication, and key conﬁrmation. Unilateral key conﬁrmation may always be added e.g., by including a one-way hash of the derived key in a ﬁnal message. 12.9 DeﬁnitionAn authenticated key establishmentprotocol is a key establishment protocol (Deﬁnition 12.2) which provides key authentication (Deﬁnition 12.6). §12.2 Classiﬁcation and framework 493 12.10 Remark (combining entity authentication and key establishment) In a key establishment protocol which involves entity authentication, it is critical that the protocol be constructed to guarantee that the party whose identity is thereby corroborated is the same party with which the key is established. When this is not so, an adversary may enlist the aid of an unsuspecting authorized party to carry out the authentication aspect, and then impersonate that party in key establishment (and subsequent communications). Identity-based and non-interactive protocols Motivation for identity-based systems is provided in§13.4.3. 12.11 DeﬁnitionA key establishment protocol is said to beidentity-basedif identity informa- tion (e.g., name and address, or an identifying index) of the party involved is used as the party’s public key. A related idea (see§13.4.4) involves use of identity information as an input to the function which determines the established key. Identity-based authentication protocols may be deﬁned similarly. 12.12 DeﬁnitionA two-party key establishment protocol is said to bemessage-independentif the messages sent by each party are independent of any per-session time-variant data (dy- namic data) received from other parties. Message-independent protocols which furthermore involve no dynamic data in the key computation are simply key pre-distribution schemes (Deﬁnition 12.5). In general, dynamic data (e.g., that received from another party) is involved in the key computation, even in message-independent protocols. 12.13 Remark (message-independent vs. non-interactive) Message-independent protocols incl- ude non-interactive protocols (zero-pass and one-pass protocols, i.e., those involving zero or one message but no reply), as well as some two-pass protocols. Regarding inter-party communications, some speciﬁcation (explicit or otherwise) of the parties involved in key establishment is necessary even in zero-pass protocols. More subtlely, in protocols involv- ingtusers identiﬁed by a vector(i 1,...,i t), the ordering of indices may determine distinct keys. In other protocols (e.g., basic Difﬁe-Hellman key agreement or Protocol 12.53), the cryptographic data in one party’s message is independent of both dynamic data in other par- ties’ messages and of all party-speciﬁc data including public keys and identity information. 12.2.2 Objectives and properties Cryptographic protocols involving message exchanges require precise deﬁnition of both the messages to be exchanged and the actions to be taken by each party. The following types of protocols may be distinguished, based on objectives as indicated: 1. authentication protocol– to provide to one party some degree of assurance regarding the identity of another with which it is purportedly communicating; 2. key establishment protocol– to establish a shared secret; 3. authenticated key establishment protocol– to establish a shared secret with a party whose identity has been (or can be) corroborated. 494 Ch. 12 Key Establishment Protocols Motivation for use of session keys Key establishment protocols result in shared secrets which are typically called, or used to derive,session keys. Ideally, a session key is anephemeralsecret, i.e., one whose use is restricted to a short time period such as a single telecommunications connection (or ses- sion), after which all trace of it is eliminated. Motivation for ephemeral keys includes the following: 1. to limit available ciphertext (under a ﬁxed key) for cryptanalytic attack; 2. to limit exposure, with respect to both time period and quantity of data, in the event of (session) key compromise; 3. to avoid long-term storage of a large number of distinct secret keys (in the case where one terminal communicates with a large number of others), by creating keys only when actually required; 4. to create independence across communications sessions or applications. It is also desirable in practice to avoid the requirement of maintaining state information across sessions. Types of assurances and distinguishing protocol characteristics When designing or selecting a key establishment technique for use, it is important to con- sider what assurances and properties an intended application requires. Distinction should be made between functionality provided to a user, and technical characteristics which dis- tinguish mechanisms at the implementation level. (The latter are typically of little interest to the user, aside from cost and performance implications.) Characteristics which differen- tiate key establishment techniques include: 1. natureof the authentication. Any combination of the following may be provided: entity authentication, key authentication, and key conﬁrmation. 2. reciprocityof authentication. When provided, each of entity authentication, key au- thentication, and key conﬁrmation may beunilateralormutual(provided to one or both parties, respectively). 3. key freshness. A key isfresh(from the viewpoint of one party) if it can be guaranteed to be new, as opposed to possibly an old key being reused through actions of either an adversary or authorized party. This is related to key control (below). 4. key control. In some protocols (key transport), one party chooses a key value. In oth- ers (key agreement), the key is derived from joint information, and it may be desirable that neither party be able to control or predict the value of the key. 5. efﬁciency. Considerations include: (a) number of message exchanges (passes) required between parties; (b) bandwidth required by messages (total number of bits transmitted); (c) complexity of computations by each party (as it affects execution time); and (d) possibility of precomputation to reduce on-line computational complexity. 6. third party requirements. Considerations include (see§13.2.4): (a) requirement of an on-line (real-time), off-line, or no third party; (b) degree of trust required in a third party (e.g., trusted to certify public keys vs. trusted not to disclose long-term secret keys). 7. type of certiﬁcate used, if any. More generally, one may consider the manner by which initial keying material is distributed, which may be related to third party re- quirements. (This is often not of direct concern to a user, being an implementation detail typically providing no additional functionality.) §12.2 Classiﬁcation and framework 495 8. non-repudiation. A protocol may provide some type of receipt that keying material has been exchanged. 12.14 Remark (efﬁciency vs. security) The efﬁciency and security of cryptographic techniques are often related. For example, in some protocols a basic step is executed repeatedly, and security increases with the number of repetitions; in this case, the level of security attainable given a ﬁxed amount of time depends on the efﬁciency of the basic step. In the description of protocol messages, it is assumed that when the claimed source identity or source network address of a message is not explicitly included as a message ﬁeld, these are known by context or otherwise available to the recipient, possibly by (unspeciﬁed) additional cleartext ﬁelds. 12.2.3 Assumptions and adversaries in key establishment protocols To clarify the threats protocols may be subject to, and to motivate the need for speciﬁc protocol characteristics, one requires (as a minimum) an informal model for key establish- ment protocols, including an understanding of underlying assumptions. Attention here is restricted to two-party protocols, although the deﬁnitions and models may be generalized. Adversaries in key establishment protocols Communicating parties or entities in key establishment protocols are formally calledprin- cipals, and assumed to have unique names. In addition to legitimate parties, the presence of an unauthorized “third” party is hypothesized, which is given many names under various circumstances, including:adversary,intruder,opponent,enemy,attacker,eavesdropper, and impersonator. When examining the security of protocols, it is assumed that the underlying crypto- graphic mechanisms used, such as encryption algorithms and digital signatures schemes, are secure. If otherwise, then there is no hope of a secure protocol. An adversary is hypoth- esized to be not acryptanalystattacking the underlying mechanisms directly, but rather one attempting to subvert the protocol objectives by defeating the manner in which such mech- anisms are combined, i.e., attacking the protocol itself. 12.15 DeﬁnitionA passive attackinvolves an adversary who attempts to defeat a cryptographic technique by simply recording data and thereafter analyzing it (e.g., in key establishment, to determine the session key). Anactive attackinvolves an adversary who modiﬁes or injects messages. It is typically assumed that protocol messages are transmitted overunprotected(open) networks, modeled by an adversary able to completely control the data therein, with the ability to record, alter, delete, insert, redirect, reorder, and reuse past or current messages, and inject new messages. To emphasize this, legitimate parties are modeled as receiv- ing messages exclusively via intervening adversaries (on every communication path, or on some subset oftofn paths), which have the option of either relaying messages unaltered to the intended recipients, or carrying out (with no noticeable delay) any of the above actions. An adversary may also be assumed capable of engaging unsuspecting authorized parties by initiating new protocol executions. An adversary in a key establishment protocol may pursue many strategies, including attempting to: 496 Ch. 12 Key Establishment Protocols 1. deduce a session key using information gained by eavesdropping; 2. participate covertly in a protocol initiated by one party with another, and inﬂuence it, e.g., by altering messages so as to be able to deduce the key; 3. initiate one or more protocol executions (possibly simultaneously), and combine (in- terleave) messages from one with another, so as to masquerade as some party or carry out one of the above attacks; 4. without being able to deduce the session key itself, deceive a legitimate party regard- ing the identity of the party with which it shares a key. A protocol susceptible to such an attack is not resilient (see Deﬁnition 12.82). In unauthenticated key establishment, impersonation is (by deﬁnition) possible. In entity authentication, where there is no session key to attack, an adversary’s objective is to ar- range that one party receives messages which satisfy that party that the protocol has been run successfully with a party other than the adversary. Distinction is sometimes made between adversaries based on the type of information available to them. Anoutsideris an adversary with no special knowledge beyond that gen- erally available, e.g., by eavesdropping on protocol messages over open channels. Anin- sideris an adversary with access to additional information (e.g., session keys or secret par- tial information), obtained by some privileged means (e.g., physical access to private com- puter resources, conspiracy, etc.). Aone-time insiderobtains such information at one point in time for use at a subsequent time; apermanent insiderhas continual access to privileged information. Perfect forward secrecy and known-key attacks In analyzing key establishment protocols, the potential impact of compromise of various types of keying material should be considered, even if such compromise is not normally expected. In particular, the effect of the following is often considered: 1. compromise of long-term secret (symmetric or asymmetric) keys, if any; 2. compromise of past session keys. 12.16 DeﬁnitionA protocol is said to haveperfect forward secrecyif compromise of long-term keys does not compromise past session keys. The idea of perfect forward secrecy (sometimes calledbreak-backward protection)i s that previous trafﬁc is locked securely in the past. It may be provided by generating session keys by Difﬁe-Hellman key agreement (e.g., Protocol 12.57), wherein the Difﬁe-Hellman exponentials are based on short-term keys. If long-term secret keys are compromised, fu- ture sessions are nonetheless subject to impersonation by an active adversary. 12.17 DeﬁnitionA protocol is said to be vulnerable to aknown-key attackif compromise of past session keys allows either a passive adversary to compromise future session keys, or impersonation by an active adversary in the future. Known-key attacks on key establishment protocols are analogous to known-plaintext attacks on encryption algorithms. One motivation for their consideration is that in some environments (e.g., due to implementation and engineering decisions), the probability of compromise of session keys may be greater than that of long-term keys. A second motiva- tion is that when using cryptographic techniques of only moderate strength, the possibility exists that over time extensive cryptanalytic effort may uncover past session keys. Finally, in some systems, past session keys may be deliberately uncovered for various reasons (e.g., §12.3 Key transport based on symmetric encryption 497 after authentication, to possibly detect use of the authentication channel as a covert or hid- den channel). 12.3 Key transport based on symmetric encryption This section presents a selection of key establishment protocols based on key transport (i.e., transfer of a speciﬁc key chosena prioriby one party) using symmetric encryption. Re- lated techniques involving non-reversible functions are also presented. Discussion is sub- divided into protocols with and without the use of a trusted server, as summarized in Ta- ble 12.2. Some of these use time-variant parameters (timestamps, sequence numbers, or random numbers) or nonces as discussed in§10.3.1. →Properties server type use of number of ↓Protocol timestamps messages point-to-point key update none optional 1-3 Shamir’s no-key protocol none no 3 Kerberos KDC yes 4 Needham-Schroeder shared-key KDC no 5 Otway-Rees KDC no 4 Protocol 13.12 KTC no 3 Table 12.2:Key transport protocols based on symmetric encryption. 12.3.1 Symmetric key transport and derivation without a server Server-less key transport based on symmetric techniques may either require that the two parties in the protocol initially share a long-term pairwise secret or not, respectively illus- trated below by point-to-point key update techniques and Shamir’s no-key algorithm. Other illustrative techniques are also given. (i) Point-to-point key update using symmetric encryption Point-to-point key updatetechniques based on symmetric encryption make use of a long- term symmetric keyKshareda prioriby two partiesAandB. This key, initially distributed over a secure channel or resulting from a key pre-distribution scheme (e.g., see Note 12.48), is used repeatedly to establish new session keysW. Representative examples of point-to- point key transport techniques follow. Notation: r A,tA, andnA, respectively, denote a random number, timestamp, and se- quence number generated byA(see§10.3.1).Edenotes a symmetric encryption algorithm (see Remark 12.19). Optional message ﬁelds are denoted by an asterisk (). 1. key transport with one pass: A→B: EK(rA)( 1 ) The session key used isW= rA, and bothAand Bobtain implicit key authentica- tion. Additional optional ﬁelds which might be transferred in the encrypted portion include: a timestamp or sequence number to provide a freshness guarantee toB(see Remark 12.18); a ﬁeld containing redundancy, to provide explicit key authentication 498 Ch. 12 Key Establishment Protocols toBor facilitate message modiﬁcation detection (see Remark 12.19); and a target identiﬁer to prevent undetectable message replay back onAimmediately. Thus: A→B: EK(rA,tA∗,B∗)( 1 ′) If it is desired that both parties contribute to the session key,Bmay sendAan analo- gous message, with the session key computed asf(rA,rB). Choosingfto be a one- way function precludes control of the ﬁnal key value by either party, or an adversary who acquires one ofrA,rB. 2. key transport with challenge-response: A←B: nB (1) A→B: EK(rA,nB,B∗)( 2 ) If a freshness guarantee is desired but reliance on timestamps is not, a random number or sequence number, denotedn B here, may be used to replace the timestamp in the one-pass technique; the cost is an additional message. The session key is againW= rA. If it is required that the session keyWbe a function of inputs from both parties,A may insert a noncenA precedingnB in (2), and a third message may be added as below. (HererA,rB are random numbers serving as keying material, whilenA,nB are nonces for freshness.) A←B: nB (1) A→B: EK(rA,nA,nB,B∗)( 2 ) A←B: EK(rB,nB,nA,A∗)( 3 ) 12.18 Remark (key update vulnerabilities) The key update techniques above do not offer perfect forward secrecy, and fail completely if the long-term keyKis compromised. For this rea- son they may be inappropriate for many applications. The one-pass protocol is also subject to replay unless a timestamp is used. 12.19 Remark (integrity guarantees within encryption) Many authentication protocols which employ encryption, including the above key update protocols and Protocols 12.24, 12.26, and 12.29, require for security reasons that the encryption function has a built-in data in- tegrity mechanism (see Figure 9.8(b) for an example, and Deﬁnition§9.75) to detect mes- sage modiﬁcation. (ii) Point-to-point key update by key derivation and non-reversible functions Key update may be achieved by key transport as above, or bykey derivationwherein the derived session key is based on per-session random input provided by one party. In this case, there is also a single message: A→B: r A (1) The session key is computed asW= EK(rA). The technique provides to bothAand B implicit key authentication. It is, however, susceptible to known-key attacks; Remark 12.18 similarly applies. The random numberr A here may be replaced by other time-variant pa- rameters; for example, a timestamptA validated by the recipient by comparison to its local clock provides an implicit key freshness property, provided the long-term key is not com- promised. §12.3 Key transport based on symmetric encryption 499 Here Acould control the value ofW, forcing it to bexby choosingrA = DK(x). Since the technique itself does not require decryption,Emay be replaced by an appropriate keyed pseudorandom functionhK, in which case the session key may be computed asW= hK(rA), withrA a time-variant parameter as noted above. In the other techniques of§12.3.1(i) employing an encryption functionE, the conﬁ- dentiality itself of the encrypted ﬁelds other than the session keyWis not critical. A key derivation protocol which entirely avoids the use of an encryption function may offer po- tential advantages with respect to export restrictions. Protocol 12.20 is such a technique, which also provides authentication guarantees as stated. It uses two distinct functionsh and h′(generating outputs of different bitlengths), respectively, for message authentication and key derivation. 12.20 ProtocolAuthenticated Key Exchange Protocol 2 (AKEP2) SUMMARY: Aand Bexchange 3 messages to derive a session keyW. RESULT: mutual entity authentication, and implicit key authentication ofW. 1. Setup:Aand Bshare long-term symmetric keysK,K′(these should differ but need not be independent).hK is a MAC (keyed hash function) used for entity authenti- cation.h′ K′is a pseudorandom permutation or keyed one-way function used for key derivation. 2. Protocol messages. DeﬁneT=( B,A,rA,rB). A→B: rA (1) A←B: T,h K(T)( 2 ) A→B:( A,rB),hK(A,rB)( 3 ) W= h′ K ′(rB) 3. Protocol actions. Perform the following steps for each shared key required. (a)Aselects and sends toBa random numberrA. (b) Bselects a random numberrB and sends toAthe values(B,A,rA,rB), along with a MAC over these quantities generated usinghwith keyK. (c) Upon receiving message (2),Achecks the identities are proper, that therA re- ceived matches that in (1), and veriﬁes the MAC. (d) Athen sends toBthe values(A,rB), along with a MAC thereon. (e) Upon receiving (3),Bveriﬁes that the MAC is correct, and that the received valuerB matches that sent earlier. (f) BothAand Bcompute the session key asW= h′ K ′(rB). 12.21 Note (AKEP1 variant of Protocol 12.20) The following modiﬁcation of AKEP2 results in AKEP1 (Authenticated Key Exchange Protocol 1).Bexplicitly generates a random ses- sion keyWand probabilistically encrypts it usingh′underK′and random numberr. The quantity(r,W⊕h′ K ′(r)) is now included as a ﬁnal extra ﬁeld withinTand hK(T) in (2), and from whichAmay recoverW. As an optimization,r= rB. (iii) Key transport without a priori shared keys Shamir’s no-key algorithm (Protocol 12.22) is a key transport protocol which, using only symmetric techniques (although involving modular exponentiation), allows key establish- ment over an open channel without requiring either shared or public keys. Each party has only its own local symmetric key. The protocol provides protection from passive adver- saries only; it does not provide authentication. It thus solves the same problem as basic 500 Ch. 12 Key Establishment Protocols Difﬁe-Hellman (Protocol 12.47) – two parties sharing noa priorikeying material end up with a shared secret key, secure against passive adversaries – although differences include that it uses three messages rather than two, and provides key transport. 12.22 ProtocolShamir’s no-key protocol SUMMARY: users Aand Bexchange 3 messages over a public channel. RESULT: secretKis transferred with privacy (but no authentication) fromAtoB. 1. One-time setup (deﬁnition and publication of system parameters). (a) Select and publish for common use a primepchosen such that computation of discrete logarithms modulopis infeasible (see Chapter 3). (b) Aand Bchoose respective secret random numbersa,b, with1 ≤a,b≤p−2, each coprime top−1. They respectively computea−1 and b−1 mod p−1. 2. Protocol messages. A→B: Ka mod p (1) A←B:( Ka)b mod p (2) A→B:( Kab)a−1 mod p (3) 3. Protocol actions. Perform the following steps for each shared key required. (a)Achooses a random keyKfor transport toB,1 ≤K≤p−1. Acomputes Ka mod pand sendsBmessage (1). (b) Bexponentiates (modp) the received value byb, and sendsAmessage (2). (c)Aexponentiates (modp) the received value bya−1 mod p−1, effectively “un- doing” its previous exponentiation and yieldingKb mod p.Asends the result toBas message (3). (d) Bexponentiates (modp) the received value byb−1 mod p−1, yielding the newly shared keyKmod p. Use of ElGamal encryption for key transport (as per§12.5.1) with an uncertiﬁed public key sent in a ﬁrst message (which would by deﬁnition be safe from passive attack) achieves in two passes the same goals as the above three-pass algorithm. In this case, the key is transportedfromthe recipient of the ﬁrst messagetothe originator. 12.23 Remark (choice of cipher in Protocol 12.22) While it might appear that any commuta- tive cipher (i.e., cipher wherein the order of encryption and decryption is interchangeable) would sufﬁce in place of modular exponentiation in Protocol 12.22, caution is advised. For example, use of the Vernam cipher (§1.5.4) would be totally insecure here, as the XOR of the three exchanged messages would equal the key itself. 12.3.2 Kerberos and related server-based protocols The key transport protocols discussed in this section are based on symmetric encryption, and involve two communicating parties,Aand B, and a trusted server with which they share long-term pairwise secret keysa priori. In such protocols, the server either plays the role of akey distribution center(KDC) and itself supplies the session key, or serves as a key translation center(KTC), and makes a key chosen by one party available to the other, by re-encrypting (translating) it under a key shared with the latter. KDCs and KTCs are discussed further in§13.2.3. §12.3 Key transport based on symmetric encryption 501 (i) Kerberos authentication protocol Kerberosis the name given to all of the following: the distributed authentication service originating from MIT’s Project Athena, which includes speciﬁcations for data integrity and encryption; the software which implements it, and the processes executing such software; and the speciﬁc authentication protocol used therein. Focus here, and use of the term “Ker- beros”, is restricted to the protocol itself, which supports both entity authentication and key establishment using symmetric techniques and a third party. The basic Kerberos protocol involvesA(theclient),B(theserverand veriﬁer), and a trusted serverT(the Kerberosauthentication server). At the outsetAandBshare no secret, whileTshares a secret with each (e.g., a user password, transformed into a cryptographic key by an appropriate function). The primary objective is forBto verifyA’s identity; the establishment of a shared key is a side effect. Options include a ﬁnal message providing mutual entity authentication and establishment of an additional secret shared byAand B (asubsession keynot chosen byT). The protocol proceeds as follows.Arequests fromTappropriatecredentials(data items) to allow it to authenticate itself toB. Tplays the role of a KDC, returning toA a session key encrypted forAand aticketencrypted forB. The ticket, whichAforwards on toB, contains the session key andA’s identity; this allows authentication ofAtoB when accompanied by an appropriate message (theauthenticator) created byAcontaining a timestamp recently encrypted under that session key. 12.24 ProtocolBasic Kerberos authentication protocol (simpliﬁed)1 SUMMARY: Ainteracts with trusted serverTand partyB. RESULT: entity authentication ofAtoB(optionally mutual), with key establishment. 1. Notation. Optional items are denoted by an asterisk (*). Eis a symmetric encryption algorithm (see Remark 12.19). NA is a nonce chosen byA;TA is a timestamp fromA’s local clock. kis the session-key chosen byT, to be shared byAand B. Lindicates a validity period (called the “lifetime”). 2. One-time setup.Aand Tshare a keyKAT ; similarly,Band TshareKBT . Deﬁne ticketB def = EKBT (k,A,L); authenticator def = Ek(A,TA,A∗ subkey). 3. Protocol messages. A→T: A,B,NA (1) A←T:t i c k e tB,EKAT (k,NA,L,B)( 2 ) A→B:t i c k e tB,authenticator (3) A←B: Ek(TA,B∗ subkey)( 4 ) 4. Protocol actions. AlgorithmEincludes a built-in integrity mechanism, and protocol failure results if any decryption yields an integrity check failure. (a)Agenerates a nonceNA and sends toTmessage (1). (b) Tgenerates a new session keyk, and deﬁnes a validity period (lifetimeL) for the ticket, consisting of an ending time and optional starting time.Tencryptsk, the received nonce, lifetime, and received identiﬁer (B) usingA’s key.Talso creates a ticket secured usingB’s key containingk, received identiﬁer (A), and lifetime.Tsends toAmessage (2). 1The basic Kerberos (version 5) protocol between client and authentication server is given, with messages simpliﬁed (some non-cryptographic ﬁelds omitted) to allow focus on cryptographic aspects. 502 Ch. 12 Key Establishment Protocols (c)Adecrypts the non-ticket part of message (2) usingKAT to recover:k,NA, lifetimeL, and the identiﬁer of the party for which the ticket was actually cre- ated.Averiﬁes that this identiﬁer andNA match those sent in message (1), and savesLfor reference.Atakes its own identiﬁer and fresh timestampTA, optionally generates a secretAsubkey, and encrypts these usingkto form the authenticator.Asends toBmessage (3). (d) Breceives message (3), decrypts the ticket usingKBT yieldingkto allow de- cryption of the authenticator.Bchecks that: i. the identiﬁer ﬁelds (A) in the ticket and authenticator match; ii. the timestampTA in the authenticator is valid (see§10.3.1); and iii.B’s local time is within the lifetimeLspeciﬁed in the ticket. If all checks pass,Bdeclares authentication ofAsuccessful, and savesAsubkey (if present) as required. (e) (Optionally for mutual entity authentication:)Bconstructs and sends toAmes- sage (4) containingA’s timestamp from the authenticator (speciﬁcally exclud- ing the identiﬁerA, to distinguish it from the authenticator), encrypted usingk. Boptionally includes a subkey to allow negotiation of a subsession key. (f) (Optionally for mutual entity authentication:)Adecrypts message (4). If the timestamp within matches that sent in message (3),Adeclares authentication ofBsuccessful and savesBsubkey (if present) as required. 12.25 Note (security and options in Kerberos protocol) (i) Since timestamps are used, the hosts on which this protocol runs must provide both secure and synchronized clocks (see§10.3.1). (ii) If, as is the case in actual implementations, the initial shared keys are password-deriv- ed, then the protocol is no more secure than the secrecy of such passwords or their resistance to password-guessing attacks. (iii) Optional parametersAsubkey and Bsubkey allow transfer of a key (other thank) from AtoBor vice-versa, or the computation of a combined key using some function f(Asubkey,Bsubkey). (iv) The lifetime within the ticket is intended to allowAto re-use the ticket over a limited time period for multiple authentications toBwithout additional interaction withT, thus eliminating messages (1) and (2). For each such re-use,Acreates a new authen- ticator with a fresh timestamp and the same session keyk; the optional subkey ﬁeld is of greater use in this case. (ii) Needham-Schroeder shared-key protocol The Needham-Schroeder shared-key protocol is important primarily for historical reasons. It is the basis for many of the server-based authentication and key distribution protocols pro- posed since 1978, including Kerberos and Otway-Rees. It is an example of a protocol inde- pendent of timestamps, providing both entity authentication assurances and key establish- ment with key conﬁrmation. However, it is no longer recommended (see Remark 12.28). §12.3 Key transport based on symmetric encryption 503 12.26 ProtocolNeedham-Schroeder shared-key protocol SUMMARY: Ainteracts with trusted serverTand partyB. RESULT: entity authentication (AwithB); key establishment with key conﬁrmation. 1. Notation.Eis a symmetric encryption algorithm (see Remark 12.19). NA and NB are nonces chosen byAand B, respectively. kis a session key chosen by the trusted serverTforAand Bto share. 2. One-time setup.Aand Tshare a symmetric keyKAT ;Band TshareKBT . 3. Protocol messages. A→T: A,B,NA (1) A←T: EKAT (NA,B,k,E KBT (k,A)) (2) A→B: EKBT (k,A)( 3 ) A←B: Ek(NB)( 4 ) A→B: Ek(NB −1) (5) 4. Protocol actions. Aside from veriﬁcation of nonces, actions are essentially analogous to those in Kerberos (Protocol 12.24), and are not detailed here. 12.27 Note (functionality and options in Needham-Schroeder shared-key protocol) (i) The protocol providesAand Bwith a shared keykwith key authentication (due to the trusted server). (ii) Messages (4) and (5) provide entity authentication ofAtoB; entity authentication ofBtoAcan be obtained providedAcan carry out some redundancy check onNB upon decrypting message (4). (iii) If it is acceptable forAto re-use a keykwithB,Amay securely cache the data sent in message (3) along withk. Upon subsequent re-use, messages (1) and (2) may then be omitted, but now to prevent replay of old messages (4), an encrypted nonceEk(NA ′) should be appended to message (3), and message (4) should be replaced byEk(NA ′− 1,NB) allowingAto verifyB’s current knowledge ofk(thereby providing entity authentication). 12.28 Remark (Needham-Schroeder weakness vs. Kerberos) The essential differences between Protocol 12.26 and Kerberos (Protocol 12.24) are as follows: the Kerberos lifetime param- eter is not present; the data of message (3), which corresponds to the Kerberos ticket, is un- necessarily double-encrypted in message (2) here; and authentication here employs nonces rather than timestamps. A weakness of the Needham-Schroeder protocol is that sinceB has no way of knowing if the keykis fresh, should a session keykever be compromised, any party knowing it may both resend message (3) and compute a correct message (5) to impersonateAtoB. This situation is ameliorated in Kerberos by the lifetime parameter which limits exposure to a ﬁxed time interval. (iii) Otway-Rees protocol The Otway-Rees protocol is a server-based protocol providing authenticated key transport (with key authentication and key freshness assurances) in only 4 messages – the same as Kerberos, but here without the requirement of timestamps. It does not, however, provide entity authentication or key conﬁrmation. 504 Ch. 12 Key Establishment Protocols 12.29 ProtocolOtway-Rees protocol SUMMARY: Binteracts with trusted serverTand partyA. RESULT: establishment of fresh shared secretKbetweenAand B. 1. Notation.Eis a symmetric encryption algorithm (see Remark 12.19).kis a session key Tgenerates forAand Bto share.NA and NB are nonces chosen byAand B, respectively, to allow veriﬁcation of key freshness (thereby detecting replay).Mis a second nonce chosen byAwhich serves as a transaction identiﬁer. 2. One-time setup.Tshares symmetric keysKAT and KBT withA,B, respectively. 3. Protocol messages. A→B: M,A,B,E KAT (NA,M,A,B )( 1 ) B→T: M,A,B,E KAT (NA,M,A,B ),EKBT (NB,M,A,B )( 2 ) B←T: EKAT (NA,k),EKBT (NB,k)( 3 ) A←B: EKAT (NA,k)( 4 ) 4. Protocol actions.Perform the following steps each time a shared key is required. (a)Aencrypts data for the server containing two nonces,NA and M, and the iden- tities of itself and the partyBto whom it wishes the server to distribute a key. Asends this and some plaintext toBin message (1). (b) Bcreates its own nonceNB and an analogous encrypted message (with the same M), and sends this along withA’s message toTin message (2). (c)Tuses the cleartext identiﬁers in message (2) to retrieveKAT and KBT , then veriﬁes the cleartext (MA,B) matches that recovered upon decrypting both parts of message (2). (VerifyingMin particular conﬁrms the encrypted parts are linked.) If so,Tinserts a new keykand the respective nonces into distinct messages encrypted forAand B, and sends both toBin message (3). (d) Bdecrypts the second part of message (3), checksNB matches that sent in mes- sage (2), and if so passes the ﬁrst part on toAin message (4). (e)Adecrypts message (4) and checksNA matches that sent in message (1). If all checks pass, each ofAand Bare assured thatkis fresh (due to their respective nonces), and trust that the other partyTsharedkwith is the party bound to their nonce in message (2).Aknows thatBis active as veriﬁcation of message (4) impliesBsent message (2) recently;Bhowever has no assurance thatAis active until subsequent use ofkby A, sinceBcannot determine if message (1) is fresh. 12.30 Remark (nonces in Otway-Rees protocol) The use of two nonces generated byAis redun- dant (NA could be eliminated in messages (1) and (2), and replaced byMin (3) and (4)), but nonetheless allowsMto serve solely as an administrative transaction identiﬁer, while keeping the format of the encrypted messages of each party identical. (The latter is gener- ally considered desirable from an implementation viewpoint, but dubious from a security viewpoint.) 12.31 Remark (extension of Otway-Rees protocol) Protocol 12.29 may be extended to provide both key conﬁrmation and entity authentication in 5 messages. Message (4) could be aug- mented to both demonstrateB’s timely knowledge ofkand transfer a nonce toA(e.g., appendingE k(NA,NB)), with a new ﬁfth message (A→B: Ek(NB)) providingBre- ciprocal assurances. §12.4 Key agreement based on symmetric techniques 505 12.4 Key agreement based on symmetric techniques This section presents ideas related to key agreement based on symmetric techniques. It also presents a key pre-distribution system which is in some ways a symmetric-key analogue to Difﬁe-Hellman key agreement with ﬁxed exponentials (Note 12.48). 12.32 DeﬁnitionA key distribution system(KDS) is a method whereby, during an initialization stage, a trusted server generates and distributes secret data values (pieces) to users, such that any pair of users may subsequently compute a shared key unknown to all others (aside from the server). For ﬁxed pairwise keys, a KDS is a key pre-distribution scheme. A trivial KDS is as follows: the trusted server chooses distinct keys for each pair among thenusers, and by some secure means initially distributes to each user itsn−1 keys appropriately labeled. This provides unconditional security (perfect security in the information-theoretic sense); an outside adversary can do no better than guess the key. However, due to the large amount of storage required, alternate methods are sought, at the price of losing unconditional secu- rity against arbitrarily large groups of colluding users. 12.33 DeﬁnitionA KDS is said to bej-secureif, given a speciﬁed pair of users, any coalition of jor fewer users (disjoint from the two), pooling their pieces, can do no better at computing the key shared by the two than a party which guesses the key without any pieces whatsoever. A j-secure KDS is thus unconditionally secure against coalitions of sizejor smaller. 12.34 Fact(Blom’ s KDS bound) In anyj-secure KDS providingm-bit pairwise session keys, the secret data stored by each user must be at leastm·(j+1 )bits. The trivial KDS described above is optimal with respect to the number of secret key bits stored, assuming collusion by all parties other than the two directly involved. This cor- responds to meeting the lower bound of Fact 12.34 forj= n−2. Blom’s symmetric key pre-distribution system Blom’s scheme (Mechanism 12.35) is a KDS which can be used to meet the bound of Fact 12.34 for valuesj<n −2. It is non-interactive; each party requires only an indexi, 1 ≤i≤n, which uniquely identiﬁes the party with which it is to form a joint key (the sch- eme is identity-based in this regard). Each user is assigned a secret vector of initial keying material (base key) from which it is then able to compute a pairwise secret (derived key) with each other user. As outlined in Remark 12.37, the scheme may be engineered to provide unconditional security against coalitions of a speciﬁed maximum size. The initial keying material as- signed to each user (a row ofS, corresponding tokkeys) allows computation of a larger number of derived keys (a row ofK, providingnkeys), one per each other user. Storage savings results from choosingkless thann. The derived keys of different user pairs, how- ever, are not statistically independent. 506 Ch. 12 Key Establishment Protocols 12.35 Mechanism Blom’s symmetric key pre-distribution system SUMMARY: each of nusers is given initial secret keying material and public data. RESULT: each pair of usersUi,Uj may compute anm-bit pairwise secret keyKi,j. 1. A k×ngenerator matrixGof an(n,k) MDS code over a ﬁnite ﬁeldFq of orderq is made known to allnsystem users (see Note 12.36). 2. A trusted partyTcreates a random secretk×ksymmetric matrixDoverFq. 3. Tgives to each userUi the secret keySi, deﬁned as rowiof then×kmatrixS= (DG)T .(Si is ak-tuple overFq ofk·lg(q) bits, allowingUi to compute any entry in rowiof(DG)T G.) 4. UsersUi and Uj compute the common secretKi,j = Kj,i of bitlengthm=l g (q) as follows. UsingSi and columnjofG,Ui computes the(i,j) entry of then×nsym- metric matrixK=( DG)T G. UsingSj and columniofG,Uj similarly computes the(j,i) entry (which is equal to the(i,j) entry sinceKis symmetric). 12.36 Note (background on MDS codes) The motivation for Mechanism 12.35 arises from well- known concepts in linear error-correcting codes, summarized here. LetG=[ IkA] be a k×nmatrix where each row is ann-tuple overFq (forqa prime or prime power).Ik is the k×kidentity matrix. The set ofn-tuples obtained by taking all linear combinations (over Fq) of rows ofGis thelinear code C. Each of theseqk n-tuples is acodeword, andC= {c: c= mG,m =( m1 m2 ...m k),mi ∈Fq}. Gis agenerator matrixfor the linear (n,k) code C. The distancebetween two codewordsc,c′is the number of components they differ in; the distancedof the code is the minimum such distance over all pairs of distinct codewords. A code of distancedcan correcte= ⌊(d−1)/2⌋component errors in a codeword, and for linear codesd≤n−k+1 (theSingleton bound). Codes meeting this bound with equality (d= n−k+1 ) have the largest possible distance for ﬁxednand k, and are calledmaximum distance separable(MDS) codes. 12.37 Remark (choice of k in Blom’ s scheme) The conditiond= n−k+1 deﬁning MDS codes can be shown equivalent to the condition that every set ofkcolumns ofGis linearly inde- pendent. From this, two facts follow about codewords of MDS codes: (i) anykcomponents uniquely deﬁne a codeword; and (ii) anyj≤k−1 components provide no information about other components. For Mechanism 12.35, the choice ofk is governed by the fact that ifk or more users conspire, they are able to recover the secret keys of all other users. (kconspirators may computekrows ofK, or equivalentlykcolumns, corresponding tok components in each row. Each row is a codeword in the MDS code generated byG, and corresponds to the key of another user, and by the above remarkkcomponents thus deﬁne all remaining components of that row.) However, if fewer thankusers conspire, they obtain no information whatsoever about the keys of any other user (by similar reasoning). Thus Blom’s scheme isj-secure forj≤k−1, and relative to Fact 12.34, is optimal with respect to the amount of initial keying material required. 12.5 Key transport based on public-key encryption Key transport based on public-key encryption involves one party choosing a symmetric key, and transferring it to a second, using that party’s encryption public key. This provides key §12.5 Key transport based on public-key encryption 507 authentication to the originator (only the intended recipient has the private key allowing de- cryption), but the originator itself obtains neither entity authentication nor key conﬁrmation. The second party receives no source authentication. Such additional assurances may be ob- tained through use of further techniques including: additional messages (§12.5.1); digital signatures (§12.5.2); and symmetric encryption in addition to signatures (§12.5.3). Authentication assurances can be provided with or without the use of digital signatures, as follows: 1. entity authentication via public-key decryption(§12.5.1). The intended recipient au- thenticates itself by returning some time-variant value which it alone may produce or recover. This may allow authentication of both the entity and a transferred key. 2. data origin authentication via digital signatures(§12.5.2). Public-key encryption is combined with a digital signature, providing key transport with source identity assur- ances. The distinction between entity authentication and data origin authentication is that the for- mer provides a timeliness assurance, whereas the latter need not. Table 12.3 summarizes the protocols presented. →Properties signatures entity number of ↓Protocol required‡ authentication messages basic PK encryption (1-pass) no no 1 Needham-Schroeder PK no mutual 3 encrypting signed keys yes data origin only† 1 separate signing, encrypting yes data origin only† 1 signing encrypted keys yes data origin only† 1 X.509 (2-pass) – timestamps yes mutual 2 X.509 (3-pass) – random #’s yes mutual 3 Beller-Yacobi (4-pass) yes mutual 4 Beller-Yacobi (2-pass) yes unilateral 2 Table 12.3:Selected key transport protocols based on public-key encryption. †Unilateral entity authentication may be achieved if timestamps are included. ‡Schemes using public keys transported by certiﬁcates require signatures for veriﬁcation thereof, but signatures are not required within protocol messages. 12.5.1 Key transport using PK encryption without signatures One-pass key transport by public-key encryption One-pass protocols are appropriate for one-way communications and store-and-forward ap- plications such as electronic mail and fax. Basic key transport using public-key encryption can be achieved in a one-pass protocol, assuming the originatorApossessesa priorian authentic copy of the encryption public key of the intended recipientB. UsingB’s pub- lic encryption key,Aencrypts a randomly generated keyk, and sends the resultP B(k) to B. Public-key encryption schemesPB of practical interest here include RSA encryption, Rabin encryption, and ElGamal encryption (see Chapter 8). The originatorAobtains no entity authentication of the intended recipientB(and in- deed, does not know ifBeven receives the message), but is assured of implicit key au- thentication – no one aside fromBcould possibly recover the key. On the other hand, Bhas no assurances regarding the source of the key, which remains true even in the case 508 Ch. 12 Key Establishment Protocols A→B: PB(k,A). A timeliness guarantee may be provided using timestamps, for ex- ample,A→B: PB(k,TA). This is necessary if security against known-key attacks is required, as this technique is otherwise vulnerable to message replay (cf. Remark 12.18). Maintaining the restriction of using public-key encryption alone (i.e., without signa- tures), assurances in addition to unilateral key authentication, namely, mutual entity au- thentication, and mutual key authentication, may be obtained through additional messages as illustrated by Protocol 12.38 below. Needham-Schroeder public-key protocol The Needham-Schroeder public-key protocol provides mutual entity authentication and mutual key transport (Aand Beach transfer a symmetric key to the other). The trans- ported keys may serve both as nonces for entity authentication and secret keys for further use. Combination of the resulting shared keys allows computation of a joint key to which both parties contribute. 12.38 ProtocolNeedham-Schroeder public-key protocol SUMMARY: Aand Bexchange 3 messages. RESULT: entity authentication, key authentication, and key transport (all mutual). 1. Notation. PX(Y) denotes public-key encryption (e.g., RSA) of dataYusing party X’s public key;PX(Y1,Y2) denotes the encryption of the concatenation ofY1 and Y2.k1,k2 are secret symmetric session keys chosen byA,B, respectively. 2. One-time setup. Assume A,Bpossess each other’s authentic public-key. (If this is not the case, but each party has a certiﬁcate carrying its own public key, then one additional message is required for certiﬁcate transport.) 3. Protocol messages. A→B: P B(k1,A)( 1 ) A←B: PA(k1,k2)( 2 ) A→B: PB(k2)( 3 ) 4. Protocol actions. (a)AsendsBmessage (1). (b) Brecoversk1 upon receiving message (1), and returns toAmessage (2). (c) Upon decrypting message (2),Achecks the keyk1 recovered agrees with that sent in message (1). (Providedk1 has never been previously used, this givesA both entity authentication ofBand assurance thatBknows this key.)Asends Bmessage (3). (d) Upon decrypting message (3),Bchecks the keyk2 recovered agrees with that sent in message (2). The session key may be computed asf(k1,k2) using an appropriate publicly known non-reversible functionf. 12.39 Note (modiﬁcation of Needham-Schroeder protocol) Protocol 12.38 may be modiﬁed to eliminate encryption in the third message. Letr1 and r2 be random numbers generated respectively byAand B. Then, with checks analogous to those in the basic protocol, the messages in the modiﬁed protocol are: A→B: PB(k1,A,r1)( 1 ′) A←B: PA(k2,r1,r2)( 2 ′) A→B: r2 (3′) §12.5 Key transport based on public-key encryption 509 12.5.2 Protocols combining PK encryption and signatures While privacy of keying material is a requirement in key transport protocols, source au- thentication is also typically needed. Encryption and signature primitives may respectively be used to provide these properties. Key transport protocols involving both public-key en- cryption and signatures include: 1. those which sign the key, then public-key encrypt the signed key; 2. those which sign the key, and separately public-key encrypt the (unsigned) key; 3. those which public-key encrypt the key, then sign the encrypted key; and 4. those using symmetric encryption in addition to public-key encryption and signa- tures. The ﬁrst three types are discussed in this subsection (as noted in§12.5.2(ii), the second is secure only in certain circumstances); the fourth is discussed in§12.5.3. The signature sch- emes S A of greatest practical interest are RSA, Rabin signatures, and ElGamal-family sig- natures (see Chapter 11). The public-key encryption schemesPB of greatest practical in- terest are RSA, Rabin encryption, and ElGamal encryption (see Chapter 8). Notation. For data inputy, in what follows,SA(y) and PB(y) denote the data values resulting, respectively, from the signature operation onyusingA’s signature private key, and the encryption operation onyusingB’s encryption public key. As a default, it is as- sumed that the signature scheme does not provide message recovery, i.e., the inputycannot be recovered from the signatureSA(y), andymust be sent explicitly in addition toSA(y) to allow signature veriﬁcation. (This is the case for DSA, or RSA following input hashing; see Chapter 11. However, in the case of encrypting and signing separately, any secret data ymust remain conﬁdential.) Ifyconsists of multiple data valuesy=( y 1,...,y n), then the input is taken to be the bitwise concatenation of these multiple values. (i) Encrypting signed keys One option for combining signatures and public-key encryption is to encrypt signed blocks: A→B: PB(k,tA∗,SA(B,k,tA∗)) The asterisk denotes that the timestamptA ofAis optional; inclusion facilitates entity au- thentication ofAtoBand provides a freshness property. The identiﬁerBwithin the scope of the signature preventsBfrom sending the signed key on to another party and imper- sonatingA. A disadvantage of this method over the “signing encrypted keys” alternative (§12.5.2(iii)) is that here the data to be public-key encrypted is larger, implying the possible requirement of adjusting the block size of the public-key encryption scheme, or the use of techniques such as cipher-block-chaining. In the case of signature schemes with message recovery (e.g., ordinary RSA), the above can be simpliﬁed to: A→B: PB(SA(B,k,tA∗)) (ii) Encrypting and signing separately For signature schemes without message recovery, a variation of the above option is to sign the key and encrypt the key, but not to encrypt the signature itself. This is acceptable only if the signature scheme is such that no information regarding plaintext data can be deduced from the signature itself on that data (e.g., when the signature operation involves prelimi- nary one-way hashing). This is critical because, in general, data may be recovered from a signature on it (e.g., RSA without hashing). A summary of this case is then as follows: A→B: P B(k,tA∗),SA(B,k,tA∗) 510 Ch. 12 Key Establishment Protocols If the keykis used solely to encrypt a data ﬁley, then the signatureSA may be overy instead ofk. This is suitable instore-and-forwardenvironments. The encrypted ﬁle may then be transferred along with the key establishment information, in which caseyis ﬁrst recovered by usingkto decrypt the ﬁle, and then the signature onyis veriﬁed. (iii) Signing encrypted keys In contrast to encrypting signed keys, one may sign encrypted keys: A→B: tA∗,PB(A,k),SA(B,tA∗,PB(A,k)) The asterisk denotes that the timestamptA ofAis optional; inclusion facilitates entity au- thentication ofAtoB. The parameterAwithin the scope of the public-key encryption preventssignature stripping– simply signing a publicly-encrypted key, e.g.,SA(PB(k)) is vulnerable to a third partyCextracting the encrypted quantityPB(k) and then oversign- ing with its own key, thus defeating authentication (cf. Note 12.42). Furthermore, the en- cryption mechanism must ensure that an adversaryCwithout access tok, cannot change PB(A,k) toPB(C,k); see Remark 12.19. It is desirable and assumed that the combined length of the parametersAand knot exceed the blocklength of the public-key encryption scheme, to limit computation to a single block encryption. Mutual entity authentication using timestamps.The message format given above can be used for key establishment in a one-pass protocol, although this provides no entity au- thentication of the recipient to the originator. For mutual entity authentication, two mes- sages of this form may be used, yielding essentially X.509 strong two-way authentication (Protocol 12.40). Mutual entity authentication using challenge-response.The 2-pass key transport pro- tocol discussed in the previous paragraph requires the use of timestamps, in which case se- curity relies on the assumption of secure, synchronized clocks. This requirement can be eliminated by using a 3-pass protocol with random numbers for challenge-response (essen- tially the X.509 strong three-way authentication protocol; cf. Protocol 12.43): A→B: r A A←B: rB,PA(B,k1),SB(rB,rA,A,PA(B,k1)) A→B: PB(A,k2),SA(rA,rB,B,PB(A,k2)) Aand Bmay compute a joint keykas some function ofk1 and k2; alternately, one of PA(B,k1) and PB(A,k2) may be omitted from the second or third message. The iden- tiﬁers within the scope of the encryption blocks remain necessary as above; the identiﬁers within the scope of (only) the signature are, however, redundant, both here and in the case of signing encrypted keys above – it may be assumed they must match those corresponding to the public-key encryption. (iv) X.509 strong authentication protocols This subsection considers in greater detail a fully-speciﬁed protocol involving public-key transport using the general technique of§12.5.2(iii), namely, signing encrypted keys. The X.509 recommendation deﬁnes both “strong two-way” and “strong three-way” au- thentication protocols, providing mutual entity authentication with optional key transport. Here strongdistinguishes these from simpler password-based methods, andtwo-and three- way refers to protocols with two and three passes (message exchanges), using timestamps and challenge-response based on random numbers, respectively. Both protocols were designed to provide the assurances listed below to the responder B(and reciprocal assurances intended for the originatorA); heretokenrefers to crypto- graphically protected data: §12.5 Key transport based on public-key encryption 511 1. the identity ofA, and that the token received byBwas constructed byA(and not thereafter altered); 2. that the token received byBwas speciﬁcally intended forB; 3. that the token received byBhas “freshness” (has not been used previously, and orig- inated within an acceptably recent timeframe); 4. the mutual secrecy of the transferred key. 12.40 ProtocolX.509 strong two-way authentication (two-pass) SUMMARY: AsendsBone message, andBresponds with one message. RESULT: mutual entity authentication and key transport with key authentication. 1. Notation. PX(y) denotes the result of applyingX’s encryption public key to datay. SX(y) denotes the result of applyingX’s signature private key toy. rA,rB are never re-used numbers (to detect replay and impersonation). certX is a certiﬁcate binding partyXto a public key suitable for both encryption and signature veriﬁcation (see Remark 12.41). 2. System setup. (a) Each party has its public key pair for signatures and encryption. (b) Amust acquire (and authenticate) the encryption public key ofBa priori. (This may require additional messages and computation.) 3. Protocol messages. (An asterisk denotes items are optional.) LetDA =( tA,rA,B,data1 ∗,PB(k1)∗),DB =( tB,rB,A,rA,data2 ∗,PA(k2)∗). A→B: certA,DA,SA(DA)( 1 ) A←B: certB,DB,SB(DB)( 2 ) 4. Protocol actions. (a)Aobtains a timestamptA indicating an expiry time, generatesrA, optionally obtains a symmetric keyk1 and sends toBmessage (1). (data1 is optional data for which data origin authentication is desired.) (b) Bveriﬁes the authenticity ofcertA (checking the signature thereon, expiry date, etc.), extractsA’s signature public key, and veriﬁesA’s signature on the data blockDA. Bthen checks that the identiﬁer in message (1) speciﬁes itself as intended recipient, that the timestamp is valid, and checks thatrA has not been replayed. (rA includes a sequential component whichBchecks, against locally maintained state information, for uniqueness within the validity period deﬁned by tA.) (c) If all checks succeed,Bdeclares the authentication ofAsuccessful, decrypts k1 using its private decryption key, and saves this now-shared key. (This termi- nates the protocol if only unilateral authentication is desired.)Bthen obtains timestamptB, generatesrB, and sendsAmessage (2). (data2 is optional data, and k2 is an optional symmetric key provided forA.) (d) Acarries out actions analogous to those carried out byB. If all checks succeed, Adeclares the authentication ofBsuccessful, and saves keyk2 for subsequent use.Aand Bshare mutual secretsk1 and k2. 12.41 Remark (separate keys in X.509) The X.509 standard assumes a public-key scheme such as RSA, whereby the same key pair may be used for both encryption and signature function- ality. The protocol, however, is easily adapted for separate signature and encryption keys, and, indeed, it is prudent to use separate keys. See also Remark 13.32. 512 Ch. 12 Key Establishment Protocols 12.42 Note (criticism of X.509 protocol) Since Protocol 12.40 does not specify inclusion of an identiﬁer (e.g.,A) within the scope of the encryptionPB withinDA, one cannot guarantee that the signing party actually knows (or was the source of) the plaintext key. 12.43 ProtocolX.509 strong three-way authentication (three-pass) SUMMARY: Aand Bexchange 3 messages. RESULT: as in Protocol 12.40, without requiring timestamps. The protocol differs from Protocol 12.40 as follows: 1. TimestampstA and tB may be set to zero, and need not be checked. 2. Upon receiving (2),Achecks the receivedrA matches that in message (1). 3. A third message is sent fromAtoB: A→B:( rB,B),SA(rB,B)( 3 ) 4. Upon receiving (3),Bveriﬁes the signature matches the received plaintext, that plain- text identiﬁerBis correct, and that plaintextrB received matches that in (2). 12.5.3 Hybrid key transport protocols using PK encryption In contrast to the preceding key transport protocols, the Beller-Yacobi protocol uses sym- metric encryption in addition to both PK encryption and digital signatures. Such protocols using both asymmetric and symmetric techniques are calledhybrid protocols. Beller-Yacobi protocol (4-pass) The key transport protocol of Beller and Yacobi, which provides mutual entity authentica- tion and explicit key authentication, was designed speciﬁcally for applications where there is an imbalance in processing power between two parties; the goal is to minimize the com- putational requirements of the weaker party. (Candidate applications include transactions involving chipcards, and wireless communications involving a low-power telephone hand- set.) Another feature of the protocol is that the identity of one of the parties (the weaker, hereA) remains concealed from eavesdroppers. Essentially,Aauthenticates itself toBby signing a random challengem, whileBau- thenticates itself toAby demonstrating knowledge of a keyKonlyBitself could recover. For simplicity of exposition, the protocol is described using RSA with public exponent3, although Rabin’s scheme is more efﬁcient and recommended in practice (but see Note 8.13 regarding chosen-ciphertext attack). §12.5 Key transport based on public-key encryption 513 12.44 ProtocolBeller-Yacobi key transport (4-pass) SUMMARY: Atransfers keyKtoBin a 4-pass protocol. RESULT: mutual entity authentication and mutual explicit key authentication. 1. Notation. EK(y) denotes symmetric encryption ofyusing keyKand algorithmE. PX(y) denotes the result of applyingX’s public-key function toy. SX(y) denotes the result of applyingX’s private-key function toy. IX denotes an identifying string for partyX. h(y) denotes the hash ofy, used in association with the signature scheme. Ify=( y1,...,y n), the input is the concatenation of these multiple values. 2. System setup. (a)Selection of system parameters. An appropriate primenS and generatorαfor the multiplicative group of integers modulonS are ﬁxed as ElGamal system parameters. A trusted serverTchooses appropriate primespand qyielding public modulusnT = pqfor RSA signatures, then for public exponenteT =3 computes a private keydT satisfying:eT dT ≡1m o d(p−1)(q−1). (b) Distribution of system parameters. Each party (Aand B) is given an authentic copy ofT’s public key and the system parameters:nT ,(nS,α). Tassigns to each partyXa uniquedistinguished nameor identifying stringIX (e.g.,X’s name and address). (c)Initialization of terminal. Each party playing the role ofA(terminal) selects a random integera,1 <a ≤nS −2, and computes its ElGamal signature public keyuA = αa mod nS.Akeeps its corresponding private keyasecret, and transfers an authentic copy ofuA toT, identifying itself toTby out-of- band means (e.g., in person).Tconstructs and returns toAthe public-key cer- tiﬁcate:certA =( IA,u A,G A). (The certiﬁcate containsA’s identity and ElGamal signature public key, plusT’s RSA signatureGA over these:GA = ST (IA,uA)=( h(IA,uA))dT mod nT .) (d) Initialization of server. Each party playing the role ofB(server) creates an encryption private key and corresponding public key based on RSA with pub- lic exponenteB =3 . Bchooses a public-key modulusnB as the product of two appropriate secret primes, and itself computes the corresponding RSA private keyd B. BtransfersnB toT, identifying itself toTby out-of-band means. Tthen constructs and returns toBthe public-key certiﬁcate:certB = (IB,n B,G B). (The certiﬁcate containsB’s identity and RSA encryption public keynB, plusT’s RSA signature over these:GB = ST (IB,n B)= (h(IB,nB))dT mod nT .) 3. Protocol messages. A←B: certB =( IB,nB,GB)( 1 ) A→B: PB(K)= K3 mod nB (2) A←B: EK(m,{0}t)( 3 ) A→B: EK((v,w),certA)( 4 ) 4. Protocol actions. The following steps are performed each time a shared key is re- quired. The protocol is aborted (with result of failure) if any check fails. (a)Precomputation by terminal. Aselects a randomx,1 ≤ x ≤ nS −2, and computes three values:v= αx mod nS; x−1 mod (nS −1); and avmod (nS −1). (For the security of ElGamal signatures,xmust be new for each signature, and be co-prime tonS −1 to ensurex−1 exists.) 514 Ch. 12 Key Establishment Protocols (b) Bsends toAmessage (1). (c)Achecks the authenticity ofnB by conﬁrming:h(IB,n B)= GB 3 mod nT . Achooses a random key1 <K<n B −1 and sendsBmessage (2), where Y= PB(K). (d) BrecoversK = SB(Y)= YdB mod nB. (The ﬁnal two messages will be encrypted usingK.)Bchooses a random integermas a challenge, extends it witht(sayt≈50) least signiﬁcant zeros, symmetrically encrypts this using key K, and sendsAmessage (3). (e)Adecrypts the received message, and checks it hasttrailing zeros; if so,Aac- cepts that it originated fromBand thatBknows keyK.Atakes the decrypted challengem, concatenates it to the identityIB of the party whose public key it used to shareKin message (2), forming the concatenated quantityM = (m, IB), then computeswsatisfying:w≡(M−av) ·x−1 mod (nS −1), and sendsBmessage (4). (Here(v,w) isA’s ElGamal signature onM, and certA =( IA,u A,G A). The identityIB inMis essential to preclude an intruder-in-the-middle attack – see§12.9.) (f)Bdecrypts the received message, and veriﬁes the authenticity ofuA by check- ing that:h(IA,u A)= GA 3 mod nT . Finally,Bconstructs the concatenated quantityM=( m, IB) from the challengemremembered from message (3) and its own identity, then veriﬁesA’s signature on the challenge by checking that:αM ≡uAv ·vw mod nS. If all checks succeed,Baccepts the partyA associated with identityIA as the source of keyK. 12.45 Note (on Beller-Yacobi key transport protocol) (i) To achieve mutual authentication here requires that each party carry out at least one private-key operation (showing knowledge of its private key), and one or two public- key operations (related to verifying the other’s identity, and its public key if not known a priori). (ii) The novelty here is careful selection of two separate public-key schemes, each re- quiring only an inexpensive computation by the computationally limited party, in this caseA. Choosing RSA with exponent3 or Rabin with exponent2 results in an inexpensive public-key operation (2 or 1 modular multiplications, respectively), for encryption and signature veriﬁcation. Choosing ElGamal-family signatures, the private-key operation is inexpensive (a single modular multiplication, assuming pre- computation). (iii) DSA signatures (Chapter 11) or others with similar properties could be used in place of ElGamal signatures. 12.46 Remark (signature scheme used to certify public keys) Protocol 12.44 requires an ElGa- mal public key be certiﬁed using an RSA signature. This is done for reasons of efﬁciency, and highlights an advantage of allowing signature public keys from one system to be certi- ﬁed using signatures of a different type. Beller-Yacobi protocol (2-pass) Protocol 12.44 can be modiﬁed to yield a 2-pass protocol as illustrated in Figure 12.2. The modiﬁed protocol is obtained by essentially combining the pair of messages each party sends into a single message, as now described using notation as in Protocol 12.44. Bgenerates a random challengemand sends toA:m,cert B.Acomputes its ElGamal signature(v,w) on the concatenationM=( m,IB), and using partvof the signature as the §12.6 Key agreement based on asymmetric techniques 515 session keyK= v,2 sends toB: PB(v),Ev(certA,w).Brecoversv(= K) via public- key decryption, uses it to recovercertA and w, then veriﬁescertA and A’s signature(v,w) on M=( m,IB). The 2-pass protocol has slightly weaker authentication assurances:Bobtains entity au- thentication ofAand obtains a keyKthatAalone knows, whileAhas key authentication with respect toB. ForAto obtain explicit key authentication ofB(implying entity authen- tication also), a third message may be added wherebyBexhibits knowledge through use of Kon a challenge or standard message (e.g.,{0}t). All three ofA’s asymmetric operations remain inexpensive. terminalA serverB precomputex,v= αx mod nS select random challengem verifycertB viaPT (GB) ←− sendm,certB compute (v,w)= SA(m,IB) certB =( IB,nB,GB) sendPB(v),Ev(certA,w) −→ recoverv, setK= v certA =( IA,uA,GA) verifycertA, signature(v,w) Figure 12.2:Summary of Beller-Yacobi protocol (2-pass). In Figure 12.2, an alternative to usingK= vas the session key is to setK= w. This results in the property that both parties inﬂuence the value ofK(aswis a function of both mand x). 12.6 Key agreement based on asymmetric techniques Difﬁe-Hellman key agreement (also calledexponential key exchange) is a fundamental technique providing unauthenticated key agreement. This section discusses key establish- ment protocols based on exponential key agreement, as well as the concept of implicitly- certiﬁed public keys and their use in Difﬁe-Hellman protocols. 12.6.1 Difﬁe-Hellman and related key agreement protocols This section considers the basic Difﬁe-Hellman protocol and related protocols providing various authentication assurances (see Table 12.4). (i) Difﬁe-Hellman key agreement Difﬁe-Hellman key agreement provided the ﬁrst practical solution to the key distribution problem, allowing two parties, never having met in advance or shared keying material, to establish a shared secret by exchanging messages over an open channel. The security rests on the intractability of the Difﬁe-Hellman problem and the related problem of computing discrete logarithms (§3.6). The basic version (Protocol 12.47) provides protection in the form of secrecy of the resulting key from passive adversaries (eavesdroppers), but not from 2A side effect of usingK = v is thatA no longer directly controls the key value, transforming the key transport protocol into a key agreement. Alternately, a randomx could be chosen byA and used as keyK = x, andx could be sent encrypted alongsidew. 516 Ch. 12 Key Establishment Protocols →Properties key entity number of ↓Protocol authentication authentication messages Difﬁe-Hellman none none 2 ElGamal key agreement unilateral none 1 MTI/A0 mutual – implicit none 2 G¨unther (see Remark 12.63) mutual – implicit none 2 STS mutual – explicit mutual 3 Table 12.4:Selected key agreement protocols. active adversaries capable of intercepting, modifying, or injecting messages. Neither party has assurances of the source identity of the incoming message or the identity of the party which may know the resulting key, i.e., entity authentication or key authentication. 12.47 ProtocolDifﬁe-Hellman key agreement (basic version) SUMMARY: Aand Beach send the other one message over an open channel. RESULT: shared secretKknown to both partiesAand B. 1. One-time setup. An appropriate primepand generatorαofZ∗ p (2 ≤α≤p−2) are selected and published. 2. Protocol messages. A→B: αx mod p (1) A←B: αy mod p (2) 3. Protocol actions. Perform the following steps each time a shared key is required. (a)Achooses a random secretx,1 ≤x≤p−2, and sendsBmessage (1). (b) Bchooses a random secrety,1 ≤y≤p−2, and sendsAmessage (2). (c)Breceivesαx and computes the shared key asK=( αx)y mod p. (d) Areceivesαy and computes the shared key asK=( αy)x mod p. 12.48 Note (Difﬁe-Hellman with ﬁxed exponentials) A variation of Protocol 12.47 provides mu- tual key authentication. Fixαx and αy mod pas long-term public keys of the respective parties, and distribute these using signed certiﬁcates, thus ﬁxing the long-term shared key for this user pair toK= αxy. If such certiﬁcates are availablea priori, this becomes a zero- pass key agreement (no cryptographic messages need be exchanged). The time-invariant nature of this keyK, however, is a drawback; Protocol 12.53 provides one resolution. A second solution involves use of key update techniques as in§12.3.1(ii). 12.49 Remark (Difﬁe-Hellman in other groups) The Difﬁe-Hellman protocol, and those based on it, can be carried out in any group in which both the discrete logarithm problem is hard and exponentiation is efﬁcient. The most common examples of such groups used in practice are the multiplicative groupZ ∗ p ofZp, the analogous multiplicative group ofF2m, and the group of points deﬁned by an elliptic curve over a ﬁnite ﬁeld. 12.50 Note (control over Difﬁe-Hellman key) While it may appear as though Difﬁe-Hellman key agreement allows each party to guarantee key freshness and preclude key control, use of an exponential with small multiplicative order restricts the order (and thereby value) of the overall key. The most degenerate case forZ p would be selection of0 as private exponent, §12.6 Key agreement based on asymmetric techniques 517 yielding an exponential with order 1 and the multiplicative identity itself as the resulting key. Thus, either participant may force the resulting key into a subset of the original (naively assumed) range set. Relatedly, some variants of Difﬁe-Hellman involving unauthenticated exponentials are vulnerable to the following active attack. AssumeαgeneratesZ∗ p where p = Rq+1 (considerR =2 and qprime). Thenβ = αq = α(p−1)/R has orderR (β= −1 forR=2 ). IfAand Bexchange unauthenticated short-term exponentialsαx and αy, an adversary may replace these by(αx)q and (αy)q, forcing the shared key to be K= αxyq = βxy, which takes one of onlyRvalues (+1 or−1 forR=2 ).Kmay thus be found by exhaustive trial ofRvalues. A more direct attack involves simply replacing the exchanged exponentials by+1 orp−1= −1. This general class of attacks may be prevented by authenticating the exchanged exponentials, e.g., by a digital signature. (ii) ElGamal key agreement in one-pass ElGamal key agreement is a Difﬁe-Hellman variant providing a one-pass protocol with uni- lateral key authentication (of the intended recipient to the originator), provided the public key of the recipient is known to the originatora priori. While related to ElGamal encryp- tion (§8.4), the protocol is more simply Difﬁe-Hellman key agreement wherein the public exponential of the recipient is ﬁxed and has veriﬁable authenticity (e.g., is embedded in a certiﬁcate). 12.51 ProtocolElGamal key agreement (half-certiﬁed Difﬁe-Hellman) SUMMARY: Asends toBa single message allowing one-pass key agreement. RESULT: shared secretKknown to both partiesAand B. 1. One-time setup (key generation and publication). Each userBdoes the following: Pick an appropriate primepand generatorαofZ∗ p. Select a random integerb,1 ≤b≤p−2, and computeαb mod p. Bpublishes its public key(p,α,αb), keeping private keybsecret. 2. Protocol messages. A→B: αx mod p (1) 3. Protocol actions. Perform the following steps each time a shared key is required. (a)Aobtains an authentic copy ofB’s public key(p,α,αb). Achooses a random integerx,1 ≤x≤p−2, and sendsBmessage (1). Acomputes the key asK=( αb)x mod p. (b) Bcomputes the same key on receipt of message (1) asK=( αx)b mod p. 12.52 Remark (assurances in one-pass ElGamal) The recipient in Protocol 12.51 has no cor- roboration of whom it shares the secret key with, nor any key freshness assurances. Neither party obtains entity authentication or key conﬁrmation. (iii) MTI two-pass key agreement protocols The MTI/A0 variant (Protocol 12.53) of Difﬁe-Hellman key agreement yields, in two mes- sages (neither requiring signatures), time-variant session keys with mutual (implicit) key authentication against passive attacks. As in ElGamal key agreement (Protocol 12.51),A sends toBa single message, resulting in the shared keyK. Bindependently initiates an analogous protocol withA, resulting in the shared keyK′. Each ofAand Bthen computes k= KK′mod p(pand αare global parameters now). Neither entity authentication nor key conﬁrmation is provided. Although appropriate for applications where only passive attacks are possible, this protocol is vulnerable to certain active attacks (see Note 12.54). 518 Ch. 12 Key Establishment Protocols 12.53 ProtocolMTI/A0 key agreement SUMMARY: two-pass Difﬁe-Hellman key agreement secure against passive attacks. RESULT: shared secretKknown to both partiesAand B. 1. One-time setup. Select and publish (in a manner guaranteeing authenticity) an ap- propriate system primepand generatorαof Z∗ p,2 ≤α≤p−2. Aselects as a long-term private key a random integera,1 ≤a≤p−2, and computes a long-term public keyzA = αa mod p.(Bhas analogous keysb,zB.)Aand Bhave access to authenticated copies of each other’s long-term public key. 2. Protocol messages. A→B: αx mod p (1) A←B: αy mod p (2) 3. Protocol actions. Perform the following steps each time a shared key is required. (a)Achooses a random secretx,1 ≤x≤p−2, and sendsBmessage (1). (b) Bchooses a random secrety,1 ≤y≤p−2, and sendsAmessage (2). (c)Acomputes the keyk=( αy)azBx mod p. (d) Bcomputes the keyk=( αx)bzAy mod p. (Both parties now share the key k= αbx+ay mod p.) Table 12.5 summarizes Protocol 12.53 and three related two-pass protocols. All four of these MTI protocols provide mutual key authentication without key conﬁrmation or entity authentication, and are role-symmetric: each party executes directly analogous operations. The protocols are also message-independent per Deﬁnition 12.12 (neither party requires receipt of the other’s message before sending its own), although three of the four requirea prioriaccess to the other party’s authentic public key. The remaining protocol – MTI/A0 – does not, and requires no additional passes (or communications delays) if this is not true; public keys may be exchanged e.g., via certiﬁcates included with the existing protocol mes- sages. Thus in MTI/A0, the content of both messages sent is also independent (e.g., of the identity and public key) of the intended recipient. ↓Protocol mAB mBA KA KB key K MTI/A0 αx αy mBAazBx mABbzAy αbx+ay MTI/B0 zBx zAy mBAa−1 αx mABb−1 αy αx+y MTI/C0 zBx zAy mBAa−1x mABb−1y αxy MTI/C1 zBxa zAyb mBAx mABy αabxy Table 12.5:Selected MTI key agreement protocols.Aand Bhave long-term secretsaand b,r e - spectively, veriﬁably authentic corresponding long-term public keyszA = αa,zB = αb mod p, and random per-session secretsxand y, respectively.mAB denotes the messageAsends toB;mBA is analogous.KA and KB are the ﬁnal keyKas computed byAand B. 12.54 Note (source-substitution attack on MTI/A0) As a general rule in all public-key proto- cols (including Table 12.5), prior to accepting the authenticated public key of a partyA, a partyBshould have assurance (either direct or through a trusted third party) thatAactu- ally knows the corresponding private key. Otherwise, an adversaryCmay claimA’s public key as its own, allowing possible attacks, such as that on MTI/A0 as follows. Assume that §12.6 Key agreement based on asymmetric techniques 519 in a particular implementation,Asends toBits certiﬁed public key in a certiﬁcate appended to message (1).CregistersA’s public key as its own (legitimately proving its own identity to the certiﬁcate-creating party). WhenAsendsBmessage (1),CreplacesA’s certiﬁcate with its own, effectively changing the source indication (but leaving the exponentialαx sent by AtoBunchanged).CforwardsB’s responseαy toA.Bconcludes that subsequently received messages encrypted by the keyk= αbx+ay originated fromC, whereas, in fact, it is onlyAwho knows kand can originate such messages. A more complicated attack achieves the same, withC’s public key differing fromA’s public keyzA.Cselects an integere, computes(zA)e = αae, and registers the public key αae. Cthen modiﬁesαy sent byBin message (2) to(αy)e. Aand Beach compute the key k= αaeyαxb, whichAbelieves is shared withB(and is), whileBbelieves it is shared withC. In both variations,Cis not actually able to computekitself, but rather causesBto have false beliefs. Such attacks may be prevented by modifying the protocol such that the expo- nentials are authenticated (cf. Note 12.50), and binding key conﬁrmation evidence to an au- thenticated source indication, e.g., through a digital signature (cf. Remark 12.58). The MTI protocols are, however, also subject to certain theoretical known-key attacks (see p.538). 12.55 Remark (implications of message independence) Protocols such as MTI/A0 “leak” no in- formation about long-term secrets, since the exchanged messages are independent thereof. However, such protocols in which each party’s message is independent of the other’s, and yet the session key depends on fresh input from each, cannot provide mutual explicit key authentication. 12.56 Remark (computational complexity of MTI protocols) The A0 and B0 protocols require 3 exponentiations by each party, whereas the C0 and C1 protocols require only 2. C1 has the additional advantage over B0 and C0 that no inverses are needed; however, these ﬁxed long-term values may be precomputed. (iv) Station-to-Station protocol (STS) The following three-pass variation of the basic Difﬁe-Hellman protocol allows the estab- lishment of a shared secret key between two parties with mutual entity authentication and mutual explicit key authentication. The protocol also facilitates anonymity – the identities ofAand Bmay be protected from eavesdroppers. The method employs digital signatures; the description below is for the speciﬁc case of RSA signatures. 12.57 ProtocolStation-to-Station protocol (STS) SUMMARY: parties Aand Bexchange 3 messages. RESULT: key agreement, mutual entity authentication, explicit key authentication. 1. Notation.Eis a symmetric encryption algorithm. SA(m) denotesA’s signature onm, deﬁned as:SA(m)=( H(m))dA mod nA (i.e., RSA preceded by an appropriate one-way hash functionH,H(m) <nA). 2. One-time setup (deﬁnition and publication of system parameters). (a) Select and publish an appropriate system primepand generatorαofZ∗ p,2 ≤ α≤p−2. (For additional security, each party may have its own unique such parameters as part of its public key.) (b) Each userAselects RSA public and private signature keys(eA,nA) and dA, respectively (Bhas analogous keys). Assume each party has access to authentic 520 Ch. 12 Key Establishment Protocols copies of the other’s public key (if not, certiﬁcates can be included in existing messages (2) and (3)). 3. Protocol messages. A→B: αx mod p (1) A←B: αy mod p,Ek(SB(αy,αx)) (2) A→B: Ek(SA(αx,αy)) (3) 4. Protocol actions. Perform the following steps each time a shared key is required. The protocol is aborted (with failure) immediately upon any signature failure. (a)Agenerates a secret randomx,1 ≤x≤p−2, and sendsBmessage (1). (b) Bgenerates a secret randomy,1 ≤y≤p−2, and computes the shared key k=( αx)y mod p.Bsigns the concatenation of both exponentials ordered as in (2), encrypts this using the computed key, and sendsAmessage (2). (c)Acomputes the shared keyk=( αy)x mod p, decrypts the encrypted data, and usesB’s public key to verify the received value as the signature on the hash of the cleartext exponential received and the exponential sent in message (1). Upon successful veriﬁcation,Aaccepts thatkis actually shared withB, and sendsBan analogous message (3). (d) Bsimilarly decrypts the received message (3) and veriﬁesA’s signature therein. If successful,Baccepts thatkis actually shared withA. The attack of Note 12.50 is precluded in the STS protocol due to the signatures over the exchanged exponentials. 12.58 Remark (key conﬁrmation in STS protocol) Encryption under keykprovides mutual key conﬁrmation plus allows the conclusion that the party knowing the key is that which signed the exponentials. The optimal use of this protocol occurs when all subsequent messages are also to be encrypted under keyk; if this is not the case, alternate means of key conﬁrmation avoiding encryption may be preferable. One alternative is to use a MAC in messages (2) and (3), e.g., fors= S A(αx,αy),A→B:( s,MACk(s)). A second alternative is inclusion of a one-way hash ofkwithin the signed messages, e.g.,A→B: SA(αx,αy,h(k)) where hereh(k) may be replaced bykalone if the signature process itself employs an appropriate one-way hash. 12.6.2 Implicitly-certiﬁed public keys In contrast both to systems which use public-key certiﬁcates (§13.4.2) and to identity-based systems (§13.4.3), an alternate approach to distributing public keys involvesimplicitly- certiﬁed public keys, for which a framework is provided in§13.4.4. Use of the wordimplicit here is consistent with that in the term (implicit) key authentication. The current section presents several speciﬁc techniques involving implicitly-certiﬁed public keys. (i) Implicitly-certiﬁed public keys (of G ¨unther) Mechanism 12.59 provides a method by which a trusted party may create a Difﬁe-Hellman public keyr s mod pfor an entity, with the key being implicitly-certiﬁed. Such public keys, which may be reconstructed from public data, may be used in key agreement protocols re- quiring certiﬁed Difﬁe-Hellman public keys (e.g.,zA in Protocol 12.53) as an alternative to transporting these keys by public-key certiﬁcates, or in customized protocols such as Pro- tocol 12.62. §12.6 Key agreement based on asymmetric techniques 521 12.59 Mechanism G¨unther’s implicitly-certiﬁed (identity-based) public keys SUMMARY: a trusted partyTcreates an implicitly-certiﬁed, publicly-recoverable Difﬁe- Hellman public key forA, and transfers toAthe corresponding private key. 1. A trusted serverTselects an appropriate ﬁxed public primepand generatorαofZ∗ p. Tselects a random integert, with1 ≤t≤p−2 and gcd(t,p−1) = 1as its private key, and publishes its public keyu= αt mod p, along withα,p. 2. Tassigns to each partyAa uniquedistinguished nameor identifying stringIA (e.g., name and address), and a random integerkA withgcd(kA,p−1) = 1.Tthen com- putesPA = αkA mod p.(PA isA’sreconstruction public data, allowing other par- ties to compute(PA)a below. The gcd condition ensures thatPA itself is a generator.) 3. Using a suitable hash functionh,Tsolves the following equation fora(restarting with a newkA ifa=0 ): h(IA) ≡t·PA + kA ·a (mod p−1). (12.1) 4. Tsecurely transmits toAthe pair(r,s)=( PA,a), which isT’s ElGamal signature (see Chapter 11) onIA.(aisA’s private key for Difﬁe-Hellman key-agreement.) 5. Any other party can then reconstructA’s (Difﬁe-Hellman) public keyPA a (= αkAa) entirely from publicly available information (α,IA,u,PA,p) by computing (since αh(IA) ≡uPA ·PA a): PA a ≡αh(IA) ·u−PA mod p. (12.2) The above mechanism can be generalized to be independent of ElGamal signatures, by using any suitable alternate method to generate a pair(r,s) where ris used as the recon- struction public data, the secretsis used as a (key-agreement) private key, and whereby the reconstructed public keyrs mod pcan be computed from public information alone. 12.60 Remark (optimization of ElGamal signatures) Equation (12.1) can be replaced by using the following optimization of the ElGamal signature scheme, wheregcd(t,p−1) = 1: h(IA) ≡t·a+ kA ·PA (mod p−1). To solve forathen requires a one-time inverse computation (t−1 mod p−1) rather than the per-signature inverse computation ((kA)−1 mod p−1) required by the original signature scheme. With this modiﬁcation,A’s key-agreement public key isua (= αta) rather than PA a (= αkAa), correspondingly recovered by computing αh(IA) ·P−PA A mod p(= αta mod p). (12.3) (ii) Self-certiﬁed public keys (of Girault) Mechanism 12.61, which is employed in several protocols in§12.6.3, presents a technique for creating implicitly-certiﬁed public keys. It differs from that of Mechanism 12.59 in that it allows users to “self-certify” the keys, in the sense that the user itself is the only party knowing the private key (as opposed to the trusted party having access to each party’s pri- vate key). 522 Ch. 12 Key Establishment Protocols 12.61 Mechanism Girault’s self-certiﬁed public keys SUMMARY: a trusted partyTcreates an implicitly-certiﬁed, publicly-recoverable Difﬁe- Hellman public key for partyA, without learning the corresponding private key. 1. A trusted serverTselects secret primespand qfor an RSA integern= pq, an ele- ment αof maximal order inZ∗ n (see Algorithm 4.83), and appropriate integerseand das a (public, private) RSA key pair forn. 2. Tassigns to each partyAa uniquedistinguished nameor identifying stringIA (e.g., name and address). 3. PartyAitself chooses a private keya, and provides the public keyαa mod ntoT in an authenticatable manner. (αa isA’s key-agreement public key.) Moreover,A provides proof toTthat it knows the corresponding secreta. (This is necessary to prevent a certain forgery attack byAin some ways analogous to that of Note 12.54, and might be done byAproducing forTa Difﬁe-Hellman key based onαa and an exponential chosen byT.) 4. TcomputesA’s reconstruction public data (essentially replacing a certiﬁcate) asPA =( αa −IA)d mod n. (Thus(PA e + IA)m o dn= αa mod n, and from public information alone, any party can computeA’s public key,αa mod n.) 12.6.3 Difﬁe-Hellman protocols using implicitly-certiﬁed keys The authenticity of Difﬁe-Hellman exponentials used as public keys in authenticated key agreement protocols can be established by distributing them via public-key certiﬁcates, or by reconstructing them as implicitly-certiﬁed public keys (e.g., using Mechanisms of §12.6.2) from publicly available parameters. Protocol 12.62 is one example of the lat- ter. The idea may be adopted to other Difﬁe-Hellman based protocols as further illustrated by Examples 12.64, 12.65, and 12.66 respectively corresponding to the ﬁxed-key Difﬁe- Hellman, ElGamal, and MTI/A0 key agreement protocols of§12.6.1. 12.62 ProtocolG¨unther’s key agreement protocol SUMMARY: Difﬁe-Hellman based key agreement protocol betweenAand B. RESULT: Aand Bestablish shared secretKwith key authentication. 1. One-time setup (deﬁnition of global parameters). Using Mechanism 12.59, a trusted partyTconstructs ElGamal signatures(PA,a) and (PB,b) on the identitiesIA and IB ofAand B, respectively, and gives these signatures respectively toAand Bas secrets, along with the following authentic public system parameters as per Mecha- nism 12.59: a primep, generatorαofZ ∗ p, andT’s public keyu. 2. Protocol messages. A→B: IA,PA (1) A←B: IB,PB,(PA)y mod p (2) A→B:( PB)x mod p (3) 3. Protocol actions. Perform the following steps each time a shared key is required. (a)AsendsBmessage (1). (b) Bgenerates a random integery,1 ≤y≤p−2, and sendsAmessage (2). (c)Agenerates a random integerx,1 ≤x≤p−2, and sendsBmessage (3). §12.6 Key agreement based on asymmetric techniques 523 (d) Key computation.As per Mechanism 12.59,Aand Brespectively construct the other’s identity-based public key (equivalent to(PB)b and (PA)a mod p, respectively). The common key-agreement keyK(= αkAya+kBbx) is estab- lished asAand Brespectively computeK=( PA y)a ·(PB b)x,K=( PA a)y · (PB x)b mod p. Protocol 12.62 is subject to theoretical known-key attacks similar to those which apply to the MTI protocols (Note 12.54). 12.63 Remark (two-pass G¨unther protocol) In Protocol 12.62, a party’s identity information and long-term public key (respectively,IA and PA) are long-term parameters. If these are kno- wn to partiesa priori, then this three-pass protocol reduces to two passes. The reduced protocol provides the same assurances, namely, key agreement with key authentication, as Protocol 12.62 and the two-pass MTI schemes of Table 12.5, and closely resembles MTI/A0 with respect to the logarithm of the ﬁnal key. 12.64 Example (Protocol G0) Fixed-key Difﬁe-Hellman key-agreement (Note 12.48) may be modiﬁed to use implicitly-certiﬁed keys as follows. Using the setup and notation as in Gi- rault’s self-certiﬁed public keys (Mechanism 12.61),Aand Bestablish the time-invariant joint keyKby respectively computing(P B)e + IB mod n(= αb) and (PA)e + IA mod n(= αa), from which they effectively compute K=( αb)a and K=( αa)b mod n. (12.4) Alternatively, the same protocol may be modiﬁed to use G¨unther’s ID-based public keys assuming the setup and notation as in Mechanism 12.59 with modiﬁed ElGamal signatures as per Remark 12.60. In this case,AandBrespectively compute the other’s key-agreement public keysα tb and αta by (12.3), in place ofαb and αa in (12.4). □ 12.65 Example (Protocol G1) The one-pass ElGamal key agreement of Protocol 12.51 may be modiﬁed to use implicitly-certiﬁed keys as follows. Using the setup and notation as in Gi- rault’s self-certiﬁed public keys (Mechanism 12.61),Achooses a random integerxand sends toB: αx mod n. Acomputes PB e + IB mod n(= αb). Aand Bestablish the time-variant joint keyK= αbx mod n, by respectively computing, effectively, K=( αb)x and K=( αx)b mod n. (12.5) The protocol may be modiﬁed to use G¨unther’s ID-based public keys as follows: rather than sendingαx mod ntoB,AsendsPB x mod p, withPB (andp,b,u, etc.) deﬁned as in Mechanism 12.59.Bthen computesK=( PB x)b mod p, whileAeffectively computes K=( PB b)x mod p, having reconstructedPB b via equation (12.2) on page 521. The re- sulting protocol is essentially one-half of the G¨unther key agreement of Protocol 12.62. A related modiﬁcation utilizing Remark 12.60 involvesAsending toBux mod pin place of PB x, the joint key now beingK = ubx mod p, computed byAas K =( ub)x withub computed per (12.3), andBcomputingK=( ux)b mod p. This ﬁnal protocol then resem- bles (one-half of) Protocol MTI/A0 in that, since the messageAsends is independent of the recipientB, it may be computed ahead of time before the recipient is determined.□ 12.66 Example (Protocol G2) The two-pass MTI/A0 key agreement (Protocol 12.53) may be modiﬁed to use implicitly-certiﬁed keys as follows. Using the setup and notation as in Gi- rault’s self-certiﬁed public keys (Mechanism 12.61),Achooses a random integerxand sends toB: αx mod n. Analogously,Bchooses a random integeryand sends toA: αy 524 Ch. 12 Key Establishment Protocols modn.AcomputesPB e + IB mod n(= αb);BcomputesPA e + IA mod n(= αa).A and Bthen establish the time-variant common keyK= αay+bx (mod n) by respectively computingK=( αy)a(PB e + IB)x and K=( αx)b(PA e + IA)y mod n. Alternatively, this protocol may be modiﬁed to use G¨unther’s ID-based public keys in a manner directly analogous to that of Example 12.64. □ 12.67 Example (self-certiﬁed version of G¨unther’ s ID-based keys) The following modiﬁcation of Mechanism 12.59 transforms it into a “self-certiﬁed” public-key scheme (i.e., one in which the third party does not learn users’ private keys).Achooses a secret randomv, 1 ≤v≤p−1 withgcd(v,p−1) = 1, computesw= αv mod p, and giveswtoT. While vis not given toT,Ashould demonstrate knowledge ofvtoT(cf. Note 12.54).Tchooses kA as before but computesPA = wkA mod p(instead of:PA = αkA).Tsolves equa- tion (12.1) foraas before (using the newPA) and again givesAthe pair(r,s)=( PA,a). Athen calculatesa′= a·v−1 mod (p−1); it follows that(PA,a′) is nowT’s ElGamal signature onIA (it is easily veriﬁed thatuPA ·PA a′ ≡αh(IA)), andTdoes not knowa′.□ 12.7 Secret sharing Secret sharing schemes are multi-party protocols related to key establishment. The original motivation for secret sharing was the following. To safeguard cryptographic keys from loss, it is desirable to create backup copies. The greater the number of copies made, the greater the risk of security exposure; the smaller the number, the greater the risk that all are lost. Se- cret sharing schemes address this issue by allowing enhanced reliability without increased risk. They also facilitate distributed trust or shared control for critical activities (e.g., sign- ing corporate cheques; opening bank vaults), by gating the critical action on cooperation by tofnusers. The idea ofsecret sharingis to start with a secret, and divide it into pieces calledshares which are distributed amongst users such that the pooled shares of speciﬁc subsets of users allow reconstruction of the original secret. This may be viewed as a key pre-distribution technique, facilitating one-time key establishment, wherein the recovered key is pre-deter- mined (static), and, in the basic case, the same for all groups. A secret sharing scheme may serve as ashared control schemeif inputs (shares) from two or more users are required to enable a critical action (perhaps the recovered key allows this action to trigger, or the recovery itself is the critical action). In what follows, simple shared-control schemes introduced in§12.7.1 are a subset of threshold schemes discussed in §12.7.2, which are themselves a subclass of generalized secret sharing schemes as described in§12.7.3. 12.7.1 Simple shared control schemes (i) Dual control by modular addition If a secret numberS,0 ≤S≤m−1 for some integerm, must be entered into a device (e.g., a seed key), but for operational reasons, it is undesirable that any single individual (other than a trusted party) know this number, the following scheme may be used. A trusted party Tgenerates a random number1 ≤S1 ≤m−1, and gives the valuesS1 andS−S1 mod m to two partiesAand B, respectively.Aand Bthen separately enter their values into the §12.7 Secret sharing 525 device, which sums them modulomto recoverS.I fAand Bare trusted not to collude, then neither one has any information aboutS, since the value each possesses is a random number between0 andm−1. This is an example of asplit-knowledgescheme – knowledge of the secretSis split among two people. Any action requiringSis said to be underdual control– two people are required to trigger it. (ii) Unanimous consent control by modular addition The dual control scheme above is easily generalized so that the secretSmay be divided among tusers, all of whom are required in order to recoverS, as follows:Tgeneratest−1 independent random numbersSi,0 ≤Si ≤m−1,1 ≤i≤t−1. PartiesP1 through Pt−1 are givenSi, whilePt is givenSt = S−∑t−1 i=1 Si mod m. The secret is recovered as S = ∑t i=1 Si mod m. Both here and in the dual control scheme above, modulom operations may be replaced by exclusive-OR, using data valuesSand Si of ﬁxed bit-length lg(m). 12.68 Remark (technique for splitting keys) The individual key components in a split control scheme should be full-length. This provides greater security than partitioning anr-bit key intotpieces ofr/tbits each. For example, forr=5 6 and t=2 , if two parties are each given28 bits of the key, exhaustive search by one party requires only228 trials, while if each party is given a56-bit piece,256 trials are necessary. 12.7.2 Threshold schemes 12.69 DeﬁnitionA (t,n) threshold scheme(t≤n) is a method by which a trusted party com- putes secretsharesSi,1 ≤i≤nfrom an initial secretS, and securely distributesSi to userPi, such that the following is true: anytor more users who pool their shares may easily recoverS, but any group knowing onlyt−1 or fewer shares may not. Aperfectthresh- old scheme is a threshold scheme in which knowing onlyt−1 or fewer shares provide no advantage (no information aboutSwhatsoever, in the information-theoretic sense) to an opponent over knowing no pieces. The split-knowledge scheme of§12.7.1(i) is an example of a(2,2) threshold scheme, while the unanimous consent control of§12.7.1(ii) is a(t,t) threshold scheme. 12.70 Remark (use of threshold schemes) If a threshold scheme is to be reused without decreased security, controls are necessary to prevent participants from deducing the shares of other users. One method is to prevent group members themselves from accessing the value of the recovered secret, as may be done by using a trusted combining device. This is appro- priate for systems where the objective is shared control, and participants need only see that an action is triggered, rather than have access to the key itself. For example, each share might be stored on a chipcard, and each user might swipe its card through a trusted card reader which computes the secret, thereby enabling the critical action of opening an access door. Shamir’s threshold scheme Shamir’s threshold scheme is based on polynomial interpolation, and the fact that a uni- variate polynomialy= f(x) of degreet−1 is uniquely deﬁned bytpoints(x i,yi) with distinctxi (since these deﬁnetlinearly independent equations intunknowns). 526 Ch. 12 Key Establishment Protocols 12.71 Mechanism Shamir’s(t,n) threshold scheme SUMMARY: a trusted party distributes shares of a secretStonusers. RESULT: any group oftusers which pool their shares can recoverS. 1. Setup. The trusted partyTbegins with a secret integerS≥0 it wishes to distribute among nusers. (a)Tchooses a primep> max(S,n), and deﬁnesa0 = S. (b) Tselectst−1 random, independent coefﬁcientsa1,...,a t−1,0 ≤aj ≤p−1, deﬁning the random polynomial overZp,f(x)= ∑t−1 j=0 ajxj. (c)TcomputesSi = f(i)m o dp,1 ≤i≤n(or for anyndistinct pointsi,1 ≤ i≤p−1), and securely transfers the shareSi to userPi, along with public indexi. 2. Pooling of shares. Any group oftor more users pool their shares (see Remark 12.70). Their shares providetdistinct points(x,y)=( i,Si) allowing computation of the coefﬁcientsaj,1 ≤j≤t−1 off(x) by Lagrange interpolation (see below). The secret is recovered by notingf(0) =a0 = S. The coefﬁcients of an unknown polynomialf(x) of degree less thant, deﬁned by points (xi,yi),1 ≤i≤t, are given by the Lagrange interpolation formula: f(x)= t∑ i=1 yi ∏ 1≤j≤t,j̸=i x−xj xi −xj . Sincef(0) =a0 = S, the shared secret may be expressed as: S= t∑ i=1 ciyi , where ci = ∏ 1≤j≤t,j̸=i xj xj −xi . Thus each group member may computeSas a linear combination oftsharesyi, since the ci are non-secret constants (which for a ﬁxed group oftusers may be pre-computed). 12.72 Note (properties of Shamir’ s threshold scheme) Properties of Mechanism 12.71 include: 1. perfect. Given knowledge of anyt−1 or fewer shares, all values0 ≤S≤p−1 of the shared secret remain equally probable (see Deﬁnition 12.69). 2. ideal. The size of one share is the size of the secret (see Deﬁnition 12.76). 3. extendable for new users. New shares (for new users) may be computed and dis- tributed without affecting shares of existing users. 4. varying levels of control possible. Providing a single user with multiple shares be- stows more control upon that individual. (In the terminology of§12.7.3, this corre- sponds to changing the access structure.) 5. no unproven assumptions. Unlike many cryptographic schemes, its security does not rely on any unproven assumptions (e.g., about the difﬁculty of number-theoretic problems). 12.7.3 Generalized secret sharing The idea of a threshold scheme may be broadened to ageneralized secret sharing schemeas follows. Given a setPof users, deﬁneA(theaccess structure) to be a set of subsets, called §12.7 Secret sharing 527 theauthorized subsetsofP. Shares are computed and distributed such that the pooling of shares corresponding to any authorized subsetA∈A allows recovery of the secretS, but the pooling of shares corresponding to any unauthorized subsetB⊆P,B ̸∈Adoes not. Threshold schemes are a special class of generalized secret sharing schemes, in which the access structure consists of precisely allt-subsets of users. An access structure is called monotone if, whenever a particular subsetAof users is an authorized subset, then any sub- set ofPcontainingAis also authorized. Monotone access structures are a requirement in many applications, and most natural schemes are monotone. Perfect secret sharing schemes have a monotone access structure as a consequence of the entropy formulation in Deﬁni- tion 12.73. 12.73 DeﬁnitionA secret sharing scheme isperfectif the shares corresponding to each unautho- rized subset provide absolutely no information, in the information-theoretic sense, about the shared secret (cf. Deﬁnition 12.69). More formally, whereHdenotes entropy (see§2.2.1), and A,Bare sets of users using the above notation:H(S\|A)=0 for anyA∈A, while H(S\|B)= H(S) for anyB̸∈A. The efﬁciency of a secret sharing scheme is measured by its information rate. 12.74 DeﬁnitionFor secret sharing schemes, theinformation ratefor a particular user is the bit- size ratio (size of the shared secret)/(size of that user’s share). Theinformation ratefor a secret sharing scheme itself is the minimum such rate over all users. 12.75 Fact(perfect share bound) In any perfect secret sharing scheme the following holds for all user shares: (size of a user share)≥(size of the shared secret). Consequently, all perfect secret sharing schemes must have information rate≤1. Justiﬁcation. If any userP i had a share of bit-size less than that of the secret, knowledge of the shares (excepting that ofPi) corresponding to any authorized set to whichPi belonged, would reduce the uncertainty in the secret to at most that inPi’s share. Thus by deﬁnition, the scheme would not be perfect. 12.76 DeﬁnitionSecret sharing schemes of rate1 (see Deﬁnition 12.74) are calledideal. As per Note 12.72, Shamir’s threshold scheme is an example of an ideal secret sharing scheme. Examples of access structures are known for which it has been proven that ideal schemes do not exist. Secret sharing schemes with extended capabilities Secret sharing schemes with a variety of extended capabilities exist, including: 1. pre-positioned secret sharing schemes. All necessary secret information is put in place excepting a single (constant) share which must later be communicated, e.g., by broadcast, to activate the scheme. 2. dynamic secret sharing schemes. These are pre-positioned schemes wherein the se- crets reconstructed by various authorized subsets vary with the value of communi- cated activating shares. 3. multi-secret threshold schemes. In these secret sharing schemes different secrets are associated with different authorized subsets. 4. detection of cheaters, andveriﬁable secret sharing. These schemes respectively ad- dresscheatingby one or more group members, and the distributor of the shares. 528 Ch. 12 Key Establishment Protocols 5. secret sharing with disenrollment. These schemes address the issue that when a secret share of a(t,n) threshold scheme is made public, it becomes a(t−1,n) scheme. 12.8 Conference keying 12.77 DeﬁnitionA conference keying protocolis a generalization of two-party key establish- ment to provide three or more parties with a shared secret key. Despite superﬁcial resemblance, conference keying protocols differ from dynamic se- cret sharing schemes in fundamental aspects. General requirements for conference keying include that distinct groups recover distinct keys (session keys); that session keys are dy- namic (excepting key pre-distribution schemes); that the information exchanged between parties is non-secret and transferred over open channels; and that each party individually computes the session key (vs. pooling shares in a black box). A typical application is tele- phone conference calls. The group able to compute a session key is called theprivileged subset. When a central point enables members of a (typically large) privileged subset to share a key by broadcasting one or more messages, the process resembles pre-positioned secret sharing somewhat and is calledbroadcast encryption. An obvious method to establish a conference keyKfor a set oft ≥3 parties is to arrange that each party share a unique symmetric key with a common trusted party. There- after the trusted party may choose a new random key and distribute it by symmetric key transport individually to each member of the conference group. Disadvantages of this ap- proach include the requirement of an on-line trusted third party, and the communication and computational burden on this party. A related approach not requiring a trusted party involves a designated group member (thechair) choosing a keyK, computing pairwise Difﬁe-Hellman keys with each other group member, and using such keys to securely sendKindividually to each. A drawback of this approach is the communication and computational burden on the chair, and the lack of protocol symmetry (balance). Protocol 12.78 offers an efﬁcient alternative, albeit more complex in design. Burmester-Desmedt conference keying protocol The following background is of use in understanding Protocol 12.78.tusersU 0 through Ut−1 with individual Difﬁe-Hellman exponentialszi = αri will form a conference key K= αr0r1+r1r2+r2r3+···+rt−1r0 . DeﬁneAj = αrjrj+1 = zrj+1 j andXj = αrj+1rj−rjrj−1 . NotingAj = Aj−1Xj,Kmay equivalently be written as (with subscripts taken modulot) Ki = A0A1 ···At−1 = Ai−1AiAi+1 ···Ai+(t−2) = Ai−1 ·(Ai−1Xi) ·(Ai−1XiXi+1) ···(Ai−1XiXi+1 ···Xi+(t−2)). NotingAi−1 t =( zi−1)tri, this is seen to be equivalent toKi as in equation (12.6) of Pro- tocol 12.78. 12.78 ProtocolBurmester-Desmedt conference keying SUMMARY: t≥2 users derive a common conference keyK. RESULT: Kis secure from attack by passive adversaries. 1. One-time setup. An appropriate primepand generatorαofZ∗ p are selected, and au- thentic copies of these are provided to each ofnsystem users. §12.8 Conference keying 529 2. Conference key generation. Any group oft ≤ nusers (typicallyt ≪ n), derive a common conference keyKas follows. (Without loss of generality, the users are labeledU0 throughUt−1, and all indicesjindicating users are taken modulot.) (a) EachUi selects a random integerri,1 ≤ri ≤p−2, computeszi = αri mod p, and sendszi to each of the othert−1 group members. (Assume thatUi has been notiﬁeda priori, of the indicesjidentifying other conference members.) (b) EachUi, after receivingzi−1 and zi+1, computesXi =( zi+1/zi−1)ri mod p (noteXi = αri+1ri−riri−1 ), and sendsXi to each of the othert−1 group members. (c) After receivingXj,1 ≤j≤texcludingj= i,Ui computesK= Ki as Ki =( zi−1)tri ·Xi t−1 ·Xi+1 t−2 ··· Xi+(t−3) 2 ·Xi+(t−2) 1 mod p (12.6) For small conferences (smallt), the computation required by each party is small, since all but one exponentiation in equation (12.6) involves an exponent between1 and t. The protocol requires an order be established among users in the privileged subset (for index- ing). Fort =2 , the resulting key isK =( αr1r2)2, the square of the standard Difﬁe- Hellman key. It is provably as difﬁcult for a passive adversary to deduce the conference key Kin Protocol 12.78 as to solve the Difﬁe-Hellman problem. Attention above has been restricted to unauthenticated conference keying; additional measures are required to provide authentication in the presence of active adversaries. Pro- tocol 12.78 as presented assumes a broadcast model (each user exchanges messages with all others); it may also be adapted for a bi-directional ring (wherein each user transmits only to two neighbors). Unconditionally secure conference keying While conference keying schemes such as Protocol 12.78 provide computational security, protocols with the goal of unconditional security are also of theoretical interest. Related to this, a generalization of Fact 12.34 is given below, for conferences of ﬁxed size (tpartici- pants from amongnusers) which are information-theoretically secure against conspiracies of up tojnon-participants. The model for this result is a non-interactive protocol, and more speciﬁcally a key pre-distribution scheme: each conference member computes the confer- ence key solely from its own secret data (pre-distributed by a server) and an identity vector specifying (an ordered sequence of) indices of the other conference members. 12.79 Fact(Blundo’ s conference KDS bound) In anyj-secure conference KDS providingm-bit conference keys to privileged subsets of ﬁxed sizet, the secret data stored by each user must be at leastm· ( j+t−1 t−1 ) bits. Fact 12.79 witht=2 and j= n−2 corresponds to the trivial scheme (see p.505) where each user hasn−1 shared keys each ofmbits, one for each other user. A non- trivial scheme meeting the bound of Fact 12.79 can be constructed as a generalization of Mechanism 12.35 (see p.540). 12.80 Remark (reﬁnement of Fact 12.79) A more precise statement of Fact 12.79 requires con- sideration of entropy; the statement holds if the conference keys in question havembits of entropy. 530 Ch. 12 Key Establishment Protocols 12.9 Analysis of key establishment protocols The main objective of this section is to highlight the delicate nature of authenticated key establishment protocols, and the subtlety of design ﬂaws. Examples of ﬂawed protocols are included to illustrate typical attack strategies, and to discourage protocol design by the novice. 12.9.1 Attack strategies and classic protocol ﬂaws The study of successful attacks which have uncovered ﬂaws in protocols allows one to learn from previous design errors, understand general attack methods and strategies, and formu- late design principles. This both motivates and allows an understanding of various design features of protocols. General attack strategies are discussed in§12.2.3. In the speciﬁc ex- amples below,Aand Bare the legitimate parties, andEis an adversary (enemy). Two of the protocols discussed are, in fact, authentication-only protocols (i.e., do not involve key establishment), but are included in this discussion because common principles apply. Attack 1: Intruder-in-the-middle The classic “intruder-in-the-middle” attack on unauthenticated Difﬁe-Hellman key agree- ment is as follows. AEB → α x → αx′ → ← αy′ ← αy ← Aand Bhave private keysxand y, respectively.Ecreates keysx′and y′. Eintercepts A’s exponential and replaces it byαx′ ; and interceptsB’s exponential, replacing it with αy′ .Aforms session keyKA = αxy′ , whileBforms session keyKB = αx′y.Eis able to compute both these keys. WhenAsubsequently sends a message toBencrypted under KA,Edeciphers it, re-enciphers underKB, and forwards it toB. SimilarlyEdeciphers messages encrypted byB(forA) underKB, and re-enciphers them underKA. Aand B believe they communicate securely, whileEreads all trafﬁc. Attack 2: Reﬂection attack Suppose Aand Bshare a symmetric keyK, and authenticate one another on the basis of demonstrating knowledge of this key by encrypting or decrypting a challenge as follows. AB → rA (1) EK(rA,rB) ← (2) → rB (3) An adversaryEcan impersonateBas follows. UponAsending (1),Eintercepts it, and initiates a new protocol, sending the identical messagerA back toAas message (1) purport- edly fromB. In this second protocol,Aresponds with message (2′):EK(rA,rA′), which Eagain intercepts and simply replays back onAas the answer (2) in response to the chal- lengerA in the original protocol.Athen completes the ﬁrst protocol, and believes it has §12.9 Analysis of key establishment protocols 531 successfully authenticatedB, while in factBhas not been involved in any communications. AE → rA (1) rA ← (1′) → EK(rA,rA′)( 2 ′) EK(rA,rB = rA′) ← (2) → rB (3) The attack can be prevented by using distinct keysKand K′for encryptions fromAto Band BtoA, respectively. An alternate solution is to avoid message symmetry, e.g., by including the identiﬁer of the originating party within the encrypted portion of (2). Attack 3: Interleaving attack Consider the following (ﬂawed) authentication protocol, wheresA denotes the signature operation of partyA, and it is assumed that all parties have authentic copies of all others’ public keys. AB → rA (1) rB,sB(rB,rA,A) ← (2) → rA′,sA(rA′,rB,B)( 3 ) The intention is that the random numbers chosen byAandB, respectively, together with the signatures, provide a guarantee of freshness and entity authentication. However, an enemy Ecan initiate one protocol withB(pretending to beA), and another withA(pretending to be B), as shown below, and use a message from the latter protocol to successfully complete the former, thereby deceivingBinto believingEisA(and thatAinitiated the protocol). AEB → rA (1) rB,sB(rB,rA,A) ← (2) rB ← (1′) → rA′,sA(rA′,rB,B)( 2 ′) → rA′,sA(rA′,rB,B)( 3 ) This attack is possible due to the message symmetry of (2) and (3), and may be prevented by making their structures differ, securely binding an identiﬁer to each message indicating a message number, or simply requiring the originalrA take the place ofrA′in (3). The implications of this attack depend on the speciﬁc objectives the protocol was as- sumed to provide. Such speciﬁc objectives are, however, (unfortunately) often not explic- itly stated. Attack 4: Misplaced trust in server The Otway-Rees protocol (Protocol 12.29) has messages as follows: A→B: M,A,B,E KAT (NA,M,A,B )( 1 ) B→T: M,A,B,E KAT (NA,M,A,B ),EKBT (NB,M,A,B )( 2 ) B←T: EKAT (NA,k),EKBT (NB,k)( 3 ) A←B: EKAT (NA,k)( 4 ) Upon receiving message (2), the server must verify that the encrypted ﬁelds(M,A,B) in both parts of (2) match, and in addition that these ﬁelds match the cleartext(M,A,B). If the latter check is not carried out, the protocol is open to attack by an enemyE(who is another authorized system user) impersonatingBas follows.Emodiﬁes (2), replacing cleartextB 532 Ch. 12 Key Establishment Protocols by E(but leaving both enciphered versions of both identiﬁersAand Bintact), replacing nonce NB by its own nonceNE, and using keyKET (whichEsharesa prioriwithT) in place ofKBT . Based on the cleartext identiﬁerE,Tthen encrypts part of message (3) underKET allowingEto recoverk; butAbelieves, as in the original protocol, thatkis shared withB. The attack is summarized as follows. A→B: M,A,B,E KAT (NA,M,A,B )( 1 ) B→E: M,A,B,E KAT (NA,M,A,B ),EKBT (NB,M,A,B )( 2 ) E→T: M,A,E,E KAT (NA,M,A,B ),EKET (NE,M,A,B )( 2 ′) E←T: EKAT (NA,k),EKET (NE,k)( 3 ) A←E: EKAT (NA,k)( 4 ) The attack is possible due to the subtle manner by whichAinfers the identity of the other party to whichkis made available: in (4),Ahas no direct indication of the other party to whichThas madekavailable, but relies on the nonceNA in (4) and its association with the pair(NA,B) within the protected part of (1). Thus,Arelies on (or delegates trust to) the server to makekavailable only to the party requested byA, and this can be assured only byTmaking use of the protected ﬁelds(M,A,B). 12.9.2 Analysis objectives and methods The primary aim of protocol analysis is to establish conﬁdence in the cryptographic security of a protocol. The following deﬁnitions aid discussion of protocol analysis. 12.81 DeﬁnitionA key establishment protocol isoperational(orcompliant) if, in the absence of active adversaries and communications errors, honest participants who comply with its speciﬁcation always complete the protocol having computed a common key and knowledge of the identities of the parties with whom the key is shared. The most obvious objectives and properties of key establishment protocols, namely authenticity and secrecy of keys, are discussed in§12.2.2. 12.82 DeﬁnitionA key establishment protocol isresilientif it is impossible for an active adver- sary to mislead honest participants as to the ﬁnal outcome. Protocol analysis should conﬁrm that a protocol meets all claimed objectives. As a minimum, for a key establishment protocol this should include being operational (note this implies no security guarantees), providing both secrecy and authenticity of the key, and being resilient. Key authenticity implies the identities of the parties sharing the key are understood and corroborated, thus addressing impersonation and substitution. Resilience differs subtlely from authentication, and is a somewhat broader requirement (e.g., see the attack of Note 12.54). Additional objectives beyond authenticated key establishment may include key conﬁrmation, perfect forward secrecy, detection of key re-use, and resistance to known-key attacks (see§12.2.3). In addition to verifying objectives are met, additional beneﬁts of analysis include: 1. explicit identiﬁcation of assumptions on which the security of a protocol is based; 2. identiﬁcation of protocol properties, and precise statement of its objectives (this fa- cilitates comparison with other protocols, and determining appropriateness); 3. examination of protocol efﬁciency (with respect to bandwidth and computation). Essentially all protocol analysis methods require the following (implicitly or explicitly): §12.9 Analysis of key establishment protocols 533 1. protocol speciﬁcation– an unambiguous speciﬁcation of protocol messages, when they are sent, and the actions to be taken upon reception thereof; 2. goals– an unambiguous statement of claimed assurances upon completion; 3. assumptions and initial state– a statement of assumptions and initial conditions; 4. proof– some form of argument that, given the assumptions and initial state, the spec- iﬁed protocol steps lead to a ﬁnal state meeting the claimed goals. Analysis methods Common approaches for analyzing cryptographic protocols include the following: 1. ad hoc and practical analysis. This approach consists of any variety of convincing arguments that any successful protocol attack requires a resource level (e.g., time or space) greater than the resources of the perceived adversary. Protocols which sur- vive such analysis are said to haveheuristic security, with security here typically in the computational sense and adversaries assumed to have ﬁxed resources. Argu- ments often presuppose secure building blocks. Protocols are typically designed to counter standard attacks, and shown to follow accepted principles. Practical argu- ments (paralleling complexity-theoretic arguments) involving constructions which assemble basic building blocks may justify security claims. While perhaps the most commonly used and practical approach, it is in some ways the least satisfying. This approach may uncover protocol ﬂaws thereby establishing that a protocol is bad. However, claims of security may remain questionable, as subtle ﬂaws in cryptographic protocols typically escape ad hoc analysis; unforeseen attacks remain a threat. 2. reducibility from hard problems. This technique consists of proving that any success- ful protocol attack leads directly to the ability to solve a well-studied reference prob- lem (Chapter 3), itself considered computationally infeasible given current knowl- edge and an adversary with bounded resources. Such analysis yields so-calledprov- ably secure protocols, although the security is conditional on the reference problem being truly (rather than presumably) difﬁcult. A challenge in this approach is to establish that all possible attacks have been taken into account, and can in fact be equated to solving the identiﬁed reference problems. This approach is considered by some to be as good a practical analysis technique as exists. Such provably secure protocols belong to the larger class of techniques which arecomputationally secure. 3. complexity-theoretic analysis. An appropriate model of computation is deﬁned, and adversaries are modeled as having polynomial computational power (they may mount attacks involving time and space polynomial in the size of appropriate security pa- rameters). A security proof relative to the model is then constructed. The existence of underlying cryptographic primitives with speciﬁed properties is typically assumed. An objective is to design cryptographic protocols which require the fewest crypto- graphic primitives, or the weakest assumptions. As the analysis is asymptotic, care is required to determine when proofs have prac- tical signiﬁcance. Polynomial attacks which are feasible under such a model may nonetheless in practice be computationally infeasible. Asymptotic analysis may be of limited relevance to concrete problems in practice, which have ﬁnite size. Despite these issues, complexity-theoretic analysis is invaluable for formulating fundamental principles and conﬁrming intuition. 4. information-theoretic analysis. This approach uses mathematical proofs involving entropy relationships to prove protocols areunconditionally secure. In some cases, 534 Ch. 12 Key Establishment Protocols this includes the case where partial secrets are disclosed (e.g., for unconditional se- curity against coalitions of ﬁxed size). Adversaries are modeled to have unbounded computing resources. While unconditional security is ultimately desirable, this approach is not applicable to most practical schemes for several reasons. These include: many schemes, such as those based on public-key techniques, can at best be computationally secure; and information-theoretic schemes typically either involve keys of impractically large size, or can only be used once. This approach cannot be combined with computa- tional complexity arguments because it allows unlimited computation. 5. formal methods. So-calledformalanalysis and veriﬁcation methods include logics of authentication (cryptographic protocol logics), term re-writing systems, expert sys- tems, and various other methods which combine algebraic and state-transition tech- niques. The most popular protocol logic is the Burrows-Abadi-Needham (BAN) log- ic. Logic-based methods attempt to reason that a protocol is correct by evolving a set of beliefs held by each party, to eventually derive a belief that the protocol goals have been obtained. This category of analysis is somewhat disjoint from the ﬁrst four. Formal meth- ods have proven to be of utility in ﬁnding ﬂaws and redundancies in protocols, and some are automatable to varying degrees. On the other hand, the “proofs” provided are proofs within the speciﬁed formal system, and cannot be interpreted as absolute proofs of security. A one-sidedness remains: the absence of discovered ﬂaws does not imply the absence of ﬂaws. Some of these techniques are also unwieldy, or ap- plicable only to a subset of protocols or classes of attack. Many require (manually) converting a concrete protocol into a formal speciﬁcation, a critical process which itself may be subject to subtle ﬂaws. 12.10 Notes and further references §12.1 While the literature is rife with proposals for key establishment protocols, few comprehen- sive treatments exist and many proposed protocols are supported only by ad hoc analysis. §12.2 Much of§12.2 builds on the survey of Rueppel and van Oorschot [1086]. Fumy and Munz- ert [431] discuss properties and principles for key establishment. While encompassing the majority of key establishment as currently used in practice, Deﬁnition 12.2 gives a some- what restricted view which excludes a rich body of research. More generally, key establish- ment may be deﬁned as a process or mechanism which provides a shared capability (rather than simply a shared secret) between speciﬁed sets of participants, facilitating some oper- ation for which the intention is that other sets of participants cannot execute. This broader deﬁnition includes many protocols in the area ofthreshold cryptography, introduced inde- pendently by Desmedt [336], Boyd [182], and Croft and Harris [288]; see the comprehen- sive survey of Desmedt [337]. The termperfect forward secrecy(Deﬁnition 12.16) was coined by G¨unther [530]; see also Difﬁe, van Oorschot, and Wiener [348]. Here “perfect” does not imply any properties of information-theoretic security (cf. Deﬁnition 12.73). The concept of known-key attacks (Deﬁnition 12.17), developed by Yacobi and Shmuely [1256] (see also Yacobi [1255]), is §12.10 Notes and further references 535 related to that of Denning and Sacco [330] on the use of timestamps to prevent message replay (see page 535). Among items not discussed in detail in this chapter isquantum cryptography, based on the uncertainty principle of quantum physics, and advanced by Bennett et al. [114] building on the idea of quantum coding ﬁrst described by Wiesner [1242]circa1970. Although not providing digital signatures or non-repudiation, quantum cryptography allows key distribu- tion (between two parties who share noa priorisecret keying material), which is provably secure against adversaries with unlimited computing power, provided the parties have ac- cess to (aside from the quantum channel) a conventional channel subject to only passive adversaries. For background on the basic quantum channel for key distribution (quantum key distribution), see Brassard [192]; Phoenix and Townsend [973] survey developments in this area including experimental implementations. Mitchell [879] presented a key agreement system based on use of a public broadcast channel transmitting data at a rate so high that an eavesdropper cannot store all data sent over a speciﬁed time interval. This is closely related to work of Maurer [815] regarding secret key agreement using only publicly available information, in turn motivated by Wyner’swire- tap channel[1254], which addresses the rate at which secret information can be conveyed to a communicating partner with security against a passive eavesdropper whose channel is subject to additional noise. §12.3 Regarding point-to-point techniques presented, those based on symmetric encryption are essentially from ISO/IEC 11770-2 [617], while AKEP1 and AKEP2 (Note 12.21; Proto- col 12.20) are derived from Bellare and Rogaway [94] (see also§12.9 below). The idea of key derivation allowing key establishment by symmetric techniques based on a one- way function (without encryption), was noted brieﬂy by Matsumoto, Takashima and Imai [800]; see also the proposals of Gong [499], and related techniques in theKryptoKnight suite [891, 141, 142]. Shamir’s no-key protocol (Protocol 12.22; also calledShamir’ s three-pass protocol), in- cluding exponentiation-based implementation, is attributed to Shamir by Konheim [705, p.345]. Massey [786, p.35] notes that Omura [792], aware of Shamir’s generic protocol, later independently proposed implementing it with an exponentiation-based cipher as per Protocol 12.22. See also Massey and Omura [956] (discussed in Chapter 15). Version 5 of Kerberos (V5), the development of which began in 1989, was speciﬁed by Kohl and Neuman [1041]; for a high-level overview, see Neuman and Ts’o [926] who also note that a typical timestamp window is 5 minutes (centered around the veriﬁer’s time). The original design of Kerberos V4 was by Miller and Neuman, with contributions by Saltzer and Schiller [877]; an overview is given by Steiner, Neuman, and Schiller [1171], while V4 issues are noted by Kohl [701] and the critique of Bellovin and Merritt [103]. The basic pro- tocol originates from the shared-key protocol of Needham and Schroeder [923], with time- stamps (which Needham and Schroeder explicitly avoided) later proposed by Denning and Sacco [330], reducing the number of messages at the expense of secure and synchronized clocks. Bauer, Berson, and Feiertag [76] addressed symmetric assurances of freshness, re- covery from single-key compromise, and reduction of messages through per-participant use of a local counter called anevent marker; they also extended the Needham-Schroeder setting to multiple security domains (each with a separate KDC) and connectionless envi- ronments. Bellare and Rogaway [96] presented an efﬁcient 4-pass server-based key trans- fer protocol with implicit key authentication, and key freshness properties secure against known-key attacks; signiﬁcantly, their treatment (the ﬁrst of its kind) shows the protocol to 536 Ch. 12 Key Establishment Protocols be provably secure (assuming a pseudorandom function). Advantages and disadvantages of using timestamps are discussed in§10.3.1. Protocol 12.29 is due to Otway and Rees [961]. Kehne, Sch¨onw¨alder, and Langend¨orfer [663] discuss a 5-message nonce-based protocol with the same features as Kerberos (Proto- col 12.24), without requiring distributed timeclocks. Excluding the optional re-authenticat- ion capability (as per Kerberos), it is essentially that of Mechanism 9 in ISO/IEC DIS 11770-2 [617], and similar to the 5-message Otway-Rees protocol as augmented per Re- mark 12.30 (with one fewer encryption by each ofAand B); but see also the analysis of Neuman and Stubblebine [925]. A 5-message authentication protocol included in ISO/IEC 9798-2 [599] provides key transport using a trusted server, with mutual entity authentication and mutual key conﬁrmation, without timestamps; Needham and Schroeder [924] propose a 7-message protocol with similar properties. §12.4 Mechanism 12.35 and Fact 12.34 are due to Blom [158]; a simpler polynomial formulation is noted under§12.8 below. For background in coding theory, see MacWilliams and Sloane [778]. Mitchell and Piper [881] consider the use of combinatorial block designs and ﬁnite incidence structures calledkey distribution patternsto construct a class of non-interactive KDS. Each user is given a set of secret subkeys (with no algebraic structure as per Blom’s scheme), from which each pair of users may compute a common key by combining appro- priate subkeys via a public function. The question of reducing key storage was considered earlier by Blom [157], including security against coalitions of ﬁxed size and the use of com- mutative functions (later generalized to symmetric functions by Blundo et al. [169]; see also §12.8 below). For related work, see Quinn [1014], Gong and Wheeler [506], and§12.7 be- low. §12.5 Protocol 12.38, the public-key protocol of Needham and Schroeder [923], was originally speciﬁed to include 4 additional messages whereby signed public keys were requested from an on-line certiﬁcation authority. Asymmetric key transport protocols involving various combinations of encryption and signatures are given in ISO/IEC CD 11770-3 [618]. The three-pass encrypt-then-sign protocol of§12.5.2(iii) originates from ISO/IEC 9798-3 [600]; it is closely related to the STS protocol (Protocol 12.57) which transfers Difﬁe-Hellman exponentials in place of random numbers. I’Anson and Mitchell [567] critique (e.g., see Note 12.42) the X.509 protocols [595]; see also the formal analysis of Gaarder and Snekken- es [433]. Protocol 12.44 and the related 2-pass key agreement of Figure 12.2 are due to Beller and Yacobi [101, 100], building on work of Beller, Chang, and Yacobi [99, 98, 97]. A two-pass key transport protocol called COMSET, based on public-key encryption, was adopted by the European community RACE Integrity Primitives Evaluation (RIPE) project [178]. Arising from zero-knowledge considerations studied by Brandt et al. [188], it em- ploys Williams’ variant of the Rabin public-key encryption (§8.3), and is similar in some aspects to the Needham-Schroeder public-key and Beller-Yacobi protocols. The protocol speciﬁed in Note 12.39 combines concepts of COMSET and the Needham-Schroeder pro- tocol. §12.6 The landmark 1976 paper of Whitﬁeld Difﬁe and Martin Hellman [345] is the standard ref- erence for both the seminal idea of public-key cryptography and the fundamental technique of exponential key agreement. An earlier conference paper of Difﬁe and Hellman [344], written in December 1975 and presented in June 1976, conceived the concept of public key agreement and the use of public-key techniques for identiﬁcation and digital signatures. §12.10 Notes and further references 537 Difﬁe [342] reports that amidst joint work on the problem for some time, Hellman distilled exponential key agreement in May 1976, and this was added to their June 1976 conference presentation (but not the written paper). Preceding this, in the fall of 1974, Merkle inde- pendently conceived a particular method for key agreement using the same abstract con- cepts. Merkle’spuzzle system[849], submitted for publication in 1975 and appearing in April 1978, is as follows. Alice constructsmpuzzles, each of which is a cryptogram Bob can solve innsteps (exhaustively tryingnkeys until a recognizable plaintext is found). Al- ice sends allmpuzzles to Bob over an unsecured channel. Bob picks one of these, solves it (cost:nsteps), and treats the plaintext therein as the agreed key, which he then uses to encrypt and send to Alice a known message. The encrypted message, now a puzzle which Alice must solve, takes Alicensteps (by exhaustively tryingnkeys). Form≈n, each of Alice and Bob requireO(n) steps for key agreement, while an opponent requiresO(n 2) steps to deduce the key. An appropriate valuenis chosen such thatnsteps is computation- ally feasible, butn2 is not. Rueppel [1078] explores the use of function composition to generalize Difﬁe-Hellman key agreement. Shmuely [1127] and McCurley [825] considercomposite Difﬁe-Hellman, i.e., Difﬁe-Hellman key agreement with a composite modulus. McCurley presents a variation thereof, with an RSA-like modulusmof speciﬁc form and particular baseαof high order inZ ∗ m, which is provably as secure (under passive attack) as the more difﬁcult of factoring mand solving the discrete logarithm problem modulo the factors ofm. Regarding Difﬁe-Hellman key agreement, van Oorschot and Wiener [1209] note that use of “short” private exponents in conjunction with a random prime modulusp(e.g., 256-bit exponents with 1024-bitp) makes computation of discrete logarithms easy. They also doc- ument the attack of Note 12.50, which is related to issues explored by Simmons [1150] con- cerning a party’s ability to control the resulting Difﬁe-Hellman key, and more general issues of unfairness in protocols. Waldvogel and Massey [1228] carefully examine the probability distribution and entropy of Difﬁe-Hellman keys under various assumptions. When private exponents are chosen independently and uniformly at random from the invertible elements ofZ p−1, theφ(p−1) keys which may result are equiprobable. When private exponents are chosen independently and uniformly at random from{0,...,p −2}(as is customary in practice), in the best case (whenpis asafe prime,p=2 q+1 ,qprime) the most probable Difﬁe-Hellman key is only 6 times more likely than the least probable, and the key entropy is less than 2 bits shy of the maximum,lg(p−1); while in the worst case (governed by a particular factorization pattern ofp−1) the distribution is still sufﬁciently good to preclude signiﬁcant cryptanalytic advantage, forpof industrial size or larger. The one-pass key agreement of Protocol 12.51 was motivated by the work of ElGamal [368]. The MTI protocols of Table 12.5 were published in 1986 by Matsumoto, Takashima, and Imai [800]. MTI/A0 is closely related to a scheme later patented by Goss [519]; in the latter, exclusive-OR is used in place of modular multiplication to combine partial keys. Matsumoto et al. equate the computational complexity of passive attacks (exclud- ing known-key attacks) on selected key agreement protocols to that of one or two Difﬁe- Hellman problems. Active attacks related to Note 12.54 are considered by Difﬁe, van Oorschot, and Wiener [348], and Menezes, Qu, and Vanstone [844]. Yacobi and Shmuely [1256] note two time-variant versions of Difﬁe-Hellman key agreement which are inse- cure against known-key attack. A similar protocol which falls to known-key attack was discussed by Yacobi [1255], subsequently rediscovered by Alexandris et al. [21], and re- examined by Nyberg and Rueppel [937]. Yacobi [1255] proves that the MTI/A0 proto- col with composite-modulus is provably secure (security equivalent to composite Difﬁe- Hellman) under known-key attack by a passive adversary; Desmedt and Burmester [339], 538 Ch. 12 Key Establishment Protocols however, note the security is only heuristic under known-key attack by an active adversary. A formal-logic security comparison of the protocols of Goss (essentially Protocol 12.53), G¨unther (Protocol 12.62), and STS (Protocol 12.57) is given by van Oorschot [1204]. Burmester [220] identiﬁes known-keytriangle attackswhich may be mounted on the for- mer two and related protocols which provide only implicit key authentication (including MTI protocols, cf. Note 12.54). Known-key attacks were also one motivation for Denning and Sacco [330] to modify the Needham-Schroeder protocol as discussed above (cf. p.534). Protocol 12.57 (STS) evolved from earlier work on ISDN telephone security as outlined by Difﬁe [342, p.568], who also reports on STU-III telephones. Variations of STS and an infor- mal model for authentication and authenticated key establishment are discussed by Difﬁe, van Oorschot, and Wiener [348]. Bellovin and Merritt [104, 105] (see also the patent [102]) propose another hybrid protocol (Encrypted Key Exchange– EKE), involving exponential key agreement with authentication based on a shared password, designed speciﬁcally to protect against password-guessing attacks by precluding easy veriﬁcation of guessed pass- words; Steiner, Tsudik, and Waidner [1172] provide further analysis and extensions. A hy- brid protocol with similar goals is given Gong et al. [504], including discussion of its rela- tionship to EKE, and expanding the earlier work of Lomas et al. [771]. Blom [157] was apparently the ﬁrst to propose an identity-based (or more accurately, index-based) key establishment protocol. Shamir [1115] proposed the more general idea of identity-based systemswherein a user’s public key may be a commonly known name and address. For further discussion of ID-based schemes, see the chapter notes on§13.4. Self- certiﬁed public keys (Mechanism 12.61) are discussed by Girault [459], who credits earlier work by others, and provides the self-certiﬁed version of G¨unther’s ID-based keys (Exam- ple 12.67). The parenthetical forgery attack mentioned in Mechanism 12.61 is outlined by Stinson [1178]. Key agreement protocols as in Examples 12.64 and 12.65, using both ID- based public keys of G¨unther [530] (Mechanism 12.59) and modiﬁed ElGamal signatures, are given by Horster and Knobloch [562]. The optimization of ElGamal signatures noted in Remark 12.60 is by Agnew, Mullin, and Vanstone [19]. Rabin’s signature scheme (Chap- ter 11) may be used in place of RSA to reduce the computations required in schemes based on Girault’s implicitly-certiﬁed public keys. Maurer and Yacobi [824] (modifying their earlier proposal [823]) propose an identity-based one-pass key pre-distribution scheme us- ing composite modulus Difﬁe-Hellman, featuring implicitly-certiﬁed public key-agreement keys essentially consisting of a user’s identity (or email address); the corresponding private key is the discrete logarithm of this, computed by a trusted authority which, knowing the factorization of an appropriately chosen modulusn, can thereby compute logarithms. Nyberg and Rueppel [936] note their signature scheme (Chapter 11) may be used to cre- ate implicitly certiﬁed, identity-based public keys with properties similar to those of Gi- rault (Mechanism 12.61), as well as key agreement protocols; Nyberg [935] presents an im- proved one-pass key agreement based on these ideas. Okamoto and Tanaka [946] propose identity-based key agreement protocols combining exponential key agreement and RSA, including one using timestamps and providing entity authentication, and a simpler protocol providing (implicit) key authentication. §12.7 The idea of split control has long been known (e.g., see Sykes [1180]). Shamir [1110] and Blakley [148] independently proposed the idea of threshold schemes, the latter based on vector subspaces. The simplest example of the Blakley’s idea is a(2,n) threshold scheme where the shares (here calledshadows) distributed to parties are non-collinear lines in a common plane; the shared secret of any two parties is the intersection of their lines. For a (3,n) scheme, the shadows consist of non-parallel planes, any two of which intersect in a §12.10 Notes and further references 539 line, and any three of which intersect in a point. While Shamir’s threshold scheme isperfect, Blakley’s vector scheme is not (the set of possible values of the shared secret narrows as subsequent shares are added). Karnin, Greene, and Hellman [662] discuss the unanimous consent control scheme of§12.7.1; see also Difﬁe and Hellman [344, p.110]. Generalized secret sharing schemes and the idea of access structures were ﬁrst studied by Ito, Saito, and Nishizeki [625], who provided a construction illustrating that any monotone access structure can be realized by a perfect secret sharing scheme. Benaloh and Leichter [112] provided more elegant constructions. A comprehensive discussion of secret shar- ing including adaptations providing shared control capabilities of arbitrary complexity, and many of the extended capabilities including pre-positioned schemes, is given by Simmons [1145, 1141, 1142], mainly with geometric illustration. An exposition by Stinson [1177] addresses information rate in particular. Ingemarsson and Simmons [570] consider secret sharing schemes which do not require a trusted party. Laih et al. [732] consider dynamic secret sharing schemes. Blundo et al. [168] consider pre-positioned schemes, dynamic secret sharing, and bounds on share sizes and broadcast messages therein; Jackson, Martin, and O’Keefe [629] examine related multi-secret thresh- old schemes. Blakley et al. [147] consider threshold schemes with disenrollment. Tompa and Woll [1195] note that an untrustworthy participantUmay cheat in Shamir’s threshold scheme by submitting a share different than its own, but carefully computed such that pooling of shares provides other participants with no information about the secretS, while allowingUto recoverS. They propose modiﬁcations which (with high probability) allow detection of cheating, and which prevent a cheaterUfrom actually obtaining the se- cret. The related problem ofveriﬁable secret sharing, which is of broader interest in secure dis- tributed computation, was introduced by Chor et al. [259]; see also Benaloh [110] and Feld- man [390], as well as Rabin and Ben-Or [1028]. Here the trusted party distributing shares might also cheat, and the goal is to verify that all distributed shares are consistent in the sense that appropriate subsets of shares deﬁne the same secret. For applications of veriﬁ- able secret sharing to key escrow, see Micali [863]. Fact 12.75 is based on the deﬁnition of perfect secret sharing and information-theoretic se- curity, as is the majority of research in secret sharing.Ramp schemes with shares shorter than the secret were examined by Blakley and Meadows [151]; while trading off per- fect security for shorter shares, their examination is nonetheless information-theoretic. In practice, a more appropriate goal may be computationally secure secret sharing; here the objective is that if one or more shares is missing, an opponent has insufﬁcient informa- tion to (computationally) recover the shared secret. This idea was elegantly addressed by Krawczyk [715] as follows. To share a larges-bit secretS = P(e.g., a plaintext ﬁle) among nusers, ﬁrst encrypt it under ak-bit symmetric keyKas C = E K(P); using a perfect secret sharing scheme such as Shamir’s(t,n) scheme, splitKintonk-bit shares K1,...,K n; then using Rabin’sinformation dispersal algorithm(IDA) [1027] splitC intonpiecesC1,...,C n each of(s/t) bits; ﬁnally, distribute to userUi the secret share Si =( Ki,Ci). Anytparticipants who pool their shares can then recoverKby secret shar- ing,Cby IDA, andP= Sby decryptingCusingK. By the remarkable property of IDA, the sum of the sizes of thetpiecesCi used is exactly the size of the recovered secretSitself (which cannot be bettered); globally, the only space overhead is that for the short keysKi, whose sizekis independent of the large secretS. The clever idea ofvisual cryptographyto facilitate sharing (or encryption) of pictures is due to Naor and Shamir [919]. The pixels of a (secret) picture are treated as individual secrets 540 Ch. 12 Key Establishment Protocols to be shared. The picture is split into two or more images each of which contains one share for each original pixel. Each original pixel is split into shares by subdivision into subpixels of appropriate size, with selection of appropriate combinations of subpixel shadings (black and white) such that stacking the images on transparencies reveals the original, while each individual image appears random. Picture recovery requires no computation (it is visual); anyone with all but one of the images still has (provably) no information. §12.8 An early investigation of conference keying schemes based on Difﬁe-Hellman key agree- ment was undertaken by Ingemarsson, Tang and Wong [571]. The protocol of Burmester and Desmedt [222] (Protocol 12.78) is the most efﬁcient of those which have been proposed and are provably secure; their work includes a review of alternate proposals and a thorough bibliography. Research in this area with particular emphasis on digital telephony includes that of Brickell, Lee, and Yacobi [205]; Steer et al. [1169]; and Heiman [547]. Matsumoto and Imai [799] systematically deﬁne (symmetric-key) key pre-distribution sch- emes, based on symmetric functions, for conferences of two or more parties. Their propos- als are non-interactive and ID-based, following the original idea of two-party non-interact- ive ID-based schemes by Blom [157, 158], including consideration of information-theoretic security against coalitions of ﬁxed size. Tsujii and Chao [1197], among many others, pro- pose schemes in a similar setting. Blundo et al. [169] both specialize the work of Mat- sumoto and Imai, and generalize Blom’s symmetric key distribution (Mechanism 12.35) and bounds from two-party key pre-distribution to non-interactivej-secure conference key- ing schemes of ﬁxed size; prove Fact 12.79; and provide a scheme meeting this bound. Their generalization uses symmetric polynomials intvariables for privileged subsets of size t, yielding in the two-party case (t=2 ) an equivalent but simpler formulation of Blom’s scheme: the trusted party selects an appropriate secret symmetric polynomialf(x,y) and gives partyithe secret univariate polynomialf(i,y), allowing partiesiand jto share the pairwise keyf(i,j)= f(j,i). They also consider an interactive model. Further examina- tion of interactive vs. non-interactive conferencing is undertaken by Beimel and Chor [83]. Fiat and Naor [394] considerj-secure broadcast encryption schemes, and practical schemes requiring less storage; for the former, Blundo and Cresti [167] establish lower bounds on the number of keys held and the size of user secrets. Berkovits [116] gives constructions for creatingsecret broadcasting schemes(conference keying schemes where all messages are broadcast) from(t,n) threshold schemes. Essen- tially, for conferences withtmembers, a new(t+1 ,2t+1 )threshold scheme with secret Kis created from the old, andtnew shares are publicly broadcast such that each of thet pre-assigned secret shares of the intended conference members serves as sharet+1, allow- ing recovery of the conference keyKin the new scheme. For related work involving use of polynomial interpolation, key distribution involving a trusted party, and broadcasting keys, see Gong [502] and Just et al. [647]. §12.9 The intruder-in-the-middle attack (Attack 1) is discussed by Rivest and Shamir [1057], who propose an “interlock protocol” to allow its detection; but see also Bellovin and Mer- ritt [106]. The reﬂection attack (Attack 2) is discussed by Mitchell [880]. Attack 4 on the Otway-Rees protocol is discussed by Boyd and Mao [183] and van Oorschot [1205]. The interleaving attack (Attack 3) is due to Wiener circa June 1991 (document ISO/IEC JTC1/SC27 N313, 2 October 1991), and discussed by Difﬁe, van Oorschot, and Wiener [348] along with attacks on sundry variations of Difﬁe-Hellman key agreement. Bird et al. [140] systematically examine interleaving attacks on symmetric-key protocols, consider §12.10 Notes and further references 541 exhaustive analysis to detect such attacks, and propose a protocol resistant thereto (namely 2PP, included in the IBM prototypeKryptoKnight[891]; see also [141, 142]). Bellare and Rogaway [94], building on the work of earlier informal models, present a complexity-theoretic communications model and formal deﬁnitions for secure symmetric- key two-party mutual authentication and authenticated key establishment, taking known- key attacks into account. They prove AKEP1 (Note 12.21) and AKEP2 (Protocol 12.20) secure relative to this model, for parameters of appropriate size and assuminghand h ′are pseudorandom functions or pseudorandom permutations; they also suggest practical con- structions for pseudorandom functions based on DES and MD5. Gong [503] examines the efﬁciency of various authentication protocols and proposes lower bounds (e.g., on the num- ber of message-passes required). The examples illustrating attacks on ﬂawed protocols are only a few of countless docu- mented in the literature. Moore [898] provides an excellent survey on protocol failure; see also Anderson and Needham [31] and Abadi and Needham [1] for sound engineering prin- ciples. A large number of authenticated key establishment protocols with weaknesses are analyzed using the BAN logic in the highly recommended report of Burrows, Abadi, and Needham [227] (and by the same title: [224, 226, 225]). Gligor et al. [463] discuss the lim- itations of authentication logics. Syverson [1181] examines the goals of formal logics for protocol analysis and the utility of formal semantics as a reasoning tool. Among the au- thentication logics evolving from BAN are those of Abadi and Tuttle [2], Gong, Needham, and Yahalom [505], and Syverson and van Oorschot [1183]. The work of Abadi and Tuttle is notable for its model of computation and formal semantics relative to this model. Lamp- son et al. [740] both provide a theory of authentication in distributed systems (including delegation and revocation) and discuss a practical system based on this theory. One of the ﬁrst contributions to formal protocol analysis was that of Dolev and Yao [359], whose formal model, which focuses on two-party protocols for transmitting secret plain- texts, facilitates precise discussion of security issues. This approach was augmented with respect to message authentication and information leakage by Book and Otto [170]. Three general approaches to protocol analysis are discussed by Kemmerer, Meadows, and Millen [664] (see also Simmons [1148]): an algebraic approach, a state transition approach, and a logical approach (which can be given a state-transition semantics). They illustrate sev- eral methods on a protocol with known ﬂaws (the infamous TMN protocol of Tatebayashi, Matsuzaki, and Newman [1188]). Other recent surveys on formal methods include that of Meadows [831], and the comprehensive survey of Rubin and Honeyman [1073]. An exten- sive bibliographic tour of authentication literature is provided by Liebl [765]. Chapter /1/3 Key Management Techniques Contents in Brief 13.1 Introduction............................. 543 13.2 Background and basic concepts................... 544 13.3 Techniques for distributing conﬁdential keys............ 551 13.4 Techniques for distributing public keys.............. 555 13.5 Techniques for controlling key usage................ 567 13.6 Key management involving multiple domains........... 570 13.7 Key life cycle issues......................... 577 13.8 Advanced trusted third party services............... 581 13.9 Notes and further references.................... 586 13.1 Introduction This chapter considers key management techniques for controlling the distribution, use, and update of cryptographic keys. Whereas Chapter 12 focuses on details of speciﬁc key estab- lishment protocols which provide shared secret keys, here the focus is on communications models for key establishment and use, classiﬁcation and control of keys based on their in- tended use, techniques for the distribution of public keys, architectures supporting auto- mated key updates in distributed systems, and the roles of trusted third parties. Systems providing cryptographic services require techniques for initialization and key distribution as well as protocols to support on-line update of keying material, key backup/recovery, re- vocation, and for managing certiﬁcates in certiﬁcate-based systems. This chapter examines techniques related to these issues. Chapter outline The remainder of this chapter is organized as follows.§13.2 provides context including background deﬁnitions, classiﬁcation of cryptographic keys, simple models for key estab- lishment, and a discussion of third party roles.§13.3 considers techniques for distributing conﬁdential keys, including key layering, key translation centers, and symmetric-key cer- tiﬁcates.§13.4 summarizes techniques for distributing and authenticating public keys in- cluding authentication trees, public-key certiﬁcates, the use of identity-based systems, and implicitly-certiﬁed keys.§13.5 presents techniques for controlling the use of keying mate- rial, including key notarization and control vectors.§13.6 considers methods for establish- ing trust in systems involving multiple domains, certiﬁcation authority trust models, and 543 544 Ch. 13 Key Management Techniques certiﬁcation chains. The key management life cycle is summarized in§13.7, while§13.8 discusses selected specialized third party services, including trusted timestamping and no- tary services supporting non-repudiation of digital signatures, and key escrow. Notes and sources for further information are provided in§13.9. 13.2 Background and basic concepts A keying relationshipis the state wherein communicating entities share common data (key- ing material) to facilitate cryptographic techniques. This data may include public or secret keys, initialization values, and additional non-secret parameters. 13.1 DeﬁnitionKey managementis the set of techniques and procedures supporting the estab- lishment and maintenance of keying relationships between authorized parties. Key management encompasses techniques and procedures supporting: 1. initialization of system users within a domain; 2. generation, distribution, and installation of keying material; 3. controlling the use of keying material; 4. update, revocation, and destruction of keying material; and 5. storage, backup/recovery, and archival of keying material. 13.2.1 Classifying keys by algorithm type and intended use The terminology of Table 13.1 is used in reference to keying material. Asymmetric cryp- tographic systemis a system involving two transformations – one for the originator and one for the recipient – both of which make use of either the same secret key (symmetric key) or two keys easily computed from each other. Anasymmetric cryptographic system is a system involving two related transformations – one deﬁned by a public key (the public transformation), and another deﬁned by a private key (the private transformation) – with the property that it is computationally infeasible to determine the private transformation from the public transformation. Term Meaning private key, public key paired keys in an asymmetric cryptographic system symmetric key key in a symmetric (single-key) cryptographic system secret adjective used to describe private or symmetric key Table 13.1:Private, public, symmetric, and secret keys. Table 13.2 indicates various types of algorithms commonly used to achieve the spec- iﬁed cryptographic objectives. Keys associated with these algorithms may be correspond- ingly classiﬁed, for the purpose of controlling key usage (§13.5). The classiﬁcation given requires speciﬁcation of both the type of algorithm (e.g., encryption vs. signature) and the intended use (e.g., conﬁdentiality vs. entity authentication). §13.2 Background and basic concepts 545 Algorithm type ↓Cryptographic objective (usage) public-key symmetric-key conﬁdentiality† encryption encryption data origin authentication‡ signature MAC key agreement Difﬁe-Hellman various methods entity authentication 1. signature 1. MAC (by challenge-response protocols) 2. decryption 2. encryption 3. customized Table 13.2:Types of algorithms commonly used to meet speciﬁed objectives. †May include data integrity, and includes key transport; see also§13.3.1. ‡Includes data integrity; and in the public-key case, non-repudiation. 13.2.2 Key management objectives, threats, and policy Key management plays a fundamental role in cryptography as the basis for securing cryp- tographic techniques providing conﬁdentiality, entity authentication, data origin authenti- cation, data integrity, and digital signatures. The goal of a good cryptographic design is to reduce more complex problems to the proper management and safe-keeping of a small number of cryptographic keys, ultimately secured through trust in hardware or software by physical isolation or procedural controls. Reliance on physical and procedural secu- rity (e.g., secured rooms with isolated equipment), tamper-resistant hardware, and trust in a large number of individuals is minimized by concentrating trust in a small number of easily monitored, controlled, and trustworthy elements. Keying relationships in a communications environment involve at least two parties (a sender and a receiver) in real-time. In a storage environment, there may be only a single party, which stores and retrieves data at distinct points in time. The objective of key management is to maintain keying relationships and keying ma- terial in a manner which counters relevant threats, such as: 1. compromise of conﬁdentiality of secret keys. 2. compromise of authenticity of secret or public keys. Authenticity requirements in- clude knowledge or veriﬁability of the true identity of the party a key is shared or associated with. 3. unauthorized use of secret or public keys. Examples include using a key which is no longer valid, or for other than an intended purpose (see Remark 13.32). In practice, an additional objective is conformance to a relevant security policy. Security policy and key management Key management is usually provided within the context of a speciﬁcsecurity policy. A se- curity policy explicitly or implicitly deﬁnes the threats a system is intended to address. The policy may affect the stringency of cryptographic requirements, depending on the suscepti- bility of the environment in question to various types of attack. Security policies typically also specify: 1. practices and procedures to be followed in carrying out technical and administrative aspects of key management, both automated and manual; 2. the responsibilities and accountability of each party involved; and 3. the types of records (audit trail information) to be kept, to support subsequent reports or reviews of security-related events. 546 Ch. 13 Key Management Techniques 13.2.3 Simple key establishment models The following key distribution problem motivates more efﬁcient key establishment models. The n2 key distribution problem In a system withn users involving symmetric-key techniques, if each pair of users may potentially need to communicate securely, then each pair must share a distinct secret key. In this case, each party must haven −1 secret keys; the overall number of keys in the system, which may need to be centrally backed up, is thenn(n −1)/2, or approximately n2. As the size of a system increases, this number becomes unacceptably large. In systems based on symmetric-key techniques, the solution is to use centralized key servers: a star-like or spoked-wheel network is set up, with a trusted third party at the cen- ter or hub of communications (see Remark 13.3). This addresses then2 key distribution problem, at the cost of the requirement of an on-line trusted server, and additional commu- nications with it. Public-key techniques offer an alternate solution. Point-to-point and centralized key management Point-to-point communications andcentralized key management, using key distribution centers or key translation centers, are examples of simple key distribution (communica- tions) models relevant to symmetric-key systems. Here “simple” implies involving at most one third party. These are illustrated in Figure 13.1 and described below, whereK XY de- notes a symmetric key shared byX and Y . A KDC (a) Point-to-point key distribution (b) Key distribution center (KDC) (i) (ii) B (2) (3) K A KTC (i) (ii) (1) B (2) K(1) (3) (c) Key translation center (KTC) K K K K (1) (2) K (1) K (2) (3) KDC K B K KTC A AB AB Figure 13.1:Simple key distribution models (symmetric-key). §13.2 Background and basic concepts 547 1. point-to-pointmechanisms. These involve two parties communicating directly (see §12.3.1). 2. key distribution centers(KDCs). KDCs are used to distribute keys between users which share distinct keys with the KDC, but not with each other. A basic KDC protocol proceeds as follows.1 Upon request fromA to share a key with B, the KDCT generates or otherwise acquires a keyK, then sends it encrypted under KAT toA, along with a copy ofK (forB) encrypted underKBT . Alternatively,T may communicateK (secured underKBT )t oB directly. 3. key translation centers(KTCs). The assumptions and objectives of KTCs are as for KDCs above, but here one of the parties (e.g.,A) supplies the session key rather than the trusted center. A basic KTC protocol proceeds as follows.2 A sends a keyK to the KTCT encrypted underKAT . The KTC deciphers and re-enciphersK underKBT , then returns this toA (to relay toB) or sends it toB directly. KDCs provide centralized key generation, while KTCs allow distributed key genera- tion. Both are centralized techniques in that they involve an on-line trusted server. 13.2 Note (initial keying requirements) Point-to-point mechanisms require thatA and B share a secret keya priori. Centralized key management involving a trusted partyT requires that A and B each share a secret key withT. These shared long-term keys are initially estab- lished by non-cryptographic, out-of-band techniques providing conﬁdentiality and authen- ticity (e.g., in person, or by trusted courier). By comparison, with public keys conﬁdential- ity is not required; initial distribution of these need only guarantee authenticity. 13.3 Remark (centralized key management – pros and cons) Centralized key management in- volving third parties (KDCs or KTCs) offers the advantage of key-storage efﬁciency: each party need maintain only one long-term secret key with the trusted third party (rather than one for each potential communications partner). Potential disadvantages include: vulner- ability to loss of overall system security if the central node is compromised (providing an attractive target to adversaries); a performance bottleneck if the central node becomes over- loaded; loss of service if the central node fails (a critical reliability point); and the require- ment of an on-line trusted server. 13.2.4 Roles of third parties Below, trusted third parties (TTPs) are ﬁrst classiﬁed based on their real-time interactions with other entities. Key management functions provided by third parties are then discussed. (i) In-line, on-line, and off-line third parties From a communications viewpoint, three categories of third partiesT can be distinguished based on relative location to and interaction with the communicating partiesA and B (see Figure 13.2): 1. in-line: T is an intermediary, serving as the real-time means of communication be- tween A and B. 2. on-line: T is involved in real-time during each protocol instance (communicating withA orB or both), butA and B communicate directly rather than throughT. 1For speciﬁc examples of such protocols including Kerberos (Protocol 12.24), see§12.3.2. 2A speciﬁc example is the message-translation protocol, Protocol 13.12, withM = K. 548 Ch. 13 Key Management Techniques 3. off-line: T is not involved in the protocol in real-time, but prepares informationa priori, which is available toA orB or both and used during protocol execution. (a) in-line B (b) on-line B B communications carried out prior to protocol run (c) off-line on-line [optional] [optional] A A A in-line TTP TTP TTP off-line Figure 13.2:In-line, on-line, and off-line third parties. In-line third parties are of particular interest whenA and B belong to different secu- rity domains or cannot otherwise interact directly due to non-interoperable security mecha- nisms. Examples of an in-line third party include a KDC or KTC which provides the com- munications path betweenA and B, as in Figure 13.1(b)(ii) or (c)(ii). Parts (b)(i) and (c)(i) illustrate examples of on-line third parties which are not in-line. An example of an off-line third party is a certiﬁcation authority producing public-key certiﬁcates and placing them in a public directory; here, the directory may be an on-line third party, but the certiﬁcation authority is not. 13.4 Remark (pros and cons: in-line, on-line, off-line) Protocols with off-line third parties usu- ally involve fewer real-time message exchanges, and do not require real-time availability of third parties. Revocation of privileges (e.g., if a secret key is compromised) is more easily handled by in-line or on-line third parties. (ii) Third party functions related to public-key certiﬁcates Potential roles played by third parties within a key management system involving public- key certiﬁcates (§13.4.2) are listed below. Their relationship is illustrated in Figure 13.3. 1. certiﬁcation authority(CA) – responsible for establishing and vouching for the au- thenticity of public keys. In certiﬁcate-based systems (§13.4.2), this includes binding public keys to distinguished names through signed certiﬁcates, managing certiﬁcate serial numbers, and certiﬁcate revocation. 3 3Certiﬁcate creation requires veriﬁcation of the authenticity of the entity to be associated with the public key. This authentication may be delegated to a registration authority. The CA may carry out the combined functions of a registration authority, name server, and key generation facility; such a combined facility is called either a CA or akey management facility. §13.2 Background and basic concepts 549 2. name server– responsible for managing a name space of unique user names (e.g., unique relative to a CA). 3. registration authority– responsible for authorizing entities, distinguished by unique names, as members of a security domain. User registration usually involves associ- ating keying material with the entity. 4. key generator– creates public/private key pairs (and symmetric keys or passwords). This may be part of the user entity, part of the CA, or an independent trusted system component. 5. certiﬁcate directory– a certiﬁcate database or server accessible for read-access by users. The CA may supply certiﬁcates to (and maintain) the database, or users may manage their own database entries (under appropriate access control). name registration authority key generator User A certiﬁcation authority certiﬁcate directory server Figure 13.3:Third party services related to public-key certiﬁcation. (iii) Other basic third party functions Additional basic functions a trusted third party may provide include: 1. key server(authentication server) – facilitates key establishment between other par- ties, including for entity authentication. Examples includeKDCs andKTCs (§13.2.3). 2. key management facility– provides a number of services including storage and arch- ival of keys, audit collection and reporting tools, and (in conjunction with a certiﬁ- cation authority or CA) enforcement of life cycle requirements including updating and revoking keys. The associated key server or certiﬁcation authority may provide a record (audit trail) of all events related to key generation and update, certiﬁcate gen- eration and revocation, etc. 13.5 Note (key access server) A key server may be generalized to akey access server, providing shared keys under controlled access to individual members of groups of two or more parties, as follows. A keyK is securely deposited with the server by partyA along with an access control list specifying entities authorized to access it. The server stores the key and the associated list. Subsequently, entities contact the server and request the key by referencing 550 Ch. 13 Key Management Techniques a key identiﬁer supplied byA. Upon entity authentication, the server grants access to the keying material (using KTC-like functionality) if the entity is authorized. 13.6 Note (digital enveloping of ﬁles) A key access server may be employed to store a keyK used to symmetrically encrypt a ﬁle. The source partyA may make the (encrypted) ﬁle available by attaching it to the encrypted key, posting it to a public site, or communicating it independently over a distinct (unsecured) channel. Retrieval of the key from the server by an authorized party then allows that party access to the (decrypted) ﬁle. The same end goal can be attained by public-key techniques directly, without key access servers, as fol- lows:A encrypts the ﬁle underK as above; asymmetrically encryptsK using the intended recipient’s public encryption key (or recipients’ keys); and includes the encrypted key(s) in a header ﬁeld preceding the encrypted ﬁle. 13.7 Remark (levels of trust vs. competency) Various third party services require different types of trust and competency in the third party. For example, a third party possessing secret de- cryption keys (or entity authentication keys) must be trusted not to disclose encrypted in- formation (or impersonate users). A third party required (only) to bind an encryption public key to an identity must still be trusted not to create false associations and thereafter imper- sonate an entity. In general, three levels of trust in a third partyT responsible for certify- ing credentials for users may be distinguished. Level 1:T knows each user’s secret key. Level 2:T does not know users’ secret keys, but can create false credentials without de- tection. Level 3:T does not know users’ secret keys, and generation of false credentials is detectable. (iv) Advanced third party functions Advanced service roles which may be provided by trusted third parties, discussed further in§13.8, include: 1. timestamp agent– used to assert the existence of a speciﬁed document at a certain point in time, or afﬁx a trusted date to a transaction or digital message. 2. notary agent– used to verify digital signatures at a given point in time to support non-repudiation, or more generally establish the truth of any statement (which it is trusted on or granted jurisdiction over) at a given point in time. 3. key escrow agent– used to provide third-party access to users’ secret keys under spe- cial circumstances. Here distinction is usually made between key types; for example, encryption private keys may need to be escrowed but not signature private keys (cf. Remark 13.32). 13.2.5 Tradeoffs among key establishment protocols A vast number of key establishment protocols are available (Chapter 12). To choose from among these for a particular application, many factors aside from cryptographic security may be relevant.§12.2.2 discusses different types of assurances provided, and characteris- tics useful in comparing protocols. In selected key management applications, hybrid protocols involving both symmet- ric and asymmetric techniques offer the best alternative (e.g., Protocol 12.44; see also Note 13.6). More generally, the optimal use of available techniques generally involves combining symmetric techniques for bulk encryption and data integrity with public-key techniques for signatures and key management. §13.3 Techniques for distributing conﬁdential keys 551 Public-key vs. symmetric-key techniques (in key management) Primary advantages offered by public-key (vs. symmetric-key) techniques for applications related to key management include: 1. simpliﬁed key management. To encrypt data for another party, only the encryption public key of that party need be obtained. This simpliﬁes key management as only authenticity of public keys is required, not their secrecy. Table 13.3 illustrates the case for encryption keys. The situation is analogous for other types of public-key pairs, e.g., signature key pairs. 2. on-line trusted server not required. Public-key techniques allow a trusted on-line server to be replaced by a trusted off-line server plus any means for delivering au- thentic public keys (e.g., public-key certiﬁcates and a public database provided by an untrusted on-line server). For applications where an on-line trusted server is not mandatory, this may make the system more amenable to scaling, to support very large numbers of users. 3. enhanced functionality. Public-key cryptography offers functionality which typically cannot be provided cost-effectively by symmetric techniques (without additional on- line trusted third parties or customized secure hardware). The most notable such fea- tures are non-repudiation of digital signatures, and true (single-source) data origin authentication. Symmetric keys Asymmetric keys secrecy authenticity secrecy authenticity encryption key yes yes no yes decryption key yes yes yes yes Table 13.3:Key protection requirements: symmetric-key vs. public-key systems. Figure 13.4 compares key management for symmetric-key and public-key encryption. The pairwise secure channel in Figure 13.4(a) is often a trusted server with which each party communicates. The pairwise authentic channel in Figure 13.4(b) may be replaced by a pub- lic directory through which public keys are available via certiﬁcates; the public key in this case is typically used to encrypt a symmetric data key (cf. Note 13.6). 13.3 Techniques for distributing conﬁdential keys Various techniques and protocols are available to distribute cryptographic keys whose con- ﬁdentiality must be preserved (both private keys and symmetric keys). These include the use of key layering (§13.3.1) and symmetric-key certiﬁcates (§13.3.2). 13.3.1 Key layering and cryptoperiods Table 13.2 (page 545) may be used to classify keys based on usage. The class “conﬁden- tiality” may be sub-classiﬁed on the nature of the information being protected: user data vs. keying material. This suggests a naturalkey layeringas follows: 1. master keys– keys at the highest level in the hierarchy, in that they themselves are not cryptographically protected. They are distributed manually or initially installed and protected by procedural controls and physical or electronic isolation. 552 Ch. 13 Key Management Techniques symmetric key generator asymmetric key pair generation (a) Symmetric-key encryption secret key secret key (b) Public-key encryption public key private key plaintext ciphertext plaintext ciphertextplaintext plaintext secure channel (privacy and authentication) unsecured channel (no protection) secure channel (authentication only) encryption decryption encryption decryption Figure 13.4:Key management: symmetric-key vs. public-key encryption. 2. key-encrypting keys– symmetric keys or encryption public keys used for key trans- port or storage of other keys, e.g., in the key transport protocols of Chapter 12. These may also be calledkey-transport keys, and may themselves be secured under other keys. 3. data keys– used to provide cryptographic operations on user data (e.g., encryption, authentication). These are generally short-term symmetric keys; however, asymmet- ric signature private keys may also be considered data keys, and these are usually longer-term keys. The keys at one layer are used to protect items at a lower level. This constraint is intended to make attacks more difﬁcult, and to limit exposure resulting from compromise of a speciﬁc key, as discussed below. 13.8 Note (protection of key-encrypting keys) Compromise of a key-encrypting key (and more- over, a master key as a special case thereof) affects all keys protected thereunder. Conse- quently, special measures are used to protect master keys, including severely limiting access and use, hardware protection, and providing access to the key only under shared control (§12.7.1). 13.9 Example (key layering with master and terminal keys) Assume each terminalX from a predeﬁned set shares a key-encrypting key (terminal key)K X with a trusted central node C, and thatC stores an encrypted list of all terminal keys under a master keyKM.C may then provide a session key to terminalsX and Y as follows.C obtains a random valueR (possibly from an external source) and deﬁnes the session key to beS = DKM (R), the decryption ofR underKM. UsingKM,C decrypts the key list to obtainKX, computesS §13.3 Techniques for distributing conﬁdential keys 553 from R, then encryptsS underKX and transmits it toX.S is analogously transmitted to Y , and can be recovered by bothX and Y . □ Cryptoperiods, long-term keys, and short-term keys 13.10 DeﬁnitionThe cryptoperiodof a key is the time period over which it is valid for use by legitimate parties. Cryptoperiods may serve to: 1. limit the information (related to a speciﬁc key) available for cryptanalysis; 2. limit exposure in the case of compromise of a single key; 3. limit the use of a particular technology to its estimated effective lifetime; and 4. limit the time available for computationally intensive cryptanalytic attacks (in appli- cations where long-term key protection is not required). In addition to the key layering hierarchy above, keys may be classiﬁed based on tem- poral considerations as follows. 1. long-term keys. These include master keys, often key-encrypting keys, and keys used to facilitate key agreement. 2. short-termkeys. These include keys established by key transport or key agreement, and often used as data keys orsession keysfor a single communications session. See Remark 13.11. In general, communications applications involve short-term keys, while data storage applications require longer-term keys. Long-term keys typically protect short-term keys. Difﬁe-Hellman keys are an exception in some cases (see§12.6.1). Cryptoperiods limit the use of keys to ﬁxed periods, after which they must be replaced. 13.11 Remark (short-term use vs. protection) The termshortas used in short-term keys refers to the intended time of the key usage by legitimate parties, rather than theprotection lifetime (cf.§13.7.1). For example, an encryption key used for only a single session might nonethe- less be required to provide protection sufﬁcient to withstand long-term attack (perhaps 20 years), whereas if signatures are veriﬁed immediately and never checked again, a signature key may need to provide protection only for a relatively short period of time. The more severe the consequences of a secret key being disclosed, the greater the reward to an adver- sary for obtaining access to it, and the greater the time or level of effort an adversary will invest to do so. (See also§12.2.2, and§12.2.3 on perfect forward secrecy.) 13.3.2 Key translation centers and symmetric-key certiﬁcates Further to centralized key management discussed in§13.2.3, this section considers tech- niques involving key translation centers, including use of symmetric-key certiﬁcates. (i) Key translation centers A key translation center (KTC)T is a trusted server which allows two partiesA and B, which do not directly share keying material, to establish secure communications through use of long-term keysKAT and KBT they respectively share withT.A may send a conﬁ- dential messageM toB using Protocol 13.12. IfM is a keyK, this provides a key transfer protocol (cf.§13.2.3); thus, KTCs provide translation of keys or messages. 554 Ch. 13 Key Management Techniques 13.12 ProtocolMessage translation protocol using a KTC SUMMARY: A interacts with a trusted server (KTC)T and partyB. RESULT: A transfers a secret messageM (or session key) toB. See Note 13.13. 1. Notation.E is a symmetric encryption algorithm.M may be a session keyK. 2. One-time setup.A and T share keyKAT . SimilarlyB and T shareKBT . 3. Protocol messages. A →T : A, EKAT (B,M )( 1 ) A ←T : EKBT (M,A )( 2 ) A →B : EKBT (M,A )( 3 ) 4. Protocol actions. (a)A encryptsM (along with the identiﬁer of the intended recipient) underKAT , and sends this toT with its own identiﬁer (to allowT to look upKAT ). (b) Upon decrypting the message,T determines it is intended forB, looks up the key (KBT ) of the indicated recipient, and re-encryptsM forB. (c)T returns the translated message forA to send to (or post in a public site for) B; alternatively,T may send the response toB directly. Only one ofA and B need communicate withT. As an alternative to the protocol as given, A may send the ﬁrst message toB directly, whichB would then relay toT for translation, withT responding directly toB. 13.13 Note (security of Protocol 13.12) (i) The identiﬁerA, corresponding to the key under which message (1) was encrypted, is included in message (2) as a secure indication (toB) of the original source. Key notarization (§13.5.2) offers a more robust method of preventing key substitution. (ii) A recognizable distinction (e.g., re-ordering the message and identiﬁer ﬁelds) be- tween the format of messages (1) and (2) is required to prevent an adversary from reﬂecting (1) back toA as a message (3) purportedly originating fromB. (iii) Message replay is possible; attacks may be detected through the use of timestamps or sequence numbers withinM. The protocol as given provides no entity authenti- cation. (iv) An integrity check mechanism on the encrypted text should be used to allowT to detect tampering of the cleartext identiﬁerA in (1), as well as in (2) and (3). (v) A chosen-text attack on keyKBT in (2) may be prevented by an encryption mode such as CBC, and inserting an initial ﬁeld containing a random number. (ii) Symmetric-key certiﬁcates Symmetric-key certiﬁcates provide a means for a KTC to avoid the requirement of either maintaining a secure database of user secrets (or duplicating such a database for multiple servers), or retrieving such keys from a database upon translation requests. As before, associated with each partyB is a keyKBT shared withT, which is now em- bedded in asymmetric-key certiﬁcateEKT (KBT ,B ) encrypted under a symmetric master key KT known only toT. (A lifetime parameterL could also be included in the certiﬁcate as a validity period.) The certiﬁcate serves as a memo fromT to itself (who alone can open it), and is given toB so thatB may subsequently present it back toT precisely when re- quired to accessB’s symmetric keyKBT for message translation. Rather than storing all user keys,T now need securely store onlyKT . §13.4 Techniques for distributing public keys 555 Symmetric-key certiﬁcates may be used in Protocol 13.12 by changing only the ﬁrst message as below, whereSCertA = EKT (KAT ,A ),SCertB = EKT (KBT ,B ): A →T : SCertA,E KAT (B,M ), SCertB (1) A public database may be established with an entry specifying the name of each user and its corresponding symmetric-key certiﬁcate. To construct message (1),A retrievesB’s symm- etric-key certiﬁcate and includes this along with its own.T carries out the translation as before, retrievingKAT and KBT from these certiﬁcates, but now also veriﬁes thatA’s in- tended recipientB as speciﬁed inEKAT (B,M ) matches the identiﬁer in the supplied cer- tiﬁcateSCertB. 13.14 Remark (public-key functionality via symmetric techniques) The trusted third party func- tionality required when using symmetric-key certiﬁcates may be provided by per-user tamper-resistant hardware units keyed with a common (user-inaccessible) master key KT . The trusted hardware unitHA of each userA generates a symmetric-key certiﬁcate SCertA = EKT (KAT ,A ), which is made available toB when required.HB decrypts the certiﬁcate to recoverKAT (inaccessible toB) and the identityA (accessible toB). By design,HB is constrained to use other users’ keysKAT = KA solely for veriﬁcation func- tions (e.g., MAC veriﬁcation, message decryption).KA then functions asA’s public key (cf. Example 13.36), allowing data origin authentication with non-repudiation; an adju- dicator may resolve disputes given a hardware unit containingK T , a disputed (message, signature) pair, and the authentic valueSCert A from HA. 13.15 Remark (symmetric-key vs. public-key certiﬁcates) Symmetric-key certiﬁcates differ from public-key certiﬁcates as follows: they are symmetric-key encrypted underT’s mas- ter key (vs. signed usingT’s private key); the symmetric key within may be extracted only by T (vs. many parties being able to verify a public-key certiﬁcate); andT is required to be on-line for key translation (vs. an off-line certiﬁcation authority). In both cases, certiﬁcates may be stored in a public directory. 13.4 Techniques for distributing public keys Protocols involving public-key cryptography are typically described assuminga prioripos- session of (authentic) public keys of appropriate parties. This allows full generality among various options for acquiring such keys. Alternatives for distributing explicit public keys with guaranteed or veriﬁable authenticity, including public exponentials for Difﬁe-Hellman key agreement (or more generally, public parameters), include the following. 1. Point-to-point delivery over a trusted channel. Authentic public keys of other users are obtained directly from the associated user by personal exchange, or over a di- rect channel, originating at that user, and which (procedurally) guarantees integrity and authenticity (e.g., a trusted courier or registered mail). This method is suitable if used infrequently (e.g., one-time user registration), or in small closed systems. A re- lated method is to exchange public keys and associated information over an untrusted electronic channel, and provide authentication of this information by communicating a hash thereof (using a collision-resistant hash function) via an independent, lower- bandwidth authentic channel, such as a registered mail. 556 Ch. 13 Key Management Techniques Drawbacks of this method include: inconvenience (elapsed time); the requirement of non-automated key acquisition prior to secured communications with each new party (chronological timing); and the cost of the trusted channel. 2. Direct access to a trusted public ﬁle (public-key registry). A public database, the in- tegrity of which is trusted, may be set up to contain the name and authentic public key of each system user. This may be implemented as a public-key registry operated by a trusted party. Users acquire keys directly from this registry. While remote access to the registry over unsecured channels is acceptable against passive adversaries, a secure channel is required for remote access in the presence of active adversaries. One method of authenticating a public ﬁle is by tree authentication of public keys (§13.4.1). 3. Use of an on-line trusted server. An on-line trusted server provides access to the equivalent of a public ﬁle storing authentic public keys, returning requested (individ- ual) public keys in signed transmissions; conﬁdentiality is not required. The request- ing party possesses a copy of the server’s signature veriﬁcation public key, allowing veriﬁcation of the authenticity of such transmissions. Disadvantages of this approach include: the trusted server must be on-line; the trusted server may become a bottleneck; and communications links must be established with both the intended communicant and the trusted server. 4. Use of an off-line server and certiﬁcates. In a one-time process, each partyA contacts an off-line trusted party referred to as acertiﬁcation authority(CA), to register its public key and obtain the CA’s signature veriﬁcation public key (allowing veriﬁcation of other users’ certiﬁcates). The CA certiﬁesA’s public key by binding it to a string identifyingA, thereby creating a certiﬁcate (§13.4.2). Parties obtain authentic public keys by exchanging certiﬁcates or extracting them from a public directory. 5. Use of systems implicitly guaranteeing authenticity of public parameters. In such systems, including identity-based systems (§13.4.3) and those using implicitly cer- tiﬁed keys (§13.4.4), by algorithmic design, modiﬁcation of public parameters re- sults in detectable, non-compromising failure of cryptographic techniques (see Re- mark 13.26). The following subsections discuss the above techniques in greater detail. Figure 13.7 (page 564) provides a comparison of the certiﬁcate-based approach, identity-based systems, and the use of implicitly-certiﬁed public keys. 13.4.1 Authentication trees Authentication trees provide a method for making public data available with veriﬁable au- thenticity, by using a tree structure in conjunction with a suitable hash function, and authen- ticating the root value. Applications include: 1. authentication of public keys(as an alternative to public-key certiﬁcates). An authen- tication tree created by a trusted third party, containing users’ public keys, allows au- thentication of a large number of such keys. 2. trusted timestamping service. Creation of an authentication tree by a trusted third party, in a similar way, facilitates a trusted timestamping service (see§13.8.1). 3. authentication of user validation parameters. Creation of a tree by a single user al- lows that user to publish, with veriﬁable authenticity, a large number of its own public validation parameters, such as required in one-time signature schemes (see§11.6.3). To facilitate discussion of authentication trees, binary trees are ﬁrst introduced. §13.4 Techniques for distributing public keys 557 Binary trees A binary treeis a structure consisting of vertices and directed edges. The vertices are di- vided into three types: 1. aroot vertex. The root has two edges directed towards it, a left and a right edge. 2. internal vertices. Each internal vertex has three edges incident to it – an upper edge directed away from it, and left and right edges directed towards it. 3. leaves. Each leaf vertex has one edge incident to it, and directed away from it. The vertices incident with the left and right edges of an internal vertex (or the root) are called thechildrenof the internal vertex. The internal (or root) vertex is called theparentof the associated children. Figure 13.5 illustrates a binary tree with 7 vertices and 6 edges. Root Right EdgeLeft Edge Figure 13.5:A binary tree (with 4 shaded leaves and 3 internal vertices). 13.16 FactThere is a unique directed path from any non-root vertex in a binary tree to the root vertex. Constructing and using authentication trees Consider a binary treeT which hast leaves. Leth be a collision-resistant hash function.T can be used to authenticatet public values,Y1,Y 2,... ,Y t, by constructing anauthentica- tion treeT∗as follows. 1. Label each of thet leaves by a unique public valueYi. 2. On the edge directed away from the leaf labeledYi, put the labelh(Yi). 3. If the left and right edge of an internal vertex are labeledh1 andh2, respectively, label the upper edge of the vertexh(h1∥h2). 4. If the edges directed toward the root vertex are labeledu1 andu2, label the root vertex h(u1∥u2). Once the public values are assigned to leaves of the binary tree, such a labeling is well- deﬁned. Figure 13.6 illustrates an authentication tree with 4 leaves. Assuming some means to authenticate the label on the root vertex, an authentication tree provides a means to au- thenticate any of thet public leaf valuesY i, as follows. For each public valueYi, there is a unique path (theauthentication path) fromYi to the root. Each edge on the path is a left or right edge of an internal vertex or the root. Ife is such an edge directed towards vertex x, record the label on the other edge (note) directed towardx. This sequence of labels (the authentication path values) used in the correct order provides the authentication ofYi,a si l - lustrated by Example 13.17. Note that if a single leaf value (e.g.,Y1) is altered, maliciously or otherwise, then authentication of that value will fail. 558 Ch. 13 Key Management Techniques Y3 Y4 h(Y2) h(Y3) h(Y4) R = h(h2∥h(Y4)) Y1 Y2 h(Y1) h1 = h(h(Y1)∥h(Y2)) h2 = h(h1∥h(Y3)) Figure 13.6:An authentication tree. 13.17 Example (key veriﬁcation using authentication trees) Refer to Figure 13.6. The public valueY1 can be authenticated by providing the sequence of labelsh(Y2),h(Y3),h(Y4). The authentication proceeds as follows: computeh(Y1); next computeh1 = h(h(Y1))∥h(Y2)); then computeh2 = h(h1∥h(Y3)); ﬁnally, acceptY1 as authentic ifh(h2∥h(Y4)) = R, where the root valueR is known to be authentic. □ The advantage of authentication trees is evident by considering the storage required to allow authentication oft public values using the following (very simple) alternate approach: an entityA authenticatest public valuesY1,Y 2,... ,Y t by registering each with a trusted third party. This approach requires registration oft public values, which may raise storage issues at the third party whent is large. In contrast, an authentication tree requires only a single value be registered with the third party. If a public keyYi of an entityA is the value corresponding to a leaf in an authentication tree, andA wishes to provideB with information allowingB to verify the authenticity of Yi, thenA must (store and) provide toB bothYi and all hash values associated with the authentication path fromYi to the root; in addition,B must have prior knowledge and trust in the authenticity of the root valueR. These values collectively guarantee authenticity, analogous to the signature on a public-key certiﬁcate. The number of values each party must store (and provide to others to allow veriﬁcation of its public key) islg(t), as per Fact 13.19. 13.18 Fact(depth of a binary tree) Consider the length of (or number of edges in) the path from each leaf to the root in a binary tree. The length of the longest such path is minimized when the tree isbalanced, i.e., when the tree is constructed such that all such paths differ in length by at most one. The length of the path from a leaf to the root in a balanced binary tree containingt leaves is aboutlg(t). 13.19 Fact(length of authentication paths) Using a balanced binary tree (Fact 13.18) as an au- thentication tree witht public values as leaves, authenticating a public value therein may be achieved by hashinglg(t) values along the path to the root. 13.20 Remark (time-space tradeoff) Authentication trees require only a single value (the root value) in a tree be registered as authentic, but veriﬁcation of the authenticity of any particu- lar leaf value requires access to and hashing of all values along the authentication path from leaf to root. 13.21 Remark (changing leaf values) To change a public (leaf) value or add more values to an authentication tree requires recomputation of the label on the root vertex. For large balanced §13.4 Techniques for distributing public keys 559 trees, this may involve a substantial computation. In all cases, re-establishing trust of all users in this new root value (i.e., its authenticity) is necessary. The computational cost involved in adding more values to a tree (Remark 13.21) may motivate constructing the new tree as an unbalanced tree with the new leaf value (or a sub- tree of such values) being the right child of the root, and the old tree, the left. Another motivation for allowing unbalanced trees arises when some leaf values are referenced far more frequently than others. 13.4.2 Public-key certiﬁcates Public-key certiﬁcates are a vehicle by which public keys may be stored, distributed or for- warded over unsecured media without danger of undetectable manipulation. The objective is to make one entity’s public key available to others such that its authenticity (i.e., its status as the true public key of that entity) and validity are veriﬁable. In practice, X.509 certiﬁ- cates are commonly used (see page 587). Further details regarding public-key certiﬁcates follow. 13.22 DeﬁnitionA public-key certiﬁcateis a data structure consisting of adata partand asig- nature part. The data part contains cleartext data including, as a minimum, a public key and a string identifying the party (subject entity) to be associated therewith. The signature part consists of the digital signature of a certiﬁcation authority over the data part, thereby binding the subject entity’s identity to the speciﬁed public key. The Certiﬁcation Authority(CA) is a trusted third party whose signature on the cer- tiﬁcate vouches for the authenticity of the public key bound to the subject entity. The sig- niﬁcance of this binding (e.g., what the key may be used for) must be provided by addi- tional means, such as an attribute certiﬁcate or policy statement. Within the certiﬁcate, the string which identiﬁes the subject entity must be a unique name within the system (distin- guished name), which the CA typically associates with a real-world entity. The CA requires its own signature key pair, the authentic public key of which is made available to each party upon registering as an authorized system user. This CA public key allows any system user, through certiﬁcate acquisition and veriﬁcation, to transitively acquire trust in the authentic- ity of the public key in any certiﬁcate signed by that CA. Certiﬁcates are a means for transferring trust, as opposed to establishing trust origi- nally. The authenticity of the CA’s public key may be originally provided by non-cryptogra- phic means including personal acquisition, or through trusted couriers; authenticity is re- quired, but not secrecy. Examples of additional information which the certiﬁcate data part might contain in- clude: 1. a validity period of the public key; 2. a serial number or key identiﬁer identifying the certiﬁcate or key; 3. additional information about the subject entity (e.g., street or network address); 4. additional information about the key (e.g., algorithm and intended use); 5. quality measures related to the identiﬁcation of the subject entity, the generation of the key pair, or other policy issues; 6. information facilitating veriﬁcation of the signature (e.g., a signature algorithm iden- tiﬁer, and issuing CA’s name); 7. the status of the public key (cf. revocation certiﬁcates,§13.6.3). 560 Ch. 13 Key Management Techniques (i) Creation of public-key certiﬁcates Before creating a public-key certiﬁcate for a subject entityA, the certiﬁcation authority should take appropriate measures (relative to the security level required, and customary business practices), typically non-cryptographic in nature, to verify the claimed identity of A and the fact that the public key to be certiﬁed is actually that ofA. Two cases may be distinguished. Case 1: trusted party creates key pair. The trusted party creates a public-key pair, as- signs it to a speciﬁc entity, and includes the public key and the identity of that entity in the certiﬁcate. The entity obtains a copy of the corresponding private key over a secure (au- thentic and private) channel after proving its identity (e.g., by showing a passport or trusted photo-id, in person). All parties subsequently using this certiﬁcate essentially delegate trust to this prior veriﬁcation of identity by the trusted party. Case 2: entity creates own key pair. The entity creates its own public-key pair, and se- curely transfers the public key to the trusted party in a manner which preserves authenticity (e.g., over a trusted channel, or in person). Upon veriﬁcation of the authenticity (source) of the public key, the trusted party creates the public-key certiﬁcate as above. 13.23 Remark (proof of knowledge of private key) In Case 2 above, the certiﬁcation authority should require proof of knowledge of the corresponding private key, to preclude (among other possible attacks) an otherwise legitimate party from obtaining, for malicious purposes, a public-key certiﬁcate binding its name to the public key of another party. For the case of signature public keys, this might be done by the party providing its own signature on a sub- set of the data part of the certiﬁcate; or by responding to a challenger 1 randomized by the party itself e.g., signingh(r1\|\|r2) for an appropriate hash functionh and a random number r2 chosen by the signer. (ii) Use and veriﬁcation of public-key certiﬁcates The overall process whereby a partyB uses a public-key certiﬁcate to obtain the authentic public key of a partyA may be summarized as follows: 1. (One-time) acquire the authentic public key of the certiﬁcation authority. 2. Obtain an identifying string which uniquely identiﬁes the intended partyA. 3. Acquire over some unsecured channel (e.g. from a central public database of certiﬁ- cates, or fromA directly), a public-key certiﬁcate corresponding to subject entityA and agreeing with the previous identifying string. 4. (a) Verify the current date and time against the validity period (if any) in the cer- tiﬁcate, relying on a local trusted time/day-clock; (b) Verify the current validity of the CA’s public key itself; (c) Verify the signature onA’s certiﬁcate, using the CA’s public key; (d) Verify that the certiﬁcate has not been revoked (§13.6.3). 5. If all checks succeed, accept the public key in the certiﬁcate asA’s authentic key. 13.24 Remark (life cycle reasons for single-key certiﬁcates) Due to differing life cycle require- ments for different types of keys (e.g., differing cryptoperiods, backup, archival, and other lifetime protection requirements – see§13.7), separate certiﬁcates are recommended for separate keys, as opposed to including several keys in a single certiﬁcate. See also Re- mark 13.32. §13.4 Techniques for distributing public keys 561 (iii) Attribute certiﬁcates Public-key certiﬁcates bind a public key and an identity, and include additional data ﬁelds necessary to clarify this binding, but are not intended for certifying additional information. Attribute certiﬁcatesare similar to public-key certiﬁcates, but speciﬁcally intended to allow speciﬁcation of information (attributes) other than public keys (but related to a CA, entity, or public key), such that it may also be conveyed in a trusted (veriﬁable) manner. Attribute certiﬁcates may be associated with a speciﬁc public key by binding the attribute informa- tion to the key by the method by which the key is identiﬁed, e.g., by the serial number of a corresponding public-key certiﬁcate, or to a hash-value of the public key or certiﬁcate. Attribute certiﬁcates may be signed by anattribute certiﬁcation authority, created in conjunction with anattribute registration authority, and distributed in conjunction with an attribute directory service(cf. Figure 13.3). More generally, any party with a signature key and appropriate recognizable authority may create an attribute certiﬁcate. One application is to certify authorization information related to a public key. More speciﬁcally, this may be used, for example, to limit liability resulting from a digital signature, or to constrain the use of a public key (e.g., to transactions of limited values, certain types, or during certain hours). 13.4.3 Identity-based systems Identity-based systems resemble ordinary public-key systems, involving a private transfor- mation and a public transformation, but users do not have explicit public keys as before. In- stead, the public key is effectively replaced by (or constructed from) a user’s publicly avail- able identity information (e.g., name and network or street address). Any publicly available information which uniquely identiﬁes a user and can be undeniably associated with the user, may serve as the identity information. 13.25 DeﬁnitionAn identity-based cryptographic system (ID-based system) is an asymmetric system wherein an entity’s public identiﬁcation information (unique name) plays the role of its public key, and is used as input by a trusted authorityT (along withT’s private key) to compute the entity’s corresponding private key. After computing it,T transfers the entity’s private key to the entity over a secure (au- thentic and private) channel. This private key is computed from not only the entity’s identity information, but must also be a function of some privileged information known only toT (T’s private key). This is necessary to prevent forgery and impersonation – it is essential that onlyT be able to create valid private keys corresponding to given identiﬁcation in- formation. Corresponding (authentic) publicly available system data must be incorporated in the cryptographic transformations of the ID-based system, analogous to the certiﬁcation authority’s public key in certiﬁcate-based systems. Figure 13.7(b) on page 564 illustrates the design of an identity-based system. In some cases, additional system-deﬁned public dataD A must be associated with each userA in addition to itsa prioriidentityIDA (see Remark 13.27); such systems are no longer “purely” identity-based, although neither the authenticity ofD A norIDA need be explicitly veriﬁed. 13.26 Remark (authenticity in ID-based systems) ID-based systems differ from public-key sys- tems in that the authenticity of user-speciﬁc public data is not (and need not be) explicitly veriﬁed, as is necessary for user public keys in certiﬁcate-based systems. The inherent re- dundancy of user public data in ID-based systems (derived through the dependence of the corresponding private key thereon), together with the use of authentic public system data, 562 Ch. 13 Key Management Techniques implicitly protects against forgery; if incorrect user public data is used, the cryptographic transformations simply fail. More speciﬁcally: signature veriﬁcation fails, entity authenti- cation fails, public-key encryption results in undecipherable text, and key-agreement results in parties establishing different keys, respectively, for (properly constructed) identity-based signature, authentication, encryption, and key establishment mechanisms. The motivation behind ID-based systems is to create a cryptographic system modeling an ideal mail system wherein knowledge of a person’s name alone sufﬁces to allow mail to be sent which that person alone can read, and to allow veriﬁcation of signatures that person alone could have produced. In such an ideal cryptographic system: 1. users need exchange neither symmetric keys nor public keys; 2. public directories (ﬁles of public keys or certiﬁcates) need not be kept; and 3. the services of a trusted authority are needed solely during a set-up phase (during which users acquire authentic public system parameters, to be maintained). 13.27 Remark (ideal vs. actual ID-based systems) A drawback in many concrete proposals of ID-based systems is that the required user-speciﬁc identity data includes additional data (an integer or public data value), denotedDA in Figure 13.7(b), beyond ana prioriidentity IDA. For example, see Note 10.29(ii) on Feige-Fiat-Shamir identiﬁcation. Ideally,DA is not required, as a primary motivation for identity-based schemes is to eliminate the need to transmit public keys, to allow truly non-interactive protocols with identity information itself sufﬁcing as an authentic public key. The issue is less signiﬁcant in signature and iden- tiﬁcation schemes where the public key of a claimant is not required until receiving a mes- sage from that claimant (in this caseD A is easily provided); but in this case, the advantage of identity-based schemes diminishes. It is more critical in key agreement and public-key encryption applications where another party’s public key is needed at the outset. See also Remark 13.31. 13.28 Example (ID-based system implemented using chipcards) A simpliﬁed ID-based system based on chipcards may be run as follows. A third partyT, acting as a trusted key genera- tion system, is responsible solely for providing each user a chipcard during a set-up phase, containing that party’s ID-based private key, after carrying out a thorough identity check. If no further users need be added,T may publish the public system data and cease to exist. Users are responsible for not disclosing their private keys or losing their cards.□ 13.4.4 Implicitly-certiﬁed public keys Another variation of public-key systems is asymmetric systems withimplicitly-certiﬁed public keys. Here explicit user public keys exist (see Figure 13.7(c)), but they must be re- constructed rather than transported by public-key certiﬁcates as per certiﬁcate-based sys- tems. For other advantages, see Remark 13.30. Examples of speciﬁc such mechanisms are given in§12.6.2. Systems with implicitly-certiﬁed public keys are designed such that: 1. Entities’ public keys may be reconstructed (by other parties) from public data (which essentially replace a certiﬁcate). 2. The public data from which a public key is reconstructed includes: (a) public (i.e., system) data associated with a trusted partyT; (b) the user entity’s identity (or identifying information, e.g., name and address); (c) additional per-user public data (reconstruction public data). §13.4 Techniques for distributing public keys 563 3. The integrity of a reconstructed public key is not directly veriﬁable, but a “correct” public key can be recovered only from authentic user public data. Regarding authenticity of reconstructed public keys, the system design must guarantee: 1. Alteration of either a user’s identity or reconstruction public data results in recov- ery of a corrupted public key, which causes denial of service but not cryptographic exposure (as per Remark 13.26). 2. It is computationally infeasible for an adversary (without knowledge ofT’s private data) to compute a private key corresponding to any party’s public key, or to construct a matching user identity and reconstruction public data for which a corresponding private key may also be computed. Reconstructed public keys are thus implicitly au- thenticated by construction. 13.29 Remark (applications of implicitly-certiﬁed keys) Implicitly-certiﬁed public keys may be used as an alternate means for distributing public keys (e.g., Difﬁe-Hellman keys – see §12.6.3) in various key agreement protocols, or in conjunction with identiﬁcation protocols, digital signature schemes, and public-key encryption schemes. Classes of implicitly-certiﬁed public keys Two classes of implicitly-certiﬁed public keys may be distinguished: 1. identity-based public keys (Class 1). The private key of each entityA is computed by a trusted partyT, based onA’s identifying information andT’s private key; it is also a function ofA’s user-speciﬁc reconstruction public data, which is ﬁxeda priori by T.A’s private key is then securely transferred byT toA. An example is Mecha- nism 12.59. 2. self-certiﬁed public keys (Class 2). Each entityA itself computes its private key and corresponding public key.A’s reconstruction public data (rather thanA’s private key, as in Class 1) is computed byT as a function of the public key (transferred toT by A), A’s identifying information, andT’s private key. An example is Mechanism 12.61. Class 1 requires more trust in the third party, which therein has access to users’ private keys. This differs from Class 2, as emphasized by the term “self” in “self-certiﬁed”, which refers to the knowledge of this key being restricted to the entity itself. 13.4.5 Comparison of techniques for distributing public keys §13.4 began with an overview of techniques for addressing authenticity in public key dis- tribution. The basic approaches of§13.4.2,§13.4.3, and§13.4.4 are discussed further here. Figure 13.7 illustrates corresponding classes of asymmetric signature systems, contrasting public-key systems (with explicit public keys), identity-based systems (the public key is a user’s identity information), and systems with implicitly-certiﬁed public keys (an explicit public key is reconstructed from user public data). 4 The main differences are as follows: 1. Certiﬁcate-based public-key systems have explicit public keys, while ID-based sys- tems do not; in implicitly-certiﬁed systems explicit public keys are reconstructed. The explicit public key in public-key systems (Figure 13.7(a)) is replaced by: (a) the triplet(D A, IDA,P T ) for identity-based systems (Figure 13.7(b)).IDA is an identifying string forA,DA is additional public data (deﬁned byT and re- lated toIDA and A’s private key), andPT consists of the trusted public key (or system parameters) of a trusted authorityT. 4While the ﬁgure focuses (for concreteness) on signature systems, concepts carry over analogously for asym- metric entity authentication, key establishment, and encryption systems. 564 Ch. 13 Key Management Techniques asymmetric key pair generation private key private key generation (a) Public key system (explicit public keys) (b) Identity-based system Note: See Figure 13.4 for notation ST ,PT areT’s private, public keys;DA isA’s public data signature generation veriﬁcation signature message PartyA message key public PartyB signature public key PASS/FAIL SA PA PA private key signature generation veriﬁcation signature message PartyA message PartyB signature PASS/FAIL ST PT IDA PT IDA DA SA SA Trusted PartyT Figure 13.7:Key management in different classes of asymmetric signature systems. §13.4 Techniques for distributing public keys 565 key private private key generator for id-based keys asymmetric key pair generation key public ST PT option below (i) or (ii) public-data generator for self-certiﬁed public keys signature generation veriﬁcation signature message PartyA message PartyB signature key public RA IDA PT PASA public key reconstruction Trusted PartyT RA RA IDA PT PartyA SA PA key private public data PA Trusted PartyT IDA PT Trusted PartyT PASS/FAIL (i) identity-based public keys (ii) self-certiﬁed public keys PT IDA (c) System with implicitly-certiﬁed public keys ST ,PT :T’s private, public keys See Figure 13.4 for further notation RA: reconstruction public data ofA SA IDA RA ST Figure 13.7:(cont’d) Key management in different classes of asymmetric signature systems. 566 Ch. 13 Key Management Techniques (b) the triplet(RA, IDA,P T ) for systems with implicitly-certiﬁed public keys (Fig- ure 13.7(c)). In this case, an explicit public keyPA is reconstructed from these parameters. The reconstruction public dataRA plays a role analogous to the public dataDA in Figure 13.7(b). 2. The authenticity of public keys can (and must) be explicitly veriﬁed in certiﬁcate- based systems, but not (and need not) in ID-based or implicitly-certiﬁed systems. 3. The trusted authority need not know users’ private keys in certiﬁcate-based public- key systems or implicitly-certiﬁed systems with self-certiﬁed public keys; but does in ID-based systems, and in implicitly-certiﬁed systems with ID-based keys. 4. Similar to identity-based systems (§13.4.3), implicitly-certiﬁed public keys (of both classes) depend on an entity’s identifying information, and in this sense are also “identity-based”. However, ID-based systems avoid explicit public keys entirely (a user’s identity data is essentially its public key), while implicitly-certiﬁed public keys are not restricted to user identities and may be explicitly computed (and thus more easily used in conjunction with ordinary public-key schemes). 5. The two classes of implicitly-certiﬁed public keys (Figure 13.7(c)) differ in their re- lationship between users’ reconstruction public data and private keys as follows. (a) Class 1: a user’s private key is computed as a function of the reconstruction data, and this private key is computed by the trusted authority; (b) Class 2: the reconstruction data is computed as a function of the user’s public key, and the corresponding private key is computed by the party itself. 6. In all three approaches, at some stage a third party which is trusted to some level (cf. Note 13.7) is required to provide a link transferring trust between users who may have never met each other and may share nothing in common other than authentic system parameters (and possibly knowledge of other users’ identities). 13.30 Remark (implicitly-certiﬁed public keys vs. public-key certiﬁcates) Advantages of implic- itly-certiﬁed public keys over public-key certiﬁcates include: possibly reduced space re- quirements (signed certiﬁcates require storage for signatures); possible computational sav- ings (signature veriﬁcation, as required for certiﬁcates, is avoided); and possible communi- cations savings (e.g. if identity-based and the identity is knowna priori). Countering these points, computation is actually required to reconstruct a public key; and additional recon- struction public data is typically required. 13.31 Remark (key revocation in ID-based systems) Revocation of public keys may be address- ed in ID-based schemes and systems using implicitly-certiﬁed public keys by incorporating information such as a key validity period or serial number into the identiﬁcation string used to compute an entity’s public key (cf. Remark 13.27). The revocation issue is then analo- gous to that for public-key certiﬁcates. Additional information, e.g., pertaining to key usage or an associated security policy, may similarly be incorporated. §13.5 Techniques for controlling key usage 567 13.5 Techniques for controlling key usage This section considers techniques for restricting keys to pre-authorized uses. 13.5.1 Key separation and constraints on key usage Information that may be associated with cryptographic keys includes both attributes which restrict their use, and other information of operational use. These include: 1. owner of key 2. validity period (intended cryptoperiod) 3. key identiﬁer (allowing non-cryptographic reference to the key) 4. intended use (see Table 13.2 for a coarse selection) 5. speciﬁc algorithm 6. system or environment of intended use, or authorized users of key 7. names of entities associated with key generation, registration, and certiﬁcation 8. integrity checksum on key (usually part of authenticity requirement) Key separation and the threat of key misuse In simple key management systems, information associated with keys, including authorized uses, are inferred by context. For additional clarity or control, information explicitly spec- ifying allowed uses may accompany distributed keys and be enforced by veriﬁcation, at the time of use, that the attempted uses are authorized. If control information is subject to manipulation, it should be bound to the key by a method which guarantees integrity and au- thenticity, e.g., through signatures (cf. public-key certiﬁcates) or an encryption technique providing data integrity. The principle ofkey separationis that keys for different purposes should be crypto- graphically separated (see Remark 13.32). The threat of key misuse may be addressed by techniques which ensure that keys are used only for those purposes pre-authorized at the time of key creation. Restrictions on key usage may be enforced by procedural techniques, physical protection (tamper-resistant hardware), or cryptographic techniques as discussed below. Discussion of other methods in§13.5.2 includes key tags, which allow key separation with explicitly-deﬁned uses; key variants, which separate keys without explicitly deﬁning authorized uses; and key notarization and control vectors, which bind control information into the process by which keys are derived. 13.32 Remark (cryptographic reasons for key separation) A principle of sound cryptographic design is to avoid use of the same cryptographic key for multiple purposes. A key-encrypt- ing key should not be used interchangeably as a data encryption key, since decrypted keys are not generally made available to application programs, whereas decrypted data is. Dis- tinct asymmetric encryption and signature keys are also generally used, due to both dif- fering life cycle requirements and cryptographic prudence. Flaws also potentially arise if: asymmetric keys are used for both signatures and challenge-response entity authentication (Remark 10.40); keys are used for both encryption and challenge-response entity authen- tication (chosen-text attacks); symmetric keys are used for both encryption and message authentication (Example 9.88). See also Remark 13.24. 568 Ch. 13 Key Management Techniques 13.5.2 Techniques for controlling use of symmetric keys The main technique discussed below is the use of control vectors. For historical context, key tags/key variants and key notarization are also discussed. (i) Key tags and key variants Key tagsprovide a simpliﬁed method for specifying allowed uses of keys (e.g., data-encryp- ting vs. key-encrypting keys). A key tag is a bit-vector or structured ﬁeld which accompa- nies and remains associated with a key over its lifetime. The tag bits are encrypted jointly with the key and thereby bound to it, appearing in plaintext form only when the key is de- crypted. If the combination of tag bits and key are sufﬁciently short to allow encryption in a single block operation (e.g., a 56-bit key with an 8-bit tag for a 64-bit block cipher), then the inherent integrity provided by encryption precludes meaningful manipulation of the tag. A naive method for providing key separation is to derive separate keys from a single base key(orderivation key) using additional non-secret parameters and a non-secret func- tion. The resulting keys are calledkey variantsorderived keys. One technique for varying keys iskey offsetting, whereby a key-encrypting keyK is modiﬁed on a per-use basis by a counterN incremented after each use. This may prevent replay of encrypted keys. The modiﬁed keyK⊕N is used to encrypt another (e.g., session) key. The recipient likewise modiﬁesK to decrypt the session key. A second technique, complementing alternate 4-bit blocks ofK commencing with the ﬁrst 4 bits, is a special case of ﬁxed-mask offsetting (Example 13.33). 13.33 Example (key variants using ﬁxed-mask offsets) Suppose exactly three classes of keys are desired. Construct keys by using variationsK 1 and K2 of a master keyK, withK1 = K⊕v1,K2 = K⊕v2, andv1,v2 nonsecret mask values. UsingK,K1, andK2 to encrypt other keys then allows key separation of the latter into three classes.□ If the derivation process is invertible, the base key can be recovered from the derived key. Ideally, the derivation technique is non-reversible (one-way), implying that compro- mise of one derived key would not compromise the base key or other derived keys (cf. §13.7.1 on security impacts of related keys). Yet another example of key derivation (see §12.3.1) has this property: computeK i = EK(ri) where ri is a random number, or replace the encryption functionE by a MAC, or simply hashK and ri using a hash functionh with suitable properties. (ii) Key notarization Key notarization is a technique intended to prevent key substitution by requiring explicit speciﬁcation of the identities of parties involved in a keying relationship. A key is au- thenticated with respect to these identities (preventing impersonation) by modifying a key- encrypting key such that the correct identities must be speciﬁed to properly recover the pro- tected key. The key is said to besealedwith these identities. Preventing key substitution is a requirement in all (authenticated) key establishment protocols. Notarization requires proper control information for accurate recovery of encrypted keys, providing implicit pro- tection analogous to implicitly-certiﬁed public keys (§13.4.4). The basic technique (simple key notarization) involves a trusted server (notary), or one of the parties sharing the key, using a key-encrypting keyK to encrypt a session keyS, in- tended for use with the originating partyi and the recipientj, as:E K⊕(i\|\|j)(S). Herei and j are assumed to identify unique entities in the given system. The party intending to recover §13.5 Techniques for controlling key usage 569 S from this must shareK and explicitly specifyi and j in the correct order, otherwise a ran- dom key will be recovered. The analogy to a notary originated from the assumption that the third party properly authenticates the identities of the intended parties, and then provides a session key which may only be recovered by these parties. A more involved process, key notarization with offsetting, is given in Example 13.34 13.34 Example (key notarization with offsetting) LetE be a block cipher operating on 64-bit blocks with 64-bit key,K = KL\|\|KR be a 128-bit key-encrypting key,N a 64-bit counter, and i = iL\|\|iR,j = jL\|\|jR 128-bit source and destination identiﬁers. For key notarization with offsetting, compute:K1 = EKR⊕iL(jR)⊕KL⊕N,K2 = EKL⊕jL(iR)⊕KR⊕N. The resulting 128-bitnotarized key(K1,K 2) then serves as a key-encrypting key in two- key triple-encryption. The leftmost termsf1(KR,i ,j) and f2(KL,i ,j) in the computation ofK1,K2 above are callednotary seals, which, when combined withKL and KR, respec- tively, result in quantities analogous to those used in simple key notarization (i.e., functions ofK,i,j). ForK a 64-bit (single-length) key, the process is modiﬁed as follows: using KL = KR = K, compute the notary sealsf1(KR,i ,j),f2(KL,i ,j) as above, concatenate the leftmost 32 bits off1 with the rightmost off2 to obtainf, then computef⊕K⊕N as the notarized key. □ (iii) Control vectors While key notarization may be viewed as a mechanism for establishing authenticated keys, control vectorsprovide a method for controlling the use of keys, by combining the idea of key tags with the mechanism of simple key notarization. Associated with each keyS is a control vectorC, which is a data ﬁeld (similar to a key tag) deﬁning the authorized uses of the key (effectivelytypingthe key). It is bound toS by varying a key-encrypting keyK before encryption:EK⊕C(S). Key decryption thus requires the control vector be properly speciﬁed, as well as the correct key-encrypting key; if the combined quantityK⊕C is incorrect, a spurious key of no advantage to an adversary is recovered. Cryptographically binding the control vectorC toS at the time of key generation prevents unauthorized manipulation ofC, assuming only authorized parties have access to the key-encrypting keyK. Control vectors may encompass key notarization by using one or more ﬁelds inC to specify identities. In relation to standard models for access control (Note 13.35), a control vector may be used to specify a subject’s identity (Si) and privileges (Ai,j) regarding the use of a key (Kj). At time of use for a speciﬁc cryptographic operation, the control vector is input as well as the protected key. At this time, a check is made that the requested operation complies with the control vector; if so, the key is decrypted using the control vector. If the control vector does not match that bound to the protected key (or ifK is incorrect), the recovered key S′̸=S will be spurious. Security here is dependent on the assumption that checking is inseparable from use, and done within a trusted subsystem. If the bitsize of the control vectorC differs from that of the keyK, a collision-resistant hash function may be used prior to coupling. This allows arbitrary length control vectors. Thus a 128-bit keyK and a hash functionh with 128-bit output may be used to encryptS as:EK⊕h(C)(S). 13.35 Note (models for access control) Several methods are available to control access to re- sources. Theaccess matrix modeluses a 2-dimensional matrixAi×j with a row for each subject (Si) and a column for each object (Oj), and relies on proper identiﬁcation of sub- jectsSi. Each access recordAi,j speciﬁes the privileges entitySi has on objectOj (e.g., 570 Ch. 13 Key Management Techniques an application program may have read, write, modify, or execute privileges on a ﬁle). Col- umn j may alternately serve as anaccess listfor objectOj, having entries(Si,P ij) where Pij = Ai,j speciﬁes privileges. Another method of resource protection uses the idea of capabilities: acapability(O, P) speciﬁes an objectO and privilege setP related toO, and functions as aticket– possession of capability(O, P) grants the holder the speciﬁed priv- ileges, without further validation or ticket-holder identiﬁcation. 13.36 Example (sample uses of control vectors) Control vectors may be used to provide a public-key like functionality as follows (cf. Remark 13.14). Two copies of a symmetric key are distributed, one typed to allow encryption only (or MAC generation), and a second al- lowing decryption only (or MAC veriﬁcation). Other sample uses of control ﬁelds include: allowing random number generation; allowing ciphertext translation (e.g., in KTCs); dis- tinguishing data encryption and key encryption keys; or incorporation of any ﬁeld within a public-key certiﬁcate. □ 13.37 Remark (key veriﬁcation and preventing replay) Replay of keys distributed by key transport protocols may be countered by the same techniques used to provide unique- ness/timeliness and prevent replay of messages – sequence numbers, timestamps, and challenge-response techniques (§10.3.1). Before a key resulting from a key derivation, no- tarization, or control vector technique is actually used, veriﬁcation of its integrity may be desirable (cf. key conﬁrmation,§12.2). This can be achieved using standard techniques for data integrity (Figure 9.8). A simple method involves the originator sending the encryption (under the key in question) of a data item which the recipient can recognize. 13.6 Key management involving multiple domains This section considers key management models for systems involving multiple domains or authorities, as opposed to the simpler single-domain models of§13.2.3. 13.38 DeﬁnitionA security domain(domain) is deﬁned as a (sub)system under the control of a single authority which the entities therein trust. The security policy in place over a domain is deﬁned either implicitly or explicitly by its authority. The trust that each entity in a domain has in its authority originates from, and is main- tained through, an entity-speciﬁc shared secret key or password (in the symmetric case), or possession of the authority’s authentic public key (in the asymmetric case). This allows se- cure communications channels (with guaranteed authenticity and/or conﬁdentiality) to be established between the entity and authority, or between two entities in the same domain. Security domains may be organized (e.g., hierarchically) to form larger domains. 13.6.1 Trust between two domains Two partiesA and B, belonging to distinct security domainsDA and DB with respective trusted authoritiesTA andTB, may wish to communicate securely (orA may wish to access resources from a distinct domainDB). This can be reduced to the requirement thatA and B either: §13.6 Key management involving multiple domains 571 1. (share a symmetric key) establish a shared secret keyKAB which both trust as being known only to the other (and possibly trusted authorities); or 2. (share trusted public keys) acquire trust in one or more common public keys which may be used to bridge trust between the domains, e.g., allowing veriﬁcation of the authenticity of messages purportedly from the other, or ensure the conﬁdentiality of messages sent to the other. Either of these is possible providedTA and TB have an existing trust relationship, based on either trusted public keys or shared secret keys. IfTA and TB do have an existing trust relationship, either requirement may be met by using this and other initial pairwise trust relationships, which allow secure communications channels between the pairs(A, T A),(TA,T B), and(TB,B ), to be successively used to es- tablish the objective trust relationship(A, B). This may be done byA and B essentially delegating to their respective authorities the task of acquiring trust in an entity under the other authority (as detailed below). IfT A and TB do not share an existing trust relationship directly, a third authorityTC, in which they both do trust, may be used as an intermediary to achieve the same end result. This is analogous to achain of trustin the public-key case (§13.6.2). The two numbered options beginning this subsection are now discussed in further detail. (3B) (2) Domain DA (1) ( 3A) Authority TA Authority TB Domain DB (4) PartyA PartyB Figure 13.8:Establishing trust between users in distinct domains. 1. trusted symmetric key: Trust in a shared secret key may be acquired through a variety of authenticated key establishment techniques (see§12.3 for detailed protocols). An outline of steps by which partiesA and B above may do so follows, with reference to Figure 13.8. (a)A makes a request toTA to obtain a key to share withB (1). (b) TA and TB establish a short-term secret keyKAB (2). (c)TA and TB, respectively, distributeKAB toA and B, guaranteeing secrecy and authenticity (3A, 3B). (d) A usesKAB for secure direct communications withB (4). Message (3B) may be eliminated if its contents are relayed byTB toA viaTA as part of the existing messages (2), (3A). In this case, fromA’s viewpoint the composition ofTA,TB and the trust relationship (TA,T B) may be seen as a single (composite) authority, whichA communicates with throughTA, and which plays the role of the (simple) authority in the standard case of a KDC or KTC (see§13.2.3). 572 Ch. 13 Key Management Techniques 2. trusted public key: Trust in a public key may be acquired, based on existing trust re- lationships, through data origin authentication by standard techniques such as digital signatures or message authentication codes.A may acquire the trusted public key of partyB described above as follows (cf. Figure 13.8). (a)A requests fromTA the trusted public key of userB (1). (b) TA acquires this fromTB, with guaranteed authenticity (2). (c)TA transfers this public key toA, with guaranteed authenticity (3A). (d) A uses this public key to secure direct communications withB (4). 13.39 DeﬁnitionA cross-certiﬁcate(orCA-certiﬁcate) is a certiﬁcate created by one certiﬁca- tion authority (CA), certifying the public key of another CA. 13.40 Remark (user-speciﬁc vs. domain cross-trust) Method 2 above transfers toA trust specif- ically in the public key ofB; this may be called a user-speciﬁc transfer of trust. Alterna- tively, a general transfer of trust between domains is possible as follows, assumingTB has created a certiﬁcateCB containing the identity and public key ofB. In this case,TA creates a cross-certiﬁcate containing the identity and public key ofTB. A, possessing the trusted signature veriﬁcation key ofTA, may verify the signature on this latter certiﬁcate, thereby acquiring trust inTB’s signature veriﬁcation key, and allowingA to verify and thereby trust B’s public key withinCB (or the public key in any other certiﬁcate signed byTB). Thus, userA from domainDA (with authorityTA) acquires trust in public keys certiﬁed inDB by TB. 13.6.2 Trust models involving multiple certiﬁcation authorities Many alternatives exist for organizing trust relationships between certiﬁcation authorities (CAs) in public-key systems involving multiple CAs. These are calledtrust modelsorcerti- ﬁcation topologies, and are logically distinct from (although possibly coincident with) com- munications models. (In particular, a communications link does not imply a trust relation- ship.) Trust relationships between CAs determine how certiﬁcates issued by one CA may be utilized or veriﬁed by entities certiﬁed by distinct CAs (in other domains). Before dis- cussing various trust models, certiﬁcate chains are ﬁrst introduced. (i) Certiﬁcate chains and certiﬁcation paths Public-key certiﬁcates provide a means for obtaining authenticated public keys, provided the veriﬁer has a trusted veriﬁcation public key of the CA which signed the certiﬁcate. In the case of multiple certiﬁcation authorities, a veriﬁer may wish to obtain an authentic pub- lic key by verifying a certiﬁcate signed by a CA other than one for which it (originally) possesses a trusted public key. In this case, the veriﬁer may still do so provided achain of certiﬁcatescan be constructed which corresponds to an unbroken chain of trust from the CA public key which the veriﬁer does trust, to the public key it wishes to obtain trust in. Certiﬁcate chains correspond to directed paths in the graphical representation of a CA trust model (see Figure 13.9). The goal is to ﬁnd a sequence of certiﬁcates corresponding to a directed path (certiﬁcation path) starting at the node corresponding to the CA whose public key a veriﬁer trustsa priori, and ending at the CA which has signed the certiﬁcate of the public key to be veriﬁed. 13.41 Example (illustration of certiﬁcate chain) Consider Figure 13.9(e) on page 574. Suppose an entityA in possession of the public keyP 5 ofCA5 wishes to verify the certiﬁcate of an §13.6 Key management involving multiple domains 573 entityB signed byCA3, and thereby obtain trust inPB. A directed path(CA5, CA4, CA3) exists. LetCA5{CA4}denote a certiﬁcate signed byCA5 binding the nameCA4 to the public keyP4. Then the certiﬁcate chain(CA5{CA4}, CA4{CA3}), along with initial trust inP5, allowsA to verify the signature onCA5{CA4}to extract a trusted copy ofP4, useP4 to verify the signature onCA4{CA3}to extract a trusted copy ofP3, and then use P3 to verify the authenticity of (the certiﬁcate containing)PB. □ Given an initial trusted public key and a certiﬁcate to be veriﬁed, if a certiﬁcate chain is not provided to the veriﬁer, a method is required to ﬁnd (build) the appropriate chain from publicly available data, prior to actual cryptographic chain veriﬁcation. This non- cryptographic task resembles that of routing in standard communications networks. 13.42 Example (building certiﬁcate chains using cross-certiﬁcate pairs) One search technique for ﬁnding the certiﬁcation path given in Example 13.41 involves cross-certiﬁcate pairs. In a public directory, in the directory entry for each CAX, for every CAY that either cross-certiﬁesX or thatX cross-certiﬁes, store the certiﬁcate pair (forward, reverse) = (CA Y {CAX},CAX{CAY }), called across-certiﬁcate pair. Here notation is as in Exam- ple 13.41, the pair consists of the forward and reverse certiﬁcates ofCAX (see page 575), and at least one of the two certiﬁcates is present. In the absence of more advanced tech- niques or routing tables, any existent certiﬁcation path could be found by depth-ﬁrst or breadth-ﬁrst search of the reverse certiﬁcates in cross-certiﬁcate pairs starting at the CA whose public key the veriﬁer possesses initially. □ As part of signature veriﬁcation with certiﬁcate chains, veriﬁcation of cross-certiﬁcates requires checking they themselves have not been revoked (see§13.6.3). (ii) Trust with separate domains Figure 13.9 illustrates a number of possible trust models for certiﬁcation, which are dis- cussed below, beginning with the case of separated domains. Simple public-key systems involve a single certiﬁcation authority (CA). Larger sys- tems involve two or more CAs. In this case, a trust relationship between CAs must be spec- iﬁed in order for users under different CAs to interoperate cryptographically. By default, two distinct CAs deﬁne separate security domains as in Figure 13.9(a), with no trust re- lationship between domains. Users in one domain are unable to verify the authenticity of certiﬁcates originating in a separate domain. (iii) Strict hierarchical trust model The ﬁrst solution to the lack of cryptographic interoperability between separate domains is the idea of a strict hierarchy, illustrated by Figure 13.9(b). Each entity starts with the public key of the root node – e.g., entityE (1) 1 is now givenCA5’s public key at registration, rather than that ofCA1 as in ﬁgure (a). This model is called therooted chainmodel, as all trust chains begin at the root. It is acentralized trustmodel. Several such rooted trees, each being a strict hierarchy, may be combined in a trust model supportingmultiple rooted treesas in Figure 13.9(c). In this case, a cross-certiﬁcate is allowed between the roots of the trees, illustrated by a bi-directional arrow between roots. The arrow directed fromCAX toCAY denotes a certiﬁcate for the public key ofCAY cre- ated byCAX. This allows users in the tree underCAX to obtain trust in certiﬁcates under CAY through certiﬁcate chains which start atCAX and cross over toCAY . In the strict hierarchical model, all entities are effectively in a single domain (deﬁned by the root). Despite the fact that, for example,CA1 signs the public-key certiﬁcate ofE(1) 1 , 574 Ch. 13 Key Management Techniques E ≡user entity E(2) 1 ··· E(2) sE(1) 1 ··· E(1) r ··· E(4) 1 ··· E(4) t CA3 CA1 CA2 CA4 E(2) 1 ··· E(2) sE(1) 1 ··· E(1) r CA1 CA2 CA ≡certiﬁcation authority CA5 CA4 CA4 CA5 CA3 CA1 CA2 CA5 CA3 CA1 CA2 CAYCAX (a) Separate domains (b) Strict hierarchy (c) Multiple rooted trees (d) Hierarchy with reverse certiﬁcates (e) Directed graph (digraph) trust model Figure 13.9:Trust models for certiﬁcation. §13.6 Key management involving multiple domains 575 E(1) 1 trusts the root (CA5) directly but notCA1. E(1) 1 trustsCA1 only indirectly through the root. Potential drawbacks of this model include: 1. all trust in the system is dependent on the root key 2. certiﬁcate chains are required even for two entities under the same CA 3. certiﬁcate chains become long in deep hierarchies 4. a more natural model in some organizations is for trust to begin at a local node (the parent CA) rather than a distant node (the root). (iv) Reverse certiﬁcates and the general digraph trust model A more general hierarchical model, ahierarchy with reverse certiﬁcates, is illustrated in Figure 13.9(d). This resembles the strict hierarchy of Figure 13.9(b), but now each CA lower in the hierarchy also creates certiﬁcates certifying the public keys of its directly su- perior (parent) CA. Two types of certiﬁcates may then be distinguished in a hierarchy: 1. forward certiﬁcate. A forward certiﬁcate (relative toCAX) is created by the CA di- rectly aboveCAX signing the public key ofCAX, and illustrated in the hierarchy by a downward arrow towardsCAX. 2. reverse certiﬁcate. A reverse certiﬁcate (relative toCAX) is created byCAX signing the public key of its immediately superior CA, and illustrated in the hierarchy by an upward arrow originating fromCAX. In this model, each entity starts not with the public key of the root, but rather with the public key of the CA which created its own certiﬁcate, i.e., its local CA (parent). All trust chains now begin at an entity’s local CA. The shortest trust chain from any entityA to any other entityB is now the path in the tree which travels upwards fromA to theleast-common- ancestorofA and B, and downwards from that node on toB. A drawback of the hierarchical model with reverse certiﬁcates is that long certiﬁcate chains may arise between entities which are under distinct CAs even if these entities com- municate frequently (e.g., consider entities underCA 1 and CA4 in Figure 13.9(d). This sit- uation can be ameliorated by allowingCA1 to cross-certifyCA4 directly, even though this edge is not in the hierarchy. This is the most general model, thedirected graph (digraph) trustmodel as illustrated in Figure 13.9(e). The analogy to graph theory is as follows: CAs are represented by nodes or vertices in a graph, and trust relationships by directed edges. (The complete graphon n vertices, with a directed edge from each vertex to every other, corresponds to complete trust, with each CA cross-certifying every other directly.) The digraph model is adistributed trustmodel. There is no central node or root, any CA may cross-certify any other, and each user-entity begins with the trusted public key of its local CA. The concept of a hierarchy remains useful as a reference for organizing trust relationships. This model may be used to implement the other trust models discussed above, including strict hierarchies if variation is permitted in the trusted public key(s) end-user en- tities are provided with initially. 13.43 Remark (assigning end-users to CAs) In hierarchical models, one option is to specify that only CAs at the lowest level certify end-users, while internal CAs serve (only) to cross- certify other CAs. In the general digraph model, where all CAs are considered equal, it is more natural to allow every CA to certify end-users. (v) Constraints in trust models Trust obtained through certiﬁcate chains requires successful veriﬁcation of each certiﬁcate forming a link in the chain. Once a CA (CA X) cross-certiﬁes the public key of another CA (CAY ), in the absence of additional constraints, this trust extended byCAX is transi- tively granted to all authorities which may be reached by certiﬁcate chains originating from 576 Ch. 13 Key Management Techniques CAY . To limit the scope of trust extended by a single cross-certiﬁcate, a CA may impose constraints on cross-certiﬁcates it signs. Such constraints would be enforced during veriﬁ- cation of certiﬁcate chains, and might be recorded explicitly through additional certiﬁcate ﬁelds indicating speciﬁc policies, or through attribute certiﬁcates (§13.4.2). Examples of simple constraints on cross-certiﬁcates include: 1. limiting chain length. A constraint may be imposed on the length of the certiﬁcate chain which may follow the cross-certiﬁcate in question. For example, a CA may limit the extent of trust granted to CAs which it directly cross-certiﬁes by specifying, in all cross-certiﬁcates it signs, that that certiﬁcate must be the last CA-certiﬁcate in any trust chain. 2. limiting the set of valid domains. A set of CAs (or domain names) may be speciﬁed as valid with respect to a given cross-certiﬁcate. All CAs in a certiﬁcate chain following the cross-certiﬁcate in question may be required to belong to this set. Certiﬁcation may also be carried out relative to acertiﬁcation policyspecifying the conditions under which certiﬁcation took place, including e.g., the type of authentication carried out on the certiﬁcate subject before certifying a key, and the method used to guar- antee unique subject names in certiﬁcates. 13.6.3 Certiﬁcate distribution and revocation A certiﬁcate directory (cf.§13.2.4) is a database which implements apullmodel – users extract (pull) certiﬁcates from the database as necessary. A different model of certiﬁcate distribution, thepushmodel, involves certiﬁcates being sent out (pushed) to all users upon certiﬁcate creation or periodically; this may be suitable for closed systems. Alternatively, individual users may provide their certiﬁcates to others when speciﬁcally needed, e.g., for signature veriﬁcation. In certiﬁcate-based systems with certiﬁcate revocation lists (CRLs – see below), a method for distribution of CRLs as well as certiﬁcates is required. A certiﬁcate directory is usually viewed as anunsecuredthird party. While access con- trol to the directory in the form of write and delete protection is necessary to allow mainte- nance and update without denial of service, certiﬁcates themselves are individually secured by the signatures thereon, and need not be transferred over secured channels. An exception ison-line certiﬁcates, which are created by a certiﬁcation authority in real-time on request and have no on-going lifetime, or are distributed by a trusted party which guarantees they have not been revoked. Certiﬁcate or CRLcachingmay be used, whereby frequently referenced items are sav- ed in short-term local storage to avoid the cost of repeated retrievals. Cached CRLs must be refreshed sufﬁciently often to ensure recent revocations are known. Certiﬁcate revocation and CRLs Upon compromise of a secret key, damage may be minimized by preventing subsequent use of or trust in the associated keying material. (Note the implications differ between sig- nature and encryption keys.) Herecompromise includes any situation whereby an adver- sary gains knowledge of secret data. If public keys must be obtained in real-time from a trusted on-line server, the keys in question may be immediately removed or replaced. The situation involving certiﬁcates is more difﬁcult, as all distributed copies must be effectively retracted. While (suspected or actual) key compromise may be rare, there may be other rea- sons a CA will prematurely dissolve its binding of a public key to a user name (i.e.,revoke the certiﬁcate). Reasons for early termination of keying material include the associated en- §13.7 Key life cycle issues 577 tity leaving or changing its role within an organization, or ceasing to require authorization as a user. Techniques for addressing the problem of revoked keys include: 1. expiration dates within certiﬁcates. Expiration dates limit exposure following com- promise. The extreme case of short validity periods resembles on-line certiﬁcates which expire essentially immediately. Short-term certiﬁcates without CRLs may be compared to long-term certiﬁcates with frequently updated CRLs. 2. manual notiﬁcation. All system users are informed of the revoked key by out-of-band means or special channels. This may be feasible in small or closed systems. 3. public ﬁle of revoked keys. A public ﬁle is maintained identifying revoked keys, to be checked by all users before key use. (The authenticity of data extracted from the ﬁle may be provided by similar techniques as for public keys – see§13.4.) 4. certiﬁcate revocation lists(CRLs). A CRL is one method of managing a public ﬁle of revoked keys (see below). 5. revocation certiﬁcates. An alternative to CRLs, these may be viewed as public-key certiﬁcates containing a revocation ﬂag and a time of revocation, serving to cancel the corresponding certiﬁcate. The original certiﬁcate may be removed from the cer- tiﬁcate directory and replaced by the revocation certiﬁcate. A CRL is a signed list of entries corresponding to revoked public keys, with each en- try indicating the serial number of the associated certiﬁcate, the time the revocation was ﬁrst made, and possibly other information such as the revocation reason. The list signature, guaranteeing its authenticity, is generated by the CA which originally issued the certiﬁcates; the CRL typically includes this name also. Inclusion of a date on the overall CRL provides an indication of its freshness. If CRLs are distributed using a pull model (e.g., via a public database), they should be issued at regular intervals (or intervals as advertised within the CRL itself) even if there are no changes, to prevent new CRLs being maliciously replaced by old CRLs. Revoked cross-certiﬁcates may be speciﬁed on separateauthority revocation lists (ARLs), analogous to CRLs (which are then restricted to revoked end-user certiﬁcates). 13.44 Note (CRL segmenting) For reasons of operational efﬁciency when large CRLs may arise, an option is to distribute CRLs in pieces. One technique is to usedelta-CRLs: upon each CRL update, only new entries which have been revoked since the last issued CRL are in- cluded. This requires end-users maintain (and update) secured, local images of the current CRL. A second technique is to partition a CRL into segments based on revocation reason. A third is to segment a CRL by pre-assigning each certiﬁcate (upon creation) to a speciﬁed sub-list, with a limitn max on the number of certiﬁcates pre-assigned to any segment and new segments created as required. In all cases, for each certiﬁcate, available information must indicate which CRL segment must be consulted. 13.7 Key life cycle issues Key management is simplest when all cryptographic keys are ﬁxed for all time. Cryptope- riods necessitate the update of keys. This imposes additional requirements, e.g., on certiﬁ- cation authorities which maintain and update user keys. The set of stages through which a key progresses during its existence, referred to as thelife cycleof keys, is discussed in this section. 578 Ch. 13 Key Management Techniques 13.7.1 Lifetime protection requirements Controls are necessary to protect keys both during usage (cf.§13.5.2) and storage. Regard- ing long-term storage of keys, the duration of protection required depends on the crypto- graphic function (e.g., encryption, signature, data origin authentication/integrity) and the time-sensitivity of the data in question. Security impact of dependencies in key updates Keying material should be updated prior to cryptoperiod expiry (see Deﬁnition 13.10). Up- date involves use of existing keying material to establish new keying material, through ap- propriate key establishment protocols (Chapter 12) and key layering (§13.3.1). To limit exposure in case of compromise of either long term secret keys or past ses- sion keys, dependencies among keying material should be avoided. For example, securing a new session key by encrypting it under the old session key is not recommended (since compromise of the old key compromises the new). See§12.2.3 regarding perfect forward secrecy and known-key attacks. Lifetime storage requirements for various types of keys Stored secret keys must be secured so as to provide both conﬁdentiality and authenticity. Stored public keys must be secured such that their authenticity is veriﬁable. Conﬁdentiality and authenticity guarantees, respectively countering the threats of disclosure and modiﬁca- tion, may be provided by cryptographic techniques, procedural (trust-based) techniques, or physical protection (tamper-resistant hardware). Signature veriﬁcation public keys may require archival to allow signature veriﬁcation at future points in time, including possibly after the private key ceases to be used. Some applications may require that signature private keys neither be backed up nor archived: such keys revealed to any party other than the owner potentially invalidates the property of non- repudiation. Note here that loss (without compromise) of a signature private key may be addressed by creation of a new key, and is non-critical as such a private key is not needed for access to past transactions; similarly, public encryption keys need not be archived. On the other hand, decryption private keys may require archival, since past information encrypted thereunder might otherwise be lost. Keys used for entity authentication need not be backed up or archived. All secret keys used for encryption or data origin authentication should remain secret for as long as the data secured thereunder requires continued protection (theprotection lifetime), and backup or archival is required to prevent loss of this data or veriﬁability should the key be lost. 13.7.2 Key management life cycle Except in simple systems where secret keys remain ﬁxed for all time, cryptoperiods associ- ated with keys require that keys be updated periodically. Key update necessitates additional procedures and protocols, often including communications with third parties in public-key systems. The sequence of states which keying material progresses through over its lifetime is called thekey management life cycle. Life cycle stages, as illustrated in Figure 13.10, may include: 1. user registration– an entity becomes an authorized member of a security domain. This involves acquisition, or creation and exchange, of initial keying material such as shared passwords or PINs by a secure, one-time technique (e.g., personal exchange, registered mail, trusted courier). §13.7 Key life cycle issues 579 revocation user key destruction and key de-registration key key key public key key recovery KEY STATE operational pre- operational post- operational key movement system trigger new user initial keyrecovered key compromise without key loss key compromise or early termination old key (expired) initialization generation existing user new key update new key installation registration cryptoperiod expiry archival obsolete normal use of key user registration key establishment protocol key backup directory Figure 13.10:Key management life cycle. 2. user initialization– an entity initializes its cryptographic application (e.g., installs and initializes software or hardware), involving use or installation (see below) of ini- tial keying material obtained during user registration. 3. key generation– generation of cryptographic keys should include measures to ensure appropriate properties for the intended application or algorithm and randomness in the sense of being predictable (to adversaries) with negligible probability (see Chap- ter 5). An entity may generate its own keys, or acquire keys from a trusted system component. 4. key installation– keying material is installed for operational use within an entity’s software or hardware, by a variety of techniques including one or more of the follow- ing: manual entry of a password or PIN, transfer of a disk, read-only-memory device, chipcard or other hardware token or device (e.g., key-loader). The initial keying ma- terial may serve to establish a secure on-line session through which working keys are established. During subsequent updates, new keying material is installed to replace that in use, ideally through a secure on-line update technique. 5. key registration– in association with key installation, keying material may be ofﬁ- cially recorded (by a registration authority) as associated with a unique name which distinguishes an entity. For public keys, public-key certiﬁcates may be created by a certiﬁcation authority (which serves as guarantor of this association), and made avail- able to others through a public directory or other means (see§13.4). 580 Ch. 13 Key Management Techniques 6. normal use– the objective of the life cycle is to facilitate operational availability of keying material for standard cryptographic purposes (cf.§13.5 regarding control of keys during usage). Under normal circumstances, this state continues until cryptope- riod expiry; it may also be subdivided – e.g., for encryption public-key pairs, a point may exist at which the public key is no longer deemed valid for encryption, but the private key remains in (normal) use for decryption. 7. key backup– backup of keying material in independent, secure storage media pro- vides a data source for key recovery (point 11 below). Backup refers to short-term storage during operational use. 8. key update– prior to cryptoperiod expiry, operational keying material is replaced by new material. This may involve some combination of key generation, key deriva- tion (§13.5.2), execution of two-party key establishment protocols (Chapter 12), or communications with a trusted third party. For public keys, update and registration of new keys typically involves secure communications protocols with certiﬁcation authorities. 9. archival– keying material no longer in normal use may be archived to provide a source for key retrieval under special circumstances (e.g., settling disputes involving repudiation). Archival refers to off-line long-term storage of post-operational keys. 10. key de-registration and destruction– once there are no further requirements for the value of a key or maintaining its association with an entity, the key is de-registered (removed from all ofﬁcial records of existing keys), and all copies of the key are de- stroyed. In the case of secret keys, all traces are securely erased. 11. key recovery– if keying material is lost in a manner free of compromise (e.g., due to equipment failure or forgotten passwords), it may be possible to restore the material from a secure backup copy. 12. key revocation– it may be necessary to remove keys from operational use prior to their originally scheduled expiry, for reasons including key compromise. For public keys distributed by certiﬁcates, this involves revoking certiﬁcates (see§13.6.3). Of the above stages, all are regularly scheduled, except key recovery and key revoca- tion which arise under special situations. 13.45 Remark (public-key vs. symmetric-key life cycle) The life cycle depicted in Figure 13.10 applies mainly to public-key pairs, and involves keying material of only a single party. The life cycle of symmetric keys (including key-encrypting and session keys) is generally less complex; for example, session keys are typically not registered, backed up, revoked, or archived. Key states within life cycle The typical events involving keying material over the lifetime of the key deﬁne stages of the life cycle. These may be grouped to deﬁne a smaller set ofstatesfor cryptographic keys, related to their availability for use. One classiﬁcation of key states is as follows (cf. Figure 13.10): 1. pre-operational. The key is not yet available for normal cryptographic operations. 2. operational. The key is available, and in normal use. 3. post-operational. The key is no longer in normal use, but off-line access to it is pos- sible for special purposes. 4. obsolete. The key is no longer available. All records of the key value are deleted. §13.8 Advanced trusted third party services 581 System initialization and key installation Key management systems require an initial keying relationship to provide an initial secure channel and optionally support the establishment of subsequent working keys (long-term and short-term) by automated techniques. The initialization process typically involves non- cryptographic one-time procedures such as transfer of keying material in person, by trusted courier, or over other trusted channels. The security of a properly architected system is reduced to the security of keying ma- terial, and ultimately to the security of initial key installation. For this reason, initial key installation may involve dual or split control, requiring co-operation of two or more inde- pendent trustworthy parties (cf. Note§13.8). 13.8 Advanced trusted third party services This section provides further details on trusted third party services of a more advanced na- ture, introduced brieﬂy in§13.2.4. 13.8.1 Trusted timestamping service A trusted timestamping service provides a user with a dated receipt (upon presentation of a document), which thereafter can be veriﬁed by others to conﬁrm the presentation or ex- istence of the document at the (earlier) date of receipt. Speciﬁc applications include estab- lishing the time of existence of documents such as signed contracts or lab notes related to patent claims, or to support non-repudiation of digital signatures (§13.8.2). The basic idea is as follows. A trusted third partyT (thetimestamp agent) appends a timestampt 1 to a submitted digital document or data ﬁleD, signs the composite document (thereby vouching for the time of its existence), and returns the signed document including t1 to the submitter. Subsequent veriﬁcation ofT’s signature then establishes, based on trust inT, the existence of the document at the timet1. If the data submitted for timestamping is the hash of a document, then the document content itself need not be disclosed at the time of timestamping. This also provides privacy protection from eavesdroppers in the case of submissions over an unsecured channel, and reduces bandwidth and storage costs for large documents. 13.46 Remark (non-cryptographic timestamp service) A similar service may be provided by non-cryptographic techniques as follows.T storesD along with a timestampt 1, and is trusted to maintain the integrity of this record by procedural techniques. Later some party A submits the document again (nowD ′), andT comparesD′toD on ﬁle. If these match, T declares thatD′existed at the timet1 of the retrieved timestamp. The timestamp agentT is trusted not to disclose its signing key, and also to compe- tently create proper signatures. An additional desirable feature isprevention of collusion:T should be unable to successfully collude (with any party) to undetectably back-date a doc- ument. This may be ensured using Mechanism 13.47, which combines digital signatures with tree authentication based on hashing. 582 Ch. 13 Key Management Techniques 13.47 Mechanism Trusted timestamping service based on tree authentication SUMMARY: party A interacts with a trusted timestamping agentT. RESULT: A obtains a timestamp on a digital documentD. 1. A submits the hash valueh(D) toT.(h is a collision-resistant hash function.) 2. T notes the date and timet1 of receipt, digitally signs the concatenation ofh(D) and t1, and returnst1 and the signature toA. (The signature is called thecertiﬁed time- stamp.)A may verify the signature to conﬁrmT’s competence. 3. At the end of each ﬁxed period (e.g., one day), or more frequently if there is a large number n of certiﬁed timestamps,T: (i) computes from these an authentication treeT ∗with root labelR (see§13.4.1); (ii) returns toA the authentication path values to its certiﬁed timestamp; and (iii) makes the root valueR widely available through a means allowing both veriﬁ- able authenticity and establishment of the time of creationtc ofT ∗(e.g., pub- lishing in a trusted dated medium such as a newspaper). 4. To allow any other partyB to verify (withT’s veriﬁcation public key) thatD was submitted at timet1,A produces the certiﬁed timestamp. If trust inT itself is chal- lenged (with respect to backdatingt1),A provides the authentication path values from its certiﬁed timestamp to the rootR, whichB may verify (see§13.4.1) against an in- dependently obtained authentic root valueR for the periodtc. To guarantee veriﬁability,A should itself verify the authentication path upon receiving the path values in step 3. 13.8.2 Non-repudiation and notarization of digital signatures The timestamping service of§13.8.1 is a document certiﬁcation or document notarization service. Anotary serviceis a more general service capable not only of ascertaining the ex- istence of a document at a certain time, but of vouching for the truth of more general state- ments at speciﬁed points in time. The terminology originates from the dictionary deﬁnition of anotary public– a public ofﬁcial (usually a solicitor) legally authorized to administer oaths, and attest and certify certain documents. No speciﬁc legal connotation is intended in the cryptographic use of this term. The non-repudiation aspect of digital signatures is a primary advantage of public-key cryptography. By this property, a signer is prevented from signing a document and subse- quently being able to successfully deny having done so. A non-repudiation service requires speciﬁcation of precise details including an adjudication process and adjudicator (judge), what evidence would be submitted to the adjudicator, and what precise process the adju- dicator is to follow to render judgement on disputes. The role of an adjudicator is distinct from that of a timestamp agent or notary which generates evidence. 13.48 Remark (origin authentication vs. non-repudiable signature) A fundamental distinction exists between a partyA being able to convince itself of the validity of a digital signatures at a point in timet 0, and that party being able to convince others at some timet1 ≥t0 thats was valid at timet0. The former resembles data origin authentication as typically provided by symmetric-key origin authentication mechanisms, and may be accepted by a veriﬁer as a form of authorization in an environment of mutual trust. This differs from digital signatures which are non-repudiable in the future. §13.8 Advanced trusted third party services 583 Data origin authentication as provided by a digital signature is valid only while the secrecy of the signer’s private key is maintained. A threat which must be addressed is a signer who intentionally discloses his private key, and thereafter claims that a previously valid signature was forged. (A similar problem exists with credit cards and other methods of authorization.) This threat may be addressed by: 1. preventing direct access to private keys. Preventing users from obtaining direct ac- cess to their own private keys precludes intentional disclosure. As an example, the private keys may be stored in tamper-resistant hardware, and by system design never available outside thereof. 2. use of a trusted timestamp agent. The party obtaining a signature on a critical docu- ment submits the signature to a timestamp agent, which afﬁxes a timestamp to signa- ture and then signs the concatenation of these. This establishes a timet 1 at which the critical signature may be ascertained to have existed. If the private signature key cor- responding to this signature is subsequently compromised, and the compromise oc- curred aftert1, then the critical signature may still be considered valid relative tot1. For reasons as given in Remark 13.49, use of a notary agent (below) may be prefer- able. 3. use of a trusted notary agent. The party obtaining a signature on a critical document (or hash thereof) submits the signature (and document or hash thereof) to an agent forsignature notarization. The agent veriﬁes the signature and notarizes the result by appending a statement (conﬁrming successful signature veriﬁcation) to the signa- ture, as well as a timestamp, and signing the concatenation of the three. A reasonable period of time (clearance period) may be allowed for declarations of lost private keys, after which the notary’s record of veriﬁcation must be accepted (by all parties who trust the notary and verify its signature) as the truth regarding the validity of the crit- ical signature at that point in time, 5 even should the private key corresponding to the critical signature subsequently be compromised. For signed messages having short lifetimes (i.e., whose signiﬁcance does not extend far into the future), non-repudiation is less important, and notarization may be unnecessary. For other messages, the requirement for a party to be able to re-verify signatures at a later point in time (including during or after signature keys have been updated or revoked), as well as the adjudication process related to non-repudiation of signatures, places additional demands on practical key management systems. These may include the storage or archival of keying material (e.g., keys, certiﬁcates, CRLs) possibly required as evidence at a future point in time. A related support service is that of maintaining a record (audit trail) of security-related events including registration, certiﬁcate generation, key update, and revocation. Audit trails may provide sufﬁcient information to allow resolution of disputed signatures by non-auto- mated procedures. 13.49 Remark (reconstructing past trust) Both signature re-veriﬁcation (relative to a past point in time) and resolution of disputes may require reconstruction of chains of trust from a past point in time. This requires access to keying material and related information for (re)constr- ucting past chains of trust. Direct reconstruction of such past chains is unnecessary if a notarizing agent was used. The original veriﬁcation of the notary establishes existence of a trust chain at that point in time, and subsequently its record thereof serves as proof of prior validity. It may be of interest (for audit purposes) to record the details of the original trust chain. 5More generally, the truth of the appended statement must be accepted, relative to the timestamp. 584 Ch. 13 Key Management Techniques 13.8.3 Key escrow The objective of a key escrow encryption system is to provide encryption of user trafﬁc (e.g., voice or data) such that the session keys used for trafﬁc encryption are available to properly authorized third parties under special circumstances (“emergency access”). This grants third parties which have monitored user trafﬁc the capability to decrypt such traf- ﬁc. Wide-scale public interest in such systems arose when law enforcement agencies pro- moted their use to facilitate legal wiretapping of telephone calls to combat criminal activi- ties. However, other uses in industry include recovery of encrypted data following loss of keying material by a legitimate party, or destruction of keying material due to equipment failure or malicious activities. One example of a key escrow system is given below, fol- lowed by more general issues. (i) The Clipper key escrow system The Clipper key escrow system involves use of the Clipper chip (or a similar tamper-resist- ant hardware device – generically referred to below as an escrow chip) in conjunction with certain administrative procedures and controls. The basic idea is to deposit two key com- ponents, which jointly determine an encryption key, with two trusted third parties (escrow agents), which subsequently allow (upon proper authorization) recovery of encrypted user data. More speciﬁcally, encryption of telecommunications between two users proceeds as follows. Each party has a telephone combined with a key escrow chip. The users negotiate or otherwise establish a session keyK S which is input to the escrow chip of the party en- crypting data (near end). As a function ofKS and an initialization vector (IV), the chip cre- ates by an undisclosed method a data block called alaw enforcement access ﬁeld(LEAF). The LEAF and IV are transmitted to the far end during call set-up of a communications ses- sion. The near end escrow chip then encrypts the user dataD underKS producingEKS(D), by a U.S. government classiﬁed symmetric algorithm named SKIPJACK. The far end es- crow chip decrypts the trafﬁc only if the transmitted LEAF validates properly. Such veri- ﬁcation requires that this far end chip has access to a common family keyKF (see below) with the near end chip. The LEAF (see Figure 13.11) contains a copy of the session key encrypted under a device-speciﬁc keyKU. KU is generated and data-ﬁlled into the chip at the time of chip manufacture, but prior to the chip being embedded in a security product. The system meets its objective by providing third party access under proper authorization (as deﬁned by the Key Escrow System) to the device keyK U of targeted individuals. To derive the keyKU embedded in an escrow chip with identiﬁer UID, two key com- ponents (KC1,KC2) are created whose XOR isKU. Each component is encrypted under a key KCK = KN1⊕KN2, whereKNi is input to the chip programming facility by the ﬁrst and second trustedkey escrow agent, respectively. (Used to program a number of chips, KNi is stored by the escrow agent for subsequent recovery ofKCK .) One encrypted key component is then given to each escrow agent, which stores it along with UID to service later requests. Stored data from both agents must subsequently be obtained by an autho- rized ofﬁcial to allow recovery ofKU (by recovering ﬁrstKCK , and thenKC1,KC2, and KU = KC1⊕KC2). Disclosed details of the LEAF are given in Figure 13.11. Each escrow chip contains a 32-bit device unique identiﬁer (UID), an 80-bit device unique key (KU), and an 80-bit fam- ily key (KF ) common to a larger collection of devices. The LEAF contains a copy of the 80-bit session keyKS encrypted underKU, the UID, and a 16-bit encryption authentica- §13.8 Advanced trusted third party services 585 tor (EA) created by an undisclosed method; these are then encrypted underKF . Recovery ofKS from the LEAF thus requires bothKF and KU. The encryption authenticator is a checksum designed to allow detection of LEAF tampering (e.g., by an adversary attempting to prevent authorized recovery ofKS and therebyD). Escrow Agent 1 Escrow Agent 2 recover KU Key Escrow Decrypt Processor KU D ciphertextLEAF Escrow chip 16 input from escrow agents wiretap Chip Programming Facility 80family keys components of escrowedKU KS UID processing Escrow component User component Recovery component KU = device unique key KF = family key KFC = family key component E−1 E EA to UID UID= device unique identiﬁer EA= encryption authenticator far Schematic representation: KFC1 KFC2 32 session keyKS IV user dataD output to escrow agents recovery request UID KF KU KF EKU (KS) EA end to third party UID EKU (KS) E E EKS(D) Figure 13.11:Creation and use of LEAF for key escrow data recovery. (ii) Issues related to key escrow Key escrow encryption systems may serve a wide variety of applications, and a correspond- ing range of features exists. Distinguishing properties of escrow systems include: 1. applicability to store-and-forward vs. real-time user communications 2. capability of real-time decryption of user trafﬁc 3. requirement of tamper-resistant hardware or hardware with trusted clock 4. capability of user selection of escrow agents 586 Ch. 13 Key Management Techniques 5. user input into value of escrowed key 6. varying trust requirements in escrow agents 7. extent of user data uncovered by one escrow access (e.g., limited to one session or ﬁxed time period) and implications thereof (e.g., hardware replacement necessary). Threshold systems and shared control systems may be put in place to access escrowed key- ing information, to limit the chances of unauthorized data recovery. Key escrow systems may be combined with other life cycle functions including key establishment, and key back- up and archival (cf. key access servers – Notes 13.5 and 13.6). 13.9 Notes and further references §13.1 Davies and Price [308] provide a comprehensive treatment of key management, includ- ing overviews of ISO 8732 [578] and techniques introduced in several 1978IBM Systems Journalpapers [364, 804]. Early work addressing protection in communications networks and/or key management includes that of Feistel, Notz, and Smith [388], Branstad [189], Kent [665], Needham and Schroeder [923], and the surveys of Popek and Kline [998] and V oydock and Kent [1225]. Security issues in electronic funds transfer (EFT) systems for point-of-sale (POS) terminals differ from those for remote banking machines due to the weaker physical security of the former; special key management techniques such as unique (derived) transaction keys reduce the implications of terminal key compromise – see Beker and Walker [85], Davies [305], and Davies and Price [308, Ch.10]. See Meyer and Matyas [859] for general symmetric-key techniques, EFT applications, and PIN management; and Ford [414] for directory services and standards, including the X.500 Directory and X.509 Authentication Framework [626]. For an overview of key management concepts and life cycles aspects, see Fumy and Lan- drock [429]. Fumy and Leclerc [430] consider placement of key distribution protocols with- in the ISO Open Systems Interconnection (OSI) architecture. Regarding key management principles, see Abadi and Needham [1], and Anderson and Needham [31]. See Vedder [1220] for security issues and architectures relevant to wireless communications, includ- ing European digital cellular (Global System for Mobile Communications – GSM) and the Digital European Cordless Telephone (DECT) system. Regarding key management for se- curity (authentication and encryption) in North American digital cellular systems, see IS-54 Rev B [365]. ISO 11166-1 [586] (see also comments by Rueppel [1082]) speciﬁes key man- agement techniques and life cycle principles for use in banking systems, and is used by the Society for Worldwide Interbank Financial Telecommunications (SWIFT). §13.2 Various parts of ISO/IEC 11770 [616, 617, 618] contain background material on key man- agement; Figure 13.3 is derived from an early draft of 11770-3. KDCs and KTCs were popularized by ANSI X9.17 [37]. Related to tradeoffs, Needham and Schroeder [923] compare symmetric and public-key techniques; the formalization proposed by Rueppel [1080] allows analysis of security architectures to distinguish complexity-increasing from complexity-reducing techniques. The Kerberos authentication service (§12.3.2) includes aticket-granting servicewhereby a client may re-authenticate itself multiple times using its long-term secret only once. The clientA ﬁrst acquires aticket-granting-ticketthrough a protocol with an Authentication Server (AS). Thereafter, using a variation of Protocol 12.24,A may obtain authentication §13.9 Notes and further references 587 credentials for a serverB from a Ticket-Granting-Server (TGS), extracting a TGS session key from the time-limited ticket to secure protocol messages with the TGS.A’s long-term secret (password) need neither be cached for an extended period in memory nor re-entered, reducing the threat of its compromise; compromise of a TGS session key has time-restricted impact. See RFC 1510 [1041] for details. Ford and Wiener [417] describe key access servers (Note 13.6), effectively an access control mechanism where the resource is akey package. Girault [459] mentions the three levels of trust of Remark 13.7. Digital envelopes (Note 13.6) are discussed in PKCS #7 [1072]. §13.3 Example 13.9 is from Tuchman [1198]. Davis and Swick [310] discuss symmetric-key cer- tiﬁcates as deﬁned herein under the nameprivate-key certiﬁcates(crediting Abadi, Bur- rows, and Lampson) and propose protocols for their use with trusted third parties, includ- ing a password-based initial registration protocol. Predating this, Davies and Price [308, p.259] note that tamper-resistant hardware may replace the trusted third party requirement of symmetric-key certiﬁcates (Note 13.14). A generalization of Protocol 13.12 appears as Mechanism 11 of ISO/IEC 11770-2 [617], along with related KTC protocols offering ad- ditional authenticity guarantees (cf. Note 13.13(iii)); these provide KTC variations of the KDC protocols of§12.3.2). §13.4 Difﬁe and Hellman [345] suggested using a trusted public ﬁle, maintained by a trusted au- thority with which each communicant registers once (in person), and from which authentic public keys of other users can be obtained. To secure requests by one party for the pub- lic key of another, Rivest, Shamir, and Adleman [1060] and Needham and Schroeder [923] note the trusted authority may respond via signed messages (essentially providing on-line certiﬁcates). Authentication trees were ﬁrst discussed in Merkle’s thesis [851, p.126-131] (see also [852, 853]). For security requirements on hash functions used for tree authentication, see Preneel [1003, p.38]. Public-key certiﬁcates were ﬁrst proposed in the 1978 B.Sc. thesis of Kohn- felder [703]; the overall thesis considers implementation and systems issues related to using RSA in practice. Kohnfelder’s original certiﬁcate was an ordered triple containing a party’s name, public-key information, and an authenticator, with the authenticator a signature over the value resulting from encrypting the name with the public key/algorithm in question. X.509 certiﬁcates [626] were deﬁned in 1988 and modiﬁed in 1993 (yielding Version 2 cer- tiﬁcates); anextensionsﬁeld was added by a technical corrigendum [627] in 1995 (yielding Version 3 certiﬁcates). Standard extensions for Version 3 certiﬁcates appear in an amend- ment to X.509 [628]; these accommodate information related to key identiﬁers, key usage, certiﬁcate policy, alternate names (vs. X.500 names) and name attributes, certiﬁcation path constraints, and enhancements for certiﬁcate revocation including revocation reasons and CRL partitioning. For details, see Ford [416]. ANSI X9.45 [49] addresses attribute certiﬁ- cates. The alternative of including hard-coded attribute ﬁelds within public-key certiﬁcates is proposed in PKCS #6 [1072]; suggested attributes are listed in PKCS #9 [1072]. In 1984 Shamir [1115] formulated the general idea of asymmetric systems employing user’s identities in place of public keys (identity-based systems), giving a concrete proposal for an ID-based signature system, and the model for an ID-based encryption scheme. Fiat and Shamir [395] combined this idea with that of zero-knowledge interactive proofs, yielding interactive identiﬁcation and signature protocols. T. Okamoto [947] (based on a January 1984 paper in Japanese by Okamoto, Shiraishi, and Kawaoka [954]) independently pro- posed a speciﬁc entity-authentication scheme wherein a trusted centerT distributes to a 588 Ch. 13 Key Management Techniques claimantA a secret accreditation value computed as a function ofT’s private key andA’s identity (or unique index value). The identity-based key-agreement scheme of Maurer and Yacobi [824] (cf.§12.6 notes on page 538) is an exception to Remark 13.27: extra public dataDA is avoided, as ideally desired. G¨unther [530] proposed a protocol for key agreement (Protocol 12.62) wherein users’ pri- vate keys are constructed by a trusted authorityT based on their identities, with correspond- ing Difﬁe-Hellman public keys reconstructed from public data provided byT (herein called implicitly-certiﬁed public keys, identity-based subclass). The protocol introduced by Gi- rault [459], based on the key agreement protocol of Paill`es and Girault [962] (itself up- dated by Girault and Paill`es [461] and Girault [458]) similar to a protocol of Tanaka and E. Okamoto [1184], involved what he christenedself-certiﬁed public keys(herein called implicitly-certiﬁed public keys, self-certiﬁed subclass); see Mechanism 12.61. Related to self-certiﬁed public keys, Brands [185] has proposedsecret-key certiﬁcatesfor use in so-called restrictive blind signature schemes. These involve a data triple consisting of a private key, matching public key, and an explicit (secret-key) certiﬁcate created by a trusted third party to certify the public key. Users can themselves create pairs of public keys and matching (secret-key) certiﬁcates, but cannot create valid triples. As with self-certiﬁed keys, performance of a cryptographic action relative to the public key (e.g., signing) im- plicitly demonstrates that the performing party knows the private key and hence that the corresponding public key was indeed issued by the trusted third party. §13.5 Key tags are due to Jones [642]. Key separation as in Example 13.33 is based on Ehrsam et al. [364], which outlines the use of master keys, key variants, key- and data-encrypting keys. Smid [1153] introduced key notarization in the Key Notarization System (KNS), a key management system designed by the U.S. National Bureau of Standards (now NIST), and based on a Key Notarization Facility (KNF) – a KTC-like system component trusted to handle master keys, and to generate and notarize symmetric keys. Key notarization with key offsetting (Example 13.34) is from ISO 8732 [578], which is derived from ANSI X9.17 [37]. The generalization of key notarization to control vectors is due to Matyas, Meyer, and Brachtl [806], and described by Matyas [803] (also [802]), including an efﬁcient method for allowing arbitrary length control vectors that does not penalize short vectors. The IBM proposal speciﬁesE as two-key triple-DES, as per ANSI X9.17. Matyas notes that a sec- ond approach to implement control vectors, using a MAC computed on the control vector and the key (albeit requiring additional processing), has the property that both the control vector and the recovered key may be authenticated before the key is used. The notion of a capability (Note 13.35) was introduced in 1966 by Dennis and Van Horn [332], who also considered the access matrix model. §13.6 Key distribution between domains is discussed in ISO/IEC 11770-1 [616]; see also Kohl and Neuman [1041] with respect to Kerberos V5, and Davis and Swick [310]. A Kerberos domain is called arealm; authentication of clients in one realm to servers in others is sup- ported in V5 byinter-realm keys, with a concept of authentication paths analogous to public- key certiﬁcation paths. Kent [666] overviews the design and implementation ofPrivacy Enhanced Mail(PEM) (see RFC 1421-1424 [1036, 1037, 1038, 1039]), a prototyped method for adding security to In- ternet mail. Encryption and signature capabilities are provided. The PEM infrastructure of §13.9 Notes and further references 589 RFC 1422 is based on a strict hierarchy of certiﬁcation authorities, and includes speciﬁca- tion ofPolicy Certiﬁcation Authorities(PCAs) which deﬁne policies with respect to which certiﬁcates are issued. Regarding certiﬁcation paths, see Tarah and Huitema [1185]. The 1988 version of X.509 [626] deﬁnes forward and reverse certiﬁcates, certiﬁcate chains, and cross-certiﬁcate pairs, allowing support for the general digraph trust model. The formal analysis of Gaarder and Snekkenes [433] highlights a practical difﬁculty in verifying the validity of certiﬁcates – the requirement of trusted timeclocks. For reports on implementa- tions based on X.509 certiﬁcates, see Tardo and Alagappan [1186] and others [660, 72, 839]. Techniques for segmenting CRLs (Note 13.44) are included in the above-cited work on Ver- sion 3 certiﬁcate extensions [628]. Kohnfelder [703] noted many of the issues regarding certiﬁcate revocation in 1978 when use of certiﬁcates was ﬁrst proposed, and suggested techniques to address revocation including manual notiﬁcation, maintaining a public ﬁle identifying revoked keys, and use of certiﬁcate expiration dates (cf. Denning [326, p.170]). §13.7 Matyas and Meyer [804] consider several life cycle aspects. ISO 11770-1 [616] provides a general overview of key management issues including key life cycle. ANSI X9.57 [52] provides broad discussion on certiﬁcate management, including trust models, registration, certiﬁcate chains, and life cycle aspects. ISO 10202-7 [584] speciﬁes a key management life cycle for chipcards. §13.8 Davies and Price [308] discuss practical issues related to registries of public keys, non- repudiation, and revocation, including the use of timestamps and notarization; see also the original works of Kohnfelder [703] and Merkle [851], which include discussion of notaries. Haber and Stornetta [535] propose two additional techniques for timestamping digital data (one enhanced by Bayer, Haber, and Stornetta [79]), although tree authentication, due to Merkle [852], appears to be preferable in practice. Benaloh and de Mare [111] introduce one-way accumulators to address the same problem. Although key backup/archive functionality existed in earlier commercial products, the widespread study of key escrow systems began circa 1992, and combines issues related to secret sharing, key establishment, and key life cycle. For practical aspects including com- mercial key recovery and backup, see Walker et al. [1229] and Maher [780]. Denning and Branstad [329] provide an excellent overview of the numerous proposals to date, including a taxonomy. Among such proposals and results are those of Micali [863] (see also [862]), Leighton and Micali [745], Beth et al. [125], Desmedt [338] (but see also Knudsen and Pedersen [690]), Jefferies, Mitchell, and Walker [635], Lenstra, Winkler, and Yacobi [755], Kilian and Leighton [671], Frankel and Yung [420], and Micali and Sidney [869]. In some systems, it is required that escrow agents be able to verify that (partial) keys received are authentic, raising issues of veriﬁable secret sharing (see Chor et al. [259]). The Clipper chip is a tamper-resistant hardware encryption device compliant with FIPS 185 [405], a voluntary U.S. government standard intended for sensitive but unclassiﬁed phone (voice and data) communications. FIPS 185 speciﬁes use of the SKIPJACK encryption al- gorithm (80-bit key, 64-bit blocks) and LEAF creation method, the details of both of which remain classiﬁed. The two initial key escrow agents named by the U.S. Government are the National Institute of Standards and Technology (NIST) and the Department of the Treasury, Automated Systems Division. Denning and Smid [331] describe the operation of an initial key escrow system employing a chip in accordance with FIPS 185. TheCapstonechip, a more advanced device than Clipper, implements in addition a public key agreement algo- rithm, DSA, SHA, high-speed general-purpose exponentiation, and a (pure noise source) 590 Ch. 13 Key Management Techniques random number generator; it is used in the U.S. government Multilevel Information Se- curity System Initiative (MISSI) for secure electronic mail and other applications. Blaze [152] demonstrated that a protocol attack is possible on Clipper, requiring at most216 trial LEAF values to construct a bogus LEAF with a valid EA; Denning and Smid note this is not a threat in practical systems. For a debate on issues related to U.S. digital telephony leg- islation passed in October 1994 as the Communications Assistance for Law Enforcement Act (CALEA), requiring telephone companies to provide technical assistance facilitating authorized wiretapping, see Denning [328]. Chapter /1/4 Efﬁcient Implementation Contents in Brief 14.1 Introduction............................. 591 14.2 Multiple-precision integer arithmetic................ 592 14.3 Multiple-precision modular arithmetic............... 599 14.4 Greatest common divisor algorithms................ 606 14.5 Chinese remainder theorem for integers.............. 610 14.6 Exponentiation ........................... 613 14.7 Exponent recoding ......................... 627 14.8 Notes and further references.................... 630 14.1 Introduction Many public-key encryption and digital signature schemes, and some hash functions (see §9.4.3), require computations inZm, the integers modulom(mis a large positive integer which may or may not be a prime). For example, the RSA, Rabin, and ElGamal schemes re- quire efﬁcient methods for performing multiplication and exponentiation inZm. Although Zm is prominent in many aspects of modern applied cryptography, other algebraic struc- tures are also important. These include, but are not limited to, polynomial rings, ﬁnite ﬁelds, and ﬁnite cyclic groups. For example, the group formed by the points on an elliptic curve over a ﬁnite ﬁeld has considerable appeal for various cryptographic applications. The efﬁ- ciency of a particular cryptographic scheme based on any one of these algebraic structures will depend on a number of factors, such as parameter size, time-memory tradeoffs, process- ing power available, software and/or hardware optimization, and mathematical algorithms. This chapter is concerned primarily with mathematical algorithms for efﬁciently carry- ing out computations in the underlying algebraic structure. Since many of the most widely implemented techniques rely onZ m, emphasis is placed on efﬁcient algorithms for per- forming the basic arithmetic operations in this structure (addition, subtraction, multiplica- tion, division, and exponentiation). In some cases, several algorithms will be presented which perform the same operation. For example, a number of techniques for doing modular multiplication and exponentiation are discussed in§14.3 and§14.6, respectively. Efﬁciency can be measured in numerous ways; thus, it is difﬁcult to deﬁnitively state which algorithm is the best. An algorithm may be efﬁcient in the time it takes to perform a certain algebraic operation, but quite inefﬁcient in the amount of storage it requires. One algorithm may require more code space than an- other. Depending on the environment in which computations are to be performed, one algo- rithm may be preferable over another. For example, current chipcard technology provides 591 592 Ch. 14 Efﬁcient Implementation very limited storage for both precomputed values and program code. For such applications, an algorithm which is less efﬁcient in time but very efﬁcient in memory requirements may be preferred. The algorithms described in this chapter are those which, for the most part, have re- ceived considerable attention in the literature. Although some attempt is made to point out their relative merits, no detailed comparisons are given. Chapter outline §14.2 deals with the basic arithmetic operations of addition, subtraction, multiplication, squaring, and division for multiple-precision integers.§14.3 describes the basic arithmetic operations of addition, subtraction, and multiplication inZm. Techniques described for per- forming modular reduction for an arbitrary modulusmare the classical method (§14.3.1), Montgomery’s method (§14.3.2), and Barrett’s method (§14.3.3).§14.3.4 describes a re- duction procedure ideally suited to moduli of a special form. Greatest common divisor (gcd) algorithms are the topic of§14.4, including the binary gcd algorithm (§14.4.1) and Lehmer’s gcd algorithm (§14.4.2). Efﬁcient algorithms for performing extended gcd com- putations are given in§14.4.3. Modular inverses are also considered in§14.4.3. Garner’s algorithm for implementing the Chinese remainder theorem can be found in§14.5.§14.6 is a treatment of several of the most practical exponentiation algorithms.§14.6.1 deals with exponentiation in general, without consideration of any special conditions.§14.6.2 looks at exponentiation when the base is variable and the exponent is ﬁxed.§14.6.3 considers al- gorithms which take advantage of a ﬁxed-base element and variable exponent. Techniques involving representing the exponent in non-binary form are given in§14.7; recoding the ex- ponent may allow signiﬁcant performance enhancements.§14.8 contains further notes and references. 14.2 Multiple-precision integer arithmetic This section deals with the basic operations performed on multiple-precision integers: ad- dition, subtraction, multiplication, squaring, and division. The algorithms presented in this section are commonly referred to as theclassical methods. 14.2.1 Radix representation Positive integers can be represented in various ways, the most common beingbase10. For example,a= 123base10 means a=1 ·102 +2 ·101 +3 ·100. For machine computations, base2 (binary representation) is preferable. Ifa= 1111011base2, thena=2 6 +2 5 + 24 +2 3 +0 ·22 +2 1 +2 0. 14.1 FactIfb≥2 is an integer, then any positive integeracan be expressed uniquely asa= anbn + an−1bn−1 + ···+ a1b+ a0, whereai is an integer with0 ≤ai <bfor0 ≤i≤n, and an ̸=0. 14.2 DeﬁnitionThe representation of a positive integeraas a sum of multiples of powers of b, as given in Fact 14.1, is called thebaseborradixbrepresentation ofa. §14.2 Multiple-precision integer arithmetic 593 14.3 Note (notation and terminology) (i) The basebrepresentation of a positive integeragiven in Fact 14.1 is usually written as a=( anan−1 ···a1a0)b. The integersai,0 ≤i≤n, are calleddigits. an is called themost signiﬁcant digitorhigh-order digit;a0 theleast signiﬁcant digitor low-order digit.I fb=1 0, the standard notation isa= anan−1 ···a1a0. (ii) It is sometimes convenient to pad high-order digits of a basebrepresentation with 0’s; such a padded number will also be referred to as the basebrepresentation. (iii) If(anan−1 ···a1a0)b is the basebrepresentation ofaand an ̸=0, then theprecision orlengthofaisn+1.I fn=0 , thenais called asingle-precision integer; otherwise, ais amultiple-precision integer.a=0 is also a single-precision integer. The division algorithm for integers (see Deﬁnition 2.82) provides an efﬁcient method for determining the basebrepresentation of a non-negative integer, for a given baseb. This provides the basis for Algorithm 14.4. 14.4 AlgorithmRadixb representation INPUT: integersaand b,a≥0,b≥2. OUTPUT: the basebrepresentationa=( an ···a1a0)b, wheren≥0 and an ̸=0ifn≥1. 1. i←0, x←a, q←⌊x b ⌋, ai←x−qb.(⌊·⌋is the ﬂoor function; see page 49.) 2. Whileq> 0, do the following: 2.1 i←i+1 , x←q, q←⌊x b ⌋, ai←x−qb. 3. Return((aiai−1 ···a1a0)). 14.5 FactIf(anan−1 ···a1a0)b is the basebrepresentation ofaand kis a positive integer, then(ulul−1 ···u1u0)bk is the basebk representation ofa, wherel= ⌈(n+1 )/k⌉−1, ui = ∑ k−1 j=0 aik+jbj for0 ≤i≤l−1, andul = ∑ n−lk j=0 alk+jbj. 14.6 Example (radixbrepresentation) The base2 representation ofa= 123 is(1111011)2. The base4 representation ofais easily obtained from its base2 representation by grouping digits in pairs from the right:a= ((1)2(11)2(10)2(11)2)4 = (1323)4. □ Representing negative numbers Negative integers can be represented in several ways. Two commonly used methods are: 1. signed-magnitude representation 2. complement representation. These methods are described below. The algorithms provided in this chapter all assume a signed-magnitude representation for integers, with the sign digit being implicit. (i) Signed-magnitude representation The signof an integer (i.e., either positive or negative) and itsmagnitude(i.e., absolute value) are represented separately in asigned-magnitude representation. Typically, a posi- tive integer is assigned a sign digit0, while a negative integer is assigned a sign digitb−1. Forn-digit radixbrepresentations, only2bn−1 sequences out of thebn possible sequences are utilized: preciselybn−1 −1 positive integers andbn−1 −1 negative integers can be rep- resented, and0 has two representations. Table 14.1 illustrates the binary signed-magnitude representation of the integers in the range[7,−7]. 594 Ch. 14 Efﬁcient Implementation Signed-magnitude representation has the drawback that when certain operations (such as addition and subtraction) are performed, the sign digit must be checked to determine the appropriate manner to perform the computation. Conditional branching of this type can be costly when many operations are performed. (ii) Complement representation Addition and subtraction usingcomplement representationdo not require the checking of the sign digit. Non-negative integers in the range[0,bn−1 −1] are represented by baseb sequences of lengthnwith the high-order digit being0. Supposexis a positive integer in this range represented by the sequence(xnxn−1 ···x1x0)b where xn =0 . Then−xis represented by the sequence x=( xn xn−1 ··· x1 x0)+1 where xi = b−1−xi and + is the standard addition with carry. Table 14.1 illustrates the binary complement representation of the integers in the range[−7,7]. In the binary case, complement representation is referred to astwo’ s complement representation. Sequence Signed- Two’s magnitude complement 0111 7 7 0110 6 6 0101 5 5 0100 4 4 0011 3 3 0010 2 2 0001 1 1 0000 0 0 Sequence Signed- Two’s magnitude complement 1111 −7 −1 1110 −6 −2 1101 −5 −3 1100 −4 −4 1011 −3 −5 1010 −2 −6 1001 −1 −7 1000 −0 −8 Table 14.1:Signed-magnitude and two’ s complement representations of integers in[−7, 7]. 14.2.2 Addition and subtraction Addition and subtraction are performed on two integers having the same number of baseb digits. To add or subtract two integers of different lengths, the smaller of the two integers is ﬁrst padded with0’s on the left (i.e., in the high-order positions). 14.7 AlgorithmMultiple-precision addition INPUT: positive integersxand y, each havingn+1 basebdigits. OUTPUT: the sum x+ y=( wn+1wn ···w1w0)b in radixbrepresentation. 1. c←0 (cis thecarrydigit). 2. Forifrom 0 tondo the following: 2.1 wi←(xi + yi + c)m o db. 2.2 If(xi + yi + c) <b thenc←0; otherwisec←1. 3. wn+1←c. 4. Return((wn+1wn ···w1w0)). 14.8 Note (computational efﬁciency) The basebshould be chosen so that(xi + yi + c)m o db can be computed by the hardware on the computing device. Some processors have instruc- tion sets which provide an add-with-carry to facilitate multiple-precision addition. §14.2 Multiple-precision integer arithmetic 595 14.9 AlgorithmMultiple-precision subtraction INPUT: positive integersxand y, each havingn+1 basebdigits, withx≥y. OUTPUT: the differencex−y=( wnwn−1 ···w1w0)b in radixbrepresentation. 1. c←0. 2. Forifrom 0 tondo the following: 2.1 wi←(xi −yi + c)m o db. 2.2 If(xi −yi + c) ≥0 thenc←0; otherwisec←−1. 3. Return((wnwn−1 ···w1w0)). 14.10 Note (eliminating the requirementx ≥ y) If the relative magnitudes of the integersx and yare unknown, then Algorithm 14.9 can be modiﬁed as follows. On termination of the algorithm, ifc = −1, then repeat Algorithm 14.9 withx =( 0 0···00)b and y = (wnwn−1 ···w1w0)b. Conditional checking on the relative magnitudes ofxand ycan also be avoided by using a complement representation (§14.2.1(ii)). 14.11 Example (modiﬁed subtraction) Letx= 3996879and y= 4637923in base10, so that x<y . Table 14.2 shows the steps of the modiﬁed subtraction algorithm (cf. Note 14.10).□ First execution of Algorithm 14.9 i 654 3 210 xi 399 6 879 yi 463 7 923 wi 935 8 956 c −100 −1 −100 Second execution of Algorithm 14.9 i 6543210 xi 0000000 yi 9358956 wi 0641044 c −1 −1 −1 −1 −1 −1 −1 Table 14.2:Modiﬁed subtraction (see Example 14.11). 14.2.3 Multiplication Letxand ybe integers expressed in radixbrepresentation:x=( xnxn−1 ···x1x0)b and y=( ytyt−1 ···y1y0)b. The productx·ywill have at most(n+ t+2 )basebdigits. Al- gorithm 14.12 is a reorganization of the standard pencil-and-paper method taught in grade school. Asingle-precisionmultiplication means the multiplication of two basebdigits. If xj and yi are two basebdigits, thenxj ·yi can be written asxj ·yi =( uv)b, whereuand vare basebdigits, andumay be 0. 14.12 AlgorithmMultiple-precision multiplication INPUT: positive integersxand yhavingn+1 and t+1 basebdigits, respectively. OUTPUT: the productx·y=( wn+t+1 ···w1w0)b in radixbrepresentation. 1. Forifrom 0 to(n+ t+1 )do:wi←0. 2. Forifrom 0 totdo the following: 2.1 c←0. 2.2 Forjfrom 0 tondo the following: Compute (uv)b = wi+j + xj ·yi + c, and setwi+j←v, c←u. 2.3 wi+n+1←u. 3. Return((wn+t+1 ···w1w0)). 596 Ch. 14 Efﬁcient Implementation 14.13 Example (multiple-precision multiplication) Takex = x3x2x1x0 = 9274 and y = y2y1y0 = 847 (base10 representations), so thatn =3 and t =2 . Table 14.3 shows the steps performed by Algorithm 14.12 to computex·y= 7855078. □ ij c wi+j + xj yi + c u v w6 w5 w4 w3 w2 w1 w0 00 0 0+2 8+0 2 8 0 0 0 0 0 0 8 1 2 0+4 9+2 5 1 0 0 0 0 0 1 8 2 5 0+1 4+5 1 9 0 0 0 0 9 1 8 3 1 0+6 3+1 6 4 0 0 6 4 9 1 8 10 0 1+1 6+0 1 7 0 0 6 4 9 7 8 1 1 9+2 8+1 3 8 0 0 6 4 8 7 8 2 3 4+8+3 1 5 0 0 6 5 8 7 8 3 1 6+3 6+1 4 3 0 4 3 5 8 7 8 20 0 8+3 2+0 4 0 0 4 3 5 0 7 8 1 4 5+5 6+4 6 5 0 4 3 5 0 7 8 2 6 3+1 6+6 2 5 0 4 5 5 0 7 8 3 2 4+7 2+2 7 8 7 8 5 5 0 7 8 Table 14.3:Multiple-precision multiplication (see Example 14.13). 14.14 Remark (pencil-and-paper method) The pencil-and-paper method for multiplyingx = 9274 and y= 847would appear as 9 274 × 847 6 4 918 (row 1) 37 0 96 (row 2) 741 9 2 (row 3) 785 5 078 The shaded entries in Table 14.3 correspond to row 1, row 1 + row 2, and row 1 + row 2 + row 3, respectively. 14.15 Note (computational efﬁciency of Algorithm 14.12) (i) The computationally intensive portion of Algorithm 14.12 is step 2.2. Computing wi+j + xj ·yi + cis called theinner-product operation. Sincewi+j,xj,yi and c are all basebdigits, the result of an inner-product operation is at most(b−1) + (b− 1)2 +( b−1) =b2 −1 and, hence, can be represented by two basebdigits. (ii) Algorithm 14.12 requires(n+1 ) (t+1 )single-precision multiplications. (iii) It is assumed in Algorithm 14.12 that single-precision multiplications are part of the instruction set on a processor. The quality of the implementation of this instruction is crucial to an efﬁcient implementation of Algorithm 14.12. 14.2.4 Squaring In the preceding algorithms,(uv)b has bothuand vas single-precision integers. This nota- tion is abused in this subsection by permittinguto be a double-precision integer, such that 0 ≤u≤2(b−1). The valuevwill always be single-precision. §14.2 Multiple-precision integer arithmetic 597 14.16 AlgorithmMultiple-precision squaring INPUT: positive integerx=( xt−1xt−2 ···x1x0)b. OUTPUT: x·x= x2 in radixbrepresentation. 1. Forifrom 0 to(2t−1) do:wi←0. 2. Forifrom 0 to(t−1) do the following: 2.1 (uv)b←w2i + xi ·xi, w2i←v, c←u. 2.2 Forjfrom (i+1 )to(t−1) do the following: (uv)b←wi+j +2 xj ·xi + c, wi+j←v, c←u. 2.3 wi+t←u. 3. Return((w2t−1w2t−2 ...w1w0)b). 14.17 Note (computational efﬁciency of Algorithm 14.16) (i) (overﬂow) In step 2.2,ucan be larger than a single-precision integer. Sincewi+j is always set tov,wi+j ≤ b−1.I fc ≤ 2(b−1), thenwi+j +2 xjxi + c ≤ (b−1) + 2(b−1)2 +2 (b−1) = (b−1)(2b+1 ), implying0 ≤u≤2(b−1). This value ofumay exceed single-precision, and must be accommodated. (ii) (number of operations) The computationally intensive part of the algorithm is step 2. The number of single-precision multiplications is about(t2 + t)/2, discounting the multiplication by2. This is approximately one half of the single-precision multipli- cations required by Algorithm 14.12 (cf. Note 14.15(ii)). 14.18 Note (squaring vs. multiplication in general) Squaring a positive integerx(i.e., computing x2) can at best be no more than twice as fast as multiplying distinct integersxand y.T o see this, consider the identityxy=( (x+ y)2 −(x−y)2)/4. Hence,x·ycan be computed with two squarings (i.e.,(x+ y)2 and (x−y)2). Of course, a speed-up by a factor of2 can be signiﬁcant in many applications. 14.19 Example (squaring) Table 14.4 shows the steps performed by Algorithm 14.16 in squar- ingx= 989. Here,t=3 and b=1 0. □ ij w2i + x2 i wi+j +2 xj xi + c u v w5 w4 w3 w2 w1 w0 0 − 0+8 1 − 8 1 0 0 0 0 0 1 1 − 0+2 ·8 ·9+8 15 2 0 0 0 0 2 1 2 − 0+2 ·9 ·9+1 5 17 7 0 0 0 7 2 1 17 7 0 0 17 7 2 1 1 − 7+6 4 − 7 1 0 0 17 1 2 1 2 − 17 + 2·9 ·8+7 16 8 0 0 8 1 2 1 16 8 0 16 8 1 2 1 2 − 16 + 81 − 9 7 0 7 8 1 2 1 9 7 9 7 8 1 2 1 Table 14.4:Multiple-precision squaring (see Example 14.19). 598 Ch. 14 Efﬁcient Implementation 14.2.5 Division Division is the most complicated and costly of the basic multiple-precision operations. Al- gorithm 14.20 computes the quotientqand remainderrin radixbrepresentation whenxis divided byy. 14.20 AlgorithmMultiple-precision division INPUT: positive integersx=( xn ···x1x0)b,y=( yt ···y1y0)b withn≥t≥1,yt ̸=0. OUTPUT: the quotientq =( qn−t ···q1q0)b and remainderr=( rt ···r1r0)b such that x= qy+ r,0 ≤r<y . 1. Forjfrom 0 to(n−t) do:qj←0. 2. While(x≥ybn−t) do the following:qn−t←qn−t +1 , x←x−ybn−t. 3. Forifrom ndown to(t+1 )do the following: 3.1 Ifxi = yt then setqi−t−1←b−1; otherwise setqi−t−1←⌊(xib+ xi−1)/yt)⌋. 3.2 While(qi−t−1(ytb+ yt−1) >xib2 + xi−1b+ xi−2) do:qi−t−1←qi−t−1 −1. 3.3 x←x−qi−t−1ybi−t−1. 3.4 Ifx< 0 then setx←x+ ybi−t−1 and qi−t−1←qi−t−1 −1. 4. r←x. 5. Return(q,r). 14.21 Example (multiple-precision division) Letx= 721948327,y= 84461, so thatn=8 and t=4 . Table 14.5 illustrates the steps in Algorithm 14.20. The last row gives the quotient q= 8547and the remainderr= 60160. □ i q4 q3 q2 q1 q0 x8 x7 x6 x5 x4 x3 x2 x1 x0 – 00000 721948327 8 09000 721948327 8000 46260327 7 8500 4029827 6 8550 4029827 8540 651387 5 8548 651387 8547 60160 Table 14.5:Multiple-precision division (see Example 14.21). 14.22 Note (comments on Algorithm 14.20) (i) Step 2 of Algorithm 14.20 is performed at most once ifyt ≥⌊b 2 ⌋and bis even. (ii) The conditionn≥t≥1 can be replaced byn≥t≥0, provided one takesxj = yj =0 whenever a subscriptj< 0 in encountered in the algorithm. 14.23 Note (normalization) The estimate for the quotient digitqi−t−1 in step 3.1 of Algorithm 14.20 is never less than the true value of the quotient digit. Furthermore, ifyt ≥⌊b 2 ⌋, then step 3.2 is repeated no more than twice. If step 3.1 is modiﬁed so thatqi−t−1←⌊(xib2 + xi−1b+ xi−2)/(ytb+ yt−1)⌋, then the estimate is almost always correct and step 3.2 is §14.3 Multiple-precision modular arithmetic 599 never repeated more than once. One can always guarantee thatyt ≥⌊b 2 ⌋by replacing the integersx,yby λx,λyfor some suitable choice ofλ. The quotient ofλxdivided byλyis the same as that ofxby y; the remainder isλtimes the remainder ofxdivided byy. If the basebis a power of2 (as in many applications), then the choice ofλshould be a power of2; multiplication byλis achieved by simply left-shifting the binary representations ofxand y. Multiplying by a suitable choice ofλto ensure thatyt ≥⌊b 2 ⌋is callednormalization. Example 14.24 illustrates the procedure. 14.24 Example (normalized division) Takex= 73418 and y= 267. Normalizexand yby multiplying each byλ=3 : x′=3 x= 220254 and y′=3 y= 801. Table 14.6 shows the steps of Algorithm 14.20 as applied tox′and y′. When x′is divided byy′, the quotient is274, and the remainder is780. When xis divided byy, the quotient is also274 and the remainder is780/3 = 260. □ i q3 q2 q1 q0 x5 x4 x3 x2 x1 x0 − 0000 220254 5 0200 60054 4 270 3984 3 274 780 Table 14.6:Multiple-precision division after normalization (see Example 14.24). 14.25 Note (computational efﬁciency of Algorithm 14.20 with normalization) (i)(multiplication count)Assuming that normalization extends the number of digits in xby 1, each iteration of step 3 requires1+( t+2 )= t+3 single-precision multi- plications. Hence, Algorithm 14.20 with normalization requires about(n−t)(t+3 ) single-precision multiplications. (ii)(division count)Since step 3.1 of Algorithm 14.20 is executedn−ttimes, at most n−tsingle-precision divisions are required when normalization is used. 14.3 Multiple-precision modular arithmetic §14.2 provided methods for carrying out the basic operations (addition, subtraction, multi- plication, squaring, and division) with multiple-precision integers. This section deals with these operations inZm, the integers modulom, wheremis a multiple-precision positive integer. (See§2.4.3 for deﬁnitions ofZm and related operations.) Let m=( mnmn−1 ···m1m0)b be a positive integer in radixbrepresentation. Let x=( xnxn−1 ···x1x0)b and y=( ynyn−1 ···y1y0)b be non-negative integers in baseb representation such thatx<m and y<m . Methods described in this section are for computingx+ ymod m(modular addition),x−ymod m(modular subtraction), and x·ymod m(modular multiplication). Computingx−1 mod m(modular inversion) is ad- dressed in§14.4.3. 14.26 DeﬁnitionIfzis any integer, thenzmod m(the integer remainder in the range[0,m−1] afterzis divided bym) is called themodular reductionofzwith respect to modulusm. 600 Ch. 14 Efﬁcient Implementation Modular addition and subtraction As is the case for ordinary multiple-precision operations, addition and subtraction are the simplest to compute of the modular operations. 14.27 FactLetxand ybe non-negative integers withx,y<m . Then: (i)x+ y< 2m; (ii) ifx≥y, then0 ≤x−y<m ; and (iii) ifx<y , then0 ≤x+ m−y<m . Ifx,y∈Zm, then modular addition can be performed by using Algorithm 14.7 to add xand yas multiple-precision integers, with the additional step of subtractingmif (and only if)x+ y≥m. Modular subtraction is precisely Algorithm 14.9, providedx≥y. 14.3.1 Classical modular multiplication Modular multiplication is more involved than multiple-precision multiplication (§14.2.3), requiring both multiple-precision multiplication and some method for performing modular reduction (Deﬁnition 14.26). The most straightforward method for performing modular re- duction is to compute the remainder on division bym, using a multiple-precision division algorithm such as Algorithm 14.20; this is commonly referred to as theclassical algorithm for performing modular multiplication. 14.28 AlgorithmClassical modular multiplication INPUT: two positive integersx,yand a modulusm, all in radixbrepresentation. OUTPUT: x·ymod m. 1. Compute x·y(using Algorithm 14.12). 2. Compute the remainderrwhen x·yis divided bym(using Algorithm 14.20). 3. Return(r). 14.3.2 Montgomery reduction Montgomery reduction is a technique which allows efﬁcient implementation of modular multiplication without explicitly carrying out the classical modular reduction step. Letmbe a positive integer, and letRandTbe integers such thatR>m, gcd(m,R)= 1, and0 ≤T<mR . A method is described for computingTR−1 mod mwithout using the classical method of Algorithm 14.28.TR−1 mod mis called aMontgomery reduction ofTmodulo mwith respect toR. With a suitable choice ofR, a Montgomery reduction can be efﬁciently computed. Suppose xand yare integers such that0 ≤ x,y < m. Let˜x = xRmod mand ˜y= yRmod m. The Montgomery reduction of˜x˜yis˜x˜yR−1 mod m= xyRmod m. This observation is used in Algorithm 14.94 to provide an efﬁcient method for modular exponentiation. To brieﬂy illustrate, consider computingx 5 mod mfor some integerx,1 ≤x<m . First compute˜x= xRmod m. Then compute the Montgomery reduction of˜x˜x, which is A= ˜x2R−1 mod m. The Montgomery reduction ofA2 isA2R−1 mod m= ˜x4R−3 mod m. Finally, the Montgomery reduction of(A2R−1 mod m)˜xis(A2R−1)˜xR−1 mod m= ˜x5R−4 mod m = x5Rmod m. Multiplying this value byR−1 mod mand reducing §14.3 Multiple-precision modular arithmetic 601 modulo mgivesx5 mod m. Provided that Montgomery reductions are more efﬁcient to compute than classical modular reductions, this method may be more efﬁcient than com- putingx5 mod mby repeated application of Algorithm 14.28. Ifmis represented as a basebinteger of lengthn, then a typical choice forRisbn. The conditionR>m is clearly satisﬁed, butgcd(R,m)=1 will hold only ifgcd(b,m)=1 . Thus, this choice ofRis not possible for all moduli. For those moduli of practical interest (such as RSA moduli),mwill be odd; thenbcan be a power of2 and R= bn will sufﬁce. Fact 14.29 is basic to the Montgomery reduction method. Note 14.30 then implies that R= bn is sufﬁcient (but not necessary) for efﬁcient implementation. 14.29 Fact(Montgomery reduction) Given integersmand Rwhere gcd(m,R)=1 , letm′= −m−1 mod R, and letTbe any integer such that0 ≤T<m R.I fU = Tm′mod R, then(T+ Um)/Ris an integer and(T+ Um)/R≡TR−1 (mod m). Justiﬁcation.T+ Um≡T (mod m) and, hence,(T+ Um)R−1 ≡TR−1 (mod m). To see that(T+ Um)R−1 is an integer, observe thatU= Tm′+ kRand m′m= −1+ lR for some integerskand l. It follows that(T+ Um)/R=( T+( Tm′+ kR)m)/R= (T+ T(−1+ lR)+ kRm)/R= lT+ km. 14.30 Note (implications of Fact 14.29) (i)(T+ Um)/Ris an estimate forTR−1 mod m. SinceT<m Rand U<R , then (T+Um)/R< (mR+mR)/R=2 m. Thus either(T+Um)/R= TR−1 mod m or(T+Um)/R=( TR−1 mod m)+m(i.e., the estimate is withinmof the residue). Example 14.31 illustrates that both possibilities can occur. (ii) If all integers are represented in radixband R = bn, thenTR−1 mod mcan be computed with two multiple-precision multiplications (i.e.,U= T·m′and U·m) and simple right-shifts ofT+ Umin order to divide byR. 14.31 Example (Montgomery reduction) Letm= 187,R = 190. ThenR−1 mod m= 125, m−1 mod R =6 3, andm′= 127.I fT = 563, thenU = Tm′mod R =6 1 and (T+ Um)/R=6 3= TR−1 mod m.I fT = 1125 thenU= Tm′mod R= 185 and (T+ Um)/R= 188 = (TR−1 mod m)+ m. □ Algorithm 14.32 computes the Montgomery reduction ofT=( t2n−1 ···t1t0)b when R= bn and m=( mn−1 ···m1m0)b. The algorithm makes implicit use of Fact 14.29 by computing quantities which have similar properties toU= Tm′mod Rand T+ Um, although the latter two expressions are not computed explicitly. 14.32 AlgorithmMontgomery reduction INPUT: integersm=( mn−1 ···m1m0)b withgcd(m,b)=1 ,R= bn,m′= −m−1 mod b, andT=( t2n−1 ···t1t0)b <mR. OUTPUT: TR−1 mod m. 1. A←T. (Notation:A=( a2n−1 ···a1a0)b.) 2. Forifrom 0 to(n−1) do the following: 2.1 ui←aim′mod b. 2.2 A←A+ uimbi. 3. A←A/bn. 4. IfA≥mthenA←A−m. 5. Return(A). 602 Ch. 14 Efﬁcient Implementation 14.33 Note (comments on Montgomery reduction) (i) Algorithm 14.32 does not requirem′= −m−1 mod R, as Fact 14.29 does, but rather m′= −m−1 mod b. This is due to the choice ofR= bn. (ii) At step 2.1 of the algorithm withi= l,Ahas the property thataj =0 ,0 ≤j≤l−1. Step 2.2 does not modify these values, but does replaceal by 0. It follows that in step 3,Ais divisible bybn. (iii) Going into step 3, the value ofAequalsTplus some multiple ofm(see step 2.2); hereA=( T+ km)/bn is an integer (see (ii) above) andA≡TR−1 (mod m).I t remains to show thatAis less than2m, so that at step 4, a subtraction (rather than a division) will sufﬁce. Going into step 3,A= T+∑ n−1 i=0 uibim. But∑ n−1 i=0 uibim< bnm= Rmand T<Rm ; hence,A< 2Rm. Going into step 4 (after division ofA by R),A< 2mas required. 14.34 Note (computational efﬁciency of Montgomery reduction) Step 2.1 and step 2.2 of Algo- rithm 14.32 require a total ofn+1 single-precision multiplications. Since these steps are executedntimes, the total number of single-precision multiplications isn(n+1 ). Algo- rithm 14.32 does not require any single-precision divisions. 14.35 Example (Montgomery reduction) Letm= 72639,b=1 0,R=1 05, andT= 7118368. Heren=5 ,m′= −m−1 mod 10 = 1,Tmod m= 72385, andTR−1 mod m= 39796. Table 14.7 displays the iterations of step 2 in Algorithm 14.32. □ i ui = aim′mod 10 uimbi A − − − 7118368 0 8 581112 7699480 1 8 5811120 13510600 2 6 43583400 57094000 3 4 290556000 347650000 4 5 3631950000 3979600000 Table 14.7:Montgomery reduction algorithm (see Example 14.35). Montgomery multiplication Algorithm 14.36 combines Montgomery reduction (Algorithm 14.32) and multiple-precis- ion multiplication (Algorithm 14.12) to compute the Montgomery reduction of the product of two integers. 14.36 AlgorithmMontgomery multiplication INPUT: integersm=( mn−1 ···m1m0)b,x=( xn−1 ···x1x0)b,y=( yn−1 ···y1y0)b with0 ≤x,y<m ,R= bn withgcd(m,b)=1 , andm′= −m−1 mod b. OUTPUT: xyR−1 mod m. 1. A←0. (Notation:A=( anan−1 ···a1a0)b.) 2. Forifrom 0 to(n−1) do the following: 2.1 ui←(a0 + xiy0)m′mod b. 2.2 A←(A+ xiy+ uim)/b. 3. IfA≥mthenA←A−m. 4. Return(A). §14.3 Multiple-precision modular arithmetic 603 14.37 Note (partial justiﬁcation of Algorithm 14.36) Suppose at theith iteration of step 2 that 0 ≤A< 2m−1. Step 2.2 replacesAwith(A+ xiy+ uim)/b; but(A+ xiy+ uim)/b≤ (2m−2+( b−1)(m−1) + (b−1)m)/b=2 m−1 −(1/b). Hence,A< 2m−1, justifying step 3. 14.38 Note (computational efﬁciency of Algorithm 14.36) SinceA+ xiy+ uimis a multiple of b, only a right-shift is required to perform a division bybin step 2.2. Step 2.1 requires two single-precision multiplications and step 2.2 requires2n. Since step 2 is executedntimes, the total number of single-precision multiplications isn(2 + 2n)=2 n(n+1 ). 14.39 Note (computingxymod mwith Montgomery multiplication) Supposex,y, andmare n-digit basebintegers with0 ≤x,y<m . Neglecting the cost of the precomputation in the input, Algorithm 14.36 computesxyR−1 mod mwith2n(n+1) single-precision mul- tiplications. Neglecting the cost to computeR2 mod mand applying Algorithm 14.36 to xyR−1 mod mand R2 mod m,xymod mis computed in4n(n+1) single-precision op- erations. Using classical modular multiplication (Algorithm 14.28) would require2n(n+1) single-precision operations and no precomputation. Hence, the classical algorithm is supe- rior for doing a single modular multiplication; however, Montgomery multiplication is very effective for performing modular exponentiation (Algorithm 14.94). 14.40 Remark (Montgomery reduction vs. Montgomery multiplication) Algorithm 14.36 (Mont- gomery multiplication) takes as input twon-digit numbers and then proceeds to interleave the multiplication and reduction steps. Because of this, Algorithm 14.36 is not able to take advantage of the special case where the input integers are equal (i.e., squaring). On the other hand, Algorithm 14.32 (Montgomery reduction) assumes as input the product of two inte- gers, each of which has at mostndigits. Since Algorithm 14.32 is independent of multiple- precision multiplication, a faster squaring algorithm such as Algorithm 14.16 may be used prior to the reduction step. 14.41 Example (Montgomery multiplication) In Algorithm 14.36, letm= 72639,R=1 0 5, x= 5792,y= 1229. Heren=5 ,m′= −m−1 mod 10 = 1, andxyR−1 mod m= 39796. Notice thatmand Rare the same values as in Example 14.35, as isxy= 7118368. Table 14.8 displays the steps in Algorithm 14.36. □ i xi xiy0 ui xiy uim A 0 2 18 8 2458 581112 58357 1 9 81 8 11061 581112 65053 2 7 63 6 8603 435834 50949 3 5 45 4 6145 290556 34765 4 0 0 5 0 363195 39796 Table 14.8:Montgomery multiplication (see Example 14.41). 14.3.3 Barrett reduction Barrett reduction (Algorithm 14.42) computesr= xmod mgivenxandm. The algorithm requires the precomputation of the quantityµ= ⌊b2k/m⌋; it is advantageous if many reduc- tions are performed with a single modulus. For example, each RSA encryption for one en- tity requires reduction modulo that entity’s public key modulus. The precomputation takes 604 Ch. 14 Efﬁcient Implementation a ﬁxed amount of work, which is negligible in comparison to modular exponentiation cost. Typically, the radixbis chosen to be close to the word-size of the processor. Hence, assume b> 3 in Algorithm 14.42 (see Note 14.44 (ii)). 14.42 AlgorithmBarrett modular reduction INPUT: positive integersx=( x2k−1 ···x1x0)b,m=( mk−1 ···m1m0)b (withmk−1 ̸= 0), andµ= ⌊b2k/m⌋. OUTPUT: r= xmod m. 1. q1←⌊x/bk−1⌋, q2←q1 ·µ, q3←⌊q2/bk+1⌋. 2. r1←xmod bk+1, r2←q3 ·mmod bk+1, r←r1 −r2. 3. Ifr< 0 thenr←r+ bk+1. 4. Whiler≥mdo:r←r−m. 5. Return(r). 14.43 FactBy the division algorithm (Deﬁnition 2.82), there exist integersQand Rsuch that x= Qm+ Rand 0 ≤R<m . In step 1 of Algorithm 14.42, the following inequality is satisﬁed:Q−2 ≤q3 ≤Q. 14.44 Note (partial justiﬁcation of correctness of Barrett reduction) (i) Algorithm 14.42 is based on the observation that⌊x/m⌋can be written asQ = ⌊(x/bk−1)(b2k/m)(1/bk+1)⌋. Moreover,Qcan be approximated by the quantity q3 = ⌊ ⌊x/bk−1⌋µ/bk+1⌋ . Fact 14.43 guarantees thatq3 is never larger than the true quotientQ, and is at most2 smaller. (ii) In step 2, observe that−bk+1 <r 1 −r2 <b k+1,r1 −r2 ≡ (Q−q3)m+ R (mod bk+1), and0 ≤(Q−q3)m+ R< 3m<b k+1 sincem<b k and 3 <b.I f r1 −r2 ≥0, thenr1 −r2 =( Q−q3)m+ R.I fr1 −r2 <0, thenr1 −r2 + bk+1 = (Q−q3)m+ R. In either case, step 4 is repeated at most twice since0 ≤r< 3m. 14.45 Note (computational efﬁciency of Barrett reduction) (i) All divisions performed in Algorithm 14.42 are simple right-shifts of the basebrep- resentation. (ii)q2 is only used to computeq3. Since thek+1 least signiﬁcant digits ofq2 are not needed to determineq3, only a partial multiple-precision multiplication (i.e.,q1 ·µ) is necessary. The only inﬂuence of thek+1 least signiﬁcant digits on the higher order digits is the carry from positionk+1 to positionk+2 . Provided the baseb is sufﬁciently large with respect tok, this carry can be accurately computed by only calculating the digits at positionskandk+1.1 Hence, thek−1 least signiﬁcant digits ofq2 need not be computed. Sinceµand q1 have at mostk+1 digits, determiningq3 requires at most(k+1 )2 − (k 2 ) = (k2 +5 k+2 )/2 single-precision multiplications. (iii) In step 2 of Algorithm 14.42,r2 can also be computed by a partial multiple-precision multiplication which evaluates only the least signiﬁcantk+1 digits ofq3 ·m. This can be done in at most (k+1 2 ) + ksingle-precision multiplications. 14.46 Example (Barrett reduction) Letb=4 ,k=3 ,x= (313221)b, andm= (233)b (i.e., x= 3561 and m=4 7). Thenµ= ⌊46/m⌋= 87 = (1113)b,q1 = ⌊(313221)b/42⌋= (3132)b,q2 = (3132)b ·(1113)b = (10231302)b,q3 = (1023)b,r1 = (3221)b,r2 = (1023)b ·(233)b mod b4 = (3011)b, andr= r1 −r2 = (210)b. Thusxmod m=3 6. □ 1Ifb>k , then the carry computed by simply considering the digits at positionk − 1 (and ignoring the carry from positionk − 2) will be in error by at most 1. §14.3 Multiple-precision modular arithmetic 605 14.3.4 Reduction methods for moduli of special form When the modulus has a special (customized) form, reduction techniques can be employed to allow more efﬁcient computation. Suppose that the modulusmis at-digit basebpositive integer of the formm = bt −c,where cis anl-digit basebpositive integer (for some l<t ). Algorithm 14.47 computesxmod mfor any positive integerxby using only shifts, additions, and single-precision multiplications of basebnumbers. 14.47 AlgorithmReduction modulom = bt −c INPUT: a baseb, positive integerx, and a modulusm= bt −c, wherecis anl-digit base binteger for somel<t . OUTPUT: r= xmod m. 1. q0←⌊x/bt⌋, r0←x−q0bt, r←r0, i←0. 2. Whileqi >0 do the following: 2.1 qi+1←⌊qic/bt⌋, ri+1←qic−qi+1bt. 2.2 i←i+1 , r←r+ ri. 3. Whiler≥mdo:r←r−m. 4. Return(r). 14.48 Example (reduction modulobt −c) Letb=4 ,m= 935 = (32213)4, andx= 31085 = (13211231)4. Sincem =4 5 −(1121)4, takec = (1121)4. Heret =5 and l =4 . Table 14.9 displays the quotients and remainders produced by Algorithm 14.47. At the be- ginning of step 3,r= (102031)4. Sincer>m , step 3 computesr−m= (3212)4. □ i qi−1c qi ri r 0 – (132)4 (11231)4 (11231)4 1 (221232)4 (2)4 (21232)4 (33123)4 2 (2302)4 (0)4 (2302)4 (102031)4 Table 14.9:Reduction modulom = bt −c (see Example 14.48). 14.49 Fact(termination) For some integers≥0,qs =0 ; hence, Algorithm 14.47 terminates. Justiﬁcation.qic= qi+1bt +ri+1,i≥0. Sincec<b t,qi =( qi+1bt/c)+(ri+1/c) >qi+1. Since theqi’s are non-negative integers which strictly decrease asiincreases, there is some integers≥0 such thatqs =0 . 14.50 Fact(correctness) Algorithm 14.47 terminates with the correct residue modulom. Justiﬁcation.Suppose thatsis the smallest indexifor whichqi =0 (i.e.,qs =0 ). Now, x= q0bt + r0 and qic= qi+1bt + ri+1,0 ≤i≤s−1. Adding these equations gives x+ (∑ s−1 i=0 qi ) c= (∑ s−1 i=0 qi ) bt + ∑ s i=0 ri. Sincebt ≡c(mod m), it follows that x≡∑ s i=0 ri (mod m). Hence, repeated subtraction ofmfrom r= ∑ s i=0 ri gives the correct residue. 606 Ch. 14 Efﬁcient Implementation 14.51 Note (computational efﬁciency of reduction modulobt −c) (i) Suppose thatxhas2tbasebdigits. Ifl≤t/2, then Algorithm 14.47 executes step 2 at mosts =3 times, requiring2 multiplications byc. In general, iflis approxi- mately(s−2)t/(s−1), then Algorithm 14.47 executes step 2 aboutstimes. Thus, Algorithm 14.47 requires aboutslsingle-precision multiplications. (ii) Ifchas few non-zero digits, then multiplication bycwill be relatively inexpensive. Ifcis large but has few non-zero digits, the number of iterations of Algorithm 14.47 will be greater, but each iteration requires a very simple multiplication. 14.52 Note (modiﬁcations) Algorithm 14.47 can be modiﬁed ifm= bt + cfor some positive integerc<b t: in step 2.2, replacer←r+ ri withr←r+( −1)iri. 14.53 Remark (using moduli of a special form) Selecting RSA moduli of the formbt ±cfor small values ofclimits the choices of primespand q. Care must also be exercised when selecting moduli of a special form, so that factoring is not made substantially easier; this is because numbers of this form are more susceptible to factoring by the special number ﬁeld sieve (see§3.2.7). A similar statement can be made regarding the selection of primes of a special form for cryptographic schemes based on the discrete logarithm problem. 14.4 Greatest common divisor algorithms Many situations in cryptography require the computation of the greatest common divisor (gcd) of two positive integers (see Deﬁnition 2.86). Algorithm 2.104 describes the classical Euclidean algorithm for this computation. For multiple-precision integers, Algorithm 2.104 requires a multiple-precision division at step 1.1 which is a relatively expensive operation. This section describes three methods for computing the gcd which are more efﬁcient than the classical approach using multiple-precision numbers. The ﬁrst is non-Euclidean and is referred to as thebinary gcd algorithm(§14.4.1). Although it requires more steps than the classical algorithm, the binary gcd algorithm eliminates the computationally expen- sive division and replaces it with elementary shifts and additions. Lehmer’s gcd algorithm (§14.4.2) is a variant of the classical algorithm more suited to multiple-precision computa- tions. A binary version of the extended Euclidean algorithm is given in§14.4.3. 14.4.1 Binary gcd algorithm 14.54 AlgorithmBinary gcd algorithm INPUT: two positive integersxand ywithx≥y. OUTPUT: gcd(x,y). 1. g←1. 2. While bothxand yare even do the following:x←x/2, y←y/2, g←2g. 3. Whilex̸=0do the following: 3.1 Whilexis even do:x←x/2. 3.2 Whileyis even do:y←y/2. 3.3 t←\|x−y\|/2. 3.4 Ifx≥ythenx←t; otherwise,y←t. 4. Return(g·y). §14.4 Greatest common divisor algorithms 607 14.55 Example (binary gcd algorithm) The following table displays the steps performed by Al- gorithm 14.54 for computinggcd(1764,868) = 28. □ x 1764 441 112 7 7 7 7 7 0 y 868 217 217 217 105 49 21 7 7 g 1 4 4 4 4 4 4 4 4 14.56 Note (computational efﬁciency of Algorithm 14.54) (i) Ifxandyare in radix2 representation, then the divisions by2 are simply right-shifts. (ii) Step 3.3 for multiple-precision integers can be computed using Algorithm 14.9. 14.4.2 Lehmer’s gcd algorithm Algorithm 14.57 is a variant of the classical Euclidean algorithm (Algorithm 2.104) and is suited to computations involving multiple-precision integers. It replaces many of the multiple-precision divisions by simpler single-precision operations. Letxand ybe positive integers in radixbrepresentation, withx≥y. Without loss of generality, assume thatxand yhave the same number of basebdigits throughout Algo- rithm 14.57; this may necessitate padding the high-order digits ofywith0’s. 14.57 AlgorithmLehmer’s gcd algorithm INPUT: two positive integersxand yin radixbrepresentation, withx≥y. OUTPUT: gcd(x,y). 1. Whiley≥bdo the following: 1.1 Set˜x,˜yto be the high-order digit ofx,y, respectively (˜ycould be0). 1.2 A←1, B←0, C←0, D←1. 1.3 While(˜y+ C) ̸=0and (˜y+ D) ̸=0do the following: q←⌊(˜x+ A)/(˜y+ C)⌋, q′←⌊(˜x+ B)/(˜y+ D)⌋. Ifq̸=q′then go to step 1.4. t←A−qC, A←C, C←t, t←B−qD, B←D, D←t. t←˜x−q˜y, ˜x←˜y, ˜y←t. 1.4 IfB=0 , thenT←xmod y, x←y, y←T; otherwise,T←Ax+ By, u←Cx+ Dy, x←T, y←u. 2. Compute v=g c d (x,y) using Algorithm 2.104. 3. Return(v). 14.58 Note (implementation notes for Algorithm 14.57) (i)Tis a multiple-precision variable.A,B,C,D, andtare signed single-precision variables; hence, one bit of each of these variables must be reserved for the sign. (ii) The ﬁrst operation of step 1.3 may result in overﬂow since0 ≤˜x+ A,˜y+ D≤b. This possibility needs to be accommodated. One solution is to reserve two bits more than the number of bits in a digit for each of˜xand ˜yto accommodate both the sign and the possible overﬂow. (iii) The multiple-precision additions of step 1.4 are actually subtractions, sinceAB≤0 and CD≤0. 608 Ch. 14 Efﬁcient Implementation 14.59 Note (computational efﬁciency of Algorithm 14.57) (i) Step 1.3 attempts to simulate multiple-precision divisions by much simpler single- precision operations. In each iteration of step 1.3, all computations are single preci- sion. The number of iterations of step 1.3 depends onb. (ii) The modular reduction in step 1.4 is a multiple-precision operation. The other op- erations are multiple-precision, but require only linear time since the multipliers are single precision. 14.60 Example (Lehmer’ s gcd algorithm) Letb=1 0 3,x= 768 454 923, andy= 542 167 814. Sinceb=1 03, the high-order digits ofxand yare˜x= 768 and ˜y= 542, respectively. Table 14.10 displays the values of the variables at various stages of Algorithm 14.57. The single-precision computations (Step 1.3) whenq = q ′are shown in Table 14.11. Hence gcd(x,y)=1 . □ 14.4.3 Binary extended gcd algorithm Given integersxand y, Algorithm 2.107 computes integersaand bsuch thatax+ by= v, where v=g c d (x,y). It has the drawback of requiring relatively costly multiple-precision divisions whenxand yare multiple-precision integers. Algorithm 14.61 eliminates this requirement at the expense of more iterations. 14.61 AlgorithmBinary extended gcd algorithm INPUT: two positive integersxand y. OUTPUT: integersa,b, andvsuch thatax+ by= v, wherev=g c d (x,y). 1. g←1. 2. Whilexand yare both even, do the following:x←x/2, y←y/2, g←2g. 3. u←x, v←y, A←1, B←0, C←0, D←1. 4. Whileuis even do the following: 4.1 u←u/2. 4.2 IfA≡B≡0( m o d2 )thenA←A/2, B←B/2; otherwise,A←(A+ y)/2, B←(B−x)/2. 5. Whilevis even do the following: 5.1 v←v/2. 5.2 IfC≡D≡0( m o d2 )then C←C/2, D←D/2; otherwise,C←(C+ y)/2, D←(D−x)/2. 6. Ifu≥vthenu←u−v, A←A−C,B←B−D; otherwise,v←v−u, C←C−A, D←D−B. 7. Ifu=0 , thena←C, b←D, and return(a,b,g ·v); otherwise, go to step 4. 14.62 Example (binary extended gcd algorithm) Letx= 693 and y= 609. Table 14.12 dis- plays the steps in Algorithm 14.61 for computing integersa,b,vsuch that693a+609b= v, where v= gcd(693,609). The algorithm returnsv=2 1,a= −181, andb= 206. □ §14.4 Greatest common divisor algorithms 609 x y q q′ precision reference 768 454 923 542 167 814 1 1 single Table 14.11(i) 89 593 596 47 099 917 1 1 single Table 14.11(ii) 42 493 679 4 606 238 10 8 multiple 4 606 238 1 037 537 5 2 multiple 1 037 537 456 090 – – multiple 456 090 125 357 3 3 single Table 14.11(iii) 34 681 10 657 3 3 single Table 14.11(iv) 10 657 2 710 5 3 multiple 2 710 2 527 1 0 multiple 2 527 183 Algorithm 2.104 183 148 Algorithm 2.104 148 35 Algorithm 2.104 35 8 Algorithm 2.104 8 3 Algorithm 2.104 3 2 Algorithm 2.104 2 1 Algorithm 2.104 1 0 Algorithm 2.104 Table 14.10:Lehmer’ s gcd algorithm (see Example 14.60). ˜x ˜y A B C D qq ′ (i) 768 542 1 0 0 1 11 542 226 0 1 1 −1 22 226 90 1 −1 −2 3 22 90 46 −2 3 5 −7 12 (ii) 89 47 1 0 0 1 11 47 42 0 1 1 −1 11 42 5 1 −1 −1 2 10 5 (iii) 456 125 1 0 0 1 33 125 81 0 1 1 −3 11 81 44 1 −3 −1 4 11 44 37 −1 4 2 −7 11 37 7 2 −7 −3 11 91 (iv) 34 10 1 0 0 1 33 10 4 0 1 1 −3 21 1 Table 14.11:Single-precision computations (see Example 14.60 and Table 14.10). 610 Ch. 14 Efﬁcient Implementation u v A B C D 693 609 1 0 0 1 84 609 1 −1 0 1 42 609 305 −347 0 1 21 609 457 −520 0 1 21 588 457 −520 −457 521 21 294 457 −520 76 −86 21 147 457 −520 38 −43 21 126 457 −520 −419 477 21 63 457 −520 95 −108 21 42 457 −520 −362 412 21 21 457 −520 −181 206 0 21 638 −726 −181 206 Table 14.12:The binary extended gcd algorithm withx = 693,y = 609(see Example 14.62). 14.63 Note (computational efﬁciency of Algorithm 14.61) (i) The only multiple-precision operations needed for Algorithm 14.61 are addition and subtraction. Division by2 is simply a right-shift of the binary representation. (ii) The number of bits needed to represent eitheruorvdecreases by (at least)1, after at most two iterations of steps 4 – 7; thus, the algorithm takes at most2(⌊lg x⌋+⌊lg y⌋+ 2) such iterations. 14.64 Note (multiplicative inverses) Given positive integersmand a, it is often necessary to ﬁnd an integerz∈Zm such thataz≡1( m o dm), if such an integer exists.zis called the multiplicative inverse ofamodulo m(see Deﬁnition 2.115). For example, construct- ing the private key for RSA requires the computation of an integerdsuch thated ≡ 1 (mod (p−1)(q−1)) (see Algorithm 8.1). Algorithm 14.61 provides a computation- ally efﬁcient method for determiningzgivenaand m, by settingx= mand y= a.I f gcd(x,y)=1 , then, at termination,z = DifD> 0,o rz = m+ DifD< 0;i f gcd(x,y) ̸=1, thenais not invertible modulom. Notice that ifmis odd, it is not nec- essary to compute the values ofAand C. It would appear that step 4 of Algorithm 14.61 requires bothAand Bin order to decide which case in step 4.2 is executed. But ifmis odd and Bis even, thenAmust be even; hence, the decision can be made using the parities of Band m. Example 14.65 illustrates Algorithm 14.61 for computing a multiplicative inverse. 14.65 Example (multiplicative inverse) Letm= 383and a= 271. Table 14.13 illustrates the steps of Algorithm 14.61 for computing271−1 mod 383 = 106. Notice that values for the variablesAand Cneed not be computed. □ 14.5 Chinese remainder theorem for integers Fact 2.120 introduced the Chinese remainder theorem (CRT) and Fact 2.121 outlined an al- gorithm for solving the associated system of linear congruences. Although the method de- scribed there is the one found in most textbooks on elementary number theory, it is not the §14.5 Chinese remainder theorem for integers 611 iteration: 1 2 3 4 5 6 7 8 9 10 u 383 112 56 28 14 7 7 7 7 7 v 271 271 271 271 271 271 264 132 66 33 B 0 −1 −192 −96 −48 −24 −24 −24 −24 −24 D 1 1 1 1 1 1 25 −179 −281 −332 iteration: 11 12 13 14 15 16 17 18 19 u 7 7 7 7 4 2 1 1 1 v 26 13 6 3 3 3 3 2 1 B −24 −24 −24 −24 41 −171 −277 −277 −277 D −308 −154 −130 −65 −65 −65 −65 212 106 Table 14.13:Inverse computation using the binary extended gcd algorithm (see Example 14.65). method of choice for large integers. Garner’s algorithm (Algorithm 14.71) has some com- putational advantages.§14.5.1 describes an alternate (non-radix) representation for non- negative integers, called amodular representation, that allows some computational advan- tages compared to standard radix representations. Algorithm 14.71 provides a technique for converting numbers from modular to basebrepresentation. 14.5.1 Residue number systems In previous sections, non-negative integers have been represented in radixbnotation. An alternate means is to use a mixed-radix representation. 14.66 FactLetBbe a ﬁxed positive integer. Letm1,m2,...,m t be positive integers such that gcd(mi,mj)=1 for alli̸=j, andM= ∏ t i=1 mi ≥B. Then each integerx,0 ≤x<B , can be uniquely represented by the sequence of integersv(x)=( v1,v2,...,v t), where vi = xmod mi,1 ≤i≤t. 14.67 DeﬁnitionReferring to Fact 14.66,v(x) is called themodular representationormixed- radix representationofxfor the modulim1,m2,...,m t. The set of modular representa- tions for all integersxin the range0 ≤x<B is called aresidue number system. Ifv(x)=( v1,v2,...,v t) andv(y)=( u1,u2,...,u t), deﬁnev(x)+v(y)=( w1,w2, ...,w t) where wi = vi + ui mod mi, andv(x) ·v(y)=( z1,z2,...,z t) where zi = vi ·ui mod mi. 14.68 FactIf0 ≤x,y<M , thenv((x+ y)m o dM)= v(x)+ v(y) and v((x·y)m o dM)= v(x) ·v(y). 14.69 Example (modular representation) LetM=3 0=2 ×3×5; here,t=3 ,m1 =2 ,m1 = 3, andm3 =5 . Table 14.14 displays each residue modulo30 along with its associated modular representation. As an example of Fact 14.68, note that21 + 27≡18 (mod 30) and (101) + (102) = (003). Also22 ·17 ≡14 (mod 30)and (012) ·(122) = (024). □ 14.70 Note (computational efﬁciency of modular representation for RSA decryption) Suppose thatn= pq, wherepand qare distinct primes. Fact 14.68 implies thatxd mod ncan be computed in a modular representation asvd(x); that is, ifv(x)=( v1,v2) with respect to modulim1 = p,m2 = q, thenvd(x)=( vd 1 mod p,vd 2 mod q). In general, computing 612 Ch. 14 Efﬁcient Implementation x v(x) x v(x) x v(x) x v(x) x v(x) 0 (000) 6 (001) 12 (002) 18 (003) 24 (004) 1 (111) 7 (112) 13 (113) 19 (114) 25 (110) 2 (022) 8 (023) 14 (024) 20 (020) 26 (021) 3 (103) 9 (104) 15 (100) 21 (101) 27 (102) 4 (014) 10 (010) 16 (011) 22 (012) 28 (013) 5 (120) 11 (121) 17 (122) 23 (123) 29 (124) Table 14.14:Modular representations (see Example 14.69). vd 1 mod pand vd 2 mod qis faster than computingxd mod n. For RSA, ifpand qare part of the private key, modular representation can be used to improve the performance of both decryption and signature generation (see Note 14.75). Converting an integerxfrom a basebrepresentation to a modular representation is eas- ily done by applying a modular reduction algorithm to computevi = xmod mi,1 ≤i≤t. Modular representations of integers inZM may facilitate some computational efﬁciencies, provided conversion from a standard radix to modular representation and back are relatively efﬁcient operations. Algorithm 14.71 describes one way of converting from modular rep- resentation back to a standard radix representation. 14.5.2 Garner’s algorithm Garner’s algorithm is an efﬁcient method for determiningx,0 ≤x<M , givenv(x)= (v1,v2,...,v t), the residues ofxmodulo the pairwise co-prime modulim1,m2,...,m t. 14.71 AlgorithmGarner’s algorithm for CRT INPUT: a positive integerM= ∏ t i=1 mi >1, withgcd(mi,mj)=1 for alli̸=j, and a modular representationv(x)=( v1,v2,...,v t) ofxfor themi. OUTPUT: the integerxin radixbrepresentation. 1. Forifrom 2 totdo the following: 1.1 Ci←1. 1.2 Forjfrom 1 to(i−1) do the following: u←m−1 j mod mi (use Algorithm 14.61). Ci←u·Ci mod mi. 2. u←v1, x←u. 3. Forifrom 2 totdo the following:u←(vi −x)Ci mod mi, x←x+ u·∏ i−1 j=1 mj. 4. Return(x). 14.72 Factxreturned by Algorithm 14.71 satisﬁes0 ≤x<M ,x≡vi (mod mi),1 ≤i≤t. 14.73 Example (Garner’ s algorithm) Letm1 =5 ,m2 =7 ,m3 =1 1,m4 =1 3,M =∏ 4 i=1 mi = 5005, andv(x)=( 2 ,1,3,8). The constantsCi computed areC2 =3 , C3 =6 , andC4 =5 . The values of(i,u,x) computed in step 3 of Algorithm 14.71 are (1,2,2),(2,4,22),(3,7,267), and(4,5,2192). Hence, the modular representationv(x)= (2,1,3,8) corresponds to the integerx= 2192. □ §14.6 Exponentiation 613 14.74 Note (computational efﬁciency of Algorithm 14.71) (i) If Garner’s algorithm is used repeatedly with the same modulusMand the same fac- tors ofM, then step 1 can be considered as a precomputation, requiring the storage oft−1 numbers. (ii) The classical algorithm for the CRT (Algorithm 2.121) typically requires a modular reduction with modulusM, whereas Algorithm 14.71 does not. SupposeMis akt- bit integer and eachmi is ak-bit integer. A modular reduction byMtakesO((kt)2) bit operations, whereas a modular reduction bymi takesO(k2) bit operations. Since Algorithm 14.71 only does modular reduction withmi,2 ≤i≤t, it takesO(tk2) bit operations in total for the reduction phase, and is thus more efﬁcient. 14.75 Note (RSA decryption and signature generation) (i) (special case of two moduli) Algorithm 14.71 is particularly efﬁcient for RSA moduli n= pq, wherem1 = pand m2 = qare distinct primes. Step 1 computes a single valueC2 = p−1 mod q. Step 3 is executed once:u =( v2 −v1)C2 mod qand x= v1 + up. (ii) (RSA exponentiation) Supposepand qaret-bit primes, and letn= pq. Letdbe a2t- bit RSA private key. RSA decryption and signature generation computexd mod n for somex∈Zn. Suppose that modular multiplication and squaring requirek2 bit operations fork-bit inputs, and that exponentiation with ak-bit exponent requires about3 2 kmultiplications and squarings (see Note 14.78). Then computingxd mod n requires about3 2 (2t)3 =1 2t3 bit operations. A more efﬁcient approach is to compute xdp mod pand xdq mod q(wheredp = dmod (p−1) and dq = dmod (q−1)), and then use Garner’s algorithm to constructxd mod pq. Although this procedure takes two exponentiations, each is considerably more efﬁcient because the moduli are smaller. Assuming that the cost of Algorithm 14.71 is negligible with respect to the exponentiations, computingxd mod nis about3 2 (2t)3/2(3 2 t3)=4 times faster. 14.6 Exponentiation One of the most important arithmetic operations for public-key cryptography is exponen- tiation. The RSA scheme (§8.2) requires exponentiation inZ m for some positive integer m, whereas Difﬁe-Hellman key agreement (§12.6.1) and the ElGamal encryption scheme (§8.4) use exponentiation inZp for some large primep. As pointed out in§8.4.2, ElGamal encryption can be generalized to any ﬁnite cyclic group. This section discusses methods for computing theexponentialge, where thebasegis an element of a ﬁnite groupG(§2.5.1) and theexponenteis a non-negative integer. A reader uncomfortable with the setting of a general group may considerGto beZ∗ m; that is, readge asge mod m. An efﬁcient method for multiplying two elements in the groupGis essential to per- forming efﬁcient exponentiation. The most naive way to computege is to doe−1 multi- plications in the groupG. For cryptographic applications, the order of the groupGtypically exceeds2160 elements, and may exceed21024. Most choices ofeare large enough that it would be infeasible to computege usinge−1 successive multiplications byg. There are two ways to reduce the time required to do exponentiation. One way is to decrease the time to multiply two elements in the group; the other is to reduce the number of multiplications used to computege. Ideally, one would do both. This section considers three types of exponentiation algorithms. 614 Ch. 14 Efﬁcient Implementation 1. basic techniques for exponentiation.Arbitrary choices of the basegand exponente are allowed. 2. ﬁxed-exponent exponentiation algorithms.The exponenteis ﬁxed and arbitrary choi- ces of the basegare allowed. RSA encryption and decryption schemes beneﬁt from such algorithms. 3. ﬁxed-base exponentiation algorithms.The basegis ﬁxed and arbitrary choices of the exponenteare allowed. ElGamal encryption and signatures schemes and Difﬁe- Hellman key agreement protocols beneﬁt from such algorithms. 14.6.1 Techniques for general exponentiation This section includes general-purpose exponentiation algorithms referred to asrepeated square-and-multiplyalgorithms. (i) Basic binary andk-ary exponentiation Algorithm 14.76 is simply Algorithm 2.143 restated in terms of an arbitrary ﬁnite abelian groupGwith identity element1. 14.76 AlgorithmRight-to-left binary exponentiation INPUT: an elementg∈Gand integere≥1. OUTPUT: ge. 1. A←1, S←g. 2. Whilee̸=0do the following: 2.1 Ifeis odd thenA←A·S. 2.2 e←⌊e/2⌋. 2.3 Ife̸=0thenS←S·S. 3. Return(A). 14.77 Example (right-to-left binary exponentiation) The following table displays the values of A,e, andSduring each iteration of Algorithm 14.76 for computingg283. □ A 1 gg 3 g3 g11 g27 g27 g27 g27 g283 e 283 141 70 35 17 8 4 2 1 0 S gg 2 g4 g8 g16 g32 g64 g128 g256 − 14.78 Note (computational efﬁciency of Algorithm 14.76) Lett+1 be the bitlength of the bi- nary representation ofe, and letwt(e) be the number of1’s in this representation. Algo- rithm 14.76 performstsquarings andwt(e) −1 multiplications. Ifeis randomly selected in the range0 ≤e< \|G\|= n, then about⌊lg n⌋squarings and1 2 (⌊lg n⌋+1 )multiplica- tions can be expected. (The assignment1 ·xis not counted as a multiplication, nor is the operation1 ·1 counted as a squaring.) If squaring is approximately as costly as an arbi- trary multiplication (cf. Note 14.18), then the expected amount of work is roughly3 2 ⌊lg n⌋ multiplications. Algorithm 14.76 computesA·Swhenevereis odd. For some choices ofg,A·gcan be computed more efﬁciently thanA·Sfor arbitraryS. Algorithm 14.79 is a left-to-right binary exponentiation which replaces the operationA·S(for arbitraryS) by the operation A·g(for ﬁxedg). §14.6 Exponentiation 615 14.79 AlgorithmLeft-to-right binary exponentiation INPUT: g∈Gand a positive integere=( etet−1 ···e1e0)2. OUTPUT: ge. 1. A←1. 2. Forifrom tdown to0 do the following: 2.1 A←A·A. 2.2 Ifei =1 , thenA←A·g. 3. Return(A). 14.80 Example (left-to-right binary exponentiation) The following table displays the values of Aduring each iteration of Algorithm 14.79 for computingg283. Note thatt=8 and 283 = (100011011)2. □ i 8 7 6 5 432 1 0 ei 1 0 0 0 110 1 1 A gg 2 g4 g8 g17 g35 g70 g141 g283 14.81 Note (computational efﬁciency of Algorithm 14.79) Lett+1 be the bitlength of the bi- nary representation ofe, and letwt(e) be the number of1’s in this representation. Algo- rithm 14.79 performst+1 squarings andwt(e) −1 multiplications byg. The number of squarings and multiplications is the same as in Algorithm 14.76 but, in this algorithm, mul- tiplication is always with the ﬁxed valueg.I fghas a special structure, this multiplication may be substantially easier than multiplying two arbitrary elements. For example, a fre- quent operation in ElGamal public-key schemes is the computation ofg k mod p, whereg is a generator ofZ∗ p andpis a large prime number. The multiple-precision computationA·g can be done in linear time ifgis chosen so that it can be represented by a single-precision integer (e.g.,g=2 ). If the radixbis sufﬁciently large, there is a high probability that such a generator exists. Algorithm 14.82, sometimes referred to as thewindow methodfor exponentiation, is a generalization of Algorithm 14.79 which processes more than one bit of the exponent per iteration. 14.82 AlgorithmLeft-to-rightk-ary exponentiation INPUT: gand e=( etet−1 ···e1e0)b, whereb=2 k for somek≥1. OUTPUT: ge. 1. Precomputation. 1.1 g0←1. 1.2 Forifrom 1 to(2k −1) do:gi←gi−1 ·g. (Thus,gi = gi.) 2. A←1. 3. Forifrom tdown to0 do the following: 3.1 A←A2k . 3.2 A←A·gei . 4. Return(A). 616 Ch. 14 Efﬁcient Implementation In Algorithm 14.83, Algorithm 14.82 is modiﬁed slightly to reduce the amount of pre- computation. The following notation is used: for eachi,0 ≤i≤t,i fei ̸=0, then write ei =2 hiui where ui is odd; ifei =0 , then lethi =0 and ui =0 . 14.83 AlgorithmModiﬁed left-to-rightk-ary exponentiation INPUT: gand e=( etet−1 ···e1e0)b, whereb=2 k for somek≥1. OUTPUT: ge. 1. Precomputation. 1.1 g0←1, g1←g, g2←g2. 1.2 Forifrom 1 to(2k−1 −1) do:g2i+1←g2i−1 ·g2. 2. A←1. 3. Forifrom tdown to0 do:A←(A2k−hi ·gui)2hi . 4. Return(A). 14.84 Remark (right-to-leftk-ary exponentiation) Algorithm 14.82 is a generalization of Algo- rithm 14.79. In a similar manner, Algorithm 14.76 can be generalized to thek-ary case. However, the optimization given in Algorithm 14.83 is not possible for the generalized right-to-leftk-ary exponentiation method. (ii) Sliding-window exponentiation Algorithm 14.85 also reduces the amount of precomputation compared to Algorithm 14.82 and, moreover, reduces the average number of multiplications performed (excluding squar- ings).kis called thewindow size. 14.85 AlgorithmSliding-window exponentiation INPUT: g,e=( etet−1 ···e1e0)2 withet =1 , and an integerk≥1. OUTPUT: ge. 1. Precomputation. 1.1 g1←g, g2←g2. 1.2 Forifrom 1 to(2k−1 −1) do:g2i+1←g2i−1 ·g2. 2. A←1, i←t. 3. Whilei≥0 do the following: 3.1 Ifei =0 then do:A←A2, i←i−1. 3.2 Otherwise (ei ̸=0), ﬁnd the longest bitstringeiei−1 ···el such thati−l+1 ≤k and el =1 , and do the following: A←A2i−l+1 ·g(eiei−1...el)2 , i←l−1. 4. Return(A). 14.86 Example (sliding-window exponentiation) Takee= 11749 = (10110111100101)2 and k=3 . Table 14.15 illustrates the steps of Algorithm 14.85. Notice that the sliding-window method for this exponent requires three multiplications, corresponding toi=7 ,4, and0. Algorithm 14.79 would have required four multiplications for the same values ofkande.□ §14.6 Exponentiation 617 i A Longest bitstring 13 1 101 10 g5 101 7 (g5)8g5 = g45 111 4 (g45)8g7 = g367 − 3 (g367)2 = g734 − 2 (g734)2 = g1468 101 0 (g1468)8g5 = g11749 − Table 14.15:Sliding-window exponentiation withk =3 and exponente = (10110111100101)2 . 14.87 Note (comparison of exponentiation algorithms) Lett+1 be the bitlength ofe, and let l+1 be the number ofk-bit words formed frome; that is,l= ⌈(t+1 )/k⌉−1= ⌊t/k⌋. Table 14.16 summarizes the number of squarings and multiplications required by Algo- rithms 14.76, 14.79, 14.82, and 14.83. Analysis of the number of squarings and multipli- cations for Algorithm 14.85 is more difﬁcult, although it is the recommended method. (i) (squarings for Algorithm 14.82) The number of squarings for Algorithm 14.82 islk. Observe thatlk= ⌊t/k⌋k= t−(tmod k). It follows thatt−(k−1) ≤lk≤t and that Algorithm 14.82 can save up tok−1 squarings over Algorithms 14.76 and 14.79. An optimal value forkin Algorithm 14.82 will depend ont. (ii) (squarings for Algorithm 14.83) The number of squarings for Algorithm 14.83 islk+ hl where 0 ≤hl ≤tmod k. Sincet−(k−1) ≤lk≤lk+hl ≤lk+(tmod k)= t ort−(k−1) ≤lk+hl ≤t, the number of squarings for this algorithm has the same bounds as Algorithm 14.82. Precomputation Multiplications Algorithm sq mult squarings worst case average case 14.76 0 0 t t t/2 14.79 0 0 t t t/2 14.82 1 2k −3 t −(k −1) ≤lk ≤t l −1 l(2k −1)/2k 14.83 1 2k−1 −1 t −(k −1) ≤lk + hl ≤t l −1 l(2k −1)/2k Table 14.16:Number of squarings (sq) and multiplications (mult) for exponentiation algorithms. (iii) Simultaneous multiple exponentiation There are a number of situations which require computation of the product of several ex- ponentials with distinct bases and distinct exponents (for example, veriﬁcation of ElGa- mal signatures; see Note 14.91). Rather than computing each exponential separately, Al- gorithm 14.88 presents a method to do them simultaneously. Lete 0,e1,...,e k−1 be positive integers each of bitlengtht; some of the high-order bits of some of the exponents might be0, but there is at least oneei whose high-order bit is1. Form ak×tarrayEA (called theexponent array) whose rows are the binary representations of the exponentsei,0 ≤ i ≤ k−1. LetIj be the non-negative integer whose binary representation is thejth column,1 ≤j≤t,o fEA, where low-order bits are at the top of the column. 618 Ch. 14 Efﬁcient Implementation 14.88 AlgorithmSimultaneous multiple exponentiation INPUT: group elementsg0,g1,...,g k−1 and non-negativet-bit integerse0,e1,...e k−1. OUTPUT: ge0 0 ge1 1 ···gek−1 k−1 . 1. Precomputation. Forifrom 0 to(2k −1):Gi←∏ k−1 j=0 gij j where i=( ik−1 ···i0)2. 2. A←1. 3. Forifrom 1 totdo the following:A←A·A, A←A·GIi . 4. Return(A). 14.89 Example (simultaneous multiple exponentiation) In this example,g30 0 g10 1 g24 2 is computed using Algorithm 14.88. Lete0 = 30 = (11110)2,e1 = 10 = (01010)2, ande2 =2 4= (11000)2. The3 ×5 arrayEA is: 11110 01010 11000 The next table displays precomputed values from step 1 of Algorithm 14.88. i 0 1 2 3 4 5 6 7 Gi 1 g0 g1 g0g1 g2 g0g2 g1g2 g0g1g2 Finally, the value ofAat the end of each iteration of step 3 is shown in the following table. Here,I1 =5 ,I2 =7 ,I3 =1 ,I4 =3 , andI5 =0 . i 1 2 3 4 5 A g0g2 g3 0g1g3 2 g7 0g2 1g6 2 g15 0 g5 1g12 2 g30 0 g10 1 g24 2 □ 14.90 Note (computational efﬁciency of Algorithm 14.88) (i) Algorithm 14.88 computesge0 0 ge1 1 ···gek−1 k−1 (where eachei is represented bytbits) by performingt−1 squarings and at most(2k −2) +t−1 multiplications. The multiplication is trivial for any column consisting of all0’s. (ii) Not all of theGi,0 ≤i≤2k −1, need to be precomputed, but only for thoseiwhose binary representation is a column ofEA. 14.91 Note (ElGamal signature veriﬁcation) The signature veriﬁcation equation for the ElGa- mal signature scheme (Algorithm 11.64) isαh(m)(α−a)r ≡rs (mod p) where pis a large prime,αa generator ofZ∗ p,αa is the public key, and(r,s) is a signature for messagem. It would appear that three exponentiations and one multiplication are required to verify the equation. Ift = ⌈lg p⌉and Algorithm 11.64 is applied, the number of squarings is 3(t−1) and the number of multiplications is, on average,3t/2. Hence, one would ex- pect to perform about(9t−4)/2 multiplications and squarings modulop. Algorithm 14.88 can reduce the number of computations substantially if the veriﬁcation equation is rewrit- ten asαh(m)(α−a)rr−s ≡1( m o dp). Takingg0 = α,g1 = α−a,g2 = r, ande0 = h(m)m o d(p−1),e1 = rmod (p−1),e2 = −smod (p−1) in Algorithm 14.88, the expected number of multiplications and squarings is(t−1)+(6+(7 t/8)) = (15t+40)/8. (For random exponents, one would expect that, on average,7 8 of the columns ofEA will be non-zero and necessitate a non-trivial multiplication.) This is only about 25% more costly than a single exponentiation computed by Algorithm 14.79. §14.6 Exponentiation 619 (iv) Additive notation Algorithms 14.76 and 14.79 have been described in the setting of a multiplicative group. Algorithm 14.92 uses the methodology of Algorithm 14.79 to perform efﬁcient multiplica- tion in an additive groupG. (For example, the group formed by the points on an elliptic curve over a ﬁnite ﬁeld uses additive notation.) Multiplication in an additive group corre- sponds to exponentiation in a multiplicative group. 14.92 AlgorithmLeft-to-right binary multiplication in an additive group INPUT: g∈G, whereGis an additive group, and a positive integere=( etet−1 ···e1e0)2. OUTPUT: e·g. 1. A←0. 2. Forifrom tdown to0 do the following: 2.1 A←A+ A. 2.2 Ifei =1 thenA←A+ g. 3. Return(A). 14.93 Note (the additive groupZm) (i) IfGis the additive groupZm, then Algorithm 14.92 provides a method for doing modular multiplication. For example, ifa,b ∈Zm, thena·bmod mcan be com- puted using Algorithm 14.92 by takingg= aand e= b, providedbis written in binary. (ii) Ifa,b ∈ Zm, thena<m and b<m . The accumulatorAin Algorithm 14.92 never contains an integer as large as2m; hence, modular reduction of the value in the accumulator can be performed by a simple subtraction whenA≥m; thus no divisions are required. (iii) Algorithms 14.82 and 14.83 can also be used for modular multiplication. In the case of the additive groupZm, the time required to do modular multiplication can be im- proved at the expense of precomputing a table of residues modulom. For a left-to- rightk-ary exponentiation scheme, the table will contain2k −1 residues modulom. (v) Montgomery exponentiation The introductory remarks to§14.3.2 outline an application of the Montgomery reduction method for exponentiation. Algorithm 14.94 below combines Algorithm 14.79 and Al- gorithm 14.36 to give aMontgomery exponentiation algorithmfor computingxe mod m. Note the deﬁnition ofm′requires thatgcd(m,R)=1 . For integersuand vwhere 0 ≤ u,v<m , deﬁne Mont(u,v)t ob euvR−1 mod mas computed by Algorithm 14.36. 620 Ch. 14 Efﬁcient Implementation 14.94 AlgorithmMontgomery exponentiation INPUT: m=( ml−1 ···m0)b,R= bl,m′= −m−1 mod b,e=( et ···e0)2 withet =1 , and an integerx,1 ≤x<m . OUTPUT: xe mod m. 1. ˜x←Mont(x,R2 mod m),A←Rmod m.(Rmod mand R2 mod mmay be pro- vided as inputs.) 2. Forifrom tdown to0 do the following: 2.1 A←Mont(A,A). 2.2 Ifei =1 thenA←Mont(A,˜x). 3. A←Mont(A,1). 4. Return(A). 14.95 Example (Montgomery exponentiation) Letx,m, andRbe integers suitable as inputs to Algorithm 14.94. Lete= 11 = (1011)2; here,t=3 . The following table displays the values ofAmod mat the end of each iteration of step 2, and after step 3.□ i 3 2 1 0 Step 3 A mod m ˜x ˜x2R−1 ˜x5R−4 ˜x11R−10 Mont (A, 1) =˜x11R−11 = x11 14.96 Note (computational efﬁciency of Montgomery exponentiation) (i) Table 14.17 displays the average number of single-precision multiplications required for each step of Algorithm 14.94. The expected number of single-precision multipli- cations to computexe mod mby Algorithm 14.94 is3l(l+1 ) (t+1 ). (ii) Each iteration of step 2 in Algorithm 14.94 applies Algorithm 14.36 at a cost of2l(l+ 1) single-precision multiplications but no single-precision divisions. A similar algo- rithm for modular exponentiation based on classical modular multiplication (Algo- rithm 14.28) would similarly use2l(l+1 )single-precision multiplications per iter- ation but alsolsingle-precision divisions. (iii) Any of the other exponentiation algorithms discussed in§14.6.1 can be combined with Montgomery reduction to give other Montgomery exponentiation algorithms. Step 1 2 3 Number of Montgomery multiplications 1 3 2 t 1 Number of single-precision multiplications 2l(l +1 ) 3tl(l +1 ) l(l +1 ) Table 14.17:Average number of single-precision multiplications per step of Algorithm 14.94. 14.6.2 Fixed-exponent exponentiation algorithms There are numerous situations in which a number of exponentiations by a ﬁxed exponent must be performed. Examples include RSA encryption and decryption, and ElGamal de- cryption. This subsection describes selected algorithms which improve the repeated square- and-multiply algorithms of§14.6.1 by reducing the number of multiplications. §14.6 Exponentiation 621 (i) Addition chains The purpose of an addition chain is to minimize the number of multiplications required for an exponentiation. 14.97 DeﬁnitionAn addition chainVof lengthsfor a positive integereis a sequenceu0,u1, ...,u s of positive integers, and an associated sequencew1,...,w s of pairswi =( i1,i2), 0 ≤i1,i2 <i, having the following properties: (i)u0 =1 and us = e; and (ii) for eachui,1 ≤i≤s,ui = ui1 + ui2 . 14.98 AlgorithmAddition chain exponentiation INPUT: a group elementg, an addition chainV=( u0,u1,...,u s) of lengthsfor a positive integere, and the associated sequencew1,...,w s, wherewi =( i1,i2). OUTPUT: ge. 1. g0←g. 2. Forifrom 1 tosdo:gi←gi1 ·gi2. 3. Return(gs). 14.99 Example (addition chain exponentiation) An addition chain of length5 fore =1 5 is u0 =1 ,u1 =2 ,u2 =3 ,u3 =6 ,u4 =1 2,u5 =1 5. The following table displays the values ofwi and gi during each iteration of Algorithm 14.98 for computingg15. □ i 0 1 2 3 4 5 wi − (0, 0) (0, 1) (2, 2) (3, 3) (2, 4) gi g g2 g3 g6 g12 g15 14.100 Remark (addition chains and binary representations) Given the binary representation of an exponente, it is a relatively simple task to construct an addition chain directly from this representation. Chains constructed in this way generally do not provide the shortest addition chain possible for the given exponent. The methods for exponentiation described in§14.6.1 could be phrased in terms of addition chains, but this is typically not done. 14.101 Note (computational efﬁciency of addition chain exponentiation) Given an addition chain of lengthsfor the positive integere, Algorithm 14.98 computesg e for anyg∈G,g̸=1, using exactlysmultiplications. 14.102 FactIflis the length of a shortest addition chain for a positive integere, thenl≥(lg e+ lg wt(e) −2.13), wherewt(e) is the number of1’s in the binary representation ofe.A n upper bound of(⌊lg e⌋+w t (e) −1) is obtained by constructing an addition chain fore from its binary representation. Determining a shortest addition chain foreis known to be an NP -hard problem. 622 Ch. 14 Efﬁcient Implementation (ii) Vector-addition chains Algorithms 14.88 and 14.104 are useful for computingge0 0 ge1 1 ··· gek−1 k−1 where g0,g1,... , gk−1 are arbitrary elements in a groupGand e0,e1,...,e k−1 are ﬁxed positive integers. These algorithms can also be used to advantage when the exponents are not necessarily ﬁxed values (see Note 14.91). Algorithm 14.104 makes use of vector-addition chains. 14.103 DeﬁnitionLet sand kbe positive integers and letvi denote ak-dimensional vector of non-negative integers. An ordered setV = {vi : −k+1 ≤i≤s}is called avector- addition chainof lengthsand dimensionkifVsatisﬁes the following: (i) Eachvi,−k+1 ≤i≤0, has a0 in each coordinate position, except for coordinate positioni+ k−1, which is a1. (Coordinate positions are labeled0 throughk−1.) (ii) For eachvi,1 ≤i≤s, there exists an associated pair of integerswi =( i1,i2) such that−k+1 ≤i1,i2 <i and vi = vi1 + vi2 (i1 = i2 is allowed). Example 14.105 illustrates a sample vector-addition chain. LetV = {vi : −k+1 ≤ i≤s}be a vector-addition chain of lengthsand dimensionkwith associated sequence w1,...,w s. Algorithm 14.104 computesge0 0 ge1 1 ···gek−1 k−1 where vs =( e0,e1,...,e k−1). 14.104 AlgorithmVector-addition chain exponentiation INPUT: group elementsg0,g1,...,g k−1 and a vector-addition chainVof lengthsand di- mensionkwith associated sequencew1,...,w s, wherewi =( i1,i2). OUTPUT: ge0 0 ge1 1 ···gek−1 k−1 where vs =( e0,e1,...,e k−1). 1. Forifrom (−k+1 )to0 do:ai←gi+k−1. 2. Forifrom 1 tosdo:ai←ai1 ·ai2 . 3. Return(as). 14.105 Example (vector-addition chain exponentiation) A vector-addition chainVof lengths= 9 and dimensionk=3 is displayed in the following table. v−2 v−1 v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 1 0 0 1 2 2 3 5 6 12 15 30 0 1 0 0 0 1 1 2 2 4 5 10 0 0 1 1 2 2 2 4 5 10 12 24 The following table displays the values ofwi and ai during each iteration of step 2 in Al- gorithm 14.104 for computingg30 0 g10 1 g24 2 . Nine multiplications are required. □ i 1 2 3 4 5 6 7 8 9 wi (−2, 0) (1, 1) (−1, 2) (−2, 3) (3, 4) (1, 5) (6, 6) (4, 7) (8, 8) ai g0g2 g2 0g2 2 g2 0g1g2 2 g3 0g1g2 2 g5 0g2 1g4 2 g6 0g2 1g5 2 g12 0 g4 1g10 2 g15 0 g5 1g12 2 g30 0 g10 1 g24 2 14.106 Note (computational efﬁciency of vector-addition chain exponentiation) (i) (multiplications) Algorithm 14.104 performs exactlysmultiplications for a vector- addition chain of lengths. To computege0 0 ge1 1 ···gek−1 k−1 using Algorithm 14.104, one would like to ﬁnd a vector-addition chain of lengthsand dimensionkwithvs = (e0,e1,...,e k−1), wheresis as small as possible (see Fact 14.107). §14.6 Exponentiation 623 (ii) (storage) Algorithm 14.104 requires intermediate storage for the elementsai,−k+ 1 ≤i<t ,a tt h etth iteration of step 2. If not all of these are required for succeeding iterations, then they need not be stored. Algorithm 14.88 provides a special case of Algorithm 14.104 where the intermediate storage is no larger than2k −1 vectors of dimensionk. 14.107 FactThe minimum value ofsin Note 14.106(i) satisﬁes the following bound, whereM= max{ei :0 ≤i≤k−1}and cis a constant: s≤k−1+l gM+ ck·lg M/lg lg(M+2 ). 14.108 Example (vector-addition chains from binary representations) The vector-addition chain implicit in Algorithm 14.88 is not necessarily of minimum length. The vector-addition chain associated with Example 14.89 is displayed in Table 14.18. This chain is longer than the one used in Example 14.105. The advantage of Algorithm 14.88 is that the vector- addition chain does not have to be explicitly provided to the algorithm. In view of this, Algorithm 14.88 can be applied more generally to situations where the exponents are not necessarily ﬁxed. □ v−2 v−1 v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 1 0 0 1 1 1 2 3 6 7 14 15 30 0 1 0 1 1 0 0 1 2 2 4 5 10 0 0 1 0 1 1 2 3 6 6 12 12 24 Table 14.18:Binary vector-addition chain exponentiation (see Example 14.108). 14.6.3 Fixed-base exponentiation algorithms Three methods are presented for exponentiation when the basegis ﬁxed and the exponent evaries. With a ﬁxed base, precomputation can be done once and used for many exponen- tiations. For example, Difﬁe-Hellman key agreement (Protocol 12.47) requires the compu- tation ofα x, whereαis a ﬁxed element inZ∗ p. For each of the algorithms described in this section,{b0,b1,...,b t}is a set of integers for somet≥0, such that any exponente≥1 (suitably bounded) can be written ase=∑ t i=0 eibi, where0 ≤ei <h for some ﬁxed positive integerh. For example, ifeis any (t+1 )-digit basebinteger withb≥2, thenbi = bi and h= bare possible choices. Algorithms 14.109 and 14.113 are two ﬁxed-base exponentiation methods. Both re- quire precomputation of the exponentialsgb0 ,gb1,...,g bt , e.g., using one of the algorithms from§14.6.1. The precomputation needed for Algorithm 14.117 is more involved and is ex- plicitly described in Algorithm 14.116. (i) Fixed-base windowing method Algorithm 14.109 takes as input the precomputed exponentialsgi = gbi,0 ≤i≤t, and positive integershand e= ∑ t i=0 eibi where 0 ≤ei <h,0 ≤i≤t. The basis for the algorithm is the observation thatge = ∏ t i=0 gei i = ∏ h−1 j=1 (∏ ei=j gi)j. 624 Ch. 14 Efﬁcient Implementation 14.109 AlgorithmFixed-base windowing method for exponentiation INPUT: {gb0,gb1,...,g bt },e= ∑ t i=0 eibi, andh. OUTPUT: ge. 1. A←1, B←1. 2. Forjfrom (h−1) down to1 do the following: 2.1 For eachifor whichei = jdo:B←B·gbi. 2.2 A←A·B. 3. Return(A). 14.110 Example (ﬁxed-base windowing exponentiation) Precompute the group elementsg1,g4, g16,g64,g256. To computege fore= 862 = (31132)4, taket=4 ,h=4 , andbi =4 i for 0 ≤i≤4, in Algorithm 14.109. The following table displays the values ofAand Bat the end of each iteration of step 2. □ j − 3 2 1 B 1 g4g256 = g260 g260g = g261 g261g16g64 = g341 A 1 g260 g260g261 = g521 g521g341 = g862 14.111 Note (computational efﬁciency of ﬁxed-base windowing exponentiation) (i) (number of multiplications) Supposet+ h≥2. Only multiplications where both operands are distinct from1 are counted. Step 2.2 is executedh−1 times, but at least one of these multiplications involves an operand with value1 (Ais initialized to1). SinceBis also initially1, at mosttmultiplications are done in step 2.1. Thus, Algo- rithm 14.109 computesge with at mostt+ h−2 multiplications (cf. Note 14.112). (ii) (storage) Storage is required for thet+1 group elementsgi,0 ≤i≤t. 14.112 Note (a particular case) The most obvious application of Algorithm 14.109 is the case where the exponenteis represented in radixb.I fe= ∑ t i=0 eibi, thengi = gbi ,0 ≤i≤t, are precomputed. Ifeis randomly selected from{0,1,...,m −1}, thent+1 ≤⌈logb m⌉ and, on average,1 b of the basebdigits inewill be0. In this case, the expected number of multiplications isb−1 b ⌈logb m⌉+ b−3.I fmis a512-bit integer andb=3 2, then128.8 multiplications are needed on average,132 in the worst case;103 values must be stored. (ii) Fixed-base Euclidean method Let {x0,x1,...,x t}be a set of integers witht ≥ 2. Deﬁne Mto be an integer in the interval[0,t] such thatxM ≥xi for all0 ≤i≤t. DeﬁneNto be an integer in the interval [0,t],N̸=M, such thateN ≥ei for all0 ≤i≤t,i̸=M. 14.113 AlgorithmFixed-base Euclidean method for exponentiation INPUT: {gb0,gb1,...,g bt }and e= ∑ t i=0 eibi. OUTPUT: ge. 1. Forifrom 0 totdo the following:gi←gbi, xi←ei. 2. Determine the indicesMand Nfor{x0,x1,...,x t}. 3. WhilexN ̸=0do the following: 3.1 q←⌊xM /xN ⌋, gN ←(gM )q ·gN , xM ←xM mod xN . §14.6 Exponentiation 625 3.2 Determine the indicesMand Nfor{x0,x1,...,x t}. 4. Return(gxM M ). 14.114 Example (ﬁxed-base Euclidean method) This example repeats the computation ofge,e= 862 done in Example 14.110, but now uses Algorithm 14.113. Takeb0 =1 ,b1 =1 6,b2 = 256. Thene=( 3,5,14)16. Precomputeg1,g16,g256. Table 14.19 illustrates the steps performed by Algorithm 14.113. Notice that for this example, Algorithm 14.113 does8 x0 x1 x2 M N q g0 g1 g2 14 5 3 0 1 2 g g18 g256 4 5 3 1 0 1 g19 g18 g256 4 1 3 0 2 1 g19 g18 g275 1 1 3 2 1 3 g19 g843 g275 1 1 0 0 1 1 g19 g862 g275 0 1 0 1 0 − g19 g862 g275 Table 14.19:Fixed-base Euclidean method to computeg862 (see Example 14.114). multiplications, whereas Algorithm 14.109 needs only6 to do the same computation. Stor- age requirements for Algorithm 14.113 are, however, smaller. The vector-addition chain (Deﬁnition 14.103) corresponding to this example is displayed in the following table.□ v−2 v−1 v0 v1 v2 v3 v4 v5 v6 v7 v8 1 0 0 2 2 3 3 6 9 11 14 0 1 0 0 1 1 1 2 3 4 5 0 0 1 0 0 0 1 2 3 3 3 14.115 Note (ﬁxed-base Euclidean vs. ﬁxed-base windowing methods) (i) In most cases, the quotientqcomputed in step 3.1 of Algorithm 14.113 is1. For a given baseb, the computational requirements of this algorithm are not signiﬁcantly greater than those of Algorithm 14.109. (ii) Since the division algorithm is logarithmic in the size of the inputs, Algorithm 14.113 can take advantage of a larger value ofhthan Algorithm 14.109. This results in less storage for precomputed values. (iii) Fixed-base comb method Algorithm 14.117 computesge where e=( etet−1 ···e1e0)2,t≥1. Select an integerh, 1 ≤h≤t+1 and computea= ⌈(t+1 )/h⌉. Select an integerv,1 ≤v≤a, and compute b= ⌈a/v⌉. Clearly,ah≥t+1 . LetX= Rh−1\|\|Rh−2\|\|···\|\| R0 be a bitstring formed from eby padding (if necessary)eon the left with0’s, so thatXhas bitlengthahand each Ri,0 ≤i≤h−1, is a bitstring of lengtha. Form anh×aarrayEA (called theexponent array) where rowiofEA is the bitstringRi,0 ≤i≤h−1. Algorithm 14.116 is the precomputation required for Algorithm 14.117. 626 Ch. 14 Efﬁcient Implementation 14.116 AlgorithmPrecomputation for Algorithm 14.117 INPUT: group elementgand parametersh,v,a, andb(deﬁned above). OUTPUT: {G[j][i]: 1 ≤i< 2h,0 ≤j<v }. 1. Forifrom 0 to(h−1) do:gi←g2ia . 2. Forifrom 1 to(2h −1) (wherei=( ih−1 ···i0)2), do the following: 2.1 G[0][i]←∏ h−1 j=0 gij j . 2.2 Forjfrom 1 to(v−1) do:G[j][i]←(G[0][i])2jb . 3. Return({G[j][i]: 1 ≤i< 2h,0 ≤j<v }). LetIj,k,0 ≤k<b ,0 ≤j<v , be the integer whose binary representation is column (jb+ k) ofEA, where column0 is on the right and the least signiﬁcant bits of a column are at the top. 14.117 AlgorithmFixed-base comb method for exponentiation INPUT: g,eand {G[j][i]: 1 ≤i< 2h,0 ≤j<v }(precomputed in Algorithm 14.116). OUTPUT: ge. 1. A←1. 2. Forkfrom (b−1) down to0 do the following: 2.1 A←A·A. 2.2 Forjfrom (v−1) down to0 do:A←G[j][Ij,k] ·A. 3. Return(A). 14.118 Example (ﬁxed-base comb method for exponentiation) Lett=9 and h=3 ; thena= ⌈10/3⌉=4 . Letv =2 ; thenb = ⌈a/v⌉=2 . Suppose the exponent input to Algo- rithm 14.117 ise =( e9e8 ···e1e0)2. Form the bitstringX = x11x10 ···x1x0 where xi = ei,0 ≤i≤9, andx11 = x10 =0 . The following table displays the exponent arrayEA. I1,1 I1,0 I0,1 I0,0 x3 x2 x1 x0 x7 x6 x5 x4 x11 x10 x9 x8 The precomputed values from Algorithm 14.116 are displayed below. Recall thatgi = g2ia , 0 ≤i< 3. i 12 3 4 5 6 7 G[0][i] g0 g1 g1g0 g2 g2g0 g2g1 g2g1g0 G[1][i] g4 0 g4 1 g4 1g4 0 g4 2 g4 2g4 0 g4 2g4 1 g4 2g4 1g4 0 Finally, the following table displays the steps in Algorithm 14.117 forEA. A = gl0 0 gl1 1 gl2 2 k j l0 l1 l2 1 − 0 0 0 1 1 4x3 4x7 4x11 1 0 4x3 + x1 4x7 + x5 4x11 + x9 0 − 8x3 +2 x1 8x7 +2 x5 8x11 +2 x9 0 1 8x3 +2 x1 +4 x2 8x7 +2 x5 +4 x6 8x11 +2 x9 +4 x10 0 0 8x3 +2 x1 +4 x2 + x0 8x7 +2 x5 +4 x6 + x4 8x11 +2 x9 +4 x10 + x8 §14.7 Exponent recoding 627 The last row of the table corresponds tog ∑11 i=0 xi2i = ge. □ 14.119 Note (computational efﬁciency of ﬁxed-base comb method) (i) (number of multiplications) Algorithm 14.117 requires at most one multiplication for each column ofEA. The right-most column ofEA requires a multiplication with the initial value 1 of the accumulatorA. The algorithm also requires a squaring of the accumulatorAfor eachk,0 ≤k<b , except fork = b−1 when Ahas value 1. Discounting multiplications by 1, the total number of non-trivial multiplications (including squarings) is, at most,a+ b−2. (ii) (storage) Algorithm 14.117 requires storage for thev(2h −1) precomputed group elements (Algorithm 14.116). If squaring is a relatively simple operation compared to multiplication in the group, then some space-saving can be achieved by storing only2 h −1 group elements (i.e., only those elements computed in step 2.1 of Algo- rithm 14.116). (iii) (trade-offs) Sincehand vare independent of the number of bits in the exponent, se- lection of these parameters can be made based on the amount of storage available vs. the amount of time (determined by multiplication) to do the computation. 14.7 Exponent recoding Another approach to reducing the number of multiplications in the basic repeated square- and-multiply algorithms (§14.6.1) is to replace the binary representation of the exponente with a representation which has fewer non-zero terms. Since the binary representation is unique (Fact 14.1), ﬁnding a representation with fewer non-zero components necessitates the use of digits besides0 and 1. Transforming an exponent from one representation to an- other is calledexponent recoding. Many techniques for exponent recoding have been pro- posed in the literature. This section describes two possibilities: signed-digit representation (§14.7.1) and string-replacement representation (§14.7.2). 14.7.1 Signed-digit representation 14.120 DeﬁnitionIfe= ∑ t i=0 di2i where di ∈{0,1,−1},0 ≤i≤t, then(dt ···d1d0)SD is called asigned-digit representation with radix2 for the integere. Unlike the binary representation, the signed-digit representation of an integer is not unique. The binary representation is an example of a signed-digit representation. Letebe a positive integer whose binary representation is(et+1etet−1 ···e1e0)2, withet+1 = et =0 . Algorithm 14.121 constructs a signed-digit representation forehaving at mostt+1 digits and the smallest possible number of non-zero terms. 628 Ch. 14 Efﬁcient Implementation 14.121 AlgorithmSigned-digit exponent recoding INPUT: a positive integere=( et+1etet−1 ···e1e0)2 withet+1 = et =0 . OUTPUT: a signed-digit representation(dt ···d1d0)SD fore. (See Deﬁnition 14.120.) 1. c0←0. 2. Forifrom 0 totdo the following: 2.1 ci+1←⌊(ei + ei+1 + ci)/2⌋, di←ei + ci −2ci+1. 3. Return((dt ···d1d0)SD). 14.122 Example (signed-digit exponent recoding) Table 14.20 lists all possible inputs to theith iteration of step 2, and the corresponding outputs. Ife = (1101110111) 2, then Algo- rithm 14.121 produces the signed-digit representatione= (100 1000 100 1)SD where 1= −1. Note thate=2 9 +2 8 +2 6 +2 5 +2 4 +2 2 +2+1=2 10 −27 −23 −1. □ inputs ei 0 0 00 11 1 1 ci 0 0 11 00 1 1 ei+1 0 1 01 01 0 1 outputs ci+1 0 0 01 01 1 1 di 001 −11 −100 Table 14.20:Signed-digit exponent recoding (see Example 14.122). 14.123 DeﬁnitionA signed-digit representation of an integereis said to besparseif no two non- zero entries are adjacent in the representation. 14.124 Fact(sparse signed-digit representation) (i) Every integerehas a unique sparse signed-digit representation. (ii) A sparse signed-digit representation forehas the smallest number of non-zero entries among all signed-digit representations fore. (iii) The signed-digit representation produced by Algorithm 14.121 is sparse. 14.125 Note (computational efﬁciency of signed-digit exponent recoding) (i) Signed-digit exponent recoding as per Algorithm 14.121 is very efﬁcient, and can be done by table look-up (using Table 14.20). (ii) Wheneis given in a signed-digit representation, computingge requires bothgand g−1.I fgis a ﬁxed base, theng−1 can be precomputed. For a variable baseg, unless g−1 can be computed very quickly, recoding an exponent to signed-digit representa- tion may not be worthwhile. 14.7.2 String-replacement representation 14.126 DeﬁnitionLet k ≥1 be a positive integer. A non-negative integereis said to have a k-ary string-replacement representation(ft−1ft−2 ···f1f0)SR(k), denotedSR(k),i fe=∑ t−1 i=0 fi2i and fi ∈{2j −1:0 ≤j≤k}for0 ≤i≤t−1. §14.7 Exponent recoding 629 14.127 Example (non-uniqueness of string-replacement representations) A string-replacement representation for a non-negative integer is generally not unique. The binary representa- tion is a1-ary string-replacement representation. Ifk=3 and e= 987 = (1111011011)2, then some other string-replacements ofeare(303003003)SR(3),(1007003003)SR(3), and (71003003)SR(3). □ 14.128 Algorithmk-ary string-replacement representation INPUT: e=( et−1et−2 ···e1e0)2 and positive integerk≥2. OUTPUT: e=( ft−1ft−2 ···f1f0)SR(k). 1. Forifrom kdown to2 do the following: starting with the most signiﬁcant digit of e=( et−1et−2 ···e1e0)2, replace each consecutive string ofiones with a string of lengthiconsisting ofi−1 zeros in the high-order string positions and the integer 2i −1 in the low-order position. 2. Return((ft−1ft−2 ···f1f0)SR(k)). 14.129 Example (k-ary string-replacement) Supposee= (110111110011101)2 and k=3 . The SR(3) representations ofeat the end of each of the two iterations of Algorithm 14.128 are (110007110000701)SR(3) and (030007030000701)SR(3). □ 14.130 AlgorithmExponentiation using anSR(k) representation INPUT: an integerk≥2, an elementg∈G, ande=( ft−1ft−2 ···f1f0)SR(k). OUTPUT: ge. 1. Precomputation. Setg1←g. Forifrom 2 tokdo:g2i−1←(g2i−1−1)2 ·g. 2. A←1. 3. Forifrom (t−1) down to0 do the following: 3.1 A←A·A. 3.2 Iffi ̸=0thenA←A·gfi . 4. Return(A). 14.131 Example (SR(k) vs. left-to-right binary exponentiation) Lete= 987 = (1111011011)2 and consider the3-ary string-replacement representation(0071003003)SR(3). Computing ge using Algorithm 14.79 requires9 squarings and7 multiplications. Algorithm 14.130 requires2 squarings and2 multiplications for computingg3 and g7, and then7 squarings and 3 multiplications for the main part of the algorithm. In total, theSR(3) forecomputes ge with9 squarings and5 multiplications. □ 14.132 Note (computational efﬁciency of Algorithm 14.130) The precomputation requiresk−1 squarings andk−1 multiplications. Algorithm 14.128 is not guaranteed to produce an SR(k) representation with a minimum number of non-zero entries, but in practice it seems to give representations which are close to minimal. Heuristic arguments indicate that a ran- domly selectedt-bit exponent will be encoded with a suitably chosen value ofkto anSR(k) representation having aboutt/4 non-zero entries; hence, one expects to performt−1 squar- ings in step 3, and aboutt/4 multiplications. 630 Ch. 14 Efﬁcient Implementation 14.8 Notes and further references §14.1 This chapter deals almost exclusively with methods to perform operations in the integers and the integers modulo some positive integer. Whenpis a prime number,Zp is called a ﬁnite ﬁeld (Fact 2.184). There are other ﬁnite ﬁelds which have signiﬁcance in cryptogra- phy. Of particular importance are those of characteristic two,F 2m. Perhaps the most useful property of these structures is that squaring is alinear operator(i.e., ifα,β ∈F2m , then (α+ β)2 = α2 + β2). This property leads to efﬁcient methods for exponentiation and for inversion. Characteristic two ﬁnite ﬁelds have been used extensively in connection with error-correcting codes; for example, see Berlekamp [118] and Lin and Costello [769]. For error-correcting codes,mis typically quite small (e.g.,1 ≤m≤16); for cryptographic applications,mis usually much larger (e.g.,m≥100). The majority of the algorithms presented in this chapter are best suited to software imple- mentations. There is a vast literature on methods to perform modular multiplication and other operations in hardware. The basis for most hardware implementations for modular multiplication is efﬁcient methods for integer addition. In particular,carry-save addersand delayed-carry addersare at the heart of the best methods to perform modular multiplica- tion. The concept of a delayed-carry adder was proposed by Norris and Simmons [933] to produce a hardware modular multiplier which computes the product of twot-bit operands modulo at-bit modulus in2tclock cycles. Brickell [199] improved the idea to produce a modular multiplier requiring onlyt+7 clock cycles. Enhancements of Brickell’s method were given by Walter [1230]. Koc¸ [699] gives a comprehensive survey of hardware meth- ods for modular multiplication. §14.2 For a treatment of radix representations includingmixed-radix representations, see Knuth [692]. Knuth describes efﬁcient methods for performing radix conversions. Representing and manipulating negative numbers is an important topic; for an introduction, consult the book by Koren [706]. The techniques described in§14.2 are commonly referred to as theclassical algorithmsfor multiple-precision addition, subtraction, multiplication, and division. These algorithms are the most useful for integers of the size used for cryptographic purposes. For much larger in- tegers (on the order of thousands of decimal digits), more efﬁcient methods exist. Although not of current practical interest, some of these may become more useful as security require- ments force practitioners to increase parameter sizes. The Karatsuba-Ofman method, de- scribed next, is practical in some situations. The classical algorithm for multiplication (Algorithm 14.12) takesO(n 2) bit operations for multiplying twon-bit integers. A recursive algorithm due to Karatsuba and Ofman [661] reduces the complexity of multiplying twon-bit integers toO(n1.58). Thisdivide-and- conquermethod is based on the following simple observation. Suppose thatxand yaren- bit integers andn=2 t. Thenx=2 tx1 +x0 andy=2 ty1 +y0, wherex1,y1 are thethigh- order bits ofxand y, respectively, andx0,y0 are thetlow-order bits. Furthermore,x·y= u222t +u12t +u0 whereu0 = x0 ·y0,u2 = x1 ·y1 andu1 =( x0 +x1)·(y0 +y1)−u0 −u2. It follows thatx·ycan be computed by performing three multiplications oft-bit integers (as opposed to one multiplication with2t-bit integers) along with two additions and two subtractions. For large values oft, the cost of the additions and subtractions is insigniﬁ- cant relative to the cost of the multiplications. With appropriate modiﬁcations,u0,u1 and §14.8 Notes and further references 631 (x0 + x1) ·(y0 + y1) can each be computed similarly. This procedure is continued on the intermediate values until the size of the integers reaches the word size of the computing de- vice, and multiplication can be efﬁciently accomplished. Due to the recursive nature of the algorithm, a number of intermediate results must be stored which can add signiﬁcant over- head, and detract from the algorithm’s efﬁciency for relatively small integers. Combining the Karatsuba-Ofman method with classical multiplication may have some practical signif- icance. For a more detailed treatment of the Karatsuba-Ofman algorithm, see Knuth [692], Koc¸ [698], and Geddes, Czapor, and Labahn [445]. Another commonly used method for multiple-precision integer multiplication is thediscrete Fourier transform(DFT). Although mathematically elegant and asymptotically better than the classical algorithm, it does not appear to be superior for the size of integers of practical importance to cryptography. Lipson [770] provides a well-motivated and easily readable treatment of this method. The identity given in Note 14.18 was known to Karatsuba and Ofman [661]. §14.3 There is an extensive literature on methods for multiple-precision modular arithmetic. A detailed treatment of methods for performing modular multiplication can be found in Knuth [692]. Koc¸ [698] and Bosselaers, Govaerts, and Vandewalle [176] provide comprehensive but brief descriptions of the classical method for modular multiplication. Montgomery reduction (Algorithm 14.32) is due to Montgomery [893], and is one of the most widely used methods in practice for performing modular exponentiation (Algorithm 14.94). Duss´e and Kaliski [361] discuss variants of Montgomery’s method. Montgomery reduction is a generalization of a much older technique due to Hensel (see Shand and Vuillemin [1119] and Bosselaers, Govaerts, and Vandewalle [176]). Hensel’s observation is the following. Ifmis an odd positive integer less than2 k (ka positive integer) andTis some integer such that2k ≤T< 22k, thenR0 =( T+ q0m)/2, whereq0 = Tmod 2 is an integer andR0 ≡T2−1 mod m. More generally,Ri =( Ri−1 + qim)/2, where qi = Ri−1 mod 2is an integer andRi ≡N2−i+1 mod m. SinceT< 22k, it follows that Rk−1 <2m. Barrett reduction (Algorithm 14.42) is due to Barrett [75]. Bosselaers, Govaerts, and Van- dewalle [176] provide a clear and concise description of the algorithm along with motiva- tion and justiﬁcation for various choices of parameters and steps, and compare three alter- native methods: classical (§14.3.1), Montgomery reduction (§14.3.2), and Barrett reduction (§14.3.3). This comparison indicates that there is not a signiﬁcant difference in performance between the three methods, provided the precomputation necessary for Montgomery and Barrett reduction is ignored. Montgomery exponentiation is shown to be somewhat better than the other two methods. The conclusions are based on both theoretical analysis and machine implementation for various sized moduli. Koc¸, Acar, and Kaliski [700] provide a more detailed comparison of various Montgomery multiplication algorithms; see also Nac- cache, M’Ra¨ıhi, and Raphaeli [915]. Naccache and M’silti [917] provide proofs for the correctness of Barrett reduction along with a possible optimization. Mohan and Adiga [890] describe a special case of Algorithm 14.47 whereb=2 . Hong, Oh, and Yoon [561] proposed new methods for modular multiplication and modu- lar squaring. They report improvements of50% and 30%, respectively, on execution times over Montgomery’s method for multiplication and squaring. Their approach to modular multiplication interleaves multiplication and modular reduction and uses precomputed ta- bles such that one operand is always single-precision. Squaring uses recursion and pre- 632 Ch. 14 Efﬁcient Implementation computed tables and, unlike Montgomery’s method, also integrates the multiplication and reduction steps. §14.4 The binary gcd algorithm (Algorithm 14.54) is due to Stein [1170]. An analysis of the al- gorithm is given by Knuth [692]. Harris [542] proposed an algorithm for computing gcd’s which combines the classical Euclidean algorithm (Algorithm 2.104) and binary operations; the method is called thebinary Euclidean algorithm. Lehmer’s gcd algorithm (Algorithm 14.57), due to Lehmer [743], determines the gcd of two positive multiple-precision integers using mostly single-precision operations. This has the advantage of using the hardware divide in the machine and only periodically resorting to an algorithm such as Algorithm 14.20 for a multiple-precision divide. Knuth [692] gives a comprehensive description of the algorithm along with motivation of its correctness. Co- hen [263] provides a similar discussion, but without motivation. Lehmer’s gcd algorithm is readily adapted to the extended Euclidean algorithm (Algorithm 2.107). According to Sorenson [1164], the binary gcd algorithm is the most efﬁcient method for computing the greatest common divisor. Jebelean [633] suggests that Lehmer’s gcd algo- rithm is more efﬁcient. Sorenson [1164] also describes ak-ary version of the binary gcd algorithm, and proves a worst-case running time ofO(n 2/lg n) bit operations for comput- ing the gcd of twon-bit integers. The binary extended gcd algorithm was ﬁrst described by Knuth [692], who attributes it to Penk. Algorithm 14.61 is due to Bach and Shallit [70], who also give a comprehensive and clear analysis of several gcd and extended gcd algorithms. Norton [934] described a version of the binary extended gcd algorithm which is somewhat more complicated than Algorithm 14.61. Gordon [516] proposed a method for computing modular inverses, derived from the classical extended Euclidean algorithm (Algorithm 2.107) with multiple-precision division replaced by an approximation to the quotient by an appropriate power of2; no analysis of the expected running time is given, but observed results on moduli of speciﬁc sizes are de- scribed. The Montgomery inverseofamod mis deﬁned to bea −12t mod mwheretis the bitlength ofm. Kaliski [653] extended ideas of Guyot [534] on the right-shift binary extended Eu- clidean algorithm, and presented an algorithm for computing the Montgomery inverse. §14.5 Letmi,1 ≤i≤t, be a set of pairwise relatively prime positive integers which deﬁne a residue number system (RNS). Ifn= ∏ t i=1 mi then this RNS provides an effective method for computing the product of integers modulonwhere the integers and the product are rep- resented in the RNS. Ifnis a positive integer where themi do not necessarily dividen, then a method for performing arithmetic modulonentirely within the RNS is not obvious. Couveignes [284] and Montgomery and Silverman [895] propose an interesting method for accomplishing this. Further research in the area is required to determine if this approach is competitive with or better than the modular multiplication methods described in§14.3. Algorithm 14.71 is due to Garner [443]. A detailed discussion of this algorithm and vari- ants of it are given by Knuth [692]; see also Cohen [263]. Algorithm 2.121 for applying the Chinese remainder theorem is due to Gauss; see Bach and Shallit [70]. Gauss’s algo- rithm is a special case of the following result due to Nagasaka, Shiue, and Ho [918]. The solution to the system of linear congruencesx ≡ a i (mod mi),1 ≤ i ≤ t, for pair- wise relative prime modulimi, is equivalent to the solution to the single linear congru- ence (∑ t i=1 biMi)x ≡ ∑ t i=1 aibiMi (mod M) where M = ∏ t i=1 mi,Mi = M/mi §14.8 Notes and further references 633 for1 ≤ i ≤ t, for any choice of integersbi where gcd(bi,Mi)=1 . Notice that if∑ t i=1 biMi ≡1( m o dM), thenbi ≡M−1 i (mod mi), giving the special case discussed in Algorithm 2.121. Quisquater and Couvreur [1016] were the ﬁrst to apply the Chinese remainder theorem to RSA decryption and signature generation. §14.6 Knuth [692] and Bach and Shallit [70] describe the right-to-left binary exponentiation meth- od (Algorithm 14.76). Cohen [263] provides a more comprehensive treatment of the right- to-left and left-to-right (Algorithm 14.79) binary methods along with their generalizations to thek-ary method. Koc¸ [698] discusses these algorithms in the context of the RSA public- key cryptosystem. Algorithm 14.92 is the basis for Blakley’s modular multiplication algo- rithm (see Blakley [149] and Koc¸ [698]). The generalization of Blakley’s method to process more than one bit per iteration (Note 14.93(iii)) is due to Quisquater and Couvreur [1016]. Quisquater and Couvreur describe an algorithm for modular exponentiation which makes use of the generalization and precomputed tables to accelerate multiplication inZ m. For a comprehensive and detailed discussion of addition chains, see Knuth [692], where various methods for constructing addition chains (such as thepower treeand factormeth- ods) are described. Computing the shortest addition chain for a positive integer was shown to be anNP -hard problem by Downey, Leong, and Sethi [360]. The lower bound on the length of a shortest addition chain (Fact 14.102) was proven by Sch¨onhage [1101]. An addition sequencefor positive integersa1 <a2 <··· <ak is an addition chain for ak in whicha1,a2,...,a k−1 appear. Yao [1257] proved that there exists an addition se- quence fora1 <a 2 < ··· <a k of length less thanlg ak + ck·lg ak/lg lg(ak +2 ) for some constantc. Olivos [955] established a 1-1 correspondence between addition se- quences of lengthlfora1 <a2 <··· <ak and vector-addition chains of lengthl+ k−1 where vl+k−1 =( a1,a2,...,a k). These results are the basis for the inequality given in Fact 14.107. Bos and Coster [173] described a heuristic method for computing vector- addition chains. The special case of Algorithm 14.104 (Algorithm 14.88) is attributed by ElGamal [368] to Shamir. The ﬁxed-base windowing method (Algorithm 14.109) for exponentiation is due to Brick- ell et al. [204], who describe a number of variants of the basic algorithm. Forba positive integer, letSbe a set of integers with the property that any integer can be expressed in base busing only coefﬁcients fromS.Sis called abasic digit setfor the baseb. Brickell et al. show how basic digit sets can be used to reduce the amount of work in Algorithm 14.109 without large increases in storage requirements. De Rooij [316] proposed the ﬁxed-base Euclidean method (Algorithm 14.113) for exponentiation; compares this algorithm to Algo- rithm 14.109; and provides a table of values for numbers of practical importance. The ﬁxed- base comb method (Algorithm 14.117) for exponentiation is due to Lim and Lee [767]. For a given exponent size, they discuss various possibilities for the choice of parametershand v, along with a comparison of their method to ﬁxed-base windowing. §14.7 The signed-digit exponent recoding algorithm (Algorithm 14.121) is due to Reitwiesner [1031]. A simpler description of the algorithm was given by Hwang [566]. Booth [171] described another algorithm for producing a signed-digit representation, but not necessar- ily one with the minimum possible non-zero components. It was originally given in terms of the additive group of integers where exponentiation is referred to as multiplication. In this case,−gis easily computed fromg. The additive abelian group formed from the points on an elliptic curve over a ﬁnite ﬁeld is another example where signed-digit representation is very useful (see Morain and Olivos [904]). Zhang [1267] described a modiﬁed signed-digit 634 Ch. 14 Efﬁcient Implementation representation which requires on averaget/3 multiplications for a square-and-multiply al- gorithm fort-bit exponents. A slightly more general version of Algorithm 14.121, given by Jedwab and Mitchell [634], does not require as input a binary representation of the exponent ebut simply a signed-digit representation. For binary inputs, the algorithms of Reitwiesner and Jedwab-Mitchell are the same. Fact 14.124 is due to Jedwab and Mitchell [634]. String-replacement representations were introduced by Gollmann, Han, and Mitchell [497], who describe Algorithms 14.128 and 14.130. They also provide an analysis of the expected number of non-zero entries in anSR(k) representation for a randomly selectedt-bit expo- nent (see Note 14.132), as well as a complexity analysis of Algorithm 14.130 for various values oftand k. Lam and Hui [735] proposed an alternate string-replacement algorithm. The idea is to precompute all odd powersg,g 3,g5,...,g 2k−1 for some ﬁxed positive in- tegerk. Given at-bit exponente, start at the most signiﬁcant bit, and look for the longest bitstring of bitlength at mostkwhose last digit is a1 (i.e., this substring represents an odd positive integer between1 and 2k −1). Applying a left-to-right square-and-multiply expo- nentiation algorithm based on this scanning process results in an algorithm which requires, at most,⌈t/k⌉multiplications. Lam and Hui proved that astincreases, the average number of multiplications approaches⌈t/(k+1 )⌉. Chapter /1/5 Patents and Standards Contents in Brief 15.1 Introduction............................. 635 15.2 Patents on cryptographic techniques................ 635 15.3 Cryptographic standards...................... 645 15.4 Notes and further references.................... 657 15.1 Introduction This chapter discusses two topics which have signiﬁcant impact on the use of cryptogra- phy in practice: patents and standards. At their best, cryptographic patents make details of signiﬁcant new processes and efﬁcient techniques publicly available, thereby increas- ing awareness and promoting use; at their worst, they limit or stiﬂe the use of such tech- niques due to licensing requirements. Cryptographic standards serve two important goals: facilitating widespread use of cryptographically sound and well-accepted techniques; and promoting interoperability between components involving security mechanisms in various systems. An overview of patents is given in§15.2. Standards are pursued in§15.3. Notes and further references follow in§15.4. 15.2 Patents on cryptographic techniques A vast number of cryptographic patents have been issued, of widely varying signiﬁcance and use. Here attention is focused on a subset of these with primary emphasis on unexpired patents of industrial interest, involving fundamental techniques and speciﬁc algorithms and protocols. In addition, some patents of historical interest are noted. Where appropriate, a brief description of major claims or disclosed techniques is given. Inclusion herein is intended to provide reference information to practitioners on the exis- tence and content of well-known patents, and to illustrate the nature of cryptographic pat- ents in general. There is no intention to convey any judgement on the validity of any claims. Because most patents are eventually ﬁled in the United States, U.S. patent numbers and associated details are given. Additional information including related ﬁlings in other coun- tries may be found in patent databases. For further technical details, the original patents should be consulted (see§15.2.4). Where details of patented techniques and algorithms ap- pear elsewhere in this book, cross-references are given. 635 636 Ch. 15 Patents and Standards Expiry of patents U.S. patents are valid for 17 years from the date of issue, or 20 years from the date a patent application was ﬁled. For applications ﬁled before June 8 1995 (and unexpired at that point), the longer period applies; the 20-year rule applies for applications ﬁled after this date. Priority data Many countries require that a patent be ﬁled before any public disclosure of the invention; in the USA, the ﬁling must be within one year of disclosure. A large number of countries are parties to a patent agreement which recognizespriority dates. A patent ﬁled in such a country, and ﬁled in another such country within one year thereof, may claim the date of the ﬁrst ﬁling as a priority date for the later ﬁling. Outline of patents section The discussion of patents is broken into three main subsections.§15.2.1 notes ﬁve fun- damental patents, including DES and basic patents on public-key cryptography.§15.2.2 addresses ten prominent patents including those on well-known block ciphers, hash func- tions, identiﬁcation and signature schemes.§15.2.3 includes ten additional patents address- ing various techniques, of historical or practical interest. Finally,§15.2.4 provides informa- tion on ordering patents. 15.2.1 Five fundamental patents Table 15.1 lists ﬁve basic cryptographic patents which are fundamental to current crypto- graphic practice, three involving basic ideas of public-key cryptography. These patents are discussed in chronological order. Inventors Patent # Issue date Ref. Major claim or area Ehrsam et al. 3,962,539 Jun. 08 1976 [363] DES Hellman-Difﬁe-Merkle 4,200,770 Apr. 29 1980 [551] Difﬁe-Hellman agreement Hellman-Merkle 4,218,582 Aug. 19 1980 [553] public-key systems Merkle 4,309,569 Jan. 05 1982 [848] tree authentication Rivest-Shamir-Adleman 4,405,829 Sep. 20 1983 [1059] RSA system Table 15.1:Five fundamental U.S. cryptographic patents. (i) DES block cipher The patent of Ehrsam et al. (3,962,539) covers the algorithm which later became well- known as DES (§7.4). Filed on February 24 1975 and now expired, the patent was assigned to the International Business Machines Corporation (IBM). Its background section com- ments brieﬂy on 1974 product cipher patents of Feistel (3,798,359) and Smith (3,796,830), respectively ﬁled June 30 1971 and November 2 1971. It notes that while the Feistel patent discloses a product cipher which combines key-dependent linear and nonlinear transforma- tions, it fails to disclose speciﬁc details including precisely how key bits are used, regard- ing the nonlinear transformation within S-boxes, and regarding a particular permutation. In addition, the effect of key bits is limited by the particular grouping used. The background section comments further on the cipher of Smith’s patent, noting its inherently serial nature as a performance drawback, and that both it and that of Feistel have only two types of sub- §15.2 Patents on cryptographic techniques 637 stitution boxes, which are selected as a function of a single key bit. Thus, apparently, the need for a new cipher. The patent contains ten (10) claims. (ii) Difﬁe-Hellman key agreement The ﬁrst public-key patent issued, on April 29 1980, was the Hellman-Difﬁe-Merkle patent (4,200,770). Filed on September 6 1977, it was assigned to Stanford University (Stan- ford, California). It is generally referred to as theDifﬁe-Hellman patent, as it covers Difﬁe- Hellman key agreement (§12.6.1). There are two major objects of the patent. The ﬁrst is a method for communicating securely over an insecure channel withouta priorishared keys; this can be done by Difﬁe-Hellman key agreement. The second is a method allowing au- thentication of an identity over insecure channels; this can be done using authentic, long- term Difﬁe-Hellman public keys secured in a public directory, with derivation and use of the resulting Difﬁe-Hellman secret keys providing the authentication. The patent contains eight (8) claims including the idea of establishing a session key by public-key distribution, e.g., using message exchanges as in two-pass Difﬁe-Hellman key agreement. Claim 8 is the most speciﬁc, specifying Difﬁe-Hellman using a prime modulusqand exponentsx i and xj in[1,q−1]. (iii) Merkle-Hellman knapsacks and public-key systems The Hellman-Merkle patent (4,218,582) was ﬁled October 6 1977 and assigned to the Board of Trustees of the Leland Stanford Junior University (Stanford, California). It covers public-key cryptosystems based on the subset-sum problem, i.e., Merkle-Hellman trapdoor knapsacks (now known to be insecure – see§8.6.1), in addition to various claims on public- key encryption and public-key signatures. The objects of the invention are to allow private conversations over channels subject to interception by eavesdroppers; to allow authentica- tion of a receiver’s identity (through its ability to use a key only it would be able to com- pute); and to allow data origin authentication without the threat of dispute (i.e., via public- key techniques, rather than a shared secret key). There are seventeen (17) claims, with Claims 1–6 broadly applying to public-key systems, and Claims 7–17 more narrowly fo- cused on knapsack systems. The broad claims address aspects of general methods using public-private key pairs for public-key encryption, public-key signatures, and the use of public-key encryption to provide authentication of a receiver via the receiver transmitting back to the sender a representation of the enciphered message. (iv) Tree authentication method of validating parameters Merkle’s 1982 patent (4,309,569) covers tree authentication (§13.4.1). It was ﬁled Septem- ber 5 1979, and assigned to the Board of Trustees of the Leland Stanford Junior University (Stanford, California). The main motivation cited was to eliminate the large storage require- ment inherent in prior one-time signature schemes, although the idea has wider application. The main ideas are to use a binary tree and a one-way hash function to allow authentication of leaf valuesY i associated with each useri. Modiﬁcations cited include: use of a ternary ork-ary tree in place of a binary tree; use of the tree for not only public values of one-time signatures, but for authenticating arbitrary public values for alternate purposes; and use of a distinct authentication tree for each useri, the rootRi of which replacesYi above, thereby allowing authentication of all values ini’s tree, rather than just a singleYi. The epitome of conciseness, this patent contains a single ﬁgure and just over two pages of text including four (4) claims. 638 Ch. 15 Patents and Standards (v) RSA public-key encryption and signature system The Rivest-Shamir-Adleman patent (4,405,829) was ﬁled December 14 1977, and assigned to the Massachusetts Institute of Technology. It covers the RSA public-key encryption (§8.2.1) and digital signature method (§11.3.1). Also mentioned are generalizations, includ- ing: use of a modulusnwhich is a product of three or more primes (not necessarily distinct); and using an encryption public keyeto encrypt a messageMto a ciphertextCby evaluating a polynomial∑t i=0 aiMe mod nwhere eand ai,0 ≤i≤t, are integers, and recovering the plaintextMby “utilizing conventional root-ﬁnding techniques, choosing which of any roots is the proper decoded version, for example, by the internal redundancy of the mes- sage”. Other variations mentioned include using RSA encipherment in CFB mode, or as a pseudorandom number generator to generate key pads; signing a compressed version of the message rather than the message itself; and using RSA encryption for key transfer, the key thereby transferred to be used in another encryption method. This patent has the distinction of a claims section, with forty (40) claims, which is longer than the remainder of the patent. 15.2.2 Ten prominent patents Ten prominent patents are discussed in this section, in order as per Table 15.2. Inventors Patent # Issue date Ref. Major claim or area Okamoto et al. 4,625,076 Nov. 25 1986 [952] ESIGN signatures Shamir-Fiat 4,748,668 May 31 1988 [1118] Fiat-Shamir identiﬁcation Matyas et al. 4,850,017 Jul. 18 1989 [806] control vectors Shimizu-Miyaguchi 4,850,019 Jul. 18 1989 [1125] FEAL cipher Brachtl et al. 4,908,861 Mar. 13 1990 [184] MDC-2, MDC-4 hashing Schnorr 4,995,082 Feb. 19 1991 [1095] Schnorr signatures Guillou-Quisquater 5,140,634 Aug. 18 1992 [523] GQ identiﬁcation Massey-Lai 5,214,703 May 25 1993 [791] IDEA cipher Kravitz 5,231,668 Jul. 27 1993 [711] DSA signatures Micali 5,276,737 Jan. 04 1994 [861, 862] ‘fair’ key escrow Table 15.2:Ten prominent U.S. cryptographic patents. (i) ESIGN signatures The Okamoto-Miyaguchi-Shiraishi-Kawaoka patent (4,625,076) covers the original ES- IGN signature scheme (see§11.7.2). The patent was ﬁled March 11 1985 and assigned to the Nippon Telegraph and Telephone Corporation (Tokyo), with priority data listed as March 19 1984 (Japanese patent ofﬁce). The objective is to provide a signature scheme faster than RSA. The patent contains twenty-ﬁve (25) claims. (ii) Fiat-Shamir identiﬁcation and signatures The Shamir-Fiat patent (4,748,668) covers Fiat-Shamir identiﬁcation (§10.4.2) and signa- tures (§11.4.1). It was ﬁled July 9 1986, and assigned to Yeda Research and Development Co. Ltd. (Israel). For identiﬁcation, the inventors suggest a typical number of roundstas 1 to 4, and parameter selections includingk=5 (secrets),t=4 for a2−20 probability of forgery, andk=6 ,t=5 for2−30. A range of parametersk,tforkt=7 2 is tabulated for the corresponding signature scheme, showing tradeoffs between key storage, signature size, and real-time operations required. Noted features relative to prior art include being §15.2 Patents on cryptographic techniques 639 able to pipeline computations, and being able to change the security level after the key is selected (e.g., by changingt). Generalizations noted include replacing square roots by cu- bic or higher roots. There are forty-two (42) claims. (iii) Control vectors for key management The Matyas-Meyer-Brachtl patent (4,850,017)is one of several in the area of control vectors for key management, in this case allowing a sending node to constrain the use of keys at a receiving node. It was ﬁled May 29 1987 and assigned to the IBM Corporation. Control vectors reduce the probability of key misuse. Two general methods are distinguished. In the ﬁrst method, the key and a control value are authenticated before use through veriﬁcation of a special authentication code, the key for which is part of the data being authenticated. In the second method (see§13.5.2), the key and control value are cryptographically bound at the time of key generation, such that recovery of the key requires speciﬁcation of the correct control vector. In each method, additional techniques may be employed to control which users may use the key in question. The patent contains twenty-two (22) claims. (iv) FEAL block cipher The Shimizu-Miyaguchi patent (4,850,019)gives the originally proposed ideas of the FEAL block cipher (see§7.5). It was ﬁled November 3 1986 and assigned to the Nippon Telegraph and Telephone Corporation (Tokyo), with priority data listed as November 8 1985 (Japanese patent ofﬁce). Embodiments of FEAL with various numbers of rounds are described, with ﬁgures including four- and six-round FEAL (now known to be insecure – see Note 7.100), and discussion of key lengths including 128 bits. The patent makes twenty-six (26) claims. (v) MDC-2/MDC-4 hash functions The patent of Brachtl et al. (4,908,861) covers the MDC-2 and MDC-4 hash functions (§9.4.1). It was ﬁled August 28 1987 and assigned to the IBM Corporation. The patent notes that interchanging internal key halves, as is done at a particular stage in both algorithms, is actually required for security in MDC-2 but not MDC-4; however, the common design was nonetheless used, to allow MDC-4 to be implemented using MDC-2 twice. A preliminary section of the patent discusses alternatives for providing message authentication (see§9.6), as well as estimates of the security of the new hash functions, and justiﬁcation for ﬁxing cer- tain bits within the speciﬁcation to avoid effects of weak DES keys. There are twenty-one (21) claims, mainly on building2N-bit hash functions fromN-bit block ciphers. (vi) Schnorr identiﬁcation and signatures The Schnorr patent (4,995,082) covers Schnorr’s identiﬁcation (§10.4.4) and signature (§11.5.3) schemes, and optimizations thereof involving speciﬁc pre-processing. It was ﬁled February 23 1990, with no assignee listed, and priority data given as February 24 1989 (Eu- ropean patent ofﬁce). There are eleven (11) claims. Part of Claim 6 covers a speciﬁc vari- ation of the Fiat-Shamir identiﬁcation method using a prime modulusp, such thatp−1 is divisible by a primeq, and using a baseβof orderq. (vii) GQ identiﬁcation and signatures The Guillou-Quisquater patent (5,140,634) addresses GQ identiﬁcation (Protocol 10.31) and signatures (Algorithm 11.48). It was ﬁled October 9 1991, as a continuation-in-part of two abandoned applications, the ﬁrst ﬁled September 7 1988. The original assignee was the U.S. Philips Corporation (New York). The disclosed techniques allow for authentica- tion of so-calledaccreditation information, authentication of messages, and the signing of messages. The central authentication protocol involves a commitment-challenge-response 640 Ch. 15 Patents and Standards method and is closely related to the zero-knowledge-based identiﬁcation technique of Fiat and Shamir (Protocol 10.24). However, it requires only a single protocol execution and sin- gle accreditation value, rather than a repetition of executions and a plurality of accreditation values. The cited advantages over previous methods include smaller memory requirements, and shorter overall duration due to fewer total message exchanges. The main applications cited are those involving chipcards in banking applications. There are twenty-three (23) claims, including speciﬁc claims involving the use of chipcards. (viii) IDEA block cipher The Massey-Lai patent (5,214,703) covers the IDEA block cipher (§7.6), proposed as a Eu- ropean or international alternative to DES offering greater key bitlength (and thereby, hope- fully greater security). It was ﬁled May 16 1991, and assigned to Ascom Tech AG (Bern), with priority data given as May 18 1990 from the original Swiss patent. A key concept in the cipher is the use of at least two different types of arithmetic and logical operations, with emphasis on different operations in successive stages. Three such types of operation are proposed: addition mod2 m, multiplication mod2m +1 , and bitwise exclusive-or (XOR). Symbols denoting these operations, hand-annotated in the European version of the patent (WO 91/18459, dated 28 November 1991, in German), appear absent in the text of the U.S. patent, making the latter difﬁcult to read. There are fourteen (14) ﬁgures and ten (10) multi- part claims. (ix) DSA signature scheme The patent of Kravitz (5,231,668), titled “Digital Signature Algorithm”, has become widely known and adopted as the DSA (§11.5.1). It was ﬁled July 26 1991, and assigned to “The United States of America as represented by the Secretary of Commerce, Washington, D.C.” The background section includes a detailed discussion of ElGamal signatures and Schnorr signatures, including their advantage relative to RSA – allowing more efﬁcient on-line sig- natures by using off-line precomputation. Schnorr signatures are noted as more efﬁcient than ElGamal for communication and signature veriﬁcation, although missing some “de- sirable features of ElGamal” and having the drawback that cryptanalytic experience and conﬁdence associated with the ElGamal system do not carry over. DSA is positioned as having all the efﬁciencies of the Schnorr model, while remaining compatible with the El- Gamal model from an analysis perspective. In the exemplary speciﬁcation of DSA, the hash function used was MD4. The patent makes forty-four (44) claims. (x) Fair cryptosystems and key escrow Micali’s patent (5,276,737) and its continuation-in-part (5,315,658), respectively ﬁled April 20 1992 and April 19 1993 (with no assignees listed), cover key escrow systems called “fair cryptosystems” (cf.§13.8.3). The subject of the ﬁrst is a method involving a public-key cryptosystem, for allowing third-party monitoring of communications (e.g., government wiretapping). A number of shares (see secret-sharing –§12.7) created from a user-selected private key are given to a set of trustees. By some method of veriﬁable secret sharing, the trustees independently verify the authenticity of the shares and communicate this to an au- thority, which approves a user’s public key upon receiving all such trustee approvals. Upon proper authorization (e.g., a court order), the trustees may then subsequently provide their shares to the authority to allow reconstruction of a user private key. Exemplary systems include transforming Difﬁe-Hellman (see paragraph below) and RSA public-key systems into fair cryptosystems. Modiﬁcations require onlykout ofntrustees to contribute shares to recover a user secret and prevent trustees from learning the identity of a user whose share is requested. The patent contains eighteen (18) claims, the ﬁrst 14 being restricted to public- §15.2 Patents on cryptographic techniques 641 key systems. A fair cryptosystem for Difﬁe-Hellman key agreement modulop, with a generatorg and ntrustees, may be constructed as follows. Each userAselectsnintegerss1,...,s n in the interval[1,p−1], and computess= ∑n i=1 si mod p, public sharesyi = gsi mod p, and a public keyy= gs mod p. TrusteeTi,1 ≤i≤n, is giveny, public sharesy1,...,y n, and the secret sharesi to be associated withA. Upon verifyingyi = gsi,Ti stores(A,y,si), and sends the authority a signature on(i,y,y1,...,y n). Upon receiving such valid sig- natures from allntrustees, verifying theyi in the signed messages are identical, and that y= ∏yi mod p, the authority authorizesyasA’s Difﬁe-Hellman public key. The continuation-in-part pursues time-bounded monitoring in greater detail, includ- ing use of tamper-proof chips with internal clocks. Methods are also speciﬁed allowing an authority (hereafter, the government) access to session keys, including users employing a master key to allow such access. A further method allows veriﬁcation, without monitor- ing content, that transmitted messages originated from government-approved devices. This may involve tamper-proof chips in each communicating device, containing and employing a government master keyK M. Such devices allow veriﬁcation by transmitting a redundant data string dependent on this key. The continuation-in-part has thirteen (13) claims, with the ﬁrst two (2) restricted to public-key systems. Claims 11 and 12 pursue methods for ver- ifying that messages originate from a tamper-proof device using an authorized encryption algorithm. 15.2.3 Ten selected patents Ten additional patents are discussed in this section, as listed in Table 15.3. These provide a selective sample of the wide array of existing cryptographic patents. Inventors Patent # Issue date Ref. Major claim or area Feistel 3,798,359 Mar.19 1974 [385] Lucifer cipher Smid-Branstad 4,386,233 May 31 1983 [1154] key notarization Hellman-Pohlig 4,424,414 Jan. 03 1984 [554] Pohlig-Hellman cipher Massey, Omura 4,567,600 Jan. 28 1986 [792, 956] normal basis arithmetic Hellman-Bach 4,633,036 Dec. 30 1986 [550] generating strong primes Merkle 4,881,264 Nov. 14 1989 [846] one-time signatures Goss 4,956,863 Sep. 11 1990 [519] Difﬁe-Hellman variation Merkle 5,003,597 Mar. 26 1991 [847] Khufu, Khafre ciphers Micali et al. 5,016,274 May 14 1991 [864] on-line/off-line signing Brickell et al. 5,299,262 Mar. 29 1994 [203] exponentiation method Table 15.3:Ten selected U.S. cryptographic patents. (i) Lucifer cipher Feistel’s patent (3,798,359) is of historical interest. Filed June 30 1971 and assigned to the IBM Corporation, it has now expired. The background section cites a number of earlier cipher patents including ciphering wheel devices and key stream generators. The patent discloses a block cipher, more speciﬁcally a product cipher noted as being under the control of subscriber keys, and designed to resist cryptanalysis “not withstanding ... knowledge of the structure of the system” (see Chapter 7 notes on§7.4). It is positioned as distinct from prior art systems, none of which “utilized the advantages of a digital processor and its 642 Ch. 15 Patents and Standards inherent speed.” The patent has 31 ﬁgures supporting (only) six pages of text plus one page of thirteen (13) claims. (ii) Key notarization The Smid-Branstad patent (4,386,233) addresses key notarization (§13.5.2). It was ﬁled September 29 1980, with no assignee listed. A primary objective of key notarization is to prevent key substitution attacks. The patent contains twenty-one (21) claims. (iii) Pohlig-Hellman exponentiation cipher The Hellman-Pohlig patent (4,424,414) was ﬁled May 1 1978 (four and one-half months after the RSA patent), and assigned to the Board of Trustees of the Leland Stanford Junior University (Stanford, California). It covers the Pohlig-Hellman symmetric-key exponenti- ation cipher, wherein a primeqis chosen, along with a secret keyK,1 ≤K≤q−2, from which a second keyD,1 ≤D≤q−2, is computed such thatKD ≡1m o d(q−1). A message Mis enciphered asC= MK mod q, and the plaintext is recovered by com- putingCD mod q= M. Two parties make use of this by arranging,a priori, to share the symmetric-keysKand D. The patent contains two (2) claims, specifying a method and an apparatus for implementing this block cipher. Although of limited practical signiﬁcance, this patent is often confused with the three well-known public-key patents of Table 15.1. (iv) Arithmetic inFFF 2m using normal bases Two patents of Massey and Omura are discussed here. The Omura-Massey patent (4,587,627) teaches a method for efﬁcient multiplication of elements of a ﬁnite ﬁeldF2m by exploiting normal bases representations. It was ﬁled September 14 1982, with prior- ity data November 30 1981 (European patent ofﬁce), and was issued May 6 1986 with the assignee being OMNET Associates (Sunnyvale, California). The customary method for representing a ﬁeld elementβ∈F 2m involves a polynomial basis1,x,x2,x3,...,x m−1, with β = ∑m−1 i=0 aixi, ai ∈{ 0,1}(see§2.6.3). Alternatively, using a normal ba- sisx,x2,x4,...,x 2m−1 (withxselected such that these are linearly independent) allows one to representβas β = ∑m−1 i=0 bix2i ,bi ∈{ 0,1}. The inventors note that this rep- resentation “is unconventional, but results in much simpler logic circuitry”. For exam- ple, squaring in this representation is particularly efﬁcient (noted already by Magleby in 1963) – it requires simply a rotation of the coordinate representation from[b m−1 ...b 1b0] to[bm−2 ...b1b0bm−1]. This follows sincex2m ≡1 and squaring inF2m is a linear opera- tion in the sense that (B+C)2 = B2+C2; furthermore,D= B×CimpliesD2 = B2×C2. From this, the main object of the patent follows directly: to multiply two elementsBand Cto yieldD= B×C=[ dm−1 ...d1d0], the same method used for computingdm−1 can be used to sequentially producedi,m−2 ≤i≤0, by applying it to one-bit rotations of the representations ofBand C. Alternatively,msuch identical processes can be used to compute themcomponentsdi in parallel. The patent makes twenty-four (24) claims. The closely related Massey-Omura patent (4,567,600) includes claims on exponentia- tion inF2m using normal bases. It was likewise ﬁled September 14 1982 and assigned to OMNET Associates (Sunnyvale, California), with priority date February 2 1982 (European patent ofﬁce). Its foundation is the observation that using a normal basis representation al- lows efﬁcient exponentiation inF 2m (Claim 16), since the cost of squaring (see above) in the customary square-and-multiply exponentiation technique is eliminated. A second subject is the implementation of Shamir’s three-pass protocol (Protocol 12.22) using modular ex- ponentiation inF 2m as the ciphering operation along with a normal basis representation for elements; and subsequently employing a shared key, established by this method, as the key in anF2m exponentiation cipher (cf. Hellman-Pohlig patent) again using normal bases. A §15.2 Patents on cryptographic techniques 643 further object is a method for computing pairs of integerse,dsuch thated≡1m o d2m−1. Whereas customarilyeis selected and, from it,dis computed via the extended Euclidean algorithm (which involves division), the new technique selects a group elementHof high order, then chooses a random integerRin[1,2m −2], and computese= HR,d= H−R. The patent includes twenty-six (26) claims in total. (v) Generation of strong primes The Hellman-Bach patent (4,633,036) covers a method for generating RSA primespand q and an RSA modulusn= pqsatisfying certain conditions such that factoringnis believed to be computationally infeasible. The patent was ﬁled May 31 1984 and assigned to Martin E. Hellman. The standard strong prime conditions (Deﬁnition 4.52) are embedded:p−1 requiring a large prime factorr;p+1 requiring a large prime factors; andr−1 requiring a large prime factorr′. A new requirement according to the invention was thats−1 have a large prime factors′, with cited justiﬁcation that the (then) best known factoring meth- ods exploiting smalls′requireds′operations. The patent includes twenty-four (24) claims, but is now apparently of historical interest only, as the best-known factoring techniques no longer depend on the cited properties (cf.§4.4.2). (vi) Efﬁcient one-time signatures using expanding trees Merkle’s 1989 patent (4,881,264), ﬁled July 30 1987 with no assignee listed on the issued patent, teaches how to construct authentication trees which may be expanded arbitrarily, without requiring a large computation when a new tree is constructed (or expanded). The primary cited use of such a tree is for making available public valuesy(corresponding to secret valuesx) of a userAin a one-time signature scheme (several of which are summa- rized). In such schemes, additional public values are continually needed over time. The key idea is to associate with each node in the tree three vectors of public information, each of which contains sufﬁcient public values to allow one one-time signature; call these the LEFT, RIGHT, and MESSAGE vectors. The combined hash valueH i of all three of these vectors serves as the hash value of the nodei. The root hash valueH1 is made widely avail- able, as per the root value of ordinary authentication trees (§13.4.1). A new messageMmay be signed by selecting a previously unused node of the tree (e.g.,H1), using the associated MESSAGE vector for a one-time signature thereon. The tree may be expanded downward from nodei(e.g.,i=1 ), to provide additional (veriﬁably authentic) public values in a new left sub-node2ior a right sub-node2i+1 , by respectively using the LEFT and RIGHT vectors at nodeito (one-time) sign the hashesH 2i and H2i+1 of the newly created public values in the respective new nodes. Full details are given in the patent; there are nine (9) claims. The one-time signatures themselves are based on a symmetric cipher such as DES; the associated one-way functionFof a private valuexmay be created by computingy= F(x)= DESx(0), i.e., encrypting a constant value usingxas key; and a hash function for the authentication tree may also be constructed using DES. Storage requirements on user Afor its own tree are further reduced by noting that onlyxvalues need be stored; and that these may be pseudorandomly generated, for example, lettingJ= 0, 1, 2 denote the LEFT, RIGHT, and MESSAGE vectors, and assuming thatKpublic values are needed per one- time signature, theK th valuexin a vector of public values at nodeImay be deﬁned as x[I,J,K]= DESKA(I\|\|J\|\|K), whereKA isA’s secret key and “\|\|” denotes concatena- tion. 644 Ch. 15 Patents and Standards (vii) Goss variation of Difﬁe-Hellman The patent of Goss (4,956,863) covers a variation of Difﬁe-Hellman key agreement essen- tially the same as Protocol 12.53. It was ﬁled April 17 1989 and assigned to TRW Inc. (Redondo Beach, California). The primary application cited is an authenticated key estab- lishment technique, completely transparent to end-users, for facsimile (FAX) machines on existing telephone networks. At the time of manufacture, a unique device identiﬁer and a signed certiﬁcate binding this to a long-term Difﬁe-Hellman public key (public exponen- tial) is embedded in each device. The identity in the certiﬁcate, upon veriﬁcation, may be used as the basis on which to accept or terminate communications channels. Such a proto- col allows new session keys for each FAX call, while basing authentication on long-term certiﬁed keys (cf. Remark 12.48; but regarding security, see also Note 12.54). The patent makes sixteen (16) claims. (viii) Khufu and Khafre block ciphers Merkle’s 1991 patent (5,003,597) covers two symmetric-key block ciphers named Khufu and Khafre (see§7.7.3). These were designed speciﬁcally as fast software-oriented alter- natives to DES, which itself was designed with hardware performance in mind. The patent was ﬁled December 21 1989 and assigned to the Xerox Corporation. Khufu and Khafre have block size 64 bits and a user-selectable number of rounds. Khufu has key bitlength up to 512 bits, and S-boxes derived from the input key; it encrypts 64-bit blocks faster than Khafre. Khafre has ﬁxed S-boxes, and a key of selectable size (with no upper bound), though larger keys impact throughput. The majority of the patent consists of C-code listings specifying the ciphers. The patent contains twenty-seven (27) claims. (ix) On-line/off-line digital signatures The Micali-Goldreich-Even patent (5,016,274) teaches on-line/off-line digital signature schemes. The patent was ﬁled November 8 1988, with no assignee listed. The basic idea is to carry out a precomputation to reduce real-time requirements for signing a particular mes- sagem. The pre-computation, executed during idle time and independent ofm, involves generation of matching one-time public and private keying material for a fast (one-time) ﬁrst signature scheme, and using a second underlying signature scheme to create a signa- tures 2 over the one-time public key. This key from the ﬁrst scheme is then used to create a signatures1 on m. The overall signature onmis(s1,s2). Appropriate hash functions can be used as usual to allow signing of a hash valueh(m) rather thanm. In the exemplary method, Rabin’s scheme is the underlying signature scheme, and DES is used both to build a one-time signature scheme and for hashing. Regarding security of the overall scheme, a one-time scheme, if secure, is presumed secure against chosen-text attack (since it is used only once); the underlying scheme is secure against chosen-text attack because it signs only strings independent of a messagem. The method thus may convert any signature scheme into one secure against chosen-text attacks (should this be a concern), or convert any un- derlying signature scheme to one with smaller real-time requirements. The patent contains thirty-three (33) claims. (x) Efﬁcient exponentiation for ﬁxed base The Brickell-Gordon-McCurley patent (5,299,262) teaches a method for fast exponentia- tion for the case where a ﬁxed base is re-used; see also page 633. This has application in systems such as the ElGamal, Schnorr, and DSA signature schemes. The patent was ﬁled August 13 1992, issued March 29 1994, and assigned to “The United States of America as represented by the United States Department of Energy, Washington, D.C.” The method is presented in Algorithm 14.109. The patent contains nine (9) claims. §15.3 Cryptographic standards 645 15.2.4 Ordering and acquiring patents Any American patent may be ordered by patent number from the U.S. Patent and Trade- mark Ofﬁce (PTO). Written requests should be posted to: PTO, Washington, D.C., 20231, USA. Telephone requests may also be made at +703-305-4350, with payment by credit card. A nominal fee applies (e.g., US$3 for patents returned by postal mail; or US$6 for re- turns by fax, usually the same day). For on-line information on recent patents, consult URL http://www.micropatent.com(e.g., specifying patent class code 380 for cryptog- raphy). 15.3 Cryptographic standards This section summarizes cryptographic and security standards of practical interest. These facilitate widespread use of cryptographically sound techniques, and interoperability of sys- tems and system components. Tables 15.4–15.11 present an overview allowing relevant standards to be located and identiﬁed, and access to formal title information allowing acqui- sition of particular standards. These tables may also be used to locate standards addressing particular areas (e.g., key management). For speciﬁc details of techniques and algorithms, the original standards should be consulted. Where relevant technical details appear else- where in the book, cross-references are given. Outline of standards section §15.3.1 presents international (ISO and ISO/IEC) application-independent standards on cryptographic techniques.§15.3.2 summarizes banking security standards, subdivided into ANSI and ISO standards.§15.3.3 considers international security architectures and frame- works (ISO and X.509).§15.3.4 summarizes security-related standards for use by U.S. federal government departments.§15.3.5 addresses selected Internet speciﬁcations, while §15.3.6 notes selected de facto industry standards.§15.3.7 provides information allowing acquisition of standards. 15.3.1 International standards – cryptographic techniques The International Organization for Standardization (ISO) and the International Electrotech- nical Commission (IEC) develop standards individually and jointly. Joint standards are developed under the joint technical committee ISO/IEC JTC 1. ISO and ISO/IEC stan- dards progress through the following draft stages before maturing to the International Stan- dard status: Working Draft (WD); Committee Draft (CD); and Draft International Standard (DIS). Each ISO and ISO/IEC standard is reviewed every ﬁve years, at which time it is ei- ther reafﬁrmed, revised, or retracted. The ISO/IEC subcommittee responsible for standard- izing generic cryptographic techniques is SC 27 (ISO/IEC JTC 1 SC 27). Table 15.4 lists selected ISO and ISO/IEC standards on cryptographic techniques. ISO 8372: This standard speciﬁes the four well-known modes of operation of a block cipher – electronic codebook (ECB), cipher block chaining (CBC), cipher feedback (CFB), and output feedback (OFB). These modes were originally standardized for DES in FIPS 81 (1980) and ANSI X3.106 (1983). ISO 8372 (ﬁrst published in 1987) speciﬁes these modes for general 64-bit block ciphers (cf. ISO/IEC 10116). 646 Ch. 15 Patents and Standards ISO # Subject Ref. 8372 modes of operation for a 64-bit cipher [574] 9796 signatures with message recovery (e.g., RSA) [596] 9797 data integrity mechanism (MAC) [597] 9798–1 entity authentication – introduction [598] 9798–2 — using symmetric encipherment [599] 9798–3 — using public-key techniques [600] 9798–4 — using keyed one-way functions [601] 9798–5 — using zero-knowledge techniques [602] 9979 register of cryptographic algorithms [603] 10116 modes of operation for ann-bit cipher [604] 10118–1 hash functions – introduction [605] 10118–2 — using block ciphers [606] 10118–3 — customized algorithms [607] 10118–4 — using modular arithmetic [608] 11770–1 key management – introduction [616] 11770–2 — symmetric techniques [617] 11770–3 — asymmetric techniques [618] 13888–1 non-repudiation – introduction [619] 13888–2 — symmetric techniques [620] 13888–3 — asymmetric techniques [621] 14888–1 signatures with appendix – introduction [622] 14888–2 — identity-based mechanisms [623] 14888–3 — certiﬁcate-based mechanisms [624] Table 15.4:ISO and ISO/IEC standards for generic cryptographic techniques. ISO/IEC 9796: This standard speciﬁes a generic mechanism for digital signature sch- emes giving message recovery (see§11.3.5 and ANSI X9.31–1; cf. ISO/IEC 14888). Ex- amples are given in its Annex B corresponding to RSA and Rabin’s variant thereof (with encryption exponent 2). The main part of the standard is a redundancy scheme, intended to be generically applicable to a large class of signature schemes, although speciﬁcally de- signed to preclude attacks on schemes such as RSA and Rabin which have a multiplicative property. ISO/IEC 9797: This standard deﬁnes a message authentication code (MAC) based on the CBC mode of operation of a block cipher, similar to the MAC algorithms of ISO 8731– 1, ISO 9807, ANSI X9.9, and ANSI X9.19 (see Algorithm 9.58). 1 Relative to these, in 9797 them-bit MAC result is constrained only bym≤n(the leftmost or most signiﬁcant bits are retained), the block cipher is unspeciﬁed but hasn-bit blocks, and a second padding method is speciﬁed. These other MAC algorithms may be viewed as special cases of 9797; for example, the speciﬁc valuesn=6 4 and m=3 2 along with use of the ﬁrst padding method (see below) and DES as the block cipher yields the MAC of X9.9. In 9797, one of two speciﬁed padding methods must be selected (Algorithms 9.29, 9.30). The ﬁrst pads the data input by appending zero or more 0-bits, as few as necessary, to obtain a string whose bitlength is a multiple ofn. The second method always appends to the data input a single 1-bit, and then zero or more 0-bits, as few as necessary, to obtain 1Speciﬁc technical details are provided for MAC standards in this chapter moreso than for other standards, in an attempt to clarify the differences between the large number of CBC-MAC standards which differ only in ﬁne details. §15.3 Cryptographic standards 647 a string whose bitlength is a multiple ofn. Annex A speciﬁes two optional processes; An- nex B provides examples. The ﬁrst optional process is the optional process as described under ANSI X9.19 in§15.3.2; this reduces the threat of exhaustive key search and chosen- plaintext attacks, and is recommended whenm= n(see Remark 9.59). The alternative second optional process, providing protection against chosen-plaintext attacks, employs a second keyK′(possibly derived fromK) to encrypt the (previously ﬁnal) output block, before extracting them-bit MAC result. ISO/IEC 9798: Parts subsequent to the introduction (9798–1) of this standard spec- ify entity authentication mechanisms based on: symmetric encryption algorithms (9798–2); public-key signature algorithms (9798–3); a cryptographic check function or MAC (9798– 4); and other customized techniques (9798–5), historically referred to by academics as zero- knowledge techniques. The mechanisms use timestamps, sequence numbers, and random numbers as time-variant parameters (§10.3.1). The 9798-3 mechanisms are functionally analogous to those of X.509, and the 9798-3 two-pass and three-pass techniques based on random number challenge-response are the source for those in FIPS 196. 9798-2 speciﬁes four entity authentication mechanisms (as given in§10.3.2) involv- ing two partiesAand Band requiring that they share a symmetric keya priori, for use in a symmetric encryption algorithm. When timestamps or sequence numbers are used, these mechanisms require one and two messages, respectively, for unilateral and mutual entity au- thentication; using challenge-response based on random numbers, one additional message is required in each case. 9798-3 includes four analogous mechanisms (see§10.3.3) wherein the role of the symmetric encryption algorithm is replaced by a digital signature algorithm, and the requirement of shared symmetric keys is replaced by that of possession of authen- tic (or the capability to authenticate) public keys. 9798-4 speciﬁes four analogous mecha- nisms (again see§10.3.2) where symmetric encryption as used in 9798-2 is replaced by a cryptographic check function or MAC. 9798-2 speciﬁes two additional mutual authentica- tion mechanisms for the case thatAand Bdo not share a keya priori, but each does share a key with a trusted third partyT; these require two further messages (for communication withT) beyond those for the respective mutual entity authentication mechanisms above. 9798-5 (draft) includes an identity-based identiﬁcation protocol of which Fiat-Shamir (cf. Protocol 10.24) and GQ identiﬁcation (Protocol 10.31) are special cases, and a protocol based on public-key decryption with witness (see§10.3.3). ISO/IEC 9979: This standard speciﬁes procedures allowing certain entities (e.g., ISO member bodies and liaison organizations) to register encryption algorithms in an ofﬁcial ISO register of such algorithms. Registration involves no security evaluation or assessment (the policy of ISO/IEC is to not standardize encryption algorithms themselves). The stan- dard speciﬁes the formats required for such register entries, and registration results in the assignment of a unique identiﬁer to each algorithm, e.g., to allow interoperability. For fur- ther information, see page 660. ISO/IEC 10116: This standard speciﬁes the same four modes of block-cipher oper- ation as ISO 8372, but subsumes that standard by allowing generaln-bit block ciphers. ISO/IEC 10116 also provides greater detail regarding various properties of the modes, and sample calculations based on DES. ISO/IEC 10118: This is a multi-part standard on cryptographic hashing algorithms. 10118–1 speciﬁes common deﬁnitions and general requirements. 10118–2 speciﬁes two generic constructions based onn-bit block ciphers: the Matyas-Meyer-Oseas hash function (Algorithm 9.41) and a block-cipher independent MDC-2 (cf. Algorithm 9.46). The draft standard 10118–3 includes SHA–1 (Algorithm 9.53), RIPEMD-128 and RIPEMD-160 (Al- gorithm 9.55). The draft 10118–4 includes MASH-1 and MASH-2 (see Algorithm 9.56). ISO/IEC 11770: This multi-part standard addresses generic key management and spe- 648 Ch. 15 Patents and Standards ciﬁes key establishment mechanisms. 11770–1 is a key management framework and over- view including discussion of the key life cycle, protection requirements for keying mate- rial, and roles of third parties in key establishment. 11770-2 speciﬁes key establishment mechanisms based on symmetric techniques, including those wherein two parties commu- nicate point-to-point (as in§12.3.1), those similar to the Kerberos and Otway-Rees proto- cols involving a trusted server or key distribution center (§12.3.2), and those involving a key translation center (e.g., Protocol 13.12). 11770-3 speciﬁes key establishment mechanisms based on asymmetric techniques. These are divided into key agreement protocols, practi- cal instantiations of which are based on Difﬁe-Hellman and similar techniques (§12.6.1); and key transfer protocols, which typically involve both public-key encryption and digital signatures (§12.5.2) including adaptations of the random number based ISO/IEC 9798-3 mechanisms involving transfer of an embedded encrypted key. ISO/IEC 13888: This multi-part (draft) standard addresses non-repudiation services (protection against false denials) related to the transfer of a message from an originator to a recipient. Mechanisms are speciﬁed for non-repudiation of origin (denial of being the originator of a message), non-repudiation of delivery (denial of having received a mes- sage), and non-repudiation associated with the actions of a third party acting as a transfer agent on behalf of others. 13888–1 (draft) provides a non-repudiation model and overview. 13888-2 (draft) speciﬁes mechanisms involving symmetric techniques (encipherment and keyed one-way functions). 13888-3 (draft) speciﬁes mechanisms involving asymmetric techniques and the use of digital signatures. ISO/IEC 14888: This multi-part (draft) standard addresses schemes for signature with appendix (see§11.2.2 and ANSI X9.30–1; cf. ISO/IEC 9796). 14888–1 (draft) provides common deﬁnitions and a general overview including models outlining the steps required for signature generation and various classes of veriﬁcation processes. 14888–2 (draft) ad- dresses identity-based signature mechanisms, wherein the signature veriﬁcation key is a public function of the signer’s identity. 14888–3 (draft) addresses certiﬁcate-based mecha- nisms, wherein this public key is explicitly speciﬁed and, for example, distributed by means of a certiﬁcate. These may include DSA and similar signature mechanisms such as ElGa- mal, Schnorr signatures, and RSA. 15.3.2 Banking security standards (ANSI, ISO) This section considers banking security standards developed by ANSI and by ISO. Banking security standards are typically divided into wholesale and retail banking (see Table 15.5). Wholesale bankinginvolves transactions between ﬁnancial institutions.Retail bankingin- volves transactions between institutions and private individuals, including automated teller machine (ATM) and point-of-sale (POS) transactions, and credit authorizations. category transaction volume average transaction value retail high (millions per day) $50 wholesale low (thousands per day) $3 million Table 15.5:Retail vs. wholesale banking characteristics. (i) ANSI encryption standards The American National Standards Institute (ANSI) develops standards through various Ac- credited Standards Committees (ASCs). Accreditation implies that standards developed un- §15.3 Cryptographic standards 649 der a particular committee become ANSI standards. Accredited committees include ASC X3 – Information Processing Systems; ASC X9 – Financial Services; and ASC X12 – Elec- tronic Business Data Interchange. Table 15.6 lists selected ANSI encryption and banking security standards developed under X3 and X9. ANSI X3.92: This standard speciﬁes the DES algorithm, which ANSI standards refer to as the Data Encryption Algorithm (DEA). X3.92 is technically the same as FIPS 46. ANSI X3.106: This standard speciﬁes DES modes of operation, or DEA modes of op- eration as referred to in ANSI standards. X3.106 is technically the same as FIPS 81 (cf. ISO 8372). An appendix in FIPS 81 contains additional background information on the various modes. (ii) ANSI banking security standards ASC X9 subcommittee X9F develops information security standards for the ﬁnancial ser- vices industry. Banking security standards include cryptographic and operational require- ments, with a heavy emphasis on controls, audit, sound business practices, and interoper- ability. Among the working groups under X9F, most of the cryptographic work is in X9F1 (public key cryptography and cryptographic tools) and X9F3 (security in wholesale ﬁnan- cial telecommunications). ANSI # Subject Ref. X3.92 data encryption algorithm (DEA) [33] X3.106 data encryption algorithm (DEA) modes [34] X9.8 PIN management and security [35] X9.9 message authentication (wholesale) [36] X9.17 key management (wholesale; symmetric) [37] X9.19 message authentication (retail) [38] X9.23 encryption of messages (wholesale) [39] X9.24 key management (retail) [40] X9.26 sign-on authentication (wholesale) [41] X9.28 multi-center key management (wholesale) [42] X9.30–1 digital signature algorithm (DSA) [43] X9.30–2 secure hash algorithm (SHA) for DSA [44] X9.31–1 RSA signature algorithm [45] X9.31–2 hashing algorithms for RSA [46] X9.42 key management using Difﬁe-Hellman [47] X9.45 attribute certiﬁcates and other controls [49] X9.52 triple DES and modes of operation [50] X9.55 certiﬁcate extensions (v3) and CRLs [51] X9.57 certiﬁcate management [52] Table 15.6:ANSI encryption and banking security standards. ANSI X9.8: This standard addresses PIN management and security. It consists of ISO 9564 reproduced in its entirety, with clearly marked “X9 Notes” added where required to adapt the text for use as an ANSI X9 standard. A standard means for interchanging PIN data is speciﬁed. Annex A of 9564 (procedures for the approval of an encipherment algorithm) is included; the only currently speciﬁed approved algorithm is DES. Annex B (general prin- ciples for key management) is also retained from 9564, but noted as superseded by X9.24 (retail key management). 650 Ch. 15 Patents and Standards ANSI X9.9: This standard speciﬁes a DES-based message authentication code (MAC) algorithm for wholesale banking as summarized below (cf. X9.19 for retail banking). If data is protected by both authentication and encryption mechanisms, a different key is re- quired for each purpose. Message replay is precluded by use of date and message identiﬁer ﬁelds. Appendix B includes sample MAC computations. X9.9 requires key management in accordance with ANSI X9.17, and also addresses implementation issues including coded character sets and representations, ﬁeld delimiters, and message normalization (e.g., replac- ing carriage returns or line feeds by space characters, and multiple spaces by single spaces), and notes other practical concerns such as escape sequences beyond the scope of a MAC causing over-writing of authenticated data ﬁelds on display devices. The X9.9 MAC algorithm may be implemented using either the cipher-block chaining (CBC) or 64-bit cipher feedback (CFB-64) mode, initialized to produce the same result (see Note 15.1). Final data blocks with fewer than 64 bits are left-justiﬁed and zero-bits are appended to complete the block before processing. The MAC result is speciﬁed to be the leftmost 32 bits of the ﬁnal DES output. X9.9 states that the capability to generate 48-bit and 64-bit MAC values should also exist. 15.1 Note (CBC-MAC and equivalent CFB-64 MAC) For data blocksD 1,...,D t and a ﬁxed MAC key K, equivalent MACs may be generated using either the CBC or 64-bit ci- pher feedback (CFB-64) modes. In the CBC case, the MACCt is deﬁned byCi = EK(Di⊕Ci−1) for1 ≤i≤tand C0 = IV =0 . For the CFB-64 case, letOi = EK(Ii) be the output from the block encryption at stageifor1 ≤i≤t, whereIi = Di⊕Oi−1 for 2 ≤i≤tand I1 = D1 (the ﬁrst 8 data bytes serve as IV). NoteOt = Ct from above. (A blockDt+1 =0 may be introduced if the CFB implementation interface requires the ﬁnal outputOt be XORed to a data block before release.) ANSI X9.17 : This standard, which was the basis for ISO 8732, speciﬁes manual and automated methods (symmetric-based) for wholesale banking key management, including key establishment techniques and protection of keys in key management facilities. A key management hierarchy is deﬁned consisting of manually-distributed key-encrypting keys, electronically-distributed key-encrypting keys, and electronically-distributed data or trans- action keys for authentication or encryption. Key management techniques include the use of key counters, key offsetting, and key notarization. Key establishment settings include direct exchange between two nodes (point-to-point), and both key distribution centers (KDCs) and key translation centers (KTCs). ANSI X9.19 : This standard speciﬁes a DES-based message authentication code (MAC) algorithm for retail banking (cf. X9.9 for wholesale banking). Implementation and other issues are addressed as per X9.9, and the MAC algorithm itself is essentially the same as X9.9, differing in that the MAC result is the leftmostmbits of the ﬁnal 64-bit output, where mis to be speciﬁed by the application. An optional X9.19 procedure using a sec- ond keyK ′is speciﬁed for increased protection against exhaustive key determination: the (previously) ﬁnal output is decrypted usingK′and then re-encrypted under the original key. The resulting algorithm is widely referred to as theretail MAC; see Figure 9.6. ANSI X9.23: This standard addresses message formatting and representation issues re- lated to the use of DES encryption in wholesale banking transactions. These include ﬁeld delimiting and padding, as well as ﬁltering methods required to prevent ciphertext bit se- quences from interfering with communications protocols when inadvertently interpreted as control characters (e.g., end-of-transmission). ANSI X9.24 : This standard, which motivated ISO 11568, speciﬁes manual and au- tomated methods for retail key management, addressing authentication and (DES-based) §15.3 Cryptographic standards 651 encryption of PINs, keys, and other data. Guidelines include protection requirements at various stages in the key management life cycle. Appendices provide additional informa- tion, including (Appendix D) methods providing unique per-transaction keys, updated af- ter each transaction as a one-way function of the current key and transaction-speciﬁc de- tails; and (Appendix E) how to derive a large number of different terminal keys (for dis- tinct terminals) from a common base key, simplifying key management for servers which must communicate with all terminals. Such derived keys may be combined with the unique per-transaction key methods. ANSI X9.26 : This standard speciﬁes two main classes of entity authentication mech- anisms of use for access control. The ﬁrst involves user passwords. The second involves cryptographic keys used in DES-based challenge-response protocols (e.g., a time-variant parameter challenge must be ECB-encrypted). The latter class is subdivided, on the basis of granularity, into user-unique and node-unique keys. ANSI X9.28: This standard extends X9.17 to allow the distribution of keying material (using X9.17 protocols) between entities (subscriber nodes) which neither share a common key, nor share a key with a common central server (KDC or KTC). Two or more key centers form amultiple-center groupto provide a more general key distribution service allowing the establishment of keying material between any two subscribers sharing a key with at least one center in the group. As there are no known or proposed implementations of this stan- dard, it appears destined to be withdrawn from the ANSI suite. ANSI X9.30: The ﬁrst in a suite of ANSI public-key standards, X9.30–1 and X9.30–2 specify DSA and SHA for the ﬁnancial services industry, as per FIPS 186 and FIPS 180, respectively. ANSI X9.31: The (draft) standard X9.31–1 parallels X9.30–1, and speciﬁes a signature mechanism based on an RSA signature algorithm, more speciﬁcally the ISO/IEC 9796 vari- ant combined with a hashing algorithm. The (draft) standard X9.31–2 deﬁnes hash func- tions for use with Part 1, including MDC-2. ANSI X9.42 : This (draft) standard speciﬁes several variations of unauthenticated Difﬁe-Hellman key agreement, providing shared symmetric keys for subsequent crypto- graphic use. ANSI X9.45 : This (draft) standard employs a particular type of attribute certiﬁcate (§13.4.2) called anauthorization certiﬁcate, and other techniques from ANSI X9.57, to al- low a party to determine whether a received message or signed document is authorized with respect to relevant rules or limits, e.g., as speciﬁed in the authorization certiﬁcate. ANSI X9.52 : This (draft) standard for encryption offers improvements over DES se- curity by specifying a number of modes of operation for triple-DES encryption, including the four basic modes of ISO 8372, enhanced modes intended to provide additional protec- tion against advanced cryptanalytic attacks, and message-interleaved and pipelined modes intended to allow increased throughput in multi-processor systems. ANSI X9.55 : This (draft) standard speciﬁes extensions to the certiﬁcate deﬁnitions of ANSI X9.57 corresponding to, and aligned with, ISO certiﬁcate extensions for ITU-T X.509 Version 3 certiﬁcates (see page 660). ANSI X9.57 : This (draft) certiﬁcate management standard includes both technical speciﬁcations deﬁning public-key certiﬁcates (based on ITU-T X.509) for electronic com- merce, and business controls necessary to employ this technology. The initial version is deﬁned for use with DSA certiﬁcates, in conjunction with ANSI X9.30–1. (iii) ISO banking security standards ISO banking security standards are developed under the ISO technical committee TC68 – Banking and Related Financial Services. TC68 subcommittees include TC68/SC2 (whole- 652 Ch. 15 Patents and Standards sale banking security) and TC68/SC6 (retail banking security and smart card security). Ta- ble 15.7 lists selected ISO banking security standards. ISO # Subject Ref. 8730 message authentication – requirements (W) [575] 8731–1 message authentication – CBC-MAC [576] 8731–2 message authentication – MAA [577] 8732 key management/symmetric (W) [578] 9564 PIN management and security [579] 9807 message authentication – requirements (R) [581] 10126 message encipherment (W) [582] 10202–7 key management for smart cards [584] 11131 sign-on authentication [585] 11166–1 key management/asymmetric – overview [586] 11166–2 key management using RSA [587] 11568 key management (R), in 6 parts [588] Table 15.7:ISO banking security standards (W–wholesale; R–retail). ISO 8730: Together with ISO 8731, this wholesale banking standard for message authentication code (MAC) algorithms forms the international equivalent of ANSI X9.9. ISO 8730 is algorithm-independent, and speciﬁes methods and requirements for the use of MACs including data formatting and representation issues, and a method by which speciﬁc algorithms are to be approved. ISO 8731: ISO 8731–1 and 8731–2 specify particular MAC algorithms complemen- tary to the companion standard ISO 8730. 8731–1 speciﬁes a DES-based CBC-MAC with m=3 2(cf. ISO/IEC 9797). 8731–2 speciﬁes the Message Authenticator Algorithm, MAA (Algorithm 9.68). ISO 8732: This standard for key management in wholesale banking was derived from ANSI X9.17, and is its international equivalent. ISO 9564: This standard, used as the basis for ANSI X9.8, speciﬁes minimum mea- sures for the management and security of Personal Identiﬁcation Numbers (PINs). Part 1 speciﬁes principles and techniques to protect against disclosure of PINs to unauthorized par- ties during the PIN life cycle. Part 2 speciﬁes encipherment algorithms approved to protect PINs. ISO 9807: This standard for message authentication in retail banking is analogous to ANSI X9.19 (cf. ISO 8730/8731–1 vs. ANSI X9.9), but does not address data representa- tion issues, and names two approved algorithms in Annex A – the CBC-MAC of 8731–1 (allowing optional ﬁnal processing as per X9.19), and the MAA of 8731-2. ISO 10126: This multi-part standard is the international equivalent of X9.23 address- ing conﬁdentiality protection of (parts of) ﬁnancial messages. ISO 10126–1 provides gen- eral principles; 10126–2 deﬁnes a speciﬁc algorithm – DES. ISO 10202: This eight-part standard addresses security architecture issues for inte- grated circuit cards (chipcards) used for ﬁnancial transactions. In particular, ISO 10202-7 speciﬁes key management aspects. ISO 11131: This standard for sign-on authentication is the international (non-DES spe- ciﬁc) analogue of ANSI X9.26. ISO 11166: This multi-part standard for banking key management speciﬁes asymmet- ric techniques for distributing keys for symmetric algorithms. It was developed from ISO §15.3 Cryptographic standards 653 8732, which uses symmetric techniques only. Part 1 speciﬁes general principles, proce- dures, and formats, including background regarding key protection during its life cycle, cer- tiﬁcation of keying material, key distribution by either key exchange (e.g., Difﬁe-Hellman) or key transport, and cryptographic service messages. Further parts are intended to deﬁne approved algorithms for use with the procedures of Part 1. Part 2 speciﬁes the RSA al- gorithm for both encipherment and digital signatures; RSA formatting differs from both ISO/IEC 9796 and PKCS #1. ISO 11568: This multi-part standard addresses retail key management and life cycle issues. It originated from X9.24, but is generalized for international use (e.g., it is no longer DES-speciﬁc), and addresses both symmetric and public-key techniques. 15.3.3 International security architectures and frameworks Table 15.8 lists selected ISO standards on security frameworks and architectures. Some of these are developed by SC21 (ISO/IEC JTC 1 SC21), which includes activities on Open Systems Interconnection (OSI) projects. The International Telecommunication Union (ITU) develops common-text speciﬁcations with JTC 1 for some standards in this area. ISO # Subject Ref. 7498-2 OSI security architecture [573] 9594-8 authentication framework (X.509) [595] 10181 OSI security frameworks [609] Table 15.8:ISO and ISO/IEC security architectures and frameworks. ISO 7498-2(X.800): The OSI basic reference model of ISO 7498 deﬁnes a commu- nications protocol stack with seven layers: application (layer 7), presentation (6), session (5), transport (4), network (3), data-link (2), and physical layers (1). ISO 7498-2 speciﬁes the security architecture for the basic reference model, including the placement of secu- rity services and mechanisms within these layers. It also provides a general description of the basic OSI security services: authentication (peer-entity and data-origin); access con- trol; data conﬁdentiality; data integrity; and non-repudiation (with proof of origin, or with proof of delivery). Speciﬁc mechanisms are used to implement these services; for example, encipherment is a mechanism for providing conﬁdentiality. ISO/IEC 9594-8(X.509): This standard is the same as ITU-T (formerly CCITT) Rec- ommendation X.509. It deﬁnes both simple authentication techniques (based on passwords) and so-called strong authentication techniques (wherein secret values themselves are not revealed to the veriﬁer). The strong techniques included are the two-pass and three-pass X.509 exchanges (see§12.5.2) based on digital signatures and the use of time-variant pa- rameters. An implicit assumption is the use of an algorithm such as RSA which may serve as both an encryption and a signature mechanism; the speciﬁcation may, however, be modi- ﬁed (e.g., to use DSA). The standard also speciﬁes techniques, including X.509 certiﬁcates, for acquiring or distributing authentic public keys; and addresses cross-certiﬁcates, and the use of certiﬁcate chains (§13.6.2(i)). ISO/IEC 10181 (X.810 through X.816): This speciﬁcation is a series of security frameworks intended to provide context and background, consisting of the following parts: security frameworks overview (1); authentication framework (2); access control framework (3); non-repudiation framework (4); conﬁdentiality framework (5); integrity framework (6); security audit and alarms framework (7). 654 Ch. 15 Patents and Standards 15.3.4 U.S. government standards (FIPS) Table 15.9 lists selected security-related Federal Information Processing Standards (FIPS) publications. These are developed under the National Institute of Standards and Technology (NIST), for use by U.S. federal government departments. FIPS # Subject Ref. FIPS 46–2 DES [396] FIPS 74 guidelines for using DES [397] FIPS 81 DES modes of operation [398] FIPS 112 password usage [399] FIPS 113 data authentication (CBC-MAC) [400] FIPS 140–1 cryptomodule security requirements [401] FIPS 171 key management using X9.17 [402] FIPS 180–1 secure hash standard (SHA–1) [404] FIPS 185 key escrow (Clipper & SKIPJACK) [405] FIPS 186 digital signature standard (DSA) [406] FIPS 196 entity authentication (asymmetric) [407] Table 15.9:Selected security-related U.S. FIPS Publications. FIPS 46: This standard speciﬁes the DES algorithm (cf. ANSI X3.92). FIPS 74: This standard provides guidelines for implementing and using DES. FIPS 81: This standard speciﬁes 4 basic DES modes of operation (cf. ANSI X3.106). FIPS 112: This standard provides guidelines on password management and usage. FIPS 113: This standard speciﬁes the customary DES-based CBC-MAC algorithm (see ISO/IEC 9797), referring to it as the Data Authentication Algorithm (DAA). The MAC result is called a Data Authentication Code (DAC). The last data bock, if incomplete, is left- justiﬁed and zero-padded before processing; the result is the leftmostmoutput bits, where mis a multiple of 8, and16 ≤m≤64. Implementation may be either by the CBC mode withIV =0 , or CFB-64 mode withIV = D 1, the ﬁrst data block (see Note 15.1). 7-bit ASCII-coded data to be authenticated by the DAA is preprocessed into 8-bit characters with leading bit 0. FIPS 140–1: This standard speciﬁes security requirements for the design and imple- mentation of cryptographic modules for protecting (U.S. government) unclassiﬁed infor- mation, including hardware, ﬁrmware, software modules, and combinations thereof. Four grades of increasing security are speciﬁed as Levels 1 through 4, covering a wide range of security applications and environments. A FIPS 140–1 validation program is run by NIST to determine if cryptomodules meet the stated requirements. FIPS 171: FIPS 171 speciﬁes, for use by (U.S.) federal government departments, a subset of the key distribution techniques of ANSI X9.17. The objective of specifying a subset is to increase interoperability and decrease system costs. FIPS 180 and 180–1: The hash algorithm speciﬁed in the original standard FIPS 180 is the Secure Hash Algorithm, SHA. A revised version was speciﬁed shortly thereafter in FIPS 180–1 (Algorithm 9.53), and denoted SHA–1. SHA–1 differs from SHA as noted in §9.8. FIPS 185: This Escrowed Encryption Standard (EES) speciﬁes the parameters and use of the SKIPJACK symmetric-key block cipher, and a method of creating Law Enforcement Access Fields (LEAFs) for use with the Clipper key escrow system (§13.8.3). The purpose §15.3 Cryptographic standards 655 is to allow wiretapping under lawful authorization. Internal details of the SKIPJACK algo- rithm are not publicly available, although its interface speciﬁcation is (§13.8.3(i)). FIPS 186: This standard is the Digital Signature Standard (DSS), which speciﬁes the Digital Signature Algorithm (DSA). The hash function originally mandated for use with DSA is deﬁned in FIPS 180 (SHA), which was superseded by FIPS 180–1 (SHA–1). FIPS 196: This standard on entity authentication using asymmetric techniques was derived from the two-pass and three-pass random-number based mechanisms of ISO/IEC 9798-3. It includes additional expository and implementation details. 15.3.5 Internet standards and RFCs Documents calledRequests for Comments(RFCs) are ofﬁcial working notes of the Inter- net research and development community. A subset of these are speciﬁcations which are candidates for standardization within the community as Internet Standards. The Internet Engineering Steering Group (IESG) of the Internet Engineering Task Force (IETF) is responsible for making recommendations regarding progression of “standards-track” speciﬁcations from Proposed Standard (PS) to Draft Standard (DS) to Standard (STD). RFCs may also correspond to the following types of documents: Experi- mental (E) protocols which may be part of early research efforts; Informational (I) protocols published for convenience of the community; and Historical (H) protocols which have been superseded, expired, or abandoned. The E, I, and H categories are not on the standards track, and the IESG does not make recommendations on these. Less mature, less stable, or less widely circulated doc- uments are typically available as an Internet-Draft (I-D); these are considered to be “work in progress”, and should be cited as such. RFC Status Subject Ref. 1319 I MD2 hash function [1033] 1320 I MD4 hash function [1034] 1321 I MD5 hash function [1035] 1421 PS PEM – encryption, authentication [1036] 1422 PS PEM – certiﬁcates, key management [1037] 1423 PS PEM – algorithms, modes, identiﬁers [1038] 1424 PS PEM – key certiﬁcation and services [1039] 1508 PS Generic Security Service API (GSS-API) [1040] 1510 PS Kerberos V5 network authentication [1041] 1828 PS keyed MD5 (as a MAC) [1044] 1847 PS security multiparts for MIME [1045] 1848 PS MIME Object Security Services (MOSS) [1046] 1938 PS one-time password system [1047] Table 15.10:Selected Internet RFCs (May 1996 status). Table 15.10 lists selected security-related Internet RFCs. The hashing algorithms MD2, MD4, and MD5 are speciﬁed in RFCs 1319-1321, respectively. The Internet Privacy- Enhanced Mail (PEM) speciﬁcations are given in RFCs 1421-1424. The Generic Security Service Application Program Interface (GSS-API) of RFC 1508 is a high-level security API which isolates application code from implementation details; for example, the interface provides functions such assignand seal(e.g., as opposed to 656 Ch. 15 Patents and Standards “seal using a 32-bit DES CBC-MAC and this particular key”). Speciﬁc implementation mechanisms must be provided beneath GSS-API; options include Kerberos V5 as per RFC 1510 for symmetric-based techniques, and SPKM for public-key based techniques (see page 661). RFC 1828 speciﬁes a method for using keyed MD5 as a MAC (cf.§9.5.2). RFC 1848 deﬁnes MIME Object Security Services (MOSS), where MIME denotes Multipurpose In- ternet Mail Extensions. MOSS makes use of the RFC 1847 framework of multipart/signed and multipart/encrypted MIME messages, and facilitates encryption and signature services for MIME including key management based on asymmetric techniques. RFC 1938 speciﬁes an authentication technique based on Lamport’s one-time password scheme (Protocol 10.6). 15.3.6 De facto standards Various security speciﬁcations arising through informal processes become de facto stan- dards. This section mentions one such class of speciﬁcations: the PKCS suite. PKCS speciﬁcations A suite of speciﬁcations calledThe Public-Key Cryptography Standards(PKCS) has parts as listed in Table 15.11. The original PKCS #2 and PKCS #4 have been incorporated into PKCS #1. PKCS #11 is referred to asCRYPTOKI . No. PKCS title 1 RSA encryption standard 3 Difﬁe-Hellman key-agreement standard 5 Password-based encryption standard 6 Extended-certiﬁcate syntax standard 7 Cryptographic message syntax standard 8 Private-key information syntax standard 9 Selected attribute types 10 Certiﬁcation request syntax standard 11 Cryptographic token interface standard Table 15.11:PKCS speciﬁcations. 15.3.7 Ordering and acquiring standards ISO and ISO/IEC standards may be obtained from (member body) national standards orga- nizations such as ANSI, the British Standards Institution (BSI), and the Standards Council of Canada (SCC). To purchase standards directly from ISO, contact ISO Central Secretariat, Case postale 56, CH-1211 Geneva 20, Switzerland; telephone +41.22.749.01.11. ANSI X9 standards are published by EDI Support Services Incorporated; to purchase standards, telephone 1-800-334-4912 (from within the USA) or +216-974-7650 (from out- side the USA). FIPS PUBS may be purchased from the National Technical Information Service, U.S. Department of Commerce, 5285 Port Royal Road, Springﬁeld, Virginia 22161 (USA); tele- phone +703-487-4650, fax +703-321-8547. To obtain copies of speciﬁcations of proposed §15.4 Notes and further references 657 (draft) FIPS, contact the Standards Processing Coordinator, National Institute of Standards and Technology, Technology Building, Room B–64, Gaithersburg, Maryland 20899 (USA); telephone +301-975-2816. Alternatively, consult URLhttp://csrc.ncsl. nist.gov/. Internet RFCs and Internet-Drafts are available on-line via anonymous FTP from numerous ftp sites (e.g.,ds.internic.net); further information can be obtained by sending an email message [email protected] the message body “help: ways to get rfcs”. RFCs are typically under the directoryrfc/asrfcXXXX.txt(e.g. rfc1321.txt), and an RFC index is available asrfc-index.txt. RFCs can also be ob- tained via electronic mail by sending an email message [email protected] body includes “Retrieve: RFC” and “Doc-ID: RFCnnnn” on separate lines. The PKCS suite is published by RSA Laboratories, 100 Marine Parkway, Suite 500, Redwood City, California 94065-1031 (telephone +415-595-7703), and is available by anonymous FTP fromrsa.comunder the directorypub/pkcs/. 15.4 Notes and further references §15.1 Levine [762] compiled a comprehensive list of American cryptographic patents issued be- tween 1861 and 1981, citing patent number, name of principal inventor, date granted, and patent title; this provides an insightful perspective of the history of cryptography over this period. Kahn [648] discusses many patents in his historical tour, including many related to rotor machines (cf. Chapter 7). Contact information regarding the current assignees of some cryptographic patents may be found throughout the book of Schneier [1094]. Davies and Price [308] provide both general discussion of standards, and detailed techni- cal discussion of selected standards. Preneel [1001] gives background on worldwide, Eu- ropean, and North American standardization organizations, and an overview of activities therein. Ford [414] provides a comprehensive overview of information security standards including extensive background information on various standardization processes and or- ganizations, including technical committees ISO TC 68 and ISO/IEC JTC 1 and their sub- committees; ITU; ANSI; and national, regional, and international standardization bodies. For a more recent overview of security standards for open systems, see Fumy and Rieten- spiess [432]. A status update of selected standards is also provided by Ford [415]. §15.2 One of the earliest and most important cryptographic patents was U.S. Patent No. 1,310,719 [1221] issued to Vernam on July 22 1919 for theVernam cipher(cf. the one-time pad, Chap- ter 7; see also Kahn [648, p.401]). Two other patents by Vernam, titled “Ciphering device”, were granted May 23 1922 (1,416,765) and January 8 1924 (1,479,846). In consideration of ANSI making DES a standard, IBM made the DES patent of Ehrsam et al. (3,962,539) [363] available free of license fees in the U.S. when used to implement ANSI standards. The ﬁrst widespread published disclosure of public-key cryptography was through the con- ference paper of Difﬁe and Hellman [344], presented June 8 1976, ﬁfteen months prior to the ﬁling of the Hellman-Difﬁe-Merkle patent [551]. Merkle independently conceived the idea of deriving a secret key over a public channel in 1974 (see§12.10); his paper [849], ﬁrst submitted toCommunications of the ACMin 1975, was rejected several times before ﬁ- nal publication in 1978. Meanwhile, the 1976 Difﬁe-Hellman conference paper introduced 658 Ch. 15 Patents and Standards the concept of a digital signature as well as public-key cryptography and public-key au- thentication. Although Difﬁe and Hellman noted: “At present we have neither a proof that public key systems exist, nor a demonstration system”, the existence of public-key systems was postulated, and three suggestions were offered supporting the general idea. The ﬁrst involved matrix inversion, which is more difﬁcult than multiplication by a factorO(n) for n×nmatrices; this offers a degree of security for very largen. The second involved compil- ing a function described in a high-level language into machine code; this makes it difﬁcult to recover the original function. The third suggestion involved obscuring the input-output relationships between, e.g., 100 input and 100 output bits (wires) in an invertible hardware circuit originally implementing the identity mapping, by, e.g., inserting 4-by-4 bit invert- ible S-boxes into randomly selected sets of 4 wires; re-arranging the particular mappings of input lines into S-boxes then makes inverting the resulting circuit difﬁcult. The Hellman-Merkle patent [553] was ﬁled sixteen months after the above Difﬁe-Hellman conference paper was presented. A major reason why the RSA patent [1059] took almost 6 years from application ﬁling to issue date was so-called interference proceedings between it and some of the Stanford patents. The subject of the authentication trees patent of Merkle [848] is discussed in his thesis [851, p.126-131] and in the open literature [852, 853]. The signature technique of the ESIGN patent [952] is discussed in the literature by Okamoto [948]; see also Fujioka, Okamoto, and Miyaguchi [428]. The identiﬁcation and signature technique of the Shamir-Fiat patent [1118] is described by Fiat and Shamir [395]. Regard- ing the Guillou-Quisquater patent [523], see Guillou and Quisquater [524]. The identiﬁ- cation and signature schemes patented by Schnorr [1095] are discussed in the literature by Schnorr [1097, 1098]; the preprocessing scheme proposed therein, however, was shown to be insecure by de Rooij [314, 315]. In its announcement of the proposed FIPS for DSS (Federal Registervol.56 no.169, August 30 1991, 42980-42982), NIST noted its intent to make the DSA patent of Kravitz [711] available world-wide on a royalty-free basis. In a letter to the Director of the Computer System Laboratories at NIST dated October 30 1991, Schnorr stated that DSA infringed on Claim 6 of his patent (4,995,082). FIPS 186 itself (1994) states that “The Department of Commerce is not aware of any patents that would be infringed by this standard”. MDC-2 and MDC-4 [184] (see also Bosselaers and Preneel [178]) are discussed in§9.4.1. For further discussion of FEAL [1125], see§7.5. A patent on IDEA was originally ﬁled in Switzerland and subsequently as a European patent [790], prior to being ﬁled as a U.S. patent [791]; for literature references, see Chapter 7. Related to the Matyas-Meyer-Brachtl patent [806] on control vectors, the October 7 1980 patent of Ehrsam et al. (4,227,253), “Cryptographic communication security for multiple domain networks”, describes use of a master key and two variants obtained by inverting designated bits of the master key, equivalent to an XOR of the master with ﬁxed mask val- ues. Also related is the key notarization method of the patent by Smid and Branstad [1154], which controls which parties use a key, but not the uses. The key notarization technique is essentially identical – involving concatenation of various quantities (user identities), which are then XOR’d with a key-encryption key – but control vectors have broader functionality. Fair cryptosystems [861, 862] are discussed in the literature by Micali [863]; but see also Kilian and Leighton [671], who remark on a critical weakness. Interest in product cipher systems was stimulated by the product ciphers described in Shan- non’s 1949 paper [1121]. Meyer and Matyas [859] note that Lucifer was the name of the cryptographic system in which the product cipher of Feistel’s patent (3,798,359) [385] was implemented, and from which the IBM team lead by Tuchman derived DES. The 1974 §15.4 Notes and further references 659 patent of Smith [1159] is also related to Lucifer. A second 1974 patent of Feistel [386] on a “step code ciphering system” was ﬁled and issued with dates matching the Lucifer al- gorithm patent. Sorkin [1165] states that Lucifer is the subject of all three of these patents, plus a fourth: “Centralized veriﬁcation system” (3,798,605) granted March 19 1974 to H. Feistel. Feistel gives a high-level background discussion on a ﬁrst variation of Lucifer in his 1973Scientiﬁc Americanarticle [387], which appeared prior to his 1974 patents being issued. A description of the second variation of Lucifer (which lead to the design of DES) is given by Sorkin [1165]; see also Biham and Shamir [138] Related to the Massey-Omura [792] and Omura-Massey [956] patents is that of Onyszchuk, Mullin, and Vanstone [959]. It was ﬁled May 30 1985 and issued May 17 1988 with no as- signee listed. The patent teaches the construction of a multiplier for elements inF2m, stated to be a signiﬁcant improvement over the method of Omura-Massey. The patent also tabu- lates those valuesm,2 ≤m≤2493, for which so-called optimal normal bases exist; in these ﬁelds, the disclosed normal-basis multipliers forF2m are more efﬁcient than in oth- ers. Shamir’s three-pass protocol was ﬁrst proposed by Shamir, as indicated by Konheim [705]. Massey [786] notes that Shamir also speciﬁcally proposed implementing the three- pass protocol using exponentiation as the ciphering operation, an idea later independently proposed by Omura (cf.§12.3 notes on page 535). In contrast to the prime generation methods of Shawe-Taylor and Maurer (§4.4.4) which result in guaranteed primes, the prime generation method of the Hellman-Bach patent [550] uses probabilistic primality tests, and is related to that presented by Gordon at Eurocrypt in April of 1984 [514], and which appeared (dated April 26 1984) in the June 7 1984 issue (vol.20 no.12) ofElectronics Letters[513]. The protocol patented by Goss [519], ﬁled April 17 1989, combines exponentials by an XOR operation. An essentially identical protocol published in 1986 by Matsumoto, Takashima, and Imai [800] uses modular multiplication (cf. Protocol 12.53). The exponentiation cipher of the Hellman-Pohlig patent [554] is discussed in the literature by Pohlig and Hellman [982]. The ciphers Khufu and Khafre [847] are similarly discussed by Merkle [856]; on-line/off-line digital signatures [864] by Even, Goldreich, and Micali [377, 378]; and the techniques of the patent on efﬁcient exponentiation [203] are presented by Brickell et al. [204] (for more recent work, see Hong, Oh, and Yoon [561]). A patent by Crandall (5,159,632) [286] includes twelve (12) claims on speciﬁc implementa- tions of elliptic curves using primespof special form (e.g.,p=2 q−CforCsmall) allowing fast multiplication using shifts and adds alone (cf. Mohan and Adiga, 1985), and speciﬁc use of Fast Fourier Transforms (FFT) for optimized modular multiplication in this case. The patent, ﬁled September 17 1991, was issued October 27 1992 and assigned to NeXT Com- puter, Inc. (Redwood City, California); see also its continuation-in-part, (5,271,061) [287]. Another patent in this area is the Miyaji-Tatebayashi patent (5,272,755) [888] ﬁled June 26 1992, with priority data June 28 1991 (Japanese patent ofﬁce). Issued December 21 1993, and assigned to the Matsushita Electric Industrial Co. (Osaka), it contains six (6) claims in the area of selecting elliptic curves overF p whose order is preciselyp. This covers a small subset of possible curves of this order overFp, and one particular method for selecting from among these; see also its continuation-in-part, (5,351,297) [889]. Regarding other block ciphers discussed in this book, a patent application has been ﬁled for the RC5 cipher (§7.7.2). Adams [3] is the inventor for a patent on the CAST block cipher design procedure (see p.281); the assignee, Northern Telecom Limited (Montreal), will, however, make a CAST cipher available free of license fees. The SEAL stream cipher (§6.4.1) of Coppersmith and Rogaway is also patented [281]. 660 Ch. 15 Patents and Standards §15.3 A draft standard in development under the IEEE Microprocessor Standards Committee group isIEEE P1363: Standard for RSA, Difﬁe-Hellman and related public-key cryptog- raphy, which includes speciﬁcations for elliptic curve systems. Theoretical justiﬁcation for the redundancy scheme used in ISO/IEC 9796 is given by Guil- lou et al. [525]. The customary 5-year review of this standard in 1996 resulted in a title change and the creation of a second part. The original standard (with content unchanged) will be re-titledDigital signature schemes giving message recovery – Part 1: Mechanisms using redundancy. The second part, a working draft (WD) as of April 1996 titledPart 2: Mechanisms using a hash function, speciﬁes mechanisms utilizing the idea that when a sig- nature algorithm such as RSA is used with a hash function, and the RSA modulus (say 1024 bits) is much larger than a hash value (say 160 bits), the remaining bits may be used to carry message text which can be recovered upon signature veriﬁcation. Thispartial message re- coverymode of the signature algorithm decreases the amount of accompanying cleartext re- quired, which is of interest in bandwidth or memory-limited applications, and those wherein the text being signed is relatively small. The Registration Authority designated by ISO/IEC to maintain the register of cryptographic algorithms of ISO/IEC 9979 is the National Computer Centre, Oxford Road, Manchester, M1 7ED, United Kingdom (telephone +44-161-228-6333, fax +44-161-228-1636). Twelve algorithms were registered as of October 1995: BARAS, B-Crypt, CDMF, DES, FEAL, IDEA, LUC, MULTI2, RC2, RC4, SXAL/MBAL, and SKIPJACK. An alternative for ob- taining unique algorithm identiﬁers is theobject identiﬁer(OID) and registration scheme of the Abstract Syntax Notation One (ASN.1) standard ISO/IEC 8824; for more information, see Ford [414, pp.478-480]. For a history of DES-related standards from an American perspective, including ANSI stan- dards, see Smid and Branstad [1156]. ANSI X9.24, Annex C contains a convenient six-page summary of ANSI X9.17. A revision of X9.30–2:1993 is to specify FIPS 180–1 (SHA–1) in place of SHA. An ANSI standard in development, but currently “on hold” pending resolu- tion of patent issues, is (draft) X9.44 [48], which speciﬁes a key transport technique based on RSA. An enhanced mode of triple-DES encryption included in the draft ANSI X9.52 [50] iscipher block chaining with output feedback masking. The draft ANSI X9.57 [52] is intended for use with X9.30 and (draft) X9.31, although the initial version addresses X9.30 (DSA) certiﬁcates. ITU-T X.509 v3 certiﬁcates and certiﬁcate extensions to which ANSI X9.55 is aligned are discussed below. Both (draft) X9.45 and (draft) X9.55 may eventually be incorporated into X9.57. Related to attribute certiﬁcates, see Fischer [410] regarding electronic document authorization and related patents [408, 409]. The ISO 11568 retail key management project includes six parts [588, 589, 590, 591, 592, 593]. Among these, 11568-3 speciﬁes the key life cycle for symmetric encryption algo- rithms; 11568–4 addresses key management techniques for public-key cryptosystems, in- cluding certiﬁcate management and (in Annex C) attribute certiﬁcates; and 11568–5 ad- dresses key life cycle for public-key cryptosystems. ISO/IEC 9594-8 (X.509) is one part of a series of speciﬁcations outlining directory ser- vices for Open Systems Interconnection (OSI) and other systems. TheDirectoryis a logical database of information with directory entries arranged in a tree structure, theDirectory In- formation Tree(DIT), as introduced in ISO/IEC 9594–1 (ITU-T Recommendation X.500) [594], which also provides an overview of directory services. For extension discussion, see Chapter 14 of Ford [414]. The 1988 version of X.509 (equivalent to ISO/IEC 9594- 8:1990) was updated in 1993 [626] (equivalent to ISO/IEC 9594-8:1995). A 1995 tech- nical corrigendum [627] added a certiﬁcateextensionsﬁeld, yielding Version 3 (v3) cer- §15.4 Notes and further references 661 tiﬁcates. Standard extensions for v3 certiﬁcates are deﬁned in a further amendment [628] (see§13.9). The OSI security frameworks project is speciﬁed in seven parts of ISO 10181 [609, 610, 611, 612, 613, 614, 615]. FIPS 140–1 [401] supersedes FIPS 140,General Security Requirements for Equipment Us- ing the Data Encryption Standard(formerly Federal Standard 1027, April 1982). Informa- tion on FS 1027 is provided by Davies and Price [308]. In May 1994, NIST announced a weakness in SHA [403], resulting from unpublished analysis carried out by the U.S. Na- tional Security Agency; the formal revision was published as FIPS 180–1 [404]. The PKCS standards, developed by industrial collaboration lead by RSA Laboratories (a Division of RSA Data Security Inc.), are widely used in practice, and periodically updated. PKCS #1,3,5,6,7,8,9,10 [1072] and PKCS #11 [1071] are currently available (e.g., from URL http://www.rsa.com/). For an overview of Internet security standards, see Kent [667]. Linn’s GSS-API (RFC 1508) [1040] is an API suitable for session-oriented applications. An analogous speciﬁcation for store-and-forward applications is the IDUP-GSS-API (Independent Data Unit Protection GSS-API) interface. Implementation mechanisms which have been speciﬁed to plug in be- neath GSS-API include a symmetric-key mechanism based on Kerberos (the Kerberos Ver- sion 5 GSS-API mechanism), and a public-key based mechanism SPKM (Simple Public- Key Mechanism). For an overview of these work-in-progress items under development in the Common Authentication Technologies (CAT) group of the IETF, see Adams [4]. Work-in-progress in the IP Security (IPSEC) working group of the IETF includes two items using Difﬁe-Hellman key exchange for session key establishment over the Internet – the Photuris protocol of Karn and Simpson, and the SKIP protocol of Aziz. Krawczyk [718] notes these and presents an alternative (SKEME). MIME, speciﬁed in RFC 1521 [1042], is designed to facilitate multipart textual and non- textual mail, i.e., mail messages whose bodies may contain multiple objects of a variety of content types including non-ASCII text, multi-font text, and audio and image fragments. An alternative to the MOSS proposal of RFC 1848 [1046] is S/MIME [1191], which adds signature and/or encryption services to MIME messages, using PKCS speciﬁcations. Many other standards, both formal and informal, have been developed or are undergoing de- velopment. A collection of cryptographic algorithms and protocols recommended for use in Europe is that resulting from the European RACE Integrity Primitives Evaluation (RIPE) project; see Bosselaers and Preneel [178]. Pretty Good Privacy (PGP) is a popular, widely available software package originally developed by Zimmermann [1272] (see Garﬁnkel [442] for additional perspective), currently employing RSA signatures, MD5 hashing, and IDEA encipherment. Examples of pseudorandom number generators (PRNGs) which appear in U.S. standards include a DES-based PRNG in ANSI X9.17 (Appendix C), and two further methods in FIPS 186 (Appendix 3) based on both the Secure Hash Algorithm (SHA) and DES. 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Index Symbols \|S\|(cardinality of a setS), 49 ∈(set member), 49 ⊆(subset), 49 ⊂(proper subset), 49 ∩(set intersection), 49 ∪(set union), 49 −(set difference), 49 ×(Cartesian product), 49 ∅(empty set), 50 O-notation (big-O), 58 Ω-notation (big-omega), 59 Θ-notation (big-theta), 59 o-notation (little-o), 59 def =(by deﬁnition), 213 Lq[α,c](subexponential notation), 60 ≤P (polytime reduction), 61 ∼(asymptotic equivalence), 134 π(mathematical constant pi), 49 e(base of natural logarithms), 49∑ (sum), 50∏ (product), 50 !(factorial), 50 ⌊⌋(ﬂoor), 49 ⌈⌉(ceiling), 49 φ(Euler phi function), 65, 286 µ(n)(M¨obius function), 154 lg(base2logarithm), 50 ln(natural logarithm), 50 [a,b](interval of integers), 49 \| (divides relation), 63, 79 ≡(congruence relation), 67, 79 ≪(much less than), 529 ≫(much greater than), 170(n k )(binomial coefﬁcient), 52(a p ) (Legendre symbol), 72 <> (inner product), 118 ∥x∥(length of a vectorx), 118 a←b(assignment operator), 66 a∥b(concatenation of stringsa,b), 38 {0,1}k (bitstrings of bitlengthk), 447 {0,1}∗ (bitstrings of arbitrary bitlength), 447 Q (the rational numbers), 49 R (the real numbers), 49 Z (the integers), 49 Zn (integers modulon), 68 Z∗ n (multiplicative group ofZn), 69 Qn (quadratic residues modulon), 70 Qn (quadratic non-residues modulon), 70 Fq (ﬁnite ﬁeld of orderq), 81 F∗ q (multiplicative group ofFq), 81 R[x](polynomial ring), 78 ∨(inclusive-OR), 213 ⊕(exclusive-OR), 20 ∧(AND), 213 ⊞ (addition mod2n), 263 ⊟ (subtraction mod2n), 270 ⊙(modiﬁed multiplication mod2n +1), 263 ←↩(left rotation), 213 ↪→(right rotation), 213 A→B(message transfer), 396 A Abelian group, 75 Abstract Syntax Notation One (ASN.1), 660 Access control, 3 Access control matrix, 387 Access matrix model, 569 Access structure, 526 monotone, 527 Accredited Standards Committee (ASC), 648 Active adversary, 15, 37 Active attack, 41, 495 Ad hoc security, 43 Adaptive chosen-ciphertext attack, 42 Adaptive chosen-message attack, 433 Adaptive chosen-plaintext attack, 41 Addition chains, 621, 633 Adversary, 13, 495 active, 15 insider, 496 one-time, 496 permanent, 496 outsider, 496 passive, 15 Afﬁne cipher, 239 Algebraic normal form, 205 Algorithm deﬁnition of, 57 755 756 Index deterministic, 62 exponential-time, 59 polynomial-time, 59 randomized, 62 expected running time, 63 running time, 58 asymptotic, 58 average-case, 58 worst-case, 58 subexponential-time, 60 Alphabet of deﬁnition, 11 Alternating step generator, 209–211, 220 Anonymity, 3 ANSI standards, 648–651, 660 ordering and acquiring, 656 ANSI X9.17 pseudorandom bit generator, 173 Anti-palindromic keys of DES, 257 Appended authenticator, 361 Arbitrated signature scheme, 472–473 Arithmetic integer,seeMultiple-precision integer arithmetic modular,seeMultiple-precision modular arith- metic Arthur-Merlin games, 421 ASN.1, seeAbstract Syntax Notation One (ASN.1) Asymmetric cryptographic system, 544 Asymptotic running time, 58 Atkin’s primality test, 145 implementation report, 166 Attack active, 41, 495 adaptive chosen-ciphertext, 42 adaptive chosen-message, 433 adaptive chosen-plaintext, 41 chosen-ciphertext, 41, 226 chosen-message, 433 chosen-plaintext, 41, 226 chosen-text, 417 ciphertext-only, 41, 225 dictionary, 42, 392 differential cryptanalysis, 258 differential-linear, 271 exhaustive key search, 233–234 forced delay, 417 forward search, 42, 288, 420 impersonation, 42, 417 interleaving, 42, 417, 531, 540 intruder-in-the-middle, 530, 540 key-only, 432 known-key, 42, 496, 534 known-key triangle, 538 known-message, 432 known-plaintext, 41, 225 linear cryptanalysis, 258 local, 419 meet-in-the-middle, 235 misplaced trust in server, 531 non-interactive, 419 off-line, 419 on-line, 419 passive, 41, 495 pre-play, 397 reﬂection, 417, 530, 540 related-key, 226 remote, 419 replay, 42, 417 time-memory tradeoff, 236 truncated differentials, 271 universal forgery, 482 Attacker, 13 Attacker (alternate names), 495 see alsoAdversary Attribute certiﬁcate, 561 Audit trail, 549, 583 Audit trail information, 545 Authenticated key establishment, 492, 493 Authenticated key exchange protocol AKEP1/AKEP2, 499, 535, 541 Authentication data origin, 4, 361 see alsoData origin authentication entity, 4 see alsoEntity authentication explicit key, 492 key, 492 message, 361 mutual, 494 protocol, 493 transaction, 362 unilateral, 494 see alsoEntity authentication (and Identiﬁca- tion) Authentication code, 376, 382 Authentication path, 557 Authentication server, 491, 549 Authentication tree, 466–468, 485, 556–559, 587 Authority revocation list (ARL), 577 Authorization, 3 Authorized subset, 527 Auto-key cipher, 242 Autocorrelation function, 180 Autocorrelation test, 182 Auxiliary-input zero-knowledge, 423 Avalanche effect, 277 Average-case running time, 58 B Baby-step giant-step algorithm, 104–106, 128 Index 757 BAN logic, 420, 534, 541 Bandwidth efﬁciency, 437 Barrett reduction, 603–605, 631 Base brepresentation, 592 Basis, 80 Bayes’ theorem, 51 BEAR block cipher, 282 Beaufort cipher, 241 Beller-Yacobi key transport 2-pass, 514 4-pass, 513 Berlekamp’sQ-matrix algorithm, 124, 132 Berlekamp-Massey algorithm, 200–201 next discrepancy, 200 Bernoulli trial, 52 Biased, 172 Big-endian, 344 Big-O notation, 58 Big-omega notation, 59 Big-theta notation, 59 Bijection, 7, 50 Binary additive stream cipher, 194 keystream generator, 194 running key generator, 194 Binary alphabet, 11 Binary Euclidean algorithm, 632 Binary extended gcd algorithm, 608–610, 632 Binary gcd algorithm, 606–607, 632 Binary operation, 75 Binary representation, 592 Binary tree, 557 balanced, 558 children, 557 depth of, 558 internal vertex, 557 leaf, 557 parent, 557 root vertex, 557 Binomial coefﬁcient, 52 distribution, 52 theorem, 52 Biometrics, 387, 420 Birthday attack, 352, 369 Birthday problem, 53 Birthday surprise, 53 Bit commitment, 421 Bitzer’s hash function, 374 Black-box, 329, 341, 369, 378 Blakley’s threshold scheme, 538 Blind signature scheme, 475, 487 based on DSA, 487 based on Nyberg-Rueppel, 487 Chaum, 475 fair, 487 Blinded message, 475 Blinding function, 475 based on RSA, 475 Blob, 421 Block cipher, 223–282 3-WAY , 281 attacks on differential cryptanalysis, 258 differential-linear, 271 exhaustive key search, 233–234, 273 key clustering attack, 281 linear cryptanalysis, 258 meet-in-the-middle attack, 235 related-key attack, 226, 281 time-memory tradeoff, 236, 273 truncated differentials, 271, 280 BEAR, 282 Blowﬁsh, 281 CAST, 281 classical cipher, 237–250 deﬁnition of, 16, 224 DES, 250–259 double DES, 235 FEAL, 259–262 GOST, 282 IDEA, 263–265 iterated, 251 Khafre, 271 Khufu, 271 LION, 282 LOKI’91, 270 Luby-Rackoff, 282 Lucifer, 276 modes of operation, 228–233, 272 ANSI X3.106 standard, 649 ANSI X9.52 standard, 651 CBC with checksum (CBCC), 367 cipher feedback mode (CFB), 231 cipher-block chaining mode (CBC), 230 counter mode, 233 electronic codebook mode (ECB), 228– 230 FIPS 81 standard, 654 ISO 8372 standard, 645 ISO/IEC 10116 standard, 647 output feedback mode (OFB), 232–233 plaintext-ciphertext block chaining (PCBC), 368 Randomized DES (RDES), 278 RC2, 282 RC5, 269–270 round function, 251 SAFER, 266–269 758 Index semi-weak keys (of DES), 257 anti-palindromic keys (of DES), 257 SHARK, 281 SKIPJACK, 282, 584 TEA, 282 triple DES, 272 WAKE, 282 Block of a sequence, 180 Blocklength, 224 Blom’s KDS bound, 505 Blom’s key pre-distribution system, 506, 536 Blowﬁsh block cipher, 281 Blum integer, 74–75 Blum-Blum-Shub pseudorandom bit generator, 186– 187, 308 Blum-Goldwasser probabilistic public-key encryp- tion, 308–311 decryption algorithm, 309 encryption algorithm, 309 key generation, 308 security of, 310 Blum-Micali pseudorandom generator, 189 Blundo’s conference KDS bound, 529 Boolean function, 202 algebraic normal form of, 205 correlation immune, 207 nonlinear order of, 205 BPP ,6 3 Break-backward protection, 496 Brickell-McCurley identiﬁcation protocol, 423 Broadcast encryption, 528 Bucket hashing, 382 Burmester-Desmedt conference keying, 528 Burst error, 363 C CA, seeCertiﬁcation authority (CA) CA-certiﬁcate, 572 Caesar cipher, 239 CALEA, 590 Capability (access control), 570 Capstone chip, 589 Cardinality of a set, 49 Carmichael number, 137 Carry-save adder, 630 Cartesian product, 49 Cascade cipher, 234, 237 Cascade generator m-sequence, 221 p-cycle, 220 Cascading hash functions, 334 CAST block cipher, 281 patent, 659 CBC, seeCipher-block chaining mode CBC-MAC, 353–354, 367 ANSI X9.9 standard, 650 ANSI X9.19 standard, 650 FIPS 113 standard, 654 ISO 8731-1 standard, 652 ISO 9807 standard, 652 ISO/IEC 9797 standard, 646 Cellular automata stream cipher, 222 Certiﬁcate ANSI X9.45 standard, 651 ANSI X9.55 standard, 651 ANSI X9.57 standard, 651 caching, 576 chain, 572 directory, 549 pull model, 576 push model, 576 forward, 575 on-line, 576 public-key,seePublic-key certiﬁcate reverse, 575 revocation, 566, 576–577 RFC 1422, 655 secret-key,seeSecret-key certiﬁcate symmetric-key,seeSymmetric-key certiﬁcate X.509 standard, 660 Certiﬁcate of primality, 166 Certiﬁcate revocation list (CRL), 576–577 Certiﬁcation, 3 path, 572 policy, 576 topology, 572 Certiﬁcation authority (CA), 491, 548, 556, 559 Certiﬁcational attack, 236 Certiﬁcational weakness, 285 CFB, seeCipher feedback mode CFB-64 MAC, 650 Challenge, 397, 409 Challenge-response identiﬁcation, 397–405, 420– 421 public-key, 403–405 ISO/IEC 9798-3, 404–405 modiﬁed Needham-Schroeder, 404 X.509, 404 symmetric-key, 400–403 ISO/IEC 9798-2, 401–402 SKID2, 402 SKID3, 402 Channel, 13 physically secure, 13 secure, 13 secured, 13 unsecured, 13 Characteristic of a ﬁeld, 77 Index 759 Chaum’s blind signature protocol, 475 Chaum-van Antwerpen undeniable signature sch- eme, 476–478 disavowal protocol, 477 key generation, 476 security of, 478 signature generation, 476 Chebyshev’s inequality, 52 Checksum, 362, 367–368 Chi-square (χ 2) distribution, 177–179 degrees of freedom, 177 mean of, 177 variance of, 177 Chinese remainder theorem (CRT), 68 Garner’s algorithm, 612–613 Gauss’s algorithm, 68 Chipcard, 387, 424 Chor-Rivest public-key encryption, 302–306, 318 attacks on, 318 decryption algorithm, 303 encryption algorithm, 303 key generation, 303 recommended parameter sizes, 305 security of, 305 Chosen-ciphertext attack, 41, 226, 285 adaptive, 285 indifferent, 285 Chosen-message attack, 433 directed, 482 generic, 482 Chosen-plaintext attack, 41, 226 Cipher, 12 see alsoEncryption Cipher-block chaining mode (CBC), 230 integrity of IV in, 230 use in public-key encryption, 285 Cipher feedback mode (CFB), 231 as a stream cipher, 233 ISO variant of, 231 Cipher machine, 242–245 Jefferson cylinder, 243 rotor-based machine, 243–245, 276 Enigma, 245 Hagelin M-209, 245 Hebern, 244 Wheatstone disc, 274 Ciphertext, 11 Ciphertext-only attack, 41, 225 Ciphertext space, 11 Claimant, 385, 386 Classical cipher, 237–250, 273–276 cipher machines,seeCipher machine cryptanalysis, 245–250, 275–276 index of coincidence, 248 Kasiski’s method, 248 measure of roughness, 249 polyalphabetic substitution cipher,seePolyal- phabetic substitution cipher substitution cipher,seeSubstitution cipher transposition cipher,seeTransposition cipher Classical modular multiplication, 600 Classical occupancy problem, 53 Claw-resistant (claw-free), 376, 468 Clipper chip, 584, 589 key escrow, 584 law enforcement access ﬁeld (LEAF), 584 Clipper key escrow, 654 Clock-controlled generator, 209–212 co-NP,6 0 Codebook, 240 Codomain of a function, 6, 50 Collision, 321 pseudo-collision, 371 Collision resistance, 324, 325 Collision resistant hash function (CRHF), 325 Combining function, 205 Common modulus attack on RSA, 289 Commutative ring, 77 Complementation property of DES, 256–257 Complete function, 277 Complexity classes, 59–62 BPP ,6 3 co-NP,6 0 NP ,6 0 NP -complete, 61 NP -hard, 62 NPC ,6 1 P,6 0 RP ,6 3 ZPP ,6 3 Complexity measure 2-adic span, 218 linear complexity, 198–201 maximum order complexity, 217 Turing-Kolmogorov-Chaitin complexity, 217 Ziv-Lempel complexity, 217 Complexity of attacks on a block cipher, 225–227 active complexity, 226 attack complexity, 226 data complexity, 226 passive complexity, 226 processing complexity, 226 storage complexity, 226 Complexity theory, 57–63 Complexity-theoretic security, 43 Compliant, 532 Composite integer, 64 Composition of functions, 19 760 Index Computation-resistance (MAC), 325 Computational problems computationally equivalent, 88 polytime reduction, 88 Computational security, 43, 226 Computational zero-knowledge protocol, 407 Computationally equivalent decision problems, 61 COMSET, 421, 536 Conditional entropy, 56 Conditional probability, 51 Conditional transinformation, 57 Conference keying, 528–529, 540 Blundo’s conference KDS bound, 529 Burmester-Desmedt, 528 deﬁnition of, 528 Conﬁdentiality, 3, 4, 12 Conﬁrmation, 3 Confounder, 418 Confusion, 20 Congruences integers, 67 polynomials, 79 Conjugate gradient method, 129 Connection polynomial of an LFSR, 196, 204 known versus secret, 204 sparse versus dense, 205 Constrained linear equations problem, 423 Continued fraction factoring algorithm, 126 Continuous random variable, 176 Control vector, 569 patent, 639, 658 Conventional encryption, 15 Coprime, 64 Correcting-block chaining attack, 373 Correlated, 172 Correlation attack, 206, 218 Correlation immunity, 207, 218 Counter mode, 233 CRC-based MAC, 359 Credential, 501 CRHF, seeCollision resistant hash function Cross-certiﬁcate (CA-certiﬁcate), 572 Cross-certiﬁcate pair, 573 CRT, seeChinese remainder theorem Cryptanalysis, 15 Cryptanalyst, 15 Cryptographic check value, 363 Cryptographic primitives, 4 taxonomy of, 5 Cryptographically secure pseudorandom bit gener- ator (CSPRBG), 185–187 Blum-Blum-Shub generator, 186–187 Blum-Micali generator, 189 deﬁnition of, 171 Micali-Schnorr generator, 186 modiﬁed-Rabin generator, 190 RSA generator, 185–186 Cryptography deﬁnition of, 4 goals of, 4 CRYPTOKI, 656 Cryptology, 15 Cryptoperiod of a key, 553 Cryptosystem, 15 Cut-and-choose protocol, 410, 421 Cycle of a periodic sequence, 180 Cyclic group, 69, 76 generator of, 76 Cyclic redundancy code (CRC), 363 Cyclic register, 220 Cycling attacks on RSA, 289, 313 D Data Authentication Algorithm (DAA), 654 Data Encryption Standard,seeDES block cipher Data integrity, 3, 4, 33, 359–368, 383 Data key, 552 Data origin authentication, 3, 4, 25, 359–368, 491 Davies-Meyer hash function, 341 de Bruijn FSR, 203 de Bruijn sequence, 203 De-skewing, 172 DEA, 649 Decimated subsequence, 211 Decision problems, 60 computationally equivalent, 61 polytime reduction, 61 Decryption, 11 Decryption exponent for RSA, 286 Decryption function, 11 DECT, 586 Degrees of freedom, 177 Delay element of an FSR, 202 of an LFSR, 195 Delayed-carry adder, 630 Density of a knapsack set, 120 Derivative of a polynomial, 123 DES block cipher, 250–259, 276–278 ANSI X3.92 standard, 649 attacks on differential cryptanalysis, 258–259 exhaustive key search, 233–234, 272 linear cryptanalysis, 258–259 complementation property, 256–257 decryption algorithm, 255 DESX, 273 double DES,seeDouble DES Index 761 encryption algorithm, 253 expansion permutation, 252 FIPS 46 standard, 654 initial permutation (IP), 252, 277 key schedule decryption, 256 encryption, 255 modes of operation,seeBlock cipher, modes of operation patent, 636 permuted choices (PC1, PC2), 252 properties and strengths, 256–259 round, 252 S-box, 252 semi-weak key, 257 anti-ﬁxed point of, 257 test vectors, 256 triple-DES, 273 weak key, 257 ﬁxed point of, 257 Designated conﬁrmer signature, 487 Deterministic, 306 Deterministic algorithm, 62 Dickson polynomial, 314 Dickson scheme, 314 Dictionary attack, 42 Difference of sets, 49 Differential chaining attack, 375 Differential cryptanalysis of block ciphers, 258, 271, 278–280 Differential-linear cryptanalysis, 271 Difﬁe-Hellman key agreement, 515–520, 522–524 ANSI X9.42 standard, 651 composite modulus, 537 patent, 637 Difﬁe-Hellman problem, 113–114 composite moduli, 114, 131 generalized, 113 Difﬁe-Lamport one-time signature scheme, 485 Diffusion, 20 Digital envelope, 550 Digital ﬁngerprint, 321 Digital signature,seeSignature Digital Signature Algorithm (DSA), 452–454, 483 ANSI X9.30-1 standard, 651 FIPS 186 standard, 655 key generation, 452 patent, 640, 658 security of, 453 signature generation, 452 signature veriﬁcation, 453 use and throw coupons, 483 Dimension of a vector space, 80 Dirichlet theorem, 135 Disavowal protocol, 477 Discrete Fourier Transform (DFT), 631 Discrete logarithms, 103–113 baby-step giant-step algorithm, 104–106 composite moduli, 114 exhaustive search, 104 for class groups, 130 for elliptic curves, 130 for hyperelliptic curves, 130 function ﬁeld sieve, 129 generalized problem, 103 heuristic running time, 129 in subgroups ofZ ∗ p,1 1 3 index-calculus algorithms, 109–112 lambda method, 128 number ﬁeld sieve, 128 Pohlig-Hellman algorithm, 107–109 Pollard’s rho algorithm, 106–107 problem deﬁnition, 103 rigorously analyzed algorithms, 129 security of individual bits, 116 Divisible electronic coin, 487 Division of integers, 63 of polynomials, 79 Division algorithm for integers, 64 for polynomials, 78 Dixon’s algorithm, 95, 127 DNA computer, 130 Domain of a function, 6, 50 Double DES, 235 Double spending, 487 Double-length MDC, 339 DSA, seeDigital Signature Algorithm Dynamic key establishment, 491 Dynamic secret sharing scheme, 527 E E-D-E triple encryption, 235, 272 E-E-E triple encryption, 272 Eavesdropper, 13, 495 ECA, seeElliptic curve factoring algorithm ECB, seeElectronic codebook mode Effective key size, 224 Electronic cash divisible, 487 untraceable, 487 Electronic codebook mode (ECB), 228–230 ElGamal key agreement, 517 ElGamal public-key encryption, 294–298 generalized decryption algorithm, 297 encryption algorithm, 297 762 Index key generation, 297 inZ∗ p decryption algorithm, 295 encryption algorithm, 295 key generation, 294 recommended parameter sizes, 296 security of, 296 ElGamal signature scheme, 454–459, 484 generalized key generation, 458 signature generation, 458 signature veriﬁcation, 458 inZ ∗ p key generation, 454 security of, 455–456 signature generation, 454 signature veriﬁcation, 454 signature veriﬁcation, 618 variants of, 457 Elliptic curve discrete logarithm problem, 130 ElGamal public-key encryption, 297 in public-key cryptography, 316 patents, 659 RSA analogue, 315 supersingular curve, 130, 316 Elliptic curve factoring algorithm (ECA), 94, 125 implementation reports, 126 Elliptic curve primality proving algorithm, 145 Encrypted key exchange (EKE), 538 Encryption, 11 see alsoBlock cipher see alsoPublic-key encryption see alsoStream cipher Encryption exponent for RSA, 286 Encryption function, 11 Encryption scheme, 12 breakable, 14 Enemy, 13, 495 Enigma, 245, 276 Entity, 13 Entity authentication, 3, 386, 491 ANSI X9.26 standard, 651 FIPS 196 standard, 655 ISO 11131 standard, 652 ISO/IEC 9798 standard, 401–402, 404–405, 421, 647 see alsoIdentiﬁcation Entropy, 56–57, 246 Ephemeral secret, 494 Equivalence class, 68, 79 Equivocation, 56 Error-correcting code, 298, 363, 506 Escrowed Encryption Standard (EES) FIPS 185, 654 ESIGN signature scheme, 473–474, 486 key generation, 473 patent, 638, 658 security of, 474 signature generation, 473 signature veriﬁcation, 473 Euclidean algorithm for integers, 66 for polynomials, 81–83 Euler liar, 138 Euler phi function (φ), 65 Euler pseudoprime, 138 Euler witness, 137 Euler’s criterion, 137 Euler’s theorem, 69 Exclusive-or (XOR), 20 Exhaustive key search, 14, 233–234, 272 Existential forgery, 30, 326, 432 exp(exponential function), 50 Expected running time, 63 Explicit authentication, 492 Exponent array, 617 Exponent recoding,seeExponentiation Exponential-time algorithm, 59 Exponentiation, 613–629, 633–634 addition chains, 621 exponent recoding, 627–629 signed-digit representation, 627–628 string-replacement representation, 628– 629 ﬁxed-base comb method, 625–627 ﬁxed-base Euclidean method, 624–625 ﬁxed-base windowing method, 623–624 left-to-right binary method, 615 left-to-rightk-ary method, 615 modiﬁed left-to-rightk-ary method, 616 Montgomery method, 619–620 repeated square-and-multiply algorithm, 71, 84 right-to-left binary method, 614 simultaneous multiple, 617–618 sliding-window method, 616 vector-addition chains, 622–623 Extendable secret sharing scheme, 526 Extended Euclidean algorithm for integers, 67 for polynomials, 82 Extended Riemann Hypothesis (ERH), 165 Extension ﬁeld, 77 Extractor, 406 F Factor base, 94, 109 Index 763 Factoring integers,seeInteger factorization Factoring polynomials,seePolynomial factoriza- tion Fail-stop signature scheme, 478–481, 488 Heijst-Pedersen, 478–481 Fair blind signature scheme, 487 Fair cryptosystems, 640–641, 658 for Difﬁe-Hellman key agreement, 641 patent, 640 FEAL block cipher, 259–262, 278–279 attacks on, 278–279 FEAL decryption algorithm, 261 FEAL-8 encryption algorithm, 261 FEAL-8 key schedule, 261 FEAL-N, 262 FEAL-NX, 262 patent, 639 test vectors, 262 Feedback shift register (FSR), 195–203 de Bruijn, 203 deﬁnition of, 202 delay element of, 202 feedback bit of, 202 feedback function of, 202 Feedback with carry shift register (FCSR), 217– 218, 222 initial state of, 202 linear feedback shift register,seeLinear feed- back shift register (LFSR) non-singular, 203 nonlinear feedback shift register, 202 output sequence of, 202 stage of, 202 Feedback with carry shift register (FCSR), 217–218, 222 Feige-Fiat-Shamir identiﬁcation protocol, 410–412, 422 Feige-Fiat-Shamir signature scheme, 447–449, 483 identity-based modiﬁcation, 449 key generation, 447 security of, 448 signature generation, 448 signature veriﬁcation, 448 Feistel cipher, 251, 276 Fermat liar, 136 Fermat number, 143, 166 Fermat witness, 136 Fermat’s primality test, 136 Fermat’s theorem, 69 Fiat-Shamir identiﬁcation protocol basic version, 408 patent, 638, 658 Fiat-Shamir signature scheme, 483 patent, 638, 658 Field, 77 characteristic of, 77 deﬁnition of, 77 extension ﬁeld of, 77 ﬁnite,seeFinite ﬁeld subﬁeld of, 77 Filtering function, 208 Finite ﬁeld, 80–85 deﬁnition of, 80 order of, 80 polynomial basis, 83 FIPS, 654–655, 661 ordering and acquiring, 656 FIPS 186 pseudorandom bit generator, 174–175 FISH stream cipher, 222 Fixed-point chaining attack, 374 Floyd’s cycle-ﬁnding algorithm, 91, 125 Forced delay attack, 417 Formal methods, 534, 541 Forward certiﬁcate, 575 Forward error correction, 363 Forward search attack, 34, 42, 288, 420 Fractionation, 276 Frequency distribution of English digrams, 247 of single English characters, 247 Frequency test, 181 Fresh key, 494 Function, 6–10, 50 bijection, 7 composition of, 19 deﬁnition of, 6 injective, 46 inverse, 7 involution, 10 one-to-one, 7 one-way, 8 onto, 7 permutation, 10 surjective, 46 trapdoor one-way, 9 Function ﬁeld sieve, 129 Functional diagram, 6 Functional graph, 54 component size, 55 cycle length, 55 predecessors size, 55 rho-length, 55 tail length, 55 tree size, 55 Functionally trusted third party, 39 G Gap of a sequence, 180 764 Index Garner’s algorithm, 612–613 Gauss’s algorithm, 68 Gaussian integer method, 128 gcd,seeGreatest common divisor Geffe generator, 206 General-purpose factoring algorithm, 90 Generator of a cyclic group, 76, 160 algorithm for ﬁnding, 163 ofF∗ q,8 1 ofF∗ 2 m, 163 ofZ∗ n ,6 9 ofZ∗ p , 164 algorithm for selecting, 164 Generator matrix, 506 Girault self-certiﬁed public key, 522 GMR one-time signature scheme, 468–471, 486 authentication tree, 470 key generation, 469 security of, 470 signature generation, 469 signature veriﬁcation, 469 GOAL stream cipher, 219 Goldwasser-Kilian primality test, 166 Goldwasser-Micali probabilistic public-key encryp- tion, 307–308 decryption algorithm, 307 encryption algorithm, 307 key generation, 307 security of, 308 Golomb’s randomness postulates, 180 Goppa code, 299, 317 Gordon’s algorithm for strong prime generation, 150 GOST block cipher, 282 GQ identiﬁcation protocol, 412–414, 422 patent, 639, 658 GQ signature scheme, 450–451 key generation, 450 message recovery variant, 451 patent, 639, 658 security of, 451 signature generation, 450 signature veriﬁcation, 450 Grandmaster postal-chess problem, 418 Greatest common divisor binary extended gcd algorithm, 608–610, 632 binary gcd algorithm, 606–607, 632 Euclidean algorithm, 66 Lehmer’s gcd algorithm, 607–608, 632 of integers, 64 of polynomials, 81 Group, 75–76 cyclic, 76 deﬁnition of, 75 of units, 77 order of, 75 subgroup of, 76 Group signature, 488 GSM, 586 GSS-API, 655, 661 G¨unther’s implicitly-certiﬁed public key, 521 G¨unther’s key agreement, 522 H Hagelin M-209, 245, 276 Hamming weight, 105 Handwritten signature, 23 Hard predicate, 115 Hash function, 33, 321–383 alternate terminology, 325, 371 applications, 321–322, 330–331 attacks, 368–375 birthday, 369–371 chaining, 373–375 Pseudo-collisions, 371–373 based on block ciphers, 338–343 Abreast Davies-Meyer, 380 Davies-Meyer, 341 Matyas-Meyer-Oseas, 341 MDC-2, 342 MDC-4, 343 Merkle’s DES-based hash, 338, 339, 378 Miyaguchi-Preneel, 341 N-Hash, 380 Tandem Davies-Meyer, 380 based on modular arithmetic, 351–352 MASH-1, 352 MASH-2, 352 cascading, 334 collision resistant (CRHF), 325 customized, 343–351 HA V AL, 379 MD2, 380 MD4, 346 MD5, 347 RIPEMD, 380 RIPEMD-128, 339, 380 RIPEMD-160, 339, 350 Secure Hash Algorithm (SHA-1), 348 Snefru, 380 deﬁnition of, 322 ideal security, 336 initialization value (IV), 335 MD-strengthening,seeMD-strengthening Merkle’s meta-method, 333 one-way (OWHF), 325 padding, 334–335 properties of Index 765 2nd-preimage resistance, 323 collision resistance, 324 compression, 322 ease of computation, 322 local one-wayness, 331 near-collision resistance, 331 non-correlation, 331 partial-preimage resistance, 331 preimage resistance, 323 strong collision resistance, 324 weak collision resistance, 324 r-collision resistant, 424 strong one-way, 325 universal classes of, 376 universal one-way, 377 weak one-way, 325 Hash-code, 321 Hash-result, 321 Hash-value, 33, 321 HA V AL hash function, 379 Heijst-Pedersen fail-stop signature scheme, 478–481 key generation, 478 proof-of-forgery algorithm, 481 signature generation, 479 signature veriﬁcation, 479 Hellman-Merkle patent, 637, 658 Heuristic security, 43, 533 High-order digit, 593 Hill cipher, 240, 274 Historical work factor, 44 HMAC, 355 Homomorphic property of RSA, 289 Homophonic substitution cipher, 17, 240 Hybrid protocol, 512 Hyperelliptic curve discrete logarithm problem, 130 ElGamal public-key encryption, 297 Hypothesis testing, 179–180 I IC card, 387 IDEA block cipher, 263–265, 279–280 attacks on, 279–280 decryption algorithm, 264 encryption algorithm, 264 key schedule, 264 patent, 640, 658 test vectors, 265 weak keys, 279 Ideal secret sharing scheme, 526, 527 Identiﬁcation, 3, 24–25, 385–424 applications of, 387 attacks on, 417–420, 424 chosen-text, 417 forced delay, 417 impersonation, 417 interleaving, 417 local, 419 non-interactive, 419 off-line, 419 pre-play, 397, 398 reﬂection, 417 remote, 419 replay, 417 challenge-response,seeChallenge-response identiﬁcation mutual, 387 passwords,seePasswords (weak authentication) questionnaire-based, 420 relation to signatures, 388 unilateral, 387 zero-knowledge,seeZero-knowledge identiﬁ- cation see alsoEntity authentication Identiﬁcation Friend or Foe (IFF) system, 421 Identity veriﬁcation, 385 Identity-based key establishment, 493 Identity-based system, 538, 561–562, 587 IDUP, 661 IEEE P1363 standard, 660 IETF, 655 Image of a function, 6, 50 Impersonation, 27, 42, 386, 417 Impersonator, 495 Implicit key authentication,seeKey authentication Implicitly-certiﬁed public key, 520–522, 562–563, 588 Difﬁe-Hellman using, 522–524 identity-based, 563 of Girault, 522 of G¨unther, 521 self-certiﬁed, 563 Imprint, 321 Improved PES (IPES), 279 In-line trusted third party, 547 Incremental hashing, 378 Independent events, 51 Index of coincidence, 248, 275 Index-calculus algorithm, 109–112, 128 Gaussian integer method, 128 inF 2m, 111 implementation reports, 128 inZp,1 1 0 implementation reports, 128 linear sieve, 128 residue list sieve, 128 Information dispersal algorithm (IDA), 539 766 Index Information rate, 527 Information security, 2 objectives of, 3 Information security service, 14 breaking of, 15 Information theory, 56–57 Initial state of an FSR, 202 of an LFSR, 196 Injective function, 46, 50 Inner product, 118 Input size, 58 Insider, 496 one-time, 496 permanent, 496 Integer, 49 multiple-precision, 593 negative signed-magnitude representation, 593 two’s complement representation, 594 single-precision, 593 Integer arithmetic,see Multiple-precision integer arithmetic Integer factorization, 89–98 continued fraction algorithm, 126 Dixon’s algorithm, 95, 127 elliptic curve algorithm, 94 general number ﬁeld sieve, 98 general-purpose algorithms, 90 heuristic running times, 127 multiple polynomial quadratic sieve, 97 Pollard’sp−1algorithm, 92–93 Pollard’s rho algorithm, 91–92 problem deﬁnition, 89 quadratic sieve algorithm, 95–97 random square methods, 94–98 special number ﬁeld sieve, 98 special-purpose algorithms, 90 trial division, 90–91 Integers modulon, 67–71 Integrity check value (ICV), 363 Interactive proof system, 406 Arthur-Merlin games, 421 completeness, 406 soundness, 406 Interleaving attack, 42, 417, 531, 540 Interloper, 13 Internal vertex, 557 Internet security standards, 655–656, 661 Intersection of sets, 49 Intruder, 13, 495 Intruder-in-the-middle attack, 530, 540 Inverse function, 7 Inversion attack on stream ciphers, 219 Involution, 10 Irreducible polynomial, 78, 154–160 algorithm for generating, 156 algorithm for testing, 155 number of, 155 primitive polynomial,seePrimitive polynomial trinomials, 157 ISO standards,seeISO/IEC standards ISO/IEC 9796, 442–444, 482–483 ISO/IEC standards, 645–648, 651–653, 660–661 committee draft (CD), 645 draft international standard (DIS), 645 ordering and acquiring, 656 working draft (WD), 645 Isomorphic, 81, 104 Iterated block cipher, 251 ITU, 653 J Jacobi sum primality test, 144, 166 Jacobi symbol, 73 computing, 73 Jefferson cylinder, 243, 274 Joint entropy, 56 JTC1, 645 K Karatsuba-Ofman multiplication, 630 Kasiski’s method, 248, 275 KDC, seeKey distribution center (KDC) Kerberos authentication protocol, 401, 501–502, 535–536 RFC 1510, 656 Kerckhoffs’ assumption, 225 Kerckhoffs’ desiderata, 14 Key, 11 archival, 580 backup, 580 cryptoperiod of, 553 data, 552 de-registration, 580 derived, 568 destruction, 580 fresh, 494 generator, 549 installation, 579 key-encrypting, 552 key-transport, 552 layering, 551–553 long-term, 553 master, 551 notarization, 568 offsetting, 568 private, 27, 544 Index 767 public, 27, 544 public-key vs. symmetric-key, 31–32, 551 recovery, 580 registration, 579 revocation, 566, 580 secret, 544 separation, 567 short-term, 553 symmetric, 544 terminal, 552 update, 580 variant, 568 Key access server, 549 Key agreement, 34, 35, 505–506, 515–524, 536– 538 Blom’s key pre-distribution system, 506 deﬁnition of, 490 Difﬁe-Hellman, 516 ElGamal, 517 encrypted key exchange (EKE), 538 G¨unther, 522 MTI/A0, 517–519 relation to key transport, 491 Station-to-station (STS), 519 Key authentication, 492 Key clustering attack on block ciphers, 281 Key conﬁrmation, 492 Key control, 494 Key derivation, 490, 498 Key distribution conﬁdential keys, 551–555 key layering, 551–553 key translation center, 553–554 symmetric-key certiﬁcates, 554–555 public keys, 555–566 authentication trees, 556–559 certiﬁcates, 559–561 identity-based, 561–562 implicitly-certiﬁed, 562–563 Key distribution center (KDC), 491, 500, 547 Key distribution pattern, 536 Key distribution problem, 16, 546 Key distribution system (KDS), 505 Blom’s KDS bound, 505 security against coalitions, 505 Key escrow, 584–586 agent, 550, 584 Clipper, 584 Key establishment, 489–541 analysis of, 530–534, 540–541 attacks on interleaving, 531 intruder-in-the-middle, 530 misplaced trust in server, 531 reﬂection, 530 authenticated, 492, 493 compliant, 532 deﬁnition of, 35, 490 identity-based, 493 key agreement,seeKey agreement key transport,seeKey transport message-independent, 493 operational, 532 resilient, 532 simpliﬁed classiﬁcation, 491 Key life cycle, 577–581 key states, 580 Key management, 36–38, 543–590 ANSI X9.17 standard, 650 ANSI X9.24 standard, 650 ANSI X9.28 standard, 651 ANSI X9.42 standard, 651 centralized, 546 controlling key usage, 567–570 deﬁnition of, 35, 544 ISO 8732 standard, 652 ISO 10202-7 standard, 652 ISO 11166 standard, 652 ISO 11568 standard, 653 ISO/IEC 11770 standard, 647 key agreement,seeKey agreement key distribution,seeKey distribution key establishment,seeKey establishment key life cycle, 577–581 key transport,seeKey transport Key management facility, 549 Key notarization, 568 patent, 642, 658 Key pair, 12 Key pre-distribution scheme, 540 deﬁnition of, 490 Key server, 549 Key space, 11, 21, 224 Key tag, 568 Key translation center (KTC), 491, 500, 547, 553 Key transport, 35, 497–504, 506–515, 535–536 AKEP1, 499 AKEP2, 499 Beller-Yacobi (2-pass), 514 Beller-Yacobi (4-pass), 513 COMSET, 536 deﬁnition of, 490 Kerberos, 501–502 Needham-Schroeder public-key, 508 Needham-Schroeder shared-key, 503 Otway-Rees protocol, 504 relation to key agreement, 491 Shamir’s no-key protocol, 500 768 Index X.509 three-way, 512 X.509 two-way, 511 Key update, 490 Keyed hash function,seeMessage authentication code (MAC) Keying material, 544 Keying relationship, 544 Keystream, 20, 193, 194 Keystream generator, 21, 194 Khafre block cipher, 271 attacks on, 281 patent, 644 Khufu block cipher, 271 attacks on, 281 patent, 644 Knapsack generator, 209, 220 Knapsack problem, 131 Knapsack public-key encryption, 300–306 Chor-Rivest, 302–306 Merkle Hellman, 300–302 Knapsack set, 117 density of, 120 Known-key attack, 42, 496, 534 Known-key triangle attack, 538 Known-message attack, 432 Known-plaintext attack, 41, 225 KryptoKnight, 535, 541 KTC, seeKey translation center (KTC) L L3-lattice basis reduction algorithm, 118–120, 131 Lagrange’s theorem, 76 Lambda method for discrete logarithms, 128 Lamport’s one-time-password scheme, 396 Lanczos method, 129 Lattice, 118 dimension of, 118 reduced basis, 118 Lattice basis reduction algorithm, 118–120, 131, 317 Law of large numbers, 52 Law of quadratic reciprocity, 72 lcm,seeLeast common multiple Leading coefﬁcient, 78 LEAF, 584–585 Leaf of a binary tree, 557 Least common multiple, 64 Least signiﬁcant digit, 593 Legendre symbol, 72 computing, 73 Lehmer’s gcd algorithm, 607–608, 632 Length of a vector, 118 Liar, 135 Euler, 138 Fermat, 136 strong, 139 Life cycle,seeKey life cycle Linear code, 506 Linear combination, 80 Linear complexity, 198–201 algorithm for computing,seeBerlekamp- Massey algorithm of a ﬁnite sequence, 198 of a random periodic sequence, 199 of a random sequence, 198 of an inﬁnite sequence, 198 proﬁle, 199 Linear complexity proﬁle, 199–200 algorithm for computing, 201 limitations of, 200 of a random sequence, 199 Linear congruential generator, 170, 187 multivariate congruential generator, 187 truncated, 187 Linear consistency attack, 219–220 Linear cryptanalysis of block ciphers, 258, 271, 278, 280 of stream ciphers, 219 Linear feedback shift register (LFSR), 195–201 connection polynomial of, 196 deﬁnition of, 195 delay element of, 195 feedback bit of, 196 initial state of, 196 maximum-length, 197 non-singular, 196 output sequence of, 195 stage of, 195 Linear sieve, 128 Linear syndrome attack, 218 Linear system (solving large), 129 Linearly dependent, 80 Linearly independent, 80 LION block cipher, 282 Little-endian, 344 Little-o notation, 59 Lock-in, 221 Logarithm, 49 LOKI block cipher, 281 LOKI’89, 281 LOKI’91, 270, 281 Long-term key, 553 Low-order digit, 593 Luby-Rackoff block cipher, 282 LUC cryptosystem, 314 LUCDIF, 316 LUCELG, 316 Lucas-Lehmer primality test, 142 Lucifer block cipher, 276 Index 769 patent, 641, 659 M m-sequence, 197 MAC, seeMessage authentication code (MAC) Manipulation detection code,seeModiﬁcation de- tection code Mapping, 6, 50 Markov cipher, 280 MASH-1 hash function, 352 ISO/IEC 10118-4 standard, 647 MASH-2 hash function, 352 ISO/IEC 10118-4 standard, 647 Master key, 551 Matyas-Meyer-Oseas hash function, 341 ISO/IEC 10118-2 standard, 647 Maurer’s algorithm for provable prime generation, 153, 167 Maurer’s universal statistical test, 183–185, 189 Maximum order complexity, 217 Maximum-length LFSR, 197 Maximum-rank-distance (MRD) code, 317 McEliece public-key encryption, 298–299, 317 decryption algorithm, 299 encryption algorithm, 299 key generation, 298 recommended parameter sizes, 299 security of, 299 MD-strengthening, 334, 335, 337 MD2 hash function, 380 RFC 1319, 655 MD4 hash function, 346 RFC 1320, 655 MD5 hash function, 347 RFC 1321, 655 MD5-MAC, 358 MDC, seeModiﬁcation detection code MDC-2 hash function, 342 ISO/IEC 10118-2 standard, 647 patent, 639 MDC-4 hash function, 343 patent, 639 MDS code, 281, 506 Mean, 51 Measure of roughness, 249 Mechanism, 34 Meet-in-the-middle attack on double DES, 235 on double encryption, 235 time-memory tradeoff, 236 on multiple encryption time-memory tradeoff, 236 Meet-in-the-middle chaining attack, 374 Merkle channel, 48 Merkle one-time signature scheme, 464–466, 485 authentication tree, 466 key generation, 464 patent, 643 security of, 465 signature generation, 465 signature veriﬁcation, 465 Merkle puzzle scheme, 47, 537 Merkle’s DES-based hash function, 338, 339, 378 Merkle’s meta-method for hashing, 333 Merkle-Hellman knapsack encryption, 300–302, 317–318 basic decryption algorithm, 301 encryption algorithm, 301 key generation, 300 multiple-iterated key generation, 302 patent, 637 security of, 302 Mersenne number, 142 Mersenne prime, 142, 143, 160 Message authentication,seeData origin authenti- cation Message authentication code (MAC), 33, 323, 352–359, 381–383 applications of, 323, 330 based on block ciphers, 353–354 CBC-MAC, seeCBC-MAC CFB-64 MAC, 650 RIPE-MAC, seeRIPE-MAC birthday attack on, 352 customized, 356–358 bucket hashing, 382 MD5-MAC, 358 Message Authenticator Algorithm (MAA), 356 deﬁnition, 325 for stream ciphers, 358–359 CRC-based, 359 Lai-Rueppel-Woollven scheme, 383 Taylor’s scheme, 383 from MDCs, 354–355 envelope method with padding, 355 hash-based MAC, 355 HMAC, 355 secret preﬁx method, 355 secret sufﬁx method, 355 XOR MAC, 382 ISO 8730 standard, 652 ISO 9807 standard, 652 properties of compression, 325 computation-resistance, 325 770 Index ease of computation, 325 key non-recovery, 325 retail MAC, 650 types of attack adaptive chosen-text, 326 chosen-text, 326 known-text, 326 types of forgery existential, 326 selective, 326 see alsoCBC-MAC Message authentication tag system, 376 Message Authenticator Algorithm (MAA), 356 ISO 8731-2 standard, 652 Message concealing in RSA, 290, 313 Message digest, 321 Message integrity code (MIC), 323 Message space, 11 Message-independent key establishment, 493 Micali-Schnorr pseudorandom bit generator, 186 Miller-Rabin primality test, 139, 165 MIME, 656, 661 Minimum disclosure proof, 421 Minimum polynomial, 156 Mips year, 126 MISSI, 590 Mixed-radix representation, 611, 630 Mixing algebraic systems, 279 Miyaguchi-Preneel hash function, 341 M¨obius function, 154 modnotation, 64 Modes of operation multiple modes,seeMultiple encryption, modes of operation single modes,seeBlock cipher, modes of op- eration Modiﬁcation detection code (MDC), 33, 323, 324 Modiﬁed-Rabin pseudorandom bit generator, 190 Modiﬁed-Rabin signature scheme, 439–442, 482 key generation, 440 security of, 441 signature generation, 440 signature veriﬁcation, 440 Modular arithmetic,seeMultiple-precision modu- lar arithmetic Modular exponentiation,seeExponentiation Modular reduction, 599 Barrett, 603–605, 631 Montgomery, 600–602, 631 special moduli, 605–606 Modular representation,seeMixed-radix represen- tation Modulus, 67 Monic polynomial, 78 Mono-alphabetic substitution cipher,seeSubstitu- tion cipher Monobit test, 181 Monotone access structure, 527 Montgomery exponentiation, 619–620 Montgomery multiplication, 602–603 Montgomery reduction, 600–602, 631 MOSS, 656 RFC 1848, 656 Most signiﬁcant digit, 593 MTI protocols, 518, 537 MTI/A0 key agreement, 517–519, 537 Goss variant, 537 patent, 644, 659 Multi-secret threshold scheme, 527 Multiple encryption, 234–237 deﬁnition of, 234 double encryption, 234 modes of operation, 237 triple-inner-CBC mode, 237 triple-outer-CBC mode, 237 triple encryption, 235 E-D-E, 235 two-key triple-encryption, 235 Multiple polynomial quadratic sieve, 97 Multiple-precision integer, 593 Multiple-precision integer arithmetic, 592–599 addition, 594–595 division, 598–599 normalization, 599 gcd,seeGreatest common divisor multiplication, 595–596 discrete Fourier transform (DFT), 631 Karatsuba-Ofman, 630 squaring, 596–597 subtraction, 594–595 Multiple-precision modular arithmetic, 599–606 addition, 600 exponentiation,seeExponentiation inversion, 610 multiplication classical, 600 Montgomery multiplication, 602–603 reduction, 599 Barrett, 603–605, 631 Montgomery, 600–602, 631 special moduli, 605–606 subtraction, 600 Multiplexer generator, 220 Multiplicative group ofZ n,6 9 of a ﬁnite ﬁeld, 81 Multiplicative inverse, 68 computing, 71, 84, 610 Index 771 Multiplicative property in RSA, 288, 435, 482 Multiplicity of a factor, 122 Multispeed inner-product generator, 220 Multivariate polynomial congruential generator, 187 Mutual authentication, 387, 402, 405, 494 Mutual information, 57 Mutually exclusive events, 51 N N-Hash function, 380 Name server, 549 Needham-Schroeder public-key, 508, 536 Needham-Schroeder shared-key, 401, 503, 535 Next-bit test, 171 Next-discrepancy, 200 Nibble, 443 NIST, 654 Noise diode, 40 Non-interactive protocol, 493 Non-interactive ZK proof, 424 Non-malleable encryption, 311, 319 Non-repudiation, 3, 4, 582–584 ISO/IEC 13888 standard, 648 Non-singular FSR, 203 LFSR, 196 Nonce, 397, 497 Nonlinear combination generator, 205–208 combining function of, 205 Nonlinear feedback shift register,seeFeedback shift register (FSR) Nonlinear ﬁlter generator, 208–209 ﬁltering function, 208 Nonlinear order, 205 Normal basis, 168 exponentiation, 642 multiplication, 642 patents, 642–643, 659 Normal distribution, 176–177 mean of, 176 standard, 176 variance of, 176 Normal polynomial, 168 Normalization, 599 Notarized key, 569 Notary agent, 550 seal, 569 service, 582 NP ,6 0 NP -complete, 61 NP -hard, 62 NPC ,6 1 Number ﬁeld sieve for discrete logarithms, 128 for integer factorization, 98, 126 implementation reports, 126, 127 general number ﬁeld sieve, 98 special number ﬁeld sieve, 98, 126 Number theory, 63–75 Nyberg-Rueppel signature scheme, 460–462, 485 security of, 461 signature generation, 461 signature veriﬁcation, 461 O Object identiﬁer (OID), 660 OFB, seeOutput feedback mode Off-line trusted third party, 548 Ohta-Okamoto identiﬁcation protocol, 422 On-line certiﬁcate, 576 On-line trusted third party, 547 On-line/off-line signature, 486 patent, 644 One-key encryption, 15 One-sided statistical test, 179 One-time insider, 496 One-time pad, 21, 192–193, 274 patent, 657 One-time password scheme, 395–397 One-time signature scheme, 462–471 Difﬁe-Lamport, 485 GMR, 468–471 Merkle, 464–466 Rabin, 462–464 validation parameters, 462 One-to-one function, 7–8, 50 One-way cipher, 377 One-way function, 8–9, 327 DES-based, 190, 328 exponentiation modulo a prime, 115, 329 multiplication of large primes, 329 Rabin function, 115 RSA function, 115 One-way hash function (OWHF), 325 One-way permutation, 115, 328 Onto function, 7, 50 Open Systems Interconnection (OSI), 653, 660 Operational, 532 Opponent, 13, 495 see alsoAttacker Optimal normal basis, 168, 659 Oracle, 88 Order generating element of maximum order inZ ∗ n, 163 ofZ∗ n ,6 9 772 Index of a ﬁnite ﬁeld, 80 of a group, 75 of a group element, 76, 160 algorithm for determining, 162 of an element inZ∗ n,6 9 Otway-Rees protocol, 504, 536 Output feedback mode (OFB), 232–233 as a stream cipher, 233 changing IV in, 232 counter mode, 233 feedback size, 233 Outsider, 496 OWHF, seeOne-way hash function Ownership, 3 P P,6 0 Palindromic keys of DES, 257 Party, 13 Passcode generator, 402 Passive adversary, 15 Passive attack, 41, 495 Passkey, 395 Passphrase, 390 Passwords (weak authentication), 388–397, 420 aging, 390 attacks on, 391–393 dictionary, 392 exhaustive search, 391 password-guessing, 392 pre-play, 397 replay, 391 encrypted password ﬁle, 389 entropy, 392 generator, 387 one-time, 395–397 Lamport’s scheme, 396 passkey, 395 passphrase, 390 personal identiﬁcation number (PIN), 394 rules, 389 salting, 390 stored password ﬁle, 389 UNIX , 393–394 Patents, 635–645, 657–659 ordering and acquiring, 645 priority date, 636 validity period, 636 PEM, seePrivacy Enhanced Mail (PEM) Pepin’s primality test, 166 Perceptrons problem, 423 Perfect forward secrecy, 496, 534 Perfect power testing for, 89 Perfect secrecy, 42, 227, 307 Perfect secret sharing scheme, 526, 527 Perfect zero-knowledge protocol, 407 Period of a periodic sequence, 180 Periodic sequence, 180 autocorrelation function of, 180 cycle of, 180 period of, 180 Permanent insider, 496 Permutation, 10, 50 Permutation polynomial, 314 Permuted kernel problem, 423 Personal Identiﬁcation Number (PIN) ANSI X9.8 standard, 649 ISO 9564 standard, 652 PGP, seePretty Good Privacy (PGP) Phi function (φ), 65 Photuris, 661 Physically secure channel, 13 PIKE stream cipher, 222 PIN,seePasswords (weak authentication),seePer- sonal Identiﬁcation Number (PIN) PKCS standards, 656, 661 ordering and acquiring, 657 PKCS #1, 445–447, 483 Plaintext, 11 Plaintext-aware encryption scheme, 311–312 Playfair cipher, 239, 274 Pless generator, 218 PN-sequence, 181 Pocklington’s theorem, 144 Pohlig-Hellman algorithm, 107–109, 128 Pohlig-Hellman cipher, 271 patent, 642, 659 Poker test, 182, 188 Policy Certiﬁcation Authority (PCA), 589 Pollard’sp−1algorithm, 92–93, 125 Pollard’s rho algorithm for discrete logarithms, 106–107, 128 for factoring, 91–92, 125 Polyalphabetic substitution cipher, 18, 241–242, 273–274 auto-key cipher, 242 Beaufort cipher, 241 cipher machine,seeCipher machine PURPLE cipher, 276 Vigen`ere cipher auto-key, 242 compound, 241 full, 242 running-key, 242 simple, 18, 241 single mixed alphabet, 242 Polygram substitution cipher, 239 Index 773 Polynomial, 78 irreducible, 78 leading coefﬁcient of, 78 Polynomial basis, 83 Polynomial factorization, 122–124, 132 Berlekamp’sQ-matrix algorithm, 124 square-free factorization, 123 Polynomial-time algorithm, 59 Polynomial-time indistinguishability, 318 Polynomial-time statistical test, 171 Polynomially security public-key encryption, 306 Polytime reduction, 61, 88 Practical security, 43 Pre-play attack, 397, 398 Pre-positioned secret sharing scheme, 527 Precision, 593 Preimage, 6, 50 Preimage resistance, 323 Pretty Good Privacy (PGP), 661 Primality proving algorithm,seePrimality test, true primality test Primality test probabilistic primality test, 135–142 comparison, 140–142 Fermat’s test, 136 Miller-Rabin test, 139 Solovay-Strassen test, 138 true primality test, 142–145 Atkin’s test, 145 Goldwasser-Kilian test, 166 Jacobi sum test, 144 Lucas-Lehmer test, 142 Pepin’s test, 166 Prime number, 9, 64 Prime number generation, 145–154 algorithms Gordon’s algorithm, 150 Maurer’s algorithm, 153 NIST method, 151 random search, 146 DSA primes, 150–152 incremental search, 148 provable primes, 152–154 random search, 145–149 strong primes, 149–150 Prime number theorem, 64 Primitive element,seeGenerator Primitive normal polynomial, 168 Primitive polynomial, 157–160 algorithm for generating, 160 algorithm for testing, 157 deﬁnition of, 84 Primitives, 4 Principal, 495 Principal square root, 74 Privacy,seeConﬁdentiality Privacy Enhanced Mail (PEM), 588, 655 RFCs 1421–1424, 655 Private key, 26, 27, 544 Private-key certiﬁcate,seeSymmetric-key certiﬁ- cate Private-key encryption, 15 Probabilistic public-key encryption, 306–312, 318–319 Blum-Goldwasser, 308–311 Goldwasser-Micali, 307–308 security level polynomially secure, 306 semantically secure, 306 Probability, 50 Probability density function, 176 Probability distribution, 50 Probability theory, 50–55 Probable prime, 136 Product cipher, 20, 251 Proof of knowledge, 406, 421, 422 Proposed Encryption Standard (PES), 279 Protection lifetime, 553, 578 Protocol authentication, 493 cut-and-choose, 410, 421 deﬁnition of, 33, 490 failure of, 34 hybrid, 512 identiﬁcation,seeIdentiﬁcation key establishment,seeKey establishment message-independent, 493 non-interactive, 493 witness hiding, 423 zero-knowledge, 405–417 Provable prime, 134, 142 Provable security, 43, 533 Prover, 386 Pseudo-collision, 371 Pseudo-Hadamard transform, 266 Pseudo-noise sequence, 181 Pseudoprime, 136 Euler, 138 strong, 139 Pseudorandom bit generator (PRBG), 173–175 ANSI X9.17, 173 deﬁnition of, 170 FIPS 186, 174–175 linear congruential generator, 170, 187 Pseudorandom bit sequence, 170 Pseudorandom function, 331 Pseudorandom sequences, 39–41 Pseudosquares modulon, 74, 99, 308 774 Index Public key, 26, 27, 544 compared vs. symmetric-key, 31–32, 551 implicitly-certiﬁed, 520–522 Public-key certiﬁcate, 39, 559–561, 587 data part, 559 distinguished name, 559 signature part, 559 Public-key encryption, 25–27, 283–319 advantages of, 31 disadvantages of, 32 ElGamal, 294–298 knapsack, 300–306 Chor-Rivest, 302–306 Merkle-Hellman, 300–302 LUC, seeLUC cryptosystem McEliece, 298–299 non-malleable, 311 plaintext-aware, 311–312 probabilistic, 306–312 Blum-Goldwasser, 308–311 Goldwasser-Micali, 307–308 Rabin, 292–294 reversible, 28 RSA, 285–291 types of attacks, 285 Williams, 315 PURPLE cipher, 276 Puzzle system, 376, 537 Q Quadratic congruential generator, 187 Quadratic non-residues, 70 Quadratic residues, 70 Quadratic residuosity problem, 99, 127, 307 Quadratic sieve factoring algorithm, 95–97, 126 implementation reports, 126 Quantum computer, 130 Quantum cryptography, 48, 535 Quotient, 64, 78 R Rabin one-time signature scheme, 462–464 key generation, 463 resolution of disputes, 463 signature generation, 463 signature veriﬁcation, 463 Rabin public-key encryption, 292–294, 315 decryption algorithm, 292 encryption algorithm, 292 key generation, 292 security of, 293 use of redundancy, 293 Rabin signature scheme, 438–442, 482 ISO/IEC 9796, 442–444 key generation, 438 signature generation, 438 signature veriﬁcation, 439 use of redundancy, 439 Rabin’s information dispersal algorithm (IDA), 539 RACE/RIPE project, 421, 536 Radix representation, 592–593 baseb, 592 binary, 592 high-order digit, 593 least signiﬁcant digit, 593 low-order digit, 593 mixed, 611, 630 most signiﬁcant digit, 593 precision, 593 radixb, 592 Ramp schemes,seeSecret sharing Random bit generator, 39–41, 171–173 cryptographically secure pseudorandom bit generator,seeCryptographically sec- ure pseudorandom bit generator (CSPRBG) deﬁnition of, 170 hardware techniques, 172 pseudorandom bit generator,seePseudorand- om bit generator (PRBG) software techniques, 172 Random cipher, 225 Random cipher model, 246 Random function, 190 poly-random, 190 Random mappings model, 54 Random oracle model, 316 Random square methods, 94–98 Random variable, 51 continuous, 176 entropy of, 56 expected value of, 51 mean of, 51 standard deviation of, 51 variance of, 51 Randomized algorithm, 62–63 Randomized DES (RDES) block cipher, 278 Randomized encryption, 225, 296, 306 Randomized stream cipher, 216 Range of a function, 46 Rate of an iterated hash function, 340 Rational numbers, 49 RC2 block cipher, 282 RC4 stream cipher, 222, 282 RC5 block cipher, 269–270, 280–281 attacks on, 280–281 decryption algorithm, 270 encryption algorithm, 270 Index 775 key schedule, 270 patent, 659 test vectors, 270 weak keys, 281 Real number, 49 Real-time, 385 Reblocking problem in RSA, 435–436, 482 Receipt, 3 Receiver, 13 Reduced basis, 118 Redundancy, 29, 431 of English, 245 Reﬂection attack, 417, 530, 540 Registration authority, 549 Related-key attack on block ciphers, 281 Relatively prime, 64 Remainder, 64, 78 Replay attack, 42, 417 Requests for Comments,seeRFCs Residue list sieve, 128 Resilient key establishment protocol, 532 Response, 409 Retail banking, 648 Retail MAC, 650 Reverse certiﬁcate, 575 Reversible public-key encryption scheme, 28 Revocation, 3 RFCs, 655–656 ordering and acquiring, 657 Ring, 76–77 commutative, 77 deﬁnition of, 76 group of units, 77 polynomial, 78–79 Rip van Winkle cipher, 216 RIPE-MAC, 354, 381 RIPEMD hash function, 380 RIPEMD-128 hash function, 339, 380 RIPEMD-160 hash function, 339, 350 ISO/IEC 10118-3 standard, 647 Root vertex, 557 Rotor-based machine,seeCipher machine Round function, 251 Round of a product cipher, 20 RP ,6 3 RSA-129 number, 126, 130 RSA problem, 98–99, 127, 287 security of individual bits, 116 RSA pseudorandom bit generator, 185–186 RSA public-key encryption, 285–291, 312–315 decryption algorithm, 286, 611, 613 decryption exponent, 286 elliptic curve analogue, 315 encryption algorithm, 286 encryption exponent, 286 key generation, 286 modulus, 286 patent, 638 prime selection, 290 recommended modulus size, 290 security of, 287–290 adaptive chosen-ciphertext attack, 289, 313 common modulus attack, 289 cycling attacks, 289, 313 forward search attack, 288 message concealing, 290, 313 multiplicative properties, 288 polynomially related plaintext, 313 relation to factoring, 287 small decryption exponent, 288 small encryption exponent, 288, 291, 313 unbalanced, 314 RSA signature scheme, 433–438, 482 ANSI X9.31-1 standard, 651 bandwidth efﬁciency, 437 ISO/IEC 9796, 442–444 key generation, 434 patent, 638 PKCS #1, 445–447 reblocking problem, 435–436, 482 redundancy function, 437 security of, 434–435 signature generation, 434, 613 signature veriﬁcation, 434 Run of a sequence, 180 Running key generator, 194 Runs test, 182, 188 S S/MIME, 661 Safe prime, 537 algorithm for generating, 164 deﬁnition of, 164 SAFER block cipher, 266–269, 280 attacks on, 280 SAFER K-64 decryption algorithm, 269 SAFER K-64 encryption algorithm, 268 SAFER K-64 key schedule, 268 SAFER K-128, 280 SAFER SK-64 key schedule, 268 SK-128, 280 test vectors, 269 Salt, 288, 390 Schnorr identiﬁcation protocol, 414–416, 422 patent, 639 Schnorr signature scheme, 459–460, 484 Brickell-McCurley variant, 484 776 Index Okamoto variant, 484 patent, 639 signature generation, 459 signature veriﬁcation, 460 SEAL stream cipher, 213–216 implementation report, 222 patent, 222 test vectors, 215 Sealed authenticator, 361 Sealed key, 568 2nd-preimage resistance, 323, 325 Secrecy,seeConﬁdentiality Secret broadcasting scheme, 540 Secret key, 544 Secret-key certiﬁcate, 588 Secret sharing, 524–528, 538–540 access structure, 526 authorized subset, 527 dynamic, 527 extendable, 526 generalized, 526–528 ideal, 527 information rate, 527 multi-secret threshold, 527 perfect, 526, 527 pre-positioned, 527 ramp schemes, 539 shared control schemes, 524–525 threshold scheme, 525–526 veriﬁable, 527 visual cryptography, 539 with disenrollment, 528 Secure channel, 13 Secure Hash Algorithm (SHA-1), 348 ANSI X9.30-2 standard, 651 FIPS 180-1 standard, 654 ISO/IEC 10118-3 standard, 647 Secured channel, 13 Security domain, 570 Security policy, 545 Seed, 21, 170 Selective forgery, 326, 432 Self-shrinking generator, 221 Self-synchronizing stream cipher, 194–195 Semantically secure public-key encryption, 306 Semi-weak keys of DES, 257 Sender, 13 Sequence block of, 180 de Bruijn, 203 gap of, 180 m-sequence, 197 periodic, 180 pn-sequence, 181 pseudo-noise, 181 run of, 180 Sequence numbers, 399 Serial test, 181, 188 Session key, 36, 494 Session key establishment, 491 SHA-1, seeSecure Hash Algorithm (SHA-1) Shadow, 538 Shamir’s no-key protocol, 500, 535 Shamir’s threshold scheme, 526, 539 Shared control schemes, 524–525 Shares, 524–528, 538 SHARK block cipher, 281 Shift cipher, 239 Short-term key, 553 Shrinking generator, 211–212 implementation report, 221 Sieving, 97 Signature, 3, 22–23, 28–30, 425–488 arbitrated, 472–473 blind,seeBlind signature scheme designated conﬁrmer, 487 deterministic, 427 Difﬁe-Lamport, 485 Digital Signature Algorithm (DSA), 452–454 ElGamal, 454–459 ESIGN, 473–474 fail-stop,seeFail-stop signature scheme Feige-Fiat-Shamir, 447–449 framework, 426–433 generation algorithm, 426 GMR, 468–471 GQ, 450–451 group, 488 handwritten, 23 Merkle one-time, 464–466 modiﬁed-Rabin, 439–442 Nyberg-Rueppel, 460–462 on-line/off-line, 486 Ong-Schnorr-Shamir (OSS), 482, 486 Rabin, 438–442 Rabin one-time, 462–464 randomized, 427 relation to identiﬁcation, 388 resolution of disputes, 30 RSA, 433–438 Schnorr, 459–460 strongly equivalent, 485 types of attacks, 432 undeniable,seeUndeniable signature scheme veriﬁcation algorithm, 426 with appendix, 481 framework, 428–430 ISO/IEC 14888 standard, 648 Index 777 PKCS #1, 445–447 with message recovery, 29 framework, 430–432 ISO/IEC 9796 standard, 442–444, 646, 660 with redundancy, 29 Signature notarization, 583 Signature space, 427 Signature stripping, 510 Signed-digit representation, 627–628 Signed-magnitude representation, 593 Signer, 23 Signiﬁcance level, 179 Signing transformation, 22 Simple substitution cipher,seeMono-alphabetic sub- stitution cipher Simulator, 407 Simultaneous diophantine approximation, 121–122 algorithm for, 122 unusually good, 121 Simultaneous multiple exponentiation, 617 Simultaneously secure bits, 115 Single-key encryption, 15 Single-length MDC, 339 Single-precision integer, 593 Singleton bound, 506 SKEME, 661 SKID2 identiﬁcation protocol, 402, 421 SKID3 identiﬁcation protocol, 402, 421 SKIP, 661 SKIPJACK block cipher, 282, 654 Sliding-window exponentiation, 616 Small decryption exponent in RSA, 288 Small encryption exponent in RSA, 288, 291, 313 Smart card, 387 ISO 10202 standard, 652 Smooth integer, 92 polynomial, 112 Snefru hash function, 380 8×32S-boxes, 281 Solovay-Strassen primality test, 138, 165 Span, 80 Sparse linear equations, 129 conjugate gradient method, 129 Lanczos method, 129 Wiedemann algorithm, 129 Special-purpose factoring algorithm, 90 SPKM, 656, 661 Split-knowledge scheme, 525 Splitting an integer, 89 Spread spectrum, 45 Square roots, 99–102 composite modulus, 101–102, 127 prime modulus, 100–101, 127 SQROOT problem, 101 Square-free factorization, 123 algorithm for, 123, 132 Square-free integer, 137 Square-free polynomial, 123 Stage of an FSR, 202 of an LFSR, 195 Standard deviation, 51 Standard normal distribution, 176 Standards, 645–657, 660–661 ANSI, 648–651 FIPS, 654–655 IEEE, 660 Internet, 655–656 ISO/IEC, 645–648, 651–653 PKCS, 656 RFC, 655–656 X.509, 653 Station-to-station (STS) key agreement, 519, 538 Statistical test, 175–185, 188–189 autocorrelation test, 182 frequency test, 181 hypothesis, 179 Maurer’s universal statistical test, 183–185, 189 one-sided test, 179 poker test, 182 polynomial-time, 171 runs test, 182 serial test, 181 signiﬁcance level, 179 two-sided test, 180 Statistical zero-knowledge protocol, 424 Steganography, 46 Step-1/step-2 generator, 220 Stirling numbers, 53 Stirling’s formula, 59 Stop-and-go generator, 220 Stream cipher, 20–21, 191–222 A5, 222 attacks on correlation attack, 206, 218 inversion attack, 219 linear consistency attack, 219–220 linear cryptanalysis, 219 linear syndrome attack, 218 lock-in, 221 cellular automata, 222 classiﬁcation, 192–195 clock-controlled generator, 209–212 alternating step generator, 209–211 m-sequence cascade, 221 778 Index p-cycle cascade, 220 self-shrinking generator, 221 shrinking generator, 211–212 step-1/step-2 generator, 220 stop-and-go generator, 220 comparison with block ciphers, 192 FISH, 222 GOAL, 219 initial state, 193, 194 keystream, 193, 194 next-state function, 193 nonlinear combination generator, 205–208 Geffe generator, 206 multiplexer generator, 220 multispeed inner-product generator, 220 Pless generator, 218 summation generator, 207 nonlinear ﬁlter generator, 208–209 knapsack generator, 209 one-time pad, 192–193 output function, 193, 194 PIKE, 222 randomized stream cipher, 216 RC4, 222 Rip van Winkle cipher, 216 SEAL, 213–216 self-synchronizing stream cipher, 194–195 synchronous stream cipher, 193–194 Strict avalanche criterion (SAC), 277 String-replacement representation, 628–629 Strong collision resistance, 324 Strong equivalent signature schemes, 485 Strong liar, 139 Strong one-way hash function, 325 Strong prime, 149–150 algorithm for generating, 150 deﬁnition of, 149, 291 Hellman-Bach patent, 643 usage in RSA, 291 Strong pseudoprime, 139 Strong pseudoprime test,seeMiller-Rabin primal- ity test Strong witness, 139 Subexponential-time algorithm, 60 Subﬁeld, 77 Subgroup, 76 Subliminal channel, 485 broadband, 485 narrowband, 485 Subset sum problem, 61, 117–122, 190 meet-in-the-middle algorithm, 118 naive algorithm, 117 superincreasing, 300 usingL 3 algorithm, 120 Subspace of a vector space, 80 Substitution cipher, 17–18, 238–241 homophonic, 17, 240 mono-alphabetic, 17, 239 afﬁne cipher, 239 Caesar cipher, 239 shift cipher, 239 unicity distance of, 247 polyalphabetic, 18 polygram, 239 Hill cipher, 240 Playfair cipher, 239 Substitution-permutation (SP) network, 251 Summation generator, 207, 218 Superincreasing subset sum problem, 300 algorithm for solving, 300 Superuser, 389 Surjective function, 46, 50 SWIFT, 586 Symmetric cryptographic system, 544 Symmetric key, 544 compared vs. public-key, 31–32, 551 Symmetric-key certiﬁcate, 554–555, 587 Symmetric-key encryption, 15–21 advantages of, 31 block cipher, 223–282 deﬁnition of, 15 disadvantages of, 31 stream cipher, 191–222 Synchronous stream cipher, 193–194 binary additive stream cipher, 194 Syndrome decoding problem, 190, 423 T Tapper, 13 TEA block cipher, 282 TEMPEST, 45 Teraﬂop, 44 Terminal key, 552 Test vectors DES, 256 FEAL, 262 IDEA, 265 MD4, 345 MD5, 345 MD5-MAC, 358 RC5, 270 RIPEMD-160, 345 SAFER, 269 SHA-1, 345 3-WAY block cipher, 281 Threshold cryptography, 534 Threshold scheme, 525–526 Blakley, 538 Index 779 Shamir, 526, 539 Ticket, 501, 570, 586 Time-memory tradeoff, 236, 273 Time-variant parameter, 362, 397–400, 497 nonce, 397 random numbers, 398–399 sequence numbers, 399 timestamps, 399–400 Timestamp, 3, 399–400, 420, 581–582 agent, 550 Toeplitz matrix, 382 Transaction authentication, 362 Transformation, 6 Transinformation, 57 Transposition cipher, 18, 238 compound, 238 simple, 18, 238 unicity distance of, 246 Trapdoor one-way function, 9, 26 Trapdoor predicate, 318 Tree authentication, 376 patent, 637 Trinomial, 154 Triple encryption, 235–237, 272 Triple-DES, 272, 651 ANSI X9.52 standard, 651 Triple-inner-CBC mode, 237 Triple-outer-CBC mode, 237 Truncated differential analysis, 271, 280 Trust model, 572 centralized, 573 directed graph, 575 distributed, 575 hierarchy with reverse certiﬁcates, 575 rooted chain, 573 separate domains, 573 strict hierarchical, 573 Trusted server, 491 Trusted third party (TTP), 30, 36, 491, 547–550, 581–584 authentication server, 549 certiﬁcate directory, 549 certiﬁcation authority (CA), 548 functionally trusted, 39 in-line, 547 KDC, seeKey distribution center (KDC) key access server, 549 key escrow agent, 550 key generator, 549 key management facility, 549 key server, 549 KTC, seeKey translation center (KTC) name server, 549 notary agent, 550 off-line, 548 on-line, 547 registration authority, 549 timestamp agent, 550 unconditionally trusted, 39 TTP, seeTrusted third party (TTP) Turing-Kolmogorov-Chaitin complexity, 217 Two’s complement representation, 594 2-adic span, 218 Two-bit test, 181 Two-key triple-encryption, 235 chosen-plaintext attack on, 236 known-plaintext attack on, 237 Two-sided statistical test, 180 Type I error, 179 Type II error, 179 U Unbalanced RSA, 314 Unblinding function, 475 Unconcealed message, 290 Unconditional security,seePerfect secrecy, 533 Unconditionally trusted third party, 39 Undeniable signature scheme, 476–478, 487–488 Chaum-van Antwerpen, 476–478 conﬁrmer, 487 Unicity distance deﬁnition of, 246 known-plaintext, 235 of a cascade cipher, 272 of a mono-alphabetic substitution cipher, 247 of a transposition cipher, 246 Unilateral authentication, 387, 401–402, 405, 494 Union of sets, 49 Unique factorization domain, 81 Unit, 68, 77, 103, 114 Universal classes of hash function, 376 Universal exponent, 287 Universal forgery, 482 Universal one-way hash function, 377 Universal statistical test,seeMaurer’s universal statistical test UNIX passwords, 393–394 Unsecured channel, 13 Unusually good simultaneous diophantine approx- imation, 121, 317 Userid, 388 V Validation, 3 Validation parameters, 462 Variance, 51 Vector space, 79–80 dimension of, 80 standard basis, 80 780 Index subspace of, 80 Vector-addition chains, 622–623 Veriﬁable secret sharing, 527, 539 Veriﬁcation algorithm, 426 Veriﬁcation transformation, 22 Veriﬁer, 23, 385, 386 Vernam cipher,seeOne-time pad Vigen`ere cipher,seePolyalphabetic substitution ci- pher Visual cryptography, 539 W WAKE block cipher, 282 Weak collision resistance, 324 Weak keys of DES, 257 Weak one-way hash function, 325 Wheatstone disc, 274 Wholesale banking, 648 Wiedemann algorithm, 129 Williams’ public-key encryption, 315 Witness, 135, 409 Euler, 137 Fermat, 136 strong, 139 Witness hiding protocol, 423 Witness indistinguishability, 423 Witnessing, 3 Work factor, 44 historical, 44 Worst-case running time, 58 Wyner’s wire-tap channel, 535 X X.509 authentication protocol, 536 three-way, 512 two-way, 511 X.509 certiﬁcate, 587 X.509 standard, 653 XOR, seeExclusive-or Y Yuval’s birthday attack, 369 Z Zero-knowledge identiﬁcation, 405–417, 421–424 Brickell-McCurley, 423 comparison of protocols, 416–417 constrained linear equations problem, 423 extended Fiat-Shamir, 422 Feige-Fiat-Shamir, 410–412 Fiat-Shamir (basic version), 408 Fischer-Micali-Rackoff, 422 GQ, 412–414 Ohta-Okamoto, 422 permuted kernel problem, 423 Schnorr, 414–416 syndrome decoding problem, 423 Zero-knowledge protocol, 405–417, 421–424 auxiliary-input, 423 black-box simulation, 423 challenge, 409 completeness, 406 computational, 407 extracting secret, 406 for possession of discrete log, 422 parallel version, 412 perfect, 407 proof of knowledge, 406, 421, 422 proof of membership, 421 response, 409 simulator, 407 soundness, 406 statistical, 424 witness, 409 Ziv-Lempel complexity, 217 Z p-operation, 82 ZPP ,6 3
Kernel-Fuzzing	<+41>: lock cmpxchg %ecx,0x54(%rbx) <+46>: setne %r15b <+50>: sete %sil <+54>: xor %edi,%edi <+56>: callq 0xffffffff81167ea0 <__sanitizer_cov_trace_const_cmp1> <+61>: test %r15b,%r15b <+64>: jne 0xffffffff815b78f9 <vm_page_remove+73> <+66>: callq 0xffffffff81167dc0 <__sanitizer_cov_trace_pc> <+71>: jmp 0xffffffff815b790d <vm_page_remove+93> <+73>: callq 0xffffffff81167dc0 <__sanitizer_cov_trace_pc> <+78>: movl $0x1,0x54(%rbx) <+85>: mov %rbx,%rdi <+88>: callq 0xffffffff81107950 <wakeup> <+93>: mov %r14d,%eax <+96>: add $0x8,%rsp <+100>: pop %rbx <+101>: pop %r14 16 FreeBSD Journal • November/December 2020 If you have ever been unlucky enough to fall victim to a FreeBSD kernel panic, you would be well-justified in asking just how those sloppy kernel programmers test their code. The kernel is the backbone of the entire system and changes to it should of course have been meticulously tested before users can boot up the latest and greatest build. In our defense, however, kernel programmers work in a harsh, inhospitable environment. The FreeBSD kernel is written in C, a programming language infamous for its subtle pitfalls and lack of amenities. The kernel also has to deal with several adversaries: first, it executes and provides services to all sorts of soft- ware, some of which may have malicious goals; second, it interacts with the computer’s hard- ware and all of its associated warts, convoluted designs and outright bugs. Many kernel devel- opers have spent sleepless nights debugging memory corruption that ultimately was the result of buggy device firmware that overwrites system memory when prodded a certain way. Finally, like any modern OS kernel, FreeBSD’s makes use of all of the CPUs available in the computer, and kernel developers have to grapple with all of the intrinsic complexity of writing efficient, scalable and correct software for multi-core systems. In short, it’s a tricky problem. FreeBSD’s developers put a great deal of effort into shipping stable, well-tested releases. It is worth thinking for a while about how one might test, say, a change to an existing system call, or a new system call. System calls are in a sense the front-end of the kernel: they provide the low-level abstractions used by all programs, and the invocation of a single system call may cause the kernel to execute thousands of lines of code on the invoker’s behalf. A developer adding a new system call will certainly write some test programs to verify that it behaves ac- cording to its specification, but generally it is not possible to exhaustively test all possible inputs to a lone system call. Furthermore, test programs cannot prove the absence of a bug; even if the system call produced a correct result, a bug may have corrupted a piece of kernel memory in a way that is not detectable for a long time after the fact. System calls may also interact with each other: a multi-threaded program will often execute multiple system calls simultaneous- ly, each updating some kernel state, so our hypothetical kernel developer must think carefully about the synchronization of these calls and how the hundreds of existing system calls might interact with the one in question. These kinds of problems are not specific to kernel programming and we have many concep- tual and technological tools that let us attack the stark complexity of writing bug-free kernel code, and deliver stable FreeBSD releases with confidence. Over the past several years a new such tool, syzkaller, has been extraordinarily successful at finding severe bugs in all major oper- ating systems, including FreeBSD. BY MARK JOHNSTON 1 of 12 Kernel Fuzzing with syzkaller 17 FreeBSD Journal • November/December 2020 Coverage-guided Fuzzing One important testing method for software that accepts untrusted input is fuzzing. Roughly speaking, fuzzing is the technique of programmatically generating inputs for the software un- der test, feeding that input to the software, and monitoring for unexpected results or side ef- fects. This is an effective technique for finding bugs in the code that handles input validation, and has become an indispensable software testing tool. For instance your PDF reader, which you presumably use to open files found on the world wide web, will hopefully have been test- ed using a fuzzer among other things: the PDF specification is rather complicated and the code which parses it will be correspondingly so, making PDFs an attractive vector for malware au- thors. Indeed, fuzzers are often used by security researchers and malware authors to find secu- rity holes. Fuzzing is one technique of many used to test software. One of its significant limitations is that it cannot generally verify that software is behaving correctly, only that it is not misbehaving according to some set of criteria. For instance, a fuzzer for a language parser would try to find input that causes the parser to crash, but the absence of a crash for a given input does not im- ply that the input was handled correctly according to the parser’s specification. Fuzzers instead excel at finding corner cases and rarely executed code paths overlooked by other software test- ing methods and which are therefore quite likely to contain bugs. To maximize effectiveness, the software under test should use assertions and other forms of runtime checking to detect invalid states as early as possible. Fuzzers vary in their level of sophistication. A naive fuzzer might generate purely random data and feed it directly to the software under test. While this approach may yield some fruit, it is unlikely to find anything other than very basic input validation bugs while consuming a large amount of computing resources. Consider a compiler fuzzer which simply generates random ASCII strings: most such strings are not valid programs and so will be rejected very quickly by the compiler’s parser, and as a result many components of the compiler, such as optimization and code generation logic, will not be exercised. Intelligent fuzzers have some knowledge of the input format so that they can generate valid-looking inputs that pass basic verification log- ic. For instance, a fuzzer which aims to test an IPv6 packet processor would ensure that inputs at least start with the 4-bit version number that begins all valid IPv6 packet headers. It could achieve this by using a corpus of valid IPv6 packets as a starting point, or with some built-in knowledge of the IPv6 packet header layout, or likely some combination of the two. A second effective optimization involves providing feedback to the fuzzer. A naive fuzzer would, in a loop, generate an input, feed it to the software under test, and wait for either a crash or graceful termination of the program. It has no general way to determine whether a given input helped improve test coverage of the software or not, and so cannot focus on “in- teresting” inputs. Consider a fuzzer target which performs input validation in two stages: InputFuzzer Stage 1 veriﬁer Stage 2 veriﬁer Stage 1 might simply verify that various components of the input have the correct length, while stage 2 verifies that the individual components contain valid values. If most input fails stage 1 validation, then stage 2 validation is left largely untested. However, if the fuzzer can 2 of 12 18 FreeBSD Journal • November/December 2020 dynamically learn which inputs pass stage 1 validation, it can improve its coverage of stage 2 validation by prioritizing inputs known to pass stage 1. There are multiple ways for a fuzzer to obtain feedback. For instance, it might measure the amount of time taken to process a given input and use a heuristic which discards inputs that are processed very quickly, under the assumption that such inputs are failing basic validity tests. Another technique, used by state-of-the-art fuzzing frameworks such as libFuzzer, AFL and syzkaller, measures code coverage. By leveraging software instrumentation facilities, a cover- age-guided fuzzer can “trace” the code paths executed when processing a given input, and use that information to try and generate inputs which uncover previously unexecuted code. Fuzzers use this technique to achieve high levels of test coverage very efficiently, and indeed, the aforementioned fuzzers have been used to find thousands of severe bugs in all sorts of software projects, even those considered mature and well-tested. syzkaller Operating system kernels handle input from a variety of untrusted sources: unprivileged pro- cesses will invoke system calls and may be trying to take control over the computer; a system connected to the internet processes network packets from untrusted sources; the kernel may be asked to mount a file system with invalid contents; a computer may support pluggable pe- ripheral devices which can communicate directly with the kernel. In short, a useful kernel pres- ents a massive attack surface, and years of high-profile kernel security holes show that there is much room for improvement among popular operatings systems. yzkaller seeks to improve this state of affairs. syzkaller is an open-source coverage-guided kernel fuzzer by Dmitry Vyukov. It originally tar- geted Linux but has since expanded to support nearly a dozen other operating systems. syz- kaller is sometimes described as a system call fuzzer but is flexible enough to target other oper- ating system interfaces; for example, it has been used to fuzz Linux’s USB stack and has found dozens of bugs in the USB subsystem alone. The details are complicated but the idea is simple: generate a program which invokes one or more system calls (or injects a packet into the net- work, etc.), run it, and check to see if the system diagnosed an error (for example by panick- ing). If not, collect kernel code coverage information and decide whether to try iterating upon the previous test program, or start anew. If so, collect information about the crash and try to discover a minimal test case that triggers the crash. syzkaller is written mostly in Go and consists of a dozen or so loosely-coupled programs, all prefixed with syz-, that together provide a self-contained system to do all of the following: • Start and run a set of operating system instances, typically in virtual machines. • Monitor those virtual machines for crashes or other diagnostic reports, typically by monitor- ing console output. • Generate programs to run under the target operating system, using coverage information to drive decisions about what to try next, and run them. • Maintain a database of observed crashes and diagnostic reports, to try and classify distinct bugs found. • Provide a web dashboard displaying statistics, code coverage information, and observed crashes and their reproducers if any. • Periodically update itself and the operating system under test without any manual interven- tion. • Attempt to bisect new crashes down to the commit introducing the bug. 3 of 12 19 FreeBSD Journal • November/December 2020 The high-level components of this system as it might run on FreeBSD are depicted here: syz-manager syz-ci syz-fuzzer netdumpd syz-executor /dev/kcov buildkernel gmake syz-prog2c VMs SSH, SCP bhyve,ZFS corpus,crash reports :80 .c ﬁles vmcoressyscalls Thanks to Google, the syzkaller developers provide numerous public syzkaller instances run- ning in continuous integration mode, wherein syzkaller updates itself and the target operating system regularly. These “syzbot” instances find bugs in the latest builds of their targets, so re- gressions are reported quickly and completely automatically. The FreeBSD instances have found numerous bugs and reproducers, enabling developers to both diagnose and fix bugs quickly and to provide higher-quality releases. kcov(4) syzkaller is not the first kernel fuzzer but is undoubtedly the most prominent. Newsgroup posts from the early 1990s describe programs which bombard UNIX kernels with random sys- tem calls to great effect. Peter Holm’s stress2 test suite for FreeBSD performs some target- ed fuzzing of certain system calls (among many other things). However, syzkaller introduces a key innovation in its use of code coverage to drive test case generation. This makes use of the kcov(4) kernel subsystem, written also by Dmitry Vyukov for Linux but later ported to other operating systems by their respective developers. While syzkaller does not strictly require code coverage information, it is much more effective with this extra feedback from the kernel. In FreeBSD, kcov(4) is a wrapper for LLVM’s SanitizerCoverage. Sanitizers are compiler fea- tures which inject bits of code enabling certain types of introspection into the compiled result. For example, LLVM’s AddressSanitizer inserts special function calls before every single memory access by the generated machine code; the calls can be used to determine whether the memory access is somehow invalid, for example because it corresponds to a use-after-free. This provides powerful bug-detection facilities similar to Valgrind but using different mechanisms: Valgrind works by running the unmodified target program in a software virtual machine which can inter- cept memory accesses and perform validation, whereas sanitizers are implemented by the com- piler itself and require special compilation flags. Sanitizers and Valgrind both introduce significant performance overhead and are generally used only in testing scenarios. Sanitizers have the add- ed benefit that they can sometimes be used to validate a kernel, while Valgrind cannot. SanitizerCoverage inserts function calls according to the control flow of the generated code. Most CPU instructions do not modify control flow: once the instruction is completed, the CPU fetches and executes the subsequent instruction from RAM. Control flow instructions cause 4 of 12 20 FreeBSD Journal • November/December 2020 the CPU to jump to a different address and begin execution there instead. This is how basic programming language constructs like if-statements, loops and goto work under the hood. A compiled program can thus be broken down into a set of “basic blocks,” where a basic block is a sequence of non-control flow instructions. Following the end of each basic block is a con- trol flow instruction. Then, if the goal is to figure out which pieces of code get executed in re- sponse to a given input, it suffices to trace out which basic blocks get executed. SanitizerCoverage, roughly speaking, inserts function calls in between each basic block, as in this machine code for the FreeBSD kernel function vm_page_remove(): <+0>: push %rbp <+1>: mov %rsp,%rbp <+4>: push %r15 <+6>: push %r14 <+8>: push %rbx <+9>: push %rax <+10>: mov %rdi,%rbx <+13>: callq 0xffffffff81167dc0 <__sanitizer_cov_trace_pc> <+18>: mov %rbx,%rdi <+21>: callq 0xffffffff815b7c50 <vm_page_remove_xbusy> <+26>: mov %eax,%r14d <+29>: mov $0x1,%ecx <+34>: xor %esi,%esi <+36>: mov $0x2,%eax <+41>: lock cmpxchg %ecx,0x54(%rbx) <+46>: setne %r15b <+50>: sete %sil <+54>: xor %edi,%edi <+56>: callq 0xffffffff81167ea0 <__sanitizer_cov_trace_const_cmp1> <+61>: test %r15b,%r15b <+64>: jne 0xffffffff815b78f9 <vm_page_remove+73> <+66>: callq 0xffffffff81167dc0 <__sanitizer_cov_trace_pc> <+71>: jmp 0xffffffff815b790d <vm_page_remove+93> <+73>: callq 0xffffffff81167dc0 <__sanitizer_cov_trace_pc> <+78>: movl $0x1,0x54(%rbx) <+85>: mov %rbx,%rdi <+88>: callq 0xffffffff81107950 <wakeup> <+93>: mov %r14d,%eax <+96>: add $0x8,%rsp <+100>: pop %rbx <+101>: pop %r14 <+103>: pop %r15 <+105>: pop %rbp <+106>: retq Here the __sanitizer_cov_trace_* function calls are inserted by SanitizerCoverage and can be implemented by the kernel. kcov(4) works by implementing these functions. 5 of 12 21 FreeBSD Journal • November/December 2020 In typical usage, a user program allocates a buffer to store coverageinformation, opens / dev/kcov and uses ioctl(2) to map the buffer into the kernel and to enable tracing of the current thread. When the thread subsequently enters the kernel, perhaps to execute a system call, the coverage tracing hooks log the address of each basic block into the buffer. When the thread disables tracing, again using an ioctl(2) call, it can make use of the information provid- ed in the buffer. For instance, the recorded addresses could be piped into the addr2line(1) program to find the file and line number of the traced C code. The kcov(4) manual page con- tains the details of this ioctl(2) interface as well as some example code. syzlang Earlier we pointed out that fuzzers work better when they have some knowledge of the software’s input format, rather than treating it as a black box. While syzkaller could theoretically invoke system calls without any knowledge of what they do or what parameters they take — using only coverage information to try and “learn” which parameter values result in more code execution — this is both inefficient and potentially counter-productive. Consider what happens if a fuzzer invokes kill(-1, SIGKILL): the kernel will do what it was asked to do and immedi- ately kill the fuzzer process. Unfortunately, system call interfaces cannot be discovered programmatically. In other words, there is generally no way to ask the kernel to describe the set of system calls that it implements. Even a set of C function prototypes omits many important details. Consider read(2): read(int fd, void buf, size_t nbytes); First, fd is here represented by an integer, but really must be a valid file descriptor as well. There are 4,294,967,295 possible values for fd and all but a tiny fraction of them are invalid. Second, it is not clear what the kernel is expected to do with buf: is the kernel supposed to read data from that address, or write to it, or both, or neither? Third, nbytes is supposed to represent the size of the buffer buf but the prototype gives no indication that these two pa- rameters are related; the C language is simply not expressive enough to do so. If you are famil- iar with ioctl(2), think for a bit about how it makes a bad situation even worse. To solve these problems, syzkaller introduces syzlang: a language for modeling the kernel’s programming interfaces. It is flexible enough to define data layouts that are binary-compati- ble with C types, and expressive enough to describe inter-related C parameters, among other things. In syzlang, the read(2) prototype above becomes: read(fd fd, buf buffer[out], count len[buf]) Unlike C (but like Go), the parameter name comes first, followed by the type. Right away we can see that this definition provides more information than the C prototype: there is an fd type, to distinguish file descriptors from plain integers; buf is a pointer to a buffer mapped in the call- er’s address space, and the [out] annotation signifies that it is an “out-parameter,” i.e., the ker- nel is supposed to write data to the buffer; count is the length, in bytes, of the buffer buf. The use of specialized types to represent file descriptors and other kernel resources is import- ant for generating programs that do “interesting” things since many system calls take as input the results of previous system calls. For example, to read data from a file a program might exe- cute the following sequence of system calls: 6 of 12 22 FreeBSD Journal • November/December 2020 const size_t len = 4096; void buf = mmap(NULL, len, PROT_READ \| PROT_WRITE, MAP_ANON, -1, 0); int fd = open(“/tmp/foo”, O_RDWR); read(fd, buf, len); close(fd); munmap(buf, len); Note that the results of the mmap(2) and open(2) calls are used as input to subsequent calls. Fuzzing munmap(2) and close(2) does not make much sense without earlier calls to mmap(2) and open(2), so syzlang models the corresponding resources and syzkaller creates chains of system calls using these relationships to guide its choices. The use of mmap(2) also illustrates a need to define the set of valid values for “flag” param- eters such as the third and fourth arguments. syzlang has a built-in flags type for this: mmap(addr vma, len len[addr], prot flags[mma_prot], flags flags[mmap_flags], fd fd, offset fileoff) mmap_prot = PROT_EXEC, PROT_READ, PROT_WRITE mmap_flags = MAP_SHARED, MAP_PRIVATE, MAP_ANON, ... The fuzzer will chose zero or more flag values when creating an argument for a flag pa- rameter. syzlang can also use subtyping to more accurately model system call interfaces. Network connections and open files are both represented by file descriptors in the system call interface, but many system calls accept only certain types of file descriptors. For instance, one of sendfile(2)’s parameters is a file descriptor corresponding to a network connection on which to send a file’s data. Such descriptors are created using socket(2) or socketpair(2). Passing a descriptor for a regular file here will fail, so to save the fuzzer time we can define a new sock- et type. First we have the definition for fd: resource fd[int32]: 0xffffffffffffffff, AT_FDCWD This derives fd from the built-in integer type int32 and defines a couple of special values: -1, for system calls where a parameter of type fd is optional (such as mmap(2)), and AT_FDCWD, used by openat(2) and similar system calls. Then we can define a derived resource type for sockets: resource sock[fd] socket(...) sock sendfile(fd fd, s sock, ...) A similar trick is used for system calls where layout of a parameter depends on the value of another parameter. ioctl(2) is the prime example of this, but fcntl(2), setsockopt(2) and bind(2) behave similarly. For example, pf(4) defines a large set of ioctl commands, but each has its own argument type. We can describe them precisely in syzlang: 7 of 12 23 FreeBSD Journal • November/December 2020 resource fd_pf[fd] openat$pf(fd const[AT_FDCWD], file ptr[in, string[“/dev/pf”]], ...) fd_pf ioctl$DIOCRADDTABLES(fd fd_pf, cmd const[DIOCRADDTABLES], arg ptr[in, pfioc_table]) Here we define a new fd type which corresponds to a file descriptor for /dev/pf. Then, spe- cial “flavours” of openat(2) and ioctl(2) describe how the fuzzer can open a pf(4) device and issue the DIOCRADDTABLES ioctl, used by pfctl(8) to define an address table. syzlang definitions are compiled at build-time into tables used by syzkaller’s fuzzer. syzkaller maintains its own internal representation of system calls and their parameters, and its repre- sentation of a program is simply a list of calls and parameters. All fuzzing is performed using these representations; when a reproducer for a kernel bug is found, it is finally translated into a standalone C program. This can be done manually using syz-prog2c but this is typically han- dled automatically. New system call definitions are added frequently since in most cases syzkaller is still playing catch-up: the existing kernel interfaces are massive and defining them in syzlang requires time and effort. In particular, many components of FreeBSD are not yet described by syzlang and therefore do not get tested by syzbot. Adding to the FreeBSD syzlang definitions is a great way to start contributing to the syzkaller project and to help ensure that FreeBSD gets as much test coverage as possible. syz-manager So far we have looked at the mechanisms by which syzkaller addresses the generic technical problems faced by all fuzzers: obtaining feedback from the target software (via kcov(4)) and describing kernel interfaces (with syzlang). Now we can look more at some of the machinery required to fuzz an operating system kernel. A fuzzer’s goal is to “exercise” the code being tested, so it needs an environment in which to execute the code and provide input. Fuzzing a kernel poses some extra challenges: the fuzz- er needs to run in the same system as the kernel being tested, so if it achieves its goal and trig- gers a kernel panic, all of the fuzzer’s state will be lost. syzkaller’s solution is to run the target kernel in a set of virtual machines which can be safely wiped without losing anything import- ant. These VMs can run on the same host as syzkaller or in cloud environments such as Google Compute Engine. syz-manager is the main front-end program of syzkaller. It takes a configuration file as in- put and starts a number of VMs according to the configuration. It automatically installs and starts the fuzzer programs in each VM instance and communicates with them using an RPC interface over SSH. syz-manager also monitors the VM consoles to detect crashes. When a crash occurs, the VM is automatically re-created. VMs are restarted periodically even in the ab- sence of a crash; one reason for this is to enable “corpus rotation.” syz-manager also maintains the instance’s crash database. When a crash is discovered, syz-manager adds an entry to the crash database. Crashes are identified by the panic message printed by the kernel, and when a new crash is found, syz-manager dedicates a subset of the VMs to spend time attempting to reproduce the crash. Programs that were executed leading up to the crash are replayed, and if a crash can be reproduced, syzkaller also attempts to find a minimal reproducible for the crash to add to the crash database. Finally, syzkaller attempts to translate the crashing program into a standalone C program, making it easy for developers to debug crashes without needing a syzkaller installation on hand. 8 of 12 24 FreeBSD Journal • November/December 2020 Most of the work to set up syzkaller involves creating a VM image containing the target op- erating system. The VM’s kernel should have kcov(4) enabled, and an SSH key for the root user must be installed. The VM image is used as template; when syz-manager starts, it creates a snapshot of the image before starting VMs, and each VM uses a local copy of the template. When a VM is restarted, it gets a fresh copy of the template image, so any damage done by the fuzzer is discarded. syz-manager supports a number of different hypervisors and cloud APIs. On FreeBSD one can use bhyve as the back-end hypervisor. To create a VM image template syz-manager uses ZFS clones, since bhyve lacks support for creating disk image snapshots. Upon starting up syz-manager also starts a web server, providing a dashboard containing statistics and code cov- erage information, deduplicated crash reports, and crash reproducers. A sample syz-manager configuration looks like this: { “target”: “freebsd/amd64”, “http”: “0.0.0.0:8080”, “workdir”: “/data/syzkaller”, “image”: “/data/syzkaller/vm.raw”, “syzkaller”: “/home/markj/go/src/github.com/google/syzkaller”, “procs”: 4, “type”: “bhyve”, “ssh_user”: “root”, “sshkey”: “/data/syzkaller/id_rsa”, “kernel_obj”: “/usr/obj/usr/home/markj/src/freebsd/amd64.amd64/sys/SYZKALLER”, “kernel_src”: “/”, “vm”: { “bridge”: “bridge0”, “count”: 32, “cpu”: 2, “hostip”: “169.254.0.1”, “dataset”: “data/syzkaller” } } This configuration specifies 32 VM instances; in general, more VMs is better since each VM runs test programs in parallel with the others. The “cpu” parameter defines the number of vir- tual CPUs given to each VM, and the “procs” parameter defines the number of fuzzer process- es that will run in each VM. Having multiple virtual CPUs and fuzzer processes improves the odds of finding certain types of bugs, but over-subscribing the host may decrease the effective- ness of fuzzing by making it hard to reproduce bugs. It is reasonable to configure one or two virtual CPUs per host CPU, but more than that is probably too many. See the FreeBSD/syzkaller documentation for details on how to build and configure your own syzkaller setup. With a configuration file written, syzkaller can be started with: # syz-manager -config /path/to/config When using bhyve, syzkaller needs to run as root in order to create virtual machines. It is pos- sible to run syzkaller in a jail with some effort; some ongoing work aims to make this simpler. 9 of 12 25 FreeBSD Journal • November/December 2020 If you wish to run your own private syzkaller instance, do be prepared to be patient — now that much of the low-hanging fruit has been fixed, it can take days for syzkaller to find a new kernel bug. Fuzzing the Kernel Now that we have encountered most of the machinery that syzkaller uses to fuzz operating system kernels, we are equipped to start looking at the brains of syzkaller. Aside from the crash database, syzkaller’s main piece of persistent state consists of the cor- pus: a representative set of programs whose execution generates coverage of the kernel. The corpus — initially empty — is effectively a seed for the fuzzer: a new test program is generat- ed by taking a program from the corpus, mutating it in some small way, and checking to see if previously uncovered kernel code was covered by the new program. If so, the program may be added to the corpus and subsequently used as the starting point for other programs. Algorith- mically, the fuzzer does nothing except try to increase the size of the corpus. Many heuristics are applied to try and make this more effective for syzkaller’s real purpose — finding bugs — but the core idea is very simple. Several types of program mutations are possible. The fuzzer might: • splice several programs together • insert a new system call • remove an existing system call • modify one of the parameters to a call in the program If the corpus is empty, the fuzzer will create a new program by generating a random list of system calls with randomly selected arguments. This is also done periodically even when the corpus is non-empty. Inside each VM managed by syzkaller runs a pair of programs, syz-fuzzer and syz-executor, which communicate using a shared memory interface. As their names suggest, syz-fuzzer generates test programs and syz-executor actually executes them. syz-fuzzer and syz-manager coordinate using a simple RPC protocol; since syz-fuzzer generates pro- grams which may crash the VM, it relies on syz-manager to store the corpus. syz-fuzzer is started by syz-manager over SSH. When it begins, it establishes an RPC con- nection with syz-manager and creates a number of work queues, each of which is managed by a thread (really, a goroutine). The worker threads each spawn a syz-executor instance and immediately begin fuzzing, yielding the following picture: syz-manager syz-fuzzer syz-executor.1syz-executor.2syz-executor.0 kernel shmem VM RPC workerworker worker 10 of 12 26 FreeBSD Journal • November/December 2020 Worker threads perform most of the work of adding to the corpus: they generate new pro- grams and mutate existing ones. They also handle special types of work: • Triage: when a program appears to generate new coverage it is placed in the triage queue for further refinement. The triage step tries to determine whether the program behaves consistently (i.e., re-runs generate the same coverage info), and if so, tries to minimize the size of the program while maintaining its coverage. • Smashing: when a program has been triaged and appears worthy of being added to the corpus, the worker spends extra time mutating it to look for new coverage. • Candidate processing: syz-manager may send candidate programs to the fuzzers in some cases. The worker executes them, potentially creating triage or smash work. In steady-state operation, syz-fuzzer uses two RPCs to communicate with the manager: Poll and NewInput. Poll is invoked periodically to update the fuzzer’s snapshot of the corpus and global cover- age information, and to collect candidate programs for fuzzing. It also serves to re-process the existing corpus when syzkaller starts up; a typical syzkaller installation will periodically update itself and the target kernel, and must subsequently restart. The saved corpus is immediately dis- tributed among the fuzzers for execution and triage since the updated kernel may handle exist- ing corpus items differently from when they were last evaluated. NewInput is used to send triaged programs back to syz-manager as possible candidates for the global corpus. syz-manager will reject new inputs in some cases, for instance to avoid blowing up the size of the corpus, or if another fuzzer had already discovered a similar pro- gram. If accepted, new corpus programs eventually become visible to other fuzzer instances via Poll. Unfortunately, code coverage is not an ideal metric: 100% code coverage of a program does not preclude the existence of detectable bugs, especially in multi-threaded code such as a modern operating system kernel. Optimizing for an imperfect metric tends to yield suboptimal results — we (hopefully!) do not evaluate programmers based on the number of lines of code they have written. In syzkaller’s case, valuable test programs may be discarded if they do not add to the corpus’ code coverage. To try and alleviate this problem, syzkaller performs corpus “rotation”: some system calls and corpus programs are hidden from individual fuzzers to force them to find programs with equivalent coverage but hopefully new characteristics. This can re- sult in duplicated effort but helps to ensure that the system does not become “stuck” by find- ing local maxima. Program Execution To round off our examination of syzkaller a look at syz-executor is in order. syzkaller uses an internal representation of system call programs for the purpose of fuzzing, but of course has to actually run them somehow. syz-executor is the component of syzkaller that performs this task; unlike the rest of syzkaller, it is written in C++. The executor is spawned by syz-fuzzer worker threads and uses a simple shared memo- ry interface to communicate with the worker. It first creates a pool of threads to actually exe- cute system calls, and then opens /dev/kcov and uses ioctl(2) to enable collection of code coverage information that is returned to the worker. Quite a lot of additional initialization may happen at this point, depending on how syzkaller is configured. For instance, the executor may enter a software sandbox in an attempt to limit the effects of the test program: a pro- gram which sends signals to unsuspecting processes is likely to wedge the VM and trigger a 11 of 12 27 FreeBSD Journal • November/December 2020 costly timeout and restart. It may also initialize devices or network facilities as part of a target- ed fuzzing regime. When it comes time to execute system calls, syz-executor iterates over the call list and as- signs an idle thread to each one, waiting for threads to become free if necessary. Initially, the main thread waits for a short period after each call is dispatched. Once the input program has finished, it is executed a second time in “collision mode”: rather than waiting for a short period after each call is dispatched, pairs of system calls are allowed to execute concurrently, helping to trigger race conditions in the kernel that would otherwise be left unexercised. Actual system call execution is achieved using the handy syscall(2) system call, a generic system call which takes a system call number and variable list of parameters as arguments. In- ternally the kernel uses the system call number to route the call to the requested handler. The system call’s result is also recorded for use in prioritizing and triaging programs: a successful sys- tem call is weighted more favorably than a failed system call. Conclusion If you managed to get this far, please don’t stop here! syzkaller is the subject of quite few talks, articles and even research papers — check out syzkaller’s documentation for some cu- rated links. This article only scratches the surface of syzkaller’s internals, and the sources are as usual the authoritative reference on how syzkaller actually works. Fuzzing is a fascinating subject and there is a certain thrill to watching a fuzzer in action — particularly when it finds bugs in your favorite operating system. We encourage you to give it a try. MARK JOHNSTON is a contractor and FreeBSD src committer based in Toronto, Canada. He is particularly interested in kernel debugging and in finding new ways to help improve the stabili- ty of FreeBSD. In his spare time he enjoys cooking, playing Bach’s cello suites, and impeding his productivity by experimenting with custom keyboard layouts. 12 of 12 Thank you! The FreesBSD Foundation would like to acknowledge the following companies for their continued support of the Project. Because of generous donations such as these we are able to continue moving the Project forward. Uranium Iridium Silver Please check out the full list of generous community investors at freebsdfoundation.org/donors/ Are you a fan of FreeBSD? Help us give back to the Project and donate today! freebsdfoundation.org/donate/ TM TM Platinum Gold Koum Family Foundation
P487	Fast Algorithms for Mining Association Rules Rakesh Agrawal Ramakrishnan S&ant* IBM Almaden Research Center 650 Harry Road, San Jose, CA 95120 Abstract We consider the problem of discovering association rules between items in a large database of sales transactions. We present two new algorithms for solving thii problem that are fundamentally different from the known algo- rithms. Empirical evaluation shows that these algorithms outperform the known algorithms by factors ranging from three for small problems to more than an order of mag- nitude for large problems. We also show how the best features of the two proposed algorithms can be combined into a hybrid algorithm, called AprioriHybrid. Scale-up experiments show that AprioriHybrid scales linearly with the number of transactions. AprioriHybrid also has ex- cellent scale-up properties with respect to the transaction size and the number of items in the database. 1 Introduction Progress in bar-code technology has made it possi- ble for retail organizations to collect and store mas- sive amounts of sales data, referred to as the basket data. A record in such data typically consists of the transaction date and the items bought in the trans- action. Successful organizations view such databases as important pieces of the marketing infrastructure. They are interested in instituting information-driven marketing processes, managed by database technol- ogy, that enable marketers to develop and implement customized marketing programs and strategies [S]. The problem of mining association rules over basket data was introduced in [4]. An example of such a rule might be that 98% of customers that purchase Visiting from the Department of Computer Science, Uni- versity of Wisconsin, Madison. Permission to copy without fee all or part of this material is granted provided that the copies are not made OT distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and itr date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copq otherwise, or to republish, nquins a fee and/or special permisrion from the Endowment. Proceedings of the 20th VLDB Conference Santiago, Chile, 1994 tires and auto accessories also get automotive services done. Finding all such rules is valuable for cross- marketing and attached mailing applications. Other applications include catalog design, add-on sales, store layout, and customer segmentation based on buying patterns. The databases involved in these applications are very large. It is imperative, therefore, to have fast algorithms for this task. The following is a formal statement of the problem [4]: Let Z = {ir,iz, . . . , im} be a set of literals, called items. Let 2) be a set of transactions, where each transaction T is a set of items such that T c Z. Associated with each transaction is a unique identifier, called its TID. We say that a transaction T contains X, a set of some items in Z, if X c T. An association rzle is an implication of the form X q Y, where X C Z, Y c 2, and X rl Y = 0. The rule X a Y holds in the transaction set ‘D with confidence c if c% of transactions in D that contain X also contain Y. The rule X _ Y has support s in the transaction set V if s% of transactions in V contain X U Y. Our rules are somewhat more general than in [4] in that we allow a consequent to have more than one item. Given a set of transactions ‘D, the problem of min- ing association rules is to generate all association rules that have support and confidence greater than the user-specified minimum support (called minsup) and minimum confidence (called minconf) respectively. Our discussion is neutral with respect to the repre- sentation of V. For example, V could be a data file, a relational table, or the result of a relational expres- sion. An algorithm for finding all association rules, henceforth referred to as the AIS algorithm, was pre- sented in [4]. Another algorithm for this task, called the SETM algorithm, has been proposed in [13]. In this paper, we present two new algorithms, Apriori and AprioriTid, that differ fundamentally from these algorithms. We present experimental results showing 487 that the proposed algorithms always outperform the earlier algorithms. The performance gap is shown to increase with problem size, and ranges from a fac- tor of three for small problems to more than an or- der of magnitude for large problems. We then dis- cuss how the best features of Apriori and Apriori- Tid can be combined into a hybrid algorithm, called AprioriHybrid. Experiments show that the Apriori- Hybrid has excellent scale-up properties, opening up the feasibility of mining association rules over very large databases. The problem of finding association rules falls within the purview of database mining [3] [12], also called knowledge discovery in databases [21]. Related, but not directly applicable, work includes the induction of classification rules [S] [ll] [22], discovery of causal rules [19], learning of logical definitions [18], fitting of functions to data [15], and clustering [9] [lo]. The closest work in the machine learning literature is the KID3 algorithm presented in [20]. If used for finding all association rules, this algorithm will make as many passes over the data as the number of combinations of items in the antecedent, which is exponentially large. Related work in the database literature is the work on inferring functional dependencies from data [16]. Functional dependencies are rules requiring strict satisfaction. Consequently, having determined a dependency X + A, the algorithms in [16] consider any other dependency of the form X + Y + A redundant and do not generate it. The association rules we consider are probabilistic in nature. The presence of a rule X + A does not necessarily mean that X + Y + A also holds because the latter may not have minimumsupport. Similarly, the presence of rules X -+ Y and Y + Z does not necessarily mean that X -+ Z holds because the latter may not have minimum confidence. There has been work on quantifying the “useful- ness” or “interestingness” of a rule [20]. What is use- ful or interesting is often application-dependent. The need for a human in the loop and providing tools to allow human guidance of the rule discovery $rocess has been articulated, for example, in [7] [14]. We do not discuss these issues in this paper, except to point out that these are necessary features of a rule discov- ery system that may use our algorithms as the engine of the discovery process. 1.1 Problem Decomposition and Paper Organization The problem of discovering all association rules can be decomposed into two subproblems [4]: 1. Find all sets of items (itemseis) that have transac- tion support above minimum support. The support 2. for an itemset is the number of transactions that contain the itemset. Itemsets with minimum sup- port are called large itemsets, and all others small itemsets. In Section 2, we give new algorithms, Apriori and AprioriTid, for solving this problem. Use the large itemsets to generate the desired rules. Here is a straightforward algorithm for this task. For every large itemset 1, find all non-empty subsets of 1. For every such subset a, output a rule of the form a =+ (1 - a) if the ratio of support(l) to support(a) is at least minconf We need to consider all subsets of 1 to generate rules with multiple consequents. Due to lack of space, we do not discuss this subproblem further, but refer the reader to [5] for a fast algorithm. In Section 3, we show the relative performance of the proposed Apriori and AprioriTid algorithms against the AIS [4] and SETM [13] algorithms. To make the paper self-contained, we include an overview of the AIS and SETM algorithms in this section. We also describe how the Apriori and AprioriTid algorithms can be combined into a hybrid algorithm, AprioriHybrid, and demonstrate the scale- up properties of this algorithm. We conclude by pointing out some related open problems in Section 4. 2 Discovering Large Itemsets Algorithms for discovering large itemsets make mul- tiple passes over the data. In the first pass, we count the support of individual items and determine which of them are large, i.e. have minimumsupport. In each subsequent pass, we start with a seed set of itemsets found to be large in the previous pass. We use this seed set for generating new potentially large itemsets, called candidate itemsets, and count the actual sup port for these candidate itemsets during the pass over the data. At the end of the pass, we determine which of the candidate itemsets are actually large, and they become the seed for the next pass. This process con- tinues until no new large itemsets are found. The Apriori and AprioriTid algorithms we propose differ fundamentally from the AIS [4] and SETM [13] algorithms in terms of which candidate itemsets are counted in a pass and in the way that those candidates are generated. In both the AIS and SETM algorithms, candidate itemsets are generated on-the-fly during the pass as data is being read. Specifically, after reading a transaction, it is determined which of the itemsets found large in the previous pass are present in the transaction. New candidate itemsets are generated by extending these large itemsets with other items in the transaction. However, as we will see, the disadvantage 488 is that this results in unnecessarily generating and counting too many candidate itemsets that turn out to be small. The Apriori and AprioriTid algorithms generate the candidate itemsets to be counted in a pass by using only the itemsets found large in the previous pass - without considering the transactions in the database. The basic intuition is that any subset of a large itemset must be large. Therefore, the candidate itemsets having k items can be generated by joining large itemsets having k - 1 items, and deleting those that contain any subset that is not large. This procedure results in generation of a much smaller number of candidate itemsets. The AprioriTid algorithm has the additional prop- erty that the database is not used at all for count- ing the support of candidate itemsets after the first pass. Rather, an encoding of the candidate itemsets used in the previous pass is employed for this purpose. In later passes, the size of this encoding can become much smaller than the database, thus saving much reading effort. We will explain these points in more detail when we describe the algorithms. Notation We assume that items in each transaction are kept sorted in their lexicographic order. It is straightforward to adapt these algorithms to the case where the database 2, is kept normalized and each database record is a <TID, item> pair, where TID is the identifier of the corresponding transaction. We call the number of items in an itemset its size, and call an itemset of size k a k-itemset. Items within an itemset are kept in lexicographic order. We use the notation c[l] . c[2] . . . . . c[k] to represent a k- itemset c consisting of items c[l], c[2], . . .c[k], where c[l] < c[2] < . . . < c[k]. If c = X + Y and Y is an m-itemset, we also call Y an m-eziension of X. Associated with each itemset is a count field to store the support for this itemset. The count field is initialized to zero when the itemset is first created. We summarize in Table 1 the notation used in the algorithms. The set i?k is used by AprioriTid and will be further discussed when we describe this algorithm. 2.1 Algorithm Apriori Figure 1 gives the Apriori algorithm. The first pass of the algorithm simply counts item occurrences to determine the large l-item&s. A subsequent pass, say pass k, consists of two phases. First, the large itemsets Lk-i found in the (k-1)th pass are used to generate the candidate itemsets ck, using the apriori- gen function described in Section 2.1.1. Next, the database is scanned and the support of candidates in ck is counted. For fast counting, we need to efficiently determine the candidates in ck that are contained in a Table 1: Notation k-item& An itemset having k items. Set of large k-items& Lk (those with miniium support). Each member of this set haa two fields: i) itemset and ii) support count. Set of candidate k-item&s ck (potentiahy large itemsets). Each member of this set has two fields: i) itemset and ii) support count. 1 Set of candidate k-itemsets when the TIDs of the generating transactions are kept associated with the candidates. given transaction t. Section 2.1.2 describes the subset function used for this purpose. See [5] for a discussion of buffer management. 1) L1 = {large 1-itemsets}; 2) for ( k = 2; Lk-1 # 0; k++ ) do begin 3) ck = apIiO&geu(&l); // New candidates 4) forall transactions 1 E 2) do begin 5) C = S&S&(&, t); // Candidatea COntahd in t 6) forall candidates c E Cr do 7) c.count++; 8) end 9) Lk = {C E ck 1 C.count 2 minsup} 10) end 11) Answer = Uk Lk; Figure 1: Algorithm Apriori 2.1.1 Apriori Candidate Generation The apriori-gen function takes as argument Lk-1, the set of all large (k - 1)-itemsets. It returns a superset of the set of all large k-item&s. The function works as follows. ’ First, in the join step, we join Lk-1 with Lk-1: S&!Ct p.iteml, p.item2, . . . . p.iteIUk-1, q.itemk-1 from Lk-1 PI Lk-1 !I where p.iteml = q.iteml, . . ., phemk-2 = q.iteXUk-2, phmk-1 < q.iteIIIk-1; Next, in the prune step, we delete all itemsets c E ck such that some (k-1)-subset of c is not in Lk-1: ‘Concurrent to our work, the following two-step candidate generation procedure has been proposed in [17) C~={XuX’IX,X’~Lk-l,(XnX’(=k-2} ck = {x E CL Ix COntdM k membm of &-I} These two steps are similar to our join and pruue steps respectively. However, in general, step 1 would produce a supemet of the candidates produced by our join step. 489 forall itemsets c E Ck do forall (k-l)-subsets s of c do if (s fZ LI-1) then delete c from ck; Example Let Ls be ((1 2 31, (1 2 4}, (1 3 4}, (1 3 5}, (2 3 4)). After the join step, Cd will be { { 1 2 3 4}, (13 4 5) }. Th e p rune step will delete the itemset (1 3 4 5) because the itemset (1 4 5) is not in LB. We will then be left with only (1 2 3 4) in Cd. Contrast this candidate generation with the one used in the AIS and SETM algorithms. In pass k of these algorithms, a database transaction t is read and it is determined which of the large itemsets in Lk-i are present in t. Each of these large itemsets I is then extended with all those large items that are present in t and occur later in the lexicographic ordering than any of the items in 1. Continuing with the previous example, consider a transaction (1 2 3 4 5). In the fourth pass, AIS and SETM will generate two candidates, { 1 2 3 4) and { 1 2 3 5}, by extending the large itemset (1 2 3). Similarly, an additional three candidate itemsets will be generated by extending the other large itemsets in LB, leading to a total of 5 candidates for consideration in the fourth pass. Apriori, on the other hand, generates and counts only one itemset, { 1 3 4 5}, because it concludes a priori that the other combinations cannot possibly have minimum support. Correctness We need to show that Ck _> Lk. Clearly, any subset of a large itemset must also have minimum support. Hence, if we extended each itemset in Lk-i with all possible items and then deleted all those whose (k - 1)-subsets were not in Lk-1, we would be left with a superset of the itemsets in Lk. The join is equivalent to extending Lk-1 with each item in the database and then deleting those itemsets for which the (k-1)-itemset obtained by deleting the (h-1)th item is not in L&l. The condition p.itemk.,i < q.itemk-i simply ensures that no duplicates are generated. Thus, after the join step, ck _> Lk. By similar reasoning, the prune step, where we delete from CA! all itemsets whose (k- 1)-subsets are not in Lk-1, also does not delete any itemset that could be in Lk. Variation: Counting Candidates of Multiple Sizes in One Pass Rather than counting only candidates of size k in the kth paas, we can also count the candidates Ci+i, where Ci,, is generated from Ck, etc. Note that CL,, > ck+r since ck+i is generated from La. This variation can pay off in the later passes when the cost of counting and keeping in memory additional Ci+r - ck+r candidates becomes less than the cost of scanning the database. 2.1.2 Subset Function Candidate itemsets Ck are stored in a hash-tree. A node of the hash-tree either contains a list of itemsets (a leaf node) or a hash table (an interior node). In an interior node, each bucket of the hash table points to another node. The root of the hash-tree is defined to be at depth 1. An interior node at depth d points to nodes at depth d+ 1. Itemsets are stored in the leaves. When we add an itemset c, we start from the root and go down the tree until we reach a leaf. At an interior node at depth d, we decide which branch to follow by applying a hash function to the dth item of the itemset. All nodes are initially created as leaf nodes. When the number of itemsets in a leaf node exceeds a specified threshold, the leaf node is converted to an interior node. Starting from the root node, the subset function finds all the candidates contained in a transaction t aa follows. If we are at a leaf, we find which of the itemsets in the leaf are contained in t and add references to them to the answer set. If we are at an interior node and we have reached it by hashing the item i, we hash on each item that comes after i in t and recursively apply this procedure to the node in the corresponding bucket. For the root node, we hash on every item in t. To see why the subset function returns the desired set of references, consider what happens at the root node. For any itemset c contained in transaction t, the first item of c must be in t. At the root, by hashing on every item in t, we ensure that we only ignore itemsets that start with an item not in t. Similar arguments apply at lower depths. The only additional factor is that, since the items in any itemset are ordered, if we reach the current node by hashing the item i, we only need to consider the items in t that occur after i. 2.2 Algorithm AprioriTid The AprioriTid algorithm, shown in Figure 2, also uses the apriori-gen function (given in Section 2.1.1) to determine the candidate itemaets before the pass begins. The interesting feature of this algorithm is that the database V is not used for counting support after the first pass. Rather, the set ck is used for this purpose. Each member of the set ck is of the form < TID, {&} >, where each x1; is a potentially large k-item& present in the transaction with identifier TID. For k = 1, ??r corresponds to the database V, although conceptually each item i is replaced by the itemset (i}. For k > 1, ck is generated by the algorithm (step 10). The member 490 of ck corresponding to transaction t is <t.TID, {c E Ck IC contained in t)>. If a transaction does not contain any candidate Ic-itemset, then ??k will not have an entry for this transaction. Thus, the number of entries in Ek may be smaller than the number of transactions in the database, especially for large values of k. In addition, for large values of Ic, each entry may be smaller than the corresponding transaction because very few candidates may be contained in the transaction. However, for small values for L, each entry may be larger than the corresponding transaction because an entry in Ck includes all candidate k-itemsets contained in the transaction. In Section 2.2.1, we give the data structures used to implement the algorithm. See [5] for a proof of correctness and a discussion of buffer management. 1) L1 = {large l-item&s}; 2) G = database V; 3) for ( k = 2; Lkel # 8; k++ ) do begin 4) Ck = apriori-gen(Lk-1); // New candidates 5) Ek = B; 6) forall entries t E Ek-1 do begin 7) // determine candidate itemsets in Ck contained // in the transaction with identifier L.TID Ct = {c E Ck 1 (c - c[k]) E ‘kset-of-itemsets A (c - c[k - 11) E &set-of-itemsets}; 8) forall candidates c E Ct do 9) c.count++; 10) if (C, # 0) then ck += < t.TID, Ct >; 11) end 12) Lk = {c E ck 1 c.count 2 minsup} 13) end 14) Answer = uk Lk; Figure 2: Algorithm AprioriTid Example Consider the database in Figure 3 and assume that minimum support is 2 transactions. Calling apriori-gen with L1 at step 4 gives the candidate itemsets Cs. In steps 6 through 10, we count the support of candidates in Cz by iterating over the entries in cl and generate ea. The first entry in Cl is { (11 (31 (41 1, corresponding to transaction 100. The Ct at step 7 corresponding to this entry t is { { 1 3) }, because { 1 3) is a member of C2 and both ((1 3) - (1)) and ((1 3) - (3)) are members of t.set-of-item&s. Calling apriori-gen with Lp gives C3. Making a pass over the data with c2 and C3 generates Es. Note that there is no entry in Es for the transactions with TIDs 100 and 400, since they do not contain any of the itemsets in Cs. The candidate (2 3 5) in C3 turns out to be large and is the only member of LB. When we generate C4 using L3, it turns out to be empty, and we terminate. Database Ll Itemset Support (11 2 77 I:; 3 3 (5) 3 1 pjjpri& , , 1 I , 1 2 Support 1 2 1 2 3 2 , Figure 3: Example 2.2.1 Data Structures We assign each candidate itemset a unique number, called its ID. Each set of candidate itemsets Ck is kept in an array indexed by the IDS of the itemsets in CA?. A member of Ek is now of the form < TID, {ID} >. Each i?‘, is stored in a sequential structure. The apriori-gen function generates a candidate k- itemset Ck by joining two large (k - I)-itemsets. We maintain two additional fields for each candidate itemset: i) generators and ii) edensions. The generators field of a candidate itemset Ck stores the IDS of the two large (k - 1)-itemsets whose join generated CI: . The extensions field of an itemset Cg stores the IDS of all the (6 + 1)-candidates that are extensions of ck. Thus, when a candidate ck is generated by joining Ii-1 and Ziel, we save the IDS of I:-1 and IiD1 in the generators field for ck. At the same time, the ID of ck is added to the extensions field of Zisl. 491 We now describe how Step 7 of Figure 2 is implemented using the above data structures. Recall that the t.set-of-itemsets field of an entry t in ck-1 gives the IDS of all (k - 1)-candidates contained in transaction t.TID. For each such candidate ck-i the extensions field gives Tk, the set of IDS of all the candidate b-item&s that are extensions of c&i. For each Ck in Tk, the generators field gives the IDS of the two itemsets that generated ck. If these itemsets are present in the entry for t.set-of-itemsets, we can conclude that ck is present in transaction t.TID, and add c) to Ct. 3 Performance To assess the relative performance of the algorithms for discovering large sets, we performed several experiments on an IBM RS/SOOO 530H workstation with a CPU clock rate of 33 MHz, 64 MB of main memory, and running AIX 3.2. The data resided in the AIX file system and was stored on a 2GB SCSI 3.5” drive, with measured sequential throughput of about 2 MB/second. We first give an overview of the AIS [4] and SETM [13] algorithms against which we compare the per- formance of the Apriori and AprioriTid algorithms. We then describe the synthetic datasets used in the performance evaluation and show the performance re- sults. Finally, we describe how the best performance features of Apriori and AprioriTid can be combined into an AprioriHybrid algorithm and demonstrate its scale-up properties. 3.1 The AIS Algorithm Candidate itemsets are generated and counted on- the-fly as the database is scanned. After reading a transaction, it is determined which of the itemsets that were found to be large in the previous pass are contained in this transaction. New candidate itemsets are generated by extending these large itemsets with other items in the transaction. A large itemset 1 is extended with only those items that are large and occur later in the lexicographic ordering of items than any of the items in 1. The candidates generated from a transaction are added to the set of candidate itemsets maintained for the pass, or the counts of the corresponding entries are increased if they were created by an earlier transaction. See [4] for further details of the AIS algorithm. 3.2 The SETM Algorithm The SETM algorithm [13] was motivated by the desire to use SQL to compute large itemsets. Like AIS, the SETM algorithm also generates candidates on- the-fly based on transactions read from the database. It thus generates and counts every candidate itemset that the AIS algorithm generates. However, to use the standard SQL join operation for candidate generation, SETM separates candidate generation from counting. It saves a copy of the candidate itemset together with the TID of the generating transaction in a sequential structure. At the end of the pass, the support count of candidate itemsets is determined by sorting and aggregating this sequential structure. SETM remembers the TIDs of the generating transactions with the candidate itemsets. To avoid needing a subset operation, it uses this information to determine the large itemsets contained in the transaction read. zk s ??k and is obtained by deleting those candidates that do not have minimum support. Assuming that the database is sorted in TID order, SETM can easily find the large itemsets contained in a transaction in the next pass by sorting & on TID. In fact, it needs to visit every member of & only once in the TID order, and the candidate generation can be performed using the relational merge-join operation [131 The disadvantage of this approach is mainly due to the size of candidate sets ck. For each candidate itemset, the candidate set now has as many entries as the number of transactions in which the candidate itemset is present. Moreover, when we are ready to count the support for candidate itemsets at the end of the pass, i?k is in the wrong order and needs to be sorted on itemsets. After counting and pruning out small candidate itemsets that do not have minimum support, the resulting set &! needs another sort on TID before it can be used for generating candidates in the next pass. 3.3 Generation of Synthetic Data We generated synthetic transactions to evaluate the performance of the algorithms over a large range of data characteristics. These transactions mimic the transactions in the retailing environment. Our model of the “real” world is that people tend to buy sets of items together. Each such set is potentially a maximal large itemset. An example of such a set might be sheets, pillow case, comforter, and ruffles. However, some people may buy only some of the items from such a set. For instance, some people might buy only sheets and pillow case, and some only sheets. A transaction may contain more than one large itemset. For example, a customer might place an order for a dress and jacket when ordering sheets and pillow cases, where the dress and jacket together form another large itemset. Transaction sizes are typically clustered around a mean and a few transactions have many items. Typical sizes of large itemsets are also 492 clustered around a mean, with a few large itemsets having a large number of items, To create a dataset, our synthetic data generation program takes the parameters shown in Table 2. Table 2: Parameters ITI Average size of the transactions 111 Average size of the maximal potentially I,1 IL1 Number of maximal potentially large itemsets We first determine the size of the next transaction. The size is picked from a Poisson distribution with mean p equal to ITI. Note that if each item is chosen with the same probability p, and there are N items, the expected number of items in a transaction is given by a binomial distribution with parameters N and p, and is approximated by a Poisson distribution with mean Np. We then assign items to the transaction. Each transaction is assigned a series of potentially large itemsets. If the large itemset on hand does not fit in the transaction, the itemset is put in the transaction anyway in half the cases, and the itemset is moved to the next transaction the rest of the cases. Large itemsets are chosen from a set I of such itemsets. The number of itemsets in ‘T is set to ILj. There is an inverse relationship between IL1 and the average support for potentially large itemsets. An itemset in T is generated by first picking the size of the itemset from a Poisson distribution with mean ~1 equal to III. Items in the first itemset are chosen randomly. To model the phenomenon that large itemsets often have common items, some fraction of items in subsequent itemsets are chosen from the previous itemset generated. We use an exponentially distributed random variable with mean equal to the correlation level to decide this fraction for each itemset. The remaining items are picked at random. In the datasets used in the experiments, the correlation level was set to 0.5. We ran some experiments with the correlation level set to 0.25 and 0.75 but did not find much difference in the nature of our performance results. Each itemset in 1 has a weight associated with it, which corresponds to the probability that this itemset will be picked. This weight is picked from an exponential distribution with unit mean, and is then normalized so that the sum of the weights for all the itemsets in 7 is 1. The next itemset to be put in the transaction is chosen from 7 by tossing an ILI- sided weighted coin, where the weight for a side is the probability of picking the associated itemset. To model the phenomenon that all the items in a large itemset are not always bought together, we assign each itemset in I a corruption level c. When adding an itemset to a transaction, we keep dropping an item from the itemset as long as a uniformly distributed random number between 0 and 1 is less than c. Thus for an itemset of size 1, we will add 1 items to the transaction 1 - c of the time, I- 1 items c(1 - c) of the time, I- 2 items c2(1 - c) of the time, etc. The corruption level for an itemset is fixed and is obtained from a normal distribution with mean 0.5 and variance 0.1. We generated datasets by setting N = 1000 and IL] = 2000. We chose 3 values for ITI: 5, 10, and 20. We also chose 3 values for 111: 2,4, and 6. The number of transactions was to set to 100,000 because, as we will see in Section 3.4, SETM could not be run for larger values. However, for our scale-up experiments, we generated datasets with up to 10 million transactions (838MB for T20). Table 3 summarizes the dataset parameter settings. For the same ITI and 101 values, the size of datasets in megabytes were roughly equal for the different values of 111. Table 3: Parameter settings Name I ITI 1 111 I 101 1 Size in Megabytes T5.12.DlOOK 1 5 1 2 1 1OOK 1 2.4 3.4 Relative Performance Figure 4 shows the execution times for the six synthetic datasets given in Table 3 for decreasing values of minimum support. As the minimum support decreases, the execution times of all the algorithms increase because of increases in the total number of candidate and large itemsets. For SETM, we have only plotted the execution times for the dataset T5.12.DlOOK in Figure 4. The execution times for SETM for the two datasets with an average transaction size of 10 are given in Table 4. We did not plot the execution times in Table 4 on the corresponding graphs because they are too large compared to the execution times of the other algorithms. For the three datasets with transaction sizes of 20, SETM took too long to execute and we aborted those runs as the trends were clear. Clearly, Apriori beats SETM by more than an order of magnitude for large datasets. 493 T5.12.DlOOK T10.12.DlOOK E E F m 60 50 I I T10.14.DlOOK 3501 I AIS -- 300- 250 - 200 - 150 - loo. 50- or-- -Y - 2 1.5 1 0.33 0.25 T20.14.DlOOK “2 1.5 1 0.75 0.5 0.33 0.25 Minimum Support IV I I 140~ ii~/~.yq 2 1.5 1 0.75 Minimum supporlo.’ 0.33 0.25 T20.12.DlOOK ‘.ig -2 1.5 1 0.75 Minimum SqporloB 0.33 0.25 T20.16.DlOOK 3500 AIS * 3om- 2500- 2 1.5 1 0.33 0.25 Minimum 0.75 S@pot5 Figure 4: Execution times 494 Table 4: Execution times in seconds for SETM Algorithm Minimum Support 2.0% 1 1.5% 1 1.0% 1 0.75% J 0.5% Dataset T10.12.DlOOK SETM 74 161 838 1262 1878 Apriori 4.4 5.3 11.0 14.5 15.3 Dataset T10.14.DlOOK SETM 41 91 659 929 1639 Apriori 3.8 4.8 11.2 17.4 19.3 Apriori beats AIS for all problem sizes, by factors ranging from 2 for high minimum support to more than an order of magnitude for low levels of support. AIS always did considerably better than SETM. For small problems, AprioriTid did about as well as Apriori, but performance degraded to about twice as slow for large problems. 3.5 Explanation of the Relative Performance To explain these performance trends, we show in Figure 5 the sizes of the large and candidate sets in different passes for the T10.14.DlOOK dataset for the minimum support of 0.75%. Note that the Y-axis in this graph has a log scale. l&o7 1 2 3 F%SSti”mber 5 6 7 Figure 5: Sizes of the large and candidate sets (T10.14.DlOOK, minsup = 0.75%) The fundamental problem with the SETM algo rithm is the size of its i!?k sets. Recall that the size of the set ??k is given by c support-count(c), candidate itemsets c Thus, the sets ck are roughly S times bigger than the corresponding ck sets, where S is the average support count of the candidate itemsets. Unless the problem size is very small, the i?k sets have to be written to disk, and externally sorted twice, causing the SETM algorithm to perform poorly.’ This explains the jump in time for SETM in Table 4 when going from 1.5% support to 1.0% support for datasets with transaction size 10. The largest dataset in the scale- up experimegs for SETM in [13] was still small enough that CI: could fit in memory; hence they did not encounter this jump in execution time. Note that for the same minimum support, the support count for candidate itemsets increases linearly with the number of transactions. Thus, as we increase the number of transactions for the same values of 12’1 and 111, though the size of Ck does not change, the size of ck goes up linearly. Thus, for,datas& with more transactions, the performance gap between SETM and the other algorithms will become even larger. The problem with AIS is that it generates too many candidates that later turn out to be small, causing it to waste too much effort. Apriori also counts too many small sets in the second pass (recall that C’s is really a cross-product of Lr with Li). However, this wastage decreases dramatically from the third pass onward. Note that for the example in Figure 5, after pass 3, almost every candidate itemset counted by Apriori turns out to be a large set. AprioriTid also has the problem of SETM that ck tends to be large. However, the apriori candidate generation used by AprioriTid generates significantly fewer candidates than the transaction-baaed candi- date generation used by SETM. As a result, the ??k of AprioriTid has fewer entries than that of SETM. Apri- oriTid is also able to use a single word (ID) to store a candidate rather than requiring as many words as the number of items in the candidate.3 In addition, unlike SETM, AprioriTid does not have to sort i?‘,. Thus, AprioriTid does not suffer as much as SETM from maintaining Gk. AprioriTid has the nice feature that it replaces a pass over the original dataset by a pass over the set ??k. Hence, AprioriTid is very effective in later passes when the size of ??a becomes small compared to the 2The cost of external sorting in SETM can be reduced somewhat as follows. Before writing out entries in Ek to dish, we can sort them on itemsets using an internal sorting procedure, and write them as sorted runs. These sorted NOB can then be merged to obtain support counts. However, given the poor performance of SETM, we do not expect this optimization to aRect the algorithm choice. 3For SETM to use IDS, it would have to maintain two additional in-memory data structures: a hash table to 6nd out whether .a candidate has been generated previousIy, and a mapping from the IDS to candidates. However, this would destroy the set-oriented nature of the algorithm. Aiso, once we have the hash table which gives us the IDS of candidates, we might as weil count them at the same time and avoid the two externsl sorts. We experimented with this variant of SETM and found that, while it did better than SETM, it stiil performed much worse than Apriori or AprioriTid. 495 size of the database. Thus, we find that AprioriTid beats Apriori when its ck sets can fit in memory and the distribution of the large itemsets has a long tail. When ck doesn’t fit in memory, there is a jump in the execution time for AprioriTid, such as when going from 0.75% to 0.5% for datasets with transaction size 10 in Figure 4. In this region, Apriori starts beating AprioriTid. 3.6 Algorithm AprioriHybrid It is not necessary to use the same algorithm in all the passes over data. Figure 6 shows the execution times for Apriori and AprioriTid for different passes over the dataset T10.14.DlOOK. In the earlier passes, Apriori does better than AprioriTid. However, AprioriTid beats Apriori in later passes. We observed similar relative behavior for the other datasets, the reason for which is as follows. Apriori and AprioriTid use the same candidate generation procedure and therefore count the same itemsets. In the later passes, the number of candidate itemsets reduces (see the size of ck for Apriori and AprioriTid in Figure 5). However, Apriori still examines every transaction in the database. On the other hand, rather than scanning the database, AprioriTid scans ck for obtaining support counts, and the size of ck has become smaller than the size of the database. When the ??k sets can fit in memory, we do not even incur the cost of writing them to disk. 0' \ 1 2 3 4 5 6 7 Figure 6: Per pass executiy times of Apriori and AprioriTid (T10.14.DlOOK, minsup = 0.75%) Based on these observations, we can design a hybrid algorithm, which we call AprioriHybrid, that uses Apriori in the initial passes and switches to AprioriTid when it expects that the set ck at the end of the pass will fit in memory. We use the following heuristic to estimate if ck would fit in memory in the next pass. At the end of the current pass, we have the counts of the candidates in ck. From this, we estimate what the size of ??k would have been if it had been generated. This Size, in words, is (Ccandidates c E ck Support(C) -!- number of transactions). If ck in this pass was small enough to fit in memory, and there were fewer large candidates in the current pass than the previous pass, we switch to AprioriTid. The latter condition is added to avoid switching when ck in the current pass fits in memory but ??k in the next pass may not. Switching from Apriori to AprioriTid does involve a cost. Assume that we decide to switch from Apriori to AprioriTid at the end of the lath pass. In the (Ic + 1)th pass, after finding the candidate itemsets contained in a transaction, we will also have to add their IDS to ck+i (see the description of AprioriTid in Section 2.2). Thus there is an extra cost incurred in this pass relative to just running Apriori. It is only in the (k+2)th pass that we actually start running AprioriTid. Thus, if there are no large (&l)-itemsets, or no (h + 2)-candidates, we will incur the cost of switching without getting any of the savings of using AprioriTid. Figure 7 shows the performance of AprioriHybrid relative to Apriori and AprioriTid for three data&s. AprioriHybrid performs better than Apriori in almost all cases. For T10.12.DlOOK with 1.5% support, AprioriHybrid does a little worse than Apriori since the pass in which the switch occurred was the last pass; AprioriHybrid thus incurred the cost of switching without realizing the benefits. In general, the advantage of AprioriHybrid over Apriori depends on how the size of the Ck set decline in the later passes. If ck remains large until nearly the end and then has an abrupt drop, we will not gain much by using AprioriHybrid since we can use AprioriTid only for a short period of time after the switch. This is what happened with the T20.16.DlOOK dataset. On the other hand, if there is a gradual decline in the size Of ck, AprioriTid can be used for a while after the switch, and a significant improvement can be obtained in the execution time. 3.7 Scale-up Experiment Figure 8 shows how AprioriHybrid scales up as the number of transactions is increased from 100,000 to 10 million transactions. We used the combinations (T5.12) (T10.14), and (T20.16) for the average sizes of transactions and itemsets respectively. All other parameters were the same as for the data in Table 3. The sizes of these datasets for 10 million transactions were 239MB, 439MB and 838MB respectively. The minimum support level was set to 0.75%. The execution times are normalized with respect to the times for the 100,000 transaction datasets in the first 496 T10.12.DlOOK 40 35- 30. 0 2 1.5 1 MlninwmSlpportO*5 0.75 0.33 0.25 T10.14.DlOOK 551 I 0' I T20.16.DlOOK 1 E F "2 1.5 1 0.75 Minimum Sqqmt5 0.33 0.25 Figure 7: Execution times: AprioriHybrid graph and with respect to the 1 million transaction dataset in the second. As shown, the execution times scale quite linearly. 0' 1 1 2.5 kmberofTraktbw(n&~~ 10 Figure 8: Number of transactions scale-up Next, we examined how AprioriHybrid scaled up with the number of items. We increased the num- ber of items from 1000 to 10,000 for the three pa- rameter settings T5.12.DlOOK, T10.14.DlOOK and T20.16.DlOOK. All other parameters were the same as for the data in Table 3. We ran experiments for a minimum support at 0.7556, and obtained the results shown in Figure 9. The execution times decreased a little since the average support for an item decreased as we increased the number of items. This resulted in fewer large itemsets and, hence, faster execution times. Finally, we investigated the scale-up as we increased the average transaction size. The aim of this experiment was to see how our data structures scaled with the transaction size, independent of other factors like the physical database size and the number of large itemsets. We kept the physical size of the 497 T20.16 - T10.14 -+--- - T5.12 -a..-- Figure 9: Number of items scale-up database roughly constant by keeping the product of the average transaction size and the number of transactions constant. The number of transactions ranged from 200,000 for the database with an average transaction size of 5 to 20,000 for the database with an average transaction size 50. Fixing the minimum support as a percentage would have led to large increases in the number of large itemsets as the transaction size increased, since the probability of a itemset being present in a transaction is roughly proportional to the transaction size. We therefore fixed the minimum support level in terms of the number of transactions. The results are shown in Figure 10. The numbers in the key (e.g. 500) refer to this minimum support. As shown, the execution times increase with the transaction size, but only gradually. The main reason for the increase was that in spite of setting the minimum support in terms of the number of transactions, the number of large itemsets increased with increasing transaction length. A secondary reason was that finding the candidates present in a transaction took a little longer time. 4 Conclusions and Future Work We presented two new algorithms, Apriori and Apri- oriTid, for discovering all significant association rules between items in a large database of transactions. We compared these algorithms to the previously known algorithms, the AIS [4] and SETM [13] algo rithms. We presented experimental results, showing that the proposed algorithms always outperform AIS and SETM. The performance gap increased with the problem size, and ranged from a factor of three for small problems to more than an order of magnitude for large problems. We showed how the best features of the two pro- OL ’ I 5 10 20 30 40 50 TransactionSize Figure 10: Transaction size scale-up posed algorithms can be combined into a hybrid al- gorithm, called AprioriHybrid, which then becomes the algorithm of choice for this problem. Scale-up ex- periments showed that AprioriHybrid scales linearly with the number of transactions. In addition, the ex- ecution time decreases a little as the number of items in the database increases. As the average transaction size increases (while keeping the database size con- stant), the execution time increases only gradually. These experiments demonstrate the feasibility of u& ing AprioriHybrid in real applications involving very large databases. The algorithms presented in this paper have been implemented on several data repositories, including the AIX file system, DBS/MVS, and DB2/6000. We have also tested these algorithms against real customer data, the details of which can be found in [5]. In the future, we plan to extend this work along the following dimensions: l Multiple taxonomies (is-a hierarchies) over items are often available. An example of such a hierarchy is that a dish washer is a kitchen appliance is a heavy electric appliance, etc. We would like to be able to find association rules that use such hierarchies. l We did not consider the quantities of the items bought in a transaction, which are useful for some applications. Finding such rules needs further work. The work reported in this paper has been done in the context of the Quest project at the IBM Al- maden Research Center. In Quest, we are exploring the various aspects of the database mining problem. Besides the problem of discovering association rules, some other problems that we have looked into include 498 the enhancement of the database capability with clas- sification queries [2] and similarity queries over time sequences [l]. We believe that database mining is an important new application area for databases, com- bining commercial interest with intriguing research questions. Acknowledgment We wish to thank Mike Carey for his insightful comments and suggestions. References [l] R. Agrawal, C. Faloutsos, and A. Swami. Ef- ficient similarity search in sequence databases. In Proc. of the Fourth International Conference on Foundations of Data Organization and Algo- rithms, Chicago, October 1993. [2] R. Agrawal, S. Ghosh, T. Imielinski, B. Iyer, and A. Swami. An interval classifier for database mining applications. In Proc. of the VLDB Conference, pages 560-573, Vancouver, British Columbia, Canada, 1992. [3] R. Agrawal, T. Imielinski, and A. Swami. Database mining: A performance perspective. IEEE Bansactions on Knowledge and Data En- gineering, 5(6):914-925, December 1993. Special Issue on Learning and Discovery in Knowledge- Based Databases. [4] R. Agrawal, T. Imielinski, and A. Swami. Mining association rules between sets of items in large databases. In Proc. of the ACM SIGMOD Con- ference on Management of Data, Washington, D.C., May 1993. [5] R. Agrawal and R. Srikant. Fast, algorithms for mining association rules in large databases. Re- search Report RJ 9839, IBM Almaden Research Center, San Jose, California, June 1994. [6] D. S. Associates. The new direct marketing. Business One Irwin, Illinois, 1990. [7] R. Brachman et al. Integrated support for data archeology. In AAAI-93 Workshop on Knowledge Discovery in Databases, July 1993. [8] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone. Classification and Regression Trees. Wadsworth, Belmont, 1984. [9] P. Cheeseman et al. Autoclass: A bayesian classification system. In 5th Int’l Conf. on Machine Learning. Morgan Kaufman, June 1988. [lo] D. H. Fisher. Knowledge acquisition via incre- mental conceptual clustering. Machine Learning, 2(2), 1987. [ll] J. Han, Y. Cai, and N. Cercone. Knowledge discovery in databases: An attribute oriented approach. In Proc. of the VLDB Conference, pages 547-559, Vancouver, British Columbia, Canada, 1992. [12] M. Holsheimer and A. Siebes. Data mining: The search for knowledge in databases. Technical Report CS-R9406, CWI, Netherlands, 1994. [13] M. Ho&ma and A. Swami. Set-oriented mining of association rules. Research Report RJ 9567, IBM Almaden Research Center, San Jose, Cali- fornia, October 1993. [14] R. Krishnamurthy and T. Imielinski. Practi- tioner problems in need of database research: Re- search directions in knowledge discovery. SIG- MOD RECORD, 20(3):76-78, September 1991. [15] P. Langley, H. Simon, G. Bradshaw, and J. Zytkow. Scientific Discovery: Computational Explorations of the Creative Process. MIT Press, 1987. [16] H. Mannila and K.-J. Raiha. Dependency inference. In Proc. of the VLDB Conference, pages 155-158, Brighton, England, 1987. [17] H. Mannila, H. Toivonen, and A. I. Verkamo. Efficient algorithms for discovering association rules. In KDD-94: AAAI Workshop on Knowl- edge Discovery in Databases, July 1994. [18] S. Muggleton and C. Feng. Efficient induction of logic programs. In S. Muggleton, editor, Inductive Logic Programming. Academic Press, 1992. [19] J. Pearl. Probabilistic reasoning in intelligent systems: Networks of plausible inference, 1992. [20] G. Piatestsky-Shapiro. Discovery, analy- sis, and presentation of strong rules. In G. Piatestsky-Shapiro, editor, Knowledge Dis- covey in Databases. AAAI/MIT Press, 1991. [21] G. Piatestsky-Shapiro, editor. Knowledge Dis- covey in Databases. AAAI/MIT Press, 1991. [22] J. R. Quinlan. C4.5: Programs for Machine Learning. Morgan Kaufman, 1993. 499
Rosenblatt1958	Psychological Review Vol. 65, No. 6, 19S8 THE PERCEPTRON: A PROBABILISTIC MODEL FOR INFORMATION STORAGE AND ORGANIZATION IN THE BRAIN1 F. ROSENBLATT Cornell Aeronautical Laboratory If we are eventually to understand the capability of higher organisms for perceptual recognition, generalization, recall, and thinking, we must first have answers to three fundamental questions: 1. How is information about the physical world sensed, or detected, by the biological system? 2. In what form is information stored, or remembered? 3. How does information contained in storage, or in memory, influence recognition and behavior? The first of these questions is in the province of sensory physiology, and is the only one for which appreciable understanding has been achieved. This article will be concerned pri- marily with the second and third questions, which are still subject to a vast amount of speculation, and where the few relevant facts currently sup- plied by neurophysiology have not yet been integrated into an acceptable theory. With regard to the second question, two alternative positions have been maintained. The first suggests that storage of sensory information is in the form of coded representations or images, with some sort of one-to-one mapping between the sensory stimulus 1 The development of this theory has been carried out at the Cornell Aeronautical Lab- oratory, Inc., under the sponsorship of the Office of Naval Research, Contract Nonr- 2381(00). This article is primarily'an adap- tation of material reported in Ref. IS, which constitutes the first full report on the program. and the stored pattern. According to this hypothesis, if one understood the code or "wiring diagram" of the nerv- ous system, one should, in principle, be able to discover exactly what an organism remembers by reconstruct- ing the original sensory patterns from the "memory traces" which they have left, much as we might develop a photographic negative, or translate the pattern of electrical charges in the "memory" of a digital computer. This hypothesis is appealing in its simplicity and ready intelligibility, and a large family of theoretical brain models has been developed around the idea of a coded, representational mem- ory (2, 3, 9, 14). The alternative ap- proach, which stems from the tradi- tion of British empiricism, hazards the guess that the images of stimuli may never really be recorded at all, and that the central nervous system simply acts as an intricate switching network, where retention takes the form of new connections, or pathways, between centers of activity. In many of the more recent developments of this position (Hebb's "cell assembly," and Hull's "cortical anticipatory goal response," for example) the "re- sponses" which are associated to stimuli may be entirely contained within the CNS itself. In this case the response represents an "idea" rather than an action. The impor- tant feature of this approach is that there is never any simple mapping of the stimulus into memory, according to some code which would permit its later reconstruction. Whatever in- 386 THE PERCEPTRON 387 formation is retained must somehow be stored as a preference for a par- ticular response; i.e., the information is contained in connections or associa- tions rather than topographic repre- sentations. (The term response, for the remainder of this presentation, should be understood to mean any distinguishable state of the organism, which may or may not involve ex- ternally detectable muscular activity. The activation of some nucleus of cells in the central nervous system, for example, can constitute a response, according to this definition.) Corresponding to these two posi- tions on the method of information retention, there exist two hypotheses with regard to the third question, the manner in which stored information exerts its influence on current activity. The "coded memory theorists" are forced to conclude that recognition of any stimulus involves the matching or systematic comparison of the con- tents of storage with incoming sen- sory patterns, in order to determine whether the current stimulus has been seen before, and to determine the ap- propriate response from the organism. The theorists in the empiricist tradi- tion, on the other hand, have essen- tially combined the answer to the third question with their answer to the second: since the stored information takes the form of new connections, or transmission channels in the nervous system (or the creation of conditions which are functionally equivalent to new connections), it follows that the new stimuli will make use of these new pathways which have been created, automatically activating the appro- priate response without requiring any separate process for their recognition or identification. The theory to be presented here takes the empiricist, or "connectionist"' position with regard to these ques- tions. The theory has been developed for a hypothetical nervous system, or machine, called a perceptron. The perceptron is designed to illustrate some of the fundamental properties of intelligent systems in general, without becoming too deeply enmeshed in the special, and frequently unknown, con- ditions which hold for particular bio- logical organisms. The analogy be- tween the perceptron and biological systems should be readily apparent to the reader. During the last few decades, the development of symbolic logic, digital computers, and switching theory has impressed many theorists with the functional similarity between a neuron and the simple on-off units of which computers are constructed, and has provided the analytical methods nec- essary for representing highly complex logical functions in terms of such elements. The result has been a profusion of brain models which amount simply to logical contrivances for performing particular algorithms (representing "recall," stimulus com- parison, transformation, and various kinds of analysis) in response to sequences of stimuli—e.g., Rashevsky (14), McCulloch (10), McCulloch & Pitts (11), Culbertson (2), Kleene (8), and Minsky (13). A relatively small number of theorists, like Ashby (1) and von Neumann (17, 18), have been concerned with the problems of how an imperfect neural network, containing many random connections, can be made to perform reliably those functions which might be represented by idealized wiring diagrams. Un- fortunately, the language of symbolic logic and Boolean algebra is less well suited for such investigations. The need for a suitable language for the mathematical analysis of events in systems where only the gross organ- ization can be characterized, and the 388 F. ROSENBLATT precise structure is unknown, has led the author to formulate the current model in terms of probability theory rather than symbolic logic. The theorists referred to above were chiefly concerned with the question of how such functions as perception and recall might be achieved by a deter- ministic physical system of any sort, rather than how this is actually done by the brain. The models which have been produced all fail in some im- portant respects (absence of equi- potentiality, lack of neuroeconomy, excessive specificity of connections and synchronization requirements, unrealistic specificity of stimuli suffi- cient for cell firing;, postulation of variables or functional features with no known neurological correlates, etc.) to correspond to a biological system. The proponents of this line of ap- proach have maintained that, once it has been shown how a physical system of any variety might be made to perceive and recognize stimuli, or perform other brainlike functions, it would require only a refinement or modification of existing principles to understand the working of a more realistic nervous system, and to elim- inate the shortcomings mentioned above. The writer takes the position, on the other hand, that these short- comings are such that a mere refine- ment or improvement of the principles already suggested can never account for biological intelligence; a difference in principle is clearly indicated. The theory of statistical separability (Cf. 15), which is to be summarized here, appears to offer a solution in principle to all of these difficulties. Those theorists—Hebb (7), Milner (12), Eccles (4), Hayek (6)—who have been more directly concerned with the biological nervous system and its activity in a natural environ- ment, rather than with formally'anal- ogous machines, have generally been less exact in their formulations and far from rigorous in their analysis, so that it is frequently hard to assess whether or not the systems that they describe could actually work in a realistic nerv- ous system, and what the necessary and sufficient conditions might be. Here again, the lack of an analytic language comparable in proficiency to the Boolean algebra of the network analysts has been one of the main obstacles. The contributions of this group should perhaps be considered as suggestions of what to look for and investigate, rather than as finished theoretical systems in their own right. Seen from this viewpoint, the most suggestive work, from the standpoint of the following theory, is that of Hebb and Hayek. The position, elaborated by Hebb (7), Hayek (6), Uttley (16), and Ashby (1), in particular, upon which the theory of the perceptron is based, can be summarized by the following assumptions: 1. The physical connections of the nervous system which are involved in learning and recognition are not iden- tical from one organism to another. At birth, the construction of the most important networks is largely random, subject to a minimum number of genetic constraints. 2. The original system of connected cells is capable of a certain amount of plasticity; after a period of neural activity, the probability that a stim- ulus applied to one set of cells will cause a response in some other set is likely to change, due to some rela- tively long-lasting changes in the neurons themselves. 3. Through exposure to a large sample of stimuli, those which are most "similar" (in some sense which must be defined in terms of the particular physical system) will tend THE PEECEPTRON 389 to form pathways to the same sets of responding cells. Those which are markedly "dissimilar" will tend to develop connections to different sets of responding cells. 4. The application of positive and/ or negative reinforcement (or stimuli which serve this function) may facil- itate or hinder whatever formation of connections is currently in progress. 5. Similarity, in such a system, is represented at some level of the nerv- ous system by a tendency of similar stimuli to activate the same sets of cells. Similarity is not a necessary attribute of particular formal or geo- metrical classes of stimuli, but de- pends on the physical organization of the perceiving system, an organiza- tion which evolves through interaction with a given environment. The structure of the system, as well as the ecology of the stimulus-environment, will affect, and will largely determine, the classes of "things" into which the perceptual world is divided. THE ORGANIZATION OF A PERCEPTRON The organization of a typical photo- perceptron (a perceptron responding to optical patterns as stimuli) is shown in Fig. 1. The rules of its organiza- tion are as follows: 1. Stimuli impinge on a retina of sensory units (S-points), which are assumed to respond on an all-or- nothing basis, in some models, or with a pulse amplitude or frequency pro- portional to the stimulus intensity, in other models. In the models con- sidered here, an all-or-nothing re- sponse will be assumed. 2. Impulses are transmitted to a set of association cells (A-units) in a "projection area" (Ai). This pro- jection area may be omitted in some models, where the retina is connected directly to the association area (An). FIG. 1. Organization of a perceptron. The cells in the projection area each receive a number of connections from the sensory points. The set of S- points transmitting impulses to a par- ticular A-unit will be called the origin points of that A-unit. These origin points may be either excitatory or in- hibitory in their effect on the A-unit. If the algebraic sum of excitatory and inhibitory impulse intensities is equal to or greater than the threshold (6) of the A-unit, then the A-unit fires, again on an all-or-nothing basis (or, in some models, which will not be considered here, with a frequency which depends on the net value of the impulses received). The origin points of the A-units in the projection area tend to be clustered or focalized, about some central point, corresponding to each A-unit. The number of origin points falls off exponentially as the retinal distance from the central point for the A-unit in question increases. (Such a distribution seems to be sup- ported by physiological evidence, and serves an important functional pur- pose in contour detection.) 3. Between the projection area and the association area (An), connections are assumed to be random. That is, each A-unit in the An set receives some number of fibers from origin points in the AI set, but these origin points are scattered at random throughout the projection area. Apart from their connection distri- bution, the An units are identical with the AI units, and respond under similar conditions. 4. The "responses," Ri, R^, . . . , Rn are cells (or sets of cells) which 390 F. ROSENBLATT respond in much the same fashion as the A-units. Each response has a typically large number of origin points located at random in the An set. The set of A-units transmitting impulses to a particular response will be called the source-set for that response. (The source-set of a response is iden- tical to its set of origin points in the A-system.) The arrows in Fig. 1 indicate the direction of transmission through the network. Note that up to An all connections are forward, and there is no feedback. When we come to the last set of connections, between An and the R-units, connections are established in both directions. The rule governing feedback connections, in most models of the perceptron, can be either of the following alternatives: (a) Each response has excitatory feedback connections to the cells in its own source-set, or (&) Each response has inhibitory feedback connections to the comple- ment of its own source-set (i.e., it tends to prohibit activity in any association cells which do not transmit to it). The first of these rules seems more plausible anatomically, since the R- units might be located in the same cortical area as their respective source- flHOKfflt LIHCt >W IHHtllTOMV COHKECTIOHI FIG. 2A. Schematic representation of connections in a simple perceptron. OWHWTMV OMNICTIOH • EMITITMV CONNtcriON FIG. 2B. Venn diagram of the same per- ceptronf (shading shows active sets for RI response). sets, making mutual excitation be- tween the R-units and the A-units of the appropriate source-set highly probable. The alternative rule (6) leads to a more readily analyzed sys- tem, however, and will therefore be assumed for most of the systems to be evaluated here. Figure 2 shows the organization of a simplified perceptron, which affords a convenient entry into the theory of statistical separability. After the theory has been developed for this simplified model, we will be in a better position to discuss the advantages of the system in Fig. 1. The feedback connections shown in Fig. 2 are in- hibitory, and go to the complement of the source-set for the response from which they originate; consequently, this system is organized according to Rule b, above. The system shown here has only three stages, the first association stage having been elim- inated. Each A-unit has a set of randomly located origin points in the retina. Such a system will form simi- larity concepts on the basis of coin- cident areas of stimuli, rather than by the similarity of contours or outlines. While such a system is at a disadvan- tage in many discrimination experi- ments, its capability is still quite impressive, as will be demonstrated presently. The system shown in Fig. 2 has only two responses, but there is clearly no limit on the number that might be included. The responses in a system organized in this fashion are mutually exclusive. If RI occurs, it will tend to inhibit Rs, and will also inhibit the source-set for R2. Likewise, if Ra should occur, it will tend to inhibit RI. If the total impulse received from all the A-units in one source-set is stronger or more frequent than the impulse received by the alternative (antagonistic) re- sponse, then the first response will THE PERCEPTRON 391 tend to gain an advantage over the other, and will be the one which occurs. If such a system is to be capable of learning, then it must be possible to modify the A-units or their connections in such a way that stimuli of one class will tend to evoke a stronger impulse in the Ri source-set than in the Ra source-set, while stimuli of another (dissimilar) class will tend to evoke a^stronger impulse in the Ra source-set than in the Ri source-set. It will be assumed that the impulses delivered by each A-unit can be characterized by a value, V, which may be an amplitude, frequency, latency, or probability of completing transmission. If an A-unit has a high value, then all of its output impulses are considered to be more effective, more potent, or more likely to arrive at their endbulbs than impulses from an A-unit with a lower value. The value of an A-unit is considered to be a fairly stable characteristic, probably depending on the metabolic condition of the cell and the cell membrane, but it is not absolutely constant. It is assumed that, in general, periods of activity tend to increase a cell's value, while the value may decay (in some models) with inactivity. The most interesting models are those in which cells are assumed to compete for met- abolic materials, the more active cells gaining at the expense of the less active cells. In such a system, if there is no activity, all cells will tend to remain in a relatively constant condition, and (regardless of activity) the net value of the system, taken in TABLE 1 COMPARISON or LOGICAL CHARACTERISTICS OF a, /3, AND 7 SYSTEMS Total value-gain of source set per rein- forcement AV for A-units active for 1 unit of time AV for inactive A-units outside of domi- nant set A V for inactive A-units of dominant set Mean value of A-system Difference between mean values of source-sets a-System(UncompensatedGain System) Nar +1 0 0 Increases with number of reinforcements Proportional to differ- ences of reinforce- ment frequency (»,„—»,„) /3-System(Constant FeedSystem) K K/Nar K/NA, 0 Increases with time 0 7-System(Parasitic GainSystem) 0 + 1 0 -Nr NAr-Nar Constant 0 Note: In the /3 and 7 systems, the total value-change for any A-unit will be the sum of the AV's for all source-sets of which it is a member. N»r = Number of active units in source-set NAT — Total number of units in source-set «,„ = Number of stimuli associated to response TJ K = Arbitrary constant 392 F. ROSENBLATT its entirety, will remain constant at all times. Three types of systems, which differ in their value dynamics, have been investigated quantitatively. Their principal logical features are compared in Table 1. In the alpha system, an active cell simply gains an increment of value for every impulse, and holds this gain indefinitely. In the beta system, each source-set is allowed a certain constant rate of gain, the increments being apportioned among the cells of the source-set in proportion to their activity. In the gamma system, active cells gain in value at the expense of the inactive cells of their source-set, so that the total value of a source-set is always constant. For purposes of analysis, it is con- venient to distinguish two phases in the response of the system to a stim- ulus (Fig. 3). In the predominant phase, some proportion of A-units (represented by solid dots in the figure) responds to the stimulus, but the R-units are still inactive. This phase is transient, and quickly gives way to the postdominant phase, in which one of the responses becomes active, inhibiting activity in the com- FIG. 3A. Predominant phase. Inhibitory connectionsare not shown. Solid black units are active. FIG. 3B. Postdominant phase. Dominant subsetsuppresses rival' sets. Inhibitory connections shown only for Ri. FIG. 3. Phases of response to a stimulus, plement of its own source-set, and thus preventing the occurrence of any alternative response. The response which happens to become dominant is initially random, but if the A-units are reinforced (i.e., if the active units are allowed to gain in value), then when the same stimulus is presented again at a later time, the same response will have a stronger tendency to recur, and learning can be said to have taken place. ANALYSIS OF THE PREDOMINANT PHASE The perceptrons considered here will always assume a fixed threshold, 6, for the activation of the A-units. Such a system will be called a fixed- threshold model, in contrast to a con- tinuous transducer model, where the response of the A-unit is some con- tinuous function of the impinging stimulus energy. In order to predict the learning curves of a fixed-threshold perceptron, two variables have been found to be of primary importance. They are defined as follows : Pa = the expected proportion of A- units activated by a stimulus of a given size, PC = the conditional probability that an A-unit which responds to a given stimulus, Si, will also respond to another given stimulus, 82- It can be shown (Rosenblatt, IS) that as the size of the retina is increased, the number of S-points (Na} quickly ceases to be a significant parameter, and the values of Pa and Pc approach the value that they would have for a retina with infinitely many points. For a large retina, therefore, the equations are as follows : (1) THE PERCEPTRON 393 where P(e,i) = and X R = proportion of S-points activated by the stimulus x — number of excitatory connec- tions to each A-unit y = number of inhibitory connec- tions to each A-unit 0 = threshold of A-units. (The quantities e and i are the ex- citatory and inhibitory components of the excitation received by the A-unit from the stimulus. If the algebraic sum a = e + i is equal to or greater than 6, the A-unit is assumed to re- spond.) j x y e i • e = "~ 2~i 2-i 2—i 2- x—0 y~~^ (e - where £P(«,t,J.,/<,«.,«<) (2) - J, + /,- + ge - «,- > 0) X X > /x - e\x ( j G"«(I - G) -«-<'•\ s^ / x (';') G.,(1-G).-., and L — proportion of the S-points illumi- nated by the first stimulus, Si, which are not illuminated by S2 G — proportion of the residual S-set (left over from the first stim- ulus) which is included in the second stimulus (82). The quantities R, L, and G specify the two stimuli and their retinal overlap. le and /,• are, respectively, the numbers of excitatory and inhibitory origin points "lost" by the A-unit when stimulus Si is replaced by 82; ge and gi are the numbers of excitatory and inhibitory origin points "gained" when stimulus Si is replaced by 82. The summations in Equation 2 are between the limits indicated, subject to the side condition e — i — I, + l{ +g, - gi > 0. Some of the most important char- acteristics of Pa are illustrated in Fig. 4, which shows Pa as a function of the retinal area illuminated (R). Note that Pa can be reduced in magnitude by either increasing the threshold, 6, or by increasing the proportion of in- hibitory connections (y). A compari- son of Fig. 4b and 4c shows that if the excitation is about equal to the inhibi- tion, the curves for Pa as a function of R are flattened out, so that there is little variation in Pa for stimuli of different sizes. This fact is of great importance for systems which require Pa to be close to an optimum value in order to perform properly. The behavior of Pc is illustrated in Fig. 5 and 6. The curves in Fig. 5 can be compared with those for Pa in Fig. 4. Note that as the threshold is in- creased, there is an even sharper re- duction in the value of Pc than was the case with Pa. Pc also decreases as the proportion of inhibitory connections increases, as does Pa. Fig. 5, which is 394 F. ROSENBLATT (•] EFFECT OF INHIBITORY- EXCITATORY MIXTURE. 6 " I (b) VARIATION WITH 6 FOR X • 10, V 0 (c) VARIATION WITH M AHD 0 FOR MIXTURES ABOUT.60J INHIBITORY. SOLID LINES ARE FOR X • 5, Y » 5. 0 .1 .2 .3 .1 .5 PROPORTION OF S-POIHTS ILLUMINATED (K) .1 .2 .3 .it .B FIG. 4. P0 as function of retinal area illuminated. calculated for nonoverlapping stimuli, illustrates the fact that Pc remains greater than zero even when the stim- uli are completely disjunct, and illumi- nate no retinal points in common. In Fig. 6, the effect of varying amounts of overlap between the stimuli is shown. In all cases, the value of Pc goes to unity as the stimuli approach perfect identity. For smaller stimuli (broken line curves), the value of Pc is lower than for large stimuli. Simi- larly, the value is less for high thresh- olds than for low thresholds. The minimum value of Pc will be equal to Pcmin = (1 - L)'(l - G)». (3) In Fig. 6, Pemin corresponds to the curve for 6 = 10. Note that under these conditions the probability that the A-unit responds to both stimuli (Pc) is practically zero, except for stimuli which are quite close to identity. This condition can be of considerable help in discrimination learning. MATHEMATICAL ANALYSIS OF LEARNING IN THE PERCEPTRON The response of the perceptron in the predominant phase, where some fraction of the A-units (scattered throughout the system) responds to the stimulus, quickly gives way to the postdominant response, in which ac- tivity is limited to a single source-set, the other sets being suppressed. Two possible systems have been studied for the determination of the "dominant" response, in the postdominant phase. In one (the mean-discriminating sys- tem, or ju-system), the response whose inputs have the greatest mean value responds first, gaining a slight advan- tage over the others, so that it quickly becomes dominant. In the second case (the sum-discriminating system, or S-system), the response whose in- puts have the greatest net value gains an advantage. In most cases, sys- tems which respond to mean values have an advantage over systems which respond to sums, since the means are THE PERCEPTEON 395 .1 .2 .3 .4 .5 R (RETINAL AREA ILLUMINATED) FIG. 5. Pc as a function of R, for nonoverlapping stimuli. less influenced by random variations in Pa from one source-set to another. In the case of the -/-system (see Table 1), however, the performance of the ^-system and S-system become identical. We have indicated that the percep- tron is expected to learn, or to form associations, as a result of the changes in value that occur as a result of the activity of the association cells. In evaluating this learning, one of two types of hypothetical experiments can be considered. In the first case, the perceptron is exposed to some series of stimulus patterns (which might be presented in random positions on the retina) and is "forced" to give the desired response in each case. (This forcing of responses is assumed to be a prerogative of the experimenter. In experiments intended to evaluate trial-and-error learning, with more sophisticated perceptrons, the experi- menter does not force the system to respond in the desired fashion, but merely applies positive reinforcement when the response happens to be cor- rect, and negative reinforcement when the response is wrong.) In evaluating the learning which has taken place during this "learning series," the perceptron is assumed to be "frozen" in its current condition, no further value changes being allowed, and the same series of stimuli is presented again in precisely the same fashion, so that the stimuli fall on identical posi- tions on the retina. The probability that the perceptron will show a bias towards the "correct" response (the one which has been previously rein- forced during the learning series) in preference to any given alternative response is called Pr, the probability of correct choice of response between two alternatives. In the second type of experiment, a learning series is presented exactly as before, but instead of evaluating the perceptron's performance using the same series of stimuli which were shown before, a new series is pre- sented, in which stimuli may be drawn from the same classes that were previ- (PROPORTION OF 0¥E«L«P BETMEEK STIMULI) FIG. 6. Pc as a function of C. X - 10, Y = 0. Solid lines: R = .5; broken lines: R = .2. 396 F. ROSENBLATT ously experienced, but are not neces- sarily identical. This new test series is assumed to be composed of stimuli projected onto random retinal posi- tions, which are chosen independently of the positions selected for the learn- ing series. The stimuli of the test series may also differ in size or rota- tional position from the stimuli which were previously experienced. In this case, we are interested in the prob- ability that the perceptron will give the correct response for the class of stimuli which is represented, regard- less of whether the particular stimulus has been seen before or not. This probability is called Pt, the prob- ability of correct generalization. As with Pr, Pg is actually the probability that a bias will be found in favor of the proper response rather than any one alternative ; only one pair of responses at a time is considered, and the fact that the response bias is correct in one pair does not mean that there may not be other pairs in which the bias favors the wrong response. The prob- ability that the correct response will be preferred over all alternatives is designated PR or Pa. In all cases investigated, a single general equation gives a close ap- proximation to Pr and Pg, if the ap- propriate constants are substituted. This equation is of the form : P = P(^ar>0).<£(Z) (4) where P(Nar > 0) = 1 - (1 - P.). <t>(Z) = normal curve integral from — oo to Z and c4n, If Ri is the "correct" response, and R2 is the alternative response under con- sideration, Equation 4 is the prob- ability that Rj, will be preferred over Ra after n,r stimuli have been shown for each of the two responses, during the learning period. N, is the number of "effective" A-units in each source- set ; that is, the number of A-units in either source-set which are not con- nected in common to both responses. Those units which are connected in common contribute equally to both sides of the value balance, and con- sequently do not affect the net bias towards one response or the other. Nar is the number of active units in a source-set, which respond to the test stimulus, St-P(Nar > 0) is the prob- ability that at least one of the Ne effective units in the source-set of the correct response (designated, by con- vention, as the Ri response) will be activated by the test stimulus, St. In the case of Pg, the constant c2 is always equal to zero, the other three constants being the same as for Pr. The values of the four constants depend on the parameters of the physical nerve net (the perceptron) and also on the organization of the stimulus environment. The simplest cases to analyze are those in which the perceptron is shown stimuli drawn from an "ideal environ- ment," consisting of randomly placed points of illumination, where there is no attempt to classify stimuli accord- ing to intrinsic similarity. Thus, in a typical learning experiment, we might show the perceptron 1,000 stimuli made up of random collections of illuminated retinal points, and we might arbitrarily reinforce Ri as the "correct" response for the first 500 of these, and R.2 for the remaining 500. This environment is "ideal" only in the sense that we speak of an ideal gas in physics; it is a convenient artifact for purposes of analysis, and does not lead to the best performance from the perceptron. In the ideal environ- ment situation, the constant c\ is always equal to zero, so that, in the THE PERCEPTRON 397 case of Pg (where c2 is also zero), the value of Z will be zero, and Pg can never be any better than the random expectation of 0.5. The evaluation of Pr for these conditions, however, throws some interesting light on the differences between the alpha, beta, and gamma systems (Table 1). First consider the alpha system, which has the simplest dynamics of the three. In this system, whenever an A-unit is active for one unit of time, it gains one unit of value. We will assume an experiment, initially, in which N,r (the number of stimuli associated to each response) is con- stant for all responses. In this case, for the sum system, (5) where u> = the fraction of responses connected to each A-unit. If the source-sets are disjunct, w = I/NR, where NR is the number of responses in the system. For the ^-system, (6) The reduction of c3 to zero gives the ^-system a definite advantage over the S-system. Typical learning curves for these systems are compared in Fig. 7 and 8. Figure 9 shows the effect of variations in Pa upon the performance of the system. If n,r, instead of being fixed, is treated as a random variable, so that the number of stimuli associated to each response is drawn separately from some distribution, then the per- 100 1000 10,000 0,n (NUMBER OF STIMULI ASSOCIATED TO EACH RESPONSE) 100,000 FIG. 7. P,( 2) as function of an,, for discrete subsets. (a>« = 0, Pa = .005. Ideal environment assumed.) 398 F. ROSENBLATT 10,000 100,000 FIG. 8. Prdit as function of an,. (For Pa — .07, wc = 0. Ideal environment assumed.) formance of the a-system is consider- ably poorer than the above equations indicate. Under these conditions, the constants for the //-system are Ci — 0 C2 = 1 - Pa r MTB-I)« ]L 7Vfl-2 ^XJ C^ — 2(1 - Pa) (7) where q = ratio of 0,, to n,, NR = number of responses in the sys- tem NA = number of A-units in the sys- tem co,. = proportion of A-units common to RX and R2. For this equation (and any others in which n,r is treated as a random variable), it is necessary to define n,r in'. Equation 4 as the expected value off this variable, over the set of all responses. For the /3-system, there is an even greater deficit in performance, due to the fact that the net value continues to grow regardless of what happens to the system. The large net values of the subsets activated by a stimulus tend to amplify small statistical differ- ences, causing an unreliable perform- ance. The constants in this case (again for the /u-system) are ci = 0 c, = (1 - Pa)Ne c, = 2(PaNequNB) c« = 2(1 - (8) In both the alpha and beta systems, performance will be poorer for the sum-discriminating model than for the mean-discriminating case. In the gamma-system, however, it can be shown thatP,(s) = PK/O; i.e., it makes no difference in performance whether the S-system or ^-system is used. Moreover, the constants for the y- system, with variable nsr, are identical to the constants for the alpha jt-sys- THE PERCEPTRON 399 FIG. 9. P,tf) as function of P. (For n,T — 1,000, u, = 0. Ideal environment assumed.) tern, with nsr fixed (Equation 6). demonstrates the advantage of the The performance of the three systems 7-system. is compared in Fig. 10, which clearly Let us now replace the "ideal en- 10,000 FIG. 10. Comparison of a, /3, and 7 systems, for variable «», (NR = 100, <mr, = .5,,, NA = 10,000, Pa = ,07, w = ,2). 400 F. ROSENBLATT vironment" assumptions with a model for a "differentiated environment," in which several distinguishable classes of stimuli are present (such as squares, circles, and triangles, or the letters of the alphabet). If we then design an experiment in which the stimuli asso- ciated to each response are drawn from a different class, then the learning curves of the perceptron are drasti- cally altered. The most important difference is that the constant c\ (the coefficient of nsr in the numerator of Z) is no longer equal to zero, so that Equation 4 now has a nonrandom asymptote. Moreover, in the form for P, (the probability of correct generalization), where c% = 0, the quantity Z remains greater than zero, and Pa actually approaches the same asymptote as Pr. Thus the equation for the perceptron's performance after infinite experience with each class of stimuli is identical for PT and Pg : This means that in the limit it makes no difference whether the perceptron has seen a particular test stimulus before or not; if the stimuli are drawn from a differentiated environment, the perform- ance will be equally good in either case. In order to evaluate the perform- ance of the system in a differentiated environment, it is necessary to define the quantity PCap. This quantity is interpreted as the expected value of PC between pairs of stimuli drawn at random from classes a and j3. In particular, PcU is the expected value of Pc between members of the same class, and Peis is the expected value of PC between an Si stimulus drawn from Class 1 and an S2 stimulus drawn from Class 2. Pclx is the expected value of Pc between members of Class 1 and stimuli drawn at random from all other classes in the environment. If Pcll > Pa > PC12, the limiting performance of the perceptron (PBoo) will be better than chance, and learn- ing of some response, RI, as the proper "generalization response" for mem- bers of Class 1 should eventually occur. If the above inequality is not met, then improvement over chance performance may not occur, and the Class 2 response is likely to occur instead. It can be shown (IS) that for most simple geometrical forms, which we ordinarily regard as "simi- lar," the required inequality can be met, if the parameters of the system are properly chosen. The equation for Pr, for the sum- discriminating version of an a-percep- tron, in a differentiated environment where n,r is fixed for all responses, will have the following expressions for the four coefficients: C2=PJV,(1-P011) r=l,2 r=»l,2 where (10) ir) and a?(Pc\x) represent the variance of Pcir and Pc\x meas- ured over the set of possible test stimuli, St, and THE PERCEPTSON 401 iJ and crf(Pcix) represent the variance of Pcir and P^ meas- ured over the set of all A-units, e = covariance of PclrPclx, which is assumed to be negligible. The variances which appear in these expressions have not yielded, thus far, to a precise analysis, and can be treated as empirical variables to be determined for the classes of stimuli in question. If the sigma is set equal to half the expected value of the vari- able, in each case, a conservative estimate can be obtained. When the stimuli of a given class are all of the same shape, and uniformly distributed over the retina, the subscript s vari- ances are equal to zero. Paw will be represented by the same set of coeffi- cients, except for c2, which is equal to zero, as usual. For the mean-discriminating sys- tem, the coefficients are: \ — \Pcn~ PM) X[cr/(Prtr C4= -i p XTr— 1,2 -* a-t'e [_Pelr Pclr (ID Some covariance terms, which are considered negligible, have been omit- ted here. A set of typical learning curves for the differentiated environment model is shown in Fig. 11, for the mean- discriminating system. The param- eters are based on measurements for a 10,000 FIG. 11. P, and Pg as function of «,r. Parameters based on square-circle discrimination. 402 F. ROSENBLATT square-circle discrimination problem. Note that the curves for Pr and Pg both approach the same asymptotes, as predicted. The values of these asymptotes can be obtained by sub- stituting the proper coefficients in Equation 9. As the number of asso- ciation cells in the system increases, the asymptotic learning limit rapidly approaches unity, so that for a system of several thousand cells, the errors in performance should be negligible on a problem as simple as the one illus- trated here. As the number of responses in the system increases, the performance be- comes progressively poorer, if every response is made mutually exclusive of all alternatives. One method of avoiding this deterioration (described in detail in Rosenblatt, 15) is through the binary coding of responses. In this case, instead of representing 100 different stimulus patterns by 100 distinct, mutually exclusive responses, a limited number of discriminating features is found, each of which can be independently recognized as being present or absent, and consequently can be represented by a single pair of mutually exclusive responses. Given an ideal set of binary characteristics (such as dark, light; tall, short; straight, curved; etc.), 100 stimulus classes could be distinguished by the proper configuration of only seven response pairs. In a further modifica- tion of the system, a single response is capable of denoting by its activity or inactivity the presence or absence of each binary characteristic. The effi- ciency of such coding depends on the number of independently recognizable "earmarks" that can be found to differentiate stimuli. If the stimulus can be identified only in its entirety and is not amenable to such analysis, then ultimately a separate binary response pair, or bit, is required to denote the presence or absence of each stimulus class (e.g., "dog" or "not dog"), and nothing has been gained over a system where all responses are mutually exclusive. BIVALENT SYSTEMS In all of the systems analyzed up to this point, the increments of value gained by an active A-unit, as a result of reinforcement or experience, have always been positive, in the sense that an active unit has always gained in its power to activate the responses to which it is connected. In the gamma-system, it is true that some units lose value, but these are always the inactive units, the active ones gaining in proportion to their rate of activity. In a bivalent system, two types of reinforcement are possible (positive and negative), and an active unit may either gain or lose in value, depending on the momentary state of affairs in the system. If the positive and negative reinforcement can be controlled by the application of ex- ternal stimuli, they become essentially equivalent to "reward" and "punish- ment," and can be used in this sense by the experimenter. Under these conditions, a perceptron appears to be capable of trial-and-error learning. A bivalent system need not necessarily involve the application of reward and punishment, however. If a binary- coded response system is so organized that there is a single response or response-pair to represent each "bit," or stimulus characteristic that is learned, with positive feedback to its own source-set if the response is "on," and negative feedback (in the sense that active A-units will lose rather than gain in value) if the response is "off," then the system is still bivalent in its characteristics. Such a bivalent THE PERCEPTRON 403 system is particularly efficient in re- ducing some of the bias effects (prefer- ence for the wrong response due to greater size or frequency of its asso- ciated stimuli) which plague the alter- native systems. Several forms of bivalent systems have been considered (15, Chap. VII). The most efficient of these has the following logical characteristics. If the system is under a state of positive reinforcement, then a positive AV is added to the values of all active A-units in the source-sets of "on" responses, while a negative AV is added to the active units in the source- sets of "off" responses. If the system is currently under negative reinforce- ment, then a negative AV is added to all active units in the source-set of an "on" response, and a positive AV is added to active units in an "off" source-set. If the source-sets are disjunct (which is essential for this system to work properly), the equa- tion for a bivalent -y-system has the same coefficients as the monovalent a-system, for the /j-case (Equation 11). The performance curves for this system are shown in Fig. 12, where the asymptotic generalization probability attainable by the system is plotted for the same stimulus parameters that were used in Fig. 11. This is the probability that all bits in an «-bit response pattern will be correct. Clearly, if a majority of correct re- sponses is sufficient to identify a stim- ulus correctly, the performance will be better than these curves indicate. In a form of bivalent system which utilizes more plausible biological as- sumptions, A-units may be either excitatory or inhibitory in their effect on connected responses. A positive AV in this system corresponds to the incrementing of an excitatory unit, while a negative AV corresponds to the incrementing of an inhibitory unit. Such a system performs similarly to the one considered above, but can be shown to be less efficient. Bivalent systems similar to those illustrated in Fig. 12 have been simulated in detail in a series of ex- periments with the IBM 704 computer at the Cornell Aeronautical Lab- oratory. The results have borne out the theory in all of its main predic- tions, and will be reported separately at a later time. IMPROVED PERCEPTRONS AND SPONTANEOUS ORGANIZATION The quantitative analysis of per- ceptron performance in the preceding sections has omitted any consideration of time as a stimulus dimension. A perceptron which has no capability for temporal pattern recognition is referred to as a "momentary stimulus perceptron." It can be shown (15) that the same principles of statistical separability will permit the perceptron to distinguish velocities, sound se- quences, etc., provided the stimuli leave some temporarily persistent trace, such as an altered threshold, 2000 woo 6000 FIG. 12. Pga for a bivalent binary system (same parameters as Fig. 11). 404 F. ROSENBLATT which causes the activity in the A- system at time t to depend to some degree on the activity at time t — 1. It has also been assumed that the origin points of A-units are completely random. It can be shown that by a suitable organization of origin points, in which the spatial distribution is constrained (as in the projection area origins shown in Fig. 1), the A-units will become particularly sensitive to the location of contours, and perform- ance will be improved. In a recent development, which we hope to report in detail in the near future, it has been proven that if the values of the A-units are allowed to decay at a rate proportional to their magnitude, a striking new property emerges: the perceptron becomes cap- able of "spontaneous" concept forma- tion. That is to say, if the system is exposed to a random series of stimuli from two "dissimilar" classes, and all of its responses are automatically rein- forced without any regard to whether they are "right" or "wrong," the system will tend towards a stable terminal condition in which (for each binary response) the response will be "1" for members of one stimulus class, and "0" for members of the other class; i.e., the perceptron will spon- taneously recognize the difference between the two classes. This phe- nomenon has been successfully dem- onstrated in simulation experiments, with the 704 computer. A perceptron, even with a single logical level of A-units and response units, can be shown to have a number of interesting properties in the field of selective recall and selective attention. These properties generally depend on the intersection of the source sets for different responses, and are elsewhere discussed in detail (IS). By com- bining audio and photo inputs, it is possible to associate sounds, or audi- tory "names" to visual objects, and to get the perceptron to perform such selective responses as are designated by the command "Name the object on the left," or "Name the color of this stimulus." The question may well be raised at this point of where the perceptron's capabilities actually stop. We have seen that the system described is suffi- cient for pattern recognition, associa- tive learning, and such cognitive sets as are necessary for selective attention and selective recall. The system ap- pears to be potentially capable of temporal pattern recognition, as well as spatial recognition, involving any sensory modality or combination of modalities. It can be shown that with proper reinforcement it will be capable of trial-and-error learning, and can learn to emit ordered se- quences of responses, provided its own responses are fed back through sensory channels. Does this mean that the perceptron is capable, without further modification in principle, of such higher order func- tions as are involved in human speech, communication, and thinking? Ac- tually, the limit of the perceptron's capabilities seems to lie in the area of relative judgment, and the abstraction of relationships. In its "symbolic be- havior," the perceptron shows some striking similarities to Goldstein's brain-damaged patients (5). Re- sponses to definite, concrete stimuli can be learned, even when the proper response calls for the recognition of a number of simultaneous qualifying conditions (such as naming the color if the stimulus is on the left, the shape if it is on the right). As soon as the response calls for the recognition of a relationship between stimuli (such as "Name the object left of the square." or "Indicate the pattern that appeared before the circle."), however, the THE PESCEPTRON 405 problem generally becomes excessively difficult for the perceptron. Statis- tical separability alone does not provide a sufficient basis for higher order abstraction. Some system, more advanced in principle than the perceptron, seems to be required at this point. CONCLUSIONS AND EVALUATION The main conclusions of the theo- retical study of the perceptron can be summarized as follows: 1. In an environment of random stimuli, a system consisting of ran- domly connected units, subject to the parametric constraints discussed above, can learn to associate specific responses to specific stimuli. Even if many stimuli are associated to each response, they can still be recognized with a better-than-chance probability, although they may resemble one an- other closely and may activate many of the same sensory inputs to the system. 2. In such an "ideal environment," the probability of a correct response diminishes towards its original ran- dom level as the number of stimuli learned increases. 3. In such an environment, no basis for generalization exists. 4. In a "differentiated environ- ment," where each response is asso- ciated to a distinct class of mutually correlated, or "similar" stimuli, the probability that a learned association of some specific stimulus will be cor- rectly retained typically approaches a better-than-chance asymptote as the number of stimuli learned by the system increases. This asymptote can be made arbitrarily close to unity by increasing the number of associa- tion cells in the system. 5. In the differentiated environ- ment, the probability that a stimulus which has not been seen before will be correctly recognized and associated to its appropriate class (the probability of correct generalization) approaches the same asymptote as the probability of a correct response to a previously reinforced stimulus. This asymptote will be better than chance if the in- equality Pci2 < Pa < Pen is met, for the stimulus classes in question. 6. The performance of the system can be improved by the use of a con- tour-sensitive projection area, and by the use of a binary response system, in which each response, or "bit," corresponds to some independent fea- ture or attribute of the stimulus. 7. Trial-and-error learning is possi- ble in bivalent reinforcement systems. 8. Temporal organizations of both stimulus patterns and responses can be learned by a system which uses only an extension of the original prin- ciples of statistical separability, with- out introducing any major complica- tions in the organization of the system. 9. The memory of the perceptron is distributed, in the sense that any association may make use of a large proportion of the cells in the system, and the removal of a portion of the association system would not have an appreciable effect on the performance of any one discrimination or associa- tion, but would begin to show up as a general deficit in all learned asso- ciations. 10. Simple cognitive sets, selective recall, and spontaneous recognition of the classes present in a given en- vironment are possible. The recogni- tion of relationships in space and time, however, seems to represent a limit to the perceptron's ability to form cog- nitive abstractions. Psychologists, and learning theorists in particular, may now ask: "What has the present theory accomplished, 406 F. ROSENBLATT beyond what has already been done in the quantitative theories of Hull, Bush and Mosteller, etc., or physio- logical theories such as Hebb's?" The present theory is still too primitive, of course, to be considered as a full- fledged rival of existing theories of human learning. Nonetheless, as a first approximation, its chief accom- plishment might be stated as follows: For a given mode of organization (a, /3, or 7; S or n; monovalent or bivalent) the fundamental phenomena of learning, perceptual discrimination, and generalization can be predicted en- tirely from six basic physical param- eters, namely: x: the number of excitatory connec- tions per A-unit, y: the number of inhibitory connec- tions per A-unit, 6: the expected threshold of an A- unit, co: the proportion of R-units to which an A-unit is connected, NA: the number of A-units in the system, and NR-. the number of R-units in the system. Ns (the number of sensory units) be- comes important if it is very small. It is assumed that the system begins with all units in a uniform state of value; otherwise the initial value dis- tribution would also be required. Each of the above parameters is a clearly defined physical variable, which is meas- urable in its own right, independently of the behavioral and perceptual phe- nomena which we are trying to predict. As a direct consequence of its foun- dation on physical variables, the present system goes far beyond exist- ing learning and behavior theories in three main points: parsimony, veri- fiability, and explanatory power and generality. Let us consider each of these points in turn. 1. Parsimony. Essentially all of the basic variables and laws used in this system are already present in the structure of physical and biological science, so that we have found it necessary to postulate only one hy- pothetical variable (or construct) which we have called V, the "value" of an association cell; this is a variable which must conform to certain func- tional characteristics which can clearly be stated, and which is assumed to have a potentially measurable physical correlate. 2. Verifiability. Previous quanti- tative learning theories, apparently without exception, have had one im- portant characteristic in common: they have all been based on measure- ments of behavior, in specified situa- tions, using these measurements (after theoretical manipulation) to predict behavior in other situations. Such a procedure, in the last analysis, amounts to a process of curve fitting and extrapolation, in the hope that the constants which describe one set of curves will hold good for other curves in other situations. While such extrapolation is not necessarily circular, in the strict sense, it shares many of the logical difficulties of circu- larity, particularly when used as an "explanation" of behavior. Such ex- trapolation is difficult to justify in a new situation, and it has been shown that if the basic constants and param- eters are to be derived anew for any situation in which they break down empirically (such as change from white rats to humans), then the basic "theory" is essentially irrefutable, just as any successful curve-fitting equa- tion is irrefutable. It has, in fact, been widely conceded by psychologists that there is little point in trying to "disprove" any of the major learning theories in use today, since by exten- sion, or a change in parameters, they THE PERCEPTRON 407 have all proved capable of adapting to any specific empirical data. This is epitomized in the increasingly com- mon attitude that a choice of theo- retical model is mostly a matter of personal aesthetic preference or pre- judice, each scientist being entitled to a favorite model of his own. In con- sidering this approach, one is reminded of a remark attributed to Kistiakow- sky, that "given seven parameters, I could fit an elephant." This is clearly not the case with a system in which the independent variables, or param- eters, can be measured independently of the predicted behavior. In such a system, it is not possible to "force" a fit to empirical data, if the param- eters in current use should lead to improper results. In the current theory, a failure to fit a curve in a new situation would be a clear indication that either the theory or the empirical measurements are wrong. Conse- quently, if such a theory does hold up for repeated tests, we can be consider- ably more confident of its validity and of its generality than in the case of a theory which must be hand-tailored to meet each situation. 3. Explanatory power and generality. The present theory, being derived from basic physical variables, is not specific to any one organism or learn- ing situation. It can be generalized in principle to cover any form of be- havior in any system for which the physical parameters are known. A theory of learning, constructed on these foundations, should be consider- ably more powerful than any which has previously been proposed. It would not only tell us what behavior might occur in any known organism, but would permit the synthesis of behaving systems, to meet special requirements. Other learning theo- ries tend to become increasingly qualitative as they are generalized. Thus a set of equations describing the effects of reward on T-maze learning in a white rat reduces simply to a statement that rewarded behavior tends to occur with increasing prob- ability, when we attempt to generalize it from any species and any situation. The theory which has been presented here loses none of its precision through generality. The theory proposed by Donald Hebb (7) attempts to avoid these difficulties of behavior-based models by showing how psychological func- tioning might be derived from neuro- physiological theory. In his attempt to achieve this, Hebb's philosophy of approach seems close to our own, and his work has been a source of inspira- tion for much of what has been pro- posed here. Hebb, however, has never actually achieved a model by which behavior (or any psychological data) can be predicted from the physio- logical system. His physiology is more a suggestion as to the sort of organic substrate which might under- lie behavior, and an attempt to show the plausibility of a bridge between biophysics and psychology. The present theory represents the first actual completion of such a bridge. Through the use of the equations in the preceding sections, it is possible to predict learning curves from neurological variables, and like- wise, to predict neurological variables from learning curves. How well this bridge stands up to repeated crossings remains to be seen. In the meantime, the theory reported here clearly dem- onstrates the feasibility and fruitful- ness of a quantitative statistical ap- proach to the organization of cognitive systems. By the study of systems such as the perceptron, it is hoped that those fundamental laws of organ- ization which are common to all information handling systems, ma- 408 F. ROSENBLATT chines and men included, may even- tually be understood. REFERENCES 1. ASHBY, W. R. Design for a brain. New York: Wiley, 1952. 2. CULBERTSON, J. T. Consciousness and be- havior. Dubuque, Iowa: Wm. C. Brown, 1950. 3. CULBERTSON, J. T. Some uneconomical robots. In C. E. Shannon & J. Mc- Carthy (Eds.), Automata studies. Princeton: Princeton Univer. Press, 1956. Pp. 99-116. 4. ECCLES, J. C. The neurophysiological basis of mind. Oxford: Clarendon, 1953. 5. GOLDSTEIN, K. Human nature in the light of psychopathology. Cambridge: Harvard Univer. Press, 1940. 6. HAYEK, F. A. The sensory order. Chi- cago: Univer. Chicago Press, 1952. 7. HEBB, D. O. The organization of be- havior. New York: Wiley, 1949. 8. KLEENE, S. C. Representation of events in nerve nets and finite automata. In C. E. Shannon & J. McCarthy (Eds.), Automata studies. Princeton: Prince- ton Univer. Press, 1956. Pp. 3-41. 9. KOHLER, W. Relational determination in perception. In L. A. Jeffress (Ed.), Cerebral mechanisms in behavior. New York: Wiley, 1951. Pp. 200-243. 10. McCuLLOCH, W. S. Why the mind is in the head. In L. A. Jeffress (Ed.), Cerebral mechanisms in behavior. New York: Wiley, 1951. Pp. 42-111. 11. MCCULLOCH, W. S., & PITTS, W. A logical calculus of the ideas immanent in nervous activity. Butt. math. Bio- physics, 1943, S, 115-133. 12. MILNER, P. M. The cell assembly: Mark II. Psychol. Rev., 1957,64,242- 252. 13. MINSKY, M. L. Some universal elements for finite automata. In C. E. Shannon & J. McCarthy (Eds.), Automata studies. Princeton: Princeton Univer. Press, 1956. Pp. 117-128. 14. RASHEVSKY, N. Mathematical biophysics. Chicago: Univer. Chicago Press, 1938. 15. ROSENBLATT, F. The perceptron: A theory of statistical separability in cognitive systems. Buffalo: Cornell Aeronautical Laboratory, Inc. Rep. No. VG-1196-G-1, 1958. 16. UTTLEY, A. M. Conditional probability machines and conditioned reflexes. In C. E. Shannon & J. McCarthy (Eds.), Automata studies. Princeton: Princeton Univer. Press, 1956. Pp. 253-275. 17. VON NEUMANN, J. The general and logical theory of automata. In L. A. Jeffress (Ed.), Cerebral mechanisms in behavior. New York: Wiley, 1951. Pp. 1-41. 18. VON NEUMANN, J. Probabilistic logics and the synthesis of reliable organisms from unreliable components. In C. E. Shannon & J. McCarthy (Eds.), Automata studies. Princeton: Prince- ton Univer. Press, 1956. Pp. 43-98. (Received April 23, 1958)
Vector_Space_Model_of_Information_Retrieval_-_A_Re	See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/221300847 Vector Space Model of Information Retrieval - A Reevaluation. Conference Paper · January 1984 Source: DBLP CITATIONS 101 READS 6,877 2 authors, including: Vijay v Raghavan University of Louisiana at Lafayette 335 PUBLICATIONS 6,254 CITATIONS SEE PROFILE All content following this page was uploaded by Vijay v Raghavan on 12 January 2014. The user has requested enhancement of the downloaded file. VECTOR SPACE MODEL OF INFORMATION RETRIEVAL - A REEVALUATION* S.K.M. Wong Department of Computer Science, University of Regina, Regina, Sask. Canada, S4S 0A2 Vijay V. Raghavan + Department of Computer Science, University of Regina, Regina, Sask. Canada, S4S 0A2 Abstract. In this paper we, in essence, point out that the methods used in the current vector based systems are in conflict with the premises of the vector space model. The considerations, naturally, lead to how things might have been done differently. More importantly, it is felt that this investigation will lead to a clearer understanding of the issues and problems in using the vector space model in information retrieval. 1. INTRODUCTION Information Storage and Retrieval (ISR) is a discipline involved with the organization, structuring, retrieval and display of bibliographic information. ISR systems are designed with the objective of providing, in response to a user query, references to documents which would contain the information desired by the user. A typical application for a computerized ISR system is in a library environment where the database consists of books, journals, etc. It is common in information retrieval to represent each document by means of keywords or index terms. When a user submits a request, which is also specified in terms of the keywords, the request is compared with the document * This research was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. + This author is on leave from Univ. of Regina and is currently with Institut fur Informatik, TU Berlin, Franklinstr. 28/29, Sekr. FR 5-8, i000 Berlin i0, FRG. Wong & Raghaven: The vector space model 168 representations to determine which of the documents should be retrieved. In essence, then, the system must retrieve references that are of value, or relevant, to the user's request. A document may or may not be relevant to a user query depending on many variables concerning the document (what it is about, how is it organized, is it clear, etc.) as well as on numerous user characteristics (the reason for search, previous knowledge, does the user know what he wants, etc.). Since relevance depends in a complex way on many factors, it is recognized that an ISR system cannot precisely select only and all relevant documents. It has, therefore, been suggested that a retrieval system should attempt to rank documents in the order of their potential relevance to a user query. One approach which has been widely used over the years to provide such a ranking models documents and queries as vectors (Salton 1971; van Rijsbergen 1979; Salton & McGill 1983). The keywords used to describe the contents of documents or queries are assumed to correspond to the various elements of the vectors. Thus, if the indexing vocabulary consists of n distinct keywords, each document is an n- element vector in which the i th element represents the importance of the i th keyword to the document concerned. When a query is presented, the system formulates the query vector and matches against the documents based on a chosen method of determining similarity between vectors. For example, similarity between the query and a document may be defined as the scalar product of the corresponding vectors and the documents could be ranked in the decreasing order of this measure. 2. MOTIVATION It is clear that the notions presented above are completely informal. Formal notions from linear algebra such as basis vectors, linear independence or dependence, and orthogonality are carefully avoided. Even the question of whether we have a vector space, that is, are the axioms to be obeyed by the elements of a vector space appropriate for Wong & Raghaven: The vector space model 169 information retrieval, is not considered. In fact, it seems that in the early literature a conscious effort was made to not make any direct connection to vector spaces. Instead, a vector meant a tuple or what, in programming languages, is considered as a one-dimensional array of certain size. Thus, the notion of vector, considered above merely refers to data structure; some sort of a physical or even simply a notational aspect. Similarly, the scalar product or some other similar similarity function is simply an operation defined on the data structure. The logical model was usually quite different. For example, information retrieval objects and processes have been modelled in set theoretic terms, as well as in statistical terms involving notions such as random variables and density function. The main point here is that the concept of a vector was not intended to be a logical or a formal tool. In fact, the earliest reference we have come across to the idea of vector spaces in information retrieval literature was in Salton et al. (1975). In this work a brief mention is made to the possibility of viewing index terms as corresponding to various dimensions of a space, and the documents as vectors in such a space. But subsequent developments in that paper do not really depend on the modelling of information retrieval objects as vectors in a vector space. Moreover, in several other papers, the term vector processing model is used, and we presume by conscious choice, instead of the term vector space model (Salton 1980; Salton et al. 1983). Given the "vector" model as outlined above, all things are fine and dandy and one felt quite content. However, certain recent developments have aroused our curiosity to ask whether in the information retrieval context one should take the vector space model seriously. Salton and McGill (1983), van Rijsbergen (1979), and Koll (1979) make specific mention of vectors in a multidimensional vector space. Van Rijsbergen (p.41) infers that Salton considers document representatives as binary vectors embedded in an n- dimensional Euclidean space. Koll observes that in the SMART system environment a vector space, in which the various dimensions correspond to the different index terms in the Wong & Raghaven: The vector space model 170 vocabulary and where the terms are mutually orthogonal, is assumed. Salton and McGill (p.129-130) discuss these ideas further and point out that the assumptions involved are only first order approximations to the true situation. First, it is felt that the vector view does not adequately capture the notion of scope. Secondly, treating each index term as a separate coordinate and assuming the terms as being orthogonal, is deemed contrary to the reality where term relationships exist and index terms are not assigned independently of each other. These statements warrant careful scrutiny. The issue, at the outset, is not so much that it is not clear what precisely these statements mean. Rather it is which of the two notions of vectors we wish to adopt in information retrieval. On the one hand, we can accept the easier of the two answers: that all along the notion of vector was used only in the sense of a tuple or an array. This represents the easier answer since it appears to be consistent with most of the traditional and fairly well accepted practices in our field. It would mean, though, that any of the references that have been made to notions such as vector spaces, n-dimensional space, and orthogonality should be disregarded as casual flirtings and be not taken seriously. On the other hand, we may assert that all along the notion of vectors was intended as a logical construct and that information retrieval objects and processes can be understood in the context of vector spaces. If this had been the case, we should be able to demonstrate that the traditional practices and interpretations are either consistent with or reasonable approximations of what is correct under the vector space model. Unfortunately, we find that earlier work in information retrieval is, for the most part, not consistent with the vector space model. In this paper we, in essence, point out the ways the traditional approaches are in conflict with the premises of the vector space model. The considerations, naturally, lead to how things might have been done differently. More importantly, it is felt that this investigation will lead to a clear understanding of the issues and problems in using the Wong & Raghaven: The vector space model 171 vector space model in information retrieval. In addition to the new insight one gains about the modelling of information retrieval objects, their relationships and processes, the current work is also significant in that it lays the groundwork for a model that is reminiscent of that used in the WEIRD system by Koll (1979). More specifically, both terms and documents are represented by being a combination (or "mean" location) of term vectors or concepts that they contain. Similarly, terms may be viewed as a combination of documents or concepts. It is also possible to investigate the problem of dimensionality and identify a (vector) space of fewer dimensions than the number of distinct index terms. 3. THE VECTOR SPACE MODEL The basic premise of adopting the vector space model is that the various information retrieval objects are modelled as elements of a vector space. Specifically terms, documents, queries, concepts, and so on are all vectors in the vector space. The existence of a vector space implies that we have a system with the linear properties: the ability to add together any two elements of the system to obtain a new element of the system and the ability to multiply any element of the system by a real number. Furthermore, the vectors obey a number of basic algebraic rules or axioms (e.g. x + ~ = y + x, for any vectors x, ~). Note that a letter with underscore denotes a vector. Let us first consider the issue of representation of documents in terms of the index terms. Let tl, t2, ... t n be the terms used to represent documents. Corresponding to each term, ti, there exists a vector ~i in the space. Without loss of generality, it is assumed that ~i's are vectors of unit length. Now, suppose that each document Dr, l~r~m, is a vector expressed in terms of ~i's. Let the document vector ~r be ~r = (alr, a2r, "-" anr)" Since it is sufficient to restrict our scope of discussion to Wong & Raghaven: The vector space model 172 the subspace spanned by the term vectors, the ~i's can be thought to be the generating set. Every vector in this subspace, and in particular all document vectors, are linear combinations of the term vectors. Thus, _D r can be, equivalently, expressed as n D r = ~ airti i=l (i) The coefficients air , for l<i<n and l<r<m, are the components of Dr along the ~i's. We next introduce one of the most important concepts in vector spaces, that of linear dependence. A set of vectors ZI, Z2, "'" [k are linearly dependent if we find some scalars al, a2, ... ak, not all zero, such that al_Y 1 + a2Y 2 + .... akY k = 0 . Using several known theorems in linear algebra (Goult 1978), it can be seen that (i) {tl, t2, ... tn} being the generating set for our space implies that any set of linearly independent vectors in this space contains at most n vectors, (ii) because a basis is a generating set consisting of linearly independent vectors, any basis of this space has at most n vectors and, hence, the dimension is at most n, (iii) it is always possible to obtain a basis from a finite generating set by eliminating vectors dependent upon other s, (iv) given a basis, {tl, t2, ... tn-}, for n'<n, any vector x in the space has a unique expression of the form: n x = ~ citi, i=l (v) if {tl, t2, ... tn. } is a basis of our space, then any n" linearly independent vectors will form a basis, and III Wong & Raghaven: The vector space model 173 the dimension of the subspace is n'. Thus, not only can documents be expressed as a linear combination of terms, but also terms as a linear combination of documents. The latter is true, of course, assuming there exists the necessary number of linearly independent documents. Notationally, if {~i, ~2, "'" ~n'} is a basis, each term, ~i, has an expression of the form n t i = ~ bri D r , r=l (i = 1,2,...n) (2) Clearly, we can also have documents expressed as a linear combination of a basis consisting of only documents. In fact, a basis could be made up of documents and terms mixed together because both of them are elements in the vector space. Another important concept in this context is that of a scalar product. Given a vector space, V, by the scalar product x.y of two vectors x, y~V, we refer to the quantity Ixl IY[ cos ~, where Ix I and IYI are the lengths of the two vectors and G is the angle between x and y. It is easy to verify the usually required properties which specify how a scalar product interacts with the operations of addition, and multiplication by scalar, mentioned earlier (Goult 1978). A vector space equipped with a scalar product is called a Euclidean space. The following definitions involving scalar products are well known: ! (i) Ixl = (x . x) ~ , (ii) any vector x~0 can be normalized; i.e. replaced by a proportional vector of unit length given by x / Ixl , (iii) (x/Ixl).y is the projection of vector y onto the vector x , (iv) vectors x and y in a Euclidean space are orthogonal if x.y=O , (v) a basis such that the vectors are mutually orthogonal and each vector is normalized is called an orthonormal basis. Wong & Raghaven: The vector space model 174 4. IMPORTANT CONCEPTS AND THEIR RELEVANCE TO EARLIER WORK IN INFORMATION RETRIEVAL For reasons of clarity, in this section, it is assumed that the number of terms is equal to the dimension of the subspace of interest, and that the number of documents are exactly the same as the number of terms, i.e. n'=n=m. Recall, also, that the term vectors ~l,~2,"'~n are normalized. Furthermore, we assume the set of documents as well as the set of terms form a basis. 4.1 Computation of Correlations From eqn. (i), we have n D r = ~ airti i=l , (r = 1,2,...n) . (3) For any query q, the corresponding query vector has the expression _q = ~ qiti • i=l In the general case, the scalar product, which we suppose is the measure of correlation between two vectors, of D and q is n Dr.q = ~ airqjti.t j . (4) i,j=l 4.2 Projections vs. Components Next we consider important relationships between components, projections, and the scalar products (vector correlations). By multiplying eqn. (3) by tj, (j=l,2,...n), on both sides, we obtain a system of linear equations: Wong & Raghaven: The vector space model 175 tj.D r = ~ airtj.ti , (j,r = 1,2,...n). (5) i=l Since ~j's are unit vectors, the scalar product ~j'~r is the projections of ~r onto ~i" Eqn. (5) can be rewritten in a matrix form as follows: P = GtA , (6) where (P) jr = ~j'~r ' (Gt)ji = tj.t i , and (A) ir = air • That is, G t is the matrix of correlations between term vectors, and the r th column of A represents the components of ~r along the vector ~i's. Example I. Consider a vector space with dimension n=2. In Fig.l t I and t 2 represent the term basic vectors, and DI, D 2 the document basic vectors. Figure i. Two Dimensional Vector Space with ti's as Basis. .~/ alr-t! .. ~' '~ • ~. t_l"D r > > h Wong & Raghaven: The vector space model 176 As in eqn. (3), each document vector D r can be expressed as D r = alrtl + a2rt 2 , (r = 1,2). The projection matrix P (defined in eqn. (6)) is given by p = tl.D tl.R l t2.D 1 t2.D 2 It tl" (alltl+a21t2) t I- (a12tl+a22t 2) 2" (alltl+a21t2) t2" (a12-~l+a22-~2) = [ tl'tl tl't2 t2-_t I t2.t 2 all a21 al21 a221 = GtA If we multiply eqn. (3) by Ds, (r = 1,2,...n), on both sides, we obtain Ds'Dr = Z i=l airDs-ti , (r,s = 1,2,...n), which can be rewritten as G d = P'A , (7) where (Gd)sr = Ds.D r is the matrix of document correlations, and P" is the transpose of P. Similarly starting with eqn. (2), multiplying both sides by Ds,(S = 1,2,...n), and tj, (j = 1,2,...n), respectively, we obtain the following matrix equations: P" = GdB , (8) G t = PB , (9) where (B)r i = bri . The i th column of B represents the Wong & Raghaven: The vector space model 177 components of ~i along the directions of the various Dr'S. Example 2. Consider a two dimensional vector space as in Example I. In this case the term vector ~i is expressed as a linear combination of the document basic vectors ~i and ~2" Figure 2. Two Dimensional Vector Space with Dr-S as Basis. D -I / 2i-D2 I Pt ' -..., / '< DI t -> D 1 -I 4.3 Important Implications Within the framework presented here, the assertions listed below will be established. This list is not intended to be exhaustive. In what follows, we let ~ denote the term- document matrix obtained from empirical data. That is, ~is the matrix such that (~)ir = dir, where dir is the occurrence frequency of term i in the document r. (i) The model is usable in the general form as seen from the set of equations (6),(7),(8), and (9), if either (a) one of the correlation matrices (G t or Gd) and one of A,B, or P are known, or (b) matrix P and one of A or B are known. (ii) Recall from eqns. (6) and (9) that Gt=PA -I and Gt=PB. Therefore, in general, B=A -I. Since B=(Gd)-IP ", the correlation matrices G t and G d are related to each other as follows: Gt = p (Gd)-ip- Wong & Raghaven: The vector space model 178 (iii) For the purpose of ranking documents against a query q, it is clear from eqn. (4) that A and G t must be known. We can, in fact, represent the scalar product Dr.~, for r = 1,2,...n, as a vector Rq = (DI. q, D2. q, ... Dn.q), which can be written as D I -q • f • i I • l On-5t ] all a12 ... aln i . J I" " [ I " " 1 I ; • a 1 an2 ''' ann tl-t I tl-t 2 ... tn.t n • B tn.t I tn.t 2 ... tn.t n qll I qn = A'Gt_ q" , (i0) p where Rq and ~" (column vectors) denote the transpose of Rq and ~ respectively. Since G t is a symmetric matrix and P=GtA, eqn. (i0) is equivalent to Rq = q(GtA ) = qP . (ii) If we assume that the term occurrence frequency dir represents the projection of the document vector ~r onto the term ~i (i.e. ~ =P), then eqn. (Ii) completely specifies the ranking of the documents with respect to the query ~ as follows: n n (Rq) r = ~ qj (D)jr = ~ qjdjr , j =i j=l (r = 1,2,...n). (12) It is important to note that there is no assumption made on term independence in the derivation of eqn. (12). Term correlations are implicitly included in the term occurrence frequencies dir'S by the assumption that ~=P. This fact may explain why eqn. (12) works quite Wong & Raghaven: The vector space model 179 well for document ranking in the traditional vector model. The advantage of interpreting ~=P is that no explicit knowledge of term correlations (i.e. Gt) is required in eqn. (12) for computing correlations between the query and the documents. However, it is important to note that qj's are the components (not projections) of ~ along the ~i's, and the matrix elements of A,B,Gt, or G d are not known in this case. (iv) The special case usually mentioned in the literature is obtained when Gt=I , i.e. the term vectors are assumed to be orthogonal and normalized, and~ =A. This means that the term occurrence frequency dir is interpreted as the component of the document vector D r along t i. By eqn. (6), P=A which implies that the components of each D r are identical to its projections onto ti's. But A ~B'! For this special case, eqn. (ii) reduces to the well known form Rq = ~A = ~P = ~ , or n n (Rq) r = ~ qjajr j=l = > qjdjr j=l (13) (v) It is interesting to note that eqns. (12) and (13) give identical values for (Rq) r. However, contrary to case (iii), eqn. (13) is obtained by assuming explicitly that t i s are orthogonal vectors. However, no such assumption is made in arriving at eqn. (12). Note that from eqn. (7) the matrix of document correlations is given by G d = P'A = ~/~ . (14) A function commonly used to measure the similariy between term i and term j is Wong & Raghaven: The vector space model 180 n dirdjr (15) r=l Clearly this function is akin to the well known term co- occurrence. Within the present framework, from eqn. (2) correlation between term vectors, ~i and ~j, can be expressed as n ti.t j = ~ bribsjDr.D s . (16) r,s=l If we assume that there exists no correlation between any pair of document vectors (i.e. Gd=I) , then eqn. (16) becomes n ti.t j = ~ bribrj - r=l (17) From eqn. (8), the fact that Gd=I implies that P=B'. Hence, eqn. (17) can be rewritten as n n ti'tj = 7_ (B')ir(B)rj = ~ (P) ir(P')rj ' (18) r=l r=l or G t = PP" (19) If we further assume that ~ =P, term correlations can be computed directly from term co-occurrence frequencies as follows: Gt c20 or Wong & Raghaven: The vector space model 181 n ti'tj = Z dirdj r " r=l Therefore, the interpretation of term co-occurrence frequency as term correlation has meaning in the vector space model only when document vectors are assumed to be orthnormal, and the term occurrence frequency dir is taken to be the projection of ~r along the term ~i" (vi) We may also assume that Gt=Gd=I and ~ =P. Obviously from eqns. (6) and (8), we obtain P=A=B'. Since Gt=I, term co-occurence frequencies can no longer be interpreted as term correlations. The function defined by eqn. (15) is simply the scalar product, ti.tj, evaluated with respect to the document co-ordinates frame of reference. (vii) In view of the discussions presented in (iii), (iv), (v), and (vi), it is clear that given ~ a decision has to be made as to what meaning to be attached to it. The interpretation that ~ =P seems to be a reasonable one. If~ is assumed to be equal to A, the vector space model is not usable unless additional information is available or appropriate assumptions are made. However, if the matrix G t is known, eqn. (4) shows precisely how term correlations should be incorporated into the retrieval strategy. (viii) In current vector based models, all vector elements are conveniently assumed to be positive. Within the present framework, there is no reason to believe that all the matrix elements of A,B,P,G t or G d are necessarily positive numbers. In fact, both negative and positive vector elements are appropriate and necessary as can be seen from the following example. Consider a two dimensional vector space shown below: Wong& Raghaven:Thevectorspace model Figure 3. Negative Components in a Two Dimensional Vector Space. 182 a12 " - - f / ~ / I i I I a22/ / a21 / D I .>// / /.. / i f f / la - i 11 f The document vectors, ~i and ~2' are expressed as a linear combination of the basic term vectors, ~i and ~2: D 1 = all_t I + a21_t 2 , ~2 = a12~l + a22~2 • It can be seen from Fig. 3 that the component a21 of D 1 along t 2 and the component a12 of D 2 along t I are negative numbers. However, all the projections of the document vectors onto t I and t 2 are positive in this particular example. In this context, it is possible in a general case that term vectors may be negatively correlated. It may, therefore, be necessary in query processing to assume negative components for the query vector q. Based on the argument presented here, the need to introduce negative term (document) correlations in the vector space model is apparent. It is also intuitively clear that two index terms may have "opposite" semantic meaning. We believe that the existing vector model fails to take into consideration Wong & Raghaven: The vector space model 183 this important difference between index terms. 5. FURTHER ISSUES PERTAINING TO THE GENERAL VECTOR SPACE MODEL At this point it may be concluded that the vector space model is inappropriate for information retrieval. Alternatively, one may investigate further to see what the issues and challenges are if one decided to model information retrieval by means of the vector space model in a more rigorous sense. In the latter direction, two important issues will be addressed: (i) If Y~J can only represent A,B, or P, how might the additional information needed to fully specify the system be obtained. Based on the discussions in Section 4.3, the interpretation of the term occurrence frequency dir as the projection of ~r onto ~i (i.e.~ =P) seems to be a plausible one. In our view, the assumption that ~=P is, in fact, consistent with the well established practices in the vector based systems, provided that the document vectors, Dr'S, are assumed to be orthonormal (i.e. Gd=I). We believe that to find a method for choosing an appropriate orthonormal basis for the document vectors is a crucial step in the development of a rigorous vector space model within the context of information retrieval. This task is currently under investigation by the authors of this paper. (ii) Let {tl, t2,.., tn} be a generating set of the term vector space. In order to have a unique expansion based on this set of ti's for any vector in the vector space, a basis (i.e. a maximal subset of linearly independent term vectors) must be identified. Of course, this would be a trivial task if the term vectors were assumed to be orthogonal, because mutual orthogonality between vectors in a set implies linear independence. In contrast linear independence only implies that any redundancy in the usage of terms has been removed and the representation in terms of the resulting vectors is Wong & Raghaven: The vector space model 184 compact (unique). Thus under non-orthogonality, correlation and dependence are rather distinct notions of term "relationship". The dimension of the space, clearly, ties in with these notions. In general, when the generating set of a vector space is not orthogonal, the task of identifying a basis may not be as trivial as it seems. First of all the issue of linear independence among an arbitrary set of vectors can be resolved only if correlation between any pair of the vectors is known. The approach we have suggested in (i) of this section may offer a solution to this problem with respect to term correlations. Secondly, even if term correlations are explicitly known, we still need a method for selecting a basis from the generating set. This problem is particularly troublesome when the number of term vectors is very large in practice. We have developed an algorithm for identifying a maximal subset of linearly independent term vectors provided that term correlations are known or approximated. The method will be reported in a forthcoming paper. 6. CONCLUSION In our reevaluation of the vector space model, two main questions are raised: (i) Whether the vector space model has been taken seriously in the information retrieval context? (ii) Whether the vector space model should be taken seriously at all? It is shown that in view of the well established practices, the answer to the first question is negative. However, based on our detailed analysis in Section 4, the answer to the second question is positive because it has been demonstrated that much insight can be gained by thinking about our problem within the framework of vector spaces. We believe that if the issues we have raised in Secion 5 can be satisfactorily resolved in the future, the vector space model looks very promising indeed and it will provide a useful and formal framework for the information retrieval systems. Wong & Raghaven: The vector space model 185 Acknowledgement We are grateful to Peter Bollmann, Ulrike Reiner and other members of the LIVE project group for helpful discussions on issues addressed in this paper. References Goult, R.J. (1978). Applied Linear Algebra. Chichester, England: John Wiley & Sons. Koll, M. (1979). An approach to concept based information retrieval. ACM-SIGIR Forum, Vol. XIII (Spring), 32-50. Salton, G. (1971). The SMART Retrieval System -Experiments in Automatic Document Processing. Englewood Cliffs, N.J. : Prentice-Hall. Salton, G., C.S. Yang & A. Wong (1975). A vector space model for automic indexing. Comm. ACM, 18 (November), 613-620. Salton, G. (1980). Automatic information retrieval. IEEE Computer, 13 (September), 41-56. Salton, G. & M.H. McGill (1983). Introduction to Modern Information Retrieval. New York, N.Y.: McGraw- Hill. Salton, G., E.A. Fox & H. Wu (1983). An automatic environment for boolean information retrieval. Information Processing "83. Proceedings of the IFIP 9th World Computer Congress, Paris, France, 755-762. van Rijsbergen, C.J. (1979). Information Retrieval, Edition. Butterworth, London. 2nd View publication stats
ed3book	Speech and Language Processing An Introduction to Natural Language Processing, Computational Linguistics, and Speech Recognition with Language Models Third Edition draft Daniel Jurafsky Stanford University James H. Martin University of Colorado at Boulder Copyright ©2024. All rights reserved. Draft of January 12, 2025. Comments and typos welcome! Summary of Contents I Fundamental Algorithms for NLP 1 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Regular Expressions, Tokenization, Edit Distance . . . . . . . . . . . . . . . 4 3 N-gram Language Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Naive Bayes, Text Classiﬁcation, and Sentiment . . . . . . . . . . . . . . . . . 56 5 Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6 Vector Semantics and Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8 RNNs and LSTMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 9 The Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10 Large Language Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 11 Masked Language Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 12 Model Alignment, Prompting, and In-Context Learning . . . . . . . . . 242 II NLP Applications 261 13 Machine Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 14 Question Answering, Information Retrieval, and RAG . . . . . . . . . . 289 15 Chatbots & Dialogue Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 16 Automatic Speech Recognition and Text-to-Speech . . . . . . . . . . . . . . 331 III Annotating Linguistic Structure 359 17 Sequence Labeling for Parts of Speech and Named Entities . . . . . . 362 18 Context-Free Grammars and Constituency Parsing . . . . . . . . . . . . . 387 19 Dependency Parsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 20 Information Extraction: Relations, Events, and Time. . . . . . . . . . . . 435 21 Semantic Role Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 22 Lexicons for Sentiment, Affect, and Connotation . . . . . . . . . . . . . . . . 481 23 Coreference Resolution and Entity Linking . . . . . . . . . . . . . . . . . . . . . 501 24 Discourse Coherence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 2 Contents I Fundamental Algorithms for NLP 1 1 Introduction 3 2 Regular Expressions, Tokenization, Edit Distance 4 2.1 Regular Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Corpora . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Simple Unix Tools for Word Tokenization . . . . . . . . . . . . . 16 2.5 Word and Subword Tokenization . . . . . . . . . . . . . . . . . . 18 2.6 Word Normalization, Lemmatization and Stemming . . . . . . . . 23 2.7 Sentence Segmentation . . . . . . . . . . . . . . . . . . . . . . . 24 2.8 Minimum Edit Distance . . . . . . . . . . . . . . . . . . . . . . . 25 2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 30 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 N-gram Language Models 32 3.1 N-Grams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Evaluating Language Models: Training and Test Sets . . . . . . . 38 3.3 Evaluating Language Models: Perplexity . . . . . . . . . . . . . . 40 3.4 Sampling sentences from a language model . . . . . . . . . . . . . 42 3.5 Generalizing vs. overﬁtting the training set . . . . . . . . . . . . . 43 3.6 Smoothing, Interpolation, and Backoff . . . . . . . . . . . . . . . 45 3.7 Advanced: Perplexity’s Relation to Entropy . . . . . . . . . . . . 49 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 52 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4 Naive Bayes, Text Classiﬁcation, and Sentiment 56 4.1 Naive Bayes Classiﬁers . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Training the Naive Bayes Classiﬁer . . . . . . . . . . . . . . . . . 60 4.3 Worked example . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 Optimizing for Sentiment Analysis . . . . . . . . . . . . . . . . . 62 4.5 Naive Bayes for other text classiﬁcation tasks . . . . . . . . . . . 64 4.6 Naive Bayes as a Language Model . . . . . . . . . . . . . . . . . 65 4.7 Evaluation: Precision, Recall, F-measure . . . . . . . . . . . . . . 66 4.8 Test sets and Cross-validation . . . . . . . . . . . . . . . . . . . . 69 4.9 Statistical Signiﬁcance Testing . . . . . . . . . . . . . . . . . . . 70 4.10 Avoiding Harms in Classiﬁcation . . . . . . . . . . . . . . . . . . 73 4.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 75 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5 Logistic Regression 77 5.1 The sigmoid function . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Classiﬁcation with Logistic Regression . . . . . . . . . . . . . . . 80 5.3 Multinomial logistic regression . . . . . . . . . . . . . . . . . . . 84 5.4 Learning in Logistic Regression . . . . . . . . . . . . . . . . . . . 87 3 4 CONTENTS 5.5 The cross-entropy loss function . . . . . . . . . . . . . . . . . . . 88 5.6 Gradient Descent . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.7 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.8 Learning in Multinomial Logistic Regression . . . . . . . . . . . . 97 5.9 Interpreting models . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.10 Advanced: Deriving the Gradient Equation . . . . . . . . . . . . . 98 5.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 100 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6 Vector Semantics and Embeddings 101 6.1 Lexical Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.2 Vector Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.3 Words and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.4 Cosine for measuring similarity . . . . . . . . . . . . . . . . . . . 110 6.5 TF-IDF: Weighing terms in the vector . . . . . . . . . . . . . . . 111 6.6 Pointwise Mutual Information (PMI) . . . . . . . . . . . . . . . . 114 6.7 Applications of the tf-idf or PPMI vector models . . . . . . . . . . 116 6.8 Word2vec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.9 Visualizing Embeddings . . . . . . . . . . . . . . . . . . . . . . . 123 6.10 Semantic properties of embeddings . . . . . . . . . . . . . . . . . 124 6.11 Bias and Embeddings . . . . . . . . . . . . . . . . . . . . . . . . 126 6.12 Evaluating Vector Models . . . . . . . . . . . . . . . . . . . . . . 127 6.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 129 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7 Neural Networks 132 7.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.2 The XOR problem . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.3 Feedforward Neural Networks . . . . . . . . . . . . . . . . . . . . 138 7.4 Feedforward networks for NLP: Classiﬁcation . . . . . . . . . . . 142 7.5 Training Neural Nets . . . . . . . . . . . . . . . . . . . . . . . . 145 7.6 Feedforward Neural Language Modeling . . . . . . . . . . . . . . 152 7.7 Training the neural language model . . . . . . . . . . . . . . . . . 155 7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 157 8 RNNs and LSTMs 158 8.1 Recurrent Neural Networks . . . . . . . . . . . . . . . . . . . . . 158 8.2 RNNs as Language Models . . . . . . . . . . . . . . . . . . . . . 162 8.3 RNNs for other NLP tasks . . . . . . . . . . . . . . . . . . . . . . 165 8.4 Stacked and Bidirectional RNN architectures . . . . . . . . . . . . 168 8.5 The LSTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.6 Summary: Common RNN NLP Architectures . . . . . . . . . . . 174 8.7 The Encoder-Decoder Model with RNNs . . . . . . . . . . . . . . 174 8.8 Attention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 8.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 182 9 The Transformer 184 9.1 Attention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 CONTENTS 5 9.2 Transformer Blocks . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.3 Parallelizing computation using a single matrix X . . . . . . . . . 194 9.4 The input: embeddings for token and position . . . . . . . . . . . 197 9.5 The Language Modeling Head . . . . . . . . . . . . . . . . . . . 199 9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 202 10 Large Language Models 203 10.1 Large Language Models with Transformers . . . . . . . . . . . . . 204 10.2 Sampling for LLM Generation . . . . . . . . . . . . . . . . . . . 207 10.3 Pretraining Large Language Models . . . . . . . . . . . . . . . . 210 10.4 Evaluating Large Language Models . . . . . . . . . . . . . . . . . 214 10.5 Dealing with Scale . . . . . . . . . . . . . . . . . . . . . . . . . . 216 10.6 Potential Harms from Language Models . . . . . . . . . . . . . . 219 10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 220 11 Masked Language Models 223 11.1 Bidirectional Transformer Encoders . . . . . . . . . . . . . . . . . 223 11.2 Training Bidirectional Encoders . . . . . . . . . . . . . . . . . . . 226 11.3 Contextual Embeddings . . . . . . . . . . . . . . . . . . . . . . . 231 11.4 Fine-Tuning for Classiﬁcation . . . . . . . . . . . . . . . . . . . . 235 11.5 Fine-Tuning for Sequence Labelling: Named Entity Recognition . 237 11.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 241 12 Model Alignment, Prompting, and In-Context Learning 242 12.1 Prompting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 12.2 Post-training and Model Alignment . . . . . . . . . . . . . . . . . 248 12.3 Model Alignment: Instruction Tuning . . . . . . . . . . . . . . . . 249 12.4 Chain-of-Thought Prompting . . . . . . . . . . . . . . . . . . . . 254 12.5 Automatic Prompt Optimization . . . . . . . . . . . . . . . . . . . 254 12.6 Evaluating Prompted Language Models . . . . . . . . . . . . . . . 258 12.7 Model Alignment with Human Preferences: RLHF and DPO . . . 258 12.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 259 II NLP Applications 261 13 Machine Translation 263 13.1 Language Divergences and Typology . . . . . . . . . . . . . . . . 264 13.2 Machine Translation using Encoder-Decoder . . . . . . . . . . . . 268 13.3 Details of the Encoder-Decoder Model . . . . . . . . . . . . . . . 272 13.4 Decoding in MT: Beam Search . . . . . . . . . . . . . . . . . . . 274 13.5 Translating in low-resource situations . . . . . . . . . . . . . . . . 278 13.6 MT Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 13.7 Bias and Ethical Issues . . . . . . . . . . . . . . . . . . . . . . . 284 13.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 286 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 14 Question Answering, Information Retrieval, and RAG 289 6 CONTENTS 14.1 Information Retrieval . . . . . . . . . . . . . . . . . . . . . . . . 290 14.2 Information Retrieval with Dense Vectors . . . . . . . . . . . . . . 298 14.3 Answering Questions with RAG . . . . . . . . . . . . . . . . . . 301 14.4 Evaluating Question Answering . . . . . . . . . . . . . . . . . . . 304 14.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 306 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 15 Chatbots & Dialogue Systems 309 15.1 Properties of Human Conversation . . . . . . . . . . . . . . . . . 311 15.2 Frame-Based Dialogue Systems . . . . . . . . . . . . . . . . . . . 314 15.3 Dialogue Acts and Dialogue State . . . . . . . . . . . . . . . . . . 317 15.4 Chatbots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 15.5 Dialogue System Design . . . . . . . . . . . . . . . . . . . . . . . 325 15.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 328 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 16 Automatic Speech Recognition and Text-to-Speech 331 16.1 The Automatic Speech Recognition Task . . . . . . . . . . . . . . 332 16.2 Feature Extraction for ASR: Log Mel Spectrum . . . . . . . . . . 334 16.3 Speech Recognition Architecture . . . . . . . . . . . . . . . . . . 339 16.4 CTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 16.5 ASR Evaluation: Word Error Rate . . . . . . . . . . . . . . . . . 346 16.6 TTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 16.7 Other Speech Tasks . . . . . . . . . . . . . . . . . . . . . . . . . 353 16.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 354 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 III Annotating Linguistic Structure 359 17 Sequence Labeling for Parts of Speech and Named Entities 362 17.1 (Mostly) English Word Classes . . . . . . . . . . . . . . . . . . . 363 17.2 Part-of-Speech Tagging . . . . . . . . . . . . . . . . . . . . . . . 365 17.3 Named Entities and Named Entity Tagging . . . . . . . . . . . . . 367 17.4 HMM Part-of-Speech Tagging . . . . . . . . . . . . . . . . . . . 369 17.5 Conditional Random Fields (CRFs) . . . . . . . . . . . . . . . . . 376 17.6 Evaluation of Named Entity Recognition . . . . . . . . . . . . . . 381 17.7 Further Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 17.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 384 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 18 Context-Free Grammars and Constituency Parsing 387 18.1 Constituency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 18.2 Context-Free Grammars . . . . . . . . . . . . . . . . . . . . . . . 388 18.3 Treebanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 18.4 Grammar Equivalence and Normal Form . . . . . . . . . . . . . . 394 18.5 Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 18.6 CKY Parsing: A Dynamic Programming Approach . . . . . . . . 397 18.7 Span-Based Neural Constituency Parsing . . . . . . . . . . . . . . 403 CONTENTS 7 18.8 Evaluating Parsers . . . . . . . . . . . . . . . . . . . . . . . . . . 405 18.9 Heads and Head-Finding . . . . . . . . . . . . . . . . . . . . . . 406 18.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 408 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 19 Dependency Parsing 411 19.1 Dependency Relations . . . . . . . . . . . . . . . . . . . . . . . . 412 19.2 Transition-Based Dependency Parsing . . . . . . . . . . . . . . . 416 19.3 Graph-Based Dependency Parsing . . . . . . . . . . . . . . . . . 425 19.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 19.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 433 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 20 Information Extraction: Relations, Events, and Time 435 20.1 Relation Extraction . . . . . . . . . . . . . . . . . . . . . . . . . 436 20.2 Relation Extraction Algorithms . . . . . . . . . . . . . . . . . . . 438 20.3 Extracting Events . . . . . . . . . . . . . . . . . . . . . . . . . . 446 20.4 Representing Time . . . . . . . . . . . . . . . . . . . . . . . . . . 447 20.5 Representing Aspect . . . . . . . . . . . . . . . . . . . . . . . . . 450 20.6 Temporally Annotated Datasets: TimeBank . . . . . . . . . . . . . 451 20.7 Automatic Temporal Analysis . . . . . . . . . . . . . . . . . . . . 452 20.8 Template Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 20.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 459 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 21 Semantic Role Labeling 461 21.1 Semantic Roles . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 21.2 Diathesis Alternations . . . . . . . . . . . . . . . . . . . . . . . . 462 21.3 Semantic Roles: Problems with Thematic Roles . . . . . . . . . . 464 21.4 The Proposition Bank . . . . . . . . . . . . . . . . . . . . . . . . 465 21.5 FrameNet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 21.6 Semantic Role Labeling . . . . . . . . . . . . . . . . . . . . . . . 468 21.7 Selectional Restrictions . . . . . . . . . . . . . . . . . . . . . . . 472 21.8 Primitive Decomposition of Predicates . . . . . . . . . . . . . . . 476 21.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 478 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 22 Lexicons for Sentiment, Affect, and Connotation 481 22.1 Deﬁning Emotion . . . . . . . . . . . . . . . . . . . . . . . . . . 482 22.2 Available Sentiment and Affect Lexicons . . . . . . . . . . . . . . 484 22.3 Creating Affect Lexicons by Human Labeling . . . . . . . . . . . 485 22.4 Semi-supervised Induction of Affect Lexicons . . . . . . . . . . . 487 22.5 Supervised Learning of Word Sentiment . . . . . . . . . . . . . . 490 22.6 Using Lexicons for Sentiment Recognition . . . . . . . . . . . . . 495 22.7 Using Lexicons for Affect Recognition . . . . . . . . . . . . . . . 496 22.8 Lexicon-based methods for Entity-Centric Affect . . . . . . . . . . 497 22.9 Connotation Frames . . . . . . . . . . . . . . . . . . . . . . . . . 497 22.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 8 CONTENTS Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 500 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 23 Coreference Resolution and Entity Linking 501 23.1 Coreference Phenomena: Linguistic Background . . . . . . . . . . 504 23.2 Coreference Tasks and Datasets . . . . . . . . . . . . . . . . . . . 509 23.3 Mention Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 510 23.4 Architectures for Coreference Algorithms . . . . . . . . . . . . . 513 23.5 Classiﬁers using hand-built features . . . . . . . . . . . . . . . . . 515 23.6 A neural mention-ranking algorithm . . . . . . . . . . . . . . . . 517 23.7 Entity Linking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 23.8 Evaluation of Coreference Resolution . . . . . . . . . . . . . . . . 524 23.9 Winograd Schema problems . . . . . . . . . . . . . . . . . . . . . 525 23.10 Gender Bias in Coreference . . . . . . . . . . . . . . . . . . . . . 526 23.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 528 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 24 Discourse Coherence 531 24.1 Coherence Relations . . . . . . . . . . . . . . . . . . . . . . . . . 533 24.2 Discourse Structure Parsing . . . . . . . . . . . . . . . . . . . . . 536 24.3 Centering and Entity-Based Coherence . . . . . . . . . . . . . . . 540 24.4 Representation learning models for local coherence . . . . . . . . 544 24.5 Global Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . 546 24.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 Bibliographical and Historical Notes . . . . . . . . . . . . . . . . . . . . 550 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 Bibliography 553 Subject Index 585 Part I FUNDAMENTAL ALGORITHMS FOR NLP In the ﬁrst part of the book we introduce the fundamental suite of algorithmic tools that make up the modern neural language model that is the heart of end-to-end NLP systems. We begin with tokenization and preprocessing, as well as useful algo- rithms like computing edit distance, and then proceed to the tasks of classiﬁcation, logistic regression, neural networks, proceeding through feedforward networks, re- current networks, and then transformers. We’ll also see the role of embeddings as a model of word meaning. CHAPTER 1 Introduction La derni`ere chose qu’on trouve en faisant un ouvrage est de savoir celle qu’il faut mettre la premi`ere. [The last thing you ﬁgure out in writing a book is what to put ﬁrst.] Pascal 3 4 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE CHAPTER 2 Regular Expressions, Tok- enization, Edit Distance User: I am unhappy. ELIZA: DO YOU THINK COMING HERE WILL HELP YOU NOT TO BE UNHAPPY User: I need some help, that much seems certain. ELIZA: WHAT WOULD IT MEAN TO YOU IF YOU GOT SOME HELP User: Perhaps I could learn to get along with my mother. ELIZA: TELL ME MORE ABOUT YOUR FAMILY User: My mother takes care of me. ELIZA: WHO ELSE IN YOU FAMILY TAKES CARE OF YOU User: My father. ELIZA: YOUR FATHER User: You are like my father in some ways. Weizenbaum (1966) The dialogue above is fromELIZA, an early natural language processing systemELIZA that could carry on a limited conversation with a user by imitating the responses of a Rogerian psychotherapist (Weizenbaum, 1966). ELIZA is a surprisingly simple program that uses pattern matching to recognize phrases like “I need X” and translate them into suitable outputs like “What would it mean to you if you got X?”. This simple technique succeeds in this domain because ELIZA doesn’t actually need to know anything to mimic a Rogerian psychotherapist. As Weizenbaum notes, this is one of the few dialogue genres where listeners can act as if they know nothing of the world. ELIZA’s mimicry of human conversation was remarkably successful: many people who interacted with ELIZA came to believe that it really understood them and their problems, many continued to believe in ELIZA’s abilities even after the program’s operation was explained to them (Weizenbaum, 1976), and even today such chatbots are a fun diversion.chatbots Of course modern conversational agents are much more than a diversion; they can answer questions, book ﬂights, or ﬁnd restaurants, functions for which they rely on a much more sophisticated understanding of the user’s intent, as we will see in Chapter 15. Nonetheless, the simple pattern-based methods that powered ELIZA and other chatbots play a crucial role in natural language processing. We’ll begin with the most important tool for describing text patterns: theregular expression. Regular expressions can be used to specify strings we might want to extract from a document, from transforming “I need X” in ELIZA above, to deﬁning strings like $199 or $24.99 for extracting tables of prices from a document. We’ll then turn to a set of tasks collectively calledtext normalization, in whichtext normalization regular expressions play an important part. Normalizing text means converting it to a more convenient, standard form. For example, most of what we are going to do with language relies on ﬁrst separating out or tokenizing words or word parts from running text, the task of tokenization. English words are often separated fromtokenization each other by whitespace, but whitespace is not always sufﬁcient. New York and rock ’n’ rollare sometimes treated as large words despite the fact that they contain spaces, while sometimes we’ll need to separate I’m into the two words I and am. For processing tweets or texts we’ll need to tokenizeemoticons like :) or hashtags 2.1 • R EGULAR EXPRESSIONS 5 like #nlproc. Some languages, like Japanese, don’t have spaces between words, so word tokenization becomes more difﬁcult. And as we’ll see, for large language models we’ll use tokens that range greatly in size, from letters to subwords (parts of words) to words and even sometimes short phrases. Another part of text normalization is lemmatization, the task of determininglemmatization that two words have the same root, despite their surface differences. For example, the words sang, sung, and sings are forms of the verb sing. The word sing is the common lemma of these words, and a lemmatizer maps from all of these to sing. Lemmatization is essential for processing morphologically complex languages like Arabic. Stemming refers to a simpler version of lemmatization in which we mainlystemming just strip sufﬁxes from the end of the word. Text normalization also includes sen- tence segmentation: breaking up a text into individual sentences, using cues likesentence segmentation periods or exclamation points. Finally, we’ll need to compare words and other strings. We’ll introduce a metric called edit distance that measures how similar two strings are based on the number of edits (insertions, deletions, substitutions) it takes to change one string into the other. Edit distance is an algorithm with applications throughout language process- ing, from spelling correction to speech recognition to coreference resolution. 2.1 Regular Expressions One of the most useful tools for text processing in computer science has been the regular expression (often shortened to regex), a language for specifying text searchregular expression strings. This practical language is used in every computer language, in text process- ing tools like the Unix tools grep, and in editors like vim or Emacs. Formally, a regular expression is an algebraic notation for characterizing a set of strings. Reg- ular expressions are particularly useful for searching in texts, when we have a pat- tern to search for and a corpus of texts to search through. A regular expressioncorpus search function will search through the corpus, returning all texts that match the pattern. The corpus can be a single document or a collection. For example, the Unix command-line tool grep takes a regular expression and returns every line of the input document that matches the expression. A search can be designed to return every match on a line, if there are more than one, or just the ﬁrst match. In the following examples we generally underline the exact string that matches the regular expression and show only the ﬁrst match. We’ll show regular expressions delimited by slashes but note that slashes are not part of the regular expressions. Regular expressions come in many variants. We’ll be describingextended regu- lar expressions; different regular expression parsers may only recognize subsets of these, or treat some expressions slightly differently. Using an online regular expres- sion tester is a handy way to test out your expressions and explore these variations. 2.1.1 Basic Regular Expression Patterns The simplest kind of regular expression is a sequence of simple characters; putting characters in sequence is called concatenation. To search for woodchuck, we typeconcatenation /woodchuck/. The expression /Buttercup/ matches any string containing the substring Buttercup; grep with that expression would return the line I’m called lit- tle Buttercup. The search string can consist of a single character (like /!/) or a 6 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE sequence of characters (like /urgl/) (see Fig. 2.1). Regex Example Patterns Matched /woodchucks/ “interesting links to woodchucks and lemurs” /a/ “Mary Ann stopped by Mona’s” /!/ “You’ve left the burglar behind again!” said Nori Figure 2.1 Some simple regex searches. Regular expressions are case sensitive; lower case /s/ is distinct from upper case /S/ (/s/ matches a lower case s but not an upper case S). This means that the pattern /woodchucks/will not match the string Woodchucks. We can solve this problem with the use of the square braces[and ]. The string of characters inside the braces speciﬁes a disjunction of characters to match. For example, Fig. 2.2 shows that the pattern /[wW]/matches patterns containing either w or W. Regex Match Example Patterns /[wW]oodchuck/ Woodchuck or woodchuck “Woodchuck” /[abc]/ ‘a’, ‘b’,or ‘c’ “In uomini, in soldati” /[1234567890]/ any digit “plenty of 7 to 5” Figure 2.2 The use of the brackets []to specify a disjunction of characters. The regular expression /[1234567890]/speciﬁes any single digit. While such classes of characters as digits or letters are important building blocks in expressions, they can get awkward (e.g., it’s inconvenient to specify /[ABCDEFGHIJKLMNOPQRSTUVWXYZ]/ (2.1) to mean “any capital letter”). In cases where there is a well-deﬁned sequence asso- ciated with a set of characters, the brackets can be used with the dash ( -) to specify any one character in a range. The pattern /[2-5]/speciﬁes any one of the charac-range ters 2, 3, 4, or 5. The pattern /[b-g]/speciﬁes one of the characters b, c, d, e, f, or g. Some other examples are shown in Fig. 2.3. Regex Match Example Patterns Matched /[A-Z]/ an upper case letter “we should call it ‘Drenched Blossoms’ ” /[a-z]/ a lower case letter “my beans were impatient to be hoed!” /[0-9]/ a single digit “Chapter 1: Down the Rabbit Hole” Figure 2.3 The use of the brackets []plus the dash -to specify a range. The square braces can also be used to specify what a single character cannot be, by use of the caret ˆ. If the caret ˆis the ﬁrst symbol after the open square brace [, the resulting pattern is negated. For example, the pattern/[ˆa]/matches any single character (including special characters) except a. This is only true when the caret is the ﬁrst symbol after the open square brace. If it occurs anywhere else, it usually stands for a caret; Fig. 2.4 shows some examples. How can we talk about optional elements, like an optional s in woodchuck and woodchucks? We can’t use the square brackets, because while they allow us to say “s or S”, they don’t allow us to say “s or nothing”. For this we use the question mark /?/, which means “the preceding character or nothing”, as shown in Fig. 2.5. We can think of the question mark as meaning “zero or one instances of the previous character”. That is, it’s a way of specifying how many of something that 2.1 • R EGULAR EXPRESSIONS 7 Regex Match (single characters) Example Patterns Matched /[ˆA-Z]/ not an upper case letter “Oyfn pripetchik” /[ˆSs]/ neither ‘S’ nor ‘s’ “I have no exquisite reason for’t” /[ˆ.]/ not a period “our resident Djinn” /[eˆ]/ either ‘e’ or ‘ˆ’ “look up ˆ now” /aˆb/ the pattern ‘aˆb’ “look up aˆ bnow” Figure 2.4 The caret ˆfor negation or just to mean ˆ. See below re: the backslash for escaping the period. Regex Match Example Patterns Matched /woodchucks?/ woodchuck or woodchucks “woodchuck” /colou?r/ color or colour “color” Figure 2.5 The question mark ? marks optionality of the previous expression. we want, something that is very important in regular expressions. For example, consider the language of certain sheep, which consists of strings that look like the following: baa! baaa! baaaa! . . . This language consists of strings with a b, followed by at least two a’s, followed by an exclamation point. The set of operators that allows us to say things like “some number of as” are based on the asterisk or , commonly called the Kleene (gen-Kleene * erally pronounced “cleany star”). The Kleene star means “zero or more occurrences of the immediately previous character or regular expression”. So /a/ means “any string of zero or more as”. This will match a or aaaaaa, but it will also match the empty string at the start of Off Minor since the string Off Minor starts with zero a’s. So the regular expression for matching one or more a is /aa/, meaning one a fol- lowed by zero or moreas. More complex patterns can also be repeated. So/[ab]/ means “zero or more a’s orb’s” (not “zero or more right square braces”). This will match strings like aaaa or ababab or bbbb, as well as the empty string. For specifying multiple digits (useful for ﬁnding prices) we can extend/[0-9]/, the regular expression for a single digit. An integer (a string of digits) is thus /[0-9][0-9]/. (Why isn’t it just/[0-9]/?) Sometimes it’s annoying to have to write the regular expression for digits twice, so there is a shorter way to specify “at least one” of some character. This is the Kleene +, which means “one or more occurrences of the immediately precedingKleene + character or regular expression”. Thus, the expression /[0-9]+/is the normal way to specify “a sequence of digits”. There are thus two ways to specify the sheep language: /baaa!/or /baa+!/. One very important special character is the period (/./), a wildcard expression that matches any single character (except a carriage return), as shown in Fig. 2.6. Regex Match Example Matches /beg.n/ any character between beg and n begin, beg’n, begun Figure 2.6 The use of the period . to specify any character. The wildcard is often used together with the Kleene star to mean “any string of characters”. For example, suppose we want to ﬁnd any line in which a particular 8 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE word, for example, aardvark, appears twice. We can specify this with the regular expression /aardvark.aardvark/. Anchors are special characters that anchor regular expressions to particular placesanchors in a string. The most common anchors are the caretˆand the dollar sign$. The caret ˆ matches the start of a line. The pattern /ˆThe/ matches the word The only at the start of a line. Thus, the caret ˆ has three uses: to match the start of a line, to in- dicate a negation inside of square brackets, and just to mean a caret. (What are the contexts that allow grep or Python to know which function a given caret is supposed to have?) The dollar sign $ matches the end of a line. So the pattern ␣$ is a useful pattern for matching a space at the end of a line, and /ˆThe dog\.$/ matches a line that contains only the phrase The dog. (We have to use the backslash here since we want the .to mean “period” and not the wildcard.) Regex Match ˆ start of line $ end of line \b word boundary \B non-word boundary Figure 2.7 Anchors in regular expressions. There are also two other anchors: \bmatches a word boundary, and \Bmatches a non word-boundary. Thus, /\bthe\b/ matches the word the but not the word other. A “word” for the purposes of a regular expression is deﬁned based on the deﬁnition of words in programming languages as a sequence of digits, underscores, or letters. Thus /\b99\b/will match the string 99 in There are 99 bottles of beer on the wall (because 99 follows a space) but not 99 in There are 299 bottles of beer on the wall (since 99 follows a number). But it will match 99 in $99 (since 99 follows a dollar sign ($), which is not a digit, underscore, or letter). 2.1.2 Disjunction, Grouping, and Precedence Suppose we need to search for texts about pets; perhaps we are particularly interested in cats and dogs. In such a case, we might want to search for either the string cat or the string dog. Since we can’t use the square brackets to search for “cat or dog” (why can’t we say/[catdog]/?), we need a new operator, thedisjunction operator, alsodisjunction called the pipe symbol \|. The pattern /cat\|dog/ matches either the string cat or the string dog. Sometimes we need to use this disjunction operator in the midst of a larger se- quence. For example, suppose I want to search for information about pet ﬁsh for my cousin David. How can I specify both guppy and guppies? We cannot simply say /guppy\|ies/, because that would match only the strings guppy and ies. This is because sequences like guppy take precedence over the disjunction operator \|.precedence To make the disjunction operator apply only to a speciﬁc pattern, we need to use the parenthesis operators ( and ). Enclosing a pattern in parentheses makes it act like a single character for the purposes of neighboring operators like the pipe \| and the Kleene. So the pattern /gupp(y\|ies)/ would specify that we meant the disjunc- tion only to apply to the sufﬁxes yand ies. The parenthesis operator ( is also useful when we are using counters like the Kleene. Unlike the \| operator, the Kleene operator applies by default only to a single character, not to a whole sequence. Suppose we want to match repeated instances of a string. Perhaps we have a line that has column labels of the form 2.1 • R EGULAR EXPRESSIONS 9 Column 1 Column 2 Column 3 . The expression /Column␣[0-9]+␣/ will not match any number of columns; instead, it will match a single column followed by any number of spaces! The star here applies only to the space ␣ that precedes it, not to the whole sequence. With the parentheses, we could write the expression /(Column␣[0-9]+␣)/ to match the word Column, followed by a number and optional spaces, the whole pattern repeated zero or more times. This idea that one operator may take precedence over another, requiring us to sometimes use parentheses to specify what we mean, is formalized by the operator precedence hierarchy for regular expressions. The following table gives the orderoperator precedence of RE operator precedence, from highest precedence to lowest precedence. Parenthesis () Counters + ? {} Sequences and anchors the ˆmy end$ Disjunction \| Thus, because counters have a higher precedence than sequences, /the/ matches theeeee but not thethe. Because sequences have a higher prece- dence than disjunction, /the\|any/matches the or any but not thany or theny. Patterns can be ambiguous in another way. Consider the expression /[a-z]/ when matching against the text once upon a time. Since /[a-z]/matches zero or more letters, this expression could match nothing, or just the ﬁrst letter o, on, onc, or once. In these cases regular expressions always match the largest string they can; we say that patterns are greedy, expanding to cover as much of a string as they can.greedy There are, however, ways to enforcenon-greedy matching, using another mean-non-greedy ing of the ? qualiﬁer. The operator ? is a Kleene star that matches as little text as? possible. The operator +? is a Kleene plus that matches as little text as possible.+? 2.1.3 A Simple Example Suppose we wanted to write a RE to ﬁnd cases of the English article the. A simple (but incorrect) pattern might be: /the/ (2.2) One problem is that this pattern will miss the word when it begins a sentence and hence is capitalized (i.e., The). This might lead us to the following pattern: /[tT]he/ (2.3) But we will still overgeneralize, incorrectly return texts withtheembedded in other words (e.g., other or there). So we need to specify that we want instances with a word boundary on both sides: /\b[tT]he\b/ (2.4) The simple process we just went through was based on ﬁxing two kinds of errors: false positives, strings that we incorrectly matched like other or there, and falsefalse positives negatives, strings that we incorrectly missed, like The. Addressing these two kindsfalse negatives of errors comes up again and again in language processing. Reducing the overall error rate for an application thus involves two antagonistic efforts: • Increasing precision (minimizing false positives) • Increasing recall (minimizing false negatives) We’ll come back to precision and recall with more precise deﬁnitions in Chapter 4. 10 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE 2.1.4 More Operators Figure 2.8 shows some aliases for common ranges, which can be used mainly to save typing. Besides the Kleene and Kleene + we can also use explicit numbers as counters, by enclosing them in curly brackets. The operator /{3}/ means “exactly 3 occurrences of the previous character or expression”. So /a\.{24}z/will match a followed by 24 dots followed by z (but not a followed by 23 or 25 dots followed by a z). Regex Expansion Match First Matches \d [0-9] any digit Party␣of␣5 \D [ˆ0-9] any non-digit Blue␣moon \w [a-zA-Z0-9_] any alphanumeric/underscore Daiyu \W [ˆ\w] a non-alphanumeric !!!! \s [␣\r\t\n\f] whitespace (space, tab) in Concord \S [ˆ\s] Non-whitespace in␣Concord Figure 2.8 Aliases for common sets of characters. A range of numbers can also be speciﬁed. So /{n,m}/ speciﬁes from n to m occurrences of the previous char or expression, and /{n,}/means at least n occur- rences of the previous expression. REs for counting are summarized in Fig. 2.9. Regex Match * zero or more occurrences of the previous char or expression + one or more occurrences of the previous char or expression ? zero or one occurrence of the previous char or expression {n} exactly n occurrences of the previous char or expression {n,m} from n to m occurrences of the previous char or expression {n,} at least n occurrences of the previous char or expression {,m} up to m occurrences of the previous char or expression Figure 2.9 Regular expression operators for counting. Finally, certain special characters are referred to by special notation based on the backslash (\) (see Fig. 2.10). The most common of these are the newline characternewline \n and the tab character \t. To refer to characters that are special themselves (like ., , [, and \), precede them with a backslash, (i.e., /\./, /\/, /\[/, and /\\/). Regex Match First Patterns Matched \* an asterisk “” “K APLAN” \. a period “.” “Dr. Livingston, I presume” \? a question mark “Why don’t they come and lend a hand? ” \n a newline \t a tab Figure 2.10 Some characters that need to be escaped (via backslash). 2.1.5 A More Complex Example Let’s try out a more signiﬁcant example of the power of REs. Suppose our goal is help a user buy a computer on the Web who wants “at least 6 GHz and 500 GB of disk space for less than $1000”. To do this kind of retrieval, we ﬁrst need to be 2.1 • R EGULAR EXPRESSIONS 11 able to look for expressions like 6 GHz or 500 GB or $999.99. Let’s work out some regular expressions for this task. First, let’s complete our regular expression for prices. Here’s a regular expres- sion for a dollar sign followed by a string of digits: /$[0-9]+/ (2.5) Note that the $ character has a different function here than the end-of-line function we discussed earlier. Most regular expression parsers are smart enough to realize that $ here doesn’t mean end-of-line. (As a thought experiment, think about how regex parsers might ﬁgure out the function of $from the context.) Now we just need to deal with fractions of dollars. We’ll add a decimal point and two digits afterwards: /$[0-9]+\.[0-9][0-9]/ (2.6) This pattern only allows $199.99 but not $199. We need to make the cents optional and to make sure we’re at a word boundary: /(ˆ\|\W)$[0-9]+(\.[0-9][0-9])?\b/ (2.7) One last catch! This pattern allows prices like $199999.99 which would be far too expensive! We need to limit the dollars: /(ˆ\|\W)$[0-9]{0,3}(\.[0-9][0-9])?\b/ (2.8) Further ﬁxes (like avoiding matching a dollar sign with no price after it) are left as an exercise for the reader. How about disk space? We’ll need to allow for optional fractions again (5.5 GB); note the use of ?for making the ﬁnal soptional, and the use of /␣/to mean “zero or more spaces” since there might always be extra spaces lying around: /\b[0-9]+(\.[0-9]+)? (GB\|[Gg]igabytes?)\b/ (2.9) Modifying this regular expression so that it only matches more than 500 GB is left as an exercise for the reader. 2.1.6 Substitution, Capture Groups, and ELIZA An important use of regular expressions is insubstitutions. For example, the substi-substitution tution operator s/regexp1/pattern/ used in Python and in Unix commands like vim or sed allows a string characterized by a regular expression to be replaced by another string: s/colour/color/ (2.10) It is often useful to be able to refer to a particular subpart of the string matching the ﬁrst pattern. For example, suppose we wanted to put angle brackets around all integers in a text, for example, changing the 35 boxes to the <35> boxes. We’d like a way to refer to the integer we’ve found so that we can easily add the brackets. To do this, we put parentheses ( and ) around the ﬁrst pattern and use the number operator \1in the second pattern to refer back. Here’s how it looks: s/([0-9]+)/<\1>/ (2.11) 12 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE The parenthesis and number operators can also specify that a certain string or ex- pression must occur twice in the text. For example, suppose we are looking for the pattern “the Xer they were, the Xer they will be”, where we want to constrain the two X’s to be the same string. We do this by surrounding the ﬁrst X with the parenthesis operator, and replacing the second X with the number operator \1, as follows: /the (.)er they were, the \1er they will be/ (2.12) Here the \1will be replaced by whatever string matched the ﬁrst item in parentheses. So this will match the bigger they were, the bigger they will be but not the bigger they were, the faster they will be. This use of parentheses to store a pattern in memory is called a capture group.capture group Every time a capture group is used (i.e., parentheses surround a pattern), the re- sulting match is stored in a numbered register. If you match two different sets ofregister parentheses, \2means whatever matched the second capture group. Thus /the (.)er they (.), the \1er we \2/ (2.13) will match the faster they ran, the faster we ranbut not the faster they ran, the faster we ate. Similarly, the third capture group is stored in \3, the fourth is \4, and so on. Parentheses thus have a double function in regular expressions; they are used to group terms for specifying the order in which operators should apply, and they are used to capture something in a register. Occasionally we might want to use parentheses for grouping, but don’t want to capture the resulting pattern in a register. In that case we use anon-capturing group, which is speciﬁed by putting the specialnon-capturing group commands ?:after the open parenthesis, in the form (?: pattern ). /(?:some\|a few) (people\|cats) like some \1/ (2.14) will match some cats like some cats but not some cats like some some. Substitutions and capture groups are very useful in implementing simple chat- bots like ELIZA (Weizenbaum, 1966). Recall that ELIZA simulates a Rogerian psychologist by carrying on conversations like the following: User1: Men are all alike. ELIZA1: IN WHAT W AY User2: They’re always bugging us about something or other. ELIZA2: CAN YOU THINK OF A SPECIFIC EXAMPLE User3: Well, my boyfriend made me come here. ELIZA3: YOUR BOYFRIEND MADE YOU COME HERE User4: He says I’m depressed much of the time. ELIZA4: I AM SORRY TO HEAR YOU ARE DEPRESSED ELIZA works by having a series or cascade of regular expression substitutions each of which matches and changes some part of the input lines. After the input is uppercased, substitutions change all instances of MY to YOUR, and I’M to YOU ARE, and so on. That way when ELIZA repeats back part of the user utterance, it will seem to be referring correctly to the user. The next set of substitutions matches and replaces other patterns in the input. Here are some examples: s/. YOU ARE (depressed\|sad) ./I AM SORRY TO HEAR YOU ARE \1/ s/. YOU ARE (depressed\|sad) ./WHY DO YOU THINK YOU ARE \1/ s/. all ./IN WHAT WAY/ s/. always ./CAN YOU THINK OF A SPECIFIC EXAMPLE/ 2.2 • W ORDS 13 Since multiple substitutions can apply to a given input, substitutions are assigned a rank and applied in order. Creating patterns is the topic of Exercise 2.3, and we return to the details of the ELIZA architecture in Chapter 15. 2.1.7 Lookahead Assertions Finally, there will be times when we need to predict the future: look ahead in the text to see if some pattern matches, but not yet advance the pointer we always keep to where we are in the text, so that we can then deal with the pattern if it occurs, but if it doesn’t we can check for something else instead. These lookahead assertions make use of the (?syntax that we saw in the previ-lookahead ous section for non-capture groups. The operator (?= pattern)is true if pattern occurs, but is zero-width, i.e. the match pointer doesn’t advance. The operatorzero-width (?! pattern)only returns true if a pattern does not match, but again is zero-width and doesn’t advance the pointer. Negative lookahead is commonly used when we are parsing some complex pattern but want to rule out a special case. For example suppose we want to match, at the beginning of a line, any single word that doesn’t start with “V olcano”. We can use negative lookahead to do this: /ˆ(?!Volcano)[A-Za-z]+/ (2.15) 2.2 Words Before we talk about processing words, we need to decide what counts as a word. Let’s start by looking at one particularcorpus (plural corpora), a computer-readablecorpus corpora collection of text or speech. For example the Brown corpus is a million-word col- lection of samples from 500 written English texts from different genres (newspa- per, ﬁction, non-ﬁction, academic, etc.), assembled at Brown University in 1963–64 (Kuˇcera and Francis, 1967). How many words are in the following Brown sentence? He stepped out into the hall, was delighted to encounter a water brother. This sentence has 13 words if we don’t count punctuation marks as words, 15 if we count punctuation. Whether we treat period (“ .”), comma (“,”), and so on as words depends on the task. Punctuation is critical for ﬁnding boundaries of things (commas, periods, colons) and for identifying some aspects of meaning (question marks, exclamation marks, quotation marks). For some tasks, like part-of-speech tagging or parsing or speech synthesis, we sometimes treat punctuation marks as if they were separate words. The Switchboard corpus of American English telephone conversations between strangers was collected in the early 1990s; it contains 2430 conversations averaging 6 minutes each, totaling 240 hours of speech and about 3 million words (Godfrey et al., 1992). Such corpora of spoken language introduce other complications with regard to deﬁning words. Let’s look at one utterance from Switchboard; an utter- ance is the spoken correlate of a sentence:utterance I do uh main- mainly business data processing This utterance has two kinds of disﬂuencies. The broken-off word main- isdisﬂuency called a fragment. Words like uh and um are called ﬁllers or ﬁlled pauses. Shouldfragment ﬁlled pause we consider these to be words? Again, it depends on the application. If we are building a speech transcription system, we might want to eventually strip out the disﬂuencies. 14 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE But we also sometimes keep disﬂuencies around. Disﬂuencies like uh or um are actually helpful in speech recognition in predicting the upcoming word, because they may signal that the speaker is restarting the clause or idea, and so for speech recognition they are treated as regular words. Because different people use differ- ent disﬂuencies they can also be a cue to speaker identiﬁcation. In fact Clark and Fox Tree (2002) showed thatuh and um have different meanings. What do you think they are? Perhaps most important, in thinking about what is a word, we need to distinguish two ways of talking about words that will be useful throughout the book. Wordtypesword type are the number of distinct words in a corpus; if the set of words in the vocabulary is V , the number of types is the vocabulary size\|V \|. Word instances are the total num-word instance ber N of running words.1 If we ignore punctuation, the following Brown sentence has 14 types and 16 instances: They picnicked by the pool, then lay back on the grass and looked at the stars. We still have decisions to make! For example, should we consider a capitalized string (like They) and one that is uncapitalized (like they) to be the same word type? The answer is that it depends on the task! They and they might be lumped together as the same type in some tasks, like speech recognition, where we care more about the sequence of words and less about the formatting, while for other tasks, such as deciding whether a particular word is a name of a person or location (named- entity tagging), capitalization is a useful feature and is retained. Sometimes we keep around two versions of a particular NLP model, one with capitalization and one without capitalization. Corpus Types = \|V \| Instances = N Shakespeare 31 thousand 884 thousand Brown corpus 38 thousand 1 million Switchboard telephone conversations 20 thousand 2.4 million COCA 2 million 440 million Google n-grams 13 million 1 trillion Figure 2.11 Rough numbers of wordform types and instances for some English language corpora. The largest, the Google n-grams corpus, contains 13 million types, but this count only includes types appearing 40 or more times, so the true number would be much larger. How many words are there in English? When we speak about the number of words in the language, we are generally referring to word types. Fig. 2.11 shows the rough numbers of types and instances computed from some English corpora. The larger the corpora we look at, the more word types we ﬁnd, and in fact this relationship between the number of types \|V \|and number of instances N is called Herdan’s Law(Herdan, 1960) or Heaps’ Law (Heaps, 1978) after its discoverersHerdan’s Law Heaps’ Law (in linguistics and information retrieval respectively). It is shown in Eq. 2.16, where k and β are positive constants, and 0 <β <1. \|V \| = kNβ (2.16) The value of β depends on the corpus size and the genre, but at least for the large corpora in Fig. 2.11, β ranges from .67 to .75. Roughly then we can say that the 1 In earlier tradition, and occasionally still, you might see word instances referred to as wordtokens, but we now try to reserve the word token instead to mean the output of subword tokenization algorithms. 2.3 • C ORPORA 15 vocabulary size for a text goes up signiﬁcantly faster than the square root of its length in words. It’s sometimes useful to make a further distinction. Consider inﬂected forms like cats versus cat. We say these two words are different wordforms but have the same lemma. A lemma is a set of lexical forms having the same stem, and usually thelemma same major part-of-speech. The wordform is the full inﬂected or derived form ofwordform the word. The two wordforms cat and cats thus have the same lemma, which we can represent as cat. For morphologically complex languages like Arabic, we often need to deal with lemmatization. For most tasks in English, however, wordforms are sufﬁcient, and when we talk about words in this book we almost always mean wordforms (although we will discuss basic algorithms for lemmatization and the related task of stemming below in Section 2.6). One of the situations even in English where we talk about lemmas is when we measure the number of words in a dictionary. Dictionary en- tries or boldface forms are a very rough approximation to (an upper bound on) the number of lemmas (since some lemmas have multiple boldface forms). The 1989 edition of the Oxford English Dictionary had 615,000 entries. Finally, we should note that in practice, for many NLP applications (for example for neural language modeling) we don’t actually use words as our internal unit of representation at all! We instead tokenize the input strings into tokens, which can be words but can also be only parts of words. We’ll return to this tokenization question when we introduce the BPE algorithm in Section 2.5.2. 2.3 Corpora Words don’t appear out of nowhere. Any particular piece of text that we study is produced by one or more speciﬁc speakers or writers, in a speciﬁc dialect of a speciﬁc language, at a speciﬁc time, in a speciﬁc place, for a speciﬁc function. Perhaps the most important dimension of variation is the language. NLP algo- rithms are most useful when they apply across many languages. The world has 7097 languages at the time of this writing, according to the online Ethnologue catalog (Simons and Fennig, 2018). It is important to test algorithms on more than one lan- guage, and particularly on languages with different properties; by contrast there is an unfortunate current tendency for NLP algorithms to be developed or tested just on English (Bender, 2019). Even when algorithms are developed beyond English, they tend to be developed for the ofﬁcial languages of large industrialized nations (Chinese, Spanish, Japanese, German etc.), but we don’t want to limit tools to just these few languages. Furthermore, most languages also have multiple varieties, of- ten spoken in different regions or by different social groups. Thus, for example, if we’re processing text that uses features of African American English ( AAE) orAAE African American Vernacular English (AA VE)—the variations of English used by millions of people in African American communities (King 2020)—we must use NLP tools that function with features of those varieties. Twitter posts might use fea- tures often used by speakers of African American English, such as constructions like iont (I don’t in Mainstream American English ( MAE)), or talmbout correspondingMAE to MAE talking about, both examples that inﬂuence word segmentation (Blodgett et al. 2016, Jones 2015). It’s also quite common for speakers or writers to use multiple languages in a single communicative act, a phenomenon called code switching. Code switchingcode switching 16 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE is enormously common across the world; here are examples showing Spanish and (transliterated) Hindi code switching with English (Solorio et al. 2014, Jurgens et al. 2017): (2.17) Por primera vez veo a @username actually being hateful! it was beautiful:) [For the ﬁrst time I get to see @username actually being hateful! it was beautiful:) ] (2.18) dost tha or ra- hega ... dont wory ... but dherya rakhe [“he was and will remain a friend ... don’t worry ... but have faith”] Another dimension of variation is the genre. The text that our algorithms must process might come from newswire, ﬁction or non-ﬁction books, scientiﬁc articles, Wikipedia, or religious texts. It might come from spoken genres like telephone conversations, business meetings, police body-worn cameras, medical interviews, or transcripts of television shows or movies. It might come from work situations like doctors’ notes, legal text, or parliamentary or congressional proceedings. Text also reﬂects the demographic characteristics of the writer (or speaker): their age, gender, race, socioeconomic class can all inﬂuence the linguistic properties of the text we are processing. And ﬁnally, time matters too. Language changes over time, and for some lan- guages we have good corpora of texts from different historical periods. Because language is so situated, when developing computational models for lan- guage processing from a corpus, it’s important to consider who produced the lan- guage, in what context, for what purpose. How can a user of a dataset know all these details? The best way is for the corpus creator to build a datasheet (Gebru et al.,datasheet 2020) or data statement (Bender et al., 2021) for each corpus. A datasheet speciﬁes properties of a dataset like: Motivation: Why was the corpus collected, by whom, and who funded it? Situation: When and in what situation was the text written/spoken? For example, was there a task? Was the language originally spoken conversation, edited text, social media communication, monologue vs. dialogue? Language variety: What language (including dialect/region) was the corpus in? Speaker demographics: What was, e.g., the age or gender of the text’s authors? Collection process: How big is the data? If it is a subsample how was it sampled? Was the data collected with consent? How was the data pre-processed, and what metadata is available? Annotation process: What are the annotations, what are the demographics of the annotators, how were they trained, how was the data annotated? Distribution: Are there copyright or other intellectual property restrictions? 2.4 Simple Unix Tools for Word Tokenization Before almost any natural language processing of a text, the text has to be normal- ized, a task called text normalization. At least three tasks are commonly applied astext normalization part of any normalization process: 1. Tokenizing (segmenting) words 2. Normalizing word formats 3. Segmenting sentences 2.4 • S IMPLE UNIX TOOLS FOR WORD TOKENIZATION 17 In the next sections we walk through each of these tasks, but we’ll ﬁrst start with an easy, if somewhat naive version of word tokenization and normalization (and fre- quency computation) that can be accomplished for English solely in a single Unix command-line, inspired by Church (1994). We’ll make use of some Unix com- mands: tr, used to systematically change particular characters in the input; sort, which sorts input lines in alphabetical order; and uniq, which collapses and counts adjacent identical lines. For example let’s begin with the ‘complete words’ of Shakespeare in one ﬁle, sh.txt. We can use tr to tokenize the words by changing every sequence of non- alphabetic characters to a newline (’A-Za-z’ means alphabetic and the -c option complements to non-alphabet, so together they mean to change every non-alphabetic character into a newline. The -s (‘squeeze’) option is used to replace the result of multiple consecutive changes into a single output, so a series of non-alphabetic characters in a row would all be ‘squeezed’ into a single newline): tr -sc ’A-Za-z’ ’\n’ < sh.txt The output of this command will be: THE SONNETS by William Shakespeare From fairest creatures We ... Now that there is one word per line, we can sort the lines, and pass them touniq -cwhich will collapse and count them: tr -sc ’A-Za-z’ ’\n’ < sh.txt \| sort \| uniq -c with the following output: 1945 A 72 AARON 19 ABBESS 25 Aaron 6 Abate 1 Abates 5 Abbess 6 Abbey 3 Abbot ... Alternatively, we can collapse all the upper case to lower case: tr -sc ’A-Za-z’ ’\n’ < sh.txt \| tr A-Z a-z \| sort \| uniq -c whose output is 14725 a 97 aaron 1 abaissiez 10 abandon 18 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE 2 abandoned 2 abase 1 abash 14 abate 3 abated 3 abatement ... Now we can sort again to ﬁnd the frequent words. The -noption to sortmeans to sort numerically rather than alphabetically, and the -r option means to sort in reverse order (highest-to-lowest): tr -sc ’A-Za-z’ ’\n’ < sh.txt \| tr A-Z a-z \| sort \| uniq -c \| sort -n -r The results show that the most frequent words in Shakespeare, as in any other corpus, are the short function words like articles, pronouns, prepositions: 27378 the 26084 and 22538 i 19771 to 17481 of 14725 a 13826 you 12489 my 11318 that 11112 in ... Unix tools of this sort can be very handy in building quick word count statistics for any corpus in English. While in some versions of Unix these command-line tools also correctly handle Unicode characters and so can be used for many languages, in general for handling most languages outside English we use more sophisticated tokenization algorithms. 2.5 Word and Subword Tokenization The simple Unix tools above were ﬁne for getting rough word statistics but more sophisticated algorithms are generally necessary for tokenization, the task of seg-tokenization menting running text into words. There are roughly two classes of tokenization algorithms. In top-down tokenization, we deﬁne a standard and implement rules to implement that kind of tokenization. But more commonly instead of using words as the input to NLP algorithms we break up words into subword tokens, which can be words or parts of words orsubword tokens even individual letters. These are derived via bottom-up tokenization, in which we use simple statistics of letter sequences to come up with the vocabulary of subword tokens, and break up the input into those subwords. 2.5.1 Top-down (rule-based) tokenization While the Unix command sequence just removed all the numbers and punctuation, for most NLP applications we’ll need to keep these in our tokenization. We often 2.5 • W ORD AND SUBWORD TOKENIZATION 19 want to break off punctuation as a separate token; commas are a useful piece of infor- mation for parsers, and periods help indicate sentence boundaries. But we’ll often want to keep the punctuation that occurs word internally, in examples like m.p.h., Ph.D., AT&T, and cap’n. Special characters and numbers will need to be kept in prices ($45.55) and dates (01/02/06); we don’t want to segment that price into sepa- rate tokens of “45” and “55”. And there are URLs (https://www.stanford.edu), Twitter hashtags (#nlproc), or email addresses ([email protected]). Number expressions introduce complications; in addition to appearing at word boundaries, commas appear inside numbers in English, every three digits:555,500.50. Tokenization differs by language; languages like Spanish, French, and German, for example, use a comma to mark the decimal point, and spaces (or sometimes periods) where English puts commas, for example, 555 500,50. A tokenizer can also be used to expand clitic contractions that are marked byclitic apostrophes, converting what’re to the two tokens what are, and we’re to we are. A clitic is a part of a word that can’t stand on its own, and can only occur when it is attached to another word. Such contractions occur in other alphabetic languages, including French pronouns (j’aiand articles l’homme). Depending on the application, tokenization algorithms may also tokenize mul- tiword expressions like New York or rock ’n’ roll as a single token, which re- quires a multiword expression dictionary of some sort. Tokenization is thus inti- mately tied up with named entity recognition, the task of detecting names, dates, and organizations (Chapter 17). One commonly used tokenization standard is known as the Penn Treebank to- kenization standard, used for the parsed corpora (treebanks) released by the Lin-Penn Treebank tokenization guistic Data Consortium (LDC), the source of many useful datasets. This standard separates out clitics ( doesn’t becomes does plus n’t), keeps hyphenated words to- gether, and separates out all punctuation (to save space we’re showing visible spaces ‘ ’ between tokens, although newlines is a more common output): Input: "The San Francisco-based restaurant," they said, "doesn’t charge $10". Output: " The San Francisco-based restaurant , " they said , " does n’t charge $ 10 " . In practice, since tokenization is run before any other language processing, it needs to be very fast. For word tokenization we generally use deterministic algo- rithms based on regular expressions compiled into efﬁcient ﬁnite state automata. For example, Fig. 2.12 shows a basic regular expression that can be used to tok- enize English with the nltk.regexp tokenizefunction of the Python-based Nat- ural Language Toolkit (NLTK) (Bird et al. 2009;https://www.nltk.org). Carefully designed deterministic algorithms can deal with the ambiguities that arise, such as the fact that the apostrophe needs to be tokenized differently when used as a genitive marker (as in the book’s cover), a quotative as in ‘The other class’, she said, or in clitics like they’re. Word tokenization is more complex in languages like written Chinese, Japanese, and Thai, which do not use spaces to mark potential word-boundaries. In Chinese, for example, words are composed of characters (called hanzi in Chinese). Eachhanzi character generally represents a single unit of meaning (called a morpheme) and is pronounceable as a single syllable. Words are about 2.4 characters long on average. But deciding what counts as a word in Chinese is complex. For example, consider the following sentence: 20 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE >>> text = ’That U.S.A. poster-print costs $12.40...’ >>> pattern = r’’’(?x) # set flag to allow verbose regexps ... (?:[A-Z]\.)+ # abbreviations, e.g. U.S.A. ... \| \w+(?:-\w+) # words with optional internal hyphens ... \| \$?\d+(?:\.\d+)?%? # currency, percentages, e.g. $12.40, 82% ... \| \.\.\. # ellipsis ... \| [][.,;"’?():_‘-] # these are separate tokens; includes ], [ ... ’’’ >>> nltk.regexp_tokenize(text, pattern) [’That’, ’U.S.A.’, ’poster-print’, ’costs’, ’$12.40’, ’...’] Figure 2.12 A Python trace of regular expression tokenization in the NLTK Python-based natural language processing toolkit (Bird et al., 2009), commented for readability; the (?x) verbose ﬂag tells Python to strip comments and whitespace. Figure from Chapter 3 of Bird et al. (2009). (2.19) 姚明进入总决赛 y´ao m´ıng j`ın r`u zˇong ju´e s`ai “Yao Ming reaches the ﬁnals” As Chen et al. (2017b) point out, this could be treated as 3 words (‘Chinese Tree- bank’ segmentation): (2.20) 姚明 YaoMing 进入 reaches 总决赛 ﬁnals or as 5 words (‘Peking University’ segmentation): (2.21) 姚 Yao 明 Ming 进入 reaches 总 overall 决赛 ﬁnals Finally, it is possible in Chinese simply to ignore words altogether and use characters as the basic elements, treating the sentence as a series of 7 characters: (2.22) 姚 Yao 明 Ming 进 enter 入 enter 总 overall 决 decision 赛 game In fact, for most Chinese NLP tasks it turns out to work better to take characters rather than words as input, since characters are at a reasonable semantic level for most applications, and since most word standards, by contrast, result in a huge vo- cabulary with large numbers of very rare words (Li et al., 2019b). However, for Japanese and Thai the character is too small a unit, and so algo- rithms for word segmentation are required. These can also be useful for Chineseword segmentation in the rare situations where word rather than character boundaries are required. For these situations we can use the subword tokenization algorithms introduced in the next section. 2.5.2 Byte-Pair Encoding: A Bottom-up Tokenization Algorithm There is a third option to tokenizing text, one that is most commonly used by large language models. Instead of deﬁning tokens as words (whether delimited by spaces or more complex algorithms), or as characters (as in Chinese), we can use our data to automatically tell us what the tokens should be. This is especially useful in dealing with unknown words, an important problem in language processing. As we will see in the next chapter, NLP algorithms often learn some facts about language from one corpus (a training corpus) and then use these facts to make decisions about a separate test corpus and its language. Thus if our training corpus contains, say the 2.5 • W ORD AND SUBWORD TOKENIZATION 21 words low, new, newer, but not lower, then if the word lower appears in our test corpus, our system will not know what to do with it. To deal with this unknown word problem, modern tokenizers automatically in- duce sets of tokens that include tokens smaller than words, called subwords. Sub-subwords words can be arbitrary substrings, or they can be meaning-bearing units like the morphemes -est or -er. (A morpheme is the smallest meaning-bearing unit of a lan- guage; for example the word unwashable has the morphemes un-, wash, and -able.) In modern tokenization schemes, most tokens are words, but some tokens are fre- quently occurring morphemes or other subwords like -er. Every unseen word like lower can thus be represented by some sequence of known subword units, such as low and er, or even as a sequence of individual letters if necessary. Most tokenization schemes have two parts: a token learner, and a token seg- menter. The token learner takes a raw training corpus (sometimes roughly pre- separated into words, for example by whitespace) and induces a vocabulary, a set of tokens. The token segmenter takes a raw test sentence and segments it into the tokens in the vocabulary. Two algorithms are widely used: byte-pair encoding (Sennrich et al., 2016), and unigram language modeling (Kudo, 2018), There is also a SentencePiece library that includes implementations of both of these (Kudo and Richardson, 2018a), and people often use the name SentencePiece to simply mean unigram language modeling tokenization. In this section we introduce the simplest of the three, the byte-pair encoding or BPE algorithm (Sennrich et al., 2016); see Fig. 2.13. The BPE token learner beginsBPE with a vocabulary that is just the set of all individual characters. It then examines the training corpus, chooses the two symbols that are most frequently adjacent (say ‘A’, ‘B’), adds a new merged symbol ‘AB’ to the vocabulary, and replaces every adjacent ’A’ ’B’ in the corpus with the new ‘AB’. It continues to count and merge, creating new longer and longer character strings, until k merges have been done creating k novel tokens; k is thus a parameter of the algorithm. The resulting vocabulary consists of the original set of characters plus k new symbols. The algorithm is usually run inside words (not merging across word boundaries), so the input corpus is ﬁrst white-space-separated to give a set of strings, each corre- sponding to the characters of a word, plus a special end-of-word symbol , and its counts. Let’s see its operation on the following tiny input corpus of 18 word tokens with counts for each word (the word low appears 5 times, the word newer 6 times, and so on), which would have a starting vocabulary of 11 letters: corpus vocabulary 5 l o w , d, e, i, l, n, o, r, s, t, w 2 l o w e s t 6 n e w e r 3 w i d e r 2 n e w The BPE algorithm ﬁrst counts all pairs of adjacent symbols: the most frequent is the pair e r because it occurs in newer (frequency of 6) and wider (frequency of 3) for a total of 9 occurrences. 2 We then merge these symbols, treating er as one symbol, and count again: 2 Note that there can be ties; we could have instead chosen to merge r ﬁrst, since that also has a frequency of 9. 22 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE corpus vocabulary 5 l o w , d, e, i, l, n, o, r, s, t, w, er 2 l o w e s t 6 n e w er 3 w i d er 2 n e w Now the most frequent pair is er , which we merge; our system has learned that there should be a token for word-ﬁnal er, represented as er : corpus vocabulary 5 l o w , d, e, i, l, n, o, r, s, t, w, er, er 2 l o w e s t 6 n e w er 3 w i d er 2 n e w Next n e(total count of 8) get merged to ne: corpus vocabulary 5 l o w , d, e, i, l, n, o, r, s, t, w, er, er , ne 2 l o w e s t 6 ne w er 3 w i d er 2 ne w If we continue, the next merges are: merge current vocabulary (ne, w) , d, e, i, l, n, o, r, s, t, w, er, er , ne, new (l, o) , d, e, i, l, n, o, r, s, t, w, er, er , ne, new, lo (lo, w) , d, e, i, l, n, o, r, s, t, w, er, er , ne, new, lo, low (new, er ) , d, e, i, l, n, o, r, s, t, w, er, er , ne, new, lo, low, newer (low, ) , d, e, i, l, n, o, r, s, t, w, er, er , ne, new, lo, low, newer , low function BYTE -PAIR ENCODING (strings C, number of merges k) returns vocab V V←all unique characters in C # initial set of tokens is characters for i = 1 to k do # merge tokens k times tL, tR ←Most frequent pair of adjacent tokens in C tNEW ←tL + tR # make new token by concatenating V←V + tNEW # update the vocabulary Replace each occurrence of tL, tR in C with tNEW # and update the corpus return V Figure 2.13 The token learner part of the BPE algorithm for taking a corpus broken up into individual characters or bytes, and learning a vocabulary by iteratively merging tokens. Figure adapted from Bostrom and Durrett (2020). Once we’ve learned our vocabulary, the token segmenter is used to tokenize a test sentence. The token segmenter just runs on the merges we have learned from the training data on the test data. It runs them greedily, in the order we learned them. (Thus the frequencies in the test data don’t play a role, just the frequencies in the training data). So ﬁrst we segment each test sentence word into characters. Then we apply the ﬁrst rule: replace every instance of e rin the test corpus with er, and 2.6 • W ORD NORMALIZATION , LEMMATIZATION AND STEMMING 23 then the second rule: replace every instance of er in the test corpus with er , and so on. By the end, if the test corpus contained the character sequence n e w e r , it would be tokenized as a full word. But the characters of a new (unknown) word like l o w e r would be merged into the two tokens low er . Of course in real settings BPE is run with many thousands of merges on a very large input corpus. The result is that most words will be represented as full symbols, and only the very rare words (and unknown words) will have to be represented by their parts. 2.6 Word Normalization, Lemmatization and Stemming Word normalization is the task of putting words or tokens in a standard format. Thenormalization simplest case of word normalization is case folding. Mapping everything to lowercase folding case means that Woodchuck and woodchuck are represented identically, which is very helpful for generalization in many tasks, such as information retrieval or speech recognition. For sentiment analysis and other text classiﬁcation tasks, information extraction, and machine translation, by contrast, case can be quite helpful and case folding is generally not done. This is because maintaining the difference between, for example, US the country and us the pronoun can outweigh the advantage in generalization that case folding would have provided for other words. Sometimes we produce both cased (i.e. including both upper and lower case words or tokens) and uncased versions of language models. Systems that use BPE or other kinds of bottom-up tokenization may do no fur- ther word normalization. In other NLP systems, we may want to do further nor- malizations, like choosing a single normal form for words with multiple forms like USA and US or uh-huh and uhhuh. This standardization may be valuable, despite the spelling information that is lost in the normalization process. For information retrieval or information extraction about the US, we might want to see information from documents whether they mention the USor the USA. 2.6.1 Lemmatization For other natural language processing situations we also want two morphologically different forms of a word to behave similarly. For example in web search, someone may type the string woodchucks but a useful system might want to also return pages that mention woodchuck with no s. This is especially common in morphologically complex languages like Polish, where for example the word Warsaw has different endings when it is the subject (Warszawa), or after a preposition like “in Warsaw” (w Warszawie), or “to Warsaw” (do Warszawy), and so on. Lemmatization is the tasklemmatization of determining that two words have the same root, despite their surface differences. The words am, are, and is have the shared lemma be; the words dinner and dinners both have the lemma dinner. Lemmatizing each of these forms to the same lemma will let us ﬁnd all mentions of words in Polish like Warsaw. The lemmatized form of a sentence like He is reading detective stories would thus be He be read detective story. How is lemmatization done? The most sophisticated methods for lemmatization involve complete morphological parsing of the word. Morphology is the study of the way words are built up from smaller meaning-bearing units called morphemes.morpheme Two broad classes of morphemes can be distinguished: stems—the central mor-stem 24 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE pheme of the word, supplying the main meaning—and afﬁxes—adding “additional”afﬁx meanings of various kinds. So, for example, the word fox consists of one morpheme (the morpheme fox) and the word cats consists of two: the morpheme cat and the morpheme -s. A morphological parser takes a word like cats and parses it into the two morphemes cat and s, or parses a Spanish word like amaren (‘if in the future they would love’) into the morphemeamar ‘to love’, and the morphological features 3PL (third person plural) and future subjunctive. Stemming: The Porter Stemmer Lemmatization algorithms can be complex. For this reason we sometimes make use of a simpler but cruder method, which mainly consists of chopping off word- ﬁnal afﬁxes. This naive version of morphological analysis is called stemming. Forstemming example, the classic Porter stemmer (Porter, 1980), when applied to the followingPorter stemmer paragraph: This was not the map we found in Billy Bones’s chest, but an accurate copy, complete in all things-names and heights and soundings-with the single exception of the red crosses and the written notes. produces the following stemmed output: Thi wa not the map we found in Billi Bone s chest but an accur copi complet in all thing name and height and sound with the singl except of the red cross and the written note The algorithm is based on rewrite rules run in series, with the output of each pass fed as input to the next pass. Some sample rules (more athttps://tartarus.org/ martin/PorterStemmer/): ATIONAL → ATE (e.g., relational →relate) ING → ϵ if the stem contains a vowel (e.g., motoring →motor) SSES → SS (e.g., grasses →grass) Simple stemmers can be useful in cases where we need to collapse across dif- ferent variants of the same lemma. Nonetheless, they are less commonly used in modern systems since they commit errors of both over-generalizing (lemmatizing policyto police) and under-generalizing (not lemmatizingEuropeanto Europe) (Krovetz, 1993). 2.7 Sentence Segmentation Sentence segmentation is another important step in text processing. The most use-sentence segmentation ful cues for segmenting a text into sentences are punctuation, like periods, question marks, and exclamation points. Question marks and exclamation points are rela- tively unambiguous markers of sentence boundaries. Periods, on the other hand, are more ambiguous. The period character “.” is ambiguous between a sentence bound- ary marker and a marker of abbreviations likeMr. or Inc. The previous sentence that you just read showed an even more complex case of this ambiguity, in which the ﬁnal period of Inc. marked both an abbreviation and the sentence boundary marker. For this reason, sentence tokenization and word tokenization may be addressed jointly. 2.8 • M INIMUM EDIT DISTANCE 25 In general, sentence tokenization methods work by ﬁrst deciding (based on rules or machine learning) whether a period is part of the word or is a sentence-boundary marker. An abbreviation dictionary can help determine whether the period is part of a commonly used abbreviation; the dictionaries can be hand-built or machine- learned (Kiss and Strunk, 2006), as can the ﬁnal sentence splitter. In the Stanford CoreNLP toolkit (Manning et al., 2014), for example sentence splitting is rule-based, a deterministic consequence of tokenization; a sentence ends when a sentence-ending punctuation (., !, or ?) is not already grouped with other characters into a token (such as for an abbreviation or number), optionally followed by additional ﬁnal quotes or brackets. 2.8 Minimum Edit Distance Much of natural language processing is concerned with measuring how similar two strings are. For example in spelling correction, the user typed some erroneous string—let’s say graffe–and we want to know what the user meant. The user prob- ably intended a word that is similar to graffe. Among candidate similar words, the word giraffe, which differs by only one letter from graffe, seems intuitively to be more similar than, say grail or graf, which differ in more letters. Another example comes from coreference, the task of deciding whether two strings such as the following refer to the same entity: Stanford Arizona Cactus Garden Stanford University Arizona Cactus Garden Again, the fact that these two strings are very similar (differing by only one word) seems like useful evidence for deciding that they might be coreferent. Finally, string similarity is commonly used to measure the quality of the transcription produced by a speech recognition system, by asking how similar (in words) the transcript is to a reference transcript. A system whose transcript is off by many words is measurably worse than one which is only off by a few words. Edit distance gives us a way to quantify these intuitions about string similarity. More formally, the minimum edit distance between two strings is deﬁned as theminimum edit distance minimum number of editing operations (operations like insertion, deletion, substitu- tion) needed to transform one string into another. The gap between intention and execution, for example, is 5 (delete an i, substi- tute e for n, substitute x for t, insert c, substitute u for n). It’s much easier to see this by looking at the most important visualization for string distances, analignmentalignment between the two strings, shown in Fig. 2.14. Given two sequences, an alignment is a correspondence between substrings of the two sequences. Thus, we say I aligns with the empty string, N with E, and so on. Beneath the aligned strings is another representation; a series of symbols expressing an operation list for converting the top string into the bottom string: d for deletion, s for substitution, i for insertion. We can also assign a particular cost or weight to each of these operations. The Levenshtein distance between two sequences is the simplest weighting factor in which each of the three operations has a cost of 1 (Levenshtein, 1966)—we assume that the substitution of a letter for itself, for example,tfor t, has zero cost. The Lev- enshtein distance between intention and execution is 5. Levenshtein also proposed an alternative version of his metric in which each insertion or deletion has a cost of 1 and substitutions are not allowed. (This is equivalent to allowing substitution, but 26 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE I N T E * N T I O N \| \| \| \| \| \| \| \| \| \| * E X E C U T I O N d s s i s Figure 2.14 Representing the minimum edit distance between two strings as analignment. The ﬁnal row gives the operation list for converting the top string into the bottom string: d for deletion, s for substitution, i for insertion. giving each substitution a cost of 2 since any substitution can be represented by one insertion and one deletion). Using this version, the Levenshtein distance between intention and execution is 8. 2.8.1 The Minimum Edit Distance Algorithm How do we ﬁnd the minimum edit distance? We can think of this as a search task, in which we are searching for the shortest path—a sequence of edits—from one string to another. n t e n t i o n i n t e c n t i o n i n x e n t i o n del ins subst i n t e n t i o n Figure 2.15 Finding the edit distance viewed as a search problem The space of all possible edits is enormous, so we can’t search naively. However, lots of distinct edit paths will end up in the same state (string), so rather than recom- puting all those paths, we could just remember the shortest path to a state each time we saw it. We can do this by usingdynamic programming. Dynamic programmingdynamic programming is the name for a class of algorithms, ﬁrst introduced by Bellman (1957), that apply a table-driven method to solve problems by combining solutions to subproblems. Some of the most commonly used algorithms in natural language processing make use of dynamic programming, such as the Viterbi algorithm (Chapter 17) and the CKY algorithm for parsing (Chapter 18). The intuition of a dynamic programming problem is that a large problem can be solved by properly combining the solutions to various subproblems. Consider the shortest path of transformed words that represents the minimum edit distance between the strings intention and execution shown in Fig. 2.16. Imagine some string (perhaps it isexention) that is in this optimal path (whatever it is). The intuition of dynamic programming is that if exention is in the optimal operation list, then the optimal sequence must also include the optimal path from intention to exention. Why? If there were a shorter path from intention to exention, then we could use it instead, resulting in a shorter overall path, and the optimal sequence wouldn’t be optimal, thus leading to a contradiction. The minimum edit distance algorithm was named by Wagner and Fischer minimum edit distance algorithm (1974) but independently discovered by many people (see the Historical Notes sec- tion of Chapter 17). Let’s ﬁrst deﬁne the minimum edit distance between two strings. Given two strings, the source string X of length n, and target string Y of length m, we’ll deﬁne 2.8 • M INIMUM EDIT DISTANCE 27 n t e n t i o n i n t e n t i o n e t e n t i o n e x e n t i o n e x e n u t i o n e x e c u t i o n delete i substitute n by e substitute t by x insert u substitute n by c Figure 2.16 Path from intention to execution. D[i,j] as the edit distance between X[1..i] and Y [1..j], i.e., the ﬁrst i characters of X and the ﬁrst j characters of Y . The edit distance between X and Y is thus D[n,m]. We’ll use dynamic programming to compute D[n,m] bottom up, combining so- lutions to subproblems. In the base case, with a source substring of length i but an empty target string, going from i characters to 0 requires i deletes. With a target substring of length j but an empty source going from 0 characters to j characters requires j inserts. Having computed D[i,j] for small i,j we then compute larger D[i,j] based on previously computed smaller values. The value of D[i,j] is com- puted by taking the minimum of the three possible paths through the matrix which arrive there: D[i,j] =min    D[i −1,j]+ del-cost(source[i]) D[i,j −1]+ ins-cost(target[ j]) D[i −1,j −1]+ sub-cost(source[i],target[ j]) (2.23) We mentioned above two versions of Levenshtein distance, one in which substitu- tions cost 1 and one in which substitutions cost 2 (i.e., are equivalent to an insertion plus a deletion). Let’s here use that second version of Levenshtein distance in which the insertions and deletions each have a cost of 1 (ins-cost( ·) = del-cost(·) = 1), and substitutions have a cost of 2 (except substitution of identical letters has zero cost). Under this version of Levenshtein, the computation for D[i,j] becomes: D[i,j] =min    D[i −1,j]+ 1 D[i,j −1]+ 1 D[i −1,j −1]+ {2; if source[i] ̸= target[ j] 0; if source[i] =target[ j] (2.24) The algorithm is summarized in Fig. 2.17; Fig. 2.18 shows the results of applying the algorithm to the distance between intention and execution with the version of Levenshtein in Eq. 2.24. Alignment Knowing the minimum edit distance is useful for algorithms like ﬁnd- ing potential spelling error corrections. But the edit distance algorithm is important in another way; with a small change, it can also provide the minimum cost align- ment between two strings. Aligning two strings is useful throughout speech and language processing. In speech recognition, minimum edit distance alignment is used to compute the word error rate (Chapter 16). Alignment plays a role in ma- chine translation, in which sentences in a parallel corpus (a corpus with a text in two languages) need to be matched to each other. To extend the edit distance algorithm to produce an alignment, we can start by visualizing an alignment as a path through the edit distance matrix. Figure 2.19 28 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE function MIN-EDIT-DISTANCE (source, target) returns min-distance n←LENGTH (source) m←LENGTH (target) Create a distance matrix D[n+1,m+1] # Initialization: the zeroth row and column is the distance from the empty string D[0,0] = 0 for each row i from 1 to n do D[i,0]←D[i-1,0] + del-cost(source[i]) for each column j from 1 to m do D[0,j]←D[0, j-1] + ins-cost(target[j]) # Recurrence relation: for each row i from 1 to n do for each column j from 1 to m do D[i, j]←MIN( D[i−1, j] + del-cost(source[i]), D[i−1, j−1] + sub-cost(source[i], target[j]), D[i, j−1] + ins-cost(target[j])) # Termination return D[n,m] Figure 2.17 The minimum edit distance algorithm, an example of the class of dynamic programming algorithms. The various costs can either be ﬁxed (e.g., ∀x,ins-cost(x) =1) or can be speciﬁc to the letter (to model the fact that some letters are more likely to be in- serted than others). We assume that there is no cost for substituting a letter for itself (i.e., sub-cost(x,x) =0). Src\Tar # e x e c u t i o n # 0 1 2 3 4 5 6 7 8 9 i 1 2 3 4 5 6 7 6 7 8 n 2 3 4 5 6 7 8 7 8 7 t 3 4 5 6 7 8 7 8 9 8 e 4 3 4 5 6 7 8 9 10 9 n 5 4 5 6 7 8 9 10 11 10 t 6 5 6 7 8 9 8 9 10 11 i 7 6 7 8 9 10 9 8 9 10 o 8 7 8 9 10 11 10 9 8 9 n 9 8 9 10 11 12 11 10 9 8 Figure 2.18 Computation of minimum edit distance between intention and execution with the algorithm of Fig. 2.17, using Levenshtein distance with cost of 1 for insertions or dele- tions, 2 for substitutions. shows this path with boldfaced cells. Each boldfaced cell represents an alignment of a pair of letters in the two strings. If two boldfaced cells occur in the same row, there will be an insertion in going from the source to the target; two boldfaced cells in the same column indicate a deletion. Figure 2.19 also shows the intuition of how to compute this alignment path. The computation proceeds in two steps. In the ﬁrst step, we augment the minimum edit distance algorithm to store backpointers in each cell. The backpointer from a cell points to the previous cell (or cells) that we came from in entering the current cell. We’ve shown a schematic of these backpointers in Fig. 2.19. Some cells have mul- 2.9 • S UMMARY 29 tiple backpointers because the minimum extension could have come from multiple previous cells. In the second step, we perform a backtrace. In a backtrace, we startbacktrace from the last cell (at the ﬁnal row and column), and follow the pointers back through the dynamic programming matrix. Each complete path between the ﬁnal cell and the initial cell is a minimum distance alignment. Exercise 2.7 asks you to modify the minimum edit distance algorithm to store the pointers and compute the backtrace to output an alignment. # e x e c u t i o n # 0 ←1 ←2 ←3 ←4 ←5 ←6 ←7 ←8 ←9 i ↑1 ↖←↑2 ↖←↑3 ↖←↑4 ↖←↑5 ↖←↑6 ↖←↑7 ↖6 ←7 ←8 n ↑2 ↖←↑3 ↖←↑4 ↖←↑5 ↖←↑6 ↖←↑7 ↖←↑8 ↑7 ↖←↑8 ↖7 t ↑3 ↖←↑4 ↖←↑5 ↖←↑6 ↖←↑7 ↖←↑8 ↖7 ←↑8 ↖←↑9 ↑8 e ↑4 ↖3 ←4 ↖←5 ←6 ←7 ←↑8 ↖←↑9 ↖←↑10 ↑9 n ↑5 ↑4 ↖←↑5 ↖←↑6 ↖←↑7 ↖←↑8 ↖←↑9 ↖←↑10 ↖←↑11 ↖↑10 t ↑6 ↑5 ↖←↑6 ↖←↑7 ↖←↑8 ↖←↑9 ↖8 ←9 ←10 ←↑11 i ↑7 ↑6 ↖←↑7 ↖←↑8 ↖←↑9 ↖←↑10 ↑9 ↖8 ←9 ←10 o ↑8 ↑7 ↖←↑8 ↖←↑9 ↖←↑10 ↖←↑11 ↑10 ↑9 ↖8 ←9 n ↑9 ↑8 ↖←↑9 ↖←↑10 ↖←↑11 ↖←↑12 ↑11 ↑10 ↑9 ↖8 Figure 2.19 When entering a value in each cell, we mark which of the three neighboring cells we came from with up to three arrows. After the table is full we compute an alignment (minimum edit path) by using a backtrace, starting at the 8 in the lower-right corner and following the arrows back. The sequence of bold cells represents one possible minimum cost alignment between the two strings, again using Levenshtein distance with cost of 1 for insertions or deletions, 2 for substitutions. Diagram design after Gusﬁeld (1997). While we worked our example with simple Levenshtein distance, the algorithm in Fig. 2.17 allows arbitrary weights on the operations. For spelling correction, for example, substitutions are more likely to happen between letters that are next to each other on the keyboard. The Viterbi algorithm is a probabilistic extension of minimum edit distance. Instead of computing the “minimum edit distance” between two strings, Viterbi computes the “maximum probability alignment” of one string with another. We’ll discuss this more in Chapter 17. 2.9 Summary This chapter introduced a fundamental tool in language processing, the regular ex- pression, and showed how to perform basic text normalization tasks including word segmentation and normalization, sentence segmentation, and stemming. We also introduced the important minimum edit distance algorithm for comparing strings. Here’s a summary of the main points we covered about these ideas: • The regular expression language is a powerful tool for pattern-matching. • Basic operations in regular expressions include concatenation of symbols, disjunction of symbols ([], \|), counters (, +, and {n,m}), anchors (ˆ, $) and precedence operators ((,)). • Word tokenization and normalization are generally done by cascades of simple regular expression substitutions or ﬁnite automata. • The Porter algorithm is a simple and efﬁcient way to dostemming, stripping off afﬁxes. It does not have high accuracy but may be useful for some tasks. 30 CHAPTER 2 • R EGULAR EXPRESSIONS , TOKENIZATION , EDIT DISTANCE • The minimum edit distance between two strings is the minimum number of operations it takes to edit one into the other. Minimum edit distance can be computed by dynamic programming, which also results in an alignment of the two strings. Bibliographical and Historical Notes Kleene 1951; 1956 ﬁrst deﬁned regular expressions and the ﬁnite automaton, based on the McCulloch-Pitts neuron. Ken Thompson was one of the ﬁrst to build regular expressions compilers into editors for text searching (Thompson, 1968). His edi- tor ed included a command “g/regular expression/p”, or Global Regular Expression Print, which later became the Unix greputility. Text normalization algorithms have been applied since the beginning of the ﬁeld. One of the earliest widely used stemmers was Lovins (1968). Stemming was also applied early to the digital humanities, by Packard (1973), who built an afﬁx-stripping morphological parser for Ancient Greek. Currently a wide vari- ety of code for tokenization and normalization is available, such as the Stanford Tokenizer (https://nlp.stanford.edu/software/tokenizer.shtml) or spe- cialized tokenizers for Twitter (O’Connor et al., 2010), or for sentiment ( http: //sentiment.christopherpotts.net/tokenizing.html). See Palmer (2012) for a survey of text preprocessing. NLTK is an essential tool that offers both useful Python libraries (https://www.nltk.org) and textbook descriptions (Bird et al., 2009) of many algorithms including text normalization and corpus interfaces. For more on Herdan’s law and Heaps’ Law, see Herdan (1960, p. 28), Heaps (1978), Egghe (2007) and Baayen (2001); For more on edit distance, see Gusﬁeld (1997). Our example measuring the edit distance from ‘intention’ to ‘execution’ was adapted from Kruskal (1983). There are various publicly available packages to compute edit distance, including Unix diffand the NIST scliteprogram (NIST, 2005). In his autobiography Bellman (1984) explains how he originally came up with the term dynamic programming: “...The 1950s were not good years for mathematical research. [the] Secretary of Defense ...had a pathological fear and hatred of the word, research... I decided therefore to use the word, “programming”. I wanted to get across the idea that this was dynamic, this was multi- stage... I thought, let’s ... take a word that has an absolutely precise meaning, namely dynamic... it’s impossible to use the word, dynamic, in a pejorative sense. Try thinking of some combination that will pos- sibly give it a pejorative meaning. It’s impossible. Thus, I thought dynamic programming was a good name. It was something not even a Congressman could object to.” Exercises 2.1 Write regular expressions for the following languages. 1. the set of all alphabetic strings; 2. the set of all lower case alphabetic strings ending in a b; EXERCISES 31 3. the set of all strings from the alphabet a,b such that each a is immedi- ately preceded by and immediately followed by a b; 2.2 Write regular expressions for the following languages. By “word”, we mean an alphabetic string separated from other words by whitespace, any relevant punctuation, line breaks, and so forth. 1. the set of all strings with two consecutive repeated words (e.g., “Hum- bert Humbert” and “the the” but not “the bug” or “the big bug”); 2. all strings that start at the beginning of the line with an integer and that end at the end of the line with a word; 3. all strings that have both the word grotto and the word raven in them (but not, e.g., words like grottos that merely contain the word grotto); 4. write a pattern that places the ﬁrst word of an English sentence in a register. Deal with punctuation. 2.3 Implement an ELIZA-like program, using substitutions such as those described on page 12. You might want to choose a different domain than a Rogerian psy- chologist, although keep in mind that you would need a domain in which your program can legitimately engage in a lot of simple repetition. 2.4 Compute the edit distance (using insertion cost 1, deletion cost 1, substitution cost 1) of “leda” to “deal”. Show your work (using the edit distance grid). 2.5 Figure out whether drive is closer to brief or to divers and what the edit dis- tance is to each. You may use any version of distance that you like. 2.6 Now implement a minimum edit distance algorithm and use your hand-computed results to check your code. 2.7 Augment the minimum edit distance algorithm to output an alignment; you will need to store pointers and add a stage to compute the backtrace. 32 CHAPTER 3 • N- GRAM LANGUAGE MODELS CHAPTER 3 N-gram Language Models “You are uniformly charming!” cried he, with a smile of associating and now and then I bowed and they perceived a chaise and four to wish for. Random sentence generated from a Jane Austen trigram model Predicting is difﬁcult—especially about the future, as the old quip goes. But how about predicting something that seems much easier, like the next word someone is going to say? What word, for example, is likely to follow The water of Walden Pond is so beautifully ... You might conclude that a likely word is blue, or green, or clear, but probably not refrigerator nor this. In this chapter we formalize this intuition by intro- ducing language models or LMs. A language model is a machine learning modellanguage model LM that predicts upcoming words. More formally, a language model assigns a prob- ability to each possible next word, or equivalently gives a probability distribution over possible next works. Language models can also assign a probability to an entire sentence. Thus an LM could tell us that the following sequence has a much higher probability of appearing in a text: all of a sudden I notice three guys standing on the sidewalk than does this same set of words in a different order: on guys all I of notice sidewalk three a sudden standing the Why would we want to predict upcoming words, or know the probability of a sen- tence? One reason is for generation: choosing contextually better words. For ex- ample we can correct grammar or spelling errors like Their are two midterms, in which There was mistyped as Their, or Everything has improve, in which improve should have been improved. The phrase There are is more probable than Their are, and has improvedthan has improve, so a language model can help users select the more grammatical variant. Or for a speech system to recognize that you said I will be back soonish and not I will be bassoon dish, it helps to know that back soonish is a more probable sequence. Language models can also help in augmentative and alternative communication (Trnka et al. 2007, Kane et al. 2017). People can use AAC systems if they are physically unable toAAC speak or sign but can instead use eye gaze or other movements to select words from a menu. Word prediction can be used to suggest likely words for the menu. Word prediction is also central to NLP for another reason:large language mod- els are built just by training them to predict words!! As we’ll see in chapters 7-9, large language models learn an enormous amount about language solely from being trained to predict upcoming words from neighboring words. In this chapter we introduce the simplest kind of language model: the n-gramn-gram 3.1 • N-G RAMS 33 language model. An n-gram is a sequence of n words: a 2-gram (which we’ll call bigram) is a two-word sequence of words like The water, or water of, and a 3- gram (a trigram) is a three-word sequence of words like The water of, or water of Walden. But we also (in a bit of terminological ambiguity) use the word ‘n- gram’ to mean a probabilistic model that can estimate the probability of a word given the n-1 previous words, and thereby also to assign probabilities to entire sequences. In later chapters we will introduce the much more powerful neural large lan- guage models, based on the transformer architecture of Chapter 9. But because n-grams have a remarkably simple and clear formalization, we use them to intro- duce some major concepts of large language modeling, including training and test sets, perplexity, sampling, and interpolation. 3.1 N-Grams Let’s begin with the task of computing P(w\|h), the probability of a word w given some history h. Suppose the history h is “The water of Walden Pond is so beautifully ” and we want to know the probability that the next word isblue: P(blue\|The water of Walden Pond is so beautifully) (3.1) One way to estimate this probability is directly from relative frequency counts: take a very large corpus, count the number of times we seeThe water of Walden Pond is so beautifully, and count the number of times this is followed byblue. This would be answering the question “Out of the times we saw the history h, how many times was it followed by the word w”, as follows: P(blue\|The water of Walden Pond is so beautifully) = C(The water of Walden Pond is so beautifully blue) C(The water of Walden Pond is so beautifully) (3.2) If we had a large enough corpus, we could compute these two counts and estimate the probability from Eq. 3.2. But even the entire web isn’t big enough to give us good estimates for counts of entire sentences. This is because language is creative; new sentences are invented all the time, and we can’t expect to get accurate counts for such large objects as entire sentences. For this reason, we’ll need more clever ways to estimate the probability of a word w given a history h, or the probability of an entire word sequence W. Let’s start with some notation. First, throughout this chapter we’ll continue to refer to words, although in practice we usually compute language models over to- kens like the BPE tokens of page 20. To represent the probability of a particular random variable Xi taking on the value “the”, or P(Xi = “the”), we will use the simpliﬁcation P(the). We’ll represent a sequence of n words either as w1 ... wn or w1:n. Thus the expression w1:n−1 means the string w1,w2,...,wn−1, but we’ll also be using the equivalent notation w<n, which can be read as “all the elements of w from w1 up to and including wn−1”. For the joint probability of each word in a se- quence having a particular value P(X1 = w1,X2 = w2,X3 = w3,...,Xn = wn) we’ll use P(w1,w2,...,wn). Now, how can we compute probabilities of entire sequences likeP(w1,w2,...,wn)? One thing we can do is decompose this probability using the chain rule of proba- 34 CHAPTER 3 • N- GRAM LANGUAGE MODELS bility: P(X1...Xn) = P(X1)P(X2\|X1)P(X3\|X1:2)... P(Xn\|X1:n−1) = n∏ k=1 P(Xk\|X1:k−1) (3.3) Applying the chain rule to words, we get P(w1:n) = P(w1)P(w2\|w1)P(w3\|w1:2)... P(wn\|w1:n−1) = n∏ k=1 P(wk\|w1:k−1) (3.4) The chain rule shows the link between computing the joint probability of a sequence and computing the conditional probability of a word given previous words. Equa- tion 3.4 suggests that we could estimate the joint probability of an entire sequence of words by multiplying together a number of conditional probabilities. But using the chain rule doesn’t really seem to help us! We don’t know any way to compute the exact probability of a word given a long sequence of preceding words,P(wn\|w1:n−1). As we said above, we can’t just estimate by counting the number of times every word occurs following every long string in some corpus, because language is creative and any particular context might have never occurred before! 3.1.1 The Markov assumption The intuition of the n-gram model is that instead of computing the probability of a word given its entire history, we can approximate the history by just the last few words. The bigram model, for example, approximates the probability of a word givenbigram all the previous words P(wn\|w1:n−1) by using only the conditional probability given the preceding word P(wn\|wn−1). In other words, instead of computing the probabil- ity P(blue\|The water of Walden Pond is so beautifully) (3.5) we approximate it with the probability P(blue\|beautifully) (3.6) When we use a bigram model to predict the conditional probability of the next word, we are thus making the following approximation: P(wn\|w1:n−1) ≈P(wn\|wn−1) (3.7) The assumption that the probability of a word depends only on the previous word is called a Markov assumption. Markov models are the class of probabilistic modelsMarkov that assume we can predict the probability of some future unit without looking too far into the past. We can generalize the bigram (which looks one word into the past) to the trigram (which looks two words into the past) and thus to the n-gram (whichn-gram looks n −1 words into the past). Let’s see a general equation for this n-gram approximation to the conditional probability of the next word in a sequence. We’ll use N here to mean the n-gram 3.1 • N-G RAMS 35 size, so N = 2 means bigrams and N = 3 means trigrams. Then we approximate the probability of a word given its entire context as follows: P(wn\|w1:n−1) ≈P(wn\|wn−N+1:n−1) (3.8) Given the bigram assumption for the probability of an individual word, we can com- pute the probability of a complete word sequence by substituting Eq. 3.7 into Eq. 3.4: P(w1:n) ≈ n∏ k=1 P(wk\|wk−1) (3.9) 3.1.2 How to estimate probabilities How do we estimate these bigram or n-gram probabilities? An intuitive way to estimate probabilities is called maximum likelihood estimation or MLE. We get maximum likelihood estimation the MLE estimate for the parameters of an n-gram model by getting counts from a corpus, and normalizing the counts so that they lie between 0 and 1. For proba-normalize bilistic models, normalizing means dividing by some total count so that the resulting probabilities fall between 0 and 1 and sum to 1. For example, to compute a particular bigram probability of a word wn given a previous word wn−1, we’ll compute the count of the bigramC(wn−1wn) and normal- ize by the sum of all the bigrams that share the same ﬁrst word wn−1: P(wn\|wn−1) = C(wn−1wn)∑ w C(wn−1w) (3.10) We can simplify this equation, since the sum of all bigram counts that start with a given wordwn−1 must be equal to the unigram count for that wordwn−1 (the reader should take a moment to be convinced of this): P(wn\|wn−1) =C(wn−1wn) C(wn−1) (3.11) Let’s work through an example using a mini-corpus of three sentences. We’ll ﬁrst need to augment each sentence with a special symbol <s> at the beginning of the sentence, to give us the bigram context of the ﬁrst word. We’ll also need a special end-symbol </s>.1 <s> I am Sam </s> <s> Sam I am </s> <s> I do not like green eggs and ham </s> Here are the calculations for some of the bigram probabilities from this corpus P(I\|<s>) =2 3 = 0.67 P(Sam\|<s>) =1 3 = 0.33 P(am\|I) =2 3 = 0.67 P(</s>\|Sam) =1 2 = 0.5 P(Sam\|am) =1 2 = 0.5 P(do\|I) =1 3 = 0.33 For the general case of MLE n-gram parameter estimation: P(wn\|wn−N+1:n−1) =C(wn−N+1:n−1 wn) C(wn−N+1:n−1) (3.12) 1 We need the end-symbol to make the bigram grammar a true probability distribution. Without an end- symbol, instead of the sentence probabilities of all sentences summing to one, the sentence probabilities for all sentencesof a given lengthwould sum to one. This model would deﬁne an inﬁnite set of probability distributions, with one distribution per sentence length. See Exercise 3.5. 36 CHAPTER 3 • N- GRAM LANGUAGE MODELS Equation 3.12 (like Eq. 3.11) estimates the n-gram probability by dividing the observed frequency of a particular sequence by the observed frequency of a preﬁx. This ratio is called a relative frequency. We said above that this use of relativerelative frequency frequencies as a way to estimate probabilities is an example of maximum likelihood estimation or MLE. In MLE, the resulting parameter set maximizes the likelihood of the training set T given the model M (i.e., P(T \|M)). For example, suppose the word Chinese occurs 400 times in a corpus of a million words. What is the probability that a random word selected from some other text of, say, a million words will be the word Chinese? The MLE of its probability is 400 1000000 or 0.0004. Now 0.0004 is not the best possible estimate of the probability of Chinese occurring in all situations; it might turn out that in some other corpus or context Chinese is a very unlikely word. But it is the probability that makes it most likely that Chinese will occur 400 times in a million-word corpus. We present ways to modify the MLE estimates slightly to get better probability estimates in Section 3.6. Let’s move on to some examples from a real but tiny corpus, drawn from the now-defunct Berkeley Restaurant Project, a dialogue system from the last century that answered questions about a database of restaurants in Berkeley, California (Ju- rafsky et al., 1994). Here are some sample user queries (text-normalized, by lower casing and with punctuation striped) (a sample of 9332 sentences is on the website): can you tell me about any good cantonese restaurants close by tell me about chez panisse i’m looking for a good place to eat breakfast when is caffe venezia open during the day Figure 3.1 shows the bigram counts from part of a bigram grammar from text- normalized Berkeley Restaurant Project sentences. Note that the majority of the values are zero. In fact, we have chosen the sample words to cohere with each other; a matrix selected from a random set of eight words would be even more sparse. i want to eat chinese food lunch spend i 5 827 0 9 0 0 0 2 want 2 0 608 1 6 6 5 1 to 2 0 4 686 2 0 6 211 eat 0 0 2 0 16 2 42 0 chinese 1 0 0 0 0 82 1 0 food 15 0 15 0 1 4 0 0 lunch 2 0 0 0 0 1 0 0 spend 1 0 1 0 0 0 0 0 Figure 3.1 Bigram counts for eight of the words (out ofV = 1446) in the Berkeley Restau- rant Project corpus of 9332 sentences. Zero counts are in gray. Each cell shows the count of the column label word following the row label word. Thus the cell in row i and column want means that want followed i 827 times in the corpus. Figure 3.2 shows the bigram probabilities after normalization (dividing each cell in Fig. 3.1 by the appropriate unigram for its row, taken from the following set of unigram counts): i want to eat chinese food lunch spend 2533 927 2417 746 158 1093 341 278 3.1 • N-G RAMS 37 i want to eat chinese food lunch spend i 0.002 0.33 0 0.0036 0 0 0 0.00079 want 0.0022 0 0.66 0.0011 0.0065 0.0065 0.0054 0.0011 to 0.00083 0 0.0017 0.28 0.00083 0 0.0025 0.087 eat 0 0 0.0027 0 0.021 0.0027 0.056 0 chinese 0.0063 0 0 0 0 0.52 0.0063 0 food 0.014 0 0.014 0 0.00092 0.0037 0 0 lunch 0.0059 0 0 0 0 0.0029 0 0 spend 0.0036 0 0.0036 0 0 0 0 0 Figure 3.2 Bigram probabilities for eight words in the Berkeley Restaurant Project corpus of 9332 sentences. Zero probabilities are in gray. Here are a few other useful probabilities: P(i\|<s>) =0.25 P(english\|want) =0.0011 P(food\|english) =0.5 P(</s>\|food) =0.68 Now we can compute the probability of sentences like I want English food or I want Chinese food by simply multiplying the appropriate bigram probabilities to- gether, as follows: P(<s> i want english food </s>) = P(i\|<s>)P(want\|i)P(english\|want) P(food\|english)P(</s>\|food) = 0.25 ×0.33 ×0.0011 ×0.5 ×0.68 = 0.000031 We leave it as Exercise 3.2 to compute the probability ofi want chinese food. What kinds of linguistic phenomena are captured in these bigram statistics? Some of the bigram probabilities above encode some facts that we think of as strictly syntactic in nature, like the fact that what comes after eat is usually a noun or an adjective, or that what comes after to is usually a verb. Others might be a fact about the personal assistant task, like the high probability of sentences beginning with the words I. And some might even be cultural rather than linguistic, like the higher probability that people are looking for Chinese versus English food. 3.1.3 Dealing with scale in large n-gram models In practice, language models can be very large, leading to practical issues. Log probabilities Language model probabilities are always stored and computed in log space aslog probabilities. This is because probabilities are (by deﬁnition) lesslog probabilities than or equal to 1, and so the more probabilities we multiply together, the smaller the product becomes. Multiplying enough n-grams together would result in numerical underﬂow. Adding in log space is equivalent to multiplying in linear space, so we combine log probabilities by adding them. By adding log probabilities instead of multiplying probabilities, we get results that are not as small. We do all computation and storage in log space, and just convert back into probabilities if we need to report probabilities at the end by taking the exp of the logprob: p1 ×p2 ×p3 ×p4 = exp(log p1 +log p2 +log p3 +log p4) (3.13) In practice throughout this book, we’ll use log to mean natural log (ln) when the base is not speciﬁed. 38 CHAPTER 3 • N- GRAM LANGUAGE MODELS Longer context Although for pedagogical purposes we have only described bi- gram models, when there is sufﬁcient training data we use trigram models, whichtrigram condition on the previous two words, or4-gram or 5-gram models. For these larger4-gram 5-gram n-grams, we’ll need to assume extra contexts to the left and right of the sentence end. For example, to compute trigram probabilities at the very beginning of the sentence, we use two pseudo-words for the ﬁrst trigram (i.e., P(I\|<s><s>). Some large n-gram datasets have been created, like the million most frequent n-grams drawn from the Corpus of Contemporary American English (COCA), a curated 1 billion word corpus of American English (Davies, 2020), Google’s Web 5-gram corpus from 1 trillion words of English web text (Franz and Brants, 2006), or the Google Books Ngrams corpora (800 billion tokens from Chinese, English, French, German, Hebrew, Italian, Russian, and Spanish) (Lin et al., 2012a)). It’s even possible to use extremely long-range n-gram context. The inﬁni-gram (∞-gram) project (Liu et al., 2024) allows n-grams of any length. Their idea is to avoid the expensive (in space and time) pre-computation of huge n-gram count ta- bles. Instead, n-gram probabilities with arbitrary n are computed quickly at inference time by using an efﬁcient representation called sufﬁx arrays. This allows computing of n-grams of every length for enormous corpora of 5 trillion tokens. Efﬁciency considerations are important when building large n-gram language models. It is standard to quantize the probabilities using only 4-8 bits (instead of 8-byte ﬂoats), store the word strings on disk and represent them in memory only as a 64-bit hash, and represent n-grams in special data structures like ‘reverse tries’. It is also common to prune n-gram language models, for example by only keeping n-grams with counts greater than some threshold or using entropy to prune less- important n-grams (Stolcke, 1998). Efﬁcient language model toolkits like KenLM (Heaﬁeld 2011, Heaﬁeld et al. 2013) use sorted arrays and use merge sorts to efﬁ- ciently build the probability tables in a minimal number of passes through a large corpus. 3.2 Evaluating Language Models: Training and Test Sets The best way to evaluate the performance of a language model is to embed it in an application and measure how much the application improves. Such end-to-end evaluation is called extrinsic evaluation. Extrinsic evaluation is the only way toextrinsic evaluation know if a particular improvement in the language model (or any component) is really going to help the task at hand. Thus for evaluating n-gram language models that are a component of some task like speech recognition or machine translation, we can compare the performance of two candidate language models by running the speech recognizer or machine translator twice, once with each language model, and seeing which gives the more accurate transcription. Unfortunately, running big NLP systems end-to-end is often very expensive. In- stead, it’s helpful to have a metric that can be used to quickly evaluate potential improvements in a language model. An intrinsic evaluation metric is one that mea-intrinsic evaluation sures the quality of a model independent of any application. In the next section we’ll introduce perplexity, which is the standard intrinsic metric for measuring language model performance, both for simple n-gram language models and for the more so- phisticated neural large language models of Chapter 9. In order to evaluate any machine learning model, we need to have at least three distinct data sets: the training set, the development set, and the test set.training set development set test set 3.2 • E VALUATING LANGUAGE MODELS : T RAINING AND TEST SETS 39 The training set is the data we use to learn the parameters of our model; for simple n-gram language models it’s the corpus from which we get the counts that we normalize into the probabilities of the n-gram language model. The test set is a different, held-out set of data, not overlapping with the training set, that we use to evaluate the model. We need a separate test set to give us an unbiased estimate of how well the model we trained can generalize when we apply it to some new unknown dataset. A machine learning model that perfectly captured the training data, but performed terribly on any other data, wouldn’t be much use when it comes time to apply it to any new data or problem! We thus measure the quality of an n-gram model by its performance on this unseen test set or test corpus. How should we choose a training and test set? The test set should reﬂect the language we want to use the model for. If we’re going to use our language model for speech recognition of chemistry lectures, the test set should be text of chemistry lectures. If we’re going to use it as part of a system for translating hotel booking re- quests from Chinese to English, the test set should be text of hotel booking requests. If we want our language model to be general purpose, then the test set should be drawn from a wide variety of texts. In such cases we might collect a lot of texts from different sources, and then divide it up into a training set and a test set. It’s important to do the dividing carefully; if we’re building a general purpose model, we don’t want the test set to consist of only text from one document, or one author, since that wouldn’t be a good measure of general performance. Thus if we are given a corpus of text and want to compare the performance of two different n-gram models, we divide the data into training and test sets, and train the parameters of both models on the training set. We can then compare how well the two trained models ﬁt the test set. But what does it mean to “ﬁt the test set”? The standard answer is simple: whichever language model assigns a higher probability to the test set—which means it more accurately predicts the test set—is a better model. Given two proba- bilistic models, the better model is the one that better predicts the details of the test data, and hence will assign a higher probability to the test data. Since our evaluation metric is based on test set probability, it’s important not to let the test sentences into the training set. Suppose we are trying to compute the probability of a particular “test” sentence. If our test sentence is part of the training corpus, we will mistakenly assign it an artiﬁcially high probability when it occurs in the test set. We call this situation training on the test set . Training on the test set introduces a bias that makes the probabilities all look too high, and causes huge inaccuracies in perplexity, the probability-based metric we introduce below. Even if we don’t train on the test set, if we test our language model on the test set many times after making different changes, we might implicitly tune to its characteristics, by noticing which changes seem to make the model better. For this reason, we only want to run our model on the test set once, or a very few number of times, once we are sure our model is ready. For this reason we normally instead have a third dataset called a developmentdevelopment test test set or, devset. We do all our testing on this dataset until the very end, and then we test on the test set once to see how good our model is. How do we divide our data into training, development, and test sets? We want our test set to be as large as possible, since a small test set may be accidentally un- representative, but we also want as much training data as possible. At the minimum, we would want to pick the smallest test set that gives us enough statistical power to measure a statistically signiﬁcant difference between two potential models. It’s 40 CHAPTER 3 • N- GRAM LANGUAGE MODELS important that the devset be drawn from the same kind of text as the test set, since its goal is to measure how we would do on the test set. 3.3 Evaluating Language Models: Perplexity We said above that we evaluate language models based on which one assigns a higher probability to the test set. A better model is better at predicting upcoming words, and so it will be less surprised by (i.e., assign a higher probability to) each word when it occurs in the test set. Indeed, a perfect language model would correctly guess each next word in a corpus, assigning it a probability of 1, and all the other words a probability of zero. So given a test corpus, a better language model will assign a higher probability to it than a worse language model. But in fact, we do not use raw probability as our metric for evaluating language models. The reason is that the probability of a test set (or any sequence) depends on the number of words or tokens in it; the probability of a test set gets smaller the longer the text. We’d prefer a metric that is per-word, normalized by length, so we could compare across texts of different lengths. The metric we use is, a function of probability called perplexity, is one of the most important metrics in NLP, used for evaluating large language models as well as n-gram models. The perplexity (sometimes abbreviated as PP or PPL) of a language model on aperplexity test set is the inverse probability of the test set (one over the probability of the test set), normalized by the number of words (or tokens). For this reason it’s sometimes called the per-word or per-token perplexity. We normalize by the number of words N by taking the Nth root. For a test set W = w1w2 ... wN ,: perplexity(W) = P(w1w2 ... wN )−1 N (3.14) = N √ 1 P(w1w2 ... wN ) Or we can use the chain rule to expand the probability of W: perplexity(W) = N √ N∏ i=1 1 P(wi\|w1 ... wi−1) (3.15) Note that because of the inverse in Eq. 3.15, the higher the probability of the word sequence, the lower the perplexity. Thus thethe lower the perplexity of a model on the data, the better the model. Minimizing perplexity is equivalent to maximizing the test set probability according to the language model. Why does perplexity use the inverse probability? It turns out the inverse arises from the original deﬁnition of perplexity from cross-entropy rate in information theory; for those interested, the explanation is in the advanced Section 3.7. Meanwhile, we just have to remember that perplexity has an inverse relationship with probability. The details of computing the perplexity of a test set W depends on which lan- guage model we use. Here’s the perplexity of W with a unigram language model (just the geometric mean of the inverse of the unigram probabilities): perplexity(W) = N √ N∏ i=1 1 P(wi) (3.16) 3.3 • E VALUATING LANGUAGE MODELS : P ERPLEXITY 41 The perplexity of W computed with a bigram language model is still a geometric mean, but now of the inverse of the bigram probabilities: perplexity(W) = N √ N∏ i=1 1 P(wi\|wi−1) (3.17) What we generally use for word sequence in Eq. 3.15 or Eq. 3.17 is the entire sequence of words in some test set. Since this sequence will cross many sentence boundaries, if our vocabulary includes a between-sentence token <EOS>or separate begin- and end-sentence markers <s> and </s> then we can include them in the probability computation. If we do, then we also include one token per sentence in the total count of word tokens N.2 We mentioned above that perplexity is a function of both the text and the lan- guage model: given a text W, different language models will have different perplex- ities. Because of this, perplexity can be used to compare different language models. For example, here we trained unigram, bigram, and trigram grammars on 38 million words from the Wall Street Journalnewspaper. We then computed the perplexity of each of these models on a WSJ test set using Eq. 3.16 for unigrams, Eq. 3.17 for bigrams, and the corresponding equation for trigrams. The table below shows the perplexity of the 1.5 million word test set according to each of the language models. Unigram Bigram Trigram Perplexity 962 170 109 As we see above, the more information the n-gram gives us about the word sequence, the higher the probability the n-gram will assign to the string. A trigram model is less surprised than a unigram model because it has a better idea of what words might come next, and so it assigns them a higher probability. And the higher the probability, the lower the perplexity (since as Eq. 3.15 showed, perplexity is related inversely to the probability of the test sequence according to the model). So a lower perplexity tells us that a language model is a better predictor of the test set. Note that in computing perplexities, the language model must be constructed without any knowledge of the test set, or else the perplexity will be artiﬁcially low. And the perplexity of two language models is only comparable if they use identical vocabularies. An (intrinsic) improvement in perplexity does not guarantee an (extrinsic) im- provement in the performance of a language processing task like speech recognition or machine translation. Nonetheless, because perplexity usually correlates with task improvements, it is commonly used as a convenient evaluation metric. Still, when possible a model’s improvement in perplexity should be conﬁrmed by an end-to-end evaluation on a real task. 3.3.1 Perplexity as Weighted Average Branching Factor It turns out that perplexity can also be thought of as the weighted average branch- ing factor of a language. The branching factor of a language is the number of possible next words that can follow any word. For example consider a mini artiﬁcial 2 For example if we use both begin and end tokens, we would include the end-of-sentence marker</s> but not the beginning-of-sentence marker<s>in our count ofN; This is because the end-sentence token is followed directly by the begin-sentence token with probability almost 1, so we don’t want the probability of that fake transition to inﬂuence our perplexity. 42 CHAPTER 3 • N- GRAM LANGUAGE MODELS language that is deterministic (no probabilities), any word can follow any word, and whose vocabulary consists of only three colors: L = {red,blue,green} (3.18) The branching factor of this language is 3. Now let’s make a probabilistic version of the same LM, let’s call itA, where each word follows each other with equal probability1 3 (it was trained on a training set with equal counts for the 3 colors), and a test set T = “red red red red blue”. Let’s ﬁrst convince ourselves that if we compute the perplexity of this artiﬁcial digit language on this test set (or any such test set) we indeed get 3. By Eq. 3.15, the perplexity of A on T is: perplexityA(T ) = PA(red red red red blue)−1 5 = ((1 3 )5)−1 5 = (1 3 )−1 = 3 (3.19) But now suppose red was very likely in the training set a different LM B, and so B has the following probabilities: P(red) =0.8 P(green) =0.1 P(blue) =0.1 (3.20) We should expect the perplexity of the same test set red red red red blue for language model B to be lower since most of the time the next color will be red, which is very predictable, i.e. has a high probability. So the probability of the test set will be higher, and since perplexity is inversely related to probability, the perplexity will be lower. Thus, although the branching factor is still 3, the perplexity or weighted branching factor is smaller: perplexityB(T ) = PB(red red red red blue)−1/5 = 0.04096−1 5 = 0.527−1 = 1.89 (3.21) 3.4 Sampling sentences from a language model One important way to visualize what kind of knowledge a language model embodies is to sample from it. Sampling from a distribution means to choose random pointssampling according to their likelihood. Thus sampling from a language model—which rep- resents a distribution over sentences—means to generate some sentences, choosing each sentence according to its likelihood as deﬁned by the model. Thus we are more likely to generate sentences that the model thinks have a high probability and less likely to generate sentences that the model thinks have a low probability. This technique of visualizing a language model by sampling was ﬁrst suggested very early on by Shannon (1948) and Miller and Selfridge (1950). It’s simplest to visualize how this works for the unigram case. Imagine all the words of the English language covering the number line between 0 and 1, each word covering an interval 3.5 • G ENERALIZING VS . OVERFITTING THE TRAINING SET 43 0 1 0.06 the .06 0.03 of 0.02 a 0.02 to in .09 .11 .13 .15 … however (p=0.0003) polyphonic p=0.0000018 …0.02 .66 .99 … Figure 3.3 A visualization of the sampling distribution for sampling sentences by repeat- edly sampling unigrams. The blue bar represents the relative frequency of each word (we’ve ordered them from most frequent to least frequent, but the choice of order is arbitrary). The number line shows the cumulative probabilities. If we choose a random number between 0 and 1, it will fall in an interval corresponding to some word. The expectation for the random number to fall in the larger intervals of one of the frequent words ( the, of, a) is much higher than in the smaller interval of one of the rare words (polyphonic). proportional to its frequency. Fig. 3.3 shows a visualization, using a unigram LM computed from the text of this book. We choose a random value between 0 and 1, ﬁnd that point on the probability line, and print the word whose interval includes this chosen value. We continue choosing random numbers and generating words until we randomly generate the sentence-ﬁnal token </s>. We can use the same technique to generate bigrams by ﬁrst generating a ran- dom bigram that starts with <s>(according to its bigram probability). Let’s say the second word of that bigram is w. We next choose a random bigram starting with w (again, drawn according to its bigram probability), and so on. 3.5 Generalizing vs. overﬁtting the training set The n-gram model, like many statistical models, is dependent on the training corpus. One implication of this is that the probabilities often encode speciﬁc facts about a given training corpus. Another implication is that n-grams do a better and better job of modeling the training corpus as we increase the value of N. We can use the sampling method from the prior section to visualize both of these facts! To give an intuition for the increasing power of higher-order n-grams, Fig. 3.4 shows random sentences generated from unigram, bigram, trigram, and 4- gram models trained on Shakespeare’s works. The longer the context, the more coherent the sentences. The unigram sen- tences show no coherent relation between words nor any sentence-ﬁnal punctua- tion. The bigram sentences have some local word-to-word coherence (especially considering punctuation as words). The trigram sentences are beginning to look a lot like Shakespeare. Indeed, the 4-gram sentences look a little too much like Shake- speare. The words It cannot be but so are directly from King John. This is because, not to put the knock on Shakespeare, his oeuvre is not very large as corpora go (N = 884,647,V = 29,066), and our n-gram probability matrices are ridiculously sparse. There are V 2 = 844,000,000 possible bigrams alone, and the number of possible 4-grams is V 4 = 7 ×1017. Thus, once the generator has chosen the ﬁrst 3-gram (It cannot be), there are only seven possible next words for the 4th element (but, I, that, thus, this, and the period). To get an idea of the dependence on the training set, let’s look at LMs trained on a completely different corpus: the Wall Street Journal(WSJ) newspaper. Shakespeare 44 CHAPTER 3 • N- GRAM LANGUAGE MODELS 1 –To him swallowed confess hear both. Which. Of save on trail for are ay device and rote life have gram –Hill he late speaks; or! a more to leg less ﬁrst you enter 2 –Why dost stand forth thy canopy, forsooth; he is this palpable hit the King Henry. Live king. Follow. gram –What means, sir. I confess she? then all sorts, he is trim, captain. 3 –Fly, and will rid me these news of price. Therefore the sadness of parting, as they say, ’tis done. gram –This shall forbid it should be branded, if renown made it empty. 4 –King Henry. What! I will go seek the traitor Gloucester. Exeunt some of the watch. A great banquet serv’d in; gram –It cannot be but so. Figure 3.4 Eight sentences randomly generated from four n-grams computed from Shakespeare’s works. All characters were mapped to lower-case and punctuation marks were treated as words. Output is hand-corrected for capitalization to improve readability. and the WSJ are both English, so we might have expected some overlap between our n-grams for the two genres. Fig. 3.5 shows sentences generated by unigram, bigram, and trigram grammars trained on 40 million words from WSJ. 1 Months the my and issue of year foreign new exchange’s september were recession exchange new endorsed a acquire to six executivesgram 2 Last December through the way to preserve the Hudson corporation N. B. E. C. Taylor would seem to complete the major central planners one gram point ﬁve percent of U. S. E. has already old M. X. corporation of living on information such as more frequently ﬁshing to keep her 3 They also point to ninety nine point six billion dollars from two hundred four oh six three percent of the rates of interest stores as Mexico and gram Brazil on market conditions Figure 3.5 Three sentences randomly generated from three n-gram models computed from 40 million words of the Wall Street Journal, lower-casing all characters and treating punctua- tion as words. Output was then hand-corrected for capitalization to improve readability. Compare these examples to the pseudo-Shakespeare in Fig. 3.4. While they both model “English-like sentences”, there is no overlap in the generated sentences, and little overlap even in small phrases. Statistical models are pretty useless as predictors if the training sets and the test sets are as different as Shakespeare and the WSJ. How should we deal with this problem when we build n-gram models? One step is to be sure to use a training corpus that has a similargenre to whatever task we are trying to accomplish. To build a language model for translating legal documents, we need a training corpus of legal documents. To build a language model for a question-answering system, we need a training corpus of questions. It is equally important to get training data in the appropriate dialect or variety, especially when processing social media posts or spoken transcripts. For exam- ple some tweets will use features of African American English (AAE)— the name for the many variations of language used in African American communities (King, 2020). Such features can include words like ﬁnna—an auxiliary verb that marks immediate future tense —that don’t occur in other varieties, or spellings like den for then, in tweets like this one (Blodgett and O’Connor, 2017): 3.6 • S MOOTHING , INTERPOLATION , AND BACKOFF 45 (3.22) Bored af den my phone ﬁnna die!!! while tweets from English-based languages like Nigerian Pidgin have markedly dif- ferent vocabulary and n-gram patterns from American English (Jurgens et al., 2017): (3.23) @username R u a wizard or wat gan sef: in d mornin - u tweet, afternoon - u tweet, nyt gan u dey tweet. beta get ur IT placement wiv twitter Is it possible for the testset nonetheless to have a word we have never seen be- fore? What happens if the wordJurafsky never occurs in our training set, but pops up in the test set? The answer is that although words might be unseen, we normally run our NLP algorithms not on words but on subword tokens. With subword tokeniza- tion (like the BPE algorithm of Chapter 2) any word can be modeled as a sequence of known smaller subwords, if necessary by a sequence of tokens corresponding to individual letters. So although for convenience we’ve been referring to words in this chapter, the language model vocabulary is normally the set of tokens rather than words, and in this way the test set can never contain unseen tokens. 3.6 Smoothing, Interpolation, and Backoff There is a problem with using maximum likelihood estimates for probabilities: any ﬁnite training corpus will be missing some perfectly acceptable English word se- quences. That is, cases where a particular n-gram never occurs in the training data but appears in the test set. Perhaps our training corpus has the words ruby and slippersin it but just happens not to have the phrase ruby slippers. These unseen sequences or zeros—sequences that don’t occur in the training setzeros but do occur in the test set—are a problem for two reasons. First, their presence means we are underestimating the probability of word sequences that might occur, which hurts the performance of any application we want to run on this data. Second, if the probability of any word in the test set is 0, the probability of the whole test set is 0. Perplexity is deﬁned based on the inverse probability of the test set. Thus if some words in context have zero probability, we can’t compute perplexity at all, since we can’t divide by 0! The standard way to deal with putative “zero probability n-grams” that should re- ally have some non-zero probability is calledsmoothing or discounting. Smoothingsmoothing discounting algorithms shave off a bit of probability mass from some more frequent events and give it to unseen events. Here we’ll introduce some simple smoothing algorithms: Laplace (add-one) smoothing, stupid backoff, and n-gram interpolation. 3.6.1 Laplace Smoothing The simplest way to do smoothing is to add one to all the n-gram counts, before we normalize them into probabilities. All the counts that used to be zero will now have a count of 1, the counts of 1 will be 2, and so on. This algorithm is called Laplace smoothing. Laplace smoothing does not perform well enough to be usedLaplace smoothing in modern n-gram models, but it usefully introduces many of the concepts that we see in other smoothing algorithms, gives a useful baseline, and is also a practical smoothing algorithm for other tasks like text classiﬁcation (Chapter 4). Let’s start with the application of Laplace smoothing to unigram probabilities. Recall that the unsmoothed maximum likelihood estimate of the unigram probability 46 CHAPTER 3 • N- GRAM LANGUAGE MODELS of the word wi is its count ci normalized by the total number of word tokens N: P(wi) =ci N Laplace smoothing merely adds one to each count (hence its alternate name add- one smoothing). Since there are V words in the vocabulary and each one was in-add-one cremented, we also need to adjust the denominator to take into account the extra V observations. (What happens to our P values if we don’t increase the denominator?) PLaplace(wi) =ci +1 N +V (3.24) Now that we have the intuition for the unigram case, let’s smooth our Berkeley Restaurant Project bigrams. Figure 3.6 shows the add-one smoothed counts for the bigrams in Fig. 3.1. i want to eat chinese food lunch spend i 6 828 1 10 1 1 1 3 want 3 1 609 2 7 7 6 2 to 3 1 5 687 3 1 7 212 eat 1 1 3 1 17 3 43 1 chinese 2 1 1 1 1 83 2 1 food 16 1 16 1 2 5 1 1 lunch 3 1 1 1 1 2 1 1 spend 2 1 2 1 1 1 1 1 Figure 3.6 Add-one smoothed bigram counts for eight of the words (out of V = 1446) in the Berkeley Restaurant Project corpus of 9332 sentences. Previously-zero counts are in gray. Figure 3.7 shows the add-one smoothed probabilities for the bigrams in Fig. 3.2, computed by Eq. 3.26 below. Recall that normal bigram probabilities are computed by normalizing each row of counts by the unigram count: PMLE(wn\|wn−1) =C(wn−1wn) C(wn−1) (3.25) For add-one smoothed bigram counts, we need to augment the unigram count in the denominator by the number of total word types in the vocabulary V . We can see why this is in the following equation, which makes it explicit that the unigram count in the denominator is really the sum over all the bigrams that start with wn−1. Since we add one to each of these, and there are V of them, we add a total of V to the denominator: PLaplace(wn\|wn−1) = C(wn−1wn)+ 1∑ w (C(wn−1w)+ 1) = C(wn−1wn)+ 1 C(wn−1)+ V (3.26) Thus, each of the unigram counts given on page 36 will need to be augmented byV = 1446. The result, using Eq. 3.26, is the smoothed bigram probabilities in Fig. 3.7. One useful visualization technique is to reconstruct an adjusted count matrix so we can see how much a smoothing algorithm has changed the original counts. This adjusted count C∗ is the count that, if divided by C(wn−1), would result in the smoothed probability. This adjusted count is easier to compare directly with the MLE counts. That is, the Laplace probability can equally be expressed as the adjusted count divided by the (non-smoothed) denominator from Eq. 3.25: PLaplace(wn\|wn−1) =C(wn−1wn)+ 1 C(wn−1)+ V = C∗(wn−1wn) C(wn−1) 3.6 • S MOOTHING , INTERPOLATION , AND BACKOFF 47 i want to eat chinese food lunch spend i 0.0015 0.21 0.00025 0.0025 0.00025 0.00025 0.00025 0.00075 want 0.0013 0.00042 0.26 0.00084 0.0029 0.0029 0.0025 0.00084 to 0.00078 0.00026 0.0013 0.18 0.00078 0.00026 0.0018 0.055 eat 0.00046 0.00046 0.0014 0.00046 0.0078 0.0014 0.02 0.00046 chinese 0.0012 0.00062 0.00062 0.00062 0.00062 0.052 0.0012 0.00062 food 0.0063 0.00039 0.0063 0.00039 0.00079 0.002 0.00039 0.00039 lunch 0.0017 0.00056 0.00056 0.00056 0.00056 0.0011 0.00056 0.00056 spend 0.0012 0.00058 0.0012 0.00058 0.00058 0.00058 0.00058 0.00058 Figure 3.7 Add-one smoothed bigram probabilities for eight of the words (out of V = 1446) in the BeRP corpus of 9332 sentences computed by Eq. 3.26. Previously-zero probabilities are in gray. Rearranging terms, we can solve for C∗(wn−1wn) : C∗(wn−1wn) =[C(wn−1wn)+ 1]×C(wn−1) C(wn−1)+ V (3.27) Figure 3.8 shows the reconstructed counts, computed by Eq. 3.27. i want to eat chinese food lunch spend i 3.8 527 0.64 6.4 0.64 0.64 0.64 1.9 want 1.2 0.39 238 0.78 2.7 2.7 2.3 0.78 to 1.9 0.63 3.1 430 1.9 0.63 4.4 133 eat 0.34 0.34 1 0.34 5.8 1 15 0.34 chinese 0.2 0.098 0.098 0.098 0.098 8.2 0.2 0.098 food 6.9 0.43 6.9 0.43 0.86 2.2 0.43 0.43 lunch 0.57 0.19 0.19 0.19 0.19 0.38 0.19 0.19 spend 0.32 0.16 0.32 0.16 0.16 0.16 0.16 0.16 Figure 3.8 Add-one reconstituted counts for eight words (ofV = 1446) in the BeRP corpus of 9332 sentences, computed by Eq. 3.27. Previously-zero counts are in gray. Note that add-one smoothing has made a very big change to the counts. Com- paring Fig. 3.8 to the original counts in Fig. 3.1, we can see thatC(want to) changed from 608 to 238! We can see this in probability space as well: P(to\|want) decreases from 0.66 in the unsmoothed case to 0.26 in the smoothed case. Looking at the dis- count d, deﬁned as the ratio between new and old counts, shows us how strikingly the counts for each preﬁx word have been reduced; the discount for the bigramwant to is 0.39, while the discount for Chinese food is 0.10, a factor of 10! The sharp change occurs because too much probability mass is moved to all the zeros. 3.6.2 Add-k smoothing One alternative to add-one smoothing is to move a bit less of the probability mass from the seen to the unseen events. Instead of adding 1 to each count, we add a fractional count k (0.5? 0.01?). This algorithm is therefore called add-k smoothing.add-k P∗ Add-k(wn\|wn−1) =C(wn−1wn)+ k C(wn−1)+ kV (3.28) Add-k smoothing requires that we have a method for choosing k; this can be done, for example, by optimizing on a devset. Although add-k is useful for some tasks (including text classiﬁcation), it turns out that it still doesn’t work well for 48 CHAPTER 3 • N- GRAM LANGUAGE MODELS language modeling, generating counts with poor variances and often inappropriate discounts (Gale and Church, 1994). 3.6.3 Language Model Interpolation There is an alternative source of knowledge we can draw on to solve the problem of zero frequency n-grams. If we are trying to compute P(wn\|wn−2wn−1) but we have no examples of a particular trigram wn−2wn−1wn, we can instead estimate its probability by using the bigram probability P(wn\|wn−1). Similarly, if we don’t have counts to compute P(wn\|wn−1), we can look to the unigram P(wn). In other words, sometimes using less context can help us generalize more for contexts that the model hasn’t learned much about. The most common way to use this n-gram hierarchy is called interpolation:interpolation computing a new probability by interpolating (weighting and combining) the tri- gram, bigram, and unigram probabilities. 3 In simple linear interpolation, we com- bine different order n-grams by linearly interpolating them. Thus, we estimate the trigram probability P(wn\|wn−2wn−1) by mixing together the unigram, bigram, and trigram probabilities, each weighted by a λ: ˆP(wn\|wn−2wn−1) = λ1P(wn) +λ2P(wn\|wn−1) +λ3P(wn\|wn−2wn−1) (3.29) The λs must sum to 1, making Eq. 3.29 equivalent to a weighted average. In a slightly more sophisticated version of linear interpolation, each λ weight is com- puted by conditioning on the context. This way, if we have particularly accurate counts for a particular bigram, we assume that the counts of the trigrams based on this bigram will be more trustworthy, so we can make the λs for those trigrams higher and thus give that trigram more weight in the interpolation. Equation 3.30 shows the equation for interpolation with context-conditioned weights, where each lambda takes an argument that is the two prior word context: ˆP(wn\|wn−2wn−1) = λ1(wn−2:n−1)P(wn) +λ2(wn−2:n−1)P(wn\|wn−1) +λ3(wn−2:n−1)P(wn\|wn−2wn−1) (3.30) How are these λ values set? Both the simple interpolation and conditional interpo- lation λs are learned from a held-out corpus. A held-out corpus is an additionalheld-out training corpus, so-called because we hold it out from the training data, that we use to set these λ values.4 We do so by choosing the λ values that maximize the likeli- hood of the held-out corpus. That is, we ﬁx the n-gram probabilities and then search for the λ values that—when plugged into Eq. 3.29—give us the highest probability of the held-out set. There are various ways to ﬁnd this optimal set of λs. One way is to use the EM algorithm, an iterative learning algorithm that converges on locally optimal λs (Jelinek and Mercer, 1980). 3 We won’t discuss the less-common alternative, called backoff, in which we use the trigram if the evidence is sufﬁcient for it, but if not we instead just use the bigram, otherwise the unigram. That is, we only “back off” to a lower-order n-gram if we have zero evidence for a higher-order n-gram. 4 Held-out corpora are generally used to set hyperparameters, which are special parameters, unlike regular counts that are learned from the training data; we’ll discuss hyperparameters in Chapter 7. 3.7 • A DVANCED : P ERPLEXITY ’S RELATION TO ENTROPY 49 3.6.4 Stupid Backoff An alternative to interpolation isbackoff. In a backoff model, if the n-gram we needbackoff has zero counts, we approximate it by backing off to the (n-1)-gram. We continue backing off until we reach a history that has some counts. For a backoff model to give a correct probability distribution, we have todiscount the higher-order n-gramsdiscount to save some probability mass for the lower order n-grams. In practice, instead of discounting, it’s common to use a much simpler non-discounted backoff algorithm called stupid backoff (Brants et al., 2007).stupid backoff Stupid backoff gives up the idea of trying to make the language model a true probability distribution. There is no discounting of the higher-order probabilities. If a higher-order n-gram has a zero count, we simply backoff to a lower order n-gram, weighed by a ﬁxed (context-independent) weight. This algorithm does not produce a probability distribution, so we’ll follow Brants et al. (2007) in referring to it asS: S(wi\|wi−N+1: i−1) =    count ( w i−N + 1: i ) count ( w i−N + 1: i−1 ) if count(wi−N+1: i) >0 λS(wi\|wi−N+2: i−1) otherwise (3.31) The backoff terminates in the unigram, which has scoreS(w) =count(w) N . Brants et al. (2007) ﬁnd that a value of 0.4 worked well for λ. 3.7 Advanced: Perplexity’s Relation to Entropy We introduced perplexity in Section 3.3 as a way to evaluate n-gram models on a test set. A better n-gram model is one that assigns a higher probability to the test data, and perplexity is a normalized version of the probability of the test set. The perplexity measure actually arises from the information-theoretic concept of cross-entropy, which explains otherwise mysterious properties of perplexity (why the inverse probability, for example?) and its relationship to entropy. Entropy is aEntropy measure of information. Given a random variable X ranging over whatever we are predicting (words, letters, parts of speech), the set of which we’ll call χ, and with a particular probability function, call it p(x), the entropy of the random variable X is: H(X) =− ∑ x∈χ p(x)log2 p(x) (3.32) The log can, in principle, be computed in any base. If we use log base 2, the resulting value of entropy will be measured in bits. One intuitive way to think about entropy is as a lower bound on the number of bits it would take to encode a certain decision or piece of information in the optimal coding scheme. Consider an example from the standard information theory textbook Cover and Thomas (1991). Imagine that we want to place a bet on a horse race but it is too far to go all the way to Yonkers Racetrack, so we’d like to send a short message to the bookie to tell him which of the eight horses to bet on. One way to encode this message is just to use the binary representation of the horse’s number as the code; thus, horse 1 would be 001, horse 2 010, horse 3 011, and so on, with horse 8 coded as 000. If we spend the whole day betting and each horse is coded with 3 bits, on average we would be sending 3 bits per race. Can we do better? Suppose that the spread is the actual distribution of the bets placed and that we represent it as the prior probability of each horse as follows: 50 CHAPTER 3 • N- GRAM LANGUAGE MODELS Horse 1 1 2 Horse 5 1 64 Horse 2 1 4 Horse 6 1 64 Horse 3 1 8 Horse 7 1 64 Horse 4 1 16 Horse 8 1 64 The entropy of the random variable X that ranges over horses gives us a lower bound on the number of bits and is H(X) = − i=8∑ i=1 p(i)log2 p(i) = −1 2 log2 1 2 −1 4 log2 1 4 −1 8 log2 1 8 −1 16 log2 1 16 −4( 1 64 log2 1 64 ) = 2 bits (3.33) A code that averages 2 bits per race can be built with short encodings for more probable horses, and longer encodings for less probable horses. For example, we could encode the most likely horse with the code 0, and the remaining horses as 10, then 110, 1110, 111100, 111101, 111110, and 111111. What if the horses are equally likely? We saw above that if we used an equal- length binary code for the horse numbers, each horse took 3 bits to code, so the average was 3. Is the entropy the same? In this case each horse would have a probability of 1 8 . The entropy of the choice of horses is then H(X) =− i=8∑ i=1 1 8 log2 1 8 = −log2 1 8 = 3 bits (3.34) Until now we have been computing the entropy of a single variable. But most of what we will use entropy for involves sequences. For a grammar, for example, we will be computing the entropy of some sequence of words W = {w1,w2,..., wn}. One way to do this is to have a variable that ranges over sequences of words. For example we can compute the entropy of a random variable that ranges over all se- quences of words of length n in some language L as follows: H(w1,w2,..., wn) =− ∑ w1: n∈L p(w1: n)log p(w1: n) (3.35) We could deﬁne the entropy rate (we could also think of this as the per-wordentropy rate entropy) as the entropy of this sequence divided by the number of words: 1 nH(w1: n) =−1 n ∑ w1: n∈L p(w1: n)log p(w1: n) (3.36) But to measure the true entropy of a language, we need to consider sequences of inﬁnite length. If we think of a language as a stochastic process L that produces a sequence of words, and allowW to represent the sequence of words w1,..., wn, then L’s entropy rateH(L) is deﬁned as H(L) = lim n→∞ 1 nH(w1: n) = −lim n→∞ 1 n ∑ W∈L p(w1: n)log p(w1: n) (3.37) 3.7 • A DVANCED : P ERPLEXITY ’S RELATION TO ENTROPY 51 The Shannon-McMillan-Breiman theorem (Algoet and Cover 1988, Cover and Thomas 1991) states that if the language is regular in certain ways (to be exact, if it is both stationary and ergodic), H(L) =lim n→∞ −1 n log p(w1: n) (3.38) That is, we can take a single sequence that is long enough instead of summing over all possible sequences. The intuition of the Shannon-McMillan-Breiman theorem is that a long-enough sequence of words will contain in it many other shorter se- quences and that each of these shorter sequences will reoccur in the longer sequence according to their probabilities. A stochastic process is said to be stationary if the probabilities it assigns to aStationary sequence are invariant with respect to shifts in the time index. In other words, the probability distribution for words at timet is the same as the probability distribution at time t + 1. Markov models, and hence n-grams, are stationary. For example, in a bigram, Pi is dependent only on Pi−1. So if we shift our time index by x, Pi+x is still dependent on Pi+x−1. But natural language is not stationary, since as we show in Appendix D, the probability of upcoming words can be dependent on events that were arbitrarily distant and time dependent. Thus, our statistical models only give an approximation to the correct distributions and entropies of natural language. To summarize, by making some incorrect but convenient simplifying assump- tions, we can compute the entropy of some stochastic process by taking a very long sample of the output and computing its average log probability. Now we are ready to introducecross-entropy. The cross-entropy is useful whencross-entropy we don’t know the actual probability distribution p that generated some data. It allows us to use some m, which is a model of p (i.e., an approximation to p). The cross-entropy of m on p is deﬁned by H(p,m) =lim n→∞ −1 n ∑ W∈L p(w1,..., wn)logm(w1,..., wn) (3.39) That is, we draw sequences according to the probability distribution p, but sum the log of their probabilities according to m. Again, following the Shannon-McMillan-Breiman theorem, for a stationary er- godic process: H(p,m) =lim n→∞ −1 n logm(w1w2 ... wn) (3.40) This means that, as for entropy, we can estimate the cross-entropy of a model m on some distribution p by taking a single sequence that is long enough instead of summing over all possible sequences. What makes the cross-entropy useful is that the cross-entropy H(p,m) is an up- per bound on the entropy H(p). For any model m: H(p) ≤H(p,m) (3.41) This means that we can use some simpliﬁed model m to help estimate the true en- tropy of a sequence of symbols drawn according to probabilityp. The more accurate m is, the closer the cross-entropy H(p,m) will be to the true entropy H(p). Thus, the difference between H(p,m) and H(p) is a measure of how accurate a model is. Between two models m1 and m2, the more accurate model will be the one with the 52 CHAPTER 3 • N- GRAM LANGUAGE MODELS lower cross-entropy. (The cross-entropy can never be lower than the true entropy, so a model cannot err by underestimating the true entropy.) We are ﬁnally ready to see the relation between perplexity and cross-entropy as we saw it in Eq. 3.40. Cross-entropy is deﬁned in the limit as the length of the observed word sequence goes to inﬁnity. We approximate this cross-entropy by relying on a (sufﬁciently long) sequence of ﬁxed length. This approximation to the cross-entropy of a model M = P(wi\|wi−N+1: i−1) on a sequence of words W is H(W) =−1 N logP(w1w2 ... wN ) (3.42) The perplexity of a model P on a sequence of words W is now formally deﬁned asperplexity 2 raised to the power of this cross-entropy: Perplexity(W) = 2H(W) = P(w1w2 ... wN )−1 N = N √ 1 P(w1w2 ... wN ) 3.8 Summary This chapter introduced language modeling via the n-gram model, a classic model that allows us to introduce many of the basic concepts in language modeling. • Language models offer a way to assign a probability to a sentence or other sequence of words or tokens, and to predict a word or token from preceding words or tokens. • N-grams are perhaps the simplest kind of language model. They are Markov models that estimate words from a ﬁxed window of previous words. N-gram models can be trained by counting in a training corpus and normalizing the counts (the maximum likelihood estimate). • N-gram language models can be evaluated on a test set using perplexity. • The perplexity of a test set according to a language model is a function of the probability of the test set: the inverse test set probability according to the model, normalized by the length. • Sampling from a language model means to generate some sentences, choos- ing each sentence according to its likelihood as deﬁned by the model. • Smoothing algorithms provide a way to estimate probabilities for events that were unseen in training. Commonly used smoothing algorithms for n-grams include add-1 smoothing, or rely on lower-order n-gram counts throughinter- polation. Bibliographical and Historical Notes The underlying mathematics of the n-gram was ﬁrst proposed by Markov (1913), who used what are now called Markov chains (bigrams and trigrams) to predict whether an upcoming letter in Pushkin’sEugene Onegin would be a vowel or a con- sonant. Markov classiﬁed 20,000 letters as V or C and computed the bigram and BIBLIOGRAPHICAL AND HISTORICAL NOTES 53 trigram probability that a given letter would be a vowel given the previous one or two letters. Shannon (1948) applied n-grams to compute approximations to English word sequences. Based on Shannon’s work, Markov models were commonly used in engineering, linguistic, and psychological work on modeling word sequences by the 1950s. In a series of extremely inﬂuential papers starting with Chomsky (1956) and including Chomsky (1957) and Miller and Chomsky (1963), Noam Chomsky argued that “ﬁnite-state Markov processes”, while a possibly useful engineering heuristic, were incapable of being a complete cognitive model of human grammatical knowl- edge. These arguments led many linguists and computational linguists to ignore work in statistical modeling for decades. The resurgence of n-gram language models came from Fred Jelinek and col- leagues at the IBM Thomas J. Watson Research Center, who were inﬂuenced by Shannon, and James Baker at CMU, who was inﬂuenced by the prior, classiﬁed work of Leonard Baum and colleagues on these topics at labs like the US Institute for Defense Analyses (IDA) after they were declassiﬁed. Independently these two labs successfully used n-grams in their speech recognition systems at the same time (Baker 1975b, Jelinek et al. 1975, Baker 1975a, Bahl et al. 1983, Jelinek 1990). The terms “language model” and “perplexity” were ﬁrst used for this technology by the IBM group. Jelinek and his colleagues used the term language model in a pretty modern way, to mean the entire set of linguistic inﬂuences on word sequence prob- abilities, including grammar, semantics, discourse, and even speaker characteristics, rather than just the particular n-gram model itself. Add-one smoothing derives from Laplace’s 1812 law of succession and was ﬁrst applied as an engineering solution to the zero frequency problem by Jeffreys (1948) based on an earlier Add-K suggestion by Johnson (1932). Problems with the add- one algorithm are summarized in Gale and Church (1994). A wide variety of different language modeling and smoothing techniques were proposed in the 80s and 90s, including Good-Turing discounting—ﬁrst applied to the n-gram smoothing at IBM by Katz (N´adas 1984, Church and Gale 1991)— Witten- Bell discounting (Witten and Bell, 1991), and varieties ofclass-based n-gram mod-class-based n-gram els that used information about word classes. Starting in the late 1990s, Chen and Goodman performed a number of carefully controlled experiments comparing dif- ferent algorithms and parameters (Chen and Goodman 1999, Goodman 2006, inter alia). They showed the advantages of Modiﬁed Interpolated Kneser-Ney, which became the standard baseline for n-gram language modeling around the turn of the century, especially because they showed that caches and class-based models pro- vided only minor additional improvement. SRILM (Stolcke, 2002) and KenLM (Heaﬁeld 2011, Heaﬁeld et al. 2013) are publicly available toolkits for building n- gram language models. Large language models are based on neural networks rather than n-grams, en- abling them to solve the two major problems with n-grams: (1) the number of param- eters increases exponentially as the n-gram order increases, and (2) n-grams have no way to generalize from training examples to test set examples unless they use iden- tical words. Neural language models instead project words into a continuous space in which words with similar contexts have similar representations. We’ll introduce transformer-based large language models in Chapter 9, along the way introducing feedforward language models (Bengio et al. 2006, Schwenk 2007) in Chapter 7 and recurrent language models (Mikolov, 2012) in Chapter 8. 54 CHAPTER 3 • N- GRAM LANGUAGE MODELS Exercises 3.1 Write out the equation for trigram probability estimation (modifying Eq. 3.11). Now write out all the non-zero trigram probabilities for the I am Samcorpus on page 35. 3.2 Calculate the probability of the sentence i want chinese food. Give two probabilities, one using Fig. 3.2 and the ‘useful probabilities’ just below it on page 37, and another using the add-1 smoothed table in Fig. 3.7. Assume the additional add-1 smoothed probabilitiesP(i\|<s>) =0.19 and P(</s>\|food) = 0.40. 3.3 Which of the two probabilities you computed in the previous exercise is higher, unsmoothed or smoothed? Explain why. 3.4 We are given the following corpus, modiﬁed from the one in the chapter: <s> I am Sam </s> <s> Sam I am </s> <s> I am Sam </s> <s> I do not like green eggs and Sam </s> Using a bigram language model with add-one smoothing, what is P(Sam \| am)? Include <s>and </s>in your counts just like any other token. 3.5 Suppose we didn’t use the end-symbol </s>. Train an unsmoothed bigram grammar on the following training corpus without using the end-symbol</s>: <s> a b <s> b b <s> b a <s> a a Demonstrate that your bigram model does not assign a single probability dis- tribution across all sentence lengths by showing that the sum of the probability of the four possible 2 word sentences over the alphabet {a,b}is 1.0, and the sum of the probability of all possible 3 word sentences over the alphabet{a,b} is also 1.0. 3.6 Suppose we train a trigram language model with add-one smoothing on a given corpus. The corpus contains V word types. Express a formula for esti- mating P(w3\|w1,w2), where w3 is a word which follows the bigram (w1,w2), in terms of various n-gram counts and V . Use the notation c(w1,w2,w3) to denote the number of times that trigram (w1,w2,w3) occurs in the corpus, and so on for bigrams and unigrams. 3.7 We are given the following corpus, modiﬁed from the one in the chapter: <s> I am Sam </s> <s> Sam I am </s> <s> I am Sam </s> <s> I do not like green eggs and Sam </s> If we use linear interpolation smoothing between a maximum-likelihood bi- gram model and a maximum-likelihood unigram model withλ1 = 1 2 and λ2 = 1 2 , what is P(Sam \|am)? Include <s> and </s> in your counts just like any other token. 3.8 Write a program to compute unsmoothed unigrams and bigrams. EXERCISES 55 3.9 Run your n-gram program on two different small corpora of your choice (you might use email text or newsgroups). Now compare the statistics of the two corpora. What are the differences in the most common unigrams between the two? How about interesting differences in bigrams? 3.10 Add an option to your program to generate random sentences. 3.11 Add an option to your program to compute the perplexity of a test set. 3.12 You are given a training set of 100 numbers that consists of 91 zeros and 1 each of the other digits 1-9. Now we see the following test set: 0 0 0 0 0 3 0 0 0 0. What is the unigram perplexity? 56 CHAPTER 4 • N AIVE BAYES , TEXT CLASSIFICATION , AND SENTIMENT CHAPTER 4 Naive Bayes, Text Classiﬁca- tion, and Sentiment Classiﬁcation lies at the heart of both human and machine intelligence. Deciding what letter, word, or image has been presented to our senses, recognizing faces or voices, sorting mail, assigning grades to homeworks; these are all examples of assigning a category to an input. The potential challenges of this task are highlighted by the fabulist Jorge Luis Borges (1964), who imagined classifying animals into: (a) those that belong to the Emperor, (b) embalmed ones, (c) those that are trained, (d) suckling pigs, (e) mermaids, (f) fabulous ones, (g) stray dogs, (h) those that are included in this classiﬁcation, (i) those that tremble as if they were mad, (j) innumerable ones, (k) those drawn with a very ﬁne camel’s hair brush, (l) others, (m) those that have just broken a ﬂower vase, (n) those that resemble ﬂies from a distance. Many language processing tasks involve classiﬁcation, although luckily our classes are much easier to deﬁne than those of Borges. In this chapter we introduce the naive Bayes algorithm and apply it to text categorization, the task of assigning a label ortext categorization category to an entire text or document. We focus on one common text categorization task, sentiment analysis, the ex-sentiment analysis traction of sentiment, the positive or negative orientation that a writer expresses toward some object. A review of a movie, book, or product on the web expresses the author’s sentiment toward the product, while an editorial or political text expresses sentiment toward a candidate or political action. Extracting consumer or public sen- timent is thus relevant for ﬁelds from marketing to politics. The simplest version of sentiment analysis is a binary classiﬁcation task, and the words of the review provide excellent cues. Consider, for example, the follow- ing phrases extracted from positive and negative reviews of movies and restaurants. Words likegreat, richly, awesome, and pathetic, and awful and ridiculously are very informative cues: + ...zany characters and richly applied satire, and some great plot twists − It was pathetic. The worst part about it was the boxing scenes... + ...awesome caramel sauce and sweet toasty almonds. I love this place! − ...awful pizza and ridiculously overpriced... Spam detection is another important commercial application, the binary clas-spam detection siﬁcation task of assigning an email to one of the two classes spam or not-spam. Many lexical and other features can be used to perform this classiﬁcation. For ex- ample you might quite reasonably be suspicious of an email containing phrases like “online pharmaceutical” or “WITHOUT ANY COST” or “Dear Winner”. Another thing we might want to know about a text is the language it’s written in. Texts on social media, for example, can be in any number of languages and we’ll need to apply different processing. The task of language id is thus the ﬁrstlanguage id step in most language processing pipelines. Related text classiﬁcation tasks like au- thorship attribution— determining a text’s author— are also relevant to the digitalauthorship attribution humanities, social sciences, and forensic linguistics. 4.1 • N AIVE BAYES CLASSIFIERS 57 Finally, one of the oldest tasks in text classiﬁcation is assigning a library sub- ject category or topic label to a text. Deciding whether a research paper concerns epidemiology or instead, perhaps, embryology, is an important component of infor- mation retrieval. Various sets of subject categories exist, such as the MeSH (Medical Subject Headings) thesaurus. In fact, as we will see, subject category classiﬁcation is the task for which the naive Bayes algorithm was invented in 1961 (Maron, 1961). Classiﬁcation is essential for tasks below the level of the document as well. We’ve already seen period disambiguation (deciding if a period is the end of a sen- tence or part of a word), and word tokenization (deciding if a character should be a word boundary). Even language modeling can be viewed as classiﬁcation: each word can be thought of as a class, and so predicting the next word is classifying the context-so-far into a class for each next word. A part-of-speech tagger (Chapter 17) classiﬁes each occurrence of a word in a sentence as, e.g., a noun or a verb. The goal of classiﬁcation is to take a single observation, extract some useful features, and thereby classify the observation into one of a set of discrete classes. One method for classifying text is to use rules handwritten by humans. Handwrit- ten rule-based classiﬁers can be components of state-of-the-art systems in language processing. But rules can be fragile, as situations or data change over time, and for some tasks humans aren’t necessarily good at coming up with the rules. The most common way of doing text classiﬁcation in language processing is instead via supervised machine learning, the subject of this chapter. In supervised supervised machine learning learning, we have a data set of input observations, each associated with some correct output (a ‘supervision signal’). The goal of the algorithm is to learn how to map from a new observation to a correct output. Formally, the task of supervised classiﬁcation is to take an input x and a ﬁxed set of output classes Y = {y1,y2,...,yM}and return a predicted class y ∈Y . For text classiﬁcation, we’ll sometimes talk about c (for “class”) instead of y as our output variable, and d (for “document”) instead of x as our input variable. In the supervised situation we have a training set ofN documents that have each been hand- labeled with a class: {(d1,c1),....,(dN ,cN )}. Our goal is to learn a classiﬁer that is capable of mapping from a new document d to its correct class c ∈C, where C is some set of useful document classes. A probabilistic classiﬁer additionally will tell us the probability of the observation being in the class. This full distribution over the classes can be useful information for downstream decisions; avoiding making discrete decisions early on can be useful when combining systems. Many kinds of machine learning algorithms are used to build classiﬁers. This chapter introduces naive Bayes; the following one introduces logistic regression. These exemplify two ways of doing classiﬁcation. Generative classiﬁers like naive Bayes build a model of how a class could generate some input data. Given an ob- servation, they return the class most likely to have generated the observation. Dis- criminative classiﬁers like logistic regression instead learn what features from the input are most useful to discriminate between the different possible classes. While discriminative systems are often more accurate and hence more commonly used, generative classiﬁers still have a role. 4.1 Naive Bayes Classiﬁers In this section we introduce the multinomial naive Bayes classiﬁer , so called be-naive Bayes classiﬁer cause it is a Bayesian classiﬁer that makes a simplifying (naive) assumption about 58 CHAPTER 4 • N AIVE BAYES , TEXT CLASSIFICATION , AND SENTIMENT how the features interact. The intuition of the classiﬁer is shown in Fig. 4.1. We represent a text document as if it were a bag of words, that is, an unordered set of words with their positionbag of words ignored, keeping only their frequency in the document. In the example in the ﬁgure, instead of representing the word order in all the phrases like “I love this movie” and “I would recommend it”, we simply note that the word I occurred 5 times in the entire excerpt, the word it 6 times, the words love, recommend, and movie once, and so on. it it it it it it I I I I I love recommend movie the the the the to to to and andand seen seen yet would with who whimsical whilewhenever times sweet several scenes satirical romantic of manages humor have happy fun friend fairy dialogue but conventions areanyone adventure always again about I love this movie! It's sweet, but with satirical humor. The dialogue is great and the adventure scenes are fun... It manages to be whimsical and romantic while laughing at the conventions of the fairy tale genre. I would recommend it to just about anyone. I've seen it several times, and I'm always happy to see it again whenever I have a friend who hasn't seen it yet! it I the to and seen yet would whimsical times sweet satirical adventure genre fairy humor have great … 6 5 4 3 3 2 1 1 1 1 1 1 1 1 1 1 1 1 … Figure 4.1 Intuition of the multinomial naive Bayes classiﬁer applied to a movie review. The position of the words is ignored (the bag-of-words assumption) and we make use of the frequency of each word. Naive Bayes is a probabilistic classiﬁer, meaning that for a document d, out of all classes c ∈C the classiﬁer returns the class ˆc which has the maximum posterior probability given the document. In Eq. 4.1 we use the hat notation ˆ to mean “ourˆ estimate of the correct class”, and we use argmax to mean an operation that selectsargmax the argument (in this case the class c) that maximizes a function (in this case the probability P(c\|d). ˆc = argmax c∈C P(c\|d) (4.1) This idea of Bayesian inference has been known since the work of Bayes (1763),Bayesian inference and was ﬁrst applied to text classiﬁcation by Mosteller and Wallace (1964). The intuition of Bayesian classiﬁcation is to use Bayes’ rule to transform Eq. 4.1 into other probabilities that have some useful properties. Bayes’ rule is presented in Eq. 4.2; it gives us a way to break down any conditional probability P(x\|y) into three other probabilities: P(x\|y) =P(y\|x)P(x) P(y) (4.2) 4.1 • N AIVE BAYES CLASSIFIERS 59 We can then substitute Eq. 4.2 into Eq. 4.1 to get Eq. 4.3: ˆc = argmax c∈C P(c\|d) =argmax c∈C P(d\|c)P(c) P(d) (4.3) We can conveniently simplify Eq. 4.3 by dropping the denominator P(d). This is possible because we will be computing P(d\|c)P(c) P(d) for each possible class. But P(d) doesn’t change for each class; we are always asking about the most likely class for the same document d, which must have the same probability P(d). Thus, we can choose the class that maximizes this simpler formula: ˆc = argmax c∈C P(c\|d) =argmax c∈C P(d\|c)P(c) (4.4) We call Naive Bayes agenerative model because we can read Eq. 4.4 as stating a kind of implicit assumption about how a document is generated: ﬁrst a class is sampled from P(c), and then the words are generated by sampling from P(d\|c). (In fact we could imagine generating artiﬁcial documents, or at least their word counts, by following this process). We’ll say more about this intuition of generative models in Chapter 5. To return to classiﬁcation: we compute the most probable class ˆ c given some document d by choosing the class which has the highest product of two probabilities: the prior probability of the class P(c) and the likelihood of the document P(d\|c):prior probability likelihood ˆc = argmax c∈C likelihood   P(d\|c) prior P(c) (4.5) Without loss of generality, we can represent a document d as a set of features f1,f2,..., fn: ˆc = argmax c∈C likelihood   P( f1,f2,...., fn\|c) prior P(c) (4.6) Unfortunately, Eq. 4.6 is still too hard to compute directly: without some sim- plifying assumptions, estimating the probability of every possible combination of features (for example, every possible set of words and positions) would require huge numbers of parameters and impossibly large training sets. Naive Bayes classiﬁers therefore make two simplifying assumptions. The ﬁrst is the bag-of-words assumption discussed intuitively above: we assume position doesn’t matter, and that the word “love” has the same effect on classiﬁcation whether it occurs as the 1st, 20th, or last word in the document. Thus we assume that the features f1,f2,..., fn only encode word identity and not position. The second is commonly called the naive Bayes assumption: this is the condi-naive Bayes assumption tional independence assumption that the probabilities P( fi\|c) are independent given the class c and hence can be ‘naively’ multiplied as follows: P( f1,f2,...., fn\|c) = P( f1\|c)·P( f2\|c)·...·P( fn\|c) (4.7) The ﬁnal equation for the class chosen by a naive Bayes classiﬁer is thus: cNB = argmax c∈C P(c) ∏ f ∈F P( f \|c) (4.8) 60 CHAPTER 4 • N AIVE BAYES , TEXT CLASSIFICATION , AND SENTIMENT To apply the naive Bayes classiﬁer to text, we will use each word in the documents as a feature, as suggested above, and we consider each of the words in the document by walking an index through every word position in the document: positions ← all word positions in test document cNB = argmax c∈C P(c) ∏ i∈positions P(wi\|c) (4.9) Naive Bayes calculations, like calculations for language modeling, are done in log space, to avoid underﬂow and increase speed. Thus Eq. 4.9 is generally instead expressed1 as cNB = argmax c∈C logP(c)+ ∑ i∈positions logP(wi\|c) (4.10) By considering features in log space, Eq. 4.10 computes the predicted class as a lin- ear function of input features. Classiﬁers that use a linear combination of the inputs to make a classiﬁcation decision —like naive Bayes and also logistic regression— are called linear classiﬁers.linear classiﬁers 4.2 Training the Naive Bayes Classiﬁer How can we learn the probabilities P(c) and P( fi\|c)? Let’s ﬁrst consider the maxi- mum likelihood estimate. We’ll simply use the frequencies in the data. For the class prior P(c) we ask what percentage of the documents in our training set are in each class c. Let Nc be the number of documents in our training data with class c and Ndoc be the total number of documents. Then: ˆP(c) = Nc Ndoc (4.11) To learn the probabilityP( fi\|c), we’ll assume a feature is just the existence of a word in the document’s bag of words, and so we’ll want P(wi\|c), which we compute as the fraction of times the word wi appears among all words in all documents of topic c. We ﬁrst concatenate all documents with category c into one big “categoryc” text. Then we use the frequency of wi in this concatenated document to give a maximum likelihood estimate of the probability: ˆP(wi\|c) = count(wi,c)∑ w∈V count(w,c) (4.12) Here the vocabulary V consists of the union of all the word types in all classes, not just the words in one class c. There is a problem, however, with maximum likelihood training. Imagine we are trying to estimate the likelihood of the word “fantastic” given classpositive, but suppose there are no training documents that both contain the word “fantastic” and are classiﬁed as positive. Perhaps the word “fantastic” happens to occur (sarcasti- cally?) in the class negative. In such a case the probability for this feature will be zero: ˆP(“fantastic”\|positive) = count(“fantastic”,positive)∑ w∈V count(w,positive) = 0 (4.13) 1 In practice throughout this book, we’ll use log to mean natural log (ln) when the base is not speciﬁed. 4.3 • W ORKED EXAMPLE 61 But since naive Bayes naively multiplies all the feature likelihoods together, zero probabilities in the likelihood term for any class will cause the probability of the class to be zero, no matter the other evidence! The simplest solution is the add-one (Laplace) smoothing introduced in Chap- ter 3. While Laplace smoothing is usually replaced by more sophisticated smoothing algorithms in language modeling, it is commonly used in naive Bayes text catego- rization: ˆP(wi\|c) = count(wi,c)+ 1∑ w∈V (count(w,c)+ 1) = count(wi,c)+ 1(∑ w∈V count(w,c) ) +\|V \| (4.14) Note once again that it is crucial that the vocabulary V consists of the union of all the word types in all classes, not just the words in one class c (try to convince yourself why this must be true; see the exercise at the end of the chapter). What do we do about words that occur in our test data but are not in our vocab- ulary at all because they did not occur in any training document in any class? The solution for such unknown words is to ignore them—remove them from the testunknown word document and not include any probability for them at all. Finally, some systems choose to completely ignore another class of words: stop words, very frequent words like the and a. This can be done by sorting the vocabu-stop words lary by frequency in the training set, and deﬁning the top 10–100 vocabulary entries as stop words, or alternatively by using one of the many predeﬁned stop word lists available online. Then each instance of these stop words is simply removed from both training and test documents as if it had never occurred. In most text classiﬁca- tion applications, however, using a stop word list doesn’t improve performance, and so it is more common to make use of the entire vocabulary and not use a stop word list. Fig. 4.2 shows the ﬁnal algorithm. 4.3 Worked example Let’s walk through an example of training and testing naive Bayes with add-one smoothing. We’ll use a sentiment analysis domain with the two classes positive (+) and negative (-), and take the following miniature training and test documents simpliﬁed from actual movie reviews. Cat Documents Training - just plain boring - entirely predictable and lacks energy - no surprises and very few laughs + very powerful + the most fun ﬁlm of the summer Test ? predictable with no fun The prior P(c) for the two classes is computed via Eq. 4.11 as Nc Ndoc : P(−) =3 5 P(+) =2 5 The word with doesn’t occur in the training set, so we drop it completely (as mentioned above, we don’t use unknown word models for naive Bayes). The like- lihoods from the training set for the remaining three words “predictable”, “no”, and 62 CHAPTER 4 • N AIVE BAYES , TEXT CLASSIFICATION , AND SENTIMENT function TRAIN NAIVE BAYES(D, C) returns V, log P(c), log P(w\|c) for each class c ∈C # Calculate P(c) terms Ndoc = number of documents in D Nc = number of documents from D in class c logprior[c]←log Nc Ndoc V←vocabulary of D bigdoc[c]←append(d) for d ∈D with class c for each word w in V # Calculate P(w\|c) terms count(w,c)←# of occurrences of w in bigdoc[c] loglikelihood[w,c]← log count(w,c) + 1∑ w′ in V (count (w′,c) + 1) return logprior, loglikelihood, V function TEST NAIVE BAYES(testdoc, logprior, loglikelihood, C, V)returns best c for each class c ∈C sum[c]←logprior[c] for each position i in testdoc word←testdoc[i] if word ∈V sum[c]←sum[c]+ loglikelihood[word,c] return argmaxc sum[c] Figure 4.2 The naive Bayes algorithm, using add-1 smoothing. To use add- α smoothing instead, change the +1 to +α for loglikelihood counts in training. “fun”, are as follows, from Eq. 4.14 (computing the probabilities for the remainder of the words in the training set is left as an exercise for the reader): P(“predictable”\|−) = 1 +1 14 +20 P(“predictable”\|+) = 0 +1 9 +20 P(“no”\|−) = 1 +1 14 +20 P(“no”\|+) = 0 +1 9 +20 P(“fun”\|−) = 0 +1 14 +20 P(“fun”\|+) = 1 +1 9 +20 For the test sentence S = “predictable with no fun”, after removing the word ‘with’, the chosen class, via Eq. 4.9, is therefore computed as follows: P(−)P(S\|−) = 3 5 ×2 ×2 ×1 343 = 6.1 ×10−5 P(+)P(S\|+) = 2 5 ×1 ×1 ×2 293 = 3.2 ×10−5 The model thus predicts the class negative for the test sentence. 4.4 Optimizing for Sentiment Analysis While standard naive Bayes text classiﬁcation can work well for sentiment analysis, some small changes are generally employed that improve performance. 4.4 • O PTIMIZING FOR SENTIMENT ANALYSIS 63 First, for sentiment classiﬁcation and a number of other text classiﬁcation tasks, whether a word occurs or not seems to matter more than its frequency. Thus it often improves performance to clip the word counts in each document at 1 (see the end of the chapter for pointers to these results). This variant is called binary multino- mial naive Bayes or binary naive Bayes. The variant uses the same algorithm asbinary naive Bayes in Fig. 4.2 except that for each document we remove all duplicate words before con- catenating them into the single big document during training and we also remove duplicate words from test documents. Fig. 4.3 shows an example in which a set of four documents (shortened and text-normalized for this example) are remapped to binary, with the modiﬁed counts shown in the table on the right. The example is worked without add-1 smoothing to make the differences clearer. Note that the results counts need not be 1; the word great has a count of 2 even for binary naive Bayes, because it appears in multiple documents. Four original documents: −it was pathetic the worst part was the boxing scenes −no plot twists or great scenes + and satire and great plot twists + great scenes great ﬁlm After per-document binarization: −it was pathetic the worst part boxing scenes −no plot twists or great scenes + and satire great plot twists + great scenes ﬁlm NB Binary Counts Counts + − + − and 2 0 1 0 boxing 0 1 0 1 ﬁlm 1 0 1 0 great 3 1 2 1 it 0 1 0 1 no 0 1 0 1 or 0 1 0 1 part 0 1 0 1 pathetic 0 1 0 1 plot 1 1 1 1 satire 1 0 1 0 scenes 1 2 1 2 the 0 2 0 1 twists 1 1 1 1 was 0 2 0 1 worst 0 1 0 1 Figure 4.3 An example of binarization for the binary naive Bayes algorithm. A second important addition commonly made when doing text classiﬁcation for sentiment is to deal with negation. Consider the difference between I really like this movie (positive) and I didn’t like this movie (negative). The negation expressed by didn’tcompletely alters the inferences we draw from the predicate like. Similarly, negation can modify a negative word to produce a positive review (don’t dismiss this ﬁlm, doesn’t let us get bored). A very simple baseline that is commonly used in sentiment analysis to deal with negation is the following: during text normalization, prepend the preﬁx NOT to every word after a token of logical negation (n’t, not, no, never) until the next punc- tuation mark. Thus the phrase didn’t like this movie , but I becomes didn’t NOT_like NOT_this NOT_movie , but I Newly formed ‘words’ like NOT like, NOT recommend will thus occur more often in negative document and act as cues for negative sentiment, while words like NOT bored, NOT dismiss will acquire positive associations. Syntactic parsing (Chapter 18) can be used deal more accurately with the scope relationship between 64 CHAPTER 4 • N AIVE BAYES , TEXT CLASSIFICATION , AND SENTIMENT these negation words and the predicates they modify, but this simple baseline works quite well in practice. Finally, in some situations we might have insufﬁcient labeled training data to train accurate naive Bayes classiﬁers using all words in the training set to estimate positive and negative sentiment. In such cases we can instead derive the positive and negative word features from sentiment lexicons, lists of words that are pre-sentiment lexicons annotated with positive or negative sentiment. Four popular lexicons are theGeneral Inquirer (Stone et al., 1966), LIWC (Pennebaker et al., 2007), the opinion lexiconGeneral Inquirer LIWC of Hu and Liu (2004a) and the MPQA Subjectivity Lexicon (Wilson et al., 2005). For example the MPQA subjectivity lexicon has 6885 words each marked for whether it is strongly or weakly biased positive or negative. Some examples: + : admirable, beautiful, conﬁdent, dazzling, ecstatic, favor, glee, great −: awful, bad, bias, catastrophe, cheat, deny, envious, foul, harsh, hate A common way to use lexicons in a naive Bayes classiﬁer is to add a feature that is counted whenever a word from that lexicon occurs. Thus we might add a feature called ‘this word occurs in the positive lexicon’, and treat all instances of words in the lexicon as counts for that one feature, instead of counting each word separately. Similarly, we might add as a second feature ‘this word occurs in the negative lexicon’ of words in the negative lexicon. If we have lots of training data, and if the test data matches the training data, using just two features won’t work as well as using all the words. But when training data is sparse or not representative of the test set, using dense lexicon features instead of sparse individual-word features may generalize better. We’ll return to this use of lexicons in Chapter 22, showing how these lexicons can be learned automatically, and how they can be applied to many other tasks be- yond sentiment classiﬁcation. 4.5 Naive Bayes for other text classiﬁcation tasks In the previous section we pointed out that naive Bayes doesn’t require that our classiﬁer use all the words in the training data as features. In fact features in naive Bayes can express any property of the input text we want. Consider the task of spam detection, deciding if a particular piece of email isspam detection an example of spam (unsolicited bulk email)—one of the ﬁrst applications of naive Bayes to text classiﬁcation (Sahami et al., 1998). A common solution here, rather than using all the words as individual features, is to predeﬁne likely sets of words or phrases as features, combined with features that are not purely linguistic. For example the open-source SpamAssassin tool 2 predeﬁnes features like the phrase “one hundred percent guaranteed”, or the feature mentions millions of dollars, which is a regular expression that matches suspiciously large sums of money. But it also includes features likeHTML has a low ratio of text to image area , that aren’t purely linguistic and might require some sophisticated computation, or totally non-linguistic features about, say, the path that the email took to arrive. More sample SpamAssassin features: • Email subject line is all capital letters • Contains phrases of urgency like “urgent reply” 2 https://spamassassin.apache.org 4.6 • N AIVE BAYES AS A LANGUAGE MODEL 65 • Email subject line contains “online pharmaceutical” • HTML has unbalanced “head” tags • Claims you can be removed from the list For other tasks, like language id —determining what language a given piecelanguage id of text is written in—the most effective naive Bayes features are not words at all, but character n-grams, 2-grams (‘zw’) 3-grams (‘nya’, ‘ V o’), or 4-grams (‘ie z’, ‘thei’), or, even simplerbyte n-grams, where instead of using the multibyte Unicode character representations called codepoints, we just pretend everything is a string of raw bytes. Because spaces count as a byte, byte n-grams can model statistics about the beginning or ending of words. A widely used naive Bayes system, langid.py (Lui and Baldwin, 2012) begins with all possible n-grams of lengths 1-4, using fea- ture selection to winnow down to the most informative 7000 ﬁnal features. Language ID systems are trained on multilingual text, such as Wikipedia (Wiki- pedia text in 68 different languages was used in (Lui and Baldwin, 2011)), or newswire. To make sure that this multilingual text correctly reﬂects different regions, dialects, and socioeconomic classes, systems also add Twitter text in many languages geo- tagged to many regions (important for getting world English dialects from countries with large Anglophone populations like Nigeria or India), Bible and Quran transla- tions, slang websites like Urban Dictionary, corpora of African American Vernacular English (Blodgett et al., 2016), and so on (Jurgens et al., 2017). 4.6 Naive Bayes as a Language Model As we saw in the previous section, naive Bayes classiﬁers can use any sort of feature: dictionaries, URLs, email addresses, network features, phrases, and so on. But if, as in Section 4.3, we use only individual word features, and we use all of the words in the text (not a subset), then naive Bayes has an important similarity to language modeling. Speciﬁcally, a naive Bayes model can be viewed as a set of class-speciﬁc unigram language models, in which the model for each class instantiates a unigram language model. Since the likelihood features from the naive Bayes model assign a probability to each word P(word\|c), the model also assigns a probability to each sentence: P(s\|c) = ∏ i∈positions P(wi\|c) (4.15) Thus consider a naive Bayes model with the classespositive (+) and negative (-) and the following model parameters: w P(w\|+) P(w\|-) I 0.1 0.2 love 0.1 0.001 this 0.01 0.01 fun 0.05 0.005 ﬁlm 0.1 0.1 ... ... ... Each of the two columns above instantiates a language model that can assign a probability to the sentence “I love this fun ﬁlm”: 66 CHAPTER 4 • N AIVE BAYES , TEXT CLASSIFICATION , AND SENTIMENT P(“I love this fun ﬁlm”\|+) = 0.1 ×0.1 ×0.01 ×0.05 ×0.1 = 5 ×10−7 P(“I love this fun ﬁlm”\|−) = 0.2 ×0.001 ×0.01 ×0.005 ×0.1 = 1.0 ×10−9 As it happens, the positive model assigns a higher probability to the sentence: P(s\|pos) >P(s\|neg). Note that this is just the likelihood part of the naive Bayes model; once we multiply in the prior a full naive Bayes model might well make a different classiﬁcation decision. 4.7 Evaluation: Precision, Recall, F-measure To introduce the methods for evaluating text classiﬁcation, let’s ﬁrst consider some simple binary detection tasks. For example, in spam detection, our goal is to label every text as being in the spam category (“positive”) or not in the spam category (“negative”). For each item (email document) we therefore need to know whether our system called it spam or not. We also need to know whether the email is actually spam or not, i.e. the human-deﬁned labels for each document that we are trying to match. We will refer to these human labels as the gold labels.gold labels Or imagine you’re the CEO of theDelicious Pie Company and you need to know what people are saying about your pies on social media, so you build a system that detects tweets concerning Delicious Pie. Here the positive class is tweets about Delicious Pie and the negative class is all other tweets. In both cases, we need a metric for knowing how well our spam detector (or pie-tweet-detector) is doing. To evaluate any system for detecting things, we start by building a confusion matrix like the one shown in Fig. 4.4. A confusion matrixconfusion matrix is a table for visualizing how an algorithm performs with respect to the human gold labels, using two dimensions (system output and gold labels), and each cell labeling a set of possible outcomes. In the spam detection case, for example, true positives are documents that are indeed spam (indicated by human-created gold labels) that our system correctly said were spam. False negatives are documents that are indeed spam but our system incorrectly labeled as non-spam. To the bottom right of the table is the equation for accuracy, which asks what percentage of all the observations (for the spam or pie examples that means all emails or tweets) our system labeled correctly. Although accuracy might seem a natural metric, we generally don’t use it for text classiﬁcation tasks. That’s because accuracy doesn’t work well when the classes are unbalanced (as indeed they are with spam, which is a large majority of email, or with tweets, which are mainly not about pie). To make this more explicit, imagine that we looked at a million tweets, and let’s say that only 100 of them are discussing their love (or hatred) for our pie, while the other 999,900 are tweets about something completely unrelated. Imagine a simple classiﬁer that stupidly classiﬁed every tweet as “not about pie”. This classiﬁer would have 999,900 true negatives and only 100 false negatives for an accuracy of 999,900/1,000,000 or 99.99%! What an amazing accuracy level! Surely we should be happy with this classiﬁer? But of course this fabulous ‘no pie’ classiﬁer would be completely useless, since it wouldn’t ﬁnd a single one of the customer comments we are looking for. In other words, accuracy is not a good metric when the goal is to discover something that is rare, or at least not completely balanced in frequency, which is a very common situation in the world. 4.7 • E VALUATION : P RECISION , RECALL , F- MEASURE 67 true positive false negative false positive true negative gold positive gold negative system positive system negative gold standard labels system output labels recall = tp tp+fn precision = tp tp+fp accuracy = tp+tn tp+fp+tn+fn Figure 4.4 A confusion matrix for visualizing how well a binary classiﬁcation system per- forms against gold standard labels. That’s why instead of accuracy we generally turn to two other metrics shown in Fig. 4.4: precision and recall. Precision measures the percentage of the items thatprecision the system detected (i.e., the system labeled as positive) that are in fact positive (i.e., are positive according to the human gold labels). Precision is deﬁned as Precision = true positives true positives + false positives Recall measures the percentage of items actually present in the input that wererecall correctly identiﬁed by the system. Recall is deﬁned as Recall = true positives true positives + false negatives Precision and recall will help solve the problem with the useless “nothing is pie” classiﬁer. This classiﬁer, despite having a fabulous accuracy of 99.99%, has a terrible recall of 0 (since there are no true positives, and 100 false negatives, the recall is 0/100). You should convince yourself that the precision at ﬁnding relevant tweets is equally problematic. Thus precision and recall, unlike accuracy, emphasize true positives: ﬁnding the things that we are supposed to be looking for. There are many ways to deﬁne a single metric that incorporates aspects of both precision and recall. The simplest of these combinations is the F-measure (vanF-measure Rijsbergen, 1975) , deﬁned as: Fβ = (β2 +1)PR β2P +R The β parameter differentially weights the importance of recall and precision, based perhaps on the needs of an application. Values of β >1 favor recall, while values of β <1 favor precision. When β = 1, precision and recall are equally bal- anced; this is the most frequently used metric, and is called Fβ=1 or just F1:F1 F1 = 2PR P +R (4.16) F-measure comes from a weighted harmonic mean of precision and recall. The harmonic mean of a set of numbers is the reciprocal of the arithmetic mean of recip- rocals: HarmonicMean(a1,a2,a3,a4,...,an) = n 1 a1 + 1 a2 + 1 a3 +...+ 1 an (4.17) 68 CHAPTER 4 • N AIVE BAYES , TEXT CLASSIFICATION , AND SENTIMENT and hence F-measure is F = 1 α 1 P +(1 −α) 1 R or ( with β2 = 1 −α α ) F = (β2 +1)PR β2P +R (4.18) Harmonic mean is used because the harmonic mean of two values is closer to the minimum of the two values than the arithmetic mean is. Thus it weighs the lower of the two numbers more heavily, which is more conservative in this situation. 4.7.1 Evaluating with more than two classes Up to now we have been describing text classiﬁcation tasks with only two classes. But lots of classiﬁcation tasks in language processing have more than two classes. For sentiment analysis we generally have 3 classes (positive, negative, neutral) and even more classes are common for tasks like part-of-speech tagging, word sense disambiguation, semantic role labeling, emotion detection, and so on. Luckily the naive Bayes algorithm is already a multi-class classiﬁcation algorithm. 8 5 10 60 urgent normal gold labels system output recallu = 8 8+5+3 precisionu= 8 8+10+11 50 30 200 spam urgent normal spam 3 recalln = recalls = precisionn= 60 5+60+50 precisions= 200 3+30+200 60 10+60+30 200 1+50+200 Figure 4.5 Confusion matrix for a three-class categorization task, showing for each pair of classes (c1,c2), how many documents from c1 were (in)correctly assigned to c2. But we’ll need to slightly modify our deﬁnitions of precision and recall. Con- sider the sample confusion matrix for a hypothetical 3-way one-of email catego- rization decision (urgent, normal, spam) shown in Fig. 4.5. The matrix shows, for example, that the system mistakenly labeled one spam document as urgent, and we have shown how to compute a distinct precision and recall value for each class. In order to derive a single metric that tells us how well the system is doing, we can com- bine these values in two ways. In macroaveraging, we compute the performancemacroaveraging for each class, and then average over classes. In microaveraging, we collect the de-microaveraging cisions for all classes into a single confusion matrix, and then compute precision and recall from that table. Fig. 4.6 shows the confusion matrix for each class separately, and shows the computation of microaveraged and macroaveraged precision. As the ﬁgure shows, a microaverage is dominated by the more frequent class (in this case spam), since the counts are pooled. The macroaverage better reﬂects the statistics of the smaller classes, and so is more appropriate when performance on all the classes is equally important. 4.8 • T EST SETS AND CROSS -VALIDATION 69 8 8 11 340 true urgent true not system urgent system not 60 40 55 212 true normal true not system normal system not 200 51 33 83 true spam true not system spam system not 268 99 99 635 true yes true no system yes system no precision = 8+11 8 = .42 precision = 200+33 200 = .86precision = 60+55 60 = .52 microaverage precision 268+99 268 = .73= macroaverage precision 3 .42+.52+.86 = .60= PooledClass 3: SpamClass 2: NormalClass 1: Urgent Figure 4.6 Separate confusion matrices for the 3 classes from the previous ﬁgure, showing the pooled confu- sion matrix and the microaveraged and macroaveraged precision. 4.8 Test sets and Cross-validation The training and testing procedure for text classiﬁcation follows what we saw with language modeling (Section 3.2): we use the training set to train the model, then use the development test set (also called a devset) to perhaps tune some parameters,development test set devset and in general decide what the best model is. Once we come up with what we think is the best model, we run it on the (hitherto unseen) test set to report its performance. While the use of a devset avoids overﬁtting the test set, having a ﬁxed train- ing set, devset, and test set creates another problem: in order to save lots of data for training, the test set (or devset) might not be large enough to be representative. Wouldn’t it be better if we could somehow use all our data for training and still use all our data for test? We can do this by cross-validation.cross-validation In cross-validation, we choose a number k, and partition our data into k disjoint subsets called folds. Now we choose one of those k folds as a test set, train ourfolds classiﬁer on the remaining k −1 folds, and then compute the error rate on the test set. Then we repeat with another fold as the test set, again training on the otherk −1 folds. We do this sampling process k times and average the test set error rate from these k runs to get an average error rate. If we choose k = 10, we would train 10 different models (each on 90% of our data), test the model 10 times, and average these 10 values. This is called 10-fold cross-validation.10-fold cross-validation The only problem with cross-validation is that because all the data is used for testing, we need the whole corpus to be blind; we can’t examine any of the data to suggest possible features and in general see what’s going on, because we’d be peeking at the test set, and such cheating would cause us to overestimate the perfor- mance of our system. However, looking at the corpus to understand what’s going on is important in designing NLP systems! What to do? For this reason, it is com- mon to create a ﬁxed training set and test set, then do 10-fold cross-validation inside the training set, but compute error rate the normal way in the test set, as shown in Fig. 4.7. 70 CHAPTER 4 • N AIVE BAYES , TEXT CLASSIFICATION , AND SENTIMENT Training Iterations 1 3 4 5 2 6 7 8 9 10 Dev Dev Dev Dev Dev Dev Dev Dev Dev Dev Training Training Training Training Training Training Training Training Training Training Training Test Set Testing Figure 4.7 10-fold cross-validation 4.9 Statistical Signiﬁcance Testing In building systems we often need to compare the performance of two systems. How can we know if the new system we just built is better than our old one? Or better than some other system described in the literature? This is the domain of statistical hypothesis testing, and in this section we introduce tests for statistical signiﬁcance for NLP classiﬁers, drawing especially on the work of Dror et al. (2020) and Berg- Kirkpatrick et al. (2012). Suppose we’re comparing the performance of classiﬁers A and B on a metric M such as F1, or accuracy. Perhaps we want to know if our logistic regression senti- ment classiﬁer A (Chapter 5) gets a higher F1 score than our naive Bayes sentiment classiﬁer B on a particular test set x. Let’s call M(A,x) the score that system A gets on test set x, and δ(x) the performance difference between A and B on x: δ(x) =M(A,x)−M(B,x) (4.19) We would like to know if δ(x) >0, meaning that our logistic regression classiﬁer has a higher F1 than our naive Bayes classiﬁer on x. δ(x) is called the effect size; aeffect size bigger δ means that A seems to be way better than B; a small δ means A seems to be only a little better. Why don’t we just check if δ(x) is positive? Suppose we do, and we ﬁnd that the F1 score of A is higher than B’s by .04. Can we be certain that A is better? We cannot! That’s becauseA might just be accidentally better thanB on this particular x. We need something more: we want to know ifA’s superiority overB is likely to hold again if we checked another test set x′, or under some other set of circumstances. In the paradigm of statistical hypothesis testing, we test this by formalizing two hypotheses. H0 : δ(x) ≤0 H1 : δ(x) >0 (4.20) The hypothesis H0, called the null hypothesis, supposes that δ(x) is actually nega-null hypothesis tive or zero, meaning that A is not better than B. We would like to know if we can conﬁdently rule out this hypothesis, and instead support H1, that A is better. We do this by creating a random variable X ranging over all test sets. Now we ask how likely is it, if the null hypothesis H0 was correct, that among these test sets 4.9 • S TATISTICAL SIGNIFICANCE TESTING 71 we would encounter the value of δ(x) that we found, if we repeated the experiment a great many times. We formalize this likelihood as the p-value: the probability,p-value assuming the null hypothesis H0 is true, of seeing the δ(x) that we saw or one even greater P(δ(X) ≥δ(x)\|H0 is true) (4.21) So in our example, this p-value is the probability that we would see δ(x) assuming A is not better than B. If δ(x) is huge (let’s say A has a very respectable F 1 of .9 and B has a terrible F1 of only .2 on x), we might be surprised, since that would be extremely unlikely to occur if H0 were in fact true, and so the p-value would be low (unlikely to have such a large δ if A is in fact not better than B). But if δ(x) is very small, it might be less surprising to us even if H0 were true and A is not really better than B, and so the p-value would be higher. A very small p-value means that the difference we observed is very unlikely under the null hypothesis, and we can reject the null hypothesis. What counts as very small? It is common to use values like .05 or .01 as the thresholds. A value of .01 means that if the p-value (the probability of observing the δ we saw assuming H0 is true) is less than .01, we reject the null hypothesis and assume thatA is indeed better than B. We say that a result (e.g., “A is better than B”) is statistically signiﬁcant ifstatistically signiﬁcant the δ we saw has a probability that is below the threshold and we therefore reject this null hypothesis. How do we compute this probability we need for the p-value? In NLP we gen- erally don’t use simple parametric tests like t-tests or ANOV As that you might be familiar with. Parametric tests make assumptions about the distributions of the test statistic (such as normality) that don’t generally hold in our cases. So in NLP we usually use non-parametric tests based on sampling: we artiﬁcially create many ver- sions of the experimental setup. For example, if we had lots of different test sets x′ we could just measure all the δ(x′) for all the x′. That gives us a distribution. Now we set a threshold (like .01) and if we see in this distribution that 99% or more of those deltas are smaller than the delta we observed, i.e., that p-value(x)—the proba- bility of seeing a δ(x) as big as the one we saw—is less than .01, then we can reject the null hypothesis and agree that δ(x) was a sufﬁciently surprising difference and A is really a better algorithm than B. There are two common non-parametric tests used in NLP: approximate ran- domization (Noreen, 1989) and the bootstrap test . We will describe bootstrapapproximate randomization below, showing the paired version of the test, which again is most common in NLP. Paired tests are those in which we compare two sets of observations that are aligned:paired each observation in one set can be paired with an observation in another. This hap- pens naturally when we are comparing the performance of two systems on the same test set; we can pair the performance of system A on an individual observation xi with the performance of system B on the same xi. 4.9.1 The Paired Bootstrap Test The bootstrap test (Efron and Tibshirani, 1993) can apply to any metric; from pre-bootstrap test cision, recall, or F1 to the BLEU metric used in machine translation. The word bootstrapping refers to repeatedly drawing large numbers of samples with replace-bootstrapping ment (called bootstrap samples) from an original set. The intuition of the bootstrap test is that we can create many virtual test sets from an observed test set by repeat- edly sampling from it. The method only makes the assumption that the sample is representative of the population. 72 CHAPTER 4 • N AIVE BAYES , TEXT CLASSIFICATION , AND SENTIMENT Consider a tiny text classiﬁcation example with a test setx of 10 documents. The ﬁrst row of Fig. 4.8 shows the results of two classiﬁers (A and B) on this test set. Each document is labeled by one of the four possibilities (A and B both right, both wrong, A right and B wrong, A wrong and B right). A slash through a letter ( B) means that that classiﬁer got the answer wrong. On the ﬁrst document both A and B get the correct class (AB), while on the second document A got it right but B got it wrong (A B). If we assume for simplicity that our metric is accuracy, A has an accuracy of .70 and B of .50, so δ(x) is .20. Now we create a large numberb (perhaps 105) of virtual test setsx(i), each of size n = 10. Fig. 4.8 shows a couple of examples. To create each virtual test set x(i), we repeatedly (n = 10 times) select a cell from rowx with replacement. For example, to create the ﬁrst cell of the ﬁrst virtual test set x(1), if we happened to randomly select the second cell of the x row; we would copy the value A B into our new cell, and move on to create the second cell of x(1), each time sampling (randomly choosing) from the original x with replacement. 1 2 3 4 5 6 7 8 9 10 A% B% δ() x AB AB AB AB AB AB AB AB AB AB .70 .50 .20 x(1) AB AB AB AB AB AB AB AB AB AB .60 .60 .00 x(2) AB AB AB AB AB AB AB AB AB AB .60 .70 -.10 ... x(b) Figure 4.8 The paired bootstrap test: Examples of b pseudo test sets x(i) being created from an initial true test set x. Each pseudo test set is created by sampling n = 10 times with replacement; thus an individual sample is a single cell, a document with its gold label and the correct or incorrect performance of classiﬁers A and B. Of course real test sets don’t have only 10 examples, and b needs to be large as well. Now that we have the b test sets, providing a sampling distribution, we can do statistics on how often A has an accidental advantage. There are various ways to compute this advantage; here we follow the version laid out in Berg-Kirkpatrick et al. (2012). Assuming H0 (A isn’t better than B), we would expect that δ(X), estimated over many test sets, would be zero or negative; a much higher value would be surprising, sinceH0 speciﬁcally assumes A isn’t better thanB. To measure exactly how surprising our observed δ(x) is, we would in other circumstances compute the p-value by counting over many test sets how oftenδ(x(i)) exceeds the expected zero value by δ(x) or more: p-value(x) =1 b b∑ i=1 1 ( δ(x(i))−δ(x) ≥0 ) (We use the notation 1 (x) to mean “1 if x is true, and 0 otherwise”.) However, although it’s generally true that the expected value of δ(X) over many test sets, (again assuming A isn’t better than B) is 0, this isn’ttrue for the bootstrapped test sets we created. That’s because we didn’t draw these samples from a distribution with 0 mean; we happened to create them from the original test setx, which happens to be biased (by .20) in favor of A. So to measure how surprising is our observed δ(x), we actually compute the p-value by counting over many test sets how often 4.10 • A VOIDING HARMS IN CLASSIFICATION 73 δ(x(i)) exceeds the expected value of δ(x) by δ(x) or more: p-value(x) = 1 b b∑ i=1 1 ( δ(x(i))−δ(x) ≥δ(x) ) = 1 b b∑ i=1 1 ( δ(x(i)) ≥2δ(x) ) (4.22) So if for example we have 10,000 test setsx(i) and a threshold of .01, and in only 47 of the test sets do we ﬁnd that A is accidentally better δ(x(i)) ≥2δ(x), the resulting p-value of .0047 is smaller than .01, indicating that the delta we found,δ(x) is indeed sufﬁciently surprising and unlikely to have happened by accident, and we can reject the null hypothesis and conclude A is better than B. function BOOTSTRAP (test set x, num of samples b) returns p-value(x) Calculate δ(x) # how much better does algorithm A do than B on x s = 0 for i = 1 to b do for j = 1 to n do # Draw a bootstrap sample x(i) of size n Select a member of x at random and add it to x(i) Calculate δ(x(i)) # how much better does algorithm A do than B on x(i) s←s + 1 if δ(x(i)) ≥2δ(x) p-value(x) ≈s b # on what % of the b samples did algorithm A beat expectations? return p-value(x) # if very few did, our observed δ is probably not accidental Figure 4.9 A version of the paired bootstrap algorithm after Berg-Kirkpatrick et al. (2012). The full algorithm for the bootstrap is shown in Fig. 4.9. It is given a test setx, a number of samples b, and counts the percentage of the b bootstrap test sets in which δ(x∗(i)) >2δ(x). This percentage then acts as a one-sided empirical p-value. 4.10 Avoiding Harms in Classiﬁcation It is important to avoid harms that may result from classiﬁers, harms that exist both for naive Bayes classiﬁers and for the other classiﬁcation algorithms we introduce in later chapters. One class of harms is representational harms (Crawford 2017, Blodgett et al.representational harms 2020), harms caused by a system that demeans a social group, for example by per- petuating negative stereotypes about them. For example Kiritchenko and Moham- mad (2018) examined the performance of 200 sentiment analysis systems on pairs of sentences that were identical except for containing either a common African Amer- ican ﬁrst name (like Shaniqua) or a common European American ﬁrst name (like Stephanie), chosen from the Caliskan et al. (2017) study discussed in Chapter 6. They found that most systems assigned lower sentiment and more negative emotion to sentences with African American names, reﬂecting and perpetuating stereotypes that associate African Americans with negative emotions (Popp et al., 2003). In other tasks classiﬁers may lead to both representational harms and other harms, such as silencing. For example the important text classiﬁcation task of tox- 74 CHAPTER 4 • N AIVE BAYES , TEXT CLASSIFICATION , AND SENTIMENT icity detection is the task of detecting hate speech, abuse, harassment, or othertoxicity detection kinds of toxic language. While the goal of such classiﬁers is to help reduce soci- etal harm, toxicity classiﬁers can themselves cause harms. For example, researchers have shown that some widely used toxicity classiﬁers incorrectly ﬂag as being toxic sentences that are non-toxic but simply mention identities like women (Park et al., 2018), blind people (Hutchinson et al., 2020) or gay people (Dixon et al., 2018; Dias Oliva et al., 2021), or simply use linguistic features characteristic of varieties like African-American Vernacular English (Sap et al. 2019, Davidson et al. 2019). Such false positive errors could lead to the silencing of discourse by or about these groups. These model problems can be caused by biases or other problems in the training data; in general, machine learning systems replicate and even amplify the biases in their training data. But these problems can also be caused by the labels (for example due to biases in the human labelers), by the resources used (like lexicons, or model components like pretrained embeddings), or even by model architecture (like what the model is trained to optimize). While the mitigation of these biases (for example by carefully considering the training data sources) is an important area of research, we currently don’t have general solutions. For this reason it’s important, when introducing any NLP model, to study these kinds of factors and make them clear. One way to do this is by releasing a model card (Mitchell et al., 2019) formodel card each version of a model. A model card documents a machine learning model with information like: • training algorithms and parameters • training data sources, motivation, and preprocessing • evaluation data sources, motivation, and preprocessing • intended use and users • model performance across different demographic or other groups and envi- ronmental situations 4.11 Summary This chapter introduced the naive Bayes model for classiﬁcation and applied it to the text categorization task of sentiment analysis. • Many language processing tasks can be viewed as tasks of classiﬁcation. • Text categorization, in which an entire text is assigned a class from a ﬁnite set, includes such tasks as sentiment analysis, spam detection, language identi- ﬁcation, and authorship attribution. • Sentiment analysis classiﬁes a text as reﬂecting the positive or negative orien- tation (sentiment) that a writer expresses toward some object. • Naive Bayes is a generative model that makes the bag-of-words assumption (position doesn’t matter) and the conditional independence assumption (words are conditionally independent of each other given the class) • Naive Bayes with binarized features seems to work better for many text clas- siﬁcation tasks. • Classiﬁers are evaluated based on precision and recall. • Classiﬁers are trained using distinct training, dev, and test sets, including the use of cross-validation in the training set. BIBLIOGRAPHICAL AND HISTORICAL NOTES 75 • Statistical signiﬁcance tests should be used to determine whether we can be conﬁdent that one version of a classiﬁer is better than another. • Designers of classiﬁers should carefully consider harms that may be caused by the model, including its training data and other components, and report model characteristics in a model card. Bibliographical and Historical Notes Multinomial naive Bayes text classiﬁcation was proposed by Maron (1961) at the RAND Corporation for the task of assigning subject categories to journal abstracts. His model introduced most of the features of the modern form presented here, ap- proximating the classiﬁcation task with one-of categorization, and implementing add-δ smoothing and information-based feature selection. The conditional independence assumptions of naive Bayes and the idea of Bayes- ian analysis of text seems to have arisen multiple times. The same year as Maron’s paper, Minsky (1961) proposed a naive Bayes classiﬁer for vision and other arti- ﬁcial intelligence problems, and Bayesian techniques were also applied to the text classiﬁcation task of authorship attribution by Mosteller and Wallace (1963). It had long been known that Alexander Hamilton, John Jay, and James Madison wrote the anonymously-published Federalist papers in 1787–1788 to persuade New York to ratify the United States Constitution. Yet although some of the 85 essays were clearly attributable to one author or another, the authorship of 12 were in dispute between Hamilton and Madison. Mosteller and Wallace (1963) trained a Bayesian probabilistic model of the writing of Hamilton and another model on the writings of Madison, then computed the maximum-likelihood author for each of the disputed essays. Naive Bayes was ﬁrst applied to spam detection in Heckerman et al. (1998). Metsis et al. (2006), Pang et al. (2002), and Wang and Manning (2012) show that using boolean attributes with multinomial naive Bayes works better than full counts. Binary multinomial naive Bayes is sometimes confused with another variant of naive Bayes that also uses a binary representation of whether a term occurs in a document: Multivariate Bernoulli naive Bayes . The Bernoulli variant instead estimates P(w\|c) as the fraction of documents that contain a term, and includes a probability for whether a term is not in a document. McCallum and Nigam (1998) and Wang and Manning (2012) show that the multivariate Bernoulli variant of naive Bayes doesn’t work as well as the multinomial algorithm for sentiment or other text tasks. There are a variety of sources covering the many kinds of text classiﬁcation tasks. For sentiment analysis see Pang and Lee (2008), and Liu and Zhang (2012). Stamatatos (2009) surveys authorship attribute algorithms. On language identiﬁca- tion see Jauhiainen et al. (2019); Jaech et al. (2016) is an important early neural system. The task of newswire indexing was often used as a test case for text classi- ﬁcation algorithms, based on the Reuters-21578 collection of newswire articles. See Manning et al. (2008) and Aggarwal and Zhai (2012) on text classiﬁcation; classiﬁcation in general is covered in machine learning textbooks (Hastie et al. 2001, Witten and Frank 2005, Bishop 2006, Murphy 2012). Non-parametric methods for computing statistical signiﬁcance were used ﬁrst in NLP in the MUC competition (Chinchor et al., 1993), and even earlier in speech recognition (Gillick and Cox 1989, Bisani and Ney 2004). Our description of the bootstrap draws on the description in Berg-Kirkpatrick et al. (2012). Recent work has focused on issues including multiple test sets and multiple metrics (Søgaard et al. 76 CHAPTER 4 • N AIVE BAYES , TEXT CLASSIFICATION , AND SENTIMENT 2014, Dror et al. 2017). Feature selection is a method of removing features that are unlikely to generalize well. Features are generally ranked by how informative they are about the classiﬁca- tion decision. A very common metric, information gain, tells us how many bits ofinformation gain information the presence of the word gives us for guessing the class. Other feature selection metrics include χ2, pointwise mutual information, and GINI index; see Yang and Pedersen (1997) for a comparison and Guyon and Elisseeff (2003) for an introduction to feature selection. Exercises 4.1 Assume the following likelihoods for each word being part of a positive or negative movie review, and equal prior probabilities for each class. pos neg I 0.09 0.16 always 0.07 0.06 like 0.29 0.06 foreign 0.04 0.15 ﬁlms 0.08 0.11 What class will Naive bayes assign to the sentence “I always like foreign ﬁlms.”? 4.2 Given the following short movie reviews, each labeled with a genre, either comedy or action: 1. fun, couple, love, love comedy 2. fast, furious, shoot action 3. couple, ﬂy, fast, fun, fun comedy 4. furious, shoot, shoot, fun action 5. ﬂy, fast, shoot, love action and a new document D: fast, couple, shoot, ﬂy compute the most likely class for D. Assume a naive Bayes classiﬁer and use add-1 smoothing for the likelihoods. 4.3 Train two models, multinomial naive Bayes and binarized naive Bayes, both with add-1 smoothing, on the following document counts for key sentiment words, with positive or negative class assigned as noted. doc “good” “poor” “great” (class) d1. 3 0 3 pos d2. 0 1 2 pos d3. 1 3 0 neg d4. 1 5 2 neg d5. 0 2 0 neg Use both naive Bayes models to assign a class (pos or neg) to this sentence: A good, good plot and great characters, but poor acting. Recall from page 61 that with naive Bayes text classiﬁcation, we simply ig- nore (throw out) any word that never occurred in the training document. (We don’t throw out words that appear in some classes but not others; that’s what add-one smoothing is for.) Do the two models agree or disagree? CHAPTER 5 Logistic Regression “And how do you know that these ﬁne begonias are not of equal importance?” Hercule Poirot, in Agatha Christie’sThe Mysterious Affair at Styles Detective stories are as littered with clues as texts are with words. Yet for the poor reader it can be challenging to know how to weigh the author’s clues in order to make the crucial classiﬁcation task: deciding whodunnit. In this chapter we introduce an algorithm that is admirably suited for discovering the link between features or clues and some particular outcome:logistic regression.logistic regression Indeed, logistic regression is one of the most important analytic tools in the social and natural sciences. In natural language processing, logistic regression is the base- line supervised machine learning algorithm for classiﬁcation, and also has a very close relationship with neural networks. As we will see in Chapter 7, a neural net- work can be viewed as a series of logistic regression classiﬁers stacked on top of each other. Thus the classiﬁcation and machine learning techniques introduced here will play an important role throughout the book. Logistic regression can be used to classify an observation into one of two classes (like ‘positive sentiment’ and ‘negative sentiment’), or into one of many classes. Because the mathematics for the two-class case is simpler, we’ll describe this special case of logistic regression ﬁrst in the next few sections, and then brieﬂy summarize the use of multinomial logistic regression for more than two classes in Section 5.3. We’ll introduce the mathematics of logistic regression in the next few sections. But let’s begin with some high-level issues. Generative and Discriminative Classiﬁers: The most important difference be- tween naive Bayes and logistic regression is that logistic regression is adiscrimina- tive classiﬁer while naive Bayes is a generative classiﬁer. These are two very different frameworks for how to build a machine learning model. Consider a visual metaphor: imagine we’re trying to distinguish dog images from cat images. A generative model would have the goal of understanding what dogs look like and what cats look like. You might literally ask such a model to ‘generate’, i.e., draw, a dog. Given a test image, the system then asks whether it’s the cat model or the dog model that better ﬁts (is less surprised by) the image, and chooses that as its label. A discriminative model, by contrast, is only try- ing to learn to distinguish the classes (perhaps with- out learning much about them). So maybe all the dogs in the training data are wearing collars and the cats aren’t. If that one feature neatly separates the classes, the model is satisﬁed. If you ask such a model what it knows about cats all it can say is that they don’t wear collars. 78 CHAPTER 5 • L OGISTIC REGRESSION More formally, recall that the naive Bayes assigns a class c to a document d not by directly computing P(c\|d) but by computing a likelihood and a prior ˆc = argmax c∈C likelihood   P(d\|c) prior P(c) (5.1) A generative model like naive Bayes makes use of this likelihood term, whichgenerative model expresses how to generate the features of a document if we knew it was of class c. By contrast a discriminative model in this text categorization scenario attemptsdiscriminative model to directly compute P(c\|d). Perhaps it will learn to assign a high weight to document features that directly improve its ability to discriminate between possible classes, even if it couldn’t generate an example of one of the classes. Components of a probabilistic machine learning classiﬁer: Like naive Bayes, logistic regression is a probabilistic classiﬁer that makes use of supervised machine learning. Machine learning classiﬁers require a training corpus of m input/output pairs (x(i),y(i)). (We’ll use superscripts in parentheses to refer to individual instances in the training set—for sentiment classiﬁcation each instance might be an individual document to be classiﬁed.) A machine learning system for classiﬁcation then has four components: 1. A feature representation of the input. For each input observation x(i), this will be a vector of features [x1,x2,...,xn]. We will generally refer to feature i for input x( j) as x( j) i , sometimes simpliﬁed as xi, but we will also see the notation fi, fi(x), or, for multiclass classiﬁcation, fi(c,x). 2. A classiﬁcation function that computes ˆy, the estimated class, via p(y\|x). In the next section we will introduce the sigmoid and softmax tools for classiﬁ- cation. 3. An objective function that we want to optimize for learning, usually involving minimizing a loss function corresponding to error on training examples. We will introduce the cross-entropy loss function. 4. An algorithm for optimizing the objective function. We introduce thestochas- tic gradient descent algorithm. Logistic regression has two phases: training: We train the system (speciﬁcally the weights w and b, introduced be- low) using stochastic gradient descent and the cross-entropy loss. test: Given a test examplex we compute p(y\|x) and return the higher probability label y = 1 or y = 0. 5.1 The sigmoid function The goal of binary logistic regression is to train a classiﬁer that can make a binary decision about the class of a new input observation. Here we introduce the sigmoid classiﬁer that will help us make this decision. Consider a single input observation x, which we will represent by a vector of features [x1,x2,...,xn]. (We’ll show sample features in the next subsection.) The classiﬁer output y can be 1 (meaning the observation is a member of the class) or 0 (the observation is not a member of the class). We want to know the probability 5.1 • T HE SIGMOID FUNCTION 79 P(y = 1\|x) that this observation is a member of the class. So perhaps the decision is “positive sentiment” versus “negative sentiment”, the features represent counts of words in a document, P(y = 1\|x) is the probability that the document has positive sentiment, and P(y = 0\|x) is the probability that the document has negative senti- ment. Logistic regression solves this task by learning, from a training set, a vector of weights and a bias term. Each weight wi is a real number, and is associated with one of the input features xi. The weight wi represents how important that input feature is to the classiﬁcation decision, and can be positive (providing evidence that the in- stance being classiﬁed belongs in the positive class) or negative (providing evidence that the instance being classiﬁed belongs in the negative class). Thus we might expect in a sentiment task the word awesome to have a high positive weight, and abysmal to have a very negative weight. The bias term, also called the intercept, isbias term intercept another real number that’s added to the weighted inputs. To make a decision on a test instance—after we’ve learned the weights in training— the classiﬁer ﬁrst multiplies each xi by its weight wi, sums up the weighted features, and adds the bias term b. The resulting single number z expresses the weighted sum of the evidence for the class. z = ( n∑ i=1 wixi ) +b (5.2) In the rest of the book we’ll represent such sums using the dot product notationdot product from linear algebra. The dot product of two vectors a and b, written as a ·b, is the sum of the products of the corresponding elements of each vector. (Notice that we represent vectors using the boldface notationb). Thus the following is an equivalent formation to Eq. 5.2: z = w ·x+b (5.3) But note that nothing in Eq. 5.3 forces z to be a legal probability, that is, to lie between 0 and 1. In fact, since weights are real-valued, the output might even be negative; z ranges from −∞ to ∞. Figure 5.1 The sigmoid function σ(z) = 1 1+e−z takes a real value and maps it to the range (0,1). It is nearly linear around 0 but outlier values get squashed toward 0 or 1. To create a probability, we’ll pass z through the sigmoid function, σ(z). Thesigmoid sigmoid function (named because it looks like an s) is also called the logistic func- tion, and gives logistic regression its name. The sigmoid has the following equation,logistic function shown graphically in Fig. 5.1: σ(z) = 1 1 +e−z = 1 1 +exp(−z) (5.4) 80 CHAPTER 5 • L OGISTIC REGRESSION (For the rest of the book, we’ll use the notation exp(x) to mean ex.) The sigmoid has a number of advantages; it takes a real-valued number and maps it into the range (0,1), which is just what we want for a probability. Because it is nearly linear around 0 but ﬂattens toward the ends, it tends to squash outlier values toward 0 or 1. And it’s differentiable, which as we’ll see in Section 5.10 will be handy for learning. We’re almost there. If we apply the sigmoid to the sum of the weighted features, we get a number between 0 and 1. To make it a probability, we just need to make sure that the two cases, p(y = 1) and p(y = 0), sum to 1. We can do this as follows: P(y = 1) = σ(w ·x+b) = 1 1 +exp(−(w ·x+b)) P(y = 0) = 1 −σ(w ·x+b) = 1 − 1 1 +exp(−(w ·x+b)) = exp(−(w ·x+b)) 1 +exp(−(w ·x+b)) (5.5) The sigmoid function has the property 1 −σ(x) =σ(−x) (5.6) so we could also have expressed P(y = 0) as σ(−(w ·x+b)). Finally, one terminological point. The input to the sigmoid function, the score z = w·x+b from Eq. 5.3, is often called thelogit. This is because the logit functionlogit is the inverse of the sigmoid. The logit function is the log of the odds ratio p 1−p : logit(p) =σ−1(p) =ln p 1 −p (5.7) Using the term logit for z is a way of reminding us that by using the sigmoid to turn z (which ranges from −∞ to ∞) into a probability, we are implicitly interpreting z as not just any real-valued number, but as speciﬁcally a log odds. 5.2 Classiﬁcation with Logistic Regression The sigmoid function from the prior section thus gives us a way to take an instance x and compute the probability P(y = 1\|x). How do we make a decision about which class to apply to a test instance x? For a given x, we say yes if the probability P(y = 1\|x) is more than .5, and no otherwise. We call .5 the decision boundary:decision boundary decision(x) = {1 if P(y = 1\|x) >0.5 0 otherwise Let’s have some examples of applying logistic regression as a classiﬁer for language tasks. 5.2 • C LASSIFICATION WITH LOGISTIC REGRESSION 81 5.2.1 Sentiment Classiﬁcation Suppose we are doing binary sentiment classiﬁcation on movie review text, and we would like to know whether to assign the sentiment class + or −to a review document doc. We’ll represent each input observation by the 6 features x1 ... x6 of the input shown in the following table; Fig. 5.2 shows the features in a sample mini test document. Var Deﬁnition Value in Fig. 5.2 x1 count(positive lexicon words ∈doc) 3 x2 count(negative lexicon words ∈doc) 2 x3 {1 if “no” ∈doc 0 otherwise 1 x4 count(1st and 2nd pronouns ∈doc) 3 x5 {1 if “!” ∈doc 0 otherwise 0 x6 ln(word count of doc) ln(66) =4.19 It's hokey . There are virtually no surprises , and the writing is second-rate . So why was it so enjoyable ? For one thing , the cast is great . Another nice touch is the music . I was overcome with the urge to get off the couch and start dancing . It sucked me in , and it'll do the same to you . x1=3 x6=4.19 x3=1 x4=3x5=0 x2=2 Figure 5.2 A sample mini test document showing the extracted features in the vector x. Let’s assume for the moment that we’ve already learned a real-valued weight for each of these features, and that the 6 weights corresponding to the 6 features are [2.5,−5.0,−1.2,0.5,2.0,0.7], while b = 0.1. (We’ll discuss in the next section how the weights are learned.) The weight w1, for example indicates how important a feature the number of positive lexicon words ( great, nice, enjoyable, etc.) is to a positive sentiment decision, while w2 tells us the importance of negative lexicon words. Note that w1 = 2.5 is positive, whilew2 = −5.0, meaning that negative words are negatively associated with a positive sentiment decision, and are about twice as important as positive words. Given these 6 features and the input review x, P(+\|x) and P(−\|x) can be com- puted using Eq. 5.5: p(+\|x) =P(y = 1\|x) = σ(w ·x+b) = σ([2.5,−5.0,−1.2,0.5,2.0,0.7]·[3,2,1,3,0,4.19]+ 0.1) = σ(.833) = 0.70 (5.8) p(−\|x) =P(y = 0\|x) = 1 −σ(w ·x+b) = 0.30 82 CHAPTER 5 • L OGISTIC REGRESSION 5.2.2 Other classiﬁcation tasks and features Logistic regression is applied to all sorts of NLP tasks, and any property of the input can be a feature. Consider the task of period disambiguation: deciding if a periodperiod disambiguation is the end of a sentence or part of a word, by classifying each period into one of two classes, EOS (end-of-sentence) and not-EOS. We might use features like x1 below expressing that the current word is lower case, perhaps with a positive weight. Or a feature expressing that the current word is in our abbreviations dictionary (“Prof.”), perhaps with a negative weight. A feature can also express a combination of proper- ties. For example a period following an upper case word is likely to be an EOS, but if the word itself is St. and the previous word is capitalized then the period is likely part of a shortening of the word street following a street name. x1 = { 1 if “ Case(wi) =Lower” 0 otherwise x2 = {1 if “ wi ∈AcronymDict” 0 otherwise x3 = {1 if “ wi = St. & Case(wi−1) =Upper” 0 otherwise Designing versus learning features: In classic models, features are designed by hand by examining the training set with an eye to linguistic intuitions and literature, supplemented by insights from error analysis on the training set of an early version of a system. We can also consider (feature interactions), complex features that arefeature interactions combinations of more primitive features. We saw such a feature for period disam- biguation above, where a period on the word St. was less likely to be the end of the sentence if the previous word was capitalized. Features can be created automatically via feature templates, abstract speciﬁcations of features. For example a bigramfeature templates template for period disambiguation might create a feature for every pair of words that occurs before a period in the training set. Thus the feature space is sparse, since we only have to create a feature if that n-gram exists in that position in the training set. The feature is generally created as a hash from the string descriptions. A user description of a feature as, “bigram(American breakfast)” is hashed into a unique integer i that becomes the feature number fi. It should be clear from the prior paragraph that designing features by hand re- quires extensive human effort. For this reason, recent NLP systems avoid hand- designed features and instead focus on representation learning: ways to learn fea- tures automatically in an unsupervised way from the input. We’ll introduce methods for representation learning in Chapter 6 and Chapter 7. Scaling input features: When different input features have extremely different ranges of values, it’s common to rescale them so they have comparable ranges. We standardize input values by centering them to result in a zero mean and a standardstandardize deviation of one (this transformation is sometimes called the z-score). That is, if µiz-score is the mean of the values of feature xi across the m observations in the input dataset, and σi is the standard deviation of the values of features xi across the input dataset, we can replace each feature xi by a new feature x′ i computed as follows: µi = 1 m m∑ j=1 x( j) i σi = √1 m m∑ j=1 ( x( j) i −µi )2 x′ i = xi −µi σi (5.9) 5.2 • C LASSIFICATION WITH LOGISTIC REGRESSION 83 Alternatively, we can normalize the input features values to lie between 0 and 1:normalize x′ i = xi −min(xi) max(xi)−min(xi) (5.10) Having input data with comparable range is useful when comparing values across features. Data scaling is especially important in large neural networks, since it helps speed up gradient descent. 5.2.3 Processing many examples at once We’ve shown the equations for logistic regression for a single example. But in prac- tice we’ll of course want to process an entire test set with many examples. Let’s suppose we have a test set consisting of m test examples each of which we’d like to classify. We’ll continue to use the notation from page 78, in which a superscript value in parentheses refers to the example index in some set of data (either for train- ing or for test). So in this case each test example x(i) has a feature vector x(i), 1 ≤i ≤m. (As usual, we’ll represent vectors and matrices in bold.) One way to compute each output value ˆy(i) is just to have a for-loop, and compute each test example one at a time: foreach x(i) in input [x(1),x(2),...,x(m)] y(i) = σ(w ·x(i) +b) (5.11) For the ﬁrst 3 test examples, then, we would be separately computing the pre- dicted ˆy(i) as follows: P(y(1) = 1\|x(1)) = σ(w ·x(1) +b) P(y(2) = 1\|x(2)) = σ(w ·x(2) +b) P(y(3) = 1\|x(3)) = σ(w ·x(3) +b) But it turns out that we can slightly modify our original equation Eq. 5.5 to do this much more efﬁciently. We’ll use matrix arithmetic to assign a class to all the examples with one matrix operation! First, we’ll pack all the input feature vectors for each input x into a single input matrix X, where each row i is a row vector consisting of the feature vector for in- put example x(i) (i.e., the vector x(i)). Assuming each example has f features and weights, X will therefore be a matrix of shape [m ×f ], as follows: X =   x(1) 1 x(1) 2 ... x(1) f x(2) 1 x(2) 2 ... x(2) f x(3) 1 x(3) 2 ... x(3) f ...   (5.12) Now if we introduce b as a vector of length m which consists of the scalar bias term b repeated m times, b = [b,b,...,b], and ˆ y= [ˆy(1),ˆy(2)...,ˆy(m)] as the vector of outputs (one scalar ˆy(i) for each input x(i) and its feature vector x(i)), and represent the weight vector w as a column vector, we can compute all the outputs with a single matrix multiplication and one addition: y = Xw +b (5.13) 84 CHAPTER 5 • L OGISTIC REGRESSION You should convince yourself that Eq. 5.13 computes the same thing as our for-loop in Eq. 5.11. For example ˆy(1), the ﬁrst entry of the output vector y, will correctly be: ˆy(1) = [x(1) 1 ,x(1) 2 ,...,x(1) f ]·[w1,w2,...,wf ]+ b (5.14) Note that we had to reorder X and w from the order they appeared in in Eq. 5.5 to make the multiplications come out properly. Here is Eq. 5.13 again with the shapes shown: y = X w + b (m ×1) ( m ×f )( f ×1) (m ×1) (5.15) Modern compilers and compute hardware can compute this matrix operation very efﬁciently, making the computation much faster, which becomes important when training or testing on very large datasets. Note by the way that we could have kept X and w in the original order ( y = Xw+b) if we had chosen to deﬁne X differently as a matrix of column vectors, one vector for each input example, instead of row vectors, and then it would have shape [ f ×m]. But we conventionally represent inputs as rows. 5.2.4 Choosing a classiﬁer Logistic regression has a number of advantages over naive Bayes. Naive Bayes has overly strong conditional independence assumptions. Consider two features which are strongly correlated; in fact, imagine that we just add the same feature f1 twice. Naive Bayes will treat both copies of f1 as if they were separate, multiplying them both in, overestimating the evidence. By contrast, logistic regression is much more robust to correlated features; if two features f1 and f2 are perfectly correlated, re- gression will simply assign part of the weight to w1 and part to w2. Thus when there are many correlated features, logistic regression will assign a more accurate probability than naive Bayes. So logistic regression generally works better on larger documents or datasets and is a common default. Despite the less accurate probabilities, naive Bayes still often makes the correct classiﬁcation decision. Furthermore, naive Bayes can work extremely well (some- times even better than logistic regression) on very small datasets (Ng and Jordan, 2002) or short documents (Wang and Manning, 2012). Furthermore, naive Bayes is easy to implement and very fast to train (there’s no optimization step). So it’s still a reasonable approach to use in some situations. 5.3 Multinomial logistic regression Sometimes we need more than two classes. Perhaps we might want to do 3-way sentiment classiﬁcation (positive, negative, or neutral). Or we could be assigning some of the labels we will introduce in Chapter 17, like the part of speech of a word (choosing from 10, 30, or even 50 different parts of speech), or the named entity type of a phrase (choosing from tags like person, location, organization). In such cases we use multinomial logistic regression, also called softmax re- multinomial logistic regression gression (in older NLP literature you will sometimes see the name maxent classi- ﬁer). In multinomial logistic regression we want to label each observation with a class k from a set of K classes, under the stipulation that only one of these classes is 5.3 • M ULTINOMIAL LOGISTIC REGRESSION 85 the correct one (sometimes called hard classiﬁcation; an observation can not be in multiple classes). Let’s use the following representation: the outputy for each input x will be a vector of length K. If class c is the correct class, we’ll set yc = 1, and set all the other elements of y to be 0, i.e., yc = 1 and yj = 0 ∀j ̸= c. A vector like this y, with one value=1 and the rest 0, is called a one-hot vector. The job of the classiﬁer is to produce an estimate vector ˆ y. For each class k, the value ˆyk will be the classiﬁer’s estimate of the probability p(yk = 1\|x). 5.3.1 Softmax The multinomial logistic classiﬁer uses a generalization of the sigmoid, called the softmax function, to compute p(yk = 1\|x). The softmax function takes a vectorsoftmax z = [z1,z2,...,zK] of K arbitrary values and maps them to a probability distribution, with each value in the range [0,1], and all the values summing to 1. Like the sigmoid, it is an exponential function. For a vector z of dimensionality K, the softmax is deﬁned as: softmax(zi) = exp(zi)∑K j=1 exp(zj) 1 ≤i ≤K (5.16) The softmax of an input vector z = [z1,z2,...,zK] is thus a vector itself: softmax(z) = [ exp(z1)∑K i=1 exp(zi) , exp(z2)∑K i=1 exp(zi) ,..., exp(zK)∑K i=1 exp(zi) ] (5.17) The denominator ∑K i=1 exp(zi) is used to normalize all the values into probabilities. Thus for example given a vector: z = [0.6,1.1,−1.5,1.2,3.2,−1.1] the resulting (rounded) softmax(z) is [0.05,0.09,0.01,0.1,0.74,0.01] Like the sigmoid, the softmax has the property of squashing values toward 0 or 1. Thus if one of the inputs is larger than the others, it will tend to push its probability toward 1, and suppress the probabilities of the smaller inputs. Finally, note that, just as for the sigmoid, we refer to z, the vector of scores that is the input to the softmax, as logits (see Eq. 5.7). 5.3.2 Applying softmax in logistic regression When we apply softmax for logistic regression, the input will (just as for the sig- moid) be the dot product between a weight vector w and an input vector x (plus a bias). But now we’ll need separate weight vectors wk and bias bk for each of the K classes. The probability of each of our output classes ˆyk can thus be computed as: p(yk = 1\|x) = exp(wk ·x+bk) K∑ j=1 exp(wj ·x+bj) (5.18) 86 CHAPTER 5 • L OGISTIC REGRESSION The form of Eq. 5.18 makes it seem that we would compute each output sep- arately. Instead, it’s more common to set up the equation for more efﬁcient com- putation by modern vector processing hardware. We’ll do this by representing the set of K weight vectors as a weight matrix W and a bias vector b. Each row k of W corresponds to the vector of weights wk. W thus has shape [K ×f ], for K the number of output classes and f the number of input features. The bias vector b has one value for each of the K output classes. If we represent the weights in this way, we can compute ˆy, the vector of output probabilities for each of the K classes, by a single elegant equation: ˆy = softmax(Wx+b) (5.19) If you work out the matrix arithmetic, you can see that the estimated score of the ﬁrst output class ˆy1 (before we take the softmax) will correctly turn out to be w1 ·x+b1. One helpful interpretation of the weight matrix W is to see each row wk as a prototype of class k. The weight vector wk that is learned represents the class asprototype a kind of template. Since two vectors that are more similar to each other have a higher dot product with each other, the dot product acts as a similarity function. Logistic regression is thus learning an exemplar representation for each class, such that incoming vectors are assigned the class k they are most similar to from the K classes. Fig. 5.3 shows the difference between binary and multinomial logistic regression by illustrating the weight vector versus weight matrix in the computation of the output class probabilities. 5.3.3 Features in Multinomial Logistic Regression Features in multinomial logistic regression act like features in binary logistic regres- sion, with the difference mentioned above that we’ll need separate weight vectors and biases for each of the K classes. Recall our binary exclamation point feature x5 from page 81: x5 = { 1 if “!” ∈doc 0 otherwise In binary classiﬁcation a positive weight w5 on a feature inﬂuences the classiﬁer toward y = 1 (positive sentiment) and a negative weight inﬂuences it toward y = 0 (negative sentiment) with the absolute value indicating how important the feature is. For multinomial logistic regression, by contrast, with separate weights for each class, a feature can be evidence for or against each individual class. In 3-way multiclass sentiment classiﬁcation, for example, we must assign each document one of the 3 classes +, −, or 0 (neutral). Now a feature related to excla- mation marks might have a negative weight for 0 documents, and a positive weight for + or −documents: Feature Deﬁnition w5,+ w5,− w5,0 f5(x) {1 if “!” ∈doc 0 otherwise 3.5 3 .1 −5.3 Because these feature weights are dependent both on the input text and the output class, we sometimes make this dependence explicit and represent the features them- selves as f (x,y): a function of both the input and the class. Using such a notation 5.4 • L EARNING IN LOGISTIC REGRESSION 87 Binary Logistic Regression w [f ⨉1] Output sigmoid [1⨉f] Input words p(+) = 1- p(-) … y^ x y Input feature vector [scalar] positive lexicon words = 1 count of “no” = 0 wordcount =3 x1 x2 x3 xf dessert was great Weight vector Multinomial Logistic Regression W [f⨉1] Output softmax [K ⨉f] Input words p(+) … y1^ y2^ y3^ x y Input feature vector [K ⨉1] positive lexicon words = 1 count of “no” = 0 wordcount =3 x1 x2 x3 xf dessert was great p(-) p(neut) Weight matrix These f red weights are a row of W corresponding to weight vector w3, (= weights for class 3, = a prototype of class 3) Figure 5.3 Binary versus multinomial logistic regression. Binary logistic regression uses a single weight vector w, and has a scalar output ˆy. In multinomial logistic regression we have K separate weight vectors corresponding to the K classes, all packed into a single weight matrix W, and a vector output ˆy. We omit the biases from both ﬁgures for clarity. f5(x) above could be represented as three features f5(x,+), f5(x,−), and f5(x,0), each of which has a single weight. We’ll use this kind of notation in our description of the CRF in Chapter 17. 5.4 Learning in Logistic Regression How are the parameters of the model, the weights w and bias b, learned? Logistic regression is an instance of supervised classiﬁcation in which we know the correct label y (either 0 or 1) for each observation x. What the system produces via Eq. 5.5 is ˆy, the system’s estimate of the true y. We want to learn parameters (meaning w and b) that make ˆy for each training observation as close as possible to the true y. This requires two components that we foreshadowed in the introduction to the chapter. The ﬁrst is a metric for how close the current label ( ˆ y) is to the true gold 88 CHAPTER 5 • L OGISTIC REGRESSION label y. Rather than measure similarity, we usually talk about the opposite of this: the distance between the system output and the gold output, and we call this distance the loss function or the cost function. In the next section we’ll introduce the lossloss function that is commonly used for logistic regression and also for neural networks, the cross-entropy loss. The second thing we need is an optimization algorithm for iteratively updating the weights so as to minimize this loss function. The standard algorithm for this is gradient descent; we’ll introduce thestochastic gradient descent algorithm in the following section. We’ll describe these algorithms for the simpler case of binary logistic regres- sion in the next two sections, and then turn to multinomial logistic regression in Section 5.8. 5.5 The cross-entropy loss function We need a loss function that expresses, for an observationx, how close the classiﬁer output ( ˆy = σ(w ·x+b)) is to the correct output (y, which is 0 or 1). We’ll call this: L(ˆy,y) = How much ˆy differs from the true y (5.20) We do this via a loss function that prefers the correct class labels of the train- ing examples to be more likely. This is called conditional maximum likelihood estimation: we choose the parameters w,b that maximize the log probability of the true y labels in the training data given the observations x. The resulting loss function is the negative log likelihood loss, generally called the cross-entropy loss.cross-entropy loss Let’s derive this loss function, applied to a single observation x. We’d like to learn weights that maximize the probability of the correct label p(y\|x). Since there are only two discrete outcomes (1 or 0), this is a Bernoulli distribution, and we can express the probability p(y\|x) that our classiﬁer produces for one observation as the following (keeping in mind that if y = 1, Eq. 5.21 simpliﬁes to ˆy; if y = 0, Eq. 5.21 simpliﬁes to 1 −ˆy): p(y\|x) = ˆyy (1 −ˆy)1−y (5.21) Now we take the log of both sides. This will turn out to be handy mathematically, and doesn’t hurt us; whatever values maximize a probability will also maximize the log of the probability: log p(y\|x) = log [ ˆyy (1 −ˆy)1−y] = ylog ˆy +(1 −y)log(1 −ˆy) (5.22) Eq. 5.22 describes a log likelihood that should be maximized. In order to turn this into a loss function (something that we need to minimize), we’ll just ﬂip the sign on Eq. 5.22. The result is the cross-entropy loss LCE: LCE(ˆy,y) =−log p(y\|x) = −[ylog ˆy +(1 −y)log(1 −ˆy)] (5.23) Finally, we can plug in the deﬁnition of ˆy = σ(w ·x+b): LCE(ˆy,y) = −[ylogσ(w ·x+b)+( 1 −y)log(1 −σ(w ·x+b))] (5.24) 5.6 • G RADIENT DESCENT 89 Let’s see if this loss function does the right thing for our example from Fig. 5.2. We want the loss to be smaller if the model’s estimate is close to correct, and bigger if the model is confused. So ﬁrst let’s suppose the correct gold label for the sentiment example in Fig. 5.2 is positive, i.e.,y = 1. In this case our model is doing well, since from Eq. 5.8 it indeed gave the example a higher probability of being positive (.70) than negative (.30). If we plug σ(w ·x +b) =.70 and y = 1 into Eq. 5.24, the right side of the equation drops out, leading to the following loss (we’ll use log to mean natural log when the base is not speciﬁed): LCE(ˆy,y) = −[ylogσ(w ·x+b)+( 1 −y)log(1 −σ(w ·x+b))] = −[logσ(w ·x+b)] = −log(.70) = .36 By contrast, let’s pretend instead that the example in Fig. 5.2 was actually negative, i.e., y = 0 (perhaps the reviewer went on to say “But bottom line, the movie is terrible! I beg you not to see it!”). In this case our model is confused and we’d want the loss to be higher. Now if we plug y = 0 and 1 −σ(w ·x+b) =.30 from Eq. 5.8 into Eq. 5.24, the left side of the equation drops out: LCE(ˆy,y) = −[ylogσ(w ·x+b)+(1 −y)log(1 −σ(w ·x+b))] = −[log(1 −σ(w ·x+b))] = −log(.30) = 1.2 Sure enough, the loss for the ﬁrst classiﬁer (.36) is less than the loss for the second classiﬁer (1.2). Why does minimizing this negative log probability do what we want? A perfect classiﬁer would assign probability 1 to the correct outcome ( y = 1 or y = 0) and probability 0 to the incorrect outcome. That means if y equals 1, the higher ˆy is (the closer it is to 1), the better the classiﬁer; the lower ˆ y is (the closer it is to 0), the worse the classiﬁer. If y equals 0, instead, the higher 1 −ˆy is (closer to 1), the better the classiﬁer. The negative log of ˆ y (if the true y equals 1) or 1 −ˆy (if the true y equals 0) is a convenient loss metric since it goes from 0 (negative log of 1, no loss) to inﬁnity (negative log of 0, inﬁnite loss). This loss function also ensures that as the probability of the correct answer is maximized, the probability of the incorrect answer is minimized; since the two sum to one, any increase in the probability of the correct answer is coming at the expense of the incorrect answer. It’s called the cross- entropy loss, because Eq. 5.22 is also the formula for thecross-entropy between the true probability distribution y and our estimated distribution ˆy. Now we know what we want to minimize; in the next section, we’ll see how to ﬁnd the minimum. 5.6 Gradient Descent Our goal with gradient descent is to ﬁnd the optimal weights: minimize the loss function we’ve deﬁned for the model. In Eq. 5.25 below, we’ll explicitly represent the fact that the cross-entropy loss function LCE is parameterized by the weights. In 90 CHAPTER 5 • L OGISTIC REGRESSION machine learning in general we refer to the parameters being learned as θ; in the case of logistic regressionθ = {w,b}. So the goal is to ﬁnd the set of weights which minimizes the loss function, averaged over all examples: ˆθ = argmin θ 1 m m∑ i=1 LCE( f (x(i);θ),y(i)) (5.25) How shall we ﬁnd the minimum of this (or any) loss function? Gradient descent is a method that ﬁnds a minimum of a function by ﬁguring out in which direction (in the space of the parameters θ) the function’s slope is rising the most steeply, and moving in the opposite direction. The intuition is that if you are hiking in a canyon and trying to descend most quickly down to the river at the bottom, you might look around yourself in all directions, ﬁnd the direction where the ground is sloping the steepest, and walk downhill in that direction. For logistic regression, this loss function is convenientlyconvex. A convex func-convex tion has at most one minimum; there are no local minima to get stuck in, so gradient descent starting from any point is guaranteed to ﬁnd the minimum. (By contrast, the loss for multi-layer neural networks is non-convex, and gradient descent may get stuck in local minima for neural network training and never ﬁnd the global opti- mum.) Although the algorithm (and the concept of gradient) are designed for direction vectors, let’s ﬁrst consider a visualization of the case where the parameter of our system is just a single scalar w, shown in Fig. 5.4. Given a random initialization of w at some value w1, and assuming the loss function L happened to have the shape in Fig. 5.4, we need the algorithm to tell us whether at the next iteration we should move left (making w2 smaller than w1) or right (making w2 bigger than w1) to reach the minimum. w Loss 0 w1 wmin slope of loss at w1 is negative (goal) one step of gradient descent Figure 5.4 The ﬁrst step in iteratively ﬁnding the minimum of this loss function, by moving w in the reverse direction from the slope of the function. Since the slope is negative, we need to move w in a positive direction, to the right. Here superscripts are used for learning steps, so w1 means the initial value of w (which is 0), w2 the value at the second step, and so on. The gradient descent algorithm answers this question by ﬁnding the gradientgradient of the loss function at the current point and moving in the opposite direction. The gradient of a function of many variables is a vector pointing in the direction of the greatest increase in a function. The gradient is a multi-variable generalization of the 5.6 • G RADIENT DESCENT 91 slope, so for a function of one variable like the one in Fig. 5.4, we can informally think of the gradient as the slope. The dotted line in Fig. 5.4 shows the slope of this hypothetical loss function at point w = w1. You can see that the slope of this dotted line is negative. Thus to ﬁnd the minimum, gradient descent tells us to go in the opposite direction: moving w in a positive direction. The magnitude of the amount to move in gradient descent is the value of the slope d dw L( f (x;w),y) weighted by a learning rate η. A higher (faster) learninglearning rate rate means that we should move w more on each step. The change we make in our parameter is the learning rate times the gradient (or the slope, in our single-variable example): wt+1 = wt −η d dw L( f (x;w),y) (5.26) Now let’s extend the intuition from a function of one scalar variable w to many variables, because we don’t just want to move left or right, we want to know where in the N-dimensional space (of the N parameters that make up θ) we should move. The gradient is just such a vector; it expresses the directional components of the sharpest slope along each of thoseN dimensions. If we’re just imagining two weight dimensions (say for one weightw and one biasb), the gradient might be a vector with two orthogonal components, each of which tells us how much the ground slopes in the w dimension and in the b dimension. Fig. 5.5 shows a visualization of the value of a 2-dimensional gradient vector taken at the red point. In an actual logistic regression, the parameter vector w is much longer than 1 or 2, since the input feature vector x can be quite long, and we need a weight wi for each xi. For each dimension/variable wi in w (plus the bias b), the gradient will have a component that tells us the slope with respect to that variable. In each dimension wi, we express the slope as a partial derivative ∂ ∂wi of the loss function. Essentially we’re asking: “How much would a small change in that variable wi inﬂuence the total loss function L?” Formally, then, the gradient of a multi-variable function f is a vector in which each component expresses the partial derivative of f with respect to one of the vari- ables. We’ll use the inverted Greek delta symbol ∇ to refer to the gradient, and Cost(w,b) w b Figure 5.5 Visualization of the gradient vector at the red point in two dimensions w and b, showing a red arrow in the x-y plane pointing in the direction we will go to look for the minimum: the opposite direction of the gradient (recall that the gradient points in the direction of increase not decrease). 92 CHAPTER 5 • L OGISTIC REGRESSION represent ˆy as f (x;θ) to make the dependence on θ more obvious: ∇L( f (x;θ),y) =   ∂ ∂w1 L( f (x;θ),y) ∂ ∂w2 L( f (x;θ),y) ... ∂ ∂wn L( f (x;θ),y) ∂ ∂b L( f (x;θ),y)   (5.27) The ﬁnal equation for updating θ based on the gradient is thus θt+1 = θt −η∇L( f (x;θ),y) (5.28) 5.6.1 The Gradient for Logistic Regression In order to updateθ, we need a deﬁnition for the gradient∇L( f (x;θ),y). Recall that for logistic regression, the cross-entropy loss function is: LCE(ˆy,y) = −[ylogσ(w ·x+b)+( 1 −y)log(1 −σ(w ·x+b))] (5.29) It turns out that the derivative of this function for one observation vectorx is Eq. 5.30 (the interested reader can see Section 5.10 for the derivation of this equation): ∂LCE(ˆy,y) ∂wj = [σ(w ·x+b)−y]xj = ( ˆy −y)xj (5.30) You’ll also sometimes see this equation in the equivalent form: ∂LCE(ˆy,y) ∂wj = −(y −ˆy)xj (5.31) Note in these equations that the gradient with respect to a single weight wj rep- resents a very intuitive value: the difference between the true y and our estimated ˆy = σ(w ·x + b) for that observation, multiplied by the corresponding input value xj. 5.6.2 The Stochastic Gradient Descent Algorithm Stochastic gradient descent is an online algorithm that minimizes the loss function by computing its gradient after each training example, and nudging θ in the right direction (the opposite direction of the gradient). (An “online algorithm” is one that processes its input example by example, rather than waiting until it sees the entire input.) Stochastic gradient descent is called stochastic because it chooses a single random example at a time; in Section 5.6.4 we’ll discuss other versions of gradient descent that batch many examples at once. Fig. 5.6 shows the algorithm. The learning rate η is a hyperparameter that must be adjusted. If it’s too high,hyperparameter the learner will take steps that are too large, overshooting the minimum of the loss function. If it’s too low, the learner will take steps that are too small, and take too long to get to the minimum. It is common to start with a higher learning rate and then slowly decrease it, so that it is a function of the iteration k of training; the notation ηk can be used to mean the value of the learning rate at iteration k. 5.6 • G RADIENT DESCENT 93 function STOCHASTIC GRADIENT DESCENT (L(), f (), x, y) returns θ # where: L is the loss function # f is a function parameterized by θ # x is the set of training inputs x(1), x(2),..., x(m) # y is the set of training outputs (labels) y(1), y(2),..., y(m) θ ←0 # (or small random values) repeat til done # see caption For each training tuple (x(i), y(i)) (in random order) 1. Optional (for reporting): # How are we doing on this tuple? Compute ˆy(i) = f (x(i);θ) # What is our estimated output ˆy? Compute the loss L(ˆy(i),y(i)) # How far off is ˆy(i) from the true output y(i)? 2. g←∇θ L( f (x(i);θ),y(i)) # How should we move θ to maximize loss? 3. θ ←θ −η g # Go the other way instead return θ Figure 5.6 The stochastic gradient descent algorithm. Step 1 (computing the loss) is used mainly to report how well we are doing on the current tuple; we don’t need to compute the loss in order to compute the gradient. The algorithm can terminate when it converges (when the gradient norm <ϵ), or when progress halts (for example when the loss starts going up on a held-out set). Weights are initialized to 0 for logistic regression, but to small random values for neural networks, as we’ll see in Chapter 7. We’ll discuss hyperparameters in more detail in Chapter 7, but in short, they are a special kind of parameter for any machine learning model. Unlike regular param- eters of a model (weights like w and b), which are learned by the algorithm from the training set, hyperparameters are special parameters chosen by the algorithm designer that affect how the algorithm works. 5.6.3 Working through an example Let’s walk through a single step of the gradient descent algorithm. We’ll use a simpliﬁed version of the example in Fig. 5.2 as it sees a single observation x, whose correct value is y = 1 (this is a positive review), and with a feature vectorx = [x1,x2] consisting of these two features: x1 = 3 (count of positive lexicon words) x2 = 2 (count of negative lexicon words) Let’s assume the initial weights and bias inθ0 are all set to 0, and the initial learning rate η is 0.1: w1 = w2 = b = 0 η = 0.1 The single update step requires that we compute the gradient, multiplied by the learning rate θt+1 = θt −η∇θ L( f (x(i);θ),y(i)) 94 CHAPTER 5 • L OGISTIC REGRESSION In our mini example there are three parameters, so the gradient vector has 3 dimen- sions, for w1, w2, and b. We can compute the ﬁrst gradient as follows: ∇w,bL =   ∂LCE(ˆy,y) ∂w1 ∂LCE(ˆy,y) ∂w2 ∂LCE(ˆy,y) ∂b  =   (σ(w ·x+b)−y)x1 (σ(w ·x+b)−y)x2 σ(w ·x+b)−y  =   (σ(0)−1)x1 (σ(0)−1)x2 σ(0)−1  =   −0.5x1 −0.5x2 −0.5  =   −1.5 −1.0 −0.5   Now that we have a gradient, we compute the new parameter vector θ1 by moving θ0 in the opposite direction from the gradient: θ1 =   w1 w2 b  −η   −1.5 −1.0 −0.5  =   .15 .1 .05   So after one step of gradient descent, the weights have shifted to be: w1 = .15, w2 = .1, and b = .05. Note that this observationx happened to be a positive example. We would expect that after seeing more negative examples with high counts of negative words, that the weight w2 would shift to have a negative value. 5.6.4 Mini-batch training Stochastic gradient descent is called stochastic because it chooses a single random example at a time, moving the weights so as to improve performance on that single example. That can result in very choppy movements, so it’s common to compute the gradient over batches of training instances rather than a single instance. For example in batch training we compute the gradient over the entire dataset.batch training By seeing so many examples, batch training offers a superb estimate of which di- rection to move the weights, at the cost of spending a lot of time processing every single example in the training set to compute this perfect direction. A compromise is mini-batch training: we train on a group of m examples (per-mini-batch haps 512, or 1024) that is less than the whole dataset. (If m is the size of the dataset, then we are doing batch gradient descent; if m = 1, we are back to doing stochas- tic gradient descent.) Mini-batch training also has the advantage of computational efﬁciency. The mini-batches can easily be vectorized, choosing the size of the mini- batch based on the computational resources. This allows us to process all the exam- ples in one mini-batch in parallel and then accumulate the loss, something that’s not possible with individual or batch training. We just need to deﬁne mini-batch versions of the cross-entropy loss function we deﬁned in Section 5.5 and the gradient in Section 5.6.1. Let’s extend the cross- entropy loss for one example from Eq. 5.23 to mini-batches of sizem. We’ll continue to use the notation that x(i) and y(i) mean the ith training features and training label, respectively. We make the assumption that the training examples are independent: log p(training labels) = log m∏ i=1 p(y(i)\|x(i)) = m∑ i=1 log p(y(i)\|x(i)) = − m∑ i=1 LCE(ˆy(i),y(i)) (5.32) 5.7 • R EGULARIZATION 95 Now the cost function for the mini-batch of m examples is the average loss for each example: Cost(ˆy,y) = 1 m m∑ i=1 LCE(ˆy(i),y(i)) = −1 m m∑ i=1 y(i) logσ(w ·x(i) +b)+( 1 −y(i))log ( 1 −σ(w ·x(i) +b) ) (5.33) The mini-batch gradient is the average of the individual gradients from Eq. 5.30: ∂Cost(ˆy,y) ∂wj = 1 m m∑ i=1 [ σ(w ·x(i) +b)−y(i) ] x(i) j (5.34) Instead of using the sum notation, we can more efﬁciently compute the gradient in its matrix form, following the vectorization we saw on page 83, where we have a matrix X of size [m ×f ] representing the m inputs in the batch, and a vector y of size [m ×1] representing the correct outputs: ∂Cost(ˆy,y) ∂w = 1 m (ˆy −y)⊺ X = 1 m (σ(Xw +b)−y)⊺ X (5.35) 5.7 Regularization Numquam ponenda est pluralitas sine necessitate ‘Plurality should never be proposed unless needed’ William of Occam There is a problem with learning weights that make the model perfectly match the training data. If a feature is perfectly predictive of the outcome because it happens to only occur in one class, it will be assigned a very high weight. The weights for features will attempt to perfectly ﬁt details of the training set, in fact too perfectly, modeling noisy factors that just accidentally correlate with the class. This problem is called overﬁtting. A good model should be able togeneralize well from the trainingoverﬁtting generalize data to the unseen test set, but a model that overﬁts will have poor generalization. To avoid overﬁtting, a new regularization term R(θ) is added to the loss func-regularization tion in Eq. 5.25, resulting in the following loss for a batch of m examples (slightly rewritten from Eq. 5.25 to be maximizing log probability rather than minimizing loss, and removing the 1 m term which doesn’t affect the argmax): ˆθ = argmax θ m∑ i=1 logP(y(i)\|x(i))−αR(θ) (5.36) The new regularization term R(θ) is used to penalize large weights. Thus a setting of the weights that matches the training data perfectly— but uses many weights with 96 CHAPTER 5 • L OGISTIC REGRESSION high values to do so—will be penalized more than a setting that matches the data a little less well, but does so using smaller weights. There are two common ways to compute this regularization term R(θ). L2 regularization is a quadratic function ofL2 regularization the weight values, named because it uses the (square of the) L2 norm of the weight values. The L2 norm, \|\|θ\|\|2, is the same as the Euclidean distance of the vector θ from the origin. If θ consists of n weights, then: R(θ) = \|\|θ\|\|2 2 = n∑ j=1 θ2 j (5.37) The L2 regularized loss function becomes: ˆθ = argmax θ [ m∑ i=1 logP(y(i)\|x(i)) ] −α n∑ j=1 θ2 j (5.38) L1 regularization is a linear function of the weight values, named after the L1 normL1 regularization \|\|W\|\|1, the sum of the absolute values of the weights, or Manhattan distance (the Manhattan distance is the distance you’d have to walk between two points in a city with a street grid like New York): R(θ) = \|\|θ\|\|1 = n∑ i=1 \|θi\| (5.39) The L1 regularized loss function becomes: ˆθ = argmax θ [ m∑ i=1 logP(y(i)\|x(i)) ] −α n∑ j=1 \|θj\| (5.40) These kinds of regularization come from statistics, where L1 regularization is called lasso regression(Tibshirani, 1996) and L2 regularization is calledridge regression,lasso ridge and both are commonly used in language processing. L2 regularization is easier to optimize because of its simple derivative (the derivative of θ2 is just 2 θ), while L1 regularization is more complex (the derivative of \|θ\|is non-continuous at zero). But while L2 prefers weight vectors with many small weights, L1 prefers sparse solutions with some larger weights but many more weights set to zero. Thus L1 regularization leads to much sparser weight vectors, that is, far fewer features. Both L1 and L2 regularization have Bayesian interpretations as constraints on the prior of how weights should look. L1 regularization can be viewed as a Laplace prior on the weights. L2 regularization corresponds to assuming that weights are distributed according to a Gaussian distribution with mean µ = 0. In a Gaussian or normal distribution, the further away a value is from the mean, the lower its probability (scaled by the varianceσ). By using a Gaussian prior on the weights, we are saying that weights prefer to have the value 0. A Gaussian for a weight θj is 1√ 2πσ 2 j exp ( −(θj −µj)2 2σ2 j ) (5.41) If we multiply each weight by a Gaussian prior on the weight, we are thus maximiz- ing the following constraint: ˆθ = argmax θ m∏ i=1 P(y(i)\|x(i))× n∏ j=1 1√ 2πσ 2 j exp ( −(θj −µj)2 2σ2 j ) (5.42) 5.8 • L EARNING IN MULTINOMIAL LOGISTIC REGRESSION 97 which in log space, with µ = 0, and assuming 2σ2 = 1, corresponds to ˆθ = argmax θ m∑ i=1 logP(y(i)\|x(i))−α n∑ j=1 θ2 j (5.43) which is in the same form as Eq. 5.38. 5.8 Learning in Multinomial Logistic Regression The loss function for multinomial logistic regression generalizes the loss function for binary logistic regression from 2 to K classes. Recall that that the cross-entropy loss for binary logistic regression (repeated from Eq. 5.23) is: LCE(ˆy,y) =−log p(y\|x) = −[ylog ˆy +(1 −y)log(1 −ˆy)] (5.44) The loss function for multinomial logistic regression generalizes the two terms in Eq. 5.44 (one that is non-zero when y = 1 and one that is non-zero when y = 0) to K terms. As we mentioned above, for multinomial regression we’ll represent bothy and ˆy as vectors. The true label y is a vector with K elements, each corresponding to a class, with yc = 1 if the correct class is c, with all other elements of y being 0. And our classiﬁer will produce an estimate vector with K elements ˆy, each element ˆyk of which represents the estimated probability p(yk = 1\|x). The loss function for a single example x, generalizing from binary logistic re- gression, is the sum of the logs of the K output classes, each weighted by the indi- cator function yk (Eq. 5.45). This turns out to be just the negative log probability of the correct class c (Eq. 5.46): LCE(ˆy,y) = − K∑ k=1 yk log ˆyk (5.45) = −log ˆyc, (where c is the correct class) (5.46) = −log ˆp(yc = 1\|x) (where c is the correct class) = −log exp(wc ·x+bc)∑K j=1 exp(wj ·x+bj) (c is the correct class) (5.47) How did we get from Eq. 5.45 to Eq. 5.46? Because only one class (let’s call itc) is the correct one, the vector y takes the value 1 only for this value ofk, i.e., has yc = 1 and yj = 0 ∀j ̸= c. That means the terms in the sum in Eq. 5.45 will all be 0 except for the term corresponding to the true classc. Hence the cross-entropy loss is simply the log of the output probability corresponding to the correct class, and we therefore also call Eq. 5.46 the negative log likelihood loss.negative log likelihood loss Of course for gradient descent we don’t need the loss, we need its gradient. The gradient for a single example turns out to be very similar to the gradient for binary logistic regression, (ˆy −y)x, that we saw in Eq. 5.30. Let’s consider one piece of the gradient, the derivative for a single weight. For each class k, the weight of the ith element of input x is wk,i. What is the partial derivative of the loss with respect to wk,i? This derivative turns out to be just the difference between the true value for the class k (which is either 1 or 0) and the probability the classiﬁer outputs for class k, 98 CHAPTER 5 • L OGISTIC REGRESSION weighted by the value of the input xi corresponding to the ith element of the weight vector for class k: ∂LCE ∂wk,i = −(yk −ˆyk)xi = −(yk −p(yk = 1\|x))xi = − ( yk − exp(wk ·x+bk)∑K j=1 exp(wj ·x+bj) ) xi (5.48) We’ll return to this case of the gradient for softmax regression when we introduce neural networks in Chapter 7, and at that time we’ll also discuss the derivation of this gradient in equations Eq. 7.33–Eq. 7.41. 5.9 Interpreting models Often we want to know more than just the correct classiﬁcation of an observation. We want to know why the classiﬁer made the decision it did. That is, we want our decision to be interpretable. Interpretability can be hard to deﬁne strictly, but theinterpretable core idea is that as humans we should know why our algorithms reach the conclu- sions they do. Because the features to logistic regression are often human-designed, one way to understand a classiﬁer’s decision is to understand the role each feature plays in the decision. Logistic regression can be combined with statistical tests (the likelihood ratio test, or the Wald test); investigating whether a particular feature is signiﬁcant by one of these tests, or inspecting its magnitude (how large is the weight w associated with the feature?) can help us interpret why the classiﬁer made the decision it makes. This is enormously important for building transparent models. Furthermore, in addition to its use as a classiﬁer, logistic regression in NLP and many other ﬁelds is widely used as an analytic tool for testing hypotheses about the effect of various explanatory variables (features). In text classiﬁcation, perhaps we want to know if logically negative words (no, not, never) are more likely to be asso- ciated with negative sentiment, or if negative reviews of movies are more likely to discuss the cinematography. However, in doing so it’s necessary to control for po- tential confounds: other factors that might inﬂuence sentiment (the movie genre, the year it was made, perhaps the length of the review in words). Or we might be study- ing the relationship between NLP-extracted linguistic features and non-linguistic outcomes (hospital readmissions, political outcomes, or product sales), but need to control for confounds (the age of the patient, the county of voting, the brand of the product). In such cases, logistic regression allows us to test whether some feature is associated with some outcome above and beyond the effect of other features. 5.10 Advanced: Deriving the Gradient Equation In this section we give the derivation of the gradient of the cross-entropy loss func- tion LCE for logistic regression. Let’s start with some quick calculus refreshers. First, the derivative of ln(x): d dx ln(x) =1 x (5.49) 5.11 • S UMMARY 99 Second, the (very elegant) derivative of the sigmoid: dσ(z) dz = σ(z)(1 −σ(z)) (5.50) Finally, the chain rule of derivatives. Suppose we are computing the derivativechain rule of a composite function f (x) =u(v(x)). The derivative of f (x) is the derivative of u(x) with respect to v(x) times the derivative of v(x) with respect to x: d f dx = du dv ·dv dx (5.51) First, we want to know the derivative of the loss function with respect to a single weight wj (we’ll need to compute it for each weight, and for the bias): ∂LCE ∂wj = ∂ ∂wj −[ylogσ(w ·x+b)+( 1 −y)log(1 −σ(w ·x+b))] = − [ ∂ ∂wj ylogσ(w ·x+b)+ ∂ ∂wj (1 −y)log[1 −σ(w ·x+b)] ] (5.52) Next, using the chain rule, and relying on the derivative of log: ∂LCE ∂wj = − y σ(w ·x+b) ∂ ∂wj σ(w ·x+b)− 1 −y 1 −σ(w ·x+b) ∂ ∂wj 1 −σ(w ·x+b) (5.53) Rearranging terms: ∂LCE ∂wj = − [ y σ(w ·x+b) − 1 −y 1 −σ(w ·x+b) ] ∂ ∂wj σ(w ·x+b) (5.54) And now plugging in the derivative of the sigmoid, and using the chain rule one more time, we end up with Eq. 5.55: ∂LCE ∂wj = − [ y −σ(w ·x+b) σ(w ·x+b)[1 −σ(w ·x+b)] ] σ(w ·x+b)[1 −σ(w ·x+b)]∂(w ·x+b) ∂wj = − [ y −σ(w ·x+b) σ(w ·x+b)[1 −σ(w ·x+b)] ] σ(w ·x+b)[1 −σ(w ·x+b)]xj = −[y −σ(w ·x+b)]xj = [σ(w ·x+b)−y]xj (5.55) 5.11 Summary This chapter introduced the logistic regression model of classiﬁcation. • Logistic regression is a supervised machine learning classiﬁer that extracts real-valued features from the input, multiplies each by a weight, sums them, and passes the sum through a sigmoid function to generate a probability. A threshold is used to make a decision. 100 CHAPTER 5 • L OGISTIC REGRESSION • Logistic regression can be used with two classes (e.g., positive and negative sentiment) or with multiple classes (multinomial logistic regression, for ex- ample for n-ary text classiﬁcation, part-of-speech labeling, etc.). • Multinomial logistic regression uses the softmax function to compute proba- bilities. • The weights (vector w and bias b) are learned from a labeled training set via a loss function, such as the cross-entropy loss, that must be minimized. • Minimizing this loss function is a convex optimization problem, and iterative algorithms like gradient descent are used to ﬁnd the optimal weights. • Regularization is used to avoid overﬁtting. • Logistic regression is also one of the most useful analytic tools, because of its ability to transparently study the importance of individual features. Bibliographical and Historical Notes Logistic regression was developed in the ﬁeld of statistics, where it was used for the analysis of binary data by the 1960s, and was particularly common in medicine (Cox, 1969). Starting in the late 1970s it became widely used in linguistics as one of the formal foundations of the study of linguistic variation (Sankoff and Labov, 1979). Nonetheless, logistic regression didn’t become common in natural language pro- cessing until the 1990s, when it seems to have appeared simultaneously from two directions. The ﬁrst source was the neighboring ﬁelds of information retrieval and speech processing, both of which had made use of regression, and both of which lent many other statistical techniques to NLP. Indeed a very early use of logistic regression for document routing was one of the ﬁrst NLP applications to use (LSI) embeddings as word representations (Sch¨utze et al., 1995). At the same time in the early 1990s logistic regression was developed and ap- plied to NLP at IBM Research under the name maximum entropy modeling ormaximum entropy maxent (Berger et al., 1996), seemingly independent of the statistical literature. Un- der that name it was applied to language modeling (Rosenfeld, 1996), part-of-speech tagging (Ratnaparkhi, 1996), parsing (Ratnaparkhi, 1997), coreference resolution (Kehler, 1997b), and text classiﬁcation (Nigam et al., 1999). More on classiﬁcation can be found in machine learning textbooks (Hastie et al. 2001, Witten and Frank 2005, Bishop 2006, Murphy 2012). Exercises CHAPTER 6 Vector Semantics and Embeddings 荃者所以在鱼，得鱼而忘荃 Nets are for ﬁsh; Once you get the ﬁsh, you can forget the net. 言者所以在意，得意而忘言 Words are for meaning; Once you get the meaning, you can forget the words 庄子(Zhuangzi), Chapter 26 The asphalt that Los Angeles is famous for occurs mainly on its freeways. But in the middle of the city is another patch of asphalt, the La Brea tar pits, and this asphalt preserves millions of fossil bones from the last of the Ice Ages of the Pleis- tocene Epoch. One of these fossils is the Smilodon, or saber-toothed tiger, instantly recognizable by its long canines. Five million years ago or so, a completely different saber-tooth tiger called Thylacosmilus lived in Argentina and other parts of South Amer- ica. Thylacosmilus was a marsupial whereas Smilodon was a placental mammal, but Thy- lacosmilus had the same long upper canines and, like Smilodon, had a protective bone ﬂange on the lower jaw. The similarity of these two mammals is one of many examples of parallel or convergent evolution, in which particular contexts or environments lead to the evolution of very similar structures in different species (Gould, 1980). The role of context is also important in the similarity of a less biological kind of organism: the word. Words that occur in similar contexts tend to have similar meanings. This link between similarity in how words are distributed and similarity in what they mean is called the distributional hypothesis. The hypothesis wasdistributional hypothesis ﬁrst formulated in the 1950s by linguists like Joos (1950), Harris (1954), and Firth (1957), who noticed that words which are synonyms (like oculist and eye-doctor) tended to occur in the same environment (e.g., near words like eye or examined) with the amount of meaning difference between two words “corresponding roughly to the amount of difference in their environments” (Harris, 1954, p. 157). In this chapter we introduce vector semantics, which instantiates this linguisticvector semantics hypothesis by learning representations of the meaning of words, calledembeddings,embeddings directly from their distributions in texts. These representations are used in every nat- ural language processing application that makes use of meaning, and the static em- beddings we introduce here underlie the more powerful dynamic orcontextualized embeddings like BERT that we will see in Chapter 11. These word representations are also the ﬁrst example in this book of repre- sentation learning, automatically learning useful representations of the input text.representation learning Finding such self-supervised ways to learn representations of the input, instead of creating representations by hand via feature engineering, is an important focus of NLP research (Bengio et al., 2013). 102 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS 6.1 Lexical Semantics Let’s begin by introducing some basic principles of word meaning. How should we represent the meaning of a word? In the n-gram models of Chapter 3, and in classical NLP applications, our only representation of a word is as a string of letters, or an index in a vocabulary list. This representation is not that different from a tradition in philosophy, perhaps you’ve seen it in introductory logic classes, in which the meaning of words is represented by just spelling the word with small capital letters; representing the meaning of “dog” as DOG , and “cat” as CAT, or by using an apostrophe (DOG ’). Representing the meaning of a word by capitalizing it is a pretty unsatisfactory model. You might have seen a version of a joke due originally to semanticist Barbara Partee (Carlson, 1977): Q: What’s the meaning of life? A: LIFE ’ Surely we can do better than this! After all, we’ll want a model of word meaning to do all sorts of things for us. It should tell us that some words have similar mean- ings (cat is similar to dog), others are antonyms (cold is the opposite of hot), some have positive connotations (happy) while others have negative connotations (sad). It should represent the fact that the meanings of buy, sell, and pay offer differing per- spectives on the same underlying purchasing event. (If I buy something from you, you’ve probably sold it to me, and I likely paid you.) More generally, a model of word meaning should allow us to draw inferences to address meaning-related tasks like question-answering or dialogue. In this section we summarize some of these desiderata, drawing on results in the linguistic study of word meaning, which is called lexical semantics; we’ll return tolexical semantics and expand on this list in Appendix G and Chapter 21. Lemmas and Senses Let’s start by looking at how one word (we’ll choosemouse) might be deﬁned in a dictionary (simpliﬁed from the online dictionary WordNet): mouse (N) 1. any of numerous small rodents... 2. a hand-operated device that controls a cursor... Here the form mouse is the lemma, also called the citation form. The formlemma citation form mouse would also be the lemma for the word mice; dictionaries don’t have separate deﬁnitions for inﬂected forms like mice. Similarly sing is the lemma for sing, sang, sung. In many languages the inﬁnitive form is used as the lemma for the verb, so Spanish dormir “to sleep” is the lemma forduermes “you sleep”. The speciﬁc forms sung or carpets or sing or duermes are called wordforms.wordform As the example above shows, each lemma can have multiple meanings; the lemma mouse can refer to the rodent or the cursor control device. We call each of these aspects of the meaning of mouse a word sense. The fact that lemmas can be polysemous (have multiple senses) can make interpretation difﬁcult (is someone who types “mouse info” into a search engine looking for a pet or a tool?). Chap- ter 11 and Appendix G will discuss the problem of polysemy, and introduce word sense disambiguation, the task of determining which sense of a word is being used in a particular context. Synonymy One important component of word meaning is the relationship be- tween word senses. For example when one word has a sense whose meaning is 6.1 • L EXICAL SEMANTICS 103 identical to a sense of another word, or nearly identical, we say the two senses of those two words are synonyms. Synonyms include such pairs assynonym couch/sofa vomit/throw up ﬁlbert/hazelnut car/automobile A more formal deﬁnition of synonymy (between words rather than senses) is that two words are synonymous if they are substitutable for one another in any sentence without changing the truth conditions of the sentence, the situations in which the sentence would be true. While substitutions between some pairs of words like car / automobile or wa- ter / H2O are truth preserving, the words are still not identical in meaning. Indeed, probably no two words are absolutely identical in meaning. One of the fundamental tenets of semantics, called theprinciple of contrast(Girard 1718, Br´eal 1897, Clarkprinciple of contrast 1987), states that a difference in linguistic form is always associated with some dif- ference in meaning. For example, the word H2O is used in scientiﬁc contexts and would be inappropriate in a hiking guide— water would be more appropriate— and this genre difference is part of the meaning of the word. In practice, the word syn- onym is therefore used to describe a relationship of approximate or rough synonymy. Word Similarity While words don’t have many synonyms, most words do have lots of similar words. Cat is not a synonym of dog, but cats and dogs are certainly similar words. In moving from synonymy to similarity, it will be useful to shift from talking about relations between word senses (like synonymy) to relations between words (like similarity). Dealing with words avoids having to commit to a particular representation of word senses, which will turn out to simplify our task. The notion of word similarity is very useful in larger semantic tasks. Knowingsimilarity how similar two words are can help in computing how similar the meaning of two phrases or sentences are, a very important component of tasks like question answer- ing, paraphrasing, and summarization. One way of getting values for word similarity is to ask humans to judge how similar one word is to another. A number of datasets have resulted from such experiments. For example the SimLex-999 dataset (Hill et al., 2015) gives values on a scale from 0 to 10, like the examples below, which range from near-synonyms ( vanish, disappear) to pairs that scarcely seem to have anything in common (hole, agreement): vanish disappear 9.8 belief impression 5.95 muscle bone 3.65 modest ﬂexible 0.98 hole agreement 0.3 Word Relatedness The meaning of two words can be related in ways other than similarity. One such class of connections is called word relatedness (Budanitskyrelatedness and Hirst, 2006), also traditionally called word association in psychology.association Consider the meanings of the words coffee and cup. Coffee is not similar to cup; they share practically no features (coffee is a plant or a beverage, while a cup is a manufactured object with a particular shape). But coffee and cup are clearly related; they are associated by co-participating in an everyday event (the event of drinking coffee out of a cup). Similarly scalpel and surgeon are not similar but are related eventively (a surgeon tends to make use of a scalpel). One common kind of relatedness between words is if they belong to the same semantic ﬁeld. A semantic ﬁeld is a set of words which cover a particular semanticsemantic ﬁeld domain and bear structured relations with each other. For example, words might be 104 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS related by being in the semantic ﬁeld of hospitals ( surgeon, scalpel, nurse, anes- thetic, hospital), restaurants (waiter, menu, plate, food, chef), or houses (door, roof, kitchen, family, bed). Semantic ﬁelds are also related to topic models, like Latenttopic models Dirichlet Allocation, LDA, which apply unsupervised learning on large sets of texts to induce sets of associated words from text. Semantic ﬁelds and topic models are very useful tools for discovering topical structure in documents. In Appendix G we’ll introduce more relations between senses like hypernymy or IS-A, antonymy (opposites) and meronymy (part-whole relations). Semantic Frames and Roles Closely related to semantic ﬁelds is the idea of a semantic frame. A semantic frame is a set of words that denote perspectives orsemantic frame participants in a particular type of event. A commercial transaction, for example, is a kind of event in which one entity trades money to another entity in return for some good or service, after which the good changes hands or perhaps the service is performed. This event can be encoded lexically by using verbs like buy (the event from the perspective of the buyer), sell (from the perspective of the seller), pay (focusing on the monetary aspect), or nouns like buyer. Frames have semantic roles (like buyer, seller, goods, money), and words in a sentence can take on these roles. Knowing that buy and sell have this relation makes it possible for a system to know that a sentence like Sam bought the book from Ling could be paraphrased as Ling sold the book to Sam , and that Sam has the role of the buyer in the frame and Ling the seller. Being able to recognize such paraphrases is important for question answering, and can help in shifting perspective for machine translation. Connotation Finally, words have affective meanings or connotations. The wordconnotations connotation has different meanings in different ﬁelds, but here we use it to mean the aspects of a word’s meaning that are related to a writer or reader’s emotions, senti- ment, opinions, or evaluations. For example some words have positive connotations (wonderful) while others have negative connotations ( dreary). Even words whose meanings are similar in other ways can vary in connotation; consider the difference in connotations between fake, knockoff, forgery, on the one hand, and copy, replica, reproduction on the other, or innocent (positive connotation) and naive (negative connotation). Some words describe positive evaluation (great, love) and others neg- ative evaluation (terrible, hate). Positive or negative evaluation language is called sentiment, as we saw in Chapter 4, and word sentiment plays a role in importantsentiment tasks like sentiment analysis, stance detection, and applications of NLP to the lan- guage of politics and consumer reviews. Early work on affective meaning (Osgood et al., 1957) found that words varied along three important dimensions of affective meaning: valence: the pleasantness of the stimulus arousal: the intensity of emotion provoked by the stimulus dominance: the degree of control exerted by the stimulus Thus words like happy or satisﬁed are high on valence, while unhappy or an- noyed are low on valence. Excited is high on arousal, while calm is low on arousal. Controlling is high on dominance, while awed or inﬂuenced are low on dominance. Each word is thus represented by three numbers, corresponding to its value on each of the three dimensions: 6.2 • V ECTOR SEMANTICS 105 Valence Arousal Dominance courageous 8.05 5.5 7.38 music 7.67 5.57 6.5 heartbreak 2.45 5.65 3.58 cub 6.71 3.95 4.24 Osgood et al. (1957) noticed that in using these 3 numbers to represent the meaning of a word, the model was representing each word as a point in a three- dimensional space, a vector whose three dimensions corresponded to the word’s rating on the three scales. This revolutionary idea that word meaning could be rep- resented as a point in space (e.g., that part of the meaning of heartbreak can be represented as the point [2.45,5.65,3.58]) was the ﬁrst expression of the vector se- mantics models that we introduce next. 6.2 Vector Semantics Vector semantics is the standard way to represent word meaning in NLP, helpingvector semantics us model many of the aspects of word meaning we saw in the previous section. The roots of the model lie in the 1950s when two big ideas converged: Osgood’s 1957 idea mentioned above to use a point in three-dimensional space to represent the connotation of a word, and the proposal by linguists like Joos (1950), Harris (1954), and Firth (1957) to deﬁne the meaning of a word by its distribution in language use, meaning its neighboring words or grammatical environments. Their idea was that two words that occur in very similar distributions (whose neighboring words are similar) have similar meanings. For example, suppose you didn’t know the meaning of the word ongchoi (a re- cent borrowing from Cantonese) but you see it in the following contexts: (6.1) Ongchoi is delicious sauteed with garlic. (6.2) Ongchoi is superb over rice. (6.3) ...ongchoi leaves with salty sauces... And suppose that you had seen many of these context words in other contexts: (6.4) ...spinach sauteed with garlic over rice... (6.5) ...chard stems and leaves are delicious... (6.6) ...collard greens and other salty leafy greens The fact that ongchoi occurs with words like rice and garlic and delicious and salty, as do words likespinach, chard, and collard greensmight suggest that ongchoi is a leafy green similar to these other leafy greens. 1 We can do the same thing computationally by just counting words in the context of ongchoi. The idea of vector semantics is to represent a word as a point in a multidimen- sional semantic space that is derived (in ways we’ll see) from the distributions of word neighbors. Vectors for representing words are called embeddings (althoughembeddings the term is sometimes more strictly applied only to dense vectors like word2vec (Section 6.8), rather than sparse tf-idf or PPMI vectors (Section 6.3-Section 6.6)). The word “embedding” derives from its mathematical sense as a mapping from one space or structure to another, although the meaning has shifted; see the end of the chapter. 1 It’s in factIpomoea aquatica, a relative of morning glory sometimes called water spinach in English. 106 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS good nice bad worst not good wonderfulamazing terriﬁc dislike worse very good incredibly good fantastic incredibly badnow youi that with byto ’s are is a than Figure 6.1 A two-dimensional (t-SNE) projection of embeddings for some words and phrases, showing that words with similar meanings are nearby in space. The original 60- dimensional embeddings were trained for sentiment analysis. Simpliﬁed from Li et al. (2015) with colors added for explanation. Fig. 6.1 shows a visualization of embeddings learned for sentiment analysis, showing the location of selected words projected down from 60-dimensional space into a two dimensional space. Notice the distinct regions containing positive words, negative words, and neutral function words. The ﬁne-grained model of word similarity of vector semantics offers enormous power to NLP applications. NLP applications like the sentiment classiﬁers of Chap- ter 4 or Chapter 5 depend on the same words appearing in the training and test sets. But by representing words as embeddings, a classiﬁer can assign sentiment as long as it sees some words with similar meanings. And as we’ll see, vector semantic models can be learned automatically from text without supervision. In this chapter we’ll introduce the two most commonly used models. In thetf-idf model, an important baseline, the meaning of a word is deﬁned by a simple function of the counts of nearby words. We will see that this method results in very long vectors that are sparse, i.e. mostly zeros (since most words simply never occur in the context of others). We’ll introduce the word2vec model family for construct- ing short, dense vectors that have useful semantic properties. We’ll also introduce the cosine, the standard way to use embeddings to compute semantic similarity, be- tween two words, two sentences, or two documents, an important tool in practical applications like question answering, summarization, or automatic essay grading. 6.3 Words and Vectors “The most important attributes of a vector in 3-space are {Location, Location, Location}” Randall Munroe, https://xkcd.com/2358/ Vector or distributional models of meaning are generally based on aco-occurrence matrix, a way of representing how often words co-occur. We’ll look at two popular matrices: the term-document matrix and the term-term matrix. 6.3.1 Vectors and documents In a term-document matrix, each row represents a word in the vocabulary and eachterm-document matrix column represents a document from some collection of documents. Fig. 6.2 shows a small selection from a term-document matrix showing the occurrence of four words in four plays by Shakespeare. Each cell in this matrix represents the number of times 6.3 • W ORDS AND VECTORS 107 a particular word (deﬁned by the row) occurs in a particular document (deﬁned by the column). Thus fool appeared 58 times in Twelfth Night. As You Like It Twelfth Night Julius Caesar Henry V battle 1 0 7 13 good 114 80 62 89 fool 36 58 1 4 wit 20 15 2 3 Figure 6.2 The term-document matrix for four words in four Shakespeare plays. Each cell contains the number of times the (row) word occurs in the (column) document. The term-document matrix of Fig. 6.2 was ﬁrst deﬁned as part of the vector space model of information retrieval (Salton, 1971). In this model, a document isvector space model represented as a count vector, a column in Fig. 6.3. To review some basic linear algebra, a vector is, at heart, just a list or array ofvector numbers. So As You Like It is represented as the list [1,114,36,20] (the ﬁrst column vector in Fig. 6.3) and Julius Caesar is represented as the list [7,62,1,2] (the third column vector). A vector space is a collection of vectors, and is characterized byvector space its dimension. Vectors in a 3-dimensional vector space have an element for eachdimension dimension of the space. We will loosely refer to a vector in a 4-dimensional space as a 4-dimensional vector, with one element along each dimension. In the example in Fig. 6.3, we’ve chosen to make the document vectors of dimension 4, just so they ﬁt on the page; in real term-document matrices, the document vectors would have dimensionality \|V \|, the vocabulary size. The ordering of the numbers in a vector space indicates the different dimensions on which documents vary. The ﬁrst dimension for both these vectors corresponds to the number of times the word battle occurs, and we can compare each dimension, noting for example that the vectors for As You Like Itand Twelfth Night have similar values (1 and 0, respectively) for the ﬁrst dimension. As You Like It Twelfth Night Julius Caesar Henry V battle 1 0 7 13 good 114 80 62 89 fool 36 58 1 4 wit 20 15 2 3 Figure 6.3 The term-document matrix for four words in four Shakespeare plays. The red boxes show that each document is represented as a column vector of length four. We can think of the vector for a document as a point in \|V \|-dimensional space; thus the documents in Fig. 6.3 are points in 4-dimensional space. Since 4-dimensional spaces are hard to visualize, Fig. 6.4 shows a visualization in two dimensions; we’ve arbitrarily chosen the dimensions corresponding to the words battle and fool. Term-document matrices were originally deﬁned as a means of ﬁnding similar documents for the task of documentinformation retrieval. Two documents that are similar will tend to have similar words, and if two documents have similar words their column vectors will tend to be similar. The vectors for the comedies As You Like It [1,114,36,20] and Twelfth Night [0,80,58,15] look a lot more like each other (more fools and wit than battles) than they look like Julius Caesar [7,62,1,2] or Henry V [13,89,4,3]. This is clear with the raw numbers; in the ﬁrst dimension (battle) the comedies have low numbers and the others have high numbers, and we can see it visually in Fig. 6.4; we’ll see very shortly how to quantify this intuition more formally. 108 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS 5 10 15 20 25 30 5 10 Henry V [4,13] As You Like It [36,1] Julius Caesar [1,7] battle fool Twelfth Night [58,0] 15 40 35 40 45 50 55 60 Figure 6.4 A spatial visualization of the document vectors for the four Shakespeare play documents, showing just two of the dimensions, corresponding to the words battle and fool. The comedies have high values for thefool dimension and low values for thebattle dimension. A real term-document matrix, of course, wouldn’t just have 4 rows and columns, let alone 2. More generally, the term-document matrix has \|V \|rows (one for each word type in the vocabulary) and D columns (one for each document in the collec- tion); as we’ll see, vocabulary sizes are generally in the tens of thousands, and the number of documents can be enormous (think about all the pages on the web). Information retrieval (IR) is the task of ﬁnding the document d from the Dinformation retrieval documents in some collection that best matches a queryq. For IR we’ll therefore also represent a query by a vector, also of length \|V \|, and we’ll need a way to compare two vectors to ﬁnd how similar they are. (Doing IR will also require efﬁcient ways to store and manipulate these vectors by making use of the convenient fact that these vectors are sparse, i.e., mostly zeros). Later in the chapter we’ll introduce some of the components of this vector com- parison process: the tf-idf term weighting, and the cosine similarity metric. 6.3.2 Words as vectors: document dimensions We’ve seen that documents can be represented as vectors in a vector space. But vector semantics can also be used to represent the meaning of words. We do this by associating each word with a word vector— a row vector rather than a columnrow vector vector, hence with different dimensions, as shown in Fig. 6.5. The four dimensions of the vector for fool, [36,58,1,4], correspond to the four Shakespeare plays. Word counts in the same four dimensions are used to form the vectors for the other 3 words: wit, [20,15,2,3]; battle, [1,0,7,13]; and good [114,80,62,89]. As You Like It Twelfth Night Julius Caesar Henry V battle 1 0 7 13 good 114 80 62 89 fool 36 58 1 4 wit 20 15 2 3 Figure 6.5 The term-document matrix for four words in four Shakespeare plays. The red boxes show that each word is represented as a row vector of length four. For documents, we saw that similar documents had similar vectors, because sim- ilar documents tend to have similar words. This same principle applies to words: similar words have similar vectors because they tend to occur in similar documents. The term-document matrix thus lets us represent the meaning of a word by the doc- uments it tends to occur in. 6.3 • W ORDS AND VECTORS 109 6.3.3 Words as vectors: word dimensions An alternative to using the term-document matrix to represent words as vectors of document counts, is to use the term-term matrix, also called the word-word ma- trix or the term-context matrix, in which the columns are labeled by words ratherword-word matrix than documents. This matrix is thus of dimensionality\|V \|×\|V \|and each cell records the number of times the row (target) word and the column (context) word co-occur in some context in some training corpus. The context could be the document, in which case the cell represents the number of times the two words appear in the same document. It is most common, however, to use smaller contexts, generally a win- dow around the word, for example of 4 words to the left and 4 words to the right, in which case the cell represents the number of times (in some training corpus) the column word occurs in such a ±4 word window around the row word. Here are four examples of words in their windows: is traditionally followed by cherry pie, a traditional dessert often mixed, such as strawberry rhubarb pie. Apple pie computer peripherals and personal digital assistants. These devices usually a computer. This includes information available on the internet If we then take every occurrence of each word (say strawberry) and count the context words around it, we get a word-word co-occurrence matrix. Fig. 6.6 shows a simpliﬁed subset of the word-word co-occurrence matrix for these four words com- puted from the Wikipedia corpus (Davies, 2015). aardvark ... computer data result pie sugar ... cherry 0 ... 2 8 9 442 25 ... strawberry 0 ... 0 0 1 60 19 ... digital 0 ... 1670 1683 85 5 4 ... information 0 ... 3325 3982 378 5 13 ... Figure 6.6 Co-occurrence vectors for four words in the Wikipedia corpus, showing six of the dimensions (hand-picked for pedagogical purposes). The vector for digital is outlined in red. Note that a real vector would have vastly more dimensions and thus be much sparser. Note in Fig. 6.6 that the two words cherry and strawberry are more similar to each other (both pie and sugar tend to occur in their window) than they are to other words like digital; conversely,digital and information are more similar to each other than, say, to strawberry. Fig. 6.7 shows a spatial visualization. 1000 2000 3000 4000 1000 2000 digital [1683,1670] computer data information [3982,3325] 3000 4000 Figure 6.7 A spatial visualization of word vectors fordigital and information, showing just two of the dimensions, corresponding to the words data and computer. Note that \|V \|, the dimensionality of the vector, is generally the size of the vo- cabulary, often between 10,000 and 50,000 words (using the most frequent words 110 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS in the training corpus; keeping words after about the most frequent 50,000 or so is generally not helpful). Since most of these numbers are zero these aresparse vector representations; there are efﬁcient algorithms for storing and computing with sparse matrices. Now that we have some intuitions, let’s move on to examine the details of com- puting word similarity. Afterwards we’ll discuss methods for weighting cells. 6.4 Cosine for measuring similarity To measure similarity between two target words v and w, we need a metric that takes two vectors (of the same dimensionality, either both with words as dimensions, hence of length \|V \|, or both with documents as dimensions, of length \|D\|) and gives a measure of their similarity. By far the most common similarity metric is thecosine of the angle between the vectors. The cosine—like most measures for vector similarity used in NLP—is based on the dot product operator from linear algebra, also called the inner product:dot product inner product dot product(v,w) =v ·w = N∑ i=1 viwi = v1w1 +v2w2 +...+vN wN (6.7) The dot product acts as a similarity metric because it will tend to be high just when the two vectors have large values in the same dimensions. Alternatively, vectors that have zeros in different dimensions—orthogonal vectors—will have a dot product of 0, representing their strong dissimilarity. This raw dot product, however, has a problem as a similarity metric: it favors long vectors. The vector length is deﬁned asvector length \|v\| = √ N∑ i=1 v2 i (6.8) The dot product is higher if a vector is longer, with higher values in each dimension. More frequent words have longer vectors, since they tend to co-occur with more words and have higher co-occurrence values with each of them. The raw dot product thus will be higher for frequent words. But this is a problem; we’d like a similarity metric that tells us how similar two words are regardless of their frequency. We modify the dot product to normalize for the vector length by dividing the dot product by the lengths of each of the two vectors. Thisnormalized dot product turns out to be the same as the cosine of the angle between the two vectors, following from the deﬁnition of the dot product between two vectors a and b: a·b = \|a\|\|b\|cosθ a·b \|a\|\|b\| = cosθ (6.9) The cosine similarity metric between two vectors v and w thus can be computed as:cosine 6.5 • TF-IDF: W EIGHING TERMS IN THE VECTOR 111 cosine(v,w) = v ·w \|v\|\|w\|= N∑ i=1 viwi √ N∑ i=1 v2 i √ N∑ i=1 w2 i (6.10) For some applications we pre-normalize each vector, by dividing it by its length, creating a unit vector of length 1. Thus we could compute a unit vector from a byunit vector dividing it by \|a\|. For unit vectors, the dot product is the same as the cosine. The cosine value ranges from 1 for vectors pointing in the same direction, through 0 for orthogonal vectors, to -1 for vectors pointing in opposite directions. But since raw frequency values are non-negative, the cosine for these vectors ranges from 0–1. Let’s see how the cosine computes which of the wordscherry or digital is closer in meaning to information, just using raw counts from the following shortened table: pie data computer cherry 442 8 2 digital 5 1683 1670 information 5 3982 3325 cos(cherry,information) = 442 ∗5 +8 ∗3982 +2 ∗3325√ 4422 +82 +22√ 52 +39822 +33252 = .018 cos(digital,information) = 5 ∗5 +1683 ∗3982 +1670 ∗3325√ 52 +16832 +16702√ 52 +39822 +33252 = .996 The model decides that information is way closer to digital than it is to cherry, a result that seems sensible. Fig. 6.8 shows a visualization. 500 1000 1500 2000 2500 3000 500 digital cherry information Dimension 1: ‘pie’ Dimension 2: ‘computer’ Figure 6.8 A (rough) graphical demonstration of cosine similarity, showing vectors for three words (cherry, digital, and information) in the two dimensional space deﬁned by counts of the wordscomputer and pie nearby. The ﬁgure doesn’t show the cosine, but it highlights the angles; note that the angle between digital and information is smaller than the angle between cherry and information. When two vectors are more similar, the cosine is larger but the angle is smaller; the cosine has its maximum (1) when the angle between two vectors is smallest (0◦); the cosine of all other angles is less than 1. 6.5 TF-IDF: Weighing terms in the vector The co-occurrence matrices above represent each cell by frequencies, either of words with documents (Fig. 6.5), or words with other words (Fig. 6.6). But raw frequency 112 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS is not the best measure of association between words. Raw frequency is very skewed and not very discriminative. If we want to know what kinds of contexts are shared by cherry and strawberry but not by digital and information, we’re not going to get good discrimination from words like the, it, or they, which occur frequently with all sorts of words and aren’t informative about any particular word. We saw this also in Fig. 6.3 for the Shakespeare corpus; the dimension for the word good is not very discriminative between plays; good is simply a frequent word and has roughly equivalent high frequencies in each of the plays. It’s a bit of a paradox. Words that occur nearby frequently (maybe pie nearby cherry) are more important than words that only appear once or twice. Yet words that are too frequent—ubiquitous, like the or good— are unimportant. How can we balance these two conﬂicting constraints? There are two common solutions to this problem: in this section we’ll describe the tf-idf weighting, usually used when the dimensions are documents. In the next section we introduce the PPMI algorithm (usually used when the dimensions are words). The tf-idf weighting (the ‘-’ here is a hyphen, not a minus sign) is the product of two terms, each term capturing one of these two intuitions: The ﬁrst is the term frequency (Luhn, 1957): the frequency of the word t in theterm frequency document d. We can just use the raw count as the term frequency: tft,d = count(t,d) (6.11) More commonly we squash the raw frequency a bit, by using the log 10 of the fre- quency instead. The intuition is that a word appearing 100 times in a document doesn’t make that word 100 times more likely to be relevant to the meaning of the document. We also need to do something special with counts of 0, since we can’t take the log of 0.2 tft,d = { 1 +log10 count(t,d) if count (t,d) >0 0 otherwise (6.12) If we use log weighting, terms which occur 0 times in a document would have tf= 0, 1 times in a document tf = 1 + log10(1) =1 + 0 = 1, 10 times in a document tf = 1 +log10(10) =2, 100 times tf = 1 +log10(100) =3, 1000 times tf = 4, and so on. The second factor in tf-idf is used to give a higher weight to words that occur only in a few documents. Terms that are limited to a few documents are useful for discriminating those documents from the rest of the collection; terms that occur frequently across the entire collection aren’t as helpful. The document frequencydocument frequency dft of a term t is the number of documents it occurs in. Document frequency is not the same as the collection frequency of a term, which is the total number of times the word appears in the whole collection in any document. Consider in the collection of Shakespeare’s 37 plays the two words Romeo and action. The words have identical collection frequencies (they both occur 113 times in all the plays) but very different document frequencies, since Romeo only occurs in a single play. If our goal is to ﬁnd documents about the romantic tribulations of Romeo, the word Romeo should be highly weighted, but not action: Collection Frequency Document Frequency Romeo 113 1 action 113 31 2 We can also use this alternative formulation, which we have used in earlier editions: tf t,d = log10(count(t,d)+ 1) 6.5 • TF-IDF: W EIGHING TERMS IN THE VECTOR 113 We emphasize discriminative words like Romeo via the inverse document fre- quency or idf term weight (Sparck Jones, 1972). The idf is deﬁned using the frac-idf tion N/dft , where N is the total number of documents in the collection, and df t is the number of documents in which term t occurs. The fewer documents in which a term occurs, the higher this weight. The lowest weight of 1 is assigned to terms that occur in all the documents. It’s usually clear what counts as a document: in Shake- speare we would use a play; when processing a collection of encyclopedia articles like Wikipedia, the document is a Wikipedia page; in processing newspaper articles, the document is a single article. Occasionally your corpus might not have appropri- ate document divisions and you might need to break up the corpus into documents yourself for the purposes of computing idf. Because of the large number of documents in many collections, this measure too is usually squashed with a log function. The resulting deﬁnition for inverse document frequency (idf) is thus idft = log10 (N dft ) (6.13) Here are some idf values for some words in the Shakespeare corpus, (along with the document frequency df values on which they are based) ranging from extremely informative words which occur in only one play likeRomeo, to those that occur in a few like salad or Falstaff, to those which are very common like fool or so common as to be completely non-discriminative since they occur in all 37 plays like good or sweet.3 Word df idf Romeo 1 1.57 salad 2 1.27 Falstaff 4 0.967 forest 12 0.489 battle 21 0.246 wit 34 0.037 fool 36 0.012 good 37 0 sweet 37 0 The tf-idf weighted value wt,d for word t in document d thus combines termtf-idf frequency tft,d (deﬁned either by Eq. 6.11 or by Eq. 6.12) with idf from Eq. 6.13: wt,d = tft,d ×idft (6.14) Fig. 6.9 applies tf-idf weighting to the Shakespeare term-document matrix in Fig. 6.2, using the tf equation Eq. 6.12. Note that the tf-idf values for the dimension corre- sponding to the word good have now all become 0; since this word appears in every document, the tf-idf weighting leads it to be ignored. Similarly, the wordfool, which appears in 36 out of the 37 plays, has a much lower weight. The tf-idf weighting is the way for weighting co-occurrence matrices in infor- mation retrieval, but also plays a role in many other aspects of natural language processing. It’s also a great baseline, the simple thing to try ﬁrst. We’ll look at other weightings like PPMI (Positive Pointwise Mutual Information) in Section 6.6. 3 Sweet was one of Shakespeare’s favorite adjectives, a fact probably related to the increased use of sugar in European recipes around the turn of the 16th century (Jurafsky, 2014, p. 175). 114 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS As You Like It Twelfth Night Julius Caesar Henry V battle 0.246 0 0.454 0.520 good 0 0 0 0 fool 0.030 0.033 0.0012 0.0019 wit 0.085 0.081 0.048 0.054 Figure 6.9 A portion of the tf-idf weighted term-document matrix for four words in Shake- speare plays, showing a selection of 4 plays, using counts from Fig. 6.2. For example the 0.085 value forwit in As You Like Itis the product of tf= 1+log10(20) =2.301 and idf= .037. Note that the idf weighting has eliminated the importance of the ubiquitous word good and vastly reduced the impact of the almost-ubiquitous word fool. 6.6 Pointwise Mutual Information (PMI) An alternative weighting function to tf-idf, PPMI (positive pointwise mutual infor- mation), is used for term-term-matrices, when the vector dimensions correspond to words rather than documents. PPMI draws on the intuition that the best way to weigh the association between two words is to ask how muchmore the two words co-occur in our corpus than we would have a priori expected them to appear by chance. Pointwise mutual information (Fano, 1961)4 is one of the most important con- pointwise mutual information cepts in NLP. It is a measure of how often two events x and y occur, compared with what we would expect if they were independent: I(x,y) =log2 P(x,y) P(x)P(y) (6.16) The pointwise mutual information between a target word w and a context word c (Church and Hanks 1989, Church and Hanks 1990) is then deﬁned as: PMI(w,c) =log2 P(w,c) P(w)P(c) (6.17) The numerator tells us how often we observed the two words together (assuming we compute probability by using the MLE). The denominator tells us how often we would expect the two words to co-occur assuming they each occurred indepen- dently; recall that the probability of two independent events both occurring is just the product of the probabilities of the two events. Thus, the ratio gives us an esti- mate of how much more the two words co-occur than we expect by chance. PMI is a useful tool whenever we need to ﬁnd words that are strongly associated. PMI values range from negative to positive inﬁnity. But negative PMI values (which imply things are co-occurring less often than we would expect by chance) tend to be unreliable unless our corpora are enormous. To distinguish whether two words whose individual probability is each 10−6 occur together less often than chance, we would need to be certain that the probability of the two occurring to- gether is signiﬁcantly less than 10−12, and this kind of granularity would require an enormous corpus. Furthermore it’s not clear whether it’s even possible to evaluate such scores of ‘unrelatedness’ with human judgments. For this reason it is more 4 PMI is based on the mutual information between two random variables X and Y , deﬁned as: I(X,Y ) = ∑ x ∑ y P(x,y)log2 P(x,y) P(x)P(y) (6.15) In a confusion of terminology, Fano used the phrase mutual information to refer to what we now call pointwise mutual information and the phrase expectation of the mutual information for what we now call mutual information 6.6 • P OINTWISE MUTUAL INFORMATION (PMI) 115 common to use Positive PMI (called PPMI) which replaces all negative PMI valuesPPMI with zero (Church and Hanks 1989, Dagan et al. 1993, Niwa and Nitta 1994)5: PPMI(w,c) =max(log2 P(w,c) P(w)P(c),0) (6.18) More formally, let’s assume we have a co-occurrence matrix F with W rows (words) and C columns (contexts), where fi j gives the number of times word wi occurs with context cj. This can be turned into a PPMI matrix where PPMI i j gives the PPMI value of word wi with context cj (which we can also express as PPMI( wi,cj) or PPMI(w = i,c = j)) as follows: pi j = fi j∑W i=1 ∑C j=1 fi j , pi∗= ∑C j=1 fi j ∑W i=1 ∑C j=1 fi j , p∗j = ∑W i=1 fi j∑W i=1 ∑C j=1 fi j (6.19) PPMIi j = max(log2 pi j pi∗p∗j ,0) (6.20) Let’s see some PPMI calculations. We’ll use Fig. 6.10, which repeats Fig. 6.6 plus all the count marginals, and let’s pretend for ease of calculation that these are the only words/contexts that matter. computer data result pie sugar count(w) cherry 2 8 9 442 25 486 strawberry 0 0 1 60 19 80 digital 1670 1683 85 5 4 3447 information 3325 3982 378 5 13 7703 count(context) 4997 5673 473 512 61 11716 Figure 6.10 Co-occurrence counts for four words in 5 contexts in the Wikipedia corpus, together with the marginals, pretending for the purpose of this calculation that no other word- s/contexts matter. Thus for example we could compute PPMI(information,data), assuming we pre- tended that Fig. 6.6 encompassed all the relevant word contexts/dimensions, as fol- lows: P(w=information, c=data) = 3982 11716 = .3399 P(w=information) = 7703 11716 = .6575 P(c=data) = 5673 11716 = .4842 PPMI(information,data) = log2(.3399/(.6575 ∗.4842)) =.0944 Fig. 6.11 shows the joint probabilities computed from the counts in Fig. 6.10, and Fig. 6.12 shows the PPMI values. Not surprisingly,cherry and strawberry are highly associated with both pie and sugar, and data is mildly associated with information. PMI has the problem of being biased toward infrequent events; very rare words tend to have very high PMI values. One way to reduce this bias toward low frequency 5 Positive PMI also cleanly solves the problem of what to do with zero counts, using 0 to replace the −∞ from log(0). 116 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS p(w,context) p(w) computer data result pie sugar p(w) cherry 0.0002 0.0007 0.0008 0.0377 0.0021 0.0415 strawberry 0.0000 0.0000 0.0001 0.0051 0.0016 0.0068 digital 0.1425 0.1436 0.0073 0.0004 0.0003 0.2942 information 0.2838 0.3399 0.0323 0.0004 0.0011 0.6575 p(context) 0.4265 0.4842 0.0404 0.0437 0.0052 Figure 6.11 Replacing the counts in Fig. 6.6 with joint probabilities, showing the marginals in the right column and the bottom row. computer data result pie sugar cherry 0 0 0 4.38 3.30 strawberry 0 0 0 4.10 5.51 digital 0.18 0.01 0 0 0 information 0.02 0.09 0.28 0 0 Figure 6.12 The PPMI matrix showing the association between words and context words, computed from the counts in Fig. 6.11. Note that most of the 0 PPMI values are ones that had a negative PMI; for example PMI(cherry,computer) = -6.7, meaning thatcherry and computer co-occur on Wikipedia less often than we would expect by chance, and with PPMI we replace negative values by zero. events is to slightly change the computation forP(c), using a different functionPα (c) that raises the probability of the context word to the power of α: PPMIα (w,c) =max(log2 P(w,c) P(w)Pα (c),0) (6.21) Pα (c) = count(c)α ∑ c count(c)α (6.22) Levy et al. (2015) found that a setting of α = 0.75 improved performance of embeddings on a wide range of tasks (drawing on a similar weighting used for skip- grams described below in Eq. 6.32). This works because raising the count to α = 0.75 increases the probability assigned to rare contexts, and hence lowers their PMI (Pα (c) >P(c) when c is rare). Another possible solution is Laplace smoothing: Before computing PMI, a small constant k (values of 0.1-3 are common) is added to each of the counts, shrinking (discounting) all the non-zero values. The larger thek, the more the non-zero counts are discounted. 6.7 Applications of the tf-idf or PPMI vector models In summary, the vector semantics model we’ve described so far represents a target word as a vector with dimensions corresponding either to the documents in a large collection (the term-document matrix) or to the counts of words in some neighboring window (the term-term matrix). The values in each dimension are counts, weighted by tf-idf (for term-document matrices) or PPMI (for term-term matrices), and the vectors are sparse (since most values are zero). The model computes the similarity between two words x and y by taking the cosine of their tf-idf or PPMI vectors; high cosine, high similarity. This entire model 6.8 • W ORD 2VEC 117 is sometimes referred to as the tf-idf model or the PPMI model, after the weighting function. The tf-idf model of meaning is often used for document functions like deciding if two documents are similar. We represent a document by taking the vectors of all the words in the document, and computing the centroid of all those vectors.centroid The centroid is the multidimensional version of the mean; the centroid of a set of vectors is a single vector that has the minimum sum of squared distances to each of the vectors in the set. Given k word vectors w1,w2,...,wk, the centroid document vector d is:document vector d = w1 +w2 +...+wk k (6.23) Given two documents, we can then compute their document vectors d1 and d2, and estimate the similarity between the two documents by cos (d1,d2). Document sim- ilarity is also useful for all sorts of applications; information retrieval, plagiarism detection, news recommender systems, and even for digital humanities tasks like comparing different versions of a text to see which are similar to each other. Either the PPMI model or the tf-idf model can be used to compute word simi- larity, for tasks like ﬁnding word paraphrases, tracking changes in word meaning, or automatically discovering meanings of words in different corpora. For example, we can ﬁnd the 10 most similar words to any target word w by computing the cosines between w and each of the V −1 other words, sorting, and looking at the top 10. 6.8 Word2vec In the previous sections we saw how to represent a word as a sparse, long vector with dimensions corresponding to words in the vocabulary or documents in a collection. We now introduce a more powerful word representation: embeddings, short dense vectors. Unlike the vectors we’ve seen so far, embeddings are short, with number of dimensions d ranging from 50-1000, rather than the much larger vocabulary size \|V \|or number of documents D we’ve seen. These d dimensions don’t have a clear interpretation. And the vectors are dense: instead of vector entries being sparse, mostly-zero counts or functions of counts, the values will be real-valued numbers that can be negative. It turns out that dense vectors work better in every NLP task than sparse vectors. While we don’t completely understand all the reasons for this, we have some intu- itions. Representing words as 300-dimensional dense vectors requires our classiﬁers to learn far fewer weights than if we represented words as 50,000-dimensional vec- tors, and the smaller parameter space possibly helps with generalization and avoid- ing overﬁtting. Dense vectors may also do a better job of capturing synonymy. For example, in a sparse vector representation, dimensions for synonyms like car and automobile dimension are distinct and unrelated; sparse vectors may thus fail to capture the similarity between a word with car as a neighbor and a word with automobile as a neighbor. In this section we introduce one method for computing embeddings: skip-gramskip-gram with negative sampling, sometimes called SGNS. The skip-gram algorithm is oneSGNS of two algorithms in a software package called word2vec, and so sometimes theword2vec algorithm is loosely referred to as word2vec (Mikolov et al. 2013a, Mikolov et al. 2013b). The word2vec methods are fast, efﬁcient to train, and easily available on- 118 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS line with code and pretrained embeddings. Word2vec embeddings are static em- beddings, meaning that the method learns one ﬁxed embedding for each word in thestatic embeddings vocabulary. In Chapter 11 we’ll introduce methods for learning dynamiccontextual embeddings like the popular family of BERT representations, in which the vector for each word is different in different contexts. The intuition of word2vec is that instead of counting how often each wordw oc- curs near, say,apricot, we’ll instead train a classiﬁer on a binary prediction task: “Is word w likely to show up nearapricot?” We don’t actually care about this prediction task; instead we’ll take the learned classiﬁerweights as the word embeddings. The revolutionary intuition here is that we can just use running text as implicitly supervised training data for such a classiﬁer; a word c that occurs near the target word apricot acts as gold ‘correct answer’ to the question “Is wordc likely to show up near apricot?” This method, often called self-supervision, avoids the need forself-supervision any sort of hand-labeled supervision signal. This idea was ﬁrst proposed in the task of neural language modeling, when Bengio et al. (2003) and Collobert et al. (2011) showed that a neural language model (a neural network that learned to predict the next word from prior words) could just use the next word in running text as its supervision signal, and could be used to learn an embedding representation for each word as part of doing this prediction task. We’ll see how to do neural networks in the next chapter, but word2vec is a much simpler model than the neural network language model, in two ways. First, word2vec simpliﬁes the task (making it binary classiﬁcation instead of word pre- diction). Second, word2vec simpliﬁes the architecture (training a logistic regression classiﬁer instead of a multi-layer neural network with hidden layers that demand more sophisticated training algorithms). The intuition of skip-gram is: 1. Treat the target word and a neighboring context word as positive examples. 2. Randomly sample other words in the lexicon to get negative samples. 3. Use logistic regression to train a classiﬁer to distinguish those two cases. 4. Use the learned weights as the embeddings. 6.8.1 The classiﬁer Let’s start by thinking about the classiﬁcation task, and then turn to how to train. Imagine a sentence like the following, with a target wordapricot, and assume we’re using a window of ±2 context words: ... lemon, a [tablespoon of apricot jam, a] pinch ... c1 c2 w c3 c4 Our goal is to train a classiﬁer such that, given a tuple (w,c) of a target word w paired with a candidate context word c (for example ( apricot, jam), or perhaps (apricot, aardvark)) it will return the probability that c is a real context word (true for jam, false for aardvark): P(+\|w,c) (6.24) The probability that word c is not a real context word for w is just 1 minus Eq. 6.24: P(−\|w,c) =1 −P(+\|w,c) (6.25) 6.8 • W ORD 2VEC 119 How does the classiﬁer compute the probability P? The intuition of the skip- gram model is to base this probability on embedding similarity: a word is likely to occur near the target if its embedding vector is similar to the target embedding. To compute similarity between these dense embeddings, we rely on the intuition that two vectors are similar if they have a high dot product (after all, cosine is just a normalized dot product). In other words: Similarity(w,c) ≈c·w (6.26) The dot product c ·w is not a probability, it’s just a number ranging from −∞ to ∞ (since the elements in word2vec embeddings can be negative, the dot product can be negative). To turn the dot product into a probability, we’ll use thelogistic or sigmoid function σ(x), the fundamental core of logistic regression: σ(x) = 1 1 +exp(−x) (6.27) We model the probability that word c is a real context word for target word w as: P(+\|w,c) = σ(c·w) = 1 1 +exp(−c·w) (6.28) The sigmoid function returns a number between 0 and 1, but to make it a probability we’ll also need the total probability of the two possible events (c is a context word, and c isn’t a context word) to sum to 1. We thus estimate the probability that wordc is not a real context word for w as: P(−\|w,c) = 1 −P(+\|w,c) = σ(−c·w) = 1 1 +exp(c·w) (6.29) Equation 6.28 gives us the probability for one word, but there are many context words in the window. Skip-gram makes the simplifying assumption that all context words are independent, allowing us to just multiply their probabilities: P(+\|w,c1:L) = L∏ i=1 σ(ci ·w) (6.30) logP(+\|w,c1:L) = L∑ i=1 logσ(ci ·w) (6.31) In summary, skip-gram trains a probabilistic classiﬁer that, given a test target word w and its context window ofL words c1:L, assigns a probability based on how similar this context window is to the target word. The probability is based on applying the logistic (sigmoid) function to the dot product of the embeddings of the target word with each context word. To compute this probability, we just need embeddings for each target word and context word in the vocabulary. Fig. 6.13 shows the intuition of the parameters we’ll need. Skip-gram actually stores two embeddings for each word, one for the word as a target, and one for the word considered as context. Thus the parameters we need to learn are two matrices W and C, each containing an embedding for every one of the \|V \|words in the vocabulary V .6 Let’s now turn to learning these embeddings (which is the real goal of training this classiﬁer in the ﬁrst place). 6 In principle the target matrix and the context matrix could use different vocabularies, but we’ll simplify by assuming one shared vocabulary V . 120 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS 1 W C aardvark zebra zebra aardvark apricot apricot \|V\| \|V\|+1 2V & = target words context & noise words … … 1..d … … Figure 6.13 The embeddings learned by the skipgram model. The algorithm stores two embeddings for each word, the target embedding (sometimes called the input embedding) and the context embedding (sometimes called the output embedding). The parameter θ that the algorithm learns is thus a matrix of 2\|V \|vectors, each of dimension d, formed by concate- nating two matrices, the target embeddings W and the context+noise embeddings C. 6.8.2 Learning skip-gram embeddings The learning algorithm for skip-gram embeddings takes as input a corpus of text, and a chosen vocabulary size N. It begins by assigning a random embedding vector for each of the N vocabulary words, and then proceeds to iteratively shift the em- bedding of each word w to be more like the embeddings of words that occur nearby in texts, and less like the embeddings of words that don’t occur nearby. Let’s start by considering a single piece of training data: ... lemon, a [tablespoon of apricot jam, a] pinch ... c1 c2 w c3 c4 This example has a target word w (apricot), and 4 context words in the L = ±2 window, resulting in 4 positive training instances (on the left below): positive examples + w c pos apricot tablespoon apricot of apricot jam apricot a negative examples - w c neg w c neg apricot aardvark apricot seven apricot my apricot forever apricot where apricot dear apricot coaxial apricot if For training a binary classiﬁer we also need negative examples. In fact skip- gram with negative sampling (SGNS) uses more negative examples than positive examples (with the ratio between them set by a parameter k). So for each of these (w,cpos) training instances we’ll create k negative samples, each consisting of the target w plus a ‘noise word’cneg. A noise word is a random word from the lexicon, constrained not to be the target word w. The right above shows the setting where k = 2, so we’ll have 2 negative examples in the negative training set −for each positive example w,cpos. The noise words are chosen according to their weighted unigram frequency pα (w), where α is a weight. If we were sampling according to unweighted fre- quency p(w), it would mean that with unigram probabilityp(“the”) we would choose the word the as a noise word, with unigram probability p(“aardvark”) we would 6.8 • W ORD 2VEC 121 choose aardvark, and so on. But in practice it is common to set α = 0.75, i.e. use the weighting p3 4 (w): Pα (w) = count(w)α ∑ w′count(w′)α (6.32) Setting α = .75 gives better performance because it gives rare noise words slightly higher probability: for rare words, Pα (w) >P(w). To illustrate this intuition, it might help to work out the probabilities for an example withα = .75 and two events, P(a) =0.99 and P(b) =0.01: Pα (a) = .99.75 .99.75 +.01.75 = 0.97 Pα (b) = .01.75 .99.75 +.01.75 = 0.03 (6.33) Thus using α = .75 increases the probability of the rare event b from 0.01 to 0.03. Given the set of positive and negative training instances, and an initial set of embeddings, the goal of the learning algorithm is to adjust those embeddings to • Maximize the similarity of the target word, context word pairs(w,cpos) drawn from the positive examples • Minimize the similarity of the (w,cneg) pairs from the negative examples. If we consider one word/context pair(w,cpos) with its k noise words cneg1 ...cnegk , we can express these two goals as the following loss function L to be minimized (hence the −); here the ﬁrst term expresses that we want the classiﬁer to assign the real context word cpos a high probability of being a neighbor, and the second term expresses that we want to assign each of the noise words cnegi a high probability of being a non-neighbor, all multiplied because we assume independence: L = −log [ P(+\|w,cpos) k∏ i=1 P(−\|w,cnegi ) ] = − [ logP(+\|w,cpos)+ k∑ i=1 logP(−\|w,cnegi ) ] = − [ logP(+\|w,cpos)+ k∑ i=1 log ( 1 −P(+\|w,cnegi ) ) ] = − [ logσ(cpos ·w)+ k∑ i=1 logσ(−cnegi ·w) ] (6.34) That is, we want to maximize the dot product of the word with the actual context words, and minimize the dot products of the word with the k negative sampled non- neighbor words. We minimize this loss function using stochastic gradient descent. Fig. 6.14 shows the intuition of one step of learning. To get the gradient, we need to take the derivative of Eq. 6.34 with respect to the different embeddings. It turns out the derivatives are the following (we leave the 122 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS W C move apricot and jam closer, increasing cpos z w aardvark move apricot and matrix apart decreasing cneg1 z w “…apricot jam…” w zebra zebra aardvark jam apricot cpos matrix Tolstoy move apricot and Tolstoy apart decreasing cneg2 z w ! cneg1 cneg2 k=2 Figure 6.14 Intuition of one step of gradient descent. The skip-gram model tries to shift embeddings so the target embeddings (here for apricot) are closer to (have a higher dot prod- uct with) context embeddings for nearby words (herejam) and further from (lower dot product with) context embeddings for noise words that don’t occur nearby (hereTolstoy and matrix). proof as an exercise at the end of the chapter): ∂L ∂cpos = [σ(cpos ·w)−1]w (6.35) ∂L ∂cneg = [σ(cneg ·w)]w (6.36) ∂L ∂w = [σ(cpos ·w)−1]cpos + k∑ i=1 [σ(cnegi ·w)]cnegi (6.37) The update equations going from time step t to t + 1 in stochastic gradient descent are thus: ct+1 pos = ct pos −η[σ(ct pos ·wt )−1]wt (6.38) ct+1 neg = ct neg −η[σ(ct neg ·wt )]wt (6.39) wt+1 = wt −η [ [σ(ct pos ·wt )−1]ct pos + k∑ i=1 [σ(ct negi ·wt )]ct negi ] (6.40) Just as in logistic regression, then, the learning algorithm starts with randomly ini- tialized W and C matrices, and then walks through the training corpus using gradient descent to move W and C so as to minimize the loss in Eq. 6.34 by making the up- dates in (Eq. 6.38)-(Eq. 6.40). Recall that the skip-gram model learnstwo separate embeddings for each wordi: the target embedding wi and the context embedding ci, stored in two matrices, thetarget embeddingcontext embedding target matrix W and the context matrix C. It’s common to just add them together, representing word i with the vector wi +ci. Alternatively we can throw away the C matrix and just represent each word i by the vector wi. As with the simple count-based methods like tf-idf, the context window size L affects the performance of skip-gram embeddings, and experiments often tune the parameter L on a devset. 6.9 • V ISUALIZING EMBEDDINGS 123 6.8.3 Other kinds of static embeddings There are many kinds of static embeddings. An extension of word2vec, fasttextfasttext (Bojanowski et al., 2017), addresses a problem with word2vec as we have presented it so far: it has no good way to deal with unknown words—words that appear in a test corpus but were unseen in the training corpus. A related problem is word sparsity, such as in languages with rich morphology, where some of the many forms for each noun and verb may only occur rarely. Fasttext deals with these problems by using subword models, representing each word as itself plus a bag of constituent n-grams, with special boundary symbols<and >added to each word. For example, with n = 3 the word where would be represented by the sequence <where>plus the character n-grams: <wh, whe, her, ere, re> Then a skipgram embedding is learned for each constituent n-gram, and the word where is represented by the sum of all of the embeddings of its constituent n-grams. Unknown words can then be presented only by the sum of the constituent n-grams. A fasttext open-source library, including pretrained embeddings for 157 languages, is available at https://fasttext.cc. Another very widely used static embedding model is GloVe (Pennington et al., 2014), short for Global Vectors, because the model is based on capturing global corpus statistics. GloVe is based on ratios of probabilities from the word-word co- occurrence matrix, combining the intuitions of count-based models like PPMI while also capturing the linear structures used by methods like word2vec. It turns out that dense embeddings like word2vec actually have an elegant math- ematical relationship with sparse embeddings like PPMI, in which word2vec can be seen as implicitly optimizing a function of a PPMI matrix (Levy and Goldberg, 2014c). 6.9 Visualizing Embeddings “I see well in many dimensions as long as the dimensions are around two.” The late economist Martin Shubik Visualizing embeddings is an important goal in helping understand, apply, and improve these models of word meaning. But how can we visualize a (for example) 100-dimensional vector? Rohde, Gonnerman, Plaut Modeling Word Meaning Using Lexical Co-Occurrence HEAD HANDFACE DOG AMERICA CAT EYE EUROPE FOOT CHINA FRANCE CHICAGO ARM FINGER NOSE LEG RUSSIA MOUSE AFRICA ATLANTA EAR SHOULDER ASIA COW BULL PUPPY LION HAWAII MONTREAL TOKYO TOE MOSCOW TOOTH NASHVILLE BRAZIL WRIST KITTEN ANKLE TURTLE OYSTER Figure 8: Multidimensional scaling for three noun classes. WRIST ANKLE SHOULDER ARM LEG HAND FOOT HEAD NOSE FINGER TOE FACE EAR EYE TOOTH DOG CAT PUPPY KITTEN COW MOUSE TURTLE OYSTER LION BULL CHICAGO ATLANTA MONTREAL NASHVILLE TOKYO CHINA RUSSIA AFRICA ASIA EUROPE AMERICA BRAZIL MOSCOW FRANCE HAWAII Figure 9: Hierarchical clustering for three noun classes using distances based on vector correlations. 20 The simplest way to visualize the meaning of a word w embedded in a space is to list the most similar words to w by sorting the vectors for all words in the vocabulary by their cosine with the vector for w. For example the 7 closest words to frog using a particular embeddings computed with the GloVe algorithm are: frogs, toad, litoria, leptodactyli- dae, rana, lizard, and eleutherodactylus (Pennington et al., 2014). Yet another visualization method is to use a clustering algorithm to show a hierarchical representation of which words are similar to others in the embedding space. The uncaptioned ﬁgure on the left uses hierarchical clustering of some embedding vectors for nouns as a visualization 124 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS method (Rohde et al., 2006). Probably the most common visualization method, how- ever, is to project the 100 dimensions of a word down into 2 dimensions. Fig. 6.1 showed one such visualization, as does Fig. 6.16, using a projection method called t-SNE (van der Maaten and Hinton, 2008). 6.10 Semantic properties of embeddings In this section we brieﬂy summarize some of the semantic properties of embeddings that have been studied. Different types of similarity or association: One parameter of vector semantic models that is relevant to both sparse PPMI vectors and dense word2vec vectors is the size of the context window used to collect counts. This is generally between 1 and 10 words on each side of the target word (for a total context of 2-20 words). The choice depends on the goals of the representation. Shorter context windows tend to lead to representations that are a bit more syntactic, since the information is coming from immediately nearby words. When the vectors are computed from short context windows, the most similar words to a target word w tend to be semantically similar words with the same parts of speech. When vectors are computed from long context windows, the highest cosine words to a target word w tend to be words that are topically related but not similar. For example Levy and Goldberg (2014a) showed that using skip-gram with a window of ±2, the most similar words to the wordHogwarts (from the Harry Potter series) were names of other ﬁctional schools: Sunnydale (from Buffy the Vampire Slayer) or Evernight (from a vampire series). With a window of±5, the most similar words to Hogwarts were other words topically related to the Harry Potter series: Dumbledore, Malfoy, and half-blood. It’s also often useful to distinguish two kinds of similarity or association between words (Sch ¨utze and Pedersen, 1993). Two words have ﬁrst-order co-occurrenceﬁrst-order co-occurrence (sometimes called syntagmatic association) if they are typically nearby each other. Thus wrote is a ﬁrst-order associate ofbook or poem. Two words havesecond-order co-occurrence (sometimes called paradigmatic association) if they have similarsecond-order co-occurrence neighbors. Thus wrote is a second-order associate of words like said or remarked. Analogy/Relational Similarity: Another semantic property of embeddings is their ability to capture relational meanings. In an important early vector space model of cognition, Rumelhart and Abrahamson (1973) proposed the parallelogram modelparallelogram model for solving simple analogy problems of the form a is to b as a is to what? . In such problems, a system is given a problem like apple:tree::grape:?, i.e., apple is to tree as grape is to , and must ﬁll in the word vine. In the parallelogram model, illustrated in Fig. 6.15, the vector from the word apple to the word tree (=# »tree −# »apple) is added to the vector for grape (# »grape); the nearest word to that point is returned. In early work with sparse embeddings, scholars showed that sparse vector mod- els of meaning could solve such analogy problems (Turney and Littman, 2005), but the parallelogram method received more modern attention because of its suc- cess with word2vec or GloVe vectors (Mikolov et al. 2013c, Levy and Goldberg 2014b, Pennington et al. 2014). For example, the result of the expression # »king − 6.10 • S EMANTIC PROPERTIES OF EMBEDDINGS 125 tree apple grape vine Figure 6.15 The parallelogram model for analogy problems (Rumelhart and Abrahamson, 1973): the location of # »vine can be found by subtracting # »apple from # »tree and adding # »grape. # »man + # »woman is a vector close to # »queen. Similarly, # »Paris −# »France + # »Italy results in a vector that is close to # »Rome. The embedding model thus seems to be extract- ing representations of relations like MALE -FEMALE , or CAPITAL -CITY-OF, or even COMPARATIVE /SUPERLATIVE , as shown in Fig. 6.16 from GloVe. (a) (b) Figure 6.16 Relational properties of the GloVe vector space, shown by projecting vectors onto two dimen- sions. (a) # »king −# »man + # »woman is close to # »queen. (b) offsets seem to capture comparative and superlative morphology (Pennington et al., 2014). For a a : b :: a∗: b∗problem, meaning the algorithm is given vectors a, b, and a∗and must ﬁnd b∗, the parallelogram method is thus: ˆb∗= argmin x distance(x,b−a+a∗) (6.41) with some distance function, such as Euclidean distance. There are some caveats. For example, the closest value returned by the paral- lelogram algorithm in word2vec or GloVe embedding spaces is usually not in fact b* but one of the 3 input words or their morphological variants (i.e., cherry:red :: potato:x returns potato or potatoes instead of brown), so these must be explicitly excluded. Furthermore while embedding spaces perform well if the task involves frequent words, small distances, and certain relations (like relating countries with their capitals or verbs/nouns with their inﬂected forms), the parallelogram method with embeddings doesn’t work as well for other relations (Linzen 2016, Gladkova et al. 2016, Schluter 2018, Ethayarajh et al. 2019a), and indeed Peterson et al. (2020) argue that the parallelogram method is in general too simple to model the human cognitive process of forming analogies of this kind. 126 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS 6.10.1 Embeddings and Historical Semantics Embeddings can also be a useful tool for studying how meaning changes over time, by computing multiple embedding spaces, each from texts written in a particular time period. For example Fig. 6.17 shows a visualization of changes in meaning in English words over the last two centuries, computed by building separate embed- ding spaces for each decade from historical corpora like Google n-grams (Lin et al., 2012b) and the Corpus of Historical American English (Davies, 2012). CHAPTER 5. DYNAMIC SOCIAL REPRESENTATIONS OF WORD MEANING79 Figure 5.1: Two-dimensional visualization of semantic change in English using SGNS vectors (see Section 5.8 for the visualization algorithm).A,T h ew o r dgay shifted from meaning “cheerful” or “frolicsome” to referring to homosexuality.A,I nt h ee a r l y 20th century broadcast referred to “casting out seeds”; with the rise of television and radio its meaning shifted to “transmitting signals”.C, Awful underwent a process of pejoration, as it shifted from meaning “full of awe” to meaning “terrible or appalling” [212]. that adverbials (e.g.,actually )h a v eag e n e r a lt e n d e n c yt ou n d e r g os u b j e c t i ﬁ c a t i o n where they shift from objective statements about the world (e.g., “Sorry, the car is actually broken”) to subjective statements (e.g., “I can’t believe he actually did that”, indicating surprise/disbelief). 5.2.2 Computational linguistic studies There are also a number of recent works analyzing semantic change using computational methods. [200] use latent semantic analysis to analyze how word meanings broaden and narrow over time. [113]u s er a wc o - o c c u r r e n c ev e c t o r st op e r f o r man u m b e ro f historical case-studies on semantic change, and [252] perform a similar set of small- scale case-studies using temporal topic models. [87]c o n s t r u c tp o i n t - w i s em u t u a l information-based embeddings and found that semantic changes uncovered by their method had reasonable agreement with human judgments. [129]a n d[119]u s e“ n e u r a l ” word-embedding methods to detect linguistic change points. Finally, [257]a n a l y z e historical co-occurrences to test whether synonyms tend to change in similar ways. Figure 6.17 A t-SNE visualization of the semantic change of 3 words in English using word2vec vectors. The modern sense of each word, and the grey context words, are com- puted from the most recent (modern) time-point embedding space. Earlier points are com- puted from earlier historical embedding spaces. The visualizations show the changes in the word gay from meanings related to “cheerful” or “frolicsome” to referring to homosexuality, the development of the modern “transmission” sense of broadcast from its original sense of sowing seeds, and the pejoration of the word awful as it shifted from meaning “full of awe” to meaning “terrible or appalling” (Hamilton et al., 2016b). 6.11 Bias and Embeddings In addition to their ability to learn word meaning from text, embeddings, alas, also reproduce the implicit biases and stereotypes that were latent in the text. As the prior section just showed, embeddings can roughly model relational similar- ity: ‘queen’ as the closest word to ‘king’ - ‘man’ + ‘woman’ implies the analogy man:woman::king:queen. But these same embedding analogies also exhibit gender stereotypes. For example Bolukbasi et al. (2016) ﬁnd that the closest occupation to ‘computer programmer’ - ‘man’ + ‘woman’ in word2vec embeddings trained on news text is ‘homemaker’, and that the embeddings similarly suggest the analogy ‘father’ is to ‘doctor’ as ‘mother’ is to ‘nurse’. This could result in what Crawford (2017) and Blodgett et al. (2020) call an allocational harm, when a system allo-allocational harm cates resources (jobs or credit) unfairly to different groups. For example algorithms that use embeddings as part of a search for hiring potential programmers or doctors might thus incorrectly downweight documents with women’s names. It turns out that embeddings don’t just reﬂect the statistics of their input, but also amplify bias; gendered terms become more gendered in embedding space than theybias ampliﬁcation were in the input text statistics (Zhao et al. 2017, Ethayarajh et al. 2019b, Jia et al. 2020), and biases are more exaggerated than in actual labor employment statistics (Garg et al., 2018). Embeddings also encode the implicit associations that are a property of human reasoning. The Implicit Association Test (Greenwald et al., 1998) measures peo- 6.12 • E VALUATING VECTOR MODELS 127 ple’s associations between concepts (like ‘ﬂowers’ or ‘insects’) and attributes (like ‘pleasantness’ and ‘unpleasantness’) by measuring differences in the latency with which they label words in the various categories. 7 Using such methods, people in the United States have been shown to associate African-American names with unpleasant words (more than European-American names), male names more with mathematics and female names with the arts, and old people’s names with unpleas- ant words (Greenwald et al. 1998, Nosek et al. 2002a, Nosek et al. 2002b). Caliskan et al. (2017) replicated all these ﬁndings of implicit associations using GloVe vectors and cosine similarity instead of human latencies. For example African-American names like ‘Leroy’ and ‘Shaniqua’ had a higher GloVe cosine with unpleasant words while European-American names (‘Brad’, ‘Greg’, ‘Courtney’) had a higher cosine with pleasant words. These problems with embeddings are an example of a repre- sentational harm (Crawford 2017, Blodgett et al. 2020), which is a harm caused byrepresentational harm a system demeaning or even ignoring some social groups. Any embedding-aware al- gorithm that made use of word sentiment could thus exacerbate bias against African Americans. Recent research focuses on ways to try to remove these kinds of biases, for example by developing a transformation of the embedding space that removes gen- der stereotypes but preserves deﬁnitional gender (Bolukbasi et al. 2016, Zhao et al. 2017) or changing the training procedure (Zhao et al., 2018b). However, although these sorts of debiasing may reduce bias in embeddings, they do not eliminate itdebiasing (Gonen and Goldberg, 2019), and this remains an open problem. Historical embeddings are also being used to measure biases in the past. Garg et al. (2018) used embeddings from historical texts to measure the association be- tween embeddings for occupations and embeddings for names of various ethnici- ties or genders (for example the relative cosine similarity of women’s names versus men’s to occupation words like ‘librarian’ or ‘carpenter’) across the 20th century. They found that the cosines correlate with the empirical historical percentages of women or ethnic groups in those occupations. Historical embeddings also repli- cated old surveys of ethnic stereotypes; the tendency of experimental participants in 1933 to associate adjectives like ‘industrious’ or ‘superstitious’ with, e.g., Chinese ethnicity, correlates with the cosine between Chinese last names and those adjectives using embeddings trained on 1930s text. They also were able to document historical gender biases, such as the fact that embeddings for adjectives related to competence (‘smart’, ‘wise’, ‘thoughtful’, ‘resourceful’) had a higher cosine with male than fe- male words, and showed that this bias has been slowly decreasing since 1960. We return in later chapters to this question about the role of bias in natural language processing. 6.12 Evaluating Vector Models The most important evaluation metric for vector models is extrinsic evaluation on tasks, i.e., using vectors in an NLP task and seeing whether this improves perfor- mance over some other model. 7 Roughly speaking, if humans associate ‘ﬂowers’ with ‘pleasantness’ and ‘insects’ with ‘unpleasant- ness’, when they are instructed to push a green button for ‘ﬂowers’ (daisy, iris, lilac) and ‘pleasant words’ (love, laughter, pleasure) and a red button for ‘insects’ (ﬂea, spider, mosquito) and ‘unpleasant words’ (abuse, hatred, ugly) they are faster than in an incongruous condition where they push a red button for ‘ﬂowers’ and ‘unpleasant words’ and a green button for ‘insects’ and ‘pleasant words’. 128 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS Nonetheless it is useful to have intrinsic evaluations. The most common metric is to test their performance on similarity, computing the correlation between an algorithm’s word similarity scores and word similarity ratings assigned by humans. WordSim-353 (Finkelstein et al., 2002) is a commonly used set of ratings from 0 to 10 for 353 noun pairs; for example ( plane, car) had an average score of 5.77. SimLex-999 (Hill et al., 2015) is a more complex dataset that quantiﬁes similarity (cup, mug) rather than relatedness ( cup, coffee), and includes concrete and abstract adjective, noun and verb pairs. The TOEFL dataset is a set of 80 questions, each consisting of a target word with 4 additional word choices; the task is to choose which is the correct synonym, as in the example: Levied is closest in meaning to: imposed, believed, requested, correlated(Landauer and Dumais, 1997). All of these datasets present words without context. Slightly more realistic are intrinsic similarity tasks that include context. The Stanford Contextual Word Similarity (SCWS) dataset (Huang et al., 2012) and the Word-in-Context (WiC) dataset (Pilehvar and Camacho-Collados, 2019) offer richer evaluation scenarios. SCWS gives human judgments on 2,003 pairs of words in their sentential context, while WiC gives target words in two sentential contexts that are either in the same or different senses; see Appendix G. The semantic textual similarity task (Agirre et al. 2012, Agirre et al. 2015) evaluates the performance of sentence-level similarity algorithms, consisting of a set of pairs of sentences, each pair with human-labeled similarity scores. Another task used for evaluation is the analogy task, discussed on page 124, where the system has to solve problems of the forma is to b as a* is to b, given a, b, and a and having to ﬁnd b* (Turney and Littman, 2005). A number of sets of tuples have been created for this task (Mikolov et al. 2013a, Mikolov et al. 2013c, Gladkova et al. 2016), covering morphology ( city:cities::child:children), lexicographic rela- tions (leg:table::spout:teapot) and encyclopedia relations (Beijing:China::Dublin:Ireland), some drawing from the SemEval-2012 Task 2 dataset of 79 different relations (Jur- gens et al., 2012). All embedding algorithms suffer from inherent variability. For example because of randomness in the initialization and the random negative sampling, algorithms like word2vec may produce different results even from the same dataset, and in- dividual documents in a collection may strongly impact the resulting embeddings (Tian et al. 2016, Hellrich and Hahn 2016, Antoniak and Mimno 2018). When em- beddings are used to study word associations in particular corpora, therefore, it is best practice to train multiple embeddings with bootstrap sampling over documents and average the results (Antoniak and Mimno, 2018). 6.13 Summary • In vector semantics, a word is modeled as a vector—a point in high-dimensional space, also called an embedding. In this chapter we focus on static embed- dings, where each word is mapped to a ﬁxed embedding. • Vector semantic models fall into two classes: sparse and dense. In sparse models each dimension corresponds to a word in the vocabulary V and cells are functions of co-occurrence counts. The term-document matrix has a row for each word (term) in the vocabulary and a column for each document. The word-context or term-term matrix has a row for each (target) word in BIBLIOGRAPHICAL AND HISTORICAL NOTES 129 the vocabulary and a column for each context term in the vocabulary. Two sparse weightings are common: the tf-idf weighting which weights each cell by its term frequency and inverse document frequency, and PPMI (point- wise positive mutual information), which is most common for word-context matrices. • Dense vector models have dimensionality 50–1000. Word2vec algorithms like skip-gram are a popular way to compute dense embeddings. Skip-gram trains a logistic regression classiﬁer to compute the probability that two words are ‘likely to occur nearby in text’. This probability is computed from the dot product between the embeddings for the two words. • Skip-gram uses stochastic gradient descent to train the classiﬁer, by learning embeddings that have a high dot product with embeddings of words that occur nearby and a low dot product with noise words. • Other important embedding algorithms include GloVe, a method based on ratios of word co-occurrence probabilities. • Whether using sparse or dense vectors, word and document similarities are computed by some function of the dot product between vectors. The cosine of two vectors—a normalized dot product—is the most popular such metric. Bibliographical and Historical Notes The idea of vector semantics arose out of research in the 1950s in three distinct ﬁelds: linguistics, psychology, and computer science, each of which contributed a fundamental aspect of the model. The idea that meaning is related to the distribution of words in context was widespread in linguistic theory of the 1950s, among distributionalists like Zellig Harris, Martin Joos, and J. R. Firth, and semioticians like Thomas Sebeok. As Joos (1950) put it, the linguist’s “meaning” of a morpheme. . . is by deﬁnition the set of conditional probabilities of its occurrence in context with all other morphemes. The idea that the meaning of a word might be modeled as a point in a multi- dimensional semantic space came from psychologists like Charles E. Osgood, who had been studying how people responded to the meaning of words by assigning val- ues along scales like happy/sad or hard/soft. Osgood et al. (1957) proposed that the meaning of a word in general could be modeled as a point in a multidimensional Euclidean space, and that the similarity of meaning between two words could be modeled as the distance between these points in the space. A ﬁnal intellectual source in the 1950s and early 1960s was the ﬁeld then called mechanical indexing, now known asinformation retrieval. In what became knownmechanical indexing as the vector space model for information retrieval (Salton 1971, Sparck Jones 1986), researchers demonstrated new ways to deﬁne the meaning of words in terms of vectors (Switzer, 1965), and reﬁned methods for word similarity based on mea- sures of statistical association between words like mutual information (Giuliano, 1965) and idf (Sparck Jones, 1972), and showed that the meaning of documents could be represented in the same vector spaces used for words. Around the same time, (Cordier, 1965) showed that factor analysis of word association probabilities could be used to form dense vector representations of words. 130 CHAPTER 6 • V ECTOR SEMANTICS AND EMBEDDINGS Some of the philosophical underpinning of the distributional way of thinking came from the late writings of the philosopher Wittgenstein, who was skeptical of the possibility of building a completely formal theory of meaning deﬁnitions for each word. Wittgenstein suggested instead that “the meaning of a word is its use in the language” (Wittgenstein, 1953, PI 43). That is, instead of using some logical lan- guage to deﬁne each word, or drawing on denotations or truth values, Wittgenstein’s idea is that we should deﬁne a word by how it is used by people in speaking and un- derstanding in their day-to-day interactions, thus preﬁguring the movement toward embodied and experiential models in linguistics and NLP (Glenberg and Robertson 2000, Lake and Murphy 2021, Bisk et al. 2020, Bender and Koller 2020). More distantly related is the idea of deﬁning words by a vector of discrete fea- tures, which has roots at least as far back as Descartes and Leibniz (Wierzbicka 1992, Wierzbicka 1996). By the middle of the 20th century, beginning with the work of Hjelmslev (Hjelmslev, 1969) (originally 1943) and ﬂeshed out in early models of generative grammar (Katz and Fodor, 1963), the idea arose of representing mean- ing with semantic features, symbols that represent some sort of primitive meaning.semantic feature For example words like hen, rooster, or chick, have something in common (they all describe chickens) and something different (their age and sex), representable as: hen +female, +chicken, +adult rooster -female, +chicken, +adult chick +chicken, -adult The dimensions used by vector models of meaning to deﬁne words, however, are only abstractly related to this idea of a small ﬁxed number of hand-built dimensions. Nonetheless, there has been some attempt to show that certain dimensions of em- bedding models do contribute some speciﬁc compositional aspect of meaning like these early semantic features. The use of dense vectors to model word meaning, and indeed the term embed- ding, grew out of the latent semantic indexing (LSI) model (Deerwester et al., 1988) recast as LSA (latent semantic analysis) (Deerwester et al., 1990). In LSA singular value decomposition— SVD— is applied to a term-document matrix (eachSVD cell weighted by log frequency and normalized by entropy), and then the ﬁrst 300 dimensions are used as the LSA embedding. Singular Value Decomposition (SVD) is a method for ﬁnding the most important dimensions of a data set, those dimen- sions along which the data varies the most. LSA was then quickly widely applied: as a cognitive model Landauer and Dumais (1997), and for tasks like spell checking (Jones and Martin, 1997), language modeling (Bellegarda 1997, Coccaro and Ju- rafsky 1998, Bellegarda 2000), morphology induction (Schone and Jurafsky 2000, Schone and Jurafsky 2001b), multiword expressions (MWEs) (Schone and Juraf- sky, 2001a), and essay grading (Rehder et al., 1998). Related models were simul- taneously developed and applied to word sense disambiguation by Sch¨utze (1992b). LSA also led to the earliest use of embeddings to represent words in a probabilis- tic classiﬁer, in the logistic regression document router of Sch ¨utze et al. (1995). The idea of SVD on the term-term matrix (rather than the term-document matrix) as a model of meaning for NLP was proposed soon after LSA by Sch ¨utze (1992b). Sch¨utze applied the low-rank (97-dimensional) embeddings produced by SVD to the task of word sense disambiguation, analyzed the resulting semantic space, and also suggested possible techniques like dropping high-order dimensions. See Sch ¨utze (1997). A number of alternative matrix models followed on from the early SVD work, including Probabilistic Latent Semantic Indexing (PLSI) (Hofmann, 1999), Latent EXERCISES 131 Dirichlet Allocation (LDA) (Blei et al., 2003), and Non-negative Matrix Factoriza- tion (NMF) (Lee and Seung, 1999). The LSA community seems to have ﬁrst used the word “embedding” in Landauer et al. (1997), in a variant of its mathematical meaning as a mapping from one space or mathematical structure to another. In LSA, the word embedding seems to have described the mapping from the space of sparse count vectors to the latent space of SVD dense vectors. Although the word thus originally meant the mapping from one space to another, it has metonymically shifted to mean the resulting dense vector in the latent space, and it is in this sense that we currently use the word. By the next decade, Bengio et al. (2003) and Bengio et al. (2006) showed that neural language models could also be used to develop embeddings as part of the task of word prediction. Collobert and Weston (2007), Collobert and Weston (2008), and Collobert et al. (2011) then demonstrated that embeddings could be used to represent word meanings for a number of NLP tasks. Turian et al. (2010) compared the value of different kinds of embeddings for different NLP tasks. Mikolov et al. (2011) showed that recurrent neural nets could be used as language models. The idea of simplifying the hidden layer of these neural net language models to create the skip- gram (and also CBOW) algorithms was proposed by Mikolov et al. (2013a). The negative sampling training algorithm was proposed in Mikolov et al. (2013b). There are numerous surveys of static embeddings and their parameterizations (Bullinaria and Levy 2007, Bullinaria and Levy 2012, Lapesa and Evert 2014, Kiela and Clark 2014, Levy et al. 2015). See Manning et al. (2008) and Chapter 14 for a deeper understanding of the role of vectors in information retrieval, including how to compare queries with docu- ments, more details on tf-idf, and issues of scaling to very large datasets. See Kim (2019) for a clear and comprehensive tutorial on word2vec. Cruse (2004) is a useful introductory linguistic text on lexical semantics. Exercises 132 CHAPTER 7 • N EURAL NETWORKS CHAPTER 7 Neural Networks “[M]achines of this character can behave in a very complicated manner when the number of units is large.” Alan Turing (1948) “Intelligent Machines”, page 6 Neural networks are a fundamental computational tool for language process- ing, and a very old one. They are called neural because their origins lie in the McCulloch-Pitts neuron (McCulloch and Pitts, 1943), a simpliﬁed model of the biological neuron as a kind of computing element that could be described in terms of propositional logic. But the modern use in language processing no longer draws on these early biological inspirations. Instead, a modern neural network is a network of small computing units, each of which takes a vector of input values and produces a single output value. In this chapter we introduce the neural net applied to classiﬁcation. The architecture we introduce is called a feedforward network because the computation proceeds iter-feedforward atively from one layer of units to the next. The use of modern neural nets is often called deep learning, because modern networks are often deep (have many layers).deep learning Neural networks share much of the same mathematics as logistic regression. But neural networks are a more powerful classiﬁer than logistic regression, and indeed a minimal neural network (technically one with a single ‘hidden layer’) can be shown to learn any function. Neural net classiﬁers are different from logistic regression in another way. With logistic regression, we applied the regression classiﬁer to many different tasks by developing many rich kinds of feature templates based on domain knowledge. When working with neural networks, it is more common to avoid most uses of rich hand- derived features, instead building neural networks that take raw words as inputs and learn to induce features as part of the process of learning to classify. We saw examples of this kind of representation learning for embeddings in Chapter 6. Nets that are very deep are particularly good at representation learning. For that reason deep neural nets are the right tool for tasks that offer sufﬁcient data to learn features automatically. In this chapter we’ll introduce feedforward networks as classiﬁers, and also ap- ply them to the simple task of language modeling: assigning probabilities to word sequences and predicting upcoming words. In subsequent chapters we’ll introduce many other aspects of neural models, such as recurrent neural networks (Chap- ter 8), the Transformer (Chapter 9), and masked language modeling (Chapter 11). 7.1 • U NITS 133 7.1 Units The building block of a neural network is a single computational unit. A unit takes a set of real valued numbers as input, performs some computation on them, and produces an output. At its heart, a neural unit is taking a weighted sum of its inputs, with one addi- tional term in the sum called a bias term. Given a set of inputs x1...xn, a unit hasbias term a set of corresponding weights w1...wn and a bias b, so the weighted sum z can be represented as: z = b + ∑ i wixi (7.1) Often it’s more convenient to express this weighted sum using vector notation; recall from linear algebra that a vector is, at heart, just a list or array of numbers. Thusvector we’ll talk aboutz in terms of a weight vector w, a scalar bias b, and an input vector x, and we’ll replace the sum with the convenientdot product: z = w ·x+b (7.2) As deﬁned in Eq. 7.2, z is just a real valued number. Finally, instead of using z, a linear function of x, as the output, neural units apply a non-linear function f to z. We will refer to the output of this function as the activation value for the unit, a. Since we are just modeling a single unit, theactivation activation for the node is in fact the ﬁnal output of the network, which we’ll generally call y. So the value y is deﬁned as: y = a = f (z) We’ll discuss three popular non-linear functionsf below (the sigmoid, the tanh, and the rectiﬁed linear unit or ReLU) but it’s pedagogically convenient to start with the sigmoid function since we saw it in Chapter 5:sigmoid y = σ(z) = 1 1 +e−z (7.3) The sigmoid (shown in Fig. 7.1) has a number of advantages; it maps the output into the range (0,1), which is useful in squashing outliers toward 0 or 1. And it’s differentiable, which as we saw in Section 5.10 will be handy for learning. Figure 7.1 The sigmoid function takes a real value and maps it to the range (0,1). It is nearly linear around 0 but outlier values get squashed toward 0 or 1. Substituting Eq. 7.2 into Eq. 7.3 gives us the output of a neural unit: y = σ(w ·x+b) = 1 1 +exp(−(w ·x+b)) (7.4) 134 CHAPTER 7 • N EURAL NETWORKS Fig. 7.2 shows a ﬁnal schematic of a basic neural unit. In this example the unit takes 3 input values x1,x2, and x3, and computes a weighted sum, multiplying each value by a weight (w1, w2, and w3, respectively), adds them to a bias termb, and then passes the resulting sum through a sigmoid function to result in a number between 0 and 1. x1 x2 x3 y w1 w2 w3 ∑ b σ +1 z a Figure 7.2 A neural unit, taking 3 inputs x1, x2, and x3 (and a bias b that we represent as a weight for an input clamped at +1) and producing an output y. We include some convenient intermediate variables: the output of the summation, z, and the output of the sigmoid, a. In this case the output of the unit y is the same as a, but in deeper networks we’ll reserve y to mean the ﬁnal output of the entire network, leaving a as the activation of an individual node. Let’s walk through an example just to get an intuition. Let’s suppose we have a unit with the following weight vector and bias: w = [0.2,0.3,0.9] b = 0.5 What would this unit do with the following input vector: x = [0.5,0.6,0.1] The resulting output y would be: y = σ(w ·x+b) = 1 1 +e−(w·x+b) = 1 1 +e−(.5∗.2+.6∗.3+.1∗.9+.5) = 1 1 +e−0.87 = .70 In practice, the sigmoid is not commonly used as an activation function. A function that is very similar but almost always better is the tanh function shown in Fig. 7.3a;tanh tanh is a variant of the sigmoid that ranges from -1 to +1: y = tanh(z) =ez −e−z ez +e−z (7.5) The simplest activation function, and perhaps the most commonly used, is the rec- tiﬁed linear unit, also called the ReLU, shown in Fig. 7.3b. It’s just the same as zReLU when z is positive, and 0 otherwise: y = ReLU(z) = max(z,0) (7.6) These activation functions have different properties that make them useful for differ- ent language applications or network architectures. For example, the tanh function has the nice properties of being smoothly differentiable and mapping outlier values toward the mean. The rectiﬁer function, on the other hand, has nice properties that 7.2 • T HE XOR PROBLEM 135 (a) (b) Figure 7.3 The tanh and ReLU activation functions. result from it being very close to linear. In the sigmoid or tanh functions, very high values of z result in values ofy that are saturated, i.e., extremely close to 1, and havesaturated derivatives very close to 0. Zero derivatives cause problems for learning, because as we’ll see in Section 7.5, we’ll train networks by propagating an error signal back- wards, multiplying gradients (partial derivatives) from each layer of the network; gradients that are almost 0 cause the error signal to get smaller and smaller until it is too small to be used for training, a problem called the vanishing gradient problem.vanishing gradient Rectiﬁers don’t have this problem, since the derivative of ReLU for high values ofz is 1 rather than very close to 0. 7.2 The XOR problem Early in the history of neural networks it was realized that the power of neural net- works, as with the real neurons that inspired them, comes from combining these units into larger networks. One of the most clever demonstrations of the need for multi-layer networks was the proof by Minsky and Papert (1969) that a single neural unit cannot compute some very simple functions of its input. Consider the task of computing elementary logical functions of two inputs, like AND, OR, and XOR. As a reminder, here are the truth tables for those functions: AND OR XOR x1 x2 y x1 x2 y x1 x2 y 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 This example was ﬁrst shown for the perceptron, which is a very simple neuralperceptron unit that has a binary output and has a very simple step function as its non-linear activation function. The output y of a perceptron is 0 or 1, and is computed as follows (using the same weight w, input x, and bias b as in Eq. 7.2): y = {0, if w ·x+b ≤0 1, if w ·x+b >0 (7.7) 136 CHAPTER 7 • N EURAL NETWORKS It’s very easy to build a perceptron that can compute the logical AND and OR functions of its binary inputs; Fig. 7.4 shows the necessary weights. x1 x2 +1 -1 1 1 x1 x2 +1 0 1 1 (a) (b) Figure 7.4 The weights w and bias b for perceptrons for computing logical functions. The inputs are shown asx1 and x2 and the bias as a special node with value+1 which is multiplied with the bias weight b. (a) logical AND, with weights w1 = 1 and w2 = 1 and bias weight b = −1. (b) logical OR, with weights w1 = 1 and w2 = 1 and bias weight b = 0. These weights/biases are just one from an inﬁnite number of possible sets of weights and biases that would implement the functions. It turns out, however, that it’s not possible to build a perceptron to compute logical XOR! (It’s worth spending a moment to give it a try!) The intuition behind this important result relies on understanding that a percep- tron is a linear classiﬁer. For a two-dimensional input x1 and x2, the perceptron equation, w1x1 +w2x2 +b = 0 is the equation of a line. (We can see this by putting it in the standard linear format: x2 = (−w1/w2)x1 + (−b/w2).) This line acts as a decision boundary in two-dimensional space in which the output 0 is assigned to alldecision boundary inputs lying on one side of the line, and the output 1 to all input points lying on the other side of the line. If we had more than 2 inputs, the decision boundary becomes a hyperplane instead of a line, but the idea is the same, separating the space into two categories. Fig. 7.5 shows the possible logical inputs (00, 01, 10, and 11) and the line drawn by one possible set of parameters for an AND and an OR classiﬁer. Notice that there is simply no way to draw a line that separates the positive cases of XOR (01 and 10) from the negative cases (00 and 11). We say that XOR is not a linearly separablelinearly separable function. Of course we could draw a boundary with a curve, or some other function, but not a single line. 7.2.1 The solution: neural networks While the XOR function cannot be calculated by a single perceptron, it can be cal- culated by a layered network of perceptron units. Rather than see this with networks of simple perceptrons, however, let’s see how to compute XOR using two layers of ReLU-based units following Goodfellow et al. (2016). Fig. 7.6 shows a ﬁgure with the input being processed by two layers of neural units. The middle layer (called h) has two units, and the output layer (called y) has one unit. A set of weights and biases are shown that allows the network to correctly compute the XOR function. Let’s walk through what happens with the input x = [0, 0]. If we multiply each input value by the appropriate weight, sum, and then add the biasb, we get the vector [0, -1], and we then apply the rectiﬁed linear transformation to give the output of the h layer as [0, 0]. Now we once again multiply by the weights, sum, and add the bias (0 in this case) resulting in the value 0. The reader should work through the computation of the remaining 3 possible input pairs to see that the resultingy values are 1 for the inputs [0, 1] and [1, 0] and 0 for [0, 0] and [1, 1]. 7.2 • T HE XOR PROBLEM 137 0 0 1 1 x1 x2 0 0 1 1 x1 x2 0 0 1 1 x1 x2 a) x1 AND x2 b) x1 OR x2 c) x1 XOR x2 ? Figure 7.5 The functions AND, OR, and XOR, represented with input x1 on the x-axis and input x2 on the y-axis. Filled circles represent perceptron outputs of 1, and white circles perceptron outputs of 0. There is no way to draw a line that correctly separates the two categories for XOR. Figure styled after Russell and Norvig (2002). x1 x2 h1 h2 y1 +1 1 -1 1 1 1 -2 0 1 +1 0 Figure 7.6 XOR solution after Goodfellow et al. (2016). There are three ReLU units, in two layers; we’ve called them h1, h2 (h for “hidden layer”) and y1. As before, the numbers on the arrows represent the weights w for each unit, and we represent the bias b as a weight on a unit clamped to +1, with the bias weights/units in gray. It’s also instructive to look at the intermediate results, the outputs of the two hidden nodes h1 and h2. We showed in the previous paragraph that the h vector for the inputs x = [0, 0] was [0, 0]. Fig. 7.7b shows the values of the h layer for all 4 inputs. Notice that hidden representations of the two input points x = [0, 1] and x = [1, 0] (the two cases with XOR output = 1) are merged to the single point h = [1, 0]. The merger makes it easy to linearly separate the positive and negative cases of XOR. In other words, we can view the hidden layer of the network as forming a representation of the input. In this example we just stipulated the weights in Fig. 7.6. But for real examples the weights for neural networks are learned automatically using the error backprop- agation algorithm to be introduced in Section 7.5. That means the hidden layers will learn to form useful representations. This intuition, that neural networks can auto- matically learn useful representations of the input, is one of their key advantages, and one that we will return to again and again in later chapters. 138 CHAPTER 7 • N EURAL NETWORKS 0 0 1 1 x1 x2 a) The original x space 0 0 1 1 h1 h2 2 b) The new (linearly separable) h space Figure 7.7 The hidden layer forming a new representation of the input. (b) shows the representation of the hidden layer, h, compared to the original input representation x in (a). Notice that the input point [0, 1] has been collapsed with the input point [1, 0], making it possible to linearly separate the positive and negative cases of XOR. After Goodfellow et al. (2016). 7.3 Feedforward Neural Networks Let’s now walk through a slightly more formal presentation of the simplest kind of neural network, the feedforward network. A feedforward network is a multilayerfeedforward network network in which the units are connected with no cycles; the outputs from units in each layer are passed to units in the next higher layer, and no outputs are passed back to lower layers. (In Chapter 8 we’ll introduce networks with cycles, called recurrent neural networks.) For historical reasons multilayer networks, especially feedforward networks, are sometimes called multi-layer perceptrons (or MLPs); this is a technical misnomer,multi-layer perceptrons MLP since the units in modern multilayer networks aren’t perceptrons (perceptrons have a simple step-function as their activation function, but modern networks are made up of units with many kinds of non-linearities like ReLUs and sigmoids), but at some point the name stuck. Simple feedforward networks have three kinds of nodes: input units, hidden units, and output units. Fig. 7.8 shows a picture. The input layerx is a vector of simple scalar values just as we saw in Fig. 7.2. The core of the neural network is thehidden layer h formed of hidden units hi,hidden layer each of which is a neural unit as described in Section 7.1, taking a weighted sum of its inputs and then applying a non-linearity. In the standard architecture, each layer is fully-connected, meaning that each unit in each layer takes as input the outputsfully-connected from all the units in the previous layer, and there is a link between every pair of units from two adjacent layers. Thus each hidden unit sums over all the input units. Recall that a single hidden unit has as parameters a weight vector and a bias. We represent the parameters for the entire hidden layer by combining the weight vector and bias for each unit i into a single weight matrix W and a single bias vector b for the whole layer (see Fig. 7.8). Each element Wji of the weight matrix W represents the weight of the connection from the ith input unit xi to the jth hidden unit hj. The advantage of using a single matrix W for the weights of the entire layer is that now the hidden layer computation for a feedforward network can be done very 7.3 • F EEDFORWARD NEURAL NETWORKS 139 x1 x2 xn0 … … +1 b … UW input layer hidden layer output layer h1 y1 y2 yn2 h2 h3 hn1 Figure 7.8 A simple 2-layer feedforward network, with one hidden layer, one output layer, and one input layer (the input layer is usually not counted when enumerating layers). efﬁciently with simple matrix operations. In fact, the computation only has three steps: multiplying the weight matrix by the input vector x, adding the bias vector b, and applying the activation functiong (such as the sigmoid, tanh, or ReLU activation function deﬁned above). The output of the hidden layer, the vectorh, is thus the following (for this exam- ple we’ll use the sigmoid functionσ as our activation function): h = σ(Wx+b) (7.8) Notice that we’re applying the σ function here to a vector, while in Eq. 7.3 it was applied to a scalar. We’re thus allowing σ(·), and indeed any activation function g(·), to apply to a vector element-wise, so g[z1,z2,z3] = [g(z1),g(z2),g(z3)]. Let’s introduce some constants to represent the dimensionalities of these vectors and matrices. We’ll refer to the input layer as layer 0 of the network, and have n0 represent the number of inputs, so x is a vector of real numbers of dimension n0, or more formally x ∈Rn0 , a column vector of dimensionality [n0,1]. Let’s call the hidden layer layer 1 and the output layer layer 2. The hidden layer has dimensional- ity n1, so h ∈Rn1 and also b ∈Rn1 (since each hidden unit can take a different bias value). And the weight matrix W has dimensionality W ∈Rn1×n0 , i.e. [n1,n0]. Take a moment to convince yourself that the matrix multiplication in Eq. 7.8 will compute the value of each hj as σ (∑n0 i=1 Wjixi +bj ) . As we saw in Section 7.2, the resulting value h (for hidden but also for hypoth- esis) forms a representation of the input. The role of the output layer is to take this new representation h and compute a ﬁnal output. This output could be a real- valued number, but in many cases the goal of the network is to make some sort of classiﬁcation decision, and so we will focus on the case of classiﬁcation. If we are doing a binary task like sentiment classiﬁcation, we might have a sin- gle output node, and its scalar value y is the probability of positive versus negative sentiment. If we are doing multinomial classiﬁcation, such as assigning a part-of- speech tag, we might have one output node for each potential part-of-speech, whose output value is the probability of that part-of-speech, and the values of all the output nodes must sum to one. The output layer is thus a vector y that gives a probability distribution across the output nodes. 140 CHAPTER 7 • N EURAL NETWORKS Let’s see how this happens. Like the hidden layer, the output layer has a weight matrix (let’s call it U), but some models don’t include a bias vector b in the output layer, so we’ll simplify by eliminating the bias vector in this example. The weight matrix is multiplied by its input vector (h) to produce the intermediate output z: z = Uh There are n2 output nodes, so z ∈Rn2 , weight matrix U has dimensionality U ∈ Rn2×n1 , and element Ui j is the weight from unit j in the hidden layer to unit i in the output layer. However, z can’t be the output of the classiﬁer, since it’s a vector of real-valued numbers, while what we need for classiﬁcation is a vector of probabilities. There is a convenient function for normalizing a vector of real values, by which we meannormalizing converting it to a vector that encodes a probability distribution (all the numbers lie between 0 and 1 and sum to 1): the softmax function that we saw on page 85 ofsoftmax Chapter 5. More generally for any vector z of dimensionality d, the softmax is deﬁned as: softmax(zi) = exp(zi)∑d j=1 exp(zj) 1 ≤i ≤d (7.9) Thus for example given a vector z = [0.6,1.1,−1.5,1.2,3.2,−1.1], (7.10) the softmax function will normalize it to a probability distribution (shown rounded): softmax(z) = [0.055,0.090,0.0067,0.10,0.74,0.010] (7.11) You may recall that we used softmax to create a probability distribution from a vector of real-valued numbers (computed from summing weights times features) in the multinomial version of logistic regression in Chapter 5. That means we can think of a neural network classiﬁer with one hidden layer as building a vector h which is a hidden layer representation of the input, and then running standard multinomial logistic regression on the features that the network develops in h. By contrast, in Chapter 5 the features were mainly designed by hand via feature templates. So a neural network is like multinomial logistic regression, but (a) with many layers, since a deep neural network is like layer after layer of lo- gistic regression classiﬁers; (b) with those intermediate layers having many possible activation functions (tanh, ReLU, sigmoid) instead of just sigmoid (although we’ll continue to use σ for convenience to mean any activation function); (c) rather than forming the features by feature templates, the prior layers of the network induce the feature representations themselves. Here are the ﬁnal equations for a feedforward network with a single hidden layer, which takes an input vector x, outputs a probability distribution y, and is parameter- ized by weight matrices W and U and a bias vector b: h = σ(Wx+b) z = Uh y = softmax(z) (7.12) And just to remember the shapes of all our variables, x ∈Rn0 , h ∈Rn1 , b ∈Rn1 , W ∈Rn1×n0 , U ∈Rn2×n1 , and the output vectory ∈Rn2 . We’ll call this network a 2- layer network (we traditionally don’t count the input layer when numbering layers, but do count the output layer). So by this terminology logistic regression is a 1-layer network. 7.3 • F EEDFORWARD NEURAL NETWORKS 141 7.3.1 More details on feedforward networks Let’s now set up some notation to make it easier to talk about deeper networks of depth more than 2. We’ll use superscripts in square brackets to mean layer num- bers, starting at 0 for the input layer. So W[1] will mean the weight matrix for the (ﬁrst) hidden layer, and b[1] will mean the bias vector for the (ﬁrst) hidden layer. nj will mean the number of units at layer j. We’ll use g(·) to stand for the activation function, which will tend to be ReLU or tanh for intermediate layers and softmax for output layers. We’ll usea[i] to mean the output from layer i, and z[i] to mean the combination of previous layer output, weights and biases W[i]a[i−1] + b[i]. The 0th layer is for inputs, so we’ll refer to the inputsx more generally as a[0]. Thus we can re-represent our 2-layer net from Eq. 7.12 as follows: z[1] = W[1]a[0] +b[1] a[1] = g[1](z[1]) z[2] = W[2]a[1] +b[2] a[2] = g[2](z[2]) ˆy = a[2] (7.13) Note that with this notation, the equations for the computation done at each layer are the same. The algorithm for computing the forward step in an n-layer feedforward network, given the input vector a[0] is thus simply: for i in 1,...,n z[i] = W[i] a[i−1] + b[i] a[i] = g[i](z[i]) ˆy = a[n] It’s often useful to have a name for the ﬁnal set of activations right before the ﬁnal softmax. So however many layers we have, we’ll generally call the unnormalized values in the ﬁnal vector z[n], the vector of scores right before the ﬁnal softmax, the logits (see Eq. 5.7).logits The need for non-linear activation functions One of the reasons we use non- linear activation functions for each layer in a neural network is that if we did not, the resulting network is exactly equivalent to a single-layer network. Let’s see why this is true. Imagine the ﬁrst two layers of such a network of purely linear layers: z[1] = W[1]x+b[1] z[2] = W[2]z[1] +b[2] We can rewrite the function that the network is computing as: z[2] = W[2]z[1] +b[2] = W[2](W[1]x+b[1])+ b[2] = W[2]W[1]x+W[2]b[1] +b[2] = W′x+b′ (7.14) This generalizes to any number of layers. So without non-linear activation functions, a multilayer network is just a notational variant of a single layer network with a different set of weights, and we lose all the representational power of multilayer networks. 142 CHAPTER 7 • N EURAL NETWORKS Replacing the bias unit In describing networks, we will often use a slightly sim- pliﬁed notation that represents exactly the same function without referring to an ex- plicit bias node b. Instead, we add a dummy node a0 to each layer whose value will always be 1. Thus layer 0, the input layer, will have a dummy node a[0] 0 = 1, layer 1 will have a[1] 0 = 1, and so on. This dummy node still has an associated weight, and that weight represents the bias value b. For example instead of an equation like h = σ(Wx+b) (7.15) we’ll use: h = σ(Wx) (7.16) But now instead of our vector x having n0 values: x = x1,..., xn0 , it will have n0 + 1 values, with a new 0th dummy value x0 = 1: x = x0,..., xn0 . And instead of computing each hj as follows: hj = σ (n0∑ i=1 Wji xi +bj ) , (7.17) we’ll instead use: hj = σ (n0∑ i=0 Wji xi ) , (7.18) where the value Wj0 replaces what had been bj. Fig. 7.9 shows a visualization. x1 x2 xn0 … … +1 b … UW h1 y1 y2 yn2 h2 h3 hn1 x1 x2 xn0 … … x0=1 … UW h1 y1 y2 yn2 h2 h3 hn1 (a) (b) Figure 7.9 Replacing the bias node (shown in a) with x0 (b). We’ll continue showing the bias as b when we go over the learning algorithm in Section 7.5, but then we’ll switch to this simpliﬁed notation without explicit bias terms for the rest of the book. 7.4 Feedforward networks for NLP: Classiﬁcation Let’s see how to apply feedforward networks to NLP tasks! In this section we’ll look at classiﬁcation tasks like sentiment analysis; in the next section we’ll introduce neural language modeling. 7.4 • F EEDFORWARD NETWORKS FOR NLP: C LASSIFICATION 143 Let’s begin with a simple 2-layer sentiment classiﬁer. You might imagine taking our logistic regression classiﬁer from Chapter 5, which corresponds to a 1-layer net- work, and just adding a hidden layer. The input element xi could be scalar features like those in Fig. 5.2, e.g., x1 = count(words ∈doc), x2 = count(positive lexicon words ∈doc), x3 = 1 if “no” ∈doc, and so on. And the output layer ˆy could have two nodes (one each for positive and negative), or 3 nodes (positive, negative, neu- tral), in which case ˆy1 would be the estimated probability of positive sentiment, ˆy2 the probability of negative and ˆy3 the probability of neutral. The resulting equations would be just what we saw above for a 2-layer network (as always, we’ll continue to use the σ to stand for any non-linearity, whether sigmoid, ReLU or other). x = [x1,x2,...xN ] (each xi is a hand-designed feature) h = σ(Wx+b) z = Uh ˆy = softmax(z) (7.19) Fig. 7.10 shows a sketch of this architecture. As we mentioned earlier, adding this hidden layer to our logistic regression classiﬁer allows the network to represent the non-linear interactions between features. This alone might give us a better sentiment classiﬁer. UW [n⨉1] Hidden layer Output layer softmax [dh⨉n] [dh⨉1] [3⨉dh] Input words p(+) h1 h2 h3 hdh … y1^ y2^ y3^ x h y Input layer n=3 features [3⨉1] x1 x2 x3 dessert was great positive lexicon words = 1 count of “no” = 0 wordcount =3 p(-) p(neut) Figure 7.10 Feedforward network sentiment analysis using traditional hand-built features of the input text. Most applications of neural networks for NLP do something different, however. Instead of using hand-built human-engineered features as the input to our classiﬁer, we draw on deep learning’s ability to learn features from the data by representing words as embeddings, like the word2vec or GloVe embeddings we saw in Chapter 6. There are various ways to represent an input for classiﬁcation. One simple baseline is to apply some sort of pooling function to the embeddings of all the words in thepooling input. For example, for a text with n input words/tokens w1,...,wn, we can turn the n embeddings e(w1),...,e(wn) (each of dimensionality d) into a single embedding also of dimensionality d by just summing the embeddings, or by taking their mean (summing and then dividing by n): xmean = 1 n n∑ i=1 e(wi) (7.20) 144 CHAPTER 7 • N EURAL NETWORKS There are many other options, like taking the element-wise max. The element-wise max of a set of n vectors is a new vector whose kth element is the max of the kth elements of all the n vectors. Here are the equations for this classiﬁer assuming mean pooling; the architecture is sketched in Fig. 7.11: x = mean(e(w1),e(w2),..., e(wn)) h = σ(Wx+b) z = Uh ˆy = softmax(z) (7.21) UW [d⨉1] Hidden layer Output layer softmax [dh⨉d] [dh⨉1] [3⨉dh] Input words p(+) embedding for “great” embedding for “dessert” h1 h2 h3 hdh … y1 ^ y2^ y3^ x h y Input layer pooled embedding [3⨉1] pooling + dessert was great embedding for “was” p(-) p(neut) Figure 7.11 Feedforward network sentiment analysis using a pooled embedding of the in- put words. While Eq. 7.21 shows how to classify a single example x, in practice we want to efﬁciently classify an entire test set of m examples. We do this by vectorizing the process, just as we saw with logistic regression; instead of using for-loops to go through each example, we’ll use matrix multiplication to do the entire computation of an entire test set at once. First, we pack all the input feature vectors for each input x into a single input matrix X, with each row i a row vector consisting of the pooled embedding for input example x(i) (i.e., the vector x(i)). If the dimensionality of our pooled input embedding is d, X will be a matrix of shape [m ×d]. We will then need to slightly modify Eq. 7.21.X is of shape [m×d] and W is of shape [dh ×d], so we’ll have to reorder how we multiplyX and W and transpose W so they correctly multiply to yield a matrix H of shape [m ×dh].1 The bias vector b from Eq. 7.21 of shape [1 ×dh] will now have to be replicated into a matrix of shape [m ×dh]. We’ll need to similarly reorder the next step and transposeU. Finally, our output matrix ˆY will be of shape [m ×3] (or more generally [m ×do], where do is the number of output classes), with each row i of our output matrix ˆY consisting of the output vector ˆy(i).‘ Here are the ﬁnal equations for computing the output class 1 Note that we could have kept the original order of our products if we had instead made our input matrix X represent each input as a column vector instead of a row vector, making it of shape[d ×m]. But representing inputs as row vectors is convenient and common in neural network models. 7.5 • T RAINING NEURAL NETS 145 distribution for an entire test set: H = σ(XW⊺ +b) Z = HU⊺ ˆY = softmax(Z) (7.22) The idea of using word2vec or GloVe embeddings as our input representation— and more generally the idea of relying on another algorithm to have already learned an embedding representation for our input words—is called pretraining. Usingpretraining pretrained embedding representations, whether simple static word embeddings like word2vec or the much more powerful contextual embeddings we’ll introduce in Chapter 11, is one of the central ideas of deep learning. (It’s also possible, how- ever, to train the word embeddings as part of an NLP task; we’ll talk about how to do this in Section 7.7 in the context of the neural language modeling task.) 7.5 Training Neural Nets A feedforward neural net is an instance of supervised machine learning in which we know the correct output y for each observation x. What the system produces, via Eq. 7.13, is ˆy, the system’s estimate of the truey. The goal of the training procedure is to learn parameters W[i] and b[i] for each layer i that make ˆy for each training observation as close as possible to the true y. In general, we do all this by drawing on the methods we introduced in Chapter 5 for logistic regression, so the reader should be comfortable with that chapter before proceeding. First, we’ll need a loss function that models the distance between the system output and the gold output, and it’s common to use the loss function used for logistic regression, the cross-entropy loss. Second, to ﬁnd the parameters that minimize this loss function, we’ll use the gradient descent optimization algorithm introduced in Chapter 5. Third, gradient descent requires knowing the gradient of the loss function, the vector that contains the partial derivative of the loss function with respect to each of the parameters. In logistic regression, for each observation we could directly compute the derivative of the loss function with respect to an individual w or b. But for neural networks, with millions of parameters in many layers, it’s much harder to see how to compute the partial derivative of some weight in layer 1 when the loss is attached to some much later layer. How do we partial out the loss over all those intermediate layers? The answer is the algorithm called error backpropagation or backward differentiation. 7.5.1 Loss function The cross-entropy loss that is used in neural networks is the same one we saw forcross-entropy loss logistic regression. If the neural network is being used as a binary classiﬁer, with the sigmoid at the ﬁnal layer, the loss function is the same logistic regression loss we saw in Eq. 5.23: LCE (ˆy,y) =−log p(y\|x) = −[ylog ˆy +(1 −y)log(1 −ˆy)] (7.23) If we are using the network to classify into 3 or more classes, the loss function is exactly the same as the loss for multinomial regression that we saw in Chapter 5 on 146 CHAPTER 7 • N EURAL NETWORKS page 97. Let’s brieﬂy summarize the explanation here for convenience. First, when we have more than 2 classes we’ll need to represent both y and ˆy as vectors. Let’s assume we’re doing hard classiﬁcation , where only one class is the correct one. The true label y is then a vector with K elements, each corresponding to a class, with yc = 1 if the correct class is c, with all other elements of y being 0. Recall that a vector like this, with one value equal to 1 and the rest 0, is called aone-hot vector. And our classiﬁer will produce an estimate vector with K elements ˆy, each element ˆyk of which represents the estimated probability p(yk = 1\|x). The loss function for a single example x is the negative sum of the logs of the K output classes, each weighted by their probability yk: LCE (ˆy,y) =− K∑ k=1 yk log ˆyk (7.24) We can simplify this equation further; let’s ﬁrst rewrite the equation using the func- tion 1 {}which evaluates to 1 if the condition in the brackets is true and to 0 oth- erwise. This makes it more obvious that the terms in the sum in Eq. 7.24 will be 0 except for the term corresponding to the true class for which yk = 1: LCE (ˆy,y) = − K∑ k=1 1 {yk = 1}log ˆyk In other words, the cross-entropy loss is simply the negative log of the output proba- bility corresponding to the correct class, and we therefore also call this the negative log likelihood loss:negative log likelihood loss LCE (ˆy,y) = −log ˆyc (where c is the correct class) (7.25) Plugging in the softmax formula from Eq. 7.9, and with K the number of classes: LCE (ˆy,y) = −log exp(zc)∑K j=1 exp(zj) (where c is the correct class) (7.26) 7.5.2 Computing the Gradient How do we compute the gradient of this loss function? Computing the gradient requires the partial derivative of the loss function with respect to each parameter. For a network with one weight layer and sigmoid output (which is what logistic regression is), we could simply use the derivative of the loss that we used for logistic regression in Eq. 7.27 (and derived in Section 5.10): ∂LCE (ˆy,y) ∂wj = ( ˆy −y)xj = (σ(w ·x+b)−y)xj (7.27) Or for a network with one weight layer and softmax output (=multinomial logistic regression), we could use the derivative of the softmax loss from Eq. 5.48, shown for a particular weight wk and input xi ∂LCE(ˆy,y) ∂wk,i = −(yk −ˆyk)xi = −(yk −p(yk = 1\|x))xi = − ( yk − exp(wk ·x+bk)∑K j=1 exp(wj ·x+bj) ) xi (7.28) 7.5 • T RAINING NEURAL NETS 147 But these derivatives only give correct updates for one weight layer: the last one! For deep networks, computing the gradients for each weight is much more complex, since we are computing the derivative with respect to weight parameters that appear all the way back in the very early layers of the network, even though the loss is computed only at the very end of the network. The solution to computing this gradient is an algorithm called error backprop- agation or backprop (Rumelhart et al., 1986). While backprop was invented spe-error back- propagation cially for neural networks, it turns out to be the same as a more general procedure called backward differentiation , which depends on the notion of computation graphs. Let’s see how that works in the next subsection. 7.5.3 Computation Graphs A computation graph is a representation of the process of computing a mathematical expression, in which the computation is broken down into separate operations, each of which is modeled as a node in a graph. Consider computing the function L(a,b,c) =c(a +2b). If we make each of the component addition and multiplication operations explicit, and add names (d and e) for the intermediate outputs, the resulting series of computations is: d = 2 ∗b e = a +d L = c ∗e We can now represent this as a graph, with nodes for each operation, and di- rected edges showing the outputs from each operation as the inputs to the next, as in Fig. 7.12. The simplest use of computation graphs is to compute the value of the function with some given inputs. In the ﬁgure, we’ve assumed the inputs a = 3, b = 1, c = −2, and we’ve shown the result of the forward pass to compute the re- sult L(3,1,−2) =−10. In the forward pass of a computation graph, we apply each operation left to right, passing the outputs of each computation as the input to the next node. e=a+d d = 2b L=ce a=3 b=1 c=-2 e=5d=2 L=-10 forward pass a b c Figure 7.12 Computation graph for the functionL(a,b,c) =c(a+2b), with values for input nodes a = 3, b = 1, c = −2, showing the forward pass computation of L. 7.5.4 Backward differentiation on computation graphs The importance of the computation graph comes from the backward pass, which is used to compute the derivatives that we’ll need for the weight update. In this example our goal is to compute the derivative of the output function L with respect 148 CHAPTER 7 • N EURAL NETWORKS to each of the input variables, i.e., ∂L ∂a , ∂L ∂b , and ∂L ∂c . The derivative ∂L ∂a tells us how much a small change in a affects L. Backwards differentiation makes use of the chain rule in calculus, so let’s re-chain rule mind ourselves of that. Suppose we are computing the derivative of a composite function f (x) =u(v(x)). The derivative of f (x) is the derivative ofu(x) with respect to v(x) times the derivative of v(x) with respect to x: d f dx = du dv ·dv dx (7.29) The chain rule extends to more than two functions. If computing the derivative of a composite function f (x) =u(v(w(x))), the derivative of f (x) is: d f dx = du dv ·dv dw ·dw dx (7.30) The intuition of backward differentiation is to pass gradients back from the ﬁnal node to all the nodes in the graph. Fig. 7.13 shows part of the backward computation at one node e. Each node takes an upstream gradient that is passed in from its parent node to the right, and for each of its inputs computes a local gradient (the gradient of its output with respect to its input), and uses the chain rule to multiply these two to compute a downstream gradient to be passed on to the next earlier node. ed L ed ∂L ∂d ∂L ∂e= ∂e ∂d ∂L ∂e ∂e ∂d upstream gradientdownstream gradient local gradient Figure 7.13 Each node (like e here) takes an upstream gradient, multiplies it by the local gradient (the gradient of its output with respect to its input), and uses the chain rule to compute a downstream gradient to be passed on to a prior node. A node may have multiple local gradients if it has multiple inputs. Let’s now compute the 3 derivatives we need. Since in the computation graph L = ce, we can directly compute the derivative ∂L ∂c : ∂L ∂c = e (7.31) For the other two, we’ll need to use the chain rule: ∂L ∂a = ∂L ∂e ∂e ∂a ∂L ∂b = ∂L ∂e ∂e ∂d ∂d ∂b (7.32) Eq. 7.32 and Eq. 7.31 thus require ﬁve intermediate derivatives: ∂L ∂e , ∂L ∂c , ∂e ∂a , ∂e ∂d , and ∂d ∂b , which are as follows (making use of the fact that the derivative of a sum is the 7.5 • T RAINING NEURAL NETS 149 sum of the derivatives): L = ce : ∂L ∂e = c,∂L ∂c = e e = a +d : ∂e ∂a = 1,∂e ∂d = 1 d = 2b : ∂d ∂b = 2 In the backward pass, we compute each of these partials along each edge of the graph from right to left, using the chain rule just as we did above. Thus we begin by computing the downstream gradients from nodeL, which are ∂L ∂e and ∂L ∂c . For node e, we then multiply this upstream gradient ∂L ∂e by the local gradient (the gradient of the output with respect to the input), ∂e ∂d to get the output we send back to node d: ∂L ∂d . And so on, until we have annotated the graph all the way to all the input variables. The forward pass conveniently already will have computed the values of the forward intermediate variables we need (liked and e) to compute these derivatives. Fig. 7.14 shows the backward pass. e=d+a d = 2b L=ce a=3 b=1 e=5d=2 L=-10 a b c ∂L=5∂c ∂L =-2∂e ∂e =1∂d ∂d =2∂b ∂e =1∂a backward pass c=-2 ∂L =-2∂e ∂L =5∂c ∂L ∂d =-2∂e ∂d ∂L ∂e= ∂L ∂a =-2∂e ∂a ∂L ∂e= ∂L ∂b =-4∂d ∂b ∂L ∂d= Figure 7.14 Computation graph for the function L(a,b,c) =c(a +2b), showing the backward pass computa- tion of ∂L ∂a , ∂L ∂b , and ∂L ∂c . Backward differentiation for a neural network Of course computation graphs for real neural networks are much more complex. Fig. 7.15 shows a sample computation graph for a 2-layer neural network with n0 = 2, n1 = 2, and n2 = 1, assuming binary classiﬁcation and hence using a sigmoid output unit for simplicity. The function that the computation graph is computing is: z[1] = W[1]x+b[1] a[1] = ReLU(z[1]) z[2] = W[2]a[1] +b[2] a[2] = σ(z[2]) ˆy = a[2] (7.33) 150 CHAPTER 7 • N EURAL NETWORKS For the backward pass we’ll also need to compute the loss L. The loss function for binary sigmoid output from Eq. 7.23 is LCE (ˆy,y) = −[ylog ˆy +(1 −y)log(1 −ˆy)] (7.34) Our output ˆy = a[2], so we can rephrase this as LCE (a[2],y) = − [ yloga[2] +(1 −y)log(1 −a[2]) ] (7.35) z[2] = + a[2] = σ a[1] = ReLU z[1] = + b[1] * * * * x1 x2 a[1] = ReLU z[1] = + b[1] * * w[2] 11 w[1] 11 w[1] 12 w[1] 21 w[1] 22 b[2] w[2] 12 L (a[2],y)1 2 1 1 1 22 Figure 7.15 Sample computation graph for a simple 2-layer neural net (= 1 hidden layer) with two input units and 2 hidden units. We’ve adjusted the notation a bit to avoid long equations in the nodes by just mentioning the function that is being computed, and the resulting variable name. Thus the * to the right of node w[1] 11 means that w[1] 11 is to be multiplied by x1, and the node z[1] = +means that the value of z[1] is computed by summing the three nodes that feed into it (the two products, and the bias term b[1] i ). The weights that need updating (those for which we need to know the partial derivative of the loss function) are shown in teal. In order to do the backward pass, we’ll need to know the derivatives of all the functions in the graph. We already saw in Section 5.10 the derivative of the sigmoid σ: dσ(z) dz = σ(z)(1 −σ(z)) (7.36) We’ll also need the derivatives of each of the other activation functions. The derivative of tanh is: d tanh(z) dz = 1 −tanh2(z) (7.37) The derivative of the ReLU is2 d ReLU(z) dz = {0 f or z <0 1 f or z ≥0 (7.38) 2 The derivative is actually undeﬁned at the point z = 0, but by convention we treat it as 1. 7.5 • T RAINING NEURAL NETS 151 We’ll give the start of the computation, computing the derivative of the loss function L with respect to z, or ∂L ∂z (and leaving the rest of the computation as an exercise for the reader). By the chain rule: ∂L ∂z = ∂L ∂a[2] ∂a[2] ∂z (7.39) So let’s ﬁrst compute ∂L ∂a[2] , taking the derivative of Eq. 7.35, repeated here: LCE (a[2],y) = − [ yloga[2] +(1 −y)log(1 −a[2]) ] ∂L ∂a[2] = − (( y∂ log(a[2]) ∂a[2] ) +(1 −y)∂ log(1 −a[2]) ∂a[2] ) = − (( y 1 a[2] ) +(1 −y) 1 1 −a[2] (−1) ) = − ( y a[2] + y −1 1 −a[2] ) (7.40) Next, by the derivative of the sigmoid: ∂a[2] ∂z = a[2](1 −a[2]) Finally, we can use the chain rule: ∂L ∂z = ∂L ∂a[2] ∂a[2] ∂z = − ( y a[2] + y −1 1 −a[2] ) a[2](1 −a[2]) = a[2] −y (7.41) Continuing the backward computation of the gradients (next by passing the gra- dients over b[2] 1 and the two product nodes, and so on, back to all the teal nodes), is left as an exercise for the reader. 7.5.5 More details on learning Optimization in neural networks is a non-convex optimization problem, more com- plex than for logistic regression, and for that and other reasons there are many best practices for successful learning. For logistic regression we can initialize gradient descent with all the weights and biases having the value 0. In neural networks, by contrast, we need to initialize the weights with small random numbers. It’s also helpful to normalize the input values to have 0 mean and unit variance. Various forms of regularization are used to prevent overﬁtting. One of the most important is dropout: randomly dropping some units and their connections fromdropout the network during training (Hinton et al. 2012, Srivastava et al. 2014). At each iteration of training (whenever we update parameters, i.e. each mini-batch if we are using mini-batch gradient descent), we repeatedly choose a probability p and for each unit we replace its output with zero with probability p (and renormalize the rest of the outputs from that layer). 152 CHAPTER 7 • N EURAL NETWORKS Tuning of hyperparameters is also important. The parameters of a neural net-hyperparameter work are the weights W and biases b; those are learned by gradient descent. The hyperparameters are things that are chosen by the algorithm designer; optimal val- ues are tuned on a devset rather than by gradient descent learning on the training set. Hyperparameters include the learning rate η, the mini-batch size, the model architecture (the number of layers, the number of hidden nodes per layer, the choice of activation functions), how to regularize, and so on. Gradient descent itself also has many architectural variants such as Adam (Kingma and Ba, 2015). Finally, most modern neural networks are built using computation graph for- malisms that make it easy and natural to do gradient computation and parallelization on vector-based GPUs (Graphic Processing Units). PyTorch (Paszke et al., 2017) and TensorFlow (Abadi et al., 2015) are two of the most popular. The interested reader should consult a neural network textbook for further details; some sugges- tions are at the end of the chapter. 7.6 Feedforward Neural Language Modeling As our second application of feedforward networks, let’s consider language mod- eling: predicting upcoming words from prior words. Neural language modeling— based on the transformer architecture that we will see in Chapter 9—is the algorithm that underlies all of modern NLP. In this section and the next we’ll introduce a sim- pler version of neural language models for feedforward networks, an algorithm ﬁrst introduced by Bengio et al. (2003). The feedforward language model introduces many of the important concepts of neural language modeling, concepts we’ll return to as we describe more powerful models in Chapter 8 and Chapter 9. Neural language models have many advantages over the n-gram language mod- els of Chapter 3. Compared to n-gram models, neural language models can handle much longer histories, can generalize better over contexts of similar words, and are more accurate at word-prediction. On the other hand, neural net language models are much more complex, are slower and need more energy to train, and are less inter- pretable than n-gram models, so for some smaller tasks an n-gram language model is still the right tool. A feedforward neural language model (LM) is a feedforward network that takes as input at time t a representation of some number of previous words ( wt−1,wt−2, etc.) and outputs a probability distribution over possible next words. Thus—like the n-gram LM—the feedforward neural LM approximates the probability of a word given the entire prior context P(wt \|w1:t−1) by approximating based on the N −1 previous words: P(wt \|w1,..., wt−1) ≈P(wt \|wt−N+1,..., wt−1) (7.42) In the following examples we’ll use a 4-gram example, so we’ll show a neural net to estimate the probability P(wt = i\|wt−3,wt−2,wt−1). Neural language models represent words in this prior context by their embed- dings, rather than just by their word identity as used in n-gram language models. Using embeddings allows neural language models to generalize better to unseen data. For example, suppose we’ve seen this sentence in training: I have to make sure that the cat gets fed. 7.6 • F EEDFORWARD NEURAL LANGUAGE MODELING 153 but have never seen the words “gets fed” after the word “dog”. Our test set has the preﬁx “I forgot to make sure that the dog gets”. What’s the next word? An n-gram language model will predict “fed” after “that the cat gets”, but not after “that the dog gets”. But a neural LM, knowing that “cat” and “dog” have similar embeddings, will be able to generalize from the “cat” context to assign a high enough probability to “fed” even after seeing “dog”. 7.6.1 Forward inference in the neural language model Let’s walk through forward inference or decoding for neural language models.forward inference Forward inference is the task, given an input, of running a forward pass on the network to produce a probability distribution over possible outputs, in this case next words. We ﬁrst represent each of the N previous words as a one-hot vector of length \|V \|, i.e., with one dimension for each word in the vocabulary. A one-hot vector isone-hot vector a vector that has one element equal to 1—in the dimension corresponding to that word’s index in the vocabulary— while all the other elements are set to zero. Thus in a one-hot representation for the word “toothpaste”, supposing it is V5, i.e., index 5 in the vocabulary, x5 = 1, and xi = 0 ∀i ̸= 5, as shown here: [0 0 0 0 1 0 0 ... 0 0 0 0] 1 2 3 4 5 6 7 ... ... \|V\| The feedforward neural language model (sketched in Fig. 7.17) has a moving window that can see N words into the past. We’ll let N equal 3, so the 3 words wt−1, wt−2, and wt−3 are each represented as a one-hot vector. We then multiply these one-hot vectors by the embedding matrix E. The embedding weight matrix E has a column for each word, each a column vector of d dimensions, and hence has dimensionality d ×\|V \|. Multiplying by a one-hot vector that has only one non-zero element xi = 1 simply selects out the relevant column vector for word i, resulting in the embedding for word i, as shown in Fig. 7.16. E \|V\| d 1 \|V\| d 1 =✕ 5 5 e5 Figure 7.16 Selecting the embedding vector for word V5 by multiplying the embedding matrix E with a one-hot vector with a 1 in index 5. The 3 resulting embedding vectors are concatenated to producee, the embedding layer. This is followed by a hidden layer and an output layer whose softmax produces a probability distribution over words. For example y42, the value of output node 42, is the probability of the next word wt being V42, the vocabulary word with index 42 (which is the word ‘ﬁsh’ in our example). Here’s the algorithm in detail for our mini example: 1. Select three embeddings from E : Given the three previous words, we look up their indices, create 3 one-hot vectors, and then multiply each by the em- bedding matrix E. Consider wt−3. The one-hot vector for ‘for’ (index 35) is multiplied by the embedding matrix E, to give the ﬁrst part of the ﬁrst hidden 154 CHAPTER 7 • N EURAL NETWORKS UW embedding layer 3d⨉1 hidden layer output layer softmax dh⨉3d dh⨉1 \|V\|⨉dh input layer one-hot vectors E \|V\|⨉3 d⨉\|V\| p(do\|…) p(aardvark\|…) p(zebra\|…) p(fish\|…) \|V\|⨉1 E E h1 h2 y1 h3 hdh … … y34 y\|V\| … 00 1 0 0 1 \|V\| 35 0 0 1 0 0 1 \|V\| 451 0 0 1 0 0 1 \|V\| 9920 0 … … y42 y35102 ^ ^ ^ ^ ^ he x y for all the ? thanks and … wt-3 wt-2 wt-1 wt … Figure 7.17 Forward inference in a feedforward neural language model. At each timestep t the network computes a d-dimensional embedding for each context word (by multiplying a one-hot vector by the embedding matrix E), and concatenates the 3 resulting embeddings to get the embedding layer e. The embedding vector e is multiplied by a weight matrix W and then an activation function is applied element-wise to produce the hidden layer h, which is then multiplied by another weight matrix U. Finally, a softmax output layer predicts at each node i the probability that the next word wt will be vocabulary word Vi. layer, the embedding layer. Since each column of the input matrix E is anembedding layer embedding for a word, and the input is a one-hot column vector xi for word Vi, the embedding layer for input w will be Exi = ei, the embedding for word i. We now concatenate the three embeddings for the three context words to produce the embedding layer e. 2. Multiply by W: We multiply by W (and add b) and pass through the ReLU (or other) activation function to get the hidden layer h. 3. Multiply by U: h is now multiplied by U 4. Apply softmax: After the softmax, each node i in the output layer estimates the probability P(wt = i\|wt−1,wt−2,wt−3) In summary, the equations for a neural language model with a window size of 3, given one-hot input vectors for each input context word, are: e = [Ext−3;Ext−2;Ext−1] h = σ(We+b) z = Uh ˆy = softmax(z) (7.43) Note that we formed the embedding layer e by concatenating the 3 embeddings for the three context vectors; we’ll often use semicolons to mean concatenation of vectors. 7.7 • T RAINING THE NEURAL LANGUAGE MODEL 155 7.7 Training the neural language model The high-level intuition of training neural language models, whether the simple feedforward language models we describe here or the more powerful transformer language models of Chapter 9, is the idea of self-training or self-supervision thatself-training we saw in Chapter 6 for learning word representations. In self-training for language modeling, we take a corpus of text as training material and at each time step t ask the model to predict the next word. At ﬁrst it will do poorly at this task, but since in each case we know the correct answer (it’s the next word in the corpus!) we can easily train it to be better at predicting the correct next word. We call such a model self-supervised because we don’t have to add any special gold labels to the data; the natural sequence of words is its own supervision! We simply train the model to minimize the error in predicting the true next word in the training sequence. In practice, training the model means setting the parametersθ = E,W,U,b. For some tasks, it’s ok to freeze the embedding layer E with initial word2vec values.freeze Freezing means we use word2vec or some other pretraining algorithm to compute the initial embedding matrix E, and then hold it constant while we only modify W, U, and b, i.e., we don’t update E during language model training. However, often we’d like to learn the embeddings simultaneously with training the network. This is useful when the task the network is designed for (like sentiment classiﬁcation, trans- lation, or parsing) places strong constraints on what makes a good representation for words. Let’s see how to train the entire model includingE, i.e. to set all the parameters θ = E,W,U,b. We’ll do this via gradient descent (Fig. 5.6), using error backprop- agation on the computation graph to compute the gradient. Training thus not only sets the weights W and U of the network, but also as we’re predicting upcoming words, we’re learning the embeddings E for each word that best predict upcoming words. Fig. 7.18 shows the set up for a window size of N=3 context words. The input x consists of 3 one-hot vectors, fully connected to the embedding layer via 3 instanti- ations of the embedding matrix E. We don’t want to learn separate weight matrices for mapping each of the 3 previous words to the projection layer. We want one single embedding dictionary E that’s shared among these three. That’s because over time, many different words will appear as wt−2 or wt−1, and we’d like to just represent each word with one vector, whichever context position it appears in. Recall that the embedding weight matrix E has a column for each word, each a column vector of d dimensions, and hence has dimensionality d ×\|V \|. Generally training proceeds by taking as input a very long text, concatenating all the sentences, starting with random weights, and then iteratively moving through the text predicting each word wt . At each word wt , we use the cross-entropy (negative log likelihood) loss. Recall that the general form for this (repeated from Eq. 7.25) is: LCE (ˆy,y) = −log ˆyi, (where i is the correct class) (7.44) For language modeling, the classes are the words in the vocabulary, so ˆyi here means the probability that the model assigns to the correct next word wt : LCE = −log p(wt \|wt−1,...,wt−n+1) (7.45) The parameter update for stochastic gradient descent for this loss from steps to s+1 156 CHAPTER 7 • N EURAL NETWORKS UW embedding layer 3d⨉1 hidden layer output layer softmax dh⨉3d dh⨉1 \|V\|⨉dh input layer one-hot vectors E \|V\|⨉3 d⨉\|V\| p(do\|…) p(aardvark\|…) p(zebra\|…) p(fish\|…) \|V\|⨉1 E E h1 h2 y1 h3 hdh … … y34 y\|V\| … 00 1 0 0 1 \|V\| 35 0 0 1 0 0 1 \|V\| 451 0 0 1 0 0 1 \|V\| 9920 0 … … y42 y35102 ^ ^ ^ ^ ^ he x y for all the fish thanks and … wt-3 wt-2 wt-1 wt … L = −log P(fish \| for, all, the)wt=fish Figure 7.18 Learning all the way back to embeddings. Again, the embedding matrix E is shared among the 3 context words. is then: θs+1 = θs −η ∂ [−log p(wt \|wt−1,...,wt−n+1)] ∂θ (7.46) This gradient can be computed in any standard neural network framework which will then backpropagate through θ = E,W,U,b. Training the parameters to minimize loss will result both in an algorithm for language modeling (a word predictor) but also a new set of embeddings E that can be used as word representations for other tasks. 7.8 Summary • Neural networks are built out ofneural units, originally inspired by biological neurons but now simply an abstract computational device. • Each neural unit multiplies input values by a weight vector, adds a bias, and then applies a non-linear activation function like sigmoid, tanh, or rectiﬁed linear unit. • In a fully-connected, feedforward network, each unit in layer i is connected to each unit in layer i +1, and there are no cycles. • The power of neural networks comes from the ability of early layers to learn representations that can be utilized by later layers in the network. • Neural networks are trained by optimization algorithms like gradient de- scent. • Error backpropagation, backward differentiation on a computation graph, is used to compute the gradients of the loss function for a network. BIBLIOGRAPHICAL AND HISTORICAL NOTES 157 • Neural language models use a neural network as a probabilistic classiﬁer, to compute the probability of the next word given the previous n words. • Neural language models can use pretrained embeddings, or can learn embed- dings from scratch in the process of language modeling. Bibliographical and Historical Notes The origins of neural networks lie in the 1940s McCulloch-Pitts neuron (McCul- loch and Pitts, 1943), a simpliﬁed model of the biological neuron as a kind of com- puting element that could be described in terms of propositional logic. By the late 1950s and early 1960s, a number of labs (including Frank Rosenblatt at Cornell and Bernard Widrow at Stanford) developed research into neural networks; this phase saw the development of the perceptron (Rosenblatt, 1958), and the transformation of the threshold into a bias, a notation we still use (Widrow and Hoff, 1960). The ﬁeld of neural networks declined after it was shown that a single perceptron unit was unable to model functions as simple as XOR (Minsky and Papert, 1969). While some small amount of work continued during the next two decades, a major revival for the ﬁeld didn’t come until the 1980s, when practical tools for building deeper networks like error backpropagation became widespread (Rumelhart et al., 1986). During the 1980s a wide variety of neural network and related architec- tures were developed, particularly for applications in psychology and cognitive sci- ence (Rumelhart and McClelland 1986b, McClelland and Elman 1986, Rumelhart and McClelland 1986a, Elman 1990), for which the term connectionist or paral-connectionist lel distributed processing was often used (Feldman and Ballard 1982, Smolensky 1988). Many of the principles and techniques developed in this period are foun- dational to modern work, including the ideas of distributed representations (Hinton, 1986), recurrent networks (Elman, 1990), and the use of tensors for compositionality (Smolensky, 1990). By the 1990s larger neural networks began to be applied to many practical lan- guage processing tasks as well, like handwriting recognition (LeCun et al. 1989) and speech recognition (Morgan and Bourlard 1990). By the early 2000s, improvements in computer hardware and advances in optimization and training techniques made it possible to train even larger and deeper networks, leading to the modern term deep learning (Hinton et al. 2006, Bengio et al. 2007). We cover more related history in Chapter 8 and Chapter 16. There are a number of excellent books on the subject. Goldberg (2017) has superb coverage of neural networks for natural language processing. For neural networks in general see Goodfellow et al. (2016) and Nielsen (2015). 158 CHAPTER 8 • RNN S AND LSTM S CHAPTER 8 RNNs and LSTMs Time will explain. Jane Austen, Persuasion Language is an inherently temporal phenomenon. Spoken language is a sequence of acoustic events over time, and we comprehend and produce both spoken and written language as a sequential input stream. The temporal nature of language is reﬂected in the metaphors we use; we talk of theﬂow of conversations, news feeds, and twitter streams, all of which emphasize that language is a sequence that unfolds in time. Yet most of the machine learning approaches we’ve studied so far, like those for sentiment analysis and other text classiﬁcation tasks don’t have this temporal nature – they assume simultaneous access to all aspects of their input. The feedfor- ward networks of Chapter 7 also assumed simultaneous access, although they also had a simple model for time. Recall that we applied feedforward networks to lan- guage modeling by having them look only at a ﬁxed-size window of words, and then sliding this window over the input, making independent predictions along the way. This sliding-window approach is also used in the transformer architecture we will introduce in Chapter 9. This chapter introduces a deep learning architecture that offers an alternative way of representing time: recurrent neural networks (RNNs), and their variants like LSTMs. RNNs have a mechanism that deals directly with the sequential nature of language, allowing them to handle the temporal nature of language without the use of arbitrary ﬁxed-sized windows. The recurrent network offers a new way to represent the prior context, in its recurrent connections, allowing the model’s decision to depend on information from hundreds of words in the past. We’ll see how to apply the model to the task of language modeling, to text classiﬁcation tasks like sentiment analysis, and to sequence modeling tasks like part-of-speech tagging (a task we’ll return to in detail in Chapter 17). 8.1 Recurrent Neural Networks A recurrent neural network (RNN) is any network that contains a cycle within its network connections, meaning that the value of some unit is directly, or indirectly, dependent on its own earlier outputs as an input. While powerful, such networks are difﬁcult to reason about and to train. However, within the general class of recur- rent networks there are constrained architectures that have proven to be extremely effective when applied to language. In this section, we consider a class of recurrent networks referred to as Elman Networks (Elman, 1990) or simple recurrent net-Elman Networks works. These networks are useful in their own right and serve as the basis for more complex approaches like the Long Short-Term Memory (LSTM) networks discussed 8.1 • R ECURRENT NEURAL NETWORKS 159 later in this chapter. In this chapter when we use the term RNN we’ll be referring to these simpler more constrained networks (although you will often see the term RNN to mean any net with recurrent properties including LSTMs). xt ht yt Figure 8.1 Simple recurrent neural network after Elman (1990). The hidden layer includes a recurrent connection as part of its input. That is, the activation value of the hidden layer depends on the current input as well as the activation value of the hidden layer from the previous time step. Fig. 8.1 illustrates the structure of an RNN. As with ordinary feedforward net- works, an input vector representing the current input, xt , is multiplied by a weight matrix and then passed through a non-linear activation function to compute the val- ues for a layer of hidden units. This hidden layer is then used to calculate a cor- responding output, yt . In a departure from our earlier window-based approach, se- quences are processed by presenting one item at a time to the network. We’ll use subscripts to represent time, thus xt will mean the input vector x at time t. The key difference from a feedforward network lies in the recurrent link shown in the ﬁgure with the dashed line. This link augments the input to the computation at the hidden layer with the value of the hidden layer from the preceding point in time. The hidden layer from the previous time step provides a form of memory, or context, that encodes earlier processing and informs the decisions to be made at later points in time. Critically, this approach does not impose a ﬁxed-length limit on this prior context; the context embodied in the previous hidden layer can include information extending back to the beginning of the sequence. Adding this temporal dimension makes RNNs appear to be more complex than non-recurrent architectures. But in reality, they’re not all that different. Given an input vector and the values for the hidden layer from the previous time step, we’re still performing the standard feedforward calculation introduced in Chapter 7. To see this, consider Fig. 8.2 which clariﬁes the nature of the recurrence and how it factors into the computation at the hidden layer. The most signiﬁcant change lies in the new set of weights, U, that connect the hidden layer from the previous time step to the current hidden layer. These weights determine how the network makes use of past context in calculating the output for the current input. As with the other weights in the network, these connections are trained via backpropagation. 8.1.1 Inference in RNNs Forward inference (mapping a sequence of inputs to a sequence of outputs) in an RNN is nearly identical to what we’ve already seen with feedforward networks. To compute an output yt for an input xt , we need the activation value for the hidden layer ht . To calculate this, we multiply the input xt with the weight matrix W, and the hidden layer from the previous time step ht−1 with the weight matrix U. We add these values together and pass them through a suitable activation function, g, to arrive at the activation value for the current hidden layer, ht . Once we have the values for the hidden layer, we proceed with the usual computation to generate the 160 CHAPTER 8 • RNN S AND LSTM S + U V W yt xt ht ht-1 Figure 8.2 Simple recurrent neural network illustrated as a feedforward network. The hid- den layer ht−1 from the prior time step is multiplied by weight matrix U and then added to the feedforward component from the current time step. output vector. ht = g(Uht−1 +Wxt ) (8.1) yt = f (Vht ) (8.2) Let’s refer to the input, hidden and output layer dimensions as din, dh, and dout respectively. Given this, our three parameter matrices are:W ∈Rdh×din , U ∈Rdh×dh , and V ∈Rdout ×dh . We compute yt via a softmax computation that gives a probability distribution over the possible output classes. yt = softmax(Vht ) (8.3) The fact that the computation at time t requires the value of the hidden layer from time t −1 mandates an incremental inference algorithm that proceeds from the start of the sequence to the end as illustrated in Fig. 8.3. The sequential nature of simple recurrent networks can also be seen by unrolling the network in time as is shown in Fig. 8.4. In this ﬁgure, the various layers of units are copied for each time step to illustrate that they will have differing values over time. However, the various weight matrices are shared across time. function FORWARD RNN(x, network) returns output sequence y h0 ←0 for i←1 to LENGTH (x) do hi ←g(Uhi−1 + Wxi) yi ←f (Vhi) return y Figure 8.3 Forward inference in a simple recurrent network. The matricesU, V and W are shared across time, while new values for h and y are calculated with each time step. 8.1.2 Training As with feedforward networks, we’ll use a training set, a loss function, and back- propagation to obtain the gradients needed to adjust the weights in these recurrent networks. As shown in Fig. 8.2, we now have 3 sets of weights to update: W, the 8.1 • R ECURRENT NEURAL NETWORKS 161 U V W U V W U V W x1 x2 x3y1 y2 y3 h1 h3 h2 h0 Figure 8.4 A simple recurrent neural network shown unrolled in time. Network layers are recalculated for each time step, while the weights U, V and W are shared across all time steps. weights from the input layer to the hidden layer, U, the weights from the previous hidden layer to the current hidden layer, and ﬁnally V, the weights from the hidden layer to the output layer. Fig. 8.4 highlights two considerations that we didn’t have to worry about with backpropagation in feedforward networks. First, to compute the loss function for the output at time t we need the hidden layer from time t −1. Second, the hidden layer at time t inﬂuences both the output at time t and the hidden layer at time t +1 (and hence the output and loss at t + 1). It follows from this that to assess the error accruing to ht , we’ll need to know its inﬂuence on both the current outputas well as the ones that follow. Tailoring the backpropagation algorithm to this situation leads to a two-pass al- gorithm for training the weights in RNNs. In the ﬁrst pass, we perform forward inference, computing ht , yt , accumulating the loss at each step in time, saving the value of the hidden layer at each step for use at the next time step. In the second phase, we process the sequence in reverse, computing the required gradients as we go, computing and saving the error term for use in the hidden layer for each step backward in time. This general approach is commonly referred to as backpropaga- tion through time (Werbos 1974, Rumelhart et al. 1986, Werbos 1990). backpropaga- tion through time Fortunately, with modern computational frameworks and adequate computing resources, there is no need for a specialized approach to training RNNs. As illus- trated in Fig. 8.4, explicitly unrolling a recurrent network into a feedforward com- putational graph eliminates any explicit recurrences, allowing the network weights to be trained directly. In such an approach, we provide a template that speciﬁes the basic structure of the network, including all the necessary parameters for the input, output, and hidden layers, the weight matrices, as well as the activation and output functions to be used. Then, when presented with a speciﬁc input sequence, we can generate an unrolled feedforward network speciﬁc to that input, and use that graph to perform forward inference or training via ordinary backpropagation. 162 CHAPTER 8 • RNN S AND LSTM S For applications that involve much longer input sequences, such as speech recog- nition, character-level processing, or streaming continuous inputs, unrolling an en- tire input sequence may not be feasible. In these cases, we can unroll the input into manageable ﬁxed-length segments and treat each segment as a distinct training item. 8.2 RNNs as Language Models Let’s see how to apply RNNs to the language modeling task. Recall from Chapter 3 that language models predict the next word in a sequence given some preceding context. For example, if the preceding context is “Thanks for all the” and we want to know how likely the next word is “ﬁsh” we would compute: P(ﬁsh\|Thanks for all the) Language models give us the ability to assign such a conditional probability to every possible next word, giving us a distribution over the entire vocabulary. We can also assign probabilities to entire sequences by combining these conditional probabilities with the chain rule: P(w1:n) = n∏ i=1 P(wi\|w<i) The n-gram language models of Chapter 3 compute the probability of a word given counts of its occurrence with the n−1 prior words. The context is thus of sizen−1. For the feedforward language models of Chapter 7, the context is the window size. RNN language models (Mikolov et al., 2010) process the input sequence one word at a time, attempting to predict the next word from the current word and the previous hidden state. RNNs thus don’t have the limited context problem that n-gram models have, or the ﬁxed context that feedforward language models have, since the hidden state can in principle represent information about all of the preceding words all the way back to the beginning of the sequence. Fig. 8.5 sketches this difference between a FFN language model and an RNN language model, showing that the RNN language model uses ht−1, the hidden state from the previous time step, as a representation of the past context. 8.2.1 Forward Inference in an RNN language model Forward inference in a recurrent language model proceeds exactly as described in Section 8.1.1. The input sequence X = [x1;...;xt ;...;xN ] consists of a series of words each represented as a one-hot vector of size\|V \|×1, and the output prediction, y, is a vector representing a probability distribution over the vocabulary. At each step, the model uses the word embedding matrix E to retrieve the embedding for the current word, multiples it by the weight matrix W, and then adds it to the hidden layer from the previous step (weighted by weight matrix U) to compute a new hidden layer. This hidden layer is then used to generate an output layer which is passed through a softmax layer to generate a probability distribution over the entire vocabulary. That is, at time t: et = Ext (8.4) ht = g(Uht−1 +Wet ) (8.5) ˆ yt = softmax(Vht ) (8.6) 8.2 • RNN S AS LANGUAGE MODELS 163 V W et htUht-1 et ht et-1et-2 U W a) b)^yt et-1 ^yt ht-2 WW et-2 U Figure 8.5 Simpliﬁed sketch of two LM architectures moving through a text, showing a schematic context of three tokens: (a) a feedforward neural language model which has a ﬁxed context input to the weight matrix W, (b) an RNN language model, in which the hidden state ht−1 summarizes the prior context. When we do language modeling with RNNs (and we’ll see this again in Chapter 9 with transformers), it’s convenient to make the assumption that the embedding di- mension de and the hidden dimension dh are the same. So we’ll just call both of these the model dimension d. So the embedding matrix E is of shape [d ×\|V \|], and xt is a one-hot vector of shape [\|V \|×1]. The product et is thus of shape [d ×1]. W and U are of shape [d ×d], so ht is also of shape [d ×1]. V is of shape [\|V \|×d], so the result of Vh is a vector of shape [\|V \|×1]. This vector can be thought of as a set of scores over the vocabulary given the evidence provided in h. Passing these scores through the softmax normalizes the scores into a probability distribution. The probability that a particular word k in the vocabulary is the next word is represented by ˆ yt [k], the kth component of ˆ yt : P(wt+1 = k\|w1,..., wt ) = ˆ yt [k] (8.7) The probability of an entire sequence is just the product of the probabilities of each item in the sequence, where we’ll useˆ yi[wi] to mean the probability of the true word wi at time step i. P(w1:n) = n∏ i=1 P(wi\|w1:i−1) (8.8) = n∏ i=1 ˆ yi[wi] (8.9) 8.2.2 Training an RNN language model To train an RNN as a language model, we use the same self-supervision (or self-self-supervision training) algorithm we saw in Section 7.7: we take a corpus of text as training material and at each time step t ask the model to predict the next word. We call such a model self-supervised because we don’t have to add any special gold labels to the data; the natural sequence of words is its own supervision! We simply train the model to minimize the error in predicting the true next word in the training sequence, using cross-entropy as the loss function. Recall that the cross-entropy loss measures the difference between a predicted probability distribution and the 164 CHAPTER 8 • RNN S AND LSTM S Input Embeddings Softmax over Vocabulary So long and thanks for long and thanks forNext word all … Loss … … RNN h y Vh <latexit sha1_base64="9tru+5ysH1zS9iUXRg/IsnxmpMA=">AAAB/XicbVDLSsNAFL3xWesr6lKQwSK4sSQi1WXRjcsK9gFNCZPpJB06yYSZiRBCcOOvuBFxo+Av+Av+jUnbTVsPDBzOOcO993gxZ0pb1q+xsrq2vrFZ2apu7+zu7ZsHhx0lEklomwguZM/DinIW0bZmmtNeLCkOPU673viu9LtPVComokedxnQQ4iBiPiNYF5Jrnlw4XATIGWGdpbmbOSHWIxlmXERBnldds2bVrQnQMrFnpAYztFzzxxkKkoQ00oRjpfq2FetBhqVmhNO86iSKxpiMcUCzyfY5OiukIfKFLF6k0USdy+FQqTT0imS5nFr0SvE/r59o/2aQsShONI3IdJCfcKQFKqtAQyYp0TwtCCaSFRsiMsISE10UVp5uLx66TDqXdbtRbzxc1Zq3sxIqcAyncA42XEMT7qEFbSDwAm/wCV/Gs/FqvBsf0+iKMftzBHMwvv8ADJKVcA==</latexit> log ˆ y long <latexit sha1_base64="tuzkS/BeX/Xmg79qpWZlpeYDhtE=">AAAB/HicbVDLSsNAFL3xWesr6lKEwSK4sSQi1WXRjcsK9gFNCZPJpB06mYSZiRBC3PgrbkTcKPgN/oJ/Y9J209YDA4dzznDvPV7MmdKW9WusrK6tb2xWtqrbO7t7++bBYUdFiSS0TSIeyZ6HFeVM0LZmmtNeLCkOPU673viu9LtPVCoWiUedxnQQ4qFgASNYF5Jrnlw4PBoiZ4R1luZu5oRYj2SYYeHnedU1a1bdmgAtE3tGajBDyzV/HD8iSUiFJhwr1betWA8yLDUjnOZVJ1E0xmSMhzSbLJ+js0LyURDJ4gmNJupcDodKpaFXJMvd1KJXiv95/UQHN4OMiTjRVJDpoCDhSEeobAL5TFKieVoQTCQrNkRkhCUmuuirPN1ePHSZdC7rdqPeeLiqNW9nJVTgGE7hHGy4hibcQwvaQOAF3uATvoxn49V4Nz6m0RVj9ucI5mB8/wEiupTp</latexit> log ˆ y and <latexit sha1_base64="0zdsmbBovZ+hafWZN7Hvufo85tU=">AAAB/3icbVDLSsNAFJ3UV62vqEs3g0VwY0lEqsuiG5cV7AOaEibTSTN0kgkzN0IIWbjxV9yIuFHwD/wF/8ak7aatBwYO55zh3nu8WHANlvVrVNbWNza3qtu1nd29/QPz8KirZaIo61AppOp7RDPBI9YBDoL1Y8VI6AnW8yZ3pd97YkpzGT1CGrNhSMYR9zklUEiuiS8cIcfYCQhkae5mTkggUGEGAYkmOs9rrlm3GtYUeJXYc1JHc7Rd88cZSZqELAIqiNYD24phmBEFnAqW15xEs5jQCRmzbLp/js8KaYR9qYoXAZ6qCzkSap2GXpEs19PLXin+5w0S8G+GGY/iBFhEZ4P8RGCQuCwDj7hiFERaEEIVLzbENCCKUCgqK0+3lw9dJd3Lht1sNB+u6q3beQlVdIJO0Tmy0TVqoXvURh1E0Qt6Q5/oy3g2Xo1342MWrRjzP8doAcb3H7Aall0=</latexit> log ˆy thanks <latexit sha1_base64="D3c31Jvxp3QWPr2h4tzQWmeenDs=">AAAB/HicbVDLSsNAFL3xWesr6lKEwSK4sSQi1WXRjcsK9gFNCZPppB06yYSZiRBC3PgrbkTcKPgN/oJ/Y9Jm09YDA4dzznDvPV7EmdKW9WusrK6tb2xWtqrbO7t7++bBYUeJWBLaJoIL2fOwopyFtK2Z5rQXSYoDj9OuN7kr/O4TlYqJ8FEnER0EeBQynxGsc8k1Ty4cLkbIGWOdJpmbOgHWYxmkvpBZVnXNmlW3pkDLxC5JDUq0XPPHGQoSBzTUhGOl+rYV6UGKpWaE06zqxIpGmEzwiKbT5TN0lktDlM/LX6jRVJ3L4UCpJPDyZLGbWvQK8T+vH2v/ZpCyMIo1DclskB9zpAUqmkBDJinRPMkJJpLlGyIyxhITnfdVnG4vHrpMOpd1u1FvPFzVmrdlCRU4hlM4BxuuoQn30II2EHiBN/iEL+PZeDXejY9ZdMUo/xzBHIzvP0CJlP0=</latexit> log ˆ y for <latexit sha1_base64="PI3y1fb9LhumoVCQRh2+Y84dRkc=">AAAB/HicbVDLSsNAFL3xWesr6lKEwSK4sSQi1WXRjcsK9gFNCZPppB06yYSZiRBC3PgrbkTcKPgN/oJ/Y9Jm09YDA4dzznDvPV7EmdKW9WusrK6tb2xWtqrbO7t7++bBYUeJWBLaJoIL2fOwopyFtK2Z5rQXSYoDj9OuN7kr/O4TlYqJ8FEnER0EeBQynxGsc8k1Ty4cLkbIGWOdJpmbOgHWYxmkmPMsq7pmzapbU6BlYpekBiVarvnjDAWJAxpqwrFSfduK9CDFUjPCaVZ1YkUjTCZ4RNPp8hk6y6Uh8oXMX6jRVJ3L4UCpJPDyZLGbWvQK8T+vH2v/ZpCyMIo1DclskB9zpAUqmkBDJinRPMkJJpLlGyIyxhITnfdVnG4vHrpMOpd1u1FvPFzVmrdlCRU4hlM4BxuuoQn30II2EHiBN/iEL+PZeDXejY9ZdMUo/xzBHIzvPyumlO8=</latexit> log ˆ y all e Figure 8.6 Training RNNs as language models. correct distribution. LCE = − ∑ w∈V yt [w]log ˆyt [w] (8.10) In the case of language modeling, the correct distributionyt comes from knowing the next word. This is represented as a one-hot vector corresponding to the vocabulary where the entry for the actual next word is 1, and all the other entries are 0. Thus, the cross-entropy loss for language modeling is determined by the probability the model assigns to the correct next word. So at time t the CE loss is the negative log probability the model assigns to the next word in the training sequence. LCE ( ˆyt ,yt ) = −log ˆyt [wt+1] (8.11) Thus at each word positiont of the input, the model takes as input the correct wordwt together with ht−1, encoding information from the preceding w1:t−1, and uses them to compute a probability distribution over possible next words so as to compute the model’s loss for the next token wt+1. Then we move to the next word, we ignore what the model predicted for the next word and instead use the correct word wt+1 along with the prior history encoded to estimate the probability of token wt+2. This idea that we always give the model the correct history sequence to predict the next word (rather than feeding the model its best case from the previous time step) is called teacher forcing.teacher forcing The weights in the network are adjusted to minimize the average CE loss over the training sequence via gradient descent. Fig. 8.6 illustrates this training regimen. 8.2.3 Weight Tying Careful readers may have noticed that the input embedding matrix E and the ﬁnal layer matrix V, which feeds the output softmax, are quite similar. The columns of E represent the word embeddings for each word in the vocab- ulary learned during the training process with the goal that words that have similar meaning and function will have similar embeddings. And, since when we use RNNs for language modeling we make the assumption that the embedding dimension and 8.3 • RNN S FOR OTHER NLP TASKS 165 the hidden dimension are the same (= the model dimension d), the embedding ma- trix E has shape [d ×\|V \|]. And the ﬁnal layer matrix V provides a way to score the likelihood of each word in the vocabulary given the evidence present in the ﬁnal hidden layer of the network through the calculation of Vh. V is of shape [\|V \|×d]. That is, is, the rows of V are shaped like a transpose of E, meaning that V provides a second set of learned word embeddings. Instead of having two sets of embedding matrices, language models use a single embedding matrix, which appears at both the input and softmax layers. That is, we dispense with V and use E at the start of the computation and E⊺ (because the shape of V is the transpose of E at the end. Using the same matrix (transposed) in two places is called weight tying.1 The weight-tied equations for an RNN languageweight tying model then become: et = Ext (8.12) ht = g(Uht−1 +Wet ) (8.13) ˆ yt = softmax(E⊺ht ) (8.14) In addition to providing improved model perplexity, this approach signiﬁcantly re- duces the number of parameters required for the model. 8.3 RNNs for other NLP tasks Now that we’ve seen the basic RNN architecture, let’s consider how to apply it to three types of NLP tasks: sequence classiﬁcation tasks like sentiment analysis and topic classiﬁcation, sequence labeling tasks like part-of-speech tagging, and text generation tasks, including with a new architecture called the encoder-decoder. 8.3.1 Sequence Labeling In sequence labeling, the network’s task is to assign a label chosen from a small ﬁxed set of labels to each element of a sequence. One classic sequence labeling tasks is part-of-speech (POS) tagging (assigning grammatical tags like NOUN and VERB to each word in a sentence). We’ll discuss part-of-speech tagging in detail in Chapter 17, but let’s give a motivating example here. In an RNN approach to sequence labeling, inputs are word embeddings and the outputs are tag probabilities generated by a softmax layer over the given tagset, as illustrated in Fig. 8.7. In this ﬁgure, the inputs at each time step are pretrained word embeddings cor- responding to the input tokens. The RNN block is an abstraction that represents an unrolled simple recurrent network consisting of an input layer, hidden layer, and output layer at each time step, as well as the shared U, V and W weight matrices that comprise the network. The outputs of the network at each time step represent the distribution over the POS tagset generated by a softmax layer. To generate a sequence of tags for a given input, we run forward inference over the input sequence and select the most likely tag from the softmax at each step. Since we’re using a softmax layer to generate the probability distribution over the output tagset at each time step, we will again employ the cross-entropy loss during training. 1 We also do this for transformers (Chapter 9) where it’s common to callE⊺ the unembedding matrix. 166 CHAPTER 8 • RNN S AND LSTM S Janet will back the bill NNDTVBMDNNPArgmax Embeddings Words e h Vh y RNN Layer(s) Softmax over tags Figure 8.7 Part-of-speech tagging as sequence labeling with a simple RNN. The goal of part-of-speech (POS) tagging is to assign a grammatical label to each word in a sentence, drawn from a predeﬁned set of tags. (The tags for this sentence include NNP (proper noun), MD (modal verb) and others; we’ll give a complete description of the task of part-of-speech tagging in Chapter 17.) Pre-trained word embeddings serve as inputs and a softmax layer provides a probability distribution over the part-of-speech tags as output at each time step. 8.3.2 RNNs for Sequence Classiﬁcation Another use of RNNs is to classify entire sequences rather than the tokens within them. This is the set of tasks commonly called text classiﬁcation, like sentiment analysis or spam detection, in which we classify a text into two or three classes (like positive or negative), as well as classiﬁcation tasks with a large number of categories, like document-level topic classiﬁcation, or message routing for customer service applications. To apply RNNs in this setting, we pass the text to be classiﬁed through the RNN a word at a time generating a new hidden layer representation at each time step. We can then take the hidden layer for the last token of the text, hn, to constitute a compressed representation of the entire sequence. We can pass this representation hn to a feedforward network that chooses a class via a softmax over the possible classes. Fig. 8.8 illustrates this approach. Note that in this approach we don’t need intermediate outputs for the words in the sequence preceding the last element. Therefore, there are no loss terms associ- ated with those elements. Instead, the loss function used to train the weights in the network is based entirely on the ﬁnal text classiﬁcation task. The output from the softmax output from the feedforward classiﬁer together with a cross-entropy loss drives the training. The error signal from the classiﬁcation is backpropagated all the way through the weights in the feedforward classiﬁer through, to its input, and then through to the three sets of weights in the RNN as described earlier in Section 8.1.2. The training regimen that uses the loss from a downstream application to adjust the weights all the way through the network is referred to as end-to-end training.end-to-end training Another option, instead of using just hidden state of the last tokenhn to represent the whole sequence, is to use some sort of pooling function of all the hidden statespooling hi for each word i in the sequence. For example, we can create a representation that 8.3 • RNN S FOR OTHER NLP TASKS 167 x1 RNN hn x2 x3 xn Softmax FFN Figure 8.8 Sequence classiﬁcation using a simple RNN combined with a feedforward net- work. The ﬁnal hidden state from the RNN is used as the input to a feedforward network that performs the classiﬁcation. pools all the n hidden states by taking their element-wise mean: hmean = 1 n n∑ i=1 hi (8.15) Or we can take the element-wise max; the element-wise max of a set of n vectors is a new vector whose kth element is the max of the kth elements of all the n vectors. The long contexts of RNNs makes it quite difﬁcult to successfully backpropagate error all the way through the entire input; we’ll talk about this problem, and some standard solutions, in Section 8.5. 8.3.3 Generation with RNN-Based Language Models RNN-based language models can also be used to generate text. Text generation is of enormous practical importance, part of tasks like question answering, machine translation, text summarization, grammar correction, story generation, and conver- sational dialogue; any task where a system needs to produce text, conditioned on some other text. This use of a language model to generate text is one of the areas in which the impact of neural language models on NLP has been the largest. Text generation, along with image generation and code generation, constitute a new area of AI that is often called generative AI. Recall back in Chapter 3 we saw how to generate text from an n-gram language model by adapting asampling technique suggested at about the same time by Claude Shannon (Shannon, 1951) and the psychologists George Miller and Jennifer Self- ridge (Miller and Selfridge, 1950). We ﬁrst randomly sample a word to begin a sequence based on its suitability as the start of a sequence. We then continue to sample words conditioned on our previous choices until we reach a pre-determined length, or an end of sequence token is generated. Today, this approach of using a language model to incrementally generate words by repeatedly sampling the next word conditioned on our previous choices is called autoregressive generation or causal LM generation . The procedure is basicallyautoregressive generation the same as that described on page 43, but adapted to a neural context: • Sample a word in the output from the softmax distribution that results from using the beginning of sentence marker, <s>, as the ﬁrst input. 168 CHAPTER 8 • RNN S AND LSTM S • Use the word embedding for that ﬁrst word as the input to the network at the next time step, and then sample the next word in the same fashion. • Continue generating until the end of sentence marker, </s>, is sampled or a ﬁxed length limit is reached. Technically an autoregressive model is a model that predicts a value at timet based on a linear function of the previous values at times t −1, t −2, and so on. Although language models are not linear (since they have many layers of non-linearities), we loosely refer to this generation technique as autoregressive generation since the word generated at each time step is conditioned on the word selected by the network from the previous step. Fig. 8.9 illustrates this approach. In this ﬁgure, the details of the RNN’s hidden layers and recurrent connections are hidden within the blue block. This simple architecture underlies state-of-the-art approaches to applications such as machine translation, summarization, and question answering. The key to these approaches is to prime the generation component with an appropriate context. That is, instead of simply using <s> to get things started we can provide a richer task-appropriate context; for translation the context is the sentence in the source language; for summarization it’s the long text we want to summarize. So long <s> and So long and ?Sampled Word Softmax Embedding Input Word RNN Figure 8.9 Autoregressive generation with an RNN-based neural language model. 8.4 Stacked and Bidirectional RNN architectures Recurrent networks are quite ﬂexible. By combining the feedforward nature of un- rolled computational graphs with vectors as common inputs and outputs, complex networks can be treated as modules that can be combined in creative ways. This section introduces two of the more common network architectures used in language processing with RNNs. 8.4.1 Stacked RNNs In our examples thus far, the inputs to our RNNs have consisted of sequences of word or character embeddings (vectors) and the outputs have been vectors useful for predicting words, tags or sequence labels. However, nothing prevents us from using 8.4 • S TACKED AND BIDIRECTIONAL RNN ARCHITECTURES 169 the entire sequence of outputs from one RNN as an input sequence to another one. Stacked RNNs consist of multiple networks where the output of one layer serves asStacked RNNs the input to a subsequent layer, as shown in Fig. 8.10. y1 y2 y3 yn x1 x2 x3 xn RNN 1 RNN 2 RNN 3 Figure 8.10 Stacked recurrent networks. The output of a lower level serves as the input to higher levels with the output of the last network serving as the ﬁnal output. Stacked RNNs generally outperform single-layer networks. One reason for this success seems to be that the network induces representations at differing levels of abstraction across layers. Just as the early stages of the human visual system detect edges that are then used for ﬁnding larger regions and shapes, the initial layers of stacked networks can induce representations that serve as useful abstractions for further layers—representations that might prove difﬁcult to induce in a single RNN. The optimal number of stacked RNNs is speciﬁc to each application and to each training set. However, as the number of stacks is increased the training costs rise quickly. 8.4.2 Bidirectional RNNs The RNN uses information from the left (prior) context to make its predictions at time t. But in many applications we have access to the entire input sequence; in those cases we would like to use words from the context to the right of t. One way to do this is to run two separate RNNs, one left-to-right, and one right-to-left, and concatenate their representations. In the left-to-right RNNs we’ve discussed so far, the hidden state at a given time t represents everything the network knows about the sequence up to that point. The state is a function of the inputs x1,...,xt and represents the context of the network to the left of the current time. hf t = RNNforward(x1,..., xt ) (8.16) This new notation hf t simply corresponds to the normal hidden state at timet, repre- senting everything the network has gleaned from the sequence so far. To take advantage of context to the right of the current input, we can train an RNN on a reversed input sequence. With this approach, the hidden state at time t represents information about the sequence to the right of the current input: hb t = RNNbackward(xt ,... xn) (8.17) 170 CHAPTER 8 • RNN S AND LSTM S Here, the hidden state hb t represents all the information we have discerned about the sequence from t to the end of the sequence. A bidirectional RNN (Schuster and Paliwal, 1997) combines two independentbidirectional RNN RNNs, one where the input is processed from the start to the end, and the other from the end to the start. We then concatenate the two representations computed by the networks into a single vector that captures both the left and right contexts of an input at each point in time. Here we use either the semicolon ”;” or the equivalent symbol ⊕to mean vector concatenation: ht = [hf t ; hb t ] = hf t ⊕hb t (8.18) Fig. 8.11 illustrates such a bidirectional network that concatenates the outputs of the forward and backward pass. Other simple ways to combine the forward and backward contexts include element-wise addition or multiplication. The output at each step in time thus captures information to the left and to the right of the current input. In sequence labeling applications, these concatenated outputs can serve as the basis for a local labeling decision. RNN 2 RNN 1 x1 y2y1 y3 yn concatenated outputs x2 x3 xn Figure 8.11 A bidirectional RNN. Separate models are trained in the forward and backward directions, with the output of each model at each time point concatenated to represent the bidirectional state at that time point. Bidirectional RNNs have also proven to be quite effective for sequence classiﬁ- cation. Recall from Fig. 8.8 that for sequence classiﬁcation we used the ﬁnal hidden state of the RNN as the input to a subsequent feedforward classiﬁer. A difﬁculty with this approach is that the ﬁnal state naturally reﬂects more information about the end of the sentence than its beginning. Bidirectional RNNs provide a simple solution to this problem; as shown in Fig. 8.12, we simply combine the ﬁnal hidden states from the forward and backward passes (for example by concatenation) and use that as input for follow-on processing. 8.5 • T HE LSTM 171 RNN 2 RNN 1 x1 x2 x3 xn hn → h1 ← hn → Softmax FFN h1 ← Figure 8.12 A bidirectional RNN for sequence classiﬁcation. The ﬁnal hidden units from the forward and backward passes are combined to represent the entire sequence. This com- bined representation serves as input to the subsequent classiﬁer. 8.5 The LSTM In practice, it is quite difﬁcult to train RNNs for tasks that require a network to make use of information distant from the current point of processing. Despite having ac- cess to the entire preceding sequence, the information encoded in hidden states tends to be fairly local, more relevant to the most recent parts of the input sequence and recent decisions. Yet distant information is critical to many language applications. Consider the following example in the context of language modeling. (8.19) The ﬂights the airline was canceling were full. Assigning a high probability towas following airline is straightforward since airline provides a strong local context for the singular agreement. However, assigning an appropriate probability to were is quite difﬁcult, not only because the plural ﬂights is quite distant, but also because the singular nounairline is closer in the intervening context. Ideally, a network should be able to retain the distant information about plural ﬂights until it is needed, while still processing the intermediate parts of the sequence correctly. One reason for the inability of RNNs to carry forward critical information is that the hidden layers, and, by extension, the weights that determine the values in the hid- den layer, are being asked to perform two tasks simultaneously: provide information useful for the current decision, and updating and carrying forward information re- quired for future decisions. A second difﬁculty with training RNNs arises from the need to backpropagate the error signal back through time. Recall from Section 8.1.2 that the hidden layer at time t contributes to the loss at the next time step since it takes part in that calcula- tion. As a result, during the backward pass of training, the hidden layers are subject to repeated multiplications, as determined by the length of the sequence. A frequent result of this process is that the gradients are eventually driven to zero, a situation 172 CHAPTER 8 • RNN S AND LSTM S called the vanishing gradients problem.vanishing gradients To address these issues, more complex network architectures have been designed to explicitly manage the task of maintaining relevant context over time, by enabling the network to learn to forget information that is no longer needed and to remember information required for decisions still to come. The most commonly used such extension to RNNs is thelong short-term mem- ory (LSTM) network (Hochreiter and Schmidhuber, 1997). LSTMs divide the con-long short-term memory text management problem into two subproblems: removing information no longer needed from the context, and adding information likely to be needed for later de- cision making. The key to solving both problems is to learn how to manage this context rather than hard-coding a strategy into the architecture. LSTMs accomplish this by ﬁrst adding an explicit context layer to the architecture (in addition to the usual recurrent hidden layer), and through the use of specialized neural units that make use of gates to control the ﬂow of information into and out of the units that comprise the network layers. These gates are implemented through the use of addi- tional weights that operate sequentially on the input, and previous hidden layer, and previous context layers. The gates in an LSTM share a common design pattern; each consists of a feed- forward layer, followed by a sigmoid activation function, followed by a pointwise multiplication with the layer being gated. The choice of the sigmoid as the activation function arises from its tendency to push its outputs to either 0 or 1. Combining this with a pointwise multiplication has an effect similar to that of a binary mask. Values in the layer being gated that align with values near 1 in the mask are passed through nearly unchanged; values corresponding to lower values are essentially erased. The ﬁrst gate we’ll consider is the forget gate. The purpose of this gate isforget gate to delete information from the context that is no longer needed. The forget gate computes a weighted sum of the previous state’s hidden layer and the current in- put and passes that through a sigmoid. This mask is then multiplied element-wise by the context vector to remove the information from context that is no longer re- quired. Element-wise multiplication of two vectors (represented by the operator ⊙, and sometimes called the Hadamard product) is the vector of the same dimension as the two input vectors, where each element i is the product of element i in the two input vectors: ft = σ(Uf ht−1 +Wf xt ) (8.20) kt = ct−1 ⊙ft (8.21) The next task is to compute the actual information we need to extract from the previ- ous hidden state and current inputs—the same basic computation we’ve been using for all our recurrent networks. gt = tanh(Ught−1 +Wgxt ) (8.22) Next, we generate the mask for the add gate to select the information to add to theadd gate current context. it = σ(Uiht−1 +Wixt ) (8.23) jt = gt ⊙it (8.24) Next, we add this to the modiﬁed context vector to get our new context vector. ct = jt +kt (8.25) 8.5 • T HE LSTM 173 + xt ht-1 ct ht ct ht ct-1 ht-1 xt tanh + σ tanh σ σ +++ i g f o 㽋㽋㽋 LSTM ct-1 Figure 8.13 A single LSTM unit displayed as a computation graph. The inputs to each unit consists of the current input, x, the previous hidden state, ht−1, and the previous context, ct−1. The outputs are a new hidden state, ht and an updated context, ct . The ﬁnal gate we’ll use is theoutput gate which is used to decide what informa-output gate tion is required for the current hidden state (as opposed to what information needs to be preserved for future decisions). ot = σ(Uoht−1 +Woxt ) (8.26) ht = ot ⊙tanh(ct ) (8.27) Fig. 8.13 illustrates the complete computation for a single LSTM unit. Given the appropriate weights for the various gates, an LSTM accepts as input the context layer, and hidden layer from the previous time step, along with the current input vector. It then generates updated context and hidden vectors as output. It is the hidden state, ht , that provides the output for the LSTM at each time step. This output can be used as the input to subsequent layers in a stacked RNN, or at the ﬁnal layer of a network ht can be used to provide the ﬁnal output of the LSTM. 8.5.1 Gated Units, Layers and Networks The neural units used in LSTMs are obviously much more complex than those used in basic feedforward networks. Fortunately, this complexity is encapsulated within the basic processing units, allowing us to maintain modularity and to easily exper- iment with different architectures. To see this, consider Fig. 8.14 which illustrates the inputs and outputs associated with each kind of unit. At the far left, (a) is the basic feedforward unit where a single set of weights and a single activation function determine its output, and when arranged in a layer there are no connections among the units in the layer. Next, (b) represents the unit in a simple recurrent network. Now there are two inputs and an additional set of weights to go with it. However, there is still a single activation function and output. The increased complexity of the LSTM units is encapsulated within the unit itself. The only additional external complexity for the LSTM over the basic recurrent unit (b) is the presence of the additional context vector as an input and output. This modularity is key to the power and widespread applicability of LSTM units. LSTM units (or other varieties, like GRUs) can be substituted into any of the network architectures described in Section 8.4. And, as with simple RNNs, multi-layered networks making use of gated units can be unrolled into deep feedforward networks 174 CHAPTER 8 • RNN S AND LSTM S h x xt xtht-1 ht ht ct-1 ct ht-1 (b)(a) (c) ⌃ g z a ⌃ g z LSTM Unit a Figure 8.14 Basic neural units used in feedforward, simple recurrent networks (SRN), and long short-term memory (LSTM). and trained in the usual fashion with backpropagation. In practice, therefore, LSTMs rather than RNNs have become the standard unit for any modern system that makes use of recurrent networks. 8.6 Summary: Common RNN NLP Architectures We’ve now introduced the RNN, seen advanced components like stacking multiple layers and using the LSTM version, and seen how the RNN can be applied to various tasks. Let’s take a moment to summarize the architectures for these applications. Fig. 8.15 shows the three architectures we’ve discussed so far: sequence la- beling, sequence classiﬁcation, and language modeling. In sequence labeling (for example for part of speech tagging), we train a model to produce a label for each input word or token. In sequence classiﬁcation, for example for sentiment analysis, we ignore the output for each token, and only take the value from the end of the sequence (and similarly the model’s training signal comes from backpropagation from that last token). In language modeling, we train the model to predict the next word at each token step. In the next section we’ll introduce a fourth architecture, the encoder-decoder. 8.7 The Encoder-Decoder Model with RNNs In this section we introduce a new model, the encoder-decoder model, which is used when we are taking an input sequence and translating it to an output sequence that is of a different length than the input, and doesn’t align with it in a word-to-word way. Recall that in the sequence labeling task, we have two sequences, but they are the same length (for example in part-of-speech tagging each token gets an associated tag), each input is associated with a speciﬁc output, and the labeling for that output takes mostly local information. Thus deciding whether a word is a verb or a noun, we look mostly at the word and the neighboring words. By contrast, encoder-decoder models are used especially for tasks like machine translation, where the input sequence and output sequence can have different lengths 8.7 • T HE ENCODER -DECODER MODEL WITH RNN S 175 … Encoder RNN Decoder RNN Context … x1 x2 xn y1 y2 ym … RNN x1 x2 xn …y1 y2 yn … RNN x1 x2 xn y … RNN x1 x2 xt-1 …x2 x3 xt a) sequence labeling b) sequence classification c) language modeling d) encoder-decoder Figure 8.15 Four architectures for NLP tasks. In sequence labeling (POS or named entity tagging) we map each input token xi to an output token yi. In sequence classiﬁcation we map the entire input sequence to a single class. In language modeling we output the next token conditioned on previous tokens. In the encoder model we have two separate RNN models, one of which maps from an input sequence x to an intermediate representation we call the context, and a second of which maps from the context to an output sequence y. and the mapping between a token in the input and a token in the output can be very indirect (in some languages the verb appears at the beginning of the sentence; in other languages at the end). We’ll introduce machine translation in detail in Chap- ter 13, but for now we’ll just point out that the mapping for a sentence in English to a sentence in Tagalog or Yoruba can have very different numbers of words, and the words can be in a very different order. Encoder-decoder networks, sometimes calledsequence-to-sequence networks,encoder- decoder are models capable of generating contextually appropriate, arbitrary length, output sequences given an input sequence. Encoder-decoder networks have been applied to a very wide range of applications including summarization, question answering, and dialogue, but they are particularly popular for machine translation. The key idea underlying these networks is the use of an encoder network that takes an input sequence and creates a contextualized representation of it, often called the context. This representation is then passed to a decoder which generates a task- speciﬁc output sequence. Fig. 8.16 illustrates the architecture. Encoder-decoder networks consist of three conceptual components: 1. An encoder that accepts an input sequence, x1:n, and generates a correspond- ing sequence of contextualized representations, h1:n. LSTMs, convolutional networks, and transformers can all be employed as encoders. 2. A context vector, c, which is a function of h1:n, and conveys the essence of the input to the decoder. 3. A decoder, which accepts c as input and generates an arbitrary length se- quence of hidden states h1:m, from which a corresponding sequence of output states y1:m, can be obtained. Just as with encoders, decoders can be realized 176 CHAPTER 8 • RNN S AND LSTM S … Encoder Decoder Context … x1 x2 xn y1 y2 ym Figure 8.16 The encoder-decoder architecture. The context is a function of the hidden representations of the input, and may be used by the decoder in a variety of ways. by any kind of sequence architecture. In this section we’ll describe an encoder-decoder network based on a pair of RNNs, but we’ll see in Chapter 13 how to apply them to transformers as well. We’ll build up the equations for encoder-decoder models by starting with the conditional RNN language model p(y), the probability of a sequence y. Recall that in any language model, we can break down the probability as follows: p(y) = p(y1)p(y2\|y1)p(y3\|y1,y2)... p(ym\|y1,...,ym−1) (8.28) In RNN language modeling, at a particular time t, we pass the preﬁx of t −1 tokens through the language model, using forward inference to produce a sequence of hidden states, ending with the hidden state corresponding to the last word of the preﬁx. We then use the ﬁnal hidden state of the preﬁx as our starting point to generate the next token. More formally, if g is an activation function like tanh or ReLU, a function of the input at time t and the hidden state at time t −1, and the softmax is over the set of possible vocabulary items, then at time t the output yt and hidden state ht are computed as: ht = g(ht−1,xt ) (8.29) ˆ yt = softmax(ht ) (8.30) We only have to make one slight change to turn this language model with au- toregressive generation into an encoder-decoder model that is a translation model that can translate from a source text in one language to a target text in a second: add a sentence separation marker at the end of the source text, and then simplysentence separation concatenate the target text. Let’s use<s> for our sentence separator token, and let’s think about translating an English source text (“the green witch arrived”), to a Spanish sentence (“ lleg´o la bruja verde” (which can be glossed word-by-word as ‘arrived the witch green’). We could also illustrate encoder-decoder models with a question-answer pair, or a text-summarization pair. Let’s usex to refer to the source text (in this case in English) plus the separator token <s>, and y to refer to the target text y (in this case in Spanish). Then an encoder-decoder model computes the probability p(y\|x) as follows: p(y\|x) = p(y1\|x)p(y2\|y1,x)p(y3\|y1,y2,x)... p(ym\|y1,...,ym−1,x) (8.31) Fig. 8.17 shows the setup for a simpliﬁed version of the encoder-decoder model (we’ll see the full model, which requires the new concept of attention, in the next section). 8.7 • T HE ENCODER -DECODER MODEL WITH RNN S 177 Source Text Target Text hn embedding layer hidden layer(s) softmax the green llegó witch arrived <s> llegó la la bruja bruja verde verde </s> (output of source is ignored) Separator Figure 8.17 Translating a single sentence (inference time) in the basic RNN version of encoder-decoder ap- proach to machine translation. Source and target sentences are concatenated with a separator token in between, and the decoder uses context information from the encoder’s last hidden state. Fig. 8.17 shows an English source text (“the green witch arrived”), a sentence separator token ( <s>, and a Spanish target text (“ lleg´o la bruja verde ”). To trans- late a source text, we run it through the network performing forward inference to generate hidden states until we get to the end of the source. Then we begin autore- gressive generation, asking for a word in the context of the hidden layer from the end of the source input as well as the end-of-sentence marker. Subsequent words are conditioned on the previous hidden state and the embedding for the last word generated. Let’s formalize and generalize this model a bit in Fig. 8.18. (To help keep things straight, we’ll use the superscripts e and d where needed to distinguish the hidden states of the encoder and the decoder.) The elements of the network on the left process the input sequence x and comprise the encoder. While our simpliﬁed ﬁgure shows only a single network layer for the encoder, stacked architectures are the norm, where the output states from the top layer of the stack are taken as the ﬁnal representation, and the encoder consists of stacked biLSTMs where the hidden states from top layers from the forward and backward passes are concatenated to provide the contextualized representations for each time step. The entire purpose of the encoder is to generate a contextualized representation of the input. This representation is embodied in the ﬁnal hidden state of the encoder, he n. This representation, also called c for context, is then passed to the decoder. The simplest version of the decoder network would take this state and use it just to initialize the ﬁrst hidden state of the decoder; the ﬁrst decoder RNN cell would use c as its prior hidden state hd 0. The decoder would then autoregressively generates a sequence of outputs, an element at a time, until an end-of-sequence marker is generated. Each hidden state is conditioned on the previous hidden state and the output generated in the previous state. As Fig. 8.18 shows, we do something more complex: we make the context vector c available to more than just the ﬁrst decoder hidden state, to ensure that the inﬂuence of the context vector, c, doesn’t wane as the output sequence is generated. We do this by adding c as a parameter to the computation of the current hidden state. using the following equation: hd t = g(ˆyt−1,hd t−1,c) (8.32) 178 CHAPTER 8 • RNN S AND LSTM S Encoder Decoder hn hd 1 he 3he 2he 1 hd 2 hd 3 hd 4 embedding layer hidden layer(s) softmax x1 x2 y1 hd m x3 xn <s> y1 y2 y2 y3 y3 y4 ym </s> he n = c = hd 0 (output is ignored during encoding) Figure 8.18 A more formal version of translating a sentence at inference time in the basic RNN-based encoder-decoder architecture. The ﬁnal hidden state of the encoder RNN, hen, serves as the context for the decoder in its role as hd 0 in the decoder RNN, and is also made available to each decoder hidden state. Now we’re ready to see the full equations for this version of the decoder in the basic encoder-decoder model, with context available at each decoding timestep. Recall that g is a stand-in for some ﬂavor of RNN and ˆyt−1 is the embedding for the output sampled from the softmax at the previous step: c = he n hd 0 = c hd t = g(ˆyt−1,hd t−1,c) ˆ yt = softmax(hd t ) (8.33) Thus ˆ yt is a vector of probabilities over the vocabulary, representing the probability of each word occurring at time t. To generate text, we sample from this distribution ˆ yt . For example, the greedy choice is simply to choose the most probable word to generate at each timestep. We’ll introduce more sophisticated sampling methods in Section 10.2. 8.7.1 Training the Encoder-Decoder Model Encoder-decoder architectures are trained end-to-end. Each training example is a tuple of paired strings, a source and a target. Concatenated with a separator token, these source-target pairs can now serve as training data. For MT, the training data typically consists of sets of sentences and their transla- tions. These can be drawn from standard datasets of aligned sentence pairs, as we’ll discuss in Section 13.2.2. Once we have a training set, the training itself proceeds as with any RNN-based language model. The network is given the source text and then starting with the separator token is trained autoregressively to predict the next word, as shown in Fig. 8.19. Note the differences between training (Fig. 8.19) and inference (Fig. 8.17) with respect to the outputs at each time step. The decoder during inference uses its own estimated output ˆyt as the input for the next time step xt+1. Thus the decoder will tend to deviate more and more from the gold target sentence as it keeps generating more tokens. In training, therefore, it is more common to use teacher forcing in theteacher forcing decoder. Teacher forcing means that we force the system to use the gold target token 8.8 • A TTENTION 179 Encoder Decoder embedding layer hidden layer(s) softmax the green llegó witch arrived <s> llegó la la bruja bruja verde verde </s> gold answers L1 = -log P(y1) x1 x2 x3 x4 L2 = -log P(y2) L3 = -log P(y3) L4 = -log P(y4) L5 = -log P(y5) per-word loss y1 y2 y3 y4 y5 Total loss is the average cross-entropy loss per target word: Figure 8.19 Training the basic RNN encoder-decoder approach to machine translation. Note that in the decoder we usually don’t propagate the model’s softmax outputs ˆyt , but use teacher forcing to force each input to the correct gold value for training. We compute the softmax output distribution over ˆy in the decoder in order to compute the loss at each token, which can then be averaged to compute a loss for the sentence. This loss is then propagated through the decoder parameters and the encoder parameters. from training as the next input xt+1, rather than allowing it to rely on the (possibly erroneous) decoder output ˆyt . This speeds up training. 8.8 Attention The simplicity of the encoder-decoder model is its clean separation of the encoder— which builds a representation of the source text—from the decoder, which uses this context to generate a target text. In the model as we’ve described it so far, this context vector is hn, the hidden state of the last ( nth) time step of the source text. This ﬁnal hidden state is thus acting as a bottleneck: it must represent absolutely everything about the meaning of the source text, since the only thing the decoder knows about the source text is what’s in this context vector (Fig. 8.20). Information at the beginning of the sentence, especially for long sentences, may not be equally well represented in the context vector. Encoder Decoderbottleneck bottleneck Figure 8.20 Requiring the contextc to be only the encoder’s ﬁnal hidden state forces all the information from the entire source sentence to pass through this representational bottleneck. The attention mechanism is a solution to the bottleneck problem, a way ofattention mechanism allowing the decoder to get information from all the hidden states of the encoder, not just the last hidden state. 180 CHAPTER 8 • RNN S AND LSTM S In the attention mechanism, as in the vanilla encoder-decoder model, the context vector c is a single vector that is a function of the hidden states of the encoder. But instead of being taken from the last hidden state, it’s a weighted average of all the hidden states of the decoder. And this weighted average is also informed by part of the decoder state as well, the state of the decoder right before the current token i. That is, c = f (he 1 ... he n,hd i−1). The weights focus on (‘attend to’) a particular part of the source text that is relevant for the tokeni that the decoder is currently producing. Attention thus replaces the static context vector with one that is dynamically derived from the encoder hidden states, but also informed by and hence different for each token in decoding. This context vector, ci, is generated anew with each decoding step i and takes all of the encoder hidden states into account in its derivation. We then make this context available during decoding by conditioning the computation of the current decoder hidden state on it (along with the prior hidden state and the previous output generated by the decoder), as we see in this equation (and Fig. 8.21): hd i = g(ˆyi−1,hd i−1,ci) (8.34) hd 1 hd 2 hd i y1 y2 yi c1 c2 ci … … Figure 8.21 The attention mechanism allows each hidden state of the decoder to see a different, dynamic, context, which is a function of all the encoder hidden states. The ﬁrst step in computing ci is to compute how much to focus on each encoder state, how relevant each encoder state is to the decoder state captured in hd i−1. We capture relevance by computing— at each state i during decoding—a score(hd i−1,he j) for each encoder state j. The simplest such score, calleddot-product attention, implements relevance asdot-product attention similarity: measuring how similar the decoder hidden state is to an encoder hidden state, by computing the dot product between them: score(hd i−1,he j) = hd i−1 ·he j (8.35) The score that results from this dot product is a scalar that reﬂects the degree of similarity between the two vectors. The vector of these scores across all the encoder hidden states gives us the relevance of each encoder state to the current step of the decoder. To make use of these scores, we’ll normalize them with a softmax to create a vector of weights,αi j, that tells us the proportional relevance of each encoder hidden state j to the prior hidden decoder state, hd i−1. αi j = softmax(score(hd i−1,he j)) = exp(score(hd i−1,he j) ∑ k exp(score(hd i−1,he k)) (8.36) Finally, given the distribution inα, we can compute a ﬁxed-length context vector for the current decoder state by taking a weighted average over all the encoder hidden 8.9 • S UMMARY 181 states. ci = ∑ j αi j he j (8.37) With this, we ﬁnally have a ﬁxed-length context vector that takes into account information from the entire encoder state that is dynamically updated to reﬂect the needs of the decoder at each step of decoding. Fig. 8.22 illustrates an encoder- decoder network with attention, focusing on the computation of one context vector ci. Encoder Decoder hd i-1he 3he 2he 1 hd ihidden layer(s) x1 x2 yi-1 x3 xn yi-2 yi-1 yi he n ci .2.1.3.4attention weights ci-1 ci <latexit sha1_base64="TNdNmv/RIlrhPa6LgQyjjQLqyBA=">AAACAnicdVDLSsNAFJ3UV62vqCtxM1gEVyHpI9Vd0Y3LCvYBTQyT6bSddvJgZiKUUNz4K25cKOLWr3Dn3zhpK6jogQuHc+7l3nv8mFEhTfNDyy0tr6yu5dcLG5tb2zv67l5LRAnHpIkjFvGOjwRhNCRNSSUjnZgTFPiMtP3xRea3bwkXNAqv5SQmboAGIe1TjKSSPP3AEUngjVIHsXiIvJSOpnB4Q7zR1NOLpmGaVbtqQdOwLbtk24qY5Yp9VoOWsjIUwQINT393ehFOAhJKzJAQXcuMpZsiLilmZFpwEkFihMdoQLqKhiggwk1nL0zhsVJ6sB9xVaGEM/X7RIoCISaBrzoDJIfit5eJf3ndRPZP3ZSGcSJJiOeL+gmDMoJZHrBHOcGSTRRBmFN1K8RDxBGWKrWCCuHrU/g/aZUMyzbKV5Vi/XwRRx4cgiNwAixQA3VwCRqgCTC4Aw/gCTxr99qj9qK9zltz2mJmH/yA9vYJSymYCA==</latexit> X j ↵ ij h e j ↵ ij <latexit sha1_base64="y8s4mGdpwrGrBnuSR+p1gJJXYdo=">AAAB/nicdVDJSgNBEO2JW4zbqHjy0hgEL4YeJyQBL0EvHiOYBbIMPT09mTY9C909QhgC/ooXD4p49Tu8+Td2FkFFHxQ83quiqp6bcCYVQh9Gbml5ZXUtv17Y2Nza3jF391oyTgWhTRLzWHRcLClnEW0qpjjtJILi0OW07Y4up377jgrJ4uhGjRPaD/EwYj4jWGnJMQ+Cgedk7NSa9IgXq955MKDOrWMWUQnNAFGpYtfsakUTZNtWGUFrYRXBAg3HfO95MUlDGinCsZRdCyWqn2GhGOF0UuilkiaYjPCQdjWNcEhlP5udP4HHWvGgHwtdkYIz9ftEhkMpx6GrO0OsAvnbm4p/ed1U+bV+xqIkVTQi80V+yqGK4TQL6DFBieJjTTARTN8KSYAFJkonVtAhfH0K/yets5JVKdnX5WL9YhFHHhyCI3ACLFAFdXAFGqAJCMjAA3gCz8a98Wi8GK/z1pyxmNkHP2C8fQICDpWK</latexit> h d i 1 · h e j …… Figure 8.22 A sketch of the encoder-decoder network with attention, focusing on the computation of ci. The context value ci is one of the inputs to the computation of hd i . It is computed by taking the weighted sum of all the encoder hidden states, each weighted by their dot product with the prior decoder hidden state hd i−1. It’s also possible to create more sophisticated scoring functions for attention models. Instead of simple dot product attention, we can get a more powerful function that computes the relevance of each encoder hidden state to the decoder hidden state by parameterizing the score with its own set of weights, Ws. score(hd i−1,he j) = hd t−1Wshe j The weights Ws, which are then trained during normal end-to-end training, give the network the ability to learn which aspects of similarity between the decoder and encoder states are important to the current application. This bilinear model also allows the encoder and decoder to use different dimensional vectors, whereas the simple dot-product attention requires that the encoder and decoder hidden states have the same dimensionality. We’ll return to the concept of attention when we deﬁne the transformer archi- tecture in Chapter 9, which is based on a slight modiﬁcation of attention called self-attention. 8.9 Summary This chapter has introduced the concepts of recurrent neural networks and how they can be applied to language problems. Here’s a summary of the main points that we 182 CHAPTER 8 • RNN S AND LSTM S covered: • In simple Recurrent Neural Networks sequences are processed one element at a time, with the output of each neural unit at time t based both on the current input at t and the hidden layer from time t −1. • RNNs can be trained with a straightforward extension of the backpropagation algorithm, known as backpropagation through time (BPTT). • Simple recurrent networks fail on long inputs because of problems like van- ishing gradients; instead modern systems use more complex gated architec- tures such as LSTMs that explicitly decide what to remember and forget in their hidden and context layers. • Common language-based applications for RNNs include: – Probabilistic language modeling: assigning a probability to a sequence, or to the next element of a sequence given the preceding words. – Auto-regressive generation using a trained language model. – Sequence labeling like part-of-speech tagging, where each element of a sequence is assigned a label. – Sequence classiﬁcation, where an entire text is assigned to a category, as in spam detection, sentiment analysis or topic classiﬁcation. – Encoder-decoder architectures, where an input is mapped to an output of different length and alignment. Bibliographical and Historical Notes Inﬂuential investigations of RNNs were conducted in the context of the Parallel Dis- tributed Processing (PDP) group at UC San Diego in the 1980’s. Much of this work was directed at human cognitive modeling rather than practical NLP applications (Rumelhart and McClelland 1986c, McClelland and Rumelhart 1986). Models using recurrence at the hidden layer in a feedforward network (Elman networks) were in- troduced by Elman (1990). Similar architectures were investigated by Jordan (1986) with a recurrence from the output layer, and Mathis and Mozer (1995) with the addition of a recurrent context layer prior to the hidden layer. The possibility of unrolling a recurrent network into an equivalent feedforward network is discussed in (Rumelhart and McClelland, 1986c). In parallel with work in cognitive modeling, RNNs were investigated extensively in the continuous domain in the signal processing and speech communities (Giles et al. 1994, Robinson et al. 1996). Schuster and Paliwal (1997) introduced bidirec- tional RNNs and described results on the TIMIT phoneme transcription task. While theoretically interesting, the difﬁculty with training RNNs and manag- ing context over long sequences impeded progress on practical applications. This situation changed with the introduction of LSTMs in Hochreiter and Schmidhuber (1997) and Gers et al. (2000). Impressive performance gains were demonstrated on tasks at the boundary of signal processing and language processing including phoneme recognition (Graves and Schmidhuber, 2005), handwriting recognition (Graves et al., 2007) and most signiﬁcantly speech recognition (Graves et al., 2013). Interest in applying neural networks to practical NLP problems surged with the work of Collobert and Weston (2008) and Collobert et al. (2011). These efforts made use of learned word embeddings, convolutional networks, and end-to-end training. BIBLIOGRAPHICAL AND HISTORICAL NOTES 183 They demonstrated near state-of-the-art performance on a number of standard shared tasks including part-of-speech tagging, chunking, named entity recognition and se- mantic role labeling without the use of hand-engineered features. Approaches that married LSTMs with pretrained collections of word-embeddings based on word2vec (Mikolov et al., 2013a) and GloVe (Pennington et al., 2014) quickly came to dominate many common tasks: part-of-speech tagging (Ling et al., 2015), syntactic chunking (Søgaard and Goldberg, 2016), named entity recognition (Chiu and Nichols, 2016; Ma and Hovy, 2016), opinion mining (Irsoy and Cardie, 2014), semantic role labeling (Zhou and Xu, 2015a) and AMR parsing (Foland and Martin, 2016). As with the earlier surge of progress involving statistical machine learning, these advances were made possible by the availability of training data pro- vided by CONLL, SemEval, and other shared tasks, as well as shared resources such as Ontonotes (Pradhan et al., 2007b), and PropBank (Palmer et al., 2005). The modern neural encoder-decoder approach was pioneered by Kalchbrenner and Blunsom (2013), who used a CNN encoder and an RNN decoder. Cho et al. (2014) (who coined the name “encoder-decoder”) and Sutskever et al. (2014) then showed how to use extended RNNs for both encoder and decoder. The idea that a generative decoder should take as input a soft weighting of the inputs, the central idea of attention, was ﬁrst developed by Graves (2013) in the context of handwriting recognition. Bahdanau et al. (2015) extended the idea, named it “attention” and applied it to MT. 184 CHAPTER 9 • T HE TRANSFORMER CHAPTER 9 The Transformer “The true art of memory is the art of attention ” Samuel Johnson, Idler #74, September 1759 In this chapter we introduce thetransformer, the standard architecture for build- ing large language models. Transformer-based large language models have com- pletely changed the ﬁeld of speech and language processing. Indeed, every subse- quent chapter in this textbook will make use of them. We’ll focus for now on left- to-right (sometimes called causal or autoregressive) language modeling, in which we are given a sequence of input tokens and predict output tokens one by one by conditioning on the prior context. The transformer is a neural network with a speciﬁc structure that includes a mechanism called self-attention or multi-head attention.1 Attention can be thought of as a way to build contextual representations of a token’s meaning byattending to and integrating information from surrounding tokens, helping the model learn how tokens relate to each other over large spans. Stacked Transformer Blocks So long and thanks for long and thanks forNext token all … … … U Input tokens x1 x2 Language Modeling Head x3 x4 x5 Input Encoding E 1+ E 2+ E 3+ E 4+ E 5+ … … ……… U U U U … logits logits logits logits logits Figure 9.1 The architecture of a (left-to-right) transformer, showing how each input token get encoded, passed through a set of stacked transformer blocks, and then a language model head that predicts the next token. Fig. 9.1 sketches the transformer architecture. A transformer has three major components. At the center are columns of transformer blocks. Each block is a multilayer network (a multi-head attention layer, feedforward networks and layer normalization steps) that maps an input vectorxi in column i (corresponding to input 1 Although multi-head attention developed historically from theRNN attention mechanism (Chapter 8), we’ll deﬁne attention from scratch here for readers who haven’t yet read Chapter 8. 9.1 • A TTENTION 185 token i) to an output vector hi. The set of n blocks maps an entire context window of input vectors (x1,...,xn) to a window of output vectors (h1,...,hn) of the same length. A column might contain from 12 to 96 or more stacked blocks. The column of blocks is preceded by theinput encoding component, which pro- cesses an input token (like the wordthanks) into a contextual vector representation, using an embedding matrix E and a mechanism for encoding token position. Each column is followed by a language modeling head, which takes the embedding out- put by the ﬁnal transformer block, passes it through anunembedding matrix U and a softmax over the vocabulary to generate a single token for that column. Transformer-based language models are complex, and so the details will unfold over the next 5 chapters. In the next sections we’ll introduce multi-head attention, the rest of the transformer block, and the input encoding and language modeling head components. Chapter 10 discusses how language models are pretrained, and how tokens are generated via sampling. Chapter 11 introduces masked language modeling and the BERT family of bidirectional transformer encoder models. Chap- ter 12 shows how toprompt LLMs to perform NLP tasks by giving instructions and demonstrations, and how to align the model with human preferences. Chapter 13 will introduce machine translation with the encoder-decoder architecture. 9.1 Attention Recall from Chapter 6 that for word2vec and other static embeddings, the repre- sentation of a word’s meaning is always the same vector irrespective of the context: the word chicken, for example, is always represented by the same ﬁxed vector. So a static vector for the word it might somehow encode that this is a pronoun used for animals and inanimate entities. But in context it has a much richer meaning. Consider itin one of these two sentences: (9.1) The chicken didn’t cross the road becauseit was too tired. (9.2) The chicken didn’t cross the road becauseit was too wide. In (9.1) it is the chicken (i.e., the reader knows that the chicken was tired), while in (9.2) it is the road (and the reader knows that the road was wide). 2 That is, if we are to compute the meaning of this sentence, we’ll need the meaning ofitto be associated with the chickenin the ﬁrst sentence and associated with the roadin the second one, sensitive to the context. Furthermore, consider reading left to right like a causal language model, pro- cessing the sentence up to the word it: (9.3) The chicken didn’t cross the road becauseit At this point we don’t yet know which thingitis going to end up referring to! So a representation of it at this point might have aspects of both chicken and road as the reader is trying to guess what happens next. This fact that words have rich linguistic relationships with other words that may be far away pervades language. Consider two more examples: (9.4) The keys to the cabinet are on the table. (9.5) I walked along the pond, and noticed one of the trees along the bank. 2 We say that in the ﬁrst example it corefers with the chicken, and in the second it corefers with the road; we’ll return to this in Chapter 23. 186 CHAPTER 9 • T HE TRANSFORMER In (9.4), the phrase The keys is the subject of the sentence, and in English and many languages, must agree in grammatical number with the verbare; in this case both are plural. In English we can’t use a singular verb like is with a plural subject like keys (we’ll discuss agreement more in Chapter 18). In (9.5), we know that bank refers to the side of a pond or river and not a ﬁnancial institution because of the context, including words like pond. (We’ll discuss word senses more in Chapter 11.) The point of all these examples is that these contextual words that help us com- pute the meaning of words in context can be quite far away in the sentence or para- graph. Transformers can build contextual representations of word meaning, contex- tual embeddings, by integrating the meaning of these helpful contextual words. In acontextual embeddings transformer, layer by layer, we build up richer and richer contextualized representa- tions of the meanings of input tokens. At each layer, we compute the representation of a token i by combining information about i from the previous layer with infor- mation about the neighboring tokens to produce a contextualized representation for each word at each position. Attention is the mechanism in the transformer that weighs and combines the representations from appropriate other tokens in the context from layerk−1 to build the representation for tokens in layer k. The chicken didn’t cross the road because it was too tired The chicken didn’t cross the road because it was too tired Layer k+1 Layer k self-attention distribution columns corresponding to input tokens Figure 9.2 The self-attention weight distribution α that is part of the computation of the representation for the word it at layer k +1. In computing the representation for it, we attend differently to the various words at layer k, with darker shades indicating higher self-attention values. Note that the transformer is attending highly to the columns corresponding to the tokens chicken and road , a sensible result, since at the point whereit occurs, it could plausibly corefer with the chicken or the road, and hence we’d like the representation for it to draw on the representation for these earlier words. Figure adapted from Uszkoreit (2017). Fig. 9.2 shows a schematic example simpliﬁed from a transformer (Uszkoreit, 2017). The ﬁgure describes the situation when the current token is it and we need to compute a contextual representation for this token at layerk+1 of the transformer, drawing on the representations (from layer k) of every prior token. The ﬁgure uses color to represent the attention distribution over the contextual words: the tokens chickenand roadboth have a high attention weight, meaning that as we are com- puting the representation for it, we will draw most heavily on the representation for chicken and road. This will be useful in building the ﬁnal representation for it, since itwill end up coreferring with either chickenor road. Let’s now turn to how this attention distribution is represented and computed. 9.1 • A TTENTION 187 9.1.1 Attention more formally As we’ve said, the attention computation is a way to compute a vector representation for a token at a particular layer of a transformer, by selectively attending to and integrating information from prior tokens at the previous layer. Attention takes an input representation xi corresponding to the input token at position i, and a context window of prior inputs x1..xi−1, and produces an output ai. In causal, left-to-right language models, the context is any of the prior words. That is, when processing xi, the model has access toxi as well as the representations of all the prior tokens in the context window (context windows consist of thousands of tokens) but no tokens afteri. (By contrast, in Chapter 11 we’ll generalize attention so it can also look ahead to future words.) Fig. 9.3 illustrates this ﬂow of information in an entire causal self-attention layer, in which this same attention computation happens in parallel at each token position i. Thus a self-attention layer maps input sequences (x1,...,xn) to output sequences of the same length (a1,...,an). attentionattentionSelf-Attention Layer attentionattentionattention a1 a2 a3 a4 a5 x3 x4 x5x1 x2 Figure 9.3 Information ﬂow in causal self-attention. When processing each input xi, the model attends to all the inputs up to, and including xi. Simpliﬁed version of attention At its heart, attention is really just a weighted sum of context vectors, with a lot of complications added to how the weights are computed and what gets summed. For pedagogical purposes let’s ﬁrst describe a simpliﬁed intuition of attention, in which the attention output ai at token position i is simply the weighted sum of all the representations xj, for all j ≤i; we’ll use αi j to mean how much xj should contribute to ai: Simpliﬁed version: ai = ∑ j≤i αi jxj (9.6) Each αi j is a scalar used for weighing the value of input xj when summing up the inputs to compute ai. How shall we compute this α weighting? In attention we weight each prior embedding proportionally to howsimilar it is to the current token i. So the output of attention is a sum of the embeddings of prior tokens weighted by their similarity with the current token embedding. We compute similarity scores via dot product, which maps two vectors into a scalar value ranging from −∞ to ∞. The larger the score, the more similar the vectors that are being compared. We’ll normalize these scores with a softmax to create the vector of weights αi j,j ≤i. Simpliﬁed Version: score(xi,xj) = xi ·xj (9.7) αi j = softmax(score(xi,xj)) ∀j ≤i (9.8) Thus in Fig. 9.3 we computea3 by computing three scores: x3 ·x1, x3 ·x2 and x3 ·x3, normalizing them by a softmax, and using the resulting probabilities as weights indicating each of their proportional relevance to the current position i. Of course, 188 CHAPTER 9 • T HE TRANSFORMER the softmax weight will likely be highest for xi, since xi is very similar to itself, resulting in a high dot product. But other context words may also be similar toi, and the softmax will also assign some weight to those words. Then we use these weights as the α values in Eq. 9.6 to compute the weighted sum that is our a3. The simpliﬁed attention in equations 9.6 – 9.8 demonstrates the attention-based approach to computing ai: compare the xi to prior vectors, normalize those scores into a probability distribution used to weight the sum of the prior vector. But now we’re ready to remove the simpliﬁcations. A single attention head using query, key, and value matrices Now that we’ve seen a simple intuition of attention, let’s introduce the actual attention head, theattention head version of attention that’s used in transformers. (The word head is often used inhead transformers to refer to speciﬁc structured layers). The attention head allows us to distinctly represent three different roles that each input embedding plays during the course of the attention process: • As the current element being compared to the preceding inputs. We’ll refer to this role as a query.query • In its role as a preceding input that is being compared to the current element to determine a similarity weight. We’ll refer to this role as akey.key • And ﬁnally, as a value of a preceding element that gets weighted and summedvalue up to compute the output for the current element. To capture these three different roles, transformers introduce weight matrices WQ, WK, and WV. These weights will project each input vector xi into a represen- tation of its role as a key, query, or value: qi = xiWQ; ki = xiWK; vi = xiWV (9.9) Given these projections, when we are computing the similarity of the current ele- ment xi with some prior element xj, we’ll use the dot product between the current element’squery vector qi and the preceding element’skey vector kj. Furthermore, the result of a dot product can be an arbitrarily large (positive or negative) value, and exponentiating large values can lead to numerical issues and loss of gradients during training. To avoid this, we scale the dot product by a factor related to the size of the embeddings, via dividing by the square root of the dimensionality of the query and key vectors (dk). We thus replace the simpliﬁed Eq. 9.7 with Eq. 9.11. The ensuing softmax calculation resulting in αi j remains the same, but the output calculation for headi is now based on a weighted sum over the value vectors v (Eq. 9.13). Here’s a ﬁnal set of equations for computing self-attention for a single self- attention output vector ai from a single input vector xi. This version of attention computes ai by summing the values of the prior elements, each weighted by the similarity of its key to the query from the current element: qi = xiWQ; kj = xjWK; vj = xjWV (9.10) score(xi,xj) = qi ·kj√dk (9.11) αi j = softmax(score(xi,xj)) ∀j ≤i (9.12) headi = ∑ j≤i αi jvj (9.13) ai = headiWO (9.14) 9.1 • A TTENTION 189 6. Sum the weighted value vectors 4. Turn into 𝛼i,j weights via softmax a3 1. Generate key, query, value vectors 2. Compare x3’s query with the keys for x1, x2, and x3 8. Output of self-attention × × x1 k q v WK WQ WV 5. Weigh each value vector ÷ √dk 3. Divide scalar score by √dk ÷ √dk ÷ √dk 𝛼3,1 𝛼3,2 𝛼3,3 x2 k q v WK WQ WV x3 k q v WK WQ WV WO [1 × d] [1 × d] [dv × d] [1 × dv] [1 × dv][1 × dv][1 × dv] [1 × dv] [1 × dv] [1 x dv] 7. Reshape to [1 x d] [1 × d] [1 × d] Figure 9.4 Calculating the value of a3, the third element of a sequence using causal (left- to-right) self-attention. We illustrate this in Fig. 9.4 for the case of calculating the value of the third output a3 in a sequence. Note that we’ve also introduced one more matrix,WO, which is right-multiplied by the attention head. This is necessary to reshape the output of the head. The input to attention xi and the output from attention ai both have the same dimensionality [1 ×d]. We often call d the model dimensionality, and indeed as we’ll discuss in Section 9.2 the output hi of each transformer block, as well as the intermediate vec- tors inside the transformer block also have the same dimensionality [1 ×d]. Having everything be the same dimensionality makes the transformer very modular. So let’s talk shapes. How do we get from [1 ×d] at the input to [1 ×d] at the output? Let’s look at all the internal shapes. We’ll have a dimension dk for the key and query vectors. The query vector and the key vector are both dimensionality 1 ×dk, so we can take their dot product qi ·kj to produce a scalar. We’ll have a separate dimension dv for the value vectors. The transform matrix WQ has shape [d ×dk], WK is [d ×dk], and WV is [d ×dv]. So the output of headi in equation Eq. 9.13 is of shape [1 ×dv]. To get the desired output shape [1 ×d] we’ll need to reshape the head output, and so WO is of shape [dv ×d]. In the original transformer work (Vaswani et al., 2017),d was 512, dk and dv were both 64. Multi-head Attention Equations 9.11-9.13 describe a single attention head. But actually, transformers use multiple attention heads. The intuition is that each head might be attending to the context for different purposes: heads might be special- ized to represent different linguistic relationships between context elements and the current token, or to look for particular kinds of patterns in the context. So in multi-head attention we have A separate attention heads that reside inmulti-head attention parallel layers at the same depth in a model, each with its own set of parameters that allows the head to model different aspects of the relationships among inputs. Thus 190 CHAPTER 9 • T HE TRANSFORMER each head i in a self-attention layer has its own set of key, query and value matrices: WKi, WQi and WVi. These are used to project the inputs into separate key, value, and query embeddings for each head. When using multiple heads the model dimension d is still used for the input and output, the key and query embeddings have dimensionality dk, and the value embeddings are of dimensionality dv (again, in the original transformer paper dk = dv = 64, A = 8, and d = 512). Thus for each head i, we have weight layers WQi of shape [d ×dk], WKi of shape [d ×dk], and WVi of shape [d ×dv]. Below are the equations for attention augmented with multiple heads; Fig. 9.5 shows an intuition. qc i = xiWQc; kc j = xjWKc; vc j = xjWVc; ∀c 1 ≤c ≤A (9.15) scorec(xi,xj) = qc i ·kc j√dk (9.16) αc i j = softmax(scorec(xi,xj)) ∀j ≤i (9.17) headc i = ∑ j≤i αc i jvc j (9.18) ai = (head1 ⊕head2...⊕headA)WO (9.19) MultiHeadAttention(xi,[x1,···,xN ]) = ai (9.20) The output of each of the A heads is of shape 1 ×dv, and so the output of the multi-head layer with A heads consists of A vectors of shape 1 ×dv. These are concatenated to produce a single output with dimensionality 1 ×hdv. Then we use yet another linear projection WO ∈RAdv×d to reshape it, resulting in the multi-head attention vector ai with the correct output shape [1 ×d] at each input i. ai xi-1 xixi-2xi-3 WK 1 Head 1 WV 1 WQ 1 … … WK 2 Head 2 WV 2 WQ 2 WK 8 Head 8 WV 8 WQ 8 ai WO [hdv x d] [1 x dv ] [1 x d] [1 x d] [1 x hdv ] Project down to d Concatenate Outputs Each head attends diﬀerently to context … [1 x dv ] Figure 9.5 The multi-head attention computation for input xi, producing output ai. A multi-head attention layer has A heads, each with its own key, query and value weight matrices. The outputs from each of the heads are concatenated and then projected down to d, thus producing an output of the same size as the input. 9.2 • T RANSFORMER BLOCKS 191 9.2 Transformer Blocks The self-attention calculation lies at the core of what’s called a transformer block, which, in addition to the self-attention layer, includes three other kinds of layers: (1) a feedforward layer, (2) residual connections, and (3) normalizing layers (colloqui- ally called “layer norm”). Layer Norm xi + hi-1 Layer Norm MultiHead Attention Feedforward xi-1 xi+1 hi hi+1 + …… Residual Stream Figure 9.6 The architecture of a transformer block showing the residual stream. This ﬁgure shows the prenorm version of the architecture, in which the layer norms happen before the attention and feedforward layers rather than after. Fig. 9.6 illustrates a transformer block, sketching a common way of thinking about the block that is called the residual stream (Elhage et al., 2021). In the resid-residual stream ual stream viewpoint, we consider the processing of an individual token i through the transformer block as a single stream of d-dimensional representations for token position i. This residual stream starts with the original input vector, and the various components read their input from the residual stream and add their output back into the stream. The input at the bottom of the stream is an embedding for a token, which has dimensionality d. This initial embedding gets passed up (by residual connections), and is progressively added to by the other components of the transformer: the at- tention layer that we have seen, and the feedforward layer that we will introduce. Before the attention and feedforward layer is a computation called the layer norm. Thus the initial vector is passed through a layer norm and attention layer, and the result is added back into the stream, in this case to the original input vector xi. And then this summed vector is again passed through another layer norm and a feedforward layer, and the output of those is added back into the residual, and we’ll use hi to refer to the resulting output of the transformer block for token i. (In earlier descriptions the residual stream was often described using a different metaphor as residual connections that add the input of a component to its output, but the residual stream is a more perspicuous way of visualizing the transformer.) We’ve already seen the attention layer, so let’s now introduce the feedforward 192 CHAPTER 9 • T HE TRANSFORMER and layer norm computations in the context of processing a single input xi at token position i. Feedforward layer The feedforward layer is a fully-connected 2-layer network, i.e., one hidden layer, two weight matrices, as introduced in Chapter 7. The weights are the same for each token position i , but are different from layer to layer. It is common to make the dimensionality dff of the hidden layer of the feedforward network be larger than the model dimensionality d. (For example in the original transformer model, d = 512 and dff = 2048.) FFN(xi) =ReLU(xiW1 +b1)W2 +b2 (9.21) Layer Norm At two stages in the transformer block we normalize the vector (Ba et al., 2016). This process, called layer norm (short for layer normalization), is onelayer norm of many forms of normalization that can be used to improve training performance in deep neural networks by keeping the values of a hidden layer in a range that facilitates gradient-based training. Layer norm is a variation of the z-score from statistics, applied to a single vec- tor in a hidden layer. That is, the term layer norm is a bit confusing; layer norm is not applied to an entire transformer layer, but just to the embedding vector of a single token. Thus the input to layer norm is a single vector of dimensionality d and the output is that vector normalized, again of dimensionality d. The ﬁrst step in layer normalization is to calculate the mean, µ, and standard deviation, σ, over the elements of the vector to be normalized. Given an embedding vector x of dimen- sionality d, these values are calculated as follows. µ = 1 d d∑ i=1 xi (9.22) σ = √1 d d∑ i=1 (xi −µ)2 (9.23) Given these values, the vector components are normalized by subtracting the mean from each and dividing by the standard deviation. The result of this computation is a new vector with zero mean and a standard deviation of one. ˆ x= (x−µ) σ (9.24) Finally, in the standard implementation of layer normalization, two learnable param- eters, γ and β, representing gain and offset values, are introduced. LayerNorm(x) =γ (x−µ) σ +β (9.25) Putting it all together The function computed by a transformer block can be ex- pressed by breaking it down with one equation for each component computation, using t (of shape [1 ×d]) to stand for transformer and superscripts to demarcate 9.2 • T RANSFORMER BLOCKS 193 each computation inside the block: t1 i = LayerNorm(xi) (9.26) t2 i = MultiHeadAttention(t1 i , [ t1 1,···,t1 N ] ) (9.27) t3 i = t2 i +xi (9.28) t4 i = LayerNorm(t3 i ) (9.29) t5 i = FFN(t4 i ) (9.30) hi = t5 i +t3 i (9.31) Notice that the only component that takes as input information from other tokens (other residual streams) is multi-head attention, which (as we see from Eq. 9.28) looks at all the neighboring tokens in the context. The output from attention, how- ever, is then added into this token’s embedding stream. In fact, Elhage et al. (2021) show that we can view attention heads as literally moving information from the residual stream of a neighboring token into the current stream. The high-dimensional embedding space at each position thus contains information about the current to- ken and about neighboring tokens, albeit in different subspaces of the vector space. Fig. 9.7 shows a visualization of this movement. Token A residual stream Token B residual stream Figure 9.7 An attention head can move information from token A’s residual stream into token B’s residual stream. Crucially, the input and output dimensions of transformer blocks are matched so they can be stacked. Each token vectorxi at the input to the block has dimensionality d, and the output hi also has dimensionality d. Transformers for large language models stack many of these blocks, from 12 layers (used for the T5 or GPT-3-small language models) to 96 layers (used for GPT-3 large), to even more for more recent models. We’ll come back to this issue of stacking in a bit. Equation 9.28 and following are just the equation for a single transformer block, but the residual stream metaphor goes through all the transformer layers, from the ﬁrst transformer blocks to the 12th, in a 12-layer transformer. At the earlier trans- former blocks, the residual stream is representing the current token. At the highest transformer blocks, the residual stream is usually representing the following token, since at the very end it’s being trained to predict the next token. Once we stack many blocks, there is one more requirement: at the very end of the last (highest) transformer block, there is a single extra layer norm that is run on the last hi of each token stream (just below the language model head layer that we will deﬁne soon). 3 3 Note that we are using the most common current transformer architecture, which is called theprenorm 194 CHAPTER 9 • T HE TRANSFORMER 9.3 Parallelizing computation using a single matrix X This description of multi-head attention and the rest of the transformer block has been from the perspective of computing a single output at a single time step i in a single residual stream. But as we pointed out earlier, the attention computation performed for each token to compute ai is independent of the computation for each other token, and that’s also true for all the computation in the transformer block computing hi from the input xi. That means we can easily parallelize the entire computation, taking advantage of efﬁcient matrix multiplication routines. We do this by packing the input embeddings for the N tokens of the input se- quence into a single matrix X of size [N ×d]. Each row of X is the embedding of one token of the input. Transformers for large language models commonly have an input length N from 1K to 32K; much longer contexts of 128K or even up to millions of tokens can also be achieved with architectural changes like special long-context mechanisms that we don’t discuss here. So for vanilla transformers, we can think of X having between 1K and 32K rows, each of the dimensionality of the embedding d (the model dimension). Parallelizing attention Let’s ﬁrst see this for a single attention head and then turn to multiple heads, and then add in the rest of the components in the transformer block. For one head we multiply X by the key, query, and value matrices WQ of shape [d ×dk], WK of shape [d ×dk], and WV of shape [d ×dv], to produce matrices Q of shape [N ×dk], K of shape [N ×dk], and V of shape [N ×dv], containing all the key, query, and value vectors: Q = XWQ; K = XWK; V = XWV (9.32) Given these matrices we can compute all the requisite query-key comparisons simul- taneously by multiplying Q and K⊺ in a single matrix multiplication. The product is of shape N ×N, visualized in Fig. 9.8. q1•k1 q2•k1 q2•k2 q4•k1 q4•k2 q4•k3 q4•k4 q3•k1 q3•k2 q3•k3 N N q1•k2 q1•k3 q1•k4 q2•k3 q2•k4 q3•k4 Figure 9.8 The N ×N QK⊺ matrix showing how it computes all qi ·kj comparisons in a single matrix multiple. Once we have this QK⊺ matrix, we can very efﬁciently scale these scores, take the softmax, and then multiply the result by V resulting in a matrix of shape N ×d: a vector embedding representation for each token in the input. We’ve reduced the entire self-attention step for an entire sequence of N tokens for one head to the architecture. The original deﬁnition of the transformer in Vaswani et al. (2017) used an alternative archi- tecture called the postnorm transformer in which the layer norm happens after the attention and FFN layers; it turns out moving the layer norm beforehand works better, but does require this one extra layer at the end. 9.3 • P ARALLELIZING COMPUTATION USING A SINGLE MATRIX X 195 following computation: head = softmax ( mask (QK⊺ √dk )) V A = head WO (9.33) Masking out the future You may have noticed that we introduced a mask function in Eq. 9.33 above. This is because the self-attention computation as we’ve described it has a problem: the calculation of QK⊺ results in a score for each query value to every key value, including those that follow the query . This is inappropriate in the setting of language modeling: guessing the next word is pretty simple if you already know it! To ﬁx this, the elements in the upper-triangular portion of the matrix are set to −∞, which the softmax will turn to zero, thus eliminating any knowledge of words that follow in the sequence. This is done in practice by adding a mask matrix M in which Mi j = −∞ ∀j >i (i.e. for the upper-triangular portion) andMi j = 0 otherwise. Fig. 9.9 shows the resulting masked QK⊺ matrix. (we’ll see in Chapter 11 how to make use of words in the future for tasks that need it). q1•k1 q2•k1 q2•k2 q4•k1 q4•k2 q4•k3 q4•k4 q3•k1 q3•k2 q3•k3 N N −∞ −∞ −∞ −∞ −∞ −∞ Figure 9.9 The N ×N QK⊺ matrix showing the qi ·kj values, with the upper-triangle por- tion of the comparisons matrix zeroed out (set to −∞, which the softmax will turn to zero). Fig. 9.10 shows a schematic of all the computations for a single attention head parallelized in matrix form. Fig. 9.8 and Fig. 9.9 also make it clear that attention is quadratic in the length of the input, since at each layer we need to compute dot products between each pair of tokens in the input. This makes it expensive to compute attention over very long documents (like entire novels). Nonetheless modern large language models manage to use quite long contexts of thousands or tens of thousands of tokens. Parallelizing multi-head attention In multi-head attention, as with self-attention, the input and output have the model dimension d, the key and query embeddings have dimensionality dk, and the value embeddings are of dimensionality dv (again, in the original transformer paper dk = dv = 64, A = 8, and d = 512). Thus for each head i, we have weight layers WQi of shape [d ×dk], WKi of shape [d ×dk], and WVi of shape [d ×dv], and these get multiplied by the inputs packed into X to produce Q of shape [N ×dk], K of shape [N ×dk], and V of shape [N ×dv]. The output of each of the A heads is of shape N ×dv, and so the output of the multi-head layer with A heads consists of A matrices of shape N ×dv. To make use of these matrices in further processing, they are concatenated to produce a single output with dimensionality N ×hdv. Finally, we use a ﬁnal linear projection WO of shape [Adv ×d], that reshape it to the original output dimension for each token. Multiplying the concatenatedN ×hdv matrix output byWO of shape [Adv ×d] yields 196 CHAPTER 9 • T HE TRANSFORMER q1 q2 q3 q4 k1 k2 k3 k4 Q KT QKT v1 v2 v3 v4 V q2•k2 q4•k2 q4•k3 q4•k4 q3•k2 q3•k3 −∞ −∞ −∞ −∞ −∞ −∞q1•k1 q2•k1 q2•k2 q4•k1 q4•k2 q4•k3 q4•k4 q3•k1 q3•k2 q3•k3 q1•k2 q2•k3 q1•k3 q3•k4 q2•k4 q1•k4x = QKT masked mask = q1•k1 q2•k1 q4•k1 q3•k1 q1•k1q1•k1 =x a1 a2 a3 a4 A Query Token 1 Query Token 2 Query Token 3 Query Token 4 Q Input Token 1 Input Token 2 Input Token 3 Input Token 4 X x WQ = Value Token 1 Value Token 2 Value Token 3 Value Token 4 V x WV = Input Token 1 Input Token 2 Input Token 3 Input Token 4 X Key Token 1 Key Token 2 Key Token 3 Key Token 4 K x WK = Input Token 1 Input Token 2 Input Token 3 Input Token 4 X N x dk dk x N N x N N x N N x dv N x dv d x dk d x dk d x dv N x d N x dk N x d N x dk N x d N x dv Figure 9.10 Schematic of the attention computation for a single attention head in parallel. The ﬁrst row shows the computation of the Q, K, and V matrices. The second row shows the computation of QKT, the masking (the softmax computation and the normalizing by dimensionality are not shown) and then the weighted sum of the value vectors to get the ﬁnal attention vectors. the self-attention output A of shape [N ×d]. Qi = XWQi ; Ki = XWKi ; Vi = XWVi (9.34) headi = SelfAttention(Qi,Ki,Vi) = softmax (QiKi⊺ √dk ) Vi (9.35) MultiHeadAttention(X) = (head1 ⊕head2...⊕headA)WO (9.36) Putting it all together with the parallel input matrix X The function computed in parallel by an entire layer of N transformer block over the entire N input tokens can be expressed as: O = X+MultiHeadAttention(LayerNorm(X)) (9.37) H = O+FFN(LayerNorm(O)) (9.38) Note that in Eq. 9.37 we are using X to mean the input to the layer, wherever it comes from. For the ﬁrst layer, as we will see in the next section, that input is the initital word + positional embedding vectors that we have been describing byX. But for subsequent layers k, the input is the output from the previous layer Hk−1. We can also break down the computation performed in a transformer layer, showing one equation for each component computation. We’ll use T (of shape [N ×d]) to stand for transformer and superscripts to demarcate each computation inside the block, and again use X to mean the input to the block from the previous layer or the initial 9.4 • T HE INPUT : EMBEDDINGS FOR TOKEN AND POSITION 197 embedding: T1 = LayerNorm(X) (9.39) T2 = MultiHeadAttention(T1) (9.40) T3 = T2 +X (9.41) T4 = LayerNorm(T3) (9.42) T5 = FFN(T4) (9.43) H = T5 +T3 (9.44) Here when we use a notation like FFN (T3) we mean that the same FFN is applied in parallel to each of the N embedding vectors in the window. Similarly, each of the N tokens is normed in parallel in the LayerNorm. Crucially, the input and output dimensions of transformer blocks are matched so they can be stacked. Since each token xi at the input to the block is represented by an embedding of dimensionality [1 ×d], that means the input X and output H are both of shape [N ×d]. 9.4 The input: embeddings for token and position Let’s talk about where the inputX comes from. Given a sequence of N tokens (N is the context length in tokens), the matrix X of shape [N ×d] has an embedding forembedding each word in the context. The transformer does this by separately computing two embeddings: an input token embedding, and an input positional embedding. A token embedding, introduced in Chapter 7 and Chapter 8, is a vector of di- mension d that will be our initial representation for the input token. (As we pass vectors up through the transformer layers in the residual stream, this embedding representation will change and grow, incorporating context and playing a different role depending on the kind of language model we are building.) The set of initial embeddings are stored in the embedding matrix E, which has a row for each of the \|V \|tokens in the vocabulary. Thus each word is a row vector of d dimensions, and E has shape [\|V \|×d]. Given an input token string like Thanks for all the we ﬁrst convert the tokens into vocabulary indices (these were created when we ﬁrst tokenized the input using BPE or SentencePiece). So the representation of thanks for all the might be w = [5,4000,10532,2224]. Next we use indexing to select the corresponding rows from E, (row 5, row 4000, row 10532, row 2224). Another way to think about selecting token embeddings from the embedding matrix is to represent tokens as one-hot vectors of shape [1 ×\|V \|], i.e., with one dimension for each word in the vocabulary. Recall that in a one-hot vector all theone-hot vector elements are 0 except one, the element whose dimension is the word’s index in the vocabulary, which has value 1. So if the word “thanks” has index 5 in the vocabulary, x5 = 1, and xi = 0 ∀i ̸= 5, as shown here: [0 0 0 0 1 0 0 ... 0 0 0 0] 1 2 3 4 5 6 7 ... ... \|V\| Multiplying by a one-hot vector that has only one non-zero elementxi = 1 simply selects out the relevant row vector for wordi, resulting in the embedding for word i, as depicted in Fig. 9.11. 198 CHAPTER 9 • T HE TRANSFORMER E \|V\| d 1 \|V\| d =✕ 55 0 0 0 0 1 0 0 … 0 0 0 0 1 Figure 9.11 Selecting the embedding vector for word V5 by multiplying the embedding matrix E with a one-hot vector with a 1 in index 5. We can extend this idea to represent the entire token sequence as a matrix of one- hot vectors, one for each of the N positions in the transformer’s context window, as shown in Fig. 9.12. E \|V\| d d N =✕ \|V\| N 0 0 0 0 0 0 0 … 0 0 1 0 0 0 0 0 1 0 0 … 0 0 0 0 1 0 0 0 0 0 0 … 0 0 0 0 0 0 0 0 1 0 0 … 0 0 0 0 … Figure 9.12 Selecting the embedding matrix for the input sequence of token idsW by mul- tiplying a one-hot matrix corresponding to W by the embedding matrix E. These token embeddings are not position-dependent. To represent the position of each token in the sequence, we combine these token embeddings with positional embeddings speciﬁc to each position in an input sequence.positional embeddings Where do we get these positional embeddings? The simplest method, called absolute position, is to start with randomly initialized embeddings correspondingabsolute position to each possible input position up to some maximum length. For example, just as we have an embedding for the wordﬁsh, we’ll have an embedding for the position 3. As with word embeddings, these positional embeddings are learned along with other parameters during training. We can store them in a matrix Epos of shape [N ×d]. To produce an input embedding that captures positional information, we just add the word embedding for each input to its corresponding positional embedding. The individual token and position embeddings are both of size[1×d], so their sum is also [1×d], This new embedding serves as the input for further processing. Fig. 9.13 shows the idea. X = Composite Embeddings (word + position) Transformer Block Janet 1 will 2 back 3 Janet will back the bill the 4 bill 5 + + + + + Position Embeddings Word Embeddings Figure 9.13 A simple way to model position: add an embedding of the absolute position to the token embedding to produce a new embedding of the same dimensionality. 9.5 • T HE LANGUAGE MODELING HEAD 199 The ﬁnal representation of the input, the matrix X, is an [N ×d] matrix in which each row i is the representation of the ith token in the input, computed by adding E[id(i)]—the embedding of the id of the token that occurred at position i—, to P[i], the positional embedding of position i. A potential problem with the simple position embedding approach is that there will be plenty of training examples for the initial positions in our inputs and corre- spondingly fewer at the outer length limits. These latter embeddings may be poorly trained and may not generalize well during testing. An alternative is to choose a static function that maps integer inputs to real-valued vectors in a way that better handle sequences of arbitrary length. A combination of sine and cosine functions with differing frequencies was used in the original transformer work. Sinusoidal po- sition embeddings may also help in capturing the inherent relationships among the positions, like the fact that position 4 in an input is more closely related to position 5 than it is to position 17. A more complex style of positional embedding methods extend this idea of cap- turing relationships even further to directly represent relative position instead ofrelative position absolute position, often implemented in the attention mechanism at each layer rather than being added once at the initial input. 9.5 The Language Modeling Head The last component of the transformer we must introduce is thelanguage modeling head. Here we are using the word head to mean the additional neural circuitry welanguage modeling head head add on top of the basic transformer architecture when we apply pretrained trans- former models to various tasks. The language modeling head is the circuitry we need to do language modeling. Recall that language models, from the simple n-gram models of Chapter 3 through the feedforward and RNN language models of Chapter 7 and Chapter 8, are word predictors. Given a context of words, they assign a probability to each possible next word. For example, if the preceding context is “Thanks for all the” and we want to know how likely the next word is “ﬁsh” we would compute: P(ﬁsh\|Thanks for all the) Language models give us the ability to assign such a conditional probability to every possible next word, giving us a distribution over the entire vocabulary. The n-gram language models of Chapter 3 compute the probability of a word given counts of its occurrence with the n −1 prior words. The context is thus of size n −1. For transformer language models, the context is the size of the transformer’s context window, which can be quite large, like 32K tokens for large models (and much larger contexts of millions of words are possible with special long-context architectures). The job of the language modeling head is to take the output of the ﬁnal trans- former layer from the last token N and use it to predict the upcoming word at posi- tion N +1. Fig. 9.14 shows how to accomplish this task, taking the output of the last token at the last layer (the d-dimensional output embedding of shape [1 ×d]) and producing a probability distribution over words (from which we will choose one to generate). The ﬁrst module in Fig. 9.14 is a linear layer, whose job is to project from the output hL N , which represents the output token embedding at positionN from the ﬁnal 200 CHAPTER 9 • T HE TRANSFORMER Layer L Transformer Block Softmax over vocabulary V Unembedding layer … 1 x \|V\| Logits Word probabilities 1 x \|V\| hL 1 w1 w2 wN hL 2 hL N d x \|V\| 1 x d Unembedding layer U = ET y1 y2 y\|V\|… u1 u2 u\|V\|… Language Model Head takes hL N and outputs a distribution over vocabulary V Figure 9.14 The language modeling head: the circuit at the top of a transformer that maps from the output embedding for token N from the last transformer layer ( hL N ) to a probability distribution over words in the vocabulary V . block L, (hence of shape [1 ×d]) to the logit vector, or score vector, that will have alogit single score for each of the \|V \|possible words in the vocabularyV . The logit vector u is thus of dimensionality 1 ×\|V \|. This linear layer can be learned, but more commonly we tie this matrix to (the transpose of) the embedding matrix E. Recall that in weight tying , we use theweight tying same weights for two different matrices in the model. Thus at the input stage of the transformer the embedding matrix (of shape[\|V \|×d]) is used to map from a one-hot vector over the vocabulary (of shape [1 ×\|V \|]) to an embedding (of shape [1 ×d]). And then in the language model head,ET, the transpose of the embedding matrix (of shape [d ×\|V \|]) is used to map back from an embedding (shape [1 ×d]) to a vector over the vocabulary (shape [1×\|V \|]). In the learning process, E will be optimized to be good at doing both of these mappings. We therefore sometimes call the transpose ET the unembedding layer because it is performing this reverse mapping.unembedding A softmax layer turns the logits u into the probabilities y over the vocabulary. u = hL N ET (9.45) y = softmax(u) (9.46) We can use these probabilities to do things like help assign a probability to a given text. But the most important usage to generate text, which we do bysampling a word from these probabilities y. We might sample the highest probability word (‘greedy’ decoding), or use another of the sampling methods we’ll introduce in Sec- tion 10.2. In either case, whatever entry yk we choose from the probability vector y, we generate the word that has that index k. Fig. 9.15 shows the total stacked architecture for one tokeni. Note that the input to each transformer layer xℓ i is the same as the output from the preceding layer hℓ−1 i . Now that we see all these transformer layers spread out on the page, we can point out another useful feature of the unembedding layer: as a tool for interpretability of the internals of the transformer that we call the logit lens (Nostalgebraist, 2020).logit lens We can take a vector from any layer of the transformer and, pretending that it is the preﬁnal embedding, simply multiply it by the unembedding layer to get logits, and compute a softmax to see the distribution over words that that vector might be representing. This can be a useful window into the internal representations of 9.6 • S UMMARY 201 wi Sample token to generate at position i+1 feedforward layer norm attention layer norm U Input token Language Modeling Head Input Encoding E i+ … logits feedforward layer norm attention layer norm Layer 1 Layer 2 h1 i = x2 i x1 i h2 i = x3 i feedforward layer norm attention layer norm hL i hL-1 i = xL i y1 y2 y\|V\|…Token probabilities u1 u2 u\|V\|… softmax wi+1 Layer L Figure 9.15 A transformer language model (decoder-only), stacking transformer blocks and mapping from an input token wi to to a predicted next token wi+1. the model. Since the network wasn’t trained to make the internal representations function in this way, the logit lens doesn’t always work perfectly, but this can still be a useful trick. A terminological note before we conclude: You will sometimes see a trans- former used for this kind of unidirectional causal language model called a decoder- only model. This is because this model constitutes roughly half of the encoder-decoder-only model decoder model for transformers that we’ll see how to apply to machine translation in Chapter 13. (Confusingly, the original introduction of the transformer had an encoder-decoder architecture, and it was only later that the standard paradigm for causal language model was deﬁned by using only the decoder part of this original architecture). 9.6 Summary This chapter has introduced the transformer and its components for the task of lan- guage modeling. We’ll continue the task of language modeling including issues like training and sampling in the next chapter. 202 CHAPTER 9 • T HE TRANSFORMER Here’s a summary of the main points that we covered: • Transformers are non-recurrent networks based on multi-head attention, a kind of self-attention. A multi-head attention computation takes an input vector xi and maps it to an output ai by adding in vectors from prior tokens, weighted by how relevant they are for the processing of the current word. • A transformer block consists of a residual stream in which the input from the prior layer is passed up to the next layer, with the output of different com- ponents added to it. These components include a multi-head attention layer followed by a feedforward layer, each preceded by layer normalizations. Transformer blocks are stacked to make deeper and more powerful networks. • The input to a transformer is computed by adding an embedding (computed with an embedding matrix) to a positional encoding that represents the se- quential position of the token in the window. • Language models can be built out of stacks of transformer blocks, with a language model head at the top, which applies an unembedding matrix to the output H of the top layer to generate the logits, which are then passed through a softmax to generate word probabilities. • Transformer-based language models have a wide context window (200K to- kens or even more for very large models with special mechanisms) allowing them to draw on enormous amounts of context to predict upcoming words. Bibliographical and Historical Notes The transformer (Vaswani et al., 2017) was developed drawing on two lines of prior research: self-attention and memory networks. Encoder-decoder attention, the idea of using a soft weighting over the encodings of input words to inform a generative decoder (see Chapter 13) was developed by Graves (2013) in the context of handwriting generation, and Bahdanau et al. (2015) for MT. This idea was extended to self-attention by dropping the need for separate encoding and decoding sequences and instead seeing attention as a way of weighting the tokens in collecting information passed from lower layers to higher layers (Ling et al., 2015; Cheng et al., 2016; Liu et al., 2016). Other aspects of the transformer, including the terminology of key, query, and value, came from memory networks, a mechanism for adding an external read- write memory to networks, by using an embedding of a query to match keys rep- resenting content in an associative memory (Sukhbaatar et al., 2015; Weston et al., 2015; Graves et al., 2014). MORE HISTORY TBD IN NEXT DRAFT. CHAPTER 10 Large Language Models “How much do we know at any time? Much more, or so I believe, than we know we know. ” Agatha Christie, The Moving Finger Fluent speakers of a language bring an enormous amount of knowledge to bear dur- ing comprehension and production. This knowledge is embodied in many forms, perhaps most obviously in the vocabulary, the rich representations we have of words and their meanings and usage. This makes the vocabulary a useful lens to explore the acquisition of knowledge from text, by both people and machines. Estimates of the size of adult vocabularies vary widely both within and across languages. For example, estimates of the vocabulary size of young adult speakers of American English range from 30,000 to 100,000 depending on the resources used to make the estimate and the deﬁnition of what it means to know a word. What is agreed upon is that the vast majority of words that mature speakers use in their day-to-day interactions are acquired early in life through spoken interactions with caregivers and peers, usually well before the start of formal schooling. This active vocabulary (usually on the order of 2000 words for young speakers) is extremely limited compared to the size of the adult vocabulary, and is quite stable, with very few additional words learned via casual conversation beyond this early stage. Obvi- ously, this leaves a very large number of words to be acquired by other means. A simple consequence of these facts is that children have to learn about 7 to 10 words a day,every single day, to arrive at observed vocabulary levels by the time they are 20 years of age. And indeed empirical estimates of vocabulary growth in late el- ementary through high school are consistent with this rate. How do children achieve this rate of vocabulary growth? The bulk of this knowledge acquisition seems to happen as a by-product of reading, as part of the rich processing and reasoning that we perform when we read. Research into the average amount of time children spend reading, and the lexical diversity of the texts they read, indicate that it is possible to achieve the desired rate. But the mechanism behind this rate of learning must be remarkable indeed, since at some points during learning the rate of vocabulary growth exceeds the rate at which new words are appearing to the learner! Such facts have motivated thedistributional hypothesis of Chapter 6, which sug- gests that aspects of meaning can be learned solely from the texts we encounter over our lives, based on the complex association of words with the words they co-occur with (and with the words that those words occur with). The distributional hypothe- sis suggests both that we can acquire remarkable amounts of knowledge from text, and that this knowledge can be brought to bear long after its initial acquisition. Of course, grounding from real-world interaction or other modalities can help build even more powerful models, but even text alone is remarkably useful. In this chapter we formalize this idea ofpretraining—learning knowledge aboutpretraining language and the world from vast amounts of text—and call the resulting pretrained language models large language models. Large language models exhibit remark- 204 CHAPTER 10 • L ARGE LANGUAGE MODELS able performance on all sorts of natural language tasks because of the knowledge they learn in pretraining, and they will play a role throughout the rest of this book. They have been especially transformative for tasks where we need to produce text, like summarization, machine translation, question answering, or chatbots. We’ll start by seeing how to apply the transformer of Chapter 9 to language modeling, in a setting often called causal or autoregressive language models, in which we iteratively predict words left-to-right from earlier words. We’ll ﬁrst in- troduce training, seeing how language models are self-trained by iteratively being taught to guess the next word in the text from the prior words. We’ll then talk about the process of text generation. The application of LLMs to generate text has vastly broadened the scope of NLP,. Text generation, code- generation, and image-generation together constitute the important new area ofgen- erative AI. We’ll introduce speciﬁc algorithms for generating text from a languagegenerative AI model, like greedy decoding and sampling. And we’ll see that almost any NLP task can be modeled as word prediction in a large language model, if we think about it in the right way. We’ll work through an example of using large language mod- els to solve one classic NLP task of summarization (generating a short text that summarizes some larger document). 10.1 Large Language Models with Transformers The prior chapter introduced most of the components of a transformer in the domain of language modeling: the transformer block including multi-head attention, the language modeling head, and the positional encoding of the input. In the following sections we’ll introduce the remaining aspects of the transformer LLM: sampling and training. Before we do that, we use this section to talk about why and how we apply transformer-based large language models to NLP tasks. The tasks we will describe are all cases of conditional generation. Conditionalconditional generation generation is the task of generating text conditioned on an input piece of text. That is, we give the LLM an input piece of text, generally called aprompt, and then have the LLM continue generating text token by token, conditioned on the prompt and the previously generated tokens. The fact that transformers have such long contexts (many thousands of tokens) makes them very powerful for conditional generation, because they can look back so far into the prompting text. Consider the simple task of text completion, illustrated in Fig. 10.1. Here a language model is given a text preﬁx and is asked to generate a possible completion. Note that as the generation process proceeds, the model has direct access to the priming context as well as to all of its own subsequently generated outputs (at least as much as ﬁts in the large context window). This ability to incorporate the entirety of the earlier context and generated outputs at each time step is the key to the power of large language models built from transformers. So why should we care about predicting upcoming words or tokens? The in- sight of large language modeling is that many practical NLP tasks can be cast as word prediction, and that a powerful-enough language model can solve them with a high degree of accuracy. For example, we can cast sentiment analysis as language modeling by giving a language model a context like: The sentiment of the sentence ‘‘I like Jackie Chan" is: and comparing the following conditional probability of the words “positive” and the 10.1 • L ARGE LANGUAGE MODELS WITH TRANSFORMERS 205 Preﬁx Text Completion Text Encoder Transformer Blocks Softmax long all and thanks for all the the … U UUnencoder layer Language Modeling Head logits So E i+ E i+ E i+ E i+ E i+ E i+ E i+ … Figure 10.1 Left-to-right (also called autoregressive) text completion with transformer-based large language models. As each token is generated, it gets added onto the context as a preﬁx for generating the next token. word “negative” to see which is higher: P(positive\|The sentiment of the sentence ‘‘I like Jackie Chan" is:) P(negative\|The sentiment of the sentence ‘‘I like Jackie Chan" is:) If the word “positive” is more probable, we say the sentiment of the sentence is positive, otherwise we say the sentiment is negative. We can also cast more complex tasks as word prediction. Consider question answering, in which the system is given a question (for example a question with a simple factual answer) and must give a textual answer; we introduce this task in detail in Chapter 14. We can cast the task of question answering as word prediction by giving a language model a question and a token likeA:suggesting that an answer should come next: Q: Who wrote the book ‘‘The Origin of Species"? A: If we ask a language model to compute the probability distribution over possible next words given this preﬁx: P(w\|Q: Who wrote the book ‘‘The Origin of Species"? A: ) and look at which words w have high probabilities, we might expect to see that Charles is very likely, and then if we choose Charles and continue and ask P(w\|Q: Who wrote the book ‘‘The Origin of Species"? A: Charles ) we might now see that Darwin is the most probable token, and select it. Conditional generation can even be used to accomplish tasks that must generate longer responses. Consider the task of text summarization, which is to take a longtext summarization text, such as a full-length article, and produce an effective shorter summary of it. We can cast summarization as language modeling by giving a large language model a text, and follow the text by a token liketl;dr; this token is short for something like 206 CHAPTER 10 • L ARGE LANGUAGE MODELS ‘too long; didn’t read’ and in recent years people often use this token, especially in informal work emails, when they are going to give a short summary. Since this token is sufﬁciently frequent in language model training data, language models have seen many texts in which the token occurs before a summary, and hence will interpret the token as instructions to generate a summary. We can then do conditional generation: give the language model this preﬁx, and then have it generate the following words, one by one, and take the entire response as a summary. Fig. 10.2 shows an example of a text and a human-produced summary from a widely-used summarization corpus consisting of CNN and Daily Mail news articles. Original Article The only thing crazier than a guy in snowbound Massachusetts boxing up the powdery white stuff and offering it for sale online? People are actually buying it. For $89, self-styled entrepreneur Kyle Waring will ship you 6 pounds of Boston-area snow in an insulated Styrofoam box – enough for 10 to 15 snowballs, he says. But not if you live in New England or surrounding states. “We will not ship snow to any states in the northeast!” says Waring’s website, ShipSnowYo.com. “We’re in the business of expunging snow!” His website and social media accounts claim to have ﬁlled more than 133 orders for snow – more than 30 on Tuesday alone, his busiest day yet. With more than 45 total inches, Boston has set a record this winter for the snowiest month in its history. Most residents see the huge piles of snow choking their yards and sidewalks as a nuisance, but Waring saw an opportunity. According to Boston.com, it all started a few weeks ago, when Waring and his wife were shov- eling deep snow from their yard in Manchester-by-the-Sea, a coastal suburb north of Boston. He joked about shipping the stuff to friends and family in warmer states, and an idea was born. [...] Summary Kyle Waring will ship you 6 pounds of Boston-area snow in an insulated Styrofoam box – enough for 10 to 15 snowballs, he says. But not if you live in New England or surrounding states. Figure 10.2 Excerpt from a sample article and its summary from the CNN/Daily Mail summarization corpus (Hermann et al., 2015b), (Nallapati et al., 2016). If we take this full article and append the tokentl;dr, we can use this as the con- text to prime the generation process to produce a summary as illustrated in Fig. 10.3. Again, what makes transformers able to succeed at this task (as compared, say, to the primitive n-gram language model) is that attention can incorporate information from the large context window, giving the model access to the original article as well as to the newly generated text throughout the process. Which words do we generate at each step? One simple way to generate words is to always generate the most likely word given the context. Generating the most likely word given the context is called greedy decoding. A greedy algorithm is onegreedy decoding that make a choice that is locally optimal, whether or not it will turn out to have been the best choice with hindsight. Thus in greedy decoding, at each time step in generation, the output yt is chosen by computing the probability for each possible output (every word in the vocabulary) and then choosing the highest probability word (the argmax): ˆwt = argmaxw∈V P(w\|w<t ) (10.1) In practice, however, we don’t use greedy decoding with large language models. A major problem with greedy decoding is that because the words it chooses are (by deﬁnition) extremely predictable, the resulting text is generic and often quite repeti- tive. Indeed, greedy decoding is so predictable that it is deterministic; if the context 10.2 • S AMPLING FOR LLM G ENERATION 207 Original Story Generated Summary … idea Kyle was born. Kyle Waring WaringonlyThe … will Delimiter will U U U tl;dr LM Head E E E E E E E E … Figure 10.3 Summarization with large language models using the tl;dr token and context-based autore- gressive generation. is identical, and the probabilistic model is the same, greedy decoding will always re- sult in generating exactly the same string. We’ll see in Chapter 13 that an extension to greedy decoding calledbeam search works well in tasks like machine translation, which are very constrained in that we are always generating a text in one language conditioned on a very speciﬁc text in another language. In most other tasks, how- ever, people prefer text which has been generated by more sophisticated methods, called sampling methods, that introduce a bit more diversity into the generations. We’ll see how to do that in the next few sections. 10.2 Sampling for LLM Generation The core of the generation process for large language models is the task of choosing the single word to generate next based on the context and based on the probabilities that the model assigns to possible words. This task of choosing a word to generate based on the model’s probabilities is called decoding. Decoding from a languagedecoding model in a left-to-right manner (or right-to-left for languages like Arabic in which we read from right to left), and thus repeatedly choosing the next word conditioned on our previous choices is called autoregressive generation or causal LM genera-autoregressive generation tion.1 (As we’ll see, alternatives like the masked language models of Chapter 11 are non-causal because they can predict words based on both past and future words). The most common method for decoding in large language models is sampling. Recall from Chapter 3 that sampling from a model’s distribution over words meanssampling to choose random words according to their probability assigned by the model. That is, we iteratively choose a word to generate according to its probability in context 1 Technically an autoregressive model predicts a value at timet based on a linear function of the values at times t −1, t −2, and so on. Although language models are not linear (since they have many layers of non-linearities), we loosely refer to this generation technique as autoregressive since the word generated at each time step is conditioned on the word selected by the network from the previous step. 208 CHAPTER 10 • L ARGE LANGUAGE MODELS as deﬁned by the model. Thus we are more likely to generate words that the model thinks have a high probability in the context and less likely to generate words that the model thinks have a low probability. We saw back in Chapter 3 on page 43 how to generate text from a unigram lan- guage model , by repeatedly randomly sampling words according to their probability until we either reach a pre-determined length or select the end-of-sentence token. To generate text from a trained transformer language model we’ll just generalize this model a bit: at each step we’ll sample words according to their probability condi- tioned on our previous choices, and we’ll use a transformer language model as the probability model that tells us this probability. We can formalize this algorithm for generating a sequence of wordsW = w1,w2,..., wN until we hit the end-of-sequence token, using x ∼p(x) to mean ‘choose x by sam- pling from the distribution p(x): i←1 wi ∼p(w) while wi != EOS i←i + 1 wi ∼p(wi \|w<i) The algorithm above is called random sampling, and it turns out random sam-random sampling pling doesn’t work well enough. The problem is that even though random sampling is mostly going to generate sensible, high-probable words, there are many odd, low- probability words in the tail of the distribution, and even though each one is low- probability, if you add up all the rare words, they constitute a large enough portion of the distribution that they get chosen often enough to result in generating weird sentences. For this reason, instead of random sampling, we usually use sampling methods that avoid generating the very unlikely words. The sampling methods we introduce below each have parameters that enable trading off two important factors in generation: quality and diversity. Methods that emphasize the most probable words tend to produce generations that are rated by people as more accurate, more coherent, and more factual, but also more boring and more repetitive. Methods that give a bit more weight to the middle-probability words tend to be more creative and more diverse, but less factual and more likely to be incoherent or otherwise low-quality. 10.2.1 Top- k sampling Top-k sampling is a simple generalization of greedy decoding. Instead of choosingtop-k sampling the single most probable word to generate, we ﬁrst truncate the distribution to the top k most likely words, renormalize to produce a legitimate probability distribution, and then randomly sample from within thesek words according to their renormalized probabilities. More formally: 1. Choose in advance a number of words k 2. For each word in the vocabulary V , use the language model to compute the likelihood of this word given the context p(wt \|w<t ) 3. Sort the words by their likelihood, and throw away any word that is not one of the top k most probable words. 4. Renormalize the scores of the k words to be a legitimate probability distribu- tion. 10.2 • S AMPLING FOR LLM G ENERATION 209 5. Randomly sample a word from within these remaining k most-probable words according to its probability. When k = 1, top-k sampling is identical to greedy decoding. Setting k to a larger number than 1 leads us to sometimes select a word which is not necessarily the most probable, but is still probable enough, and whose choice results in generating more diverse but still high-enough-quality text. 10.2.2 Nucleus or top- p sampling One problem with top-k sampling is that k is ﬁxed, but the shape of the probability distribution over words differs in different contexts. If we set k = 10, sometimes the top 10 words will be very likely and include most of the probability mass, but other times the probability distribution will be ﬂatter and the top 10 words will only include a small part of the probability mass. An alternative, called top-p sampling or nucleus sampling (Holtzman et al.,top-p sampling 2020), is to keep not the top k words, but the top p percent of the probability mass. The goal is the same; to truncate the distribution to remove the very unlikely words. But by measuring probability rather than the number of words, the hope is that the measure will be more robust in very different contexts, dynamically increasing and decreasing the pool of word candidates. Given a distributionP(wt \|w<t ), we sort the distribution from most probable, and then the top-p vocabulary V (p) is the smallest set of words such that ∑ w∈V (p) P(w\|w<t ) ≥p. (10.2) 10.2.3 Temperature sampling In temperature sampling, we don’t truncate the distribution, but instead reshapetemperature sampling it. The intuition for temperature sampling comes from thermodynamics, where a system at a high temperature is very ﬂexible and can explore many possible states, while a system at a lower temperature is likely to explore a subset of lower energy (better) states. In low-temperature sampling, we smoothly increase the probability of the most probable words and decrease the probability of the rare words. We implement this intuition by simply dividing the logit by a temperature param- eter τ before we normalize it by passing it through the softmax. In low-temperature sampling, τ ∈(0,1]. Thus instead of computing the probability distribution over the vocabulary directly from the logit as in the following (repeated from Eq. 9.46): y = softmax(u) (10.3) we instead ﬁrst divide the logits by τ, computing the probability vector y as y = softmax(u/τ) (10.4) Why does this work? When τ is close to 1 the distribution doesn’t change much. But the lower τ is, the larger the scores being passed to the softmax (dividing by a smaller fraction τ ≤1 results in making each score larger). Recall that one of the useful properties of a softmax is that it tends to push high values toward 1 and low values toward 0. Thus when larger numbers are passed to a softmax the result is a distribution with increased probabilities of the most high-probability words and decreased probabilities of the low probability words, making the distribution more greedy. As τ approaches 0 the probability of the most likely word approaches 1. 210 CHAPTER 10 • L ARGE LANGUAGE MODELS Note, by the way, that there can be other situations where we may want to do something quite different and ﬂatten the word probability distribution instead of making it greedy. Temperature sampling can help with this situation too, in this case high-temperature sampling, in which case we use τ >1. 10.3 Pretraining Large Language Models How do we teach a transformer to be a language model? What is the algorithm and what data do we train on? 10.3.1 Self-supervised training algorithm To train a transformer as a language model, we use the same self-supervision (orself-supervision self-training) algorithm we saw in Section 8.2.2: we take a corpus of text as training material and at each time stept ask the model to predict the next word. We call such a model self-supervised because we don’t have to add any special gold labels to the data; the natural sequence of words is its own supervision! We simply train the model to minimize the error in predicting the true next word in the training sequence, using cross-entropy as the loss function. Recall that the cross-entropy loss measures the difference between a predicted probability distribution and the correct distribution. LCE = − ∑ w∈V yt [w]log ˆyt [w] (10.5) In the case of language modeling, the correct distributionyt comes from knowing the next word. This is represented as a one-hot vector corresponding to the vocabulary where the entry for the actual next word is 1, and all the other entries are 0. Thus, the cross-entropy loss for language modeling is determined by the probability the model assigns to the correct next word (all other words get multiplied by zero). So at time t the CE loss in Eq. 10.5 can be simpliﬁed as the negative log probability the model assigns to the next word in the training sequence. LCE (ˆyt ,yt ) = −log ˆyt [wt+1] (10.6) Thus at each word position t of the input, the model takes as input the correct se- quence of tokens w1:t , and uses them to compute a probability distribution over possible next words so as to compute the model’s loss for the next tokenwt+1. Then we move to the next word, we ignore what the model predicted for the next word and instead use the correct sequence of tokens w1:t+1 to estimate the probability of token wt+2. This idea that we always give the model the correct history sequence to predict the next word (rather than feeding the model its best case from the previous time step) is called teacher forcing.teacher forcing Fig. 10.4 illustrates the general training approach. At each step, given all the preceding words, the ﬁnal transformer layer produces an output distribution over the entire vocabulary. During training, the probability assigned to the correct word is used to calculate the cross-entropy loss for each item in the sequence. The loss for a training sequence is the average cross-entropy loss over the entire sequence. The weights in the network are adjusted to minimize the average CE loss over the training sequence via gradient descent. 10.3 • P RETRAINING LARGE LANGUAGE MODELS 211 long and thanks forNext token all Loss … = <latexit sha1_base64="AovqpaL476UmJ1EU1xZPgDZ70tQ=">AAAB9nicbVDLSsNAFL2pr1pfURcu3AwWwY0lEakui25cVrAPaEqYTCbt0EkmzEzEEvIrbkTcKPgZ/oJ/Y9Jm09YDA4dzznDvPV7MmdKW9WtU1tY3Nreq27Wd3b39A/PwqKtEIgntEMGF7HtYUc4i2tFMc9qPJcWhx2nPm9wXfu+ZSsVE9KSnMR2GeBSxgBGsc8k1Ty4dLkZo6qZOiPVYhimO/CyruWbdalgzoFVil6QOJdqu+eP4giQhjTThWKmBbcV6mGKpGeE0qzmJojEmEzyi6WztDJ3nko8CIfMXaTRTF3I4VGoaenmy2E0te4X4nzdIdHA7TFkUJ5pGZD4oSDjSAhUdIJ9JSjSf5gQTyfINERljiYnOmypOt5cPXSXdq4bdbDQfr+utu7KEKpzCGVyADTfQggdoQwcIZPAGn/BlvBivxrvxMY9WjPLPMSzA+P4DPEiSHA==</latexit> log y and Stacked Transformer Blocks So long and thanks for … … … U Input tokens x1 x2 Language Modeling Head x3 x4 x5 Input Encoding E 1+ E 2+ E 3+ E 4+ E 5+ … … ……… U U U U … logits logits logits logits logits … <latexit sha1_base64="q3ZgXDyG7qtkT7t8hT47RdlwYG4=">AAAB+XicbVDLSsNAFJ3UV62vWHe6GVsEN5bERXUlBUVcVrAPaEqYTCft0MlMmJkIIQT8AT/CTRE3Cv6Ev+DfmLTdtPXAwOGcM9x7jxcyqrRl/RqFtfWNza3idmlnd2//wDwst5WIJCYtLJiQXQ8pwignLU01I91QEhR4jHS88W3ud56JVFTwJx2HpB+gIac+xUhnkmseXzhMDGHsJk6A9EgGiR4hPlZpWnLNqlWzpoCrxJ6TauP0tXw3qdw0XfPHGQgcBYRrzJBSPdsKdT9BUlPMSFpyIkVChMdoSJLp5ik8y6QB9IXMHtdwqi7kUKBUHHhZMl9PLXu5+J/Xi7R/3U8oDyNNOJ4N8iMGtYB5DXBAJcGaxRlBWNJsQ4hHSCKss7Ly0+3lQ1dJ+7Jm12v1x6yDezBDEZyACjgHNrgCDfAAmqAFMHgBE/AJvozEeDPejY9ZtGDM/xyBBRjff79pldo=</latexit> log y thanks Figure 10.4 Training a transformer as a language model. Note the key difference between this ﬁgure and the earlier RNN-based version shown in Fig. 8.6. There the calculation of the outputs and the losses at each step was inherently serial given the recurrence in the calculation of the hidden states. With transformers, each training item can be processed in parallel since the output for each element in the sequence is computed separately. Large models are generally trained by ﬁlling the full context window (for exam- ple 4096 tokens for GPT4 or 8192 for Llama 3) with text. If documents are shorter than this, multiple documents are packed into the window with a special end-of-text token between them. The batch size for gradient descent is usually quite large (the largest GPT-3 model uses a batch size of 3.2 million tokens). 10.3.2 Training corpora for large language models Large language models are mainly trained on text scraped from the web, augmented by more carefully curated data. Because these training corpora are so large, they are likely to contain many natural examples that can be helpful for NLP tasks, such as question and answer pairs (for example from FAQ lists), translations of sentences between various languages, documents together with their summaries, and so on. Web text is usually taken from corpora of automatically-crawled web pages like the common crawl, a series of snapshots of the entire web produced by the non-common crawl proﬁt Common Crawl ( https://commoncrawl.org/) that each have billions of webpages. Various versions of common crawl data exist, such as the Colossal Clean Crawled Corpus (C4; Raffel et al. 2020), a corpus of 156 billion tokens of English that is ﬁltered in various ways (deduplicated, removing non-natural language like code, sentences with offensive words from a blocklist). This C4 corpus seems to consist in large part of patent text documents, Wikipedia, and news sites (Dodge et al., 2021). Wikipedia plays a role in lots of language model training, as do corpora of books. The Pile (Gao et al., 2020) is an 825 GB English text corpus that is constructed byThe Pile publicly released code, containing again a large amount of text scraped from the web 212 CHAPTER 10 • L ARGE LANGUAGE MODELS as well as books and Wikipedia; Fig. 10.5 shows its composition. Dolma is a larger open corpus of English, created with public tools, containing three trillion tokens, which similarly consists of web text, academic papers, code, books, encyclopedic materials, and social media (Soldaini et al., 2024). Figure 1: Treemap of Pile components by effective size. troduce a new ﬁltered subset of Common Crawl, Pile-CC, with improved extraction quality. Through our analyses, we conﬁrm that the Pile is signiﬁcantly distinct from pure Common Crawl data. Additionally, our evaluations show that the existing GPT-2 and GPT-3 models perform poorly on many components of the Pile, and that models trained on the Pile signiﬁcantly outperform both raw and ﬁltered Common Crawl models. To com- plement the performance evaluations, we also per- form an exploratory analysis of the text within the Pile to provide a detailed picture of the data. We hope that our extensive documentation of the con- struction and characteristics of the Pile will help researchers make informed decisions about poten- tial downstream applications. Finally, we make publicly available the preprocess- ing code for the constituent datasets of the Pile and the code for constructing alternative versions2. In the interest of reproducibility, we also document all processing performed on each dataset (and the Pile as a whole) in as much detail as possible. For further details about the processing of each dataset, see Section2 and AppendixC. 2https://github.com/EleutherAI/ the-pile 1.1 Contributions The core contributions of this paper are: 1. The introduction of a 825.18 GiB english- language dataset for language modeling com- bining 22 diverse sources. 2. The introduction of14 new language model- ing datasets, which we expect to be of inde- pendent interest to researchers. 3. Evaluations demonstrating signiﬁcant im- provements across many domains by GPT-2- sized models trained on this new dataset, com- pared to training on CC-100 and raw Common Crawl. 4. The investigation and documentation of this dataset, which we hope will better inform re- searchers about how to use it as well as moti- vate them to undertake similar investigations of their own data. 2 The Pile Datasets The Pile is composed of 22 constituent sub-datasets, as shown in Table1. FollowingBrown et al.(2020), we increase the weights of higher quality compo- nents, with certain high-quality datasets such as Wikipedia being seen up to 3 times (“epochs”) for 2 Figure 10.5 The Pile corpus, showing the size of different components, color coded as academic (articles from PubMed and ArXiv, patents from the USPTA; internet (webtext in- cluding a subset of the common crawl as well as Wikipedia), prose (a large corpus of books), dialogue (including movie subtitles and chat data), and misc.. Figure from Gao et al. (2020). Filtering for quality and safety Pretraining data drawn from the web is ﬁltered for both quality and safety. Quality ﬁlters are classiﬁers that assign a score to each document. Quality is of course subjective, so different quality ﬁlters are trained in different ways, but often to value high-quality reference corpora like Wikipedia, books, and particular websites and to avoid websites with lots ofPII (Personal Iden-PII tiﬁable Information) or adult content. Filters also remove boilerplate text which is very frequent on the web. Another kind of quality ﬁltering is deduplication, which can be done at various levels, so as to remove duplicate documents, duplicate web pages, or duplicate text. Quality ﬁltering generally improves language model per- formance (Longpre et al., 2024b; Llama Team, 2024). Safety ﬁltering is again a subjective decision, and often includes toxicity detec- tion based on running off-the-shelf toxicity classiﬁers. This can have mixed results. One problem is that current toxicity classiﬁers mistakenly ﬂag non-toxic data if it is generated by speakers of minority dialects like African American English (Xu et al., 2021). Another problem is that models trained on toxicity-ﬁltered data, while somewhat less toxic, are also worse at detecting toxicity themselves (Longpre et al., 2024b). These issues make the question of how to do better safety ﬁltering an im- portant open problem. Using large datasets scraped from the web to train language models poses ethical and legal questions: Copyright: Much of the text in these large datasets (like the collections of ﬁc- tion and non-ﬁction books) is copyrighted. In some countries, like the United States, the fair use doctrine may allow copyrighted content to be used for transformative uses, but it’s not clear if that remains true if the language mod- els are used to generate text that competes with the market for the text they 10.3 • P RETRAINING LARGE LANGUAGE MODELS 213 are trained on (Henderson et al., 2023). Data consent: Owners of websites can indicate that they don’t want their sites to be crawled by web crawlers (either via a robots.txt ﬁle, or via Terms of Service). Recently there has been a sharp increase in the number of web- sites that have indicated that they don’t want large language model builders crawling their sites for training data (Longpre et al., 2024a). Because it’s not clear what legal status these indications have in different countries, or whether these restrictions are retroactive, what effect this will have on large pretraining datasets is unclear. Privacy: Large web datasets also have privacy issues since they contain private information like phone numbers and IP addresses. While ﬁlters are used to try to remove websites likely to contain large amounts of personal information, such ﬁltering isn’t sufﬁcient. 10.3.3 Finetuning Although the enormous pretraining data for a large language model includes text from many domains, it’s often the case that we want to apply it in a new domain or task that might not have appeared sufﬁciently in the pre-training data. For example, we might want a language model that’s specialized to legal or medical text. Or we might have a multilingual language model that knows many languages but might beneﬁt from some more data in our particular language of interest. Or we want a language model that is specialized to a particular task. In such cases, we can simply continue training the model on relevant data from the new domain or language (Gururangan et al., 2020). This process of taking a fully pretrained model and running additional training passes on some new data is called ﬁnetuning. Fig. 10.6 sketches the paradigm.ﬁnetuning Fine- tuning Data Pretraining Data Pretraining … … … Fine-tuning … … … Pretrained LM Fine-tuned LM Figure 10.6 Pretraining and ﬁnetuning. A pre-trained model can be ﬁnetuned to a par- ticular domain, dataset, or task. There are many different ways to ﬁnetune, depending on exactly which parameters are updated from the ﬁnetuning data: all the parameters, some of the parameters, or only the parameters of speciﬁc extra circuitry. We’ll introduce four related kinds of ﬁnetuning in this chapter and the two fol- lowing chapters. In all four cases, ﬁnetuning means the process of taking a pre- trained model and further adapting some or all of its parameters to some new data. But they differ on exactly which parameters get updated. In the ﬁrst kind of ﬁnetuning we retrain all the parameters of the model on this new data, using the same method (word prediction) and loss function (cross-entropy loss) as for pretraining. In a sense it’s as if the new data were at the tail end of 214 CHAPTER 10 • L ARGE LANGUAGE MODELS the pretraining data, and so you’ll sometimes see this method calledcontinued pre- training.continued pretraining Retraining all the parameters of the model is very slow and expensive when the language model is huge. So instead we canfreeze some of the parameters (i.e., leavefreeze them unchanged from their pretrained value) and train only a subset of parameters on the new data. In Section 10.5.3 we’ll describe this second variety of ﬁnetun- ing, called parameter-efﬁcient ﬁnetuning, or PEFT. because we efﬁciently select speciﬁc parameters to update when ﬁnetuning, and leave the rest in their pretrained values. In Chapter 11 we’ll introduce a third kind of ﬁnetuning, also parameter-efﬁcient. In this version, the goal is to use a language model as a kind of classiﬁer or labeler for a speciﬁc task. For example we might train the model to be a sentiment classiﬁer. We do this by adding extra neural circuitry (an extra head) after the top layer of the model. This classiﬁcation head takes as input some of the top layer embeddings of the transformer and produces as output a classiﬁcation. In this method, most com- monly used with masked language models like BERT, we freeze the entire pretrained model and only train the classiﬁcation head on some new data, usually labeled with some class that we want to predict. Finally, in Chapter 12 we’ll introduce a fourth kind of ﬁnetuning, that is a cru- cial component of the largest language models:supervised ﬁnetuning or SFT. SFT is often used for instruction ﬁnetuning, in which we want a pretrained language model to learn to follow text instructions, for example to answer questions or follow a command to write something. Here we create a dataset of prompts and desired responses (for example questions and their answers, or commands and their ful- ﬁllments), and we train the language model using the normal cross-entropy loss to predict each token in the instruction prompt iteratively, essentially training it to pro- duce the desired response from the command in the prompt. It’s called supervised because unlike in pretraining, where we just take any data and predict the words in it, we build the special ﬁnetuning dataset by hand, creating supervised responses to each command. Often everything that happens after pretraining is lumped together aspost-training; we’ll discuss the various parts of post-training in Chapter 12. 10.4 Evaluating Large Language Models Perplexity As we ﬁrst saw in Chapter 3, one way to evaluate language models is to measure how well they predict unseen text. Intuitively, good models are those that assign higher probabilities to unseen data (are less surprised when encountering the new words). We instantiate this intuition by using perplexity to measure the quality of aperplexity language model. Recall from page 40 that the perplexity of a model θ on an unseen test set is the inverse probability that θ assigns to the test set, normalized by the test set length. For a test set of n tokens w1:n, the perplexity is Perplexityθ (w1:n) = Pθ (w1:n)−1 n = n √ 1 Pθ (w1:n) (10.7) To visualize how perplexity can be computed as a function of the probabilities the 10.4 • E VALUATING LARGE LANGUAGE MODELS 215 LM computes for each new word, we can use the chain rule to expand the computa- tion of probability of the test set: Perplexityθ (w1:n) = n √ n∏ i=1 1 Pθ (wi\|w<i) (10.8) Note that because of the inverse in Eq. 10.7, the higher the probability of the word sequence, the lower the perplexity. Thus thethe lower the perplexity of a model on the data, the better the model. Minimizing perplexity is equivalent to maximizing the test set probability according to the language model. One caveat: because perplexity depends on the length of a text, it is very sensitive to differences in the tokenization algorithm. That means that it’s hard to exactly compare perplexities produced by two language models if they have very different tokenizers. For this reason perplexity is best used when comparing language models that use the same tokenizer. Other factors While the predictive accuracy of a language model, as measured by perplexity, is a very useful metric, we also care about different kinds of accuracy, for the downstream tasks we apply our language model to. For each task like machine translation, summarization, question answering, speech recognition, and dialogue, we can measure the accuracy at those tasks. Future chapters will introduce task- speciﬁc metrics that allow us to evaluate how accuracy or correct language models are at these downstream tasks. But when evaluating models we also care about factors besides any of these kinds of accuracy (Dodge et al., 2019; Ethayarajh and Jurafsky, 2020). For example, we often care about how a big a model is, and how long it takes to train or do inference. This can matter because we have constraints on time either for training or at inference. Or we may have constraints on memory, since the GPUs we run our models on have ﬁxed memory sizes. Big models also use more energy, and we prefer models that use less energy, both to reduce the environmental impact of the model and to reduce the ﬁnancial cost of building or deploying it. We can target our evaluation to these factors by measuring performance normalized to a giving compute or memory budget. We can also directly measure the energy usage of our model in kWh or in kilograms of CO 2 emitted (Strubell et al., 2019; Henderson et al., 2020; Liang et al., 2023). Another feature that a language model evaluation can measure is fairness. We know that language models are biased, exhibiting gendered and racial stereotypes, or decreased performance for language from or about certain demographics groups. There are language model evaluation benchmarks that measure the strength of these biases, such as StereoSet (Nadeem et al., 2021), RealToxicityPrompts (Gehman et al., 2020), and BBQ (Parrish et al., 2022) among many others. We also want language models whose performance is equally fair to different groups. For exam- ple, we could chose an evaluation that is fair in a Rawlsian sense by maximizing the welfare of the worst-off group (Rawls, 2001; Hashimoto et al., 2018; Sagawa et al., 2020). Finally, there are many kinds of leaderboards like Dynabench (Kiela et al., 2021) and general evaluation protocols like HELM (Liang et al., 2023); we will return to these in later chapters when we introduce evaluation metrics for speciﬁc tasks like question answering and information retrieval. 216 CHAPTER 10 • L ARGE LANGUAGE MODELS 10.5 Dealing with Scale Large language models are large. For example the Llama 3.1 405B Instruct model from Meta has 405 billion parameters ( L=126 layers, a model dimensionality of d=16,384, A=128 attention heads) and was trained on 15.6 terabytes of text tokens (Llama Team, 2024), using a vocabulary of 128K tokens. So there is a lot of research on understanding how LLMs scale, and especially how to implement them given limited resources. In the next few sections we discuss how to think about scale (the concept of scaling laws), and important techniques for getting language models to work efﬁciently, such as the KV cache and parameter-efﬁcient ﬁne tuning. 10.5.1 Scaling laws The performance of large language models has shown to be mainly determined by 3 factors: model size (the number of parameters not counting embeddings), dataset size (the amount of training data), and the amount of compute used for training. That is, we can improve a model by adding parameters (adding more layers or having wider contexts or both), by training on more data, or by training for more iterations. The relationships between these factors and performance are known as scaling laws. Roughly speaking, the performance of a large language model (the loss) scalesscaling laws as a power-law with each of these three properties of model training. For example, Kaplan et al. (2020) found the following three relationships for loss L as a function of the number of non-embedding parameters N, the dataset size D, and the compute budget C, for models training with limited parameters, dataset, or compute budget, if in each case the other two properties are held constant: L(N) = (Nc N )αN (10.9) L(D) = (Dc D )αD (10.10) L(C) = (Cc C )αC (10.11) The number of (non-embedding) parameters N can be roughly computed as fol- lows (ignoring biases, and with d as the input and output dimensionality of the model, dattn as the self-attention layer size, anddff the size of the feedforward layer): N ≈ 2 d nlayer(2 dattn +dff) ≈ 12 nlayer d2 (10.12) (assuming dattn = dff/4 = d) Thus GPT-3, with n = 96 layers and dimensionality d = 12288, has 12 ×96 × 122882 ≈175 billion parameters. The values of Nc, Dc, Cc, αN , αD, and αC depend on the exact transformer architecture, tokenization, and vocabulary size, so rather than all the precise values, scaling laws focus on the relationship with loss.2 Scaling laws can be useful in deciding how to train a model to a particular per- formance, for example by looking at early in the training curve, or performance with 2 For the initial experiment in Kaplan et al. (2020) the precise values were αN = 0.076, Nc = 8.8 ×1013 (parameters), αD = 0.095, Dc = 5.4 ×1013 (tokens), αC = 0.050, Cc = 3.1 ×108 (petaﬂop-days). 10.5 • D EALING WITH SCALE 217 smaller amounts of data, to predict what the loss would be if we were to add more data or increase model size. Other aspects of scaling laws can also tell us how much data we need to add when scaling up a model. 10.5.2 KV Cache We saw in Fig. 9.10 and in Eq. 9.33 (repeated below) how the attention vector can be very efﬁciently computed in parallel for training, via two matrix multiplications: A = softmax (QK⊺ √dk ) V (10.13) Unfortunately we can’t do quite the same efﬁcient computation in inference as in training. That’s because at inference time, we iteratively generate the next tokens one at a time. For a new token that we have just generated, call it xi, we need to compute its query, key, and values by multiplying by WQ, WK, and WV respec- tively. But it would be a waste of computation time to recompute the key and value vectors for all the prior tokens x<i; at prior steps we already computed these key and value vectors! So instead of recomputing these, whenever we compute the key and value vectors we store them in memory in the KV cache, and then we can justKV cache grab them from the cache when we need them. Fig. 10.7 modiﬁes Fig. 9.10 to show the computation that takes place for a single new token, showing which values we can take from the cache rather than recompute. q4 k1 k2 k4 Q KT QKT v1 v2 v3 v4 V q4•k1 q4•k2 q4•k3 q4•k4 x = =x a4 A 1 x dk dk x N 1 x N N x dv 1 x dv k3 Figure 10.7 Parts of the attention computation (extracted from Fig. 9.10) showing, in black, the vectors that can be stored in the cache rather than recomputed when computing the atten- tion score for the 4th token. 10.5.3 Parameter Efﬁcient Fine Tuning As we mentioned above, it’s very common to take a language model and give it more information about a new domain by ﬁnetuning it (continuing to train it to predict upcoming words) on some additional data. Fine-tuning can be very difﬁcult with very large language models, because there are enormous numbers of parameters to train; each pass of batch gradient descent has to backpropagate through many many huge layers. This makes ﬁnetuning huge language models extremely expensive in processing power, in memory, and in time. For this reason, there are alternative methods that allow a model to be ﬁnetuned without changing all the parameters. Such methods are called parameter-efﬁcient ﬁne tuning or sometimes PEFT, because we efﬁciently select a subset of parameters parameter- efﬁcient ﬁne tuning PEFT to update when ﬁnetuning. For example we freeze some of the parameters (don’t change them), and only update some particular subset of parameters. 218 CHAPTER 10 • L ARGE LANGUAGE MODELS Here we describe one such model, calledLoRA, for Low-Rank Adaptation. TheLoRA intuition of LoRA is that transformers have many dense layers which perform matrix multiplication (for example the WQ, WK, WV, WO layers in the attention computa- tion). Instead of updating these layers during ﬁnetuning, with LoRA we freeze these layers and instead update a low-rank approximation that has fewer parameters. Consider a matrix W of dimensionality [N ×d] that needs to be updated during ﬁnetuning via gradient descent. Normally this matrix would get updates ∆W of dimensionality [N ×d], for updating the N ×d parameters after gradient descent. In LoRA, we freeze W and update instead a low-rank decomposition of W. We create two matrices A and B, where A has size [N ×r] and B has size [r×d], and we choose r to be quite small, r <<min(d,N). During ﬁnetuning we update A and B instead of W. That is, we replace W + ∆W with W + BA. Fig. 10.8 shows the intuition. For replacing the forward pass h = xW, the new forward pass is instead: h = xW +xAB (10.14) h Pretrained Weights W d k r k A Br x d 1 1 k d × Figure 10.8 The intuition of LoRA. We freezeW to its pretrained values, and instead ﬁne- tune by training a pair of matricesA and B, updating those instead of W, and just sum W and the updated AB. LoRA has a number of advantages. It dramatically reduces hardware require- ments, since gradients don’t have to be calculated for most parameters. The weight updates can be simply added in to the pretrained weights, since BA is the same size as W). That means it doesn’t add any time during inference. And it also means it’s possible to build LoRA modules for different domains and just swap them in and out by adding them in or subtracting them from W. In its original version LoRA was applied just to the matrices in the attention computation (the WQ, WK, WV, and WO layers). Many variants of LoRA exist. 10.6 • P OTENTIAL HARMS FROM LANGUAGE MODELS 219 10.6 Potential Harms from Language Models Large pretrained neural language models exhibit many of the potential harms dis- cussed in Chapter 4 and Chapter 6. Many of these harms become realized when pretrained language models are used for any downstream task, particularly those involving text generation, whether question answering, machine translation, or in assistive technologies like writing aids or web search query completion, or predic- tive typing for email (Olteanu et al., 2020). For example, language models are prone to saying things that are false, a prob- lem called hallucination. Language models are trained to generate text that is pre-hallucination dictable and coherent, but the training algorithms we have seen so far don’t have any way to enforce that the text that is generated is correct or true. This causes enormous problems for any application where the facts matter! We’ll return to this issue in Chapter 14 where we introduce proposed mitigation methods like retrieval augmented generation. A second source of harm is that language models can generate toxic language.toxic language Gehman et al. (2020) show that even completely non-toxic prompts can lead large language models to output hate speech and abuse their users. Language models also generate stereotypes (Cheng et al., 2023) and negative attitudes (Brown et al., 2020; Sheng et al., 2019) about many demographic groups. One source of biases is the training data. Gehman et al. (2020) shows that large language model training datasets include toxic text scraped from banned sites. There are other biases than toxicity: the training data is disproportionately generated by authors from the US and from developed countries. Such biased population samples likely skew the resulting generation toward the perspectives or topics of this group alone. Furthermore, language models can amplify demographic and other biases in training data, just as we saw for embedding models in Chapter 6. Datasets can be another source of harms. We already saw in Section 10.3.2 that using pretraining corpora scraped from the web can lead to harms related to copyright and data consent. We also mentioned that pretraining data can tend to have private information like phone numbers and addresses. This is problematic because large language models can leak information from their training data. That is, an adversary can extract training-data text from a language model such as a per- son’s name, phone number, and address (Henderson et al. 2017, Carlini et al. 2021). This becomes even more problematic when large language models are trained on extremely sensitive private datasets such as electronic health records. Language models can also be used by malicious actors for generating text for misinformation, phishing, or other socially harmful activities (Brown et al., 2020). McGufﬁe and Newhouse (2020) show how large language models generate text that emulates online extremists, with the risk of amplifying extremist movements and their attempt to radicalize and recruit. Finding ways to mitigate all these harms is an important current research area in NLP. At the very least, carefully analyzing the data used to pretrain large language models is important as a way of understanding issues of toxicity, bias, privacy, and fair use, making it extremely important that language models include datasheets (page 16) or model cards (page 74) giving full replicable information on the cor- pora used to train them. Open-source models can specify their exact training data. Requirements that models are transparent in such ways is also in the process of being incorporated into the regulations of various national governments. 220 CHAPTER 10 • L ARGE LANGUAGE MODELS 10.7 Summary This chapter has introduced the large language model, and how it can be built out of the transformer. Here’s a summary of the main points that we covered: • Many NLP tasks—such as question answering, summarization, sentiment, and machine translation—can be cast as tasks of word prediction and hence addressed with Large language models. • Large language models are generally pretrained on large datasets of 100s of billions of words generally scraped from the web. • These datasets need to be ﬁltered for quality and balanced for domains by upsampling and downsampling. Addressing some problems with pretraining data, like toxicity, are open research problems. • The choice of which word to generate in large language models is generally done by using a sampling algorithm. • Language models are evaluated by perplexity but there are also evaluations of accuracy downstream tasks, and ways to measure other factors like fairness and energy use. • There are various computational tricks for making large language models more efﬁcient, such as the KV cache and parameter-efﬁcient ﬁnetuning. • Because of their ability to be used in so many ways, language models also have the potential to cause harms. Some harms include hallucinations, bias, stereotypes, misinformation and propaganda, and violations of privacy and copyright. Bibliographical and Historical Notes As we discussed in Chapter 3, the earliest language models were the n-gram lan- guage models developed (roughly simultaneously and independently) by Fred Je- linek and colleagues at the IBM Thomas J. Watson Research Center, and James Baker at CMU. It was the Jelinek and the IBM team who ﬁrst coined the term lan- guage model to mean a model of the way any kind of linguistic property (grammar, semantics, discourse, speaker characteristics), inﬂuenced word sequence probabil- ities (Jelinek et al., 1975). They contrasted the language model with the acoustic model which captured acoustic/phonetic characteristics of phone sequences. N-gram language models were very widely used over the next 30 years and more, across a wide variety of NLP tasks like speech recognition and machine translations, often as one of multiple components of the model. The contexts for these n-gram models grew longer, with 5-gram models used quite commonly by very efﬁcient LM toolkits (Stolcke, 2002; Heaﬁeld, 2011). The roots of the neural language model lie in multiple places. One was the application in the 1990s, again in Jelinek’s group at IBM Research, of discrimi- native classiﬁers to language models. Roni Rosenfeld in his dissertation (Rosen- feld, 1992) ﬁrst applied logistic regression (under the name maximum entropy or maxent models) to language modeling in that IBM lab, and published a more fully formed version in Rosenfeld (1996). His model integrated various sorts of infor- mation in a logistic regression predictor, including n-gram information along with BIBLIOGRAPHICAL AND HISTORICAL NOTES 221 other features from the context, including distant n-grams and pairs of associated words called trigger pairs. Rosenfeld’s model preﬁgured modern language models by being a statistical word predictor trained in a self-supervised manner simply by learning to predict upcoming words in a corpus. Another was the ﬁrst use of pretrained embeddings to model word meaning in the LSA/LSI models (Deerwester et al., 1988). Recall from the history section of Chap- ter 6 that in LSA (latent semantic analysis) a term-document matrix was trained on a corpus and then singular value decomposition was applied and the ﬁrst 300 dimen- sions were used as a vector embedding to represent words. Landauer et al. (1997) ﬁrst used the word “embedding”. In addition to their development of the idea of pre- training and of embeddings, the LSA community also developed ways to combine LSA embeddings with n-grams in an integrated language model (Bellegarda, 1997; Coccaro and Jurafsky, 1998). In a very inﬂuential series of papers developing the idea of neural language models, (Bengio et al. 2000; Bengio et al. 2003; Bengio et al. 2006), Yoshua Ben- gio and colleagues drew on the central ideas of both these lines of self-supervised language modeling work, (the discriminatively trained word predictor, and the pre- trained embeddings). Like the maxent models of Rosenfeld, Bengio’s model used the next word in running text as its supervision signal. Like the LSA models, Ben- gio’s model learned an embedding, but unlike the LSA models did it as part of the process of language modeling. The Bengio et al. (2003) model was a neural lan- guage model: a neural network that learned to predict the next word from prior words, and did so via learning embeddings as part of the prediction process. The neural language model was extended in various ways over the years, perhaps most importantly in the form of the RNN language model of Mikolov et al. (2010) and Mikolov et al. (2011). The RNN language model was perhaps the ﬁrst neural model that was accurate enough to surpass the performance of a traditional 5-gram language model. Soon afterwards, Mikolov et al. (2013a) and Mikolov et al. (2013b) proposed to simplify the hidden layer of these neural net language models to create pretrained word2vec word embeddings. The static embedding models like LSA and word2vec instantiated a particular model of pretraining: a representation was trained on a pretraining dataset, and then the representations could be used in further tasks. ‘Dai and Le (2015) and (Peters et al., 2018) reframed this idea by proposing models that were pretrained using a language model objective, and then the identical model could be either frozen and directly applied for language modeling or further ﬁnetuned still using a language model objective. For example ELMo used a biLSTM self-supervised on a large pretrained dataset using a language model objective, then ﬁnetuned on a domain- speciﬁc dataset, and then froze the weights and added task-speciﬁc heads. The ELMo work was particularly inﬂuential and its appearance was perhaps the mo- ment when it became clear to the community that language models could be used as a general solution for NLP problems. Transformers were ﬁrst applied as encoder-decoders (Vaswani et al., 2017) and then to masked language modeling (Devlin et al., 2019) (as we’ll see in Chapter 13 and Chapter 11). Radford et al. (2019) then showed that the transformer-based au- toregressive language model GPT2 could perform zero-shot on many NLP tasks like summarization and question answering. The technology used for transformer-based language models can also be applied to other domains and tasks, like vision, speech, and genetics. the term foundation 222 CHAPTER 10 • L ARGE LANGUAGE MODELS model is sometimes used as a more general term for this use of large languagefoundation model model technology across domains and areas, when the elements we are computing over are not necessarily words. Bommasani et al. (2021) is a broad survey that sketches the opportunities and risks of foundation models, with special attention to large language models. CHAPTER 11 Masked Language Models Larvatus prodeo [Masked, I go forward] Descartes In the previous two chapters we introduced the transformer and saw how to pre- train a transformer language model as a causal or left-to-right language model. In this chapter we’ll introduce a second paradigm for pretrained language models, the bidirectional transformer encoder, and the most widely-used version, the BERTBERT model (Devlin et al., 2019). This model is trained via masked language modeling, masked language modeling where instead of predicting the following word, we mask a word in the middle and ask the model to guess the word given the words on both sides. This method thus allows the model to see both the right and left context. We also introduced ﬁnetuning in the prior chapter. Here we describe a newﬁnetuning kind of ﬁnetuning, in which we take the transformer network learned by these pre- trained models, add a neural net classiﬁer after the top layer of the network, and train it on some additional labeled data to perform some downstream task like named entity tagging or natural language inference. As before, the intuition is that the pretraining phase learns a language model that instantiates rich representations of word meaning, that thus enables the model to more easily learn (‘be ﬁnetuned to’) the requirements of a downstream language understanding task. This aspect of the pretrain-ﬁnetune paradigm is an instance of what is called transfer learning in ma-transfer learning chine learning: the method of acquiring knowledge from one task or domain, and then applying it (transferring it) to solve a new task. The second idea that we introduce in this chapter is the idea of contextual em- beddings: representations for words in context. The methods of Chapter 6 like word2vec or GloVe learned a single vector embedding for each unique word w in the vocabulary. By contrast, with contextual embeddings, such as those learned by masked language models like BERT, each wordw will be represented by a different vector each time it appears in a different context. While the causal language models of Chapter 9 also use contextual embeddings, the embeddings created by masked language models seem to function particularly well as representations. 11.1 Bidirectional Transformer Encoders Let’s begin by introducing the bidirectional transformer encoder that underlies mod- els like BERT and its descendants like RoBERTa (Liu et al., 2019) or SpanBERT (Joshi et al., 2020). In Chapter 9 we introduced causal (left-to-right) transformers and in Chapter 10 saw how they can serve as the basis for language models that can be applied to autoregressive contextual generation problems like question answering or summarization. But this left-to-right nature of these models is also a limitation, because there are tasks for which it would be useful, when processing a token, to be able to peak at future tokens. This is especially true for sequence labeling tasks 224 CHAPTER 11 • M ASKED LANGUAGE MODELS in which we want to tag each token with a label, such as the named entity tagging task we’ll introduce in Section 11.5, or tasks like part-of-speech tagging or parsing that come up in later chapters. The bidirectional encoders that we introduce here are a different kind of beast than causal models. The causal models of Chapter 9 are generative models, de- signed to easily generate the next token in a sequence. But the focus of bidirec- tional encoders is instead on computing contextualized representations of the input tokens. Bidirectional encoders use self-attention to map sequences of input embed- dings (x1,...,xn) to sequences of output embeddings of the same length (h1,...,hn), where the output vectors have been contextualized using information from the en- tire input sequence. These output embeddings are contextualized representations of each input token that are useful across a range of applications where we need to do a classiﬁcation or a decision based on the token in context. Remember that we said the models of Chapter 9 are sometimes called decoder- only, because they correspond to the decoder part of the encoder-decoder model we will introduce in Chapter 13. By contrast, the masked language models of this chap- ter are sometimes called encoder-only, because they produce an encoding for each input token but generally aren’t used to produce running text by decoding/sampling. That’s an important point: masked language models are not used for generation. They are generally instead used for interpretative tasks. 11.1.1 The architecture for bidirectional masked models Let’s ﬁrst discuss the overall architecture. Bidirectional transformer-based language models differ in two ways from the causal transformers in the previous chapters. The ﬁrst is that the attention function isn’t causal; the attention for a token i can look at following tokens i +1 and so on. The second is that the training is slightly different since we are predicting something in the middle of our text rather than at the end. We’ll discuss the ﬁrst here and the second in the following section. Fig. 11.1a, reproduced here from Chapter 9, shows the information ﬂow in the left-to-right approach of Chapter 9. The attention computation at each token is based on the preceding (and current) input tokens, ignoring potentially useful information located to the right of the token under consideration. Bidirectional encoders over- come this limitation by allowing the attention mechanism to range over the entire input, as shown in Fig. 11.1b. a) A causal self-attention layer b) A bidirectional self-attention layer attentionattentionattentionattentionattention a1 a2 a3 a4 a5 x3 x4 x5x1 x2 attentionattentionattentionattentionattention a1 a2 a3 a4 a5 x3 x4 x5x1 x2 Figure 11.1 (a) The causal transformer from Chapter 9, highlighting the attention computation at token 3. The attention value at each token is computed using only information seen earlier in the context. (b) Information ﬂow in a bidirectional attention model. In processing each token, the model attends to all inputs, both before and after the current one. So attention for token 3 can draw on information from following tokens. The implementation is very simple! We simply remove the attention masking step that we introduced in Eq. 9.33. Recall from Chapter 9 that we had to mask the QK⊺ matrix for causal transformers so that attention couldn’t look at future tokens 11.1 • B IDIRECTIONAL TRANSFORMER ENCODERS 225 (repeated from Eq. 9.33 for a single attention head): head = softmax ( mask (QK⊺ √dk )) V (11.1) q1•k1 q2•k1 q2•k2 q4•k1 q4•k2 q4•k3 q4•k4 q3•k1 q3•k2 q3•k3 N N −∞ −∞ −∞ −∞ −∞ −∞ q1•k1 q2•k1 q2•k2 q4•k1 q4•k2 q4•k3 q4•k4 q3•k1 q3•k2 q3•k3 N N q1•k2 q1•k3 q1•k4 q2•k3 q2•k4 q3•k4 (a) (b) Figure 11.2 The N ×N QK⊺ matrix showing the qi ·kj values, with the upper-triangle portion of the comparisons matrix zeroed out (set to −∞, which the softmax will turn to zero). Fig. 11.2 shows the masked version ofQK⊺ and the unmasked version. For bidi- rectional attention, we use the unmasked version of Fig. 11.2b. Thus the attention computation for bidirectional attention is exactly the same as Eq. 11.1 but with the mask removed: head = softmax (QK⊺ √dk ) V (11.2) Otherwise, the attention computation is identical to what we saw in Chapter 9, as is the transformer block architecture (the feedforward layer, layer norm, and so on). As in Chapter 9, the input is also a series of subword tokens, usually computed by one of the 3 popular tokenization algorithms (including the BPE algorithm that we already saw in Chapter 2 and two others, the WordPiece algorithm and the SentencePiece Unigram LM algorithm). That means every input sentence ﬁrst has to be tokenized, and all further processing takes place on subword tokens rather than words. This will require, as we’ll see in the third part of the textbook, that for some NLP tasks that require notions of words (like parsing) we will occasionally need to map subwords back to words. To make this more concrete, the original English-only bidirectional transformer encoder model, BERT (Devlin et al., 2019), consisted of the following: • An English-only subword vocabulary consisting of 30,000 tokens generated using the WordPiece algorithm (Schuster and Nakajima, 2012). • Input context window N=512 tokens, and model dimensionality d=768 • So X, the input to the model, is of shape [N ×d] = [512 ×768]. • L=12 layers of transformer blocks, each with A=12 (bidirectional) multihead attention layers. • The resulting model has about 100M parameters. The larger multilingual XLM-RoBERTa model, trained on 100 languages, has • A multilingual subword vocabulary with 250,000 tokens generated using the SentencePiece Unigram LM algorithm (Kudo and Richardson, 2018b). 226 CHAPTER 11 • M ASKED LANGUAGE MODELS • Input context windowN=512 tokens, and model dimensionalityd=1024, hence X, the input to the model, is of shape [N ×d] = [512 ×1024]. • L=24 layers of transformer blocks, with A=16 multihead attention layers each • The resulting model has about 550M parameters. Note that 550M parameters is relatively small as large language models go (Llama 3 has 405B parameters, so is 3 orders of magnitude bigger). Indeed, masked language models tend to be much smaller than causal language models. 11.2 Training Bidirectional Encoders We trained causal transformer language models in Chapter 9 by making them it- eratively predict the next word in a text. But eliminating the causal mask in at- tention makes the guess-the-next-word language modeling task trivial—the answer is directly available from the context—so we’re in need of a new training scheme. Instead of trying to predict the next word, the model learns to perform a ﬁll-in-the- blank task, technically called the cloze task (Taylor, 1953). To see this, let’s returncloze task to the motivating example from Chapter 3. Instead of predicting which words are likely to come next in this example: The water of Walden Pond is so beautifully we’re asked to predict a missing item given the rest of the sentence. The of Walden Pond is so beautifully ... That is, given an input sequence with one or more elements missing, the learning task is to predict the missing elements. More precisely, during training the model is deprived of one or more tokens of an input sequence and must generate a probability distribution over the vocabulary for each of the missing items. We then use the cross- entropy loss from each of the model’s predictions to drive the learning process. This approach can be generalized to any of a variety of methods that corrupt the training input and then asks the model to recover the original input. Examples of the kinds of manipulations that have been used include masks, substitutions, reorder- ings, deletions, and extraneous insertions into the training text. The general name for this kind of training is called denoising: we corrupt (add noise to) the input indenoising some way (by masking a word, or putting in an incorrect word) and the goal of the system is to remove the noise. 11.2.1 Masking Words Let’s describe theMasked Language Modeling (MLM) approach to training bidi- Masked Language Modeling rectional encoders (Devlin et al., 2019). As with the language model training meth- ods we’ve already seen,MLM uses unannotated text from a large corpus. In MLM training, the model is presented with a series of sentences from the training corpus in which a percentage of tokens (15% in the BERT model) have been randomly cho- sen to be manipulated by the masking procedure. Given an input sentence lunch was delicious and assume we randomly chose the 3rd token delicious to be manipulated, • 80% of the time: The token is replaced with the special vocabulary token named [MASK], e.g. lunch was delicious → lunch was [MASK]. 11.2 • T RAINING BIDIRECTIONAL ENCODERS 227 • 10% of the time: The token is replaced with another token, randomly sampled from the vocabulary based on token unigram probabilities. e.g. lunch was delicious → lunch was gasp. • 10% of the time: the token is left unchanged. e.g. lunch was delicious → lunch was delicious. We then train the model to guess the correct token for the manipulated tokens. Why the three possible manipulations? Adding the [MASK]token creates a mismatch be- tween pretraining and downstream ﬁne-tuning or inference, since when we employ the MLM model to perform a downstream task, we don’t use any[MASK]tokens. If we just replaced tokens with the [MASK], the model might only predict tokens when it sees a [MASK], but we want the model to try to always predict the input token. To train the model to make the prediction, the original input sequence is to- kenized using a subword model and tokens are sampled to be manipulated. Word embeddings for all of the tokens in the input are retrieved from theE embedding ma- trix and combined with positional embeddings to form the input to the transformer, passed through the stack of bidirectional transformer blocks, and then the language modeling head. The MLM training objective is to predict the original inputs for each of the masked tokens and the cross-entropy loss from these predictions drives the training process for all the parameters in the model. That is, all of the input tokens play a role in the self-attention process, but only the sampled tokens are used for learning. LM Head with Softmax over Vocabulary So [mask] and [mask] for long thanks CE Loss all apricot ﬁsh the Token + Positional Embeddings So long and thanks for all ﬁshthe Bidirectional Transformer Encoder + p1 + + + + + + + p2 p3 p4 p5 p6 p7 p8 z1 z2 z3 z4 z5 z6 z7 z8 Figure 11.3 Masked language model training. In this example, three of the input tokens are selected, two of which are masked and the third is replaced with an unrelated word. The probabilities assigned by the model to these three items are used as the training loss. The other 5 tokens don’t play a role in training loss. Fig. 11.3 illustrates this approach with a simple example. Here,long, thanks and the have been sampled from the training sequence, with the ﬁrst two masked andthe replaced with the randomly sampled token apricot. The resulting embeddings are passed through a stack of bidirectional transformer blocks. Recall from Section 9.5 in Chapter 9 that to produce a probability distribution over the vocabulary for each of the masked tokens, the language modeling head takes the output vector hL i from the ﬁnal transformer layer L for each masked token i, multiplies it by the unembed- ding layer ET to produce the logits u, and then uses softmax to turn the logits into 228 CHAPTER 11 • M ASKED LANGUAGE MODELS probabilities y over the vocabulary: ui = hL i ET (11.3) yi = softmax(ui) (11.4) With a predicted probability distribution for each masked item, we can use cross- entropy to compute the loss for each masked item—the negative log probability assigned to the actual masked word, as shown in Fig. 11.3. More formally, for a given vector of input tokens in a sentence or batch be x, let the set of tokens that are masked be M, the version of that sentence with some tokens replaced by masks be xmask, and the sequence of output vectors be h. For a given input token xi, such as the word long in Fig. 11.3, the loss is the probability of the correct word long, given xmask (as summarized in the single output vector hL i ): LMLM (xi) =−logP(xi\|hL i ) The gradients that form the basis for the weight updates are based on the average loss over the sampled learning items from a single training sequence (or batch of sequences). LMLM = − 1 \|M\| ∑ i∈M logP(xi\|hL i ) Note that only the tokens in M play a role in learning; the other words play no role in the loss function, so in that sense BERT and its descendents are inefﬁcient; only 15% of the input samples in the training data are actually used for training weights.1 11.2.2 Next Sentence Prediction The focus of mask-based learning is on predicting words from surrounding contexts with the goal of producing effective word-level representations. However, an im- portant class of applications involves determining the relationship between pairs of sentences. These include tasks like paraphrase detection (detecting if two sentences have similar meanings), entailment (detecting if the meanings of two sentences en- tail or contradict each other) or discourse coherence (deciding if two neighboring sentences form a coherent discourse). To capture the kind of knowledge required for applications such as these, some models in the BERT family include a second learning objective called Next Sen- tence Prediction (NSP). In this task, the model is presented with pairs of sentencesNext Sentence Prediction and is asked to predict whether each pair consists of an actual pair of adjacent sen- tences from the training corpus or a pair of unrelated sentences. In BERT, 50% of the training pairs consisted of positive pairs, and in the other 50% the second sen- tence of a pair was randomly selected from elsewhere in the corpus. The NSP loss is based on how well the model can distinguish true pairs from random pairs. To facilitate NSP training, BERT introduces two special tokens to the input rep- resentation (tokens that will prove useful for ﬁnetuning as well). After tokenizing the input with the subword model, the token [CLS] is prepended to the input sen- tence pair, and the token [SEP] is placed between the sentences and after the ﬁnal token of the second sentence. There are actually two more special tokens, a ‘First Segment’ token, and a ‘Second Segment’ token. These tokens are added in the in- put stage to the word and positional embeddings. That is, each token of the input 1 ELECTRA, another BERT family member, does use all examples for training (Clark et al., 2020b). 11.2 • T RAINING BIDIRECTIONAL ENCODERS 229 X is actually formed by summing 3 embeddings: word, position, and ﬁrst/second segment embeddings. During training, the output vector hL CLS from the ﬁnal layer associated with the [CLS] token represents the next sentence prediction. As with the MLM objective, we add a special head, in this case an NSP head, which consists of a learned set of classiﬁcation weights WNSP ∈Rd×2 that produces a two-class prediction from the raw [CLS]vector hL CLS: yi = softmax(hL CLSWNSP) Cross entropy is used to compute the NSP loss for each sentence pair presented to the model. Fig. 11.4 illustrates the overall NSP training setup. In BERT, the NSP loss was used in conjunction with the MLM training objective to form ﬁnal loss. Cancel my ﬂight [SEP] 1 CE Loss And the Bidirectional Transformer Encoder p1 p2 p3 p4 p5 p6 p7 p8 [CLS] + + s1 NSP Head Token + Segment + Positional Embeddings hotel p9 [SEP] ++ s1 s1 s1 s1 s2 s2 s2 s2 + + + + + + + + + + + + + + hCLS Figure 11.4 An example of the NSP loss calculation. 11.2.3 Training Regimes BERT and other early transformer-based language models were trained on about 3.3 billion words (a combination of English Wikipedia and a corpus of book texts called BooksCorpus (Zhu et al., 2015) that is no longer used for intellectual property reasons). Modern masked language models are now trained on much larger datasets of web text, ﬁltered a bit, and augmented by higher-quality data like Wikipedia, the same as those we discussed for the causal large language models of Chapter 9. Multilingual models similarly use webtext and multilingual Wikipedia. For example the XLM-R model was trained on about 300 billion tokens in 100 languages, taken from the web via Common Crawl (https://commoncrawl.org/). To train the original BERT models, pairs of text segments were selected from the training corpus according to the next sentence prediction 50/50 scheme. Pairs were sampled so that their combined length was less than the 512 token input. Tokens within these sentence pairs were then masked using the MLM approach with the combined loss from the MLM and NSP objectives used for a ﬁnal loss. Because this ﬁnal loss is backpropagated through the entire transformer, the embeddings at each transformer layer will learn representations that are useful for predicting words from their neighbors. Since the [CLS] tokens are the direct input to the NSP classiﬁer, their learned representations will tend to contain information about the sequence as 230 CHAPTER 11 • M ASKED LANGUAGE MODELS a whole. Approximately 40 passes (epochs) over the training data was required for the model to converge. Some models, like the RoBERTa model, drop the next sentence prediction ob- jective, and therefore change the training regime a bit. Instead of sampling pairs of sentence, the input is simply a series of contiguous sentences, still beginning with the special [CLS]token. If the document runs out before 512 tokens are reached, an extra separator token is added, and sentences from the next document are packed in, until we reach a total of 512 tokens. Usually large batch sizes are used, between 8K and 32K tokens. Multilingual models have an additional decision to make: what data to use to build the vocabulary? Recall that all language models use subword tokenization (BPE or SentencePiece Unigram LM are the two most common algorithms). What text should be used to learn this multilingual tokenization, given that it’s easier to get much more text in some languages than others? One option would be to create this vocabulary-learning dataset by sampling sentences from our training data (perhaps web text from Common Crawl), randomly. In that case we will choose a lot of sen- tences from languages like languages with lots of web representation like English, and the tokens will be biased toward rare English tokens instead of creating frequent tokens from languages with less data. Instead, it is common to divide the training data into subcorpora of N different languages, compute the number of sentences ni of each language i, and readjust these probabilities so as to upweight the probability of less-represented languages (Lample and Conneau, 2019). The new probability of selecting a sentence from each of the N languages (whose prior frequency is ni) is {qi}i=1...N , where: qi = pα i∑N j=1 pα j with pi = ni ∑N k=1 nk (11.5) Recall from Eq. 6.32 in Chapter 6 that an α value between 0 and 1 will give higher weight to lower probability samples. Conneau et al. (2020) show thatα = 0.3 works well to give rare languages more inclusion in the tokenization, resulting in better multilingual performance overall. The result of this pretraining process consists of both learned word embeddings, as well as all the parameters of the bidirectional encoder that are used to produce contextual embeddings for novel inputs. For many purposes, a pretrained multilingual model is more practical than a monolingual model, since it avoids the need to build many (a hundred!) separate monolingual models. And multilingual models can improve performance on low- resourced languages by leveraging linguistic information from a similar language in the training data that happens to have more resources. Nonetheless, when the num- ber of languages grows very large, multilingual models exhibit what has been called the curse of multilinguality (Conneau et al., 2020): the performance on each lan- guage degrades compared to a model training on fewer languages. Another problem with multilingual models is that they ‘have an accent’: grammatical structures in higher-resource languages (often English) bleed into lower-resource languages; the vast amount of English language in training makes the model’s representations for low-resource languages slightly more English-like (Papadimitriou et al., 2023). 11.3 • C ONTEXTUAL EMBEDDINGS 231 11.3 Contextual Embeddings Given a pretrained language model and a novel input sentence, we can think of the sequence of model outputs as constitutingcontextual embeddings for each token incontextual embeddings the input. These contextual embeddings are vectors representing some aspect of the meaning of a token in context, and can be used for any task requiring the meaning of tokens or words. More formally, given a sequence of input tokens x1,...,xn, we can use the output vector hLi from the ﬁnal layer L of the model as a representation of the meaning of token xi in the context of sentence x1,...,xn. Or instead of just using the vector hLi from the ﬁnal layer of the model, it’s common to compute a representation for xi by averaging the output tokens hi from each of the last four layers of the model, i.e., hLi, hL−1i, hL−2i, and hL−3i. [CLS] So long and thanks for all hL 1hL CLS hL 2 hL 3 hL 4 hL 5 hL 6 E i+ E i+ E i+ E i+ E i+ E i+ E i+ Figure 11.5 The output of a BERT-style model is a contextual embedding vector hL i for each input token xi. Just as we used static embeddings like word2vec in Chapter 6 to represent the meaning of words, we can use contextual embeddings as representations of word meanings in context for any task that might require a model of word meaning. Where static embeddings represent the meaning of wordtypes (vocabulary entries), contex- tual embeddings represent the meaning of word instances: instances of a particular word type in a particular context. Thus where word2vec had a single vector for each word type, contextual embeddings provide a single vector for each instance of that word type in its sentential context. Contextual embeddings can thus be used for tasks like measuring the semantic similarity of two words in context, and are useful in linguistic tasks that require models of word meaning. 11.3.1 Contextual Embeddings and Word Sense Words are ambiguous: the same word can be used to mean different things. Inambiguous Chapter 6 we saw that the word “mouse” can mean (1) a small rodent, or (2) a hand- operated device to control a cursor. The word “bank” can mean: (1) a ﬁnancial institution or (2) a sloping mound. We say that the words ‘mouse’ or ‘bank’ are 232 CHAPTER 11 • M ASKED LANGUAGE MODELS polysemous (from Greek ‘many senses’,poly- ‘many’ +sema, ‘sign, mark’).2 A sense (or word sense) is a discrete representation of one aspect of the meaningword sense of a word. We can represent each sense with a superscript: bank1 and bank2, mouse1 and mouse2. These senses can be found listed in online thesauruses (or thesauri) like WordNet (Fellbaum, 1998), which has datasets in many languagesWordNet listing the senses of many words. In context, it’s easy to see the different meanings: mouse1 : .... a mouse controlling a computer system in 1968. mouse2 : .... a quiet animal like a mouse bank1 : ...a bank can hold the investments in a custodial account ... bank2 : ...as agriculture burgeons on the east bank, the river ... This fact that context disambiguates the senses of mouse and bank above can also be visualized geometrically. Fig. 11.6 shows a two-dimensional projection of many instances of the BERT embeddings of the word die in English and German. Each point in the graph represents the use of die in one input sentence. We can clearly see at least two different English senses of die (the singular of dice and the verb to die, as well as the German article, in the BERT embedding space. Figure 4: Embeddings for the word "die" in different contexts, visualized with UMAP. Sample points are annotated with corresponding sentences. Overall annotations (blue text) are added as a guide. 4.1 Visualization of word senses Our ﬁrst experiment is an exploratory visualization of how word sense affects context embeddings. For data on different word senses, we collected all sentences used in the introductions to English- language Wikipedia articles. (Text outside of introductions was frequently fragmentary.) We created an interactive application, which we plan to make public. A user enters a word, and the system retrieves 1,000 sentences containing that word. It sends these sentences to BERT-base as input, and for each one it retrieves the context embedding for the word from a layer of the user’s choosing. The system visualizes these 1,000 context embeddings using UMAP [15], generally showing clear clusters relating to word senses. Different senses of a word are typically spatially separated, and within the clusters there is often further structure related to ﬁne shades of meaning. In Figure 4, for example, we not only see crisp, well-separated clusters for three meanings of the word “die,” but within one of these clusters there is a kind of quantitative scale, related to the number of people dying. See Appendix 6.4 for further examples. The apparent detail in the clusters we visualized raises two immediate questions. First, is it possible to ﬁnd quantitative corroboration that word senses are well-represented? Second, how can we resolve a seeming contradiction: in the previous section, we saw how position represented syntax; yet here we see position representing semantics. 4.2 Measurement of word sense disambiguation capability The crisp clusters seen in visualizations such as Figure 4 suggest that BERT may create simple, effective internal representations of word senses, putting different meanings in different locations. To test this hypothesis quantitatively, we test whether a simple classiﬁer on these internal representations can perform well at word-sense disambiguation (WSD). We follow the procedure described in [20], which performed a similar experiment with the ELMo model. For a given word withn senses, we make a nearest-neighbor classiﬁer where each neighbor is the centroid of a given word sense’s BERT-base embeddings in the training data. To classify a new word we ﬁnd the closest of these centroids, defaulting to the most commonly used sense if the word was not present in the training data. We used the data and evaluation from [21]: the training data was SemCor [17] (33,362 senses), and the testing data was the suite described in [21] (3,669 senses). The simple nearest-neighbor classiﬁer achieves an F1 score of 71.1, higher than the current state of the art (Table 1), with the accuracy monotonically increasing through the layers. This is a strong signal that context embeddings are representing word-sense information. Additionally, an even higher score of 71.5 was obtained using the technique described in the following section. 6 Figure 11.6 Each blue dot shows a BERT contextual embedding for the word die from different sentences in English and German, projected into two dimensions with the UMAP algorithm. The German and English meanings and the different English senses fall into different clusters. Some sample points are shown with the contextual sentence they came from. Figure from Coenen et al. (2019). Thus while thesauruses like WordNet give discrete lists of senses, embeddings (whether static or contextual) offer a continuous high-dimensional model of meaning that, although it can be clustered, doesn’t divide up into fully discrete senses. Word Sense Disambiguation The task of selecting the correct sense for a word is calledword sense disambigua- tion, or WSD. WSD algorithms take as input a word in context and a ﬁxed inventoryword sense disambiguation WSD of potential word senses (like the ones in WordNet) and outputs the correct word sense in context. Fig. 11.7 sketches out the task. 2 The word polysemy itself is ambiguous; you may see it used in a different way, to refer only to cases where a word’s senses are related in some structured way, reserving the wordhomonymy to mean sense ambiguities with no relation between the senses (Haber and Poesio, 2020). Here we will use ‘polysemy’ to mean any kind of sense ambiguity, and ‘structured polysemy’ for polysemy with sense relations. 11.3 • C ONTEXTUAL EMBEDDINGS 233 an electric guitar and bass player stand oﬀ to one side electric1: using electricity electric2: tense electric3: thrilling guitar1 bass1: low range … bass4: sea ﬁsh … bass7: instrument … player1: in game player2: musician player3: actor … stand1: upright … stand5: bear … stand10: put upright … side1: relative region … side3: of body … side11: slope … x1 y1 x2 y2 x3 y3 y4 y5 y6 x4 x5 x6 Figure 11.7 The all-words WSD task, mapping from input words ( x) to WordNet senses (y). Figure inspired by Chaplot and Salakhutdinov (2018). WSD can be a useful analytic tool for text analysis in the humanities and social sciences, and word senses can play a role in model interpretability for word repre- sentations. Word senses also have interesting distributional properties. For example a word often is used in roughly the same sense through a discourse, an observation called the one sense per discourse rule (Gale et al., 1992a).one sense per discourse The best performing WSD algorithm is a simple 1-nearest-neighbor algorithm using contextual word embeddings, due to Melamud et al. (2016) and Peters et al. (2018). At training time we pass each sentence in some sense-labeled dataset (like the SemCore or SenseEval datasets in various languages) through any contextual embedding (e.g., BERT) resulting in a contextual embedding for each labeled token. (There are various ways to compute this contextual embedding vi for a token i; for BERT it is common to pool multiple layers by summing the vector representations of i from the last four BERT layers). Then for each senses of any word in the corpus, for each of the n tokens of that sense, we average their n contextual representations vi to produce a contextual sense embedding vs for s: vs = 1 n ∑ i vi ∀vi ∈tokens(s) (11.6) At test time, given a token of a target word t in context, we compute its contextual embedding t and choose its nearest neighbor sense from the training set, i.e., the sense whose sense embedding has the highest cosine with t: sense(t) =argmax s∈senses(t) cosine(t,vs) (11.7) Fig. 11.8 illustrates the model. 11.3.2 Contextual Embeddings and Word Similarity In Chapter 6 we introduced the idea that we could measure the similarity of two words by considering how close they are geometrically, by using the cosine as a similarity function. The idea of meaning similarity is also clear geometrically in the meaning clusters in Fig. 11.6; the representation of a word which has a particular sense in a context is closer to other instances of the same sense of the word. Thus we 234 CHAPTER 11 • M ASKED LANGUAGE MODELS I found the jar empty cI cfound find1 v cthe cjar cempty find9 v find5 vfind4 v ENCODER Figure 11.8 The nearest-neighbor algorithm for WSD. In green are the contextual embed- dings precomputed for each sense of each word; here we just show a few of the senses for ﬁnd. A contextual embedding is computed for the target word found, and then the nearest neighbor sense (in this case ﬁnd9v ) is chosen. Figure inspired by Loureiro and Jorge (2019). often measure the similarity between two instances of two words in context (or two instances of the same word in two different contexts) by using the cosine between their contextual embeddings. Usually some transformations to the embeddings are required before computing cosine. This is because contextual embeddings (whether from masked language models or from autoregressive ones) have the property that the vectors for all words are extremely similar. If we look at the embeddings from the ﬁnal layer of BERT or other models, embeddings for instances of any two randomly chosen words will have extremely high cosines that can be quite close to 1, meaning all word vectors tend to point in the same direction. The property of vectors in a system all tending to point in the same direction is known as anisotropy. Ethayarajh (2019) deﬁnes the anisotropy of a model as the expected cosine similarity of any pair of words inanisotropy a corpus. The word ‘isotropy’ means uniformity in all directions, so in an isotropic model, the collection of vectors should point in all directions and the expected cosine between a pair of random embeddings would be zero. Timkey and van Schijndel (2021) show that one cause of anisotropy is that cosine measures are dominated by a small number of dimensions of the contextual embedding whose values are very different than the others: these rogue dimensions have very large magnitudes and very high variance. Timkey and van Schijndel (2021) shows that we can make the embeddings more isotropic by standardizing (z-scoring) the vectors, i.e., subtracting the mean and dividing by the variance. Given a set C of all the embeddings in some corpus, each with dimensionality d (i.e., x ∈Rd), the mean vector µ ∈Rd is: µ = 1 \|C\| ∑ x∈C x (11.8) The standard deviation in each dimension σ ∈Rd is: σ = √ 1 \|C\| ∑ x∈C (x−µ)2 (11.9) Then each word vector x is replaced by a standardized version z: z = x−µ σ (11.10) 11.4 • F INE -TUNING FOR CLASSIFICATION 235 One problem with cosine that is not solved by standardization is that cosine tends to underestimate human judgments on similarity of word meaning for very frequent words (Zhou et al., 2022). In the next section we’ll see the most common use of contextual representations: as representations of words or even entire sentences that can be the inputs to classi- ﬁers in the ﬁnetuning process for downstream NLP applications. 11.4 Fine-Tuning for Classiﬁcation The power of pretrained language models lies in their ability to extract generaliza- tions from large amounts of text—generalizations that are useful for myriad down- stream applications. There are two ways to make practical use of the generalizations to solve downstream tasks. The most common way is to use natural language to prompt the model, putting it in a state where it contextually generates what we want. We’ll introduce prompting in Chapter 12. In this section we explore an alternative way to use pretrained language models for downstream applications: a version of theﬁnetuning paradigm from Chapter 10.ﬁnetuning In the kind of ﬁnetuning used for masked language models, we add application- speciﬁc circuitry (often called a special head) on top of pretrained models, taking their output as its input. The ﬁnetuning process consists of using labeled data about the application to train these additional application-speciﬁc parameters. Typically, this training will either freeze or make only minimal adjustments to the pretrained language model parameters. The following sections introduce ﬁnetuning methods for the most common kinds of applications: sequence classiﬁcation, sentence-pair classiﬁcation, and sequence labeling. 11.4.1 Sequence Classiﬁcation The task of sequence classiﬁcation is to classify an entire sequence of text with a single label. This set of tasks is commonly called text classiﬁcation, like sentiment analysis or spam detection (Chapter 4) in which we classify a text into two or three classes (like positive or negative), as well as classiﬁcation tasks with a large number of categories, like document-level topic classiﬁcation. For sequence classiﬁcation we represent the entire input to be classiﬁed by a single vector. We can represent a sequence in various ways. One way is to take the sum or the mean of the last output vector from each token in the sequence. For BERT, we instead add a new unique token to the vocabulary called [CLS], and prepended it to the start of all input sequences, both during pretraining and encoding. The output vector in the ﬁnal layer of the model for the [CLS] input represents the entire input sequence and serves as the input to a classiﬁer head , a logisticclassiﬁer head regression or neural network classiﬁer that makes the relevant decision. As an example, let’s return to the problem of sentiment classiﬁcation. to ﬁnetun- ing a classiﬁer for this application involves learning a set of weights,WC, to map the output vector for the [CLS]token— hL CLS—to a set of scores over the possible senti- ment classes. Assuming a three-way sentiment classiﬁcation task (positive, negative, neutral) and dimensionality d as the model dimension,WC will be of size[d ×3]. To classify a document, we pass the input text through the pretrained language model to 236 CHAPTER 11 • M ASKED LANGUAGE MODELS generate hL CLS, multiply it by WC, and pass the resulting vector through a softmax. y = softmax(hL CLSWC) (11.11) Finetuning the values inWC requires supervised training data consisting of input sequences labeled with the appropriate sentiment class. Training proceeds in the usual way; cross-entropy loss between the softmax output and the correct answer is used to drive the learning that produces WC. A key difference from what we’ve seen earlier with neural classiﬁers is that this loss can be used to not only learn the weights of the classiﬁer, but also to update the weights for the pretrained language model itself. In practice, reasonable classiﬁca- tion performance is typically achieved with only minimal changes to the language model parameters, often limited to updates over the ﬁnal few layers of the trans- former. Fig. 11.9 illustrates this overall approach to sequence classiﬁcation. [CLS] entirely predictable and lacks energy Bidirectional Transformer Encoder hCLS E i+ E i+ E i+ E i+ E i+ E i+ sentiment classiﬁcation head WC y Figure 11.9 Sequence classiﬁcation with a bidirectional transformer encoder. The output vector for the [CLS]token serves as input to a simple classiﬁer. 11.4.2 Sequence-Pair Classiﬁcation As mentioned in Section 11.2.2, an important type of problem involves the classiﬁca- tion of pairs of input sequences. Practical applications that fall into this class include paraphrase detection (are the two sentences paraphrases of each other?), logical en- tailment (does sentence A logically entail sentence B?), and discourse coherence (how coherent is sentence B as a follow-on to sentence A?). Fine-tuning an application for one of these tasks proceeds just as with pretrain- ing using the NSP objective. During ﬁnetuning, pairs of labeled sentences from a supervised ﬁnetuning set are presented to the model, and run through all the layers of the model to produce the h outputs for each input token. As with sequence classi- ﬁcation, the output vector associated with the prepended[CLS]token represents the model’s view of the input pair. And as with NSP training, the two inputs are sepa- rated by the [SEP] token. To perform classiﬁcation, the [CLS] vector is multiplied by a set of learning classiﬁcation weights and passed through a softmax to generate label predictions, which are then used to update the weights. 11.5 • F INE -TUNING FOR SEQUENCE LABELLING : N AMED ENTITY RECOGNITION 237 As an example, let’s consider an entailment classiﬁcation task with the Multi- Genre Natural Language Inference (MultiNLI) dataset (Williams et al., 2018). In the task of natural language inference or NLI, also called recognizing textual natural language inference entailment, a model is presented with a pair of sentences and must classify the re- lationship between their meanings. For example in the MultiNLI corpus, pairs of sentences are given one of 3 labels: entails, contradicts and neutral. These labels describe a relationship between the meaning of the ﬁrst sentence (the premise) and the meaning of the second sentence (the hypothesis). Here are representative exam- ples of each class from the corpus: • Neutral a: Jon walked back to the town to the smithy. b: Jon traveled back to his hometown. • Contradicts a: Tourist Information ofﬁces can be very helpful. b: Tourist Information ofﬁces are never of any help. • Entails a: I’m confused. b: Not all of it is very clear to me. A relationship of contradicts means that the premise contradicts the hypothesis; en- tails means that the premise entails the hypothesis; neutral means that neither is necessarily true. The meaning of these labels is looser than strict logical entailment or contradiction indicating that a typical human reading the sentences would most likely interpret the meanings in this way. To ﬁnetune a classiﬁer for the MultiNLI task, we pass the premise/hypothesis pairs through a bidirectional encoder as described above and use the output vector for the [CLS]token as the input to the classiﬁcation head. As with ordinary sequence classiﬁcation, this head provides the input to a three-way classiﬁer that can be trained on the MultiNLI training corpus. 11.5 Fine-Tuning for Sequence Labelling: Named En- tity Recognition In sequence labeling, the network’s task is to assign a label chosen from a small ﬁxed set of labels to each token in the sequence. One of the most common sequence labeling task is named entity recognition. 11.5.1 Named Entities A named entity is, roughly speaking, anything that can be referred to with a propernamed entity name: a person, a location, an organization. The task of named entity recognitionnamed entity recognition (NER) is to ﬁnd spans of text that constitute proper names and tag the type of theNER entity. Four entity tags are most common: PER (person), LOC (location), ORG (organization), or GPE (geo-political entity). However, the term named entity is commonly extended to include things that aren’t entities per se, including temporal expressions like dates and times, and even numerical expressions like prices. Here’s an example of the output of an NER tagger: 238 CHAPTER 11 • M ASKED LANGUAGE MODELS Citing high fuel prices, [ ORG United Airlines] said [TIME Friday] it has increased fares by [ MONEY $6] per round trip on ﬂights to some cities also served by lower-cost carriers. [ ORG American Airlines], a unit of [ORG AMR Corp.], immediately matched the move, spokesman [PER Tim Wagner] said. [ ORG United], a unit of [ORG UAL Corp.], said the increase took effect [ TIME Thursday] and applies to most routes where it competes against discount carriers, such as [LOC Chicago] to [LOC Dallas] and [LOC Denver] to [LOC San Francisco]. The text contains 13 mentions of named entities including 5 organizations, 4 loca- tions, 2 times, 1 person, and 1 mention of money. Figure 11.10 shows typical generic named entity types. Many applications will also need to use speciﬁc entity types like proteins, genes, commercial products, or works of art. Type Tag Sample Categories Example sentences People PER people, characters Turing is a giant of computer science. Organization ORG companies, sports teams The IPCC warned about the cyclone. Location LOC regions, mountains, seas Mt. Sanitas is in Sunshine Canyon. Geo-Political Entity GPE countries, states Palo Alto is raising the fees for parking. Figure 11.10 A list of generic named entity types with the kinds of entities they refer to. Named entity recognition is a useful step in various natural language processing tasks, including linking text to information in structured knowledge sources like Wikipedia, measuring sentiment or attitudes toward a particular entity in text, or even as part of anonymizing text for privacy. The NER task is is difﬁcult because of the ambiguity of segmenting NER spans, ﬁguring out which tokens are entities and which aren’t, since most words in a text will not be named entities. Another difﬁculty is caused by type ambiguity. The mention Washington can refer to a person, a sports team, a city, or the US government, as we see in Fig. 11.11. [PER Washington] was born into slavery on the farm of James Burroughs. [ORG Washington] went up 2 games to 1 in the four-game series. Blair arrived in [LOC Washington] for what may well be his last state visit. In June, [GPE Washington] passed a primary seatbelt law. Figure 11.11 Examples of type ambiguities in the use of the name Washington. 11.5.2 BIO Tagging One standard approach to sequence labeling for a span-recognition problem like NER is BIO tagging (Ramshaw and Marcus, 1995). This is a method that allows usBIO tagging to treat NER like a word-by-word sequence labeling task, via tags that capture both the boundary and the named entity type. Consider the following sentence: [PER Jane Villanueva ] of [ORG United] , a unit of [ ORG United Airlines Holding] , said the fare applies to the [LOC Chicago ] route. Figure 11.12 shows the same excerpt represented with BIO tagging, as well asBIO variants called IO tagging and BIOES tagging. In BIO tagging we label any token that begins a span of interest with the label B, tokens that occur inside a span are tagged with an I, and any tokens outside of any span of interest are labeled O. While there is only one O tag, we’ll have distinctB and I tags for each named entity class. The number of tags is thus 2 n + 1 tags, where n is the number of entity types. BIO 11.5 • F INE -TUNING FOR SEQUENCE LABELLING : N AMED ENTITY RECOGNITION 239 tagging can represent exactly the same information as the bracketed notation, but has the advantage that we can represent the task in the same simple sequence modeling way as part-of-speech tagging: assigning a single label yi to each input word xi: Words IO Label BIO Label BIOES Label Jane I-PER B-PER B-PER Villanueva I-PER I-PER E-PER of O O O United I-ORG B-ORG B-ORG Airlines I-ORG I-ORG I-ORG Holding I-ORG I-ORG E-ORG discussed O O O the O O O Chicago I-LOC B-LOC S-LOC route O O O . O O O Figure 11.12 NER as a sequence model, showing IO, BIO, and BIOES taggings. We’ve also shown two variant tagging schemes: IO tagging, which loses some information by eliminating the B tag, and BIOES tagging, which adds an end tag E for the end of a span, and a span tag S for a span consisting of only one word. 11.5.3 Sequence Labeling In sequence labeling, we pass the ﬁnal output vector corresponding to each input token to a classiﬁer that produces a softmax distribution over the possible set of tags. For a single feedforward layer classiﬁer, the set of weights to be learned is WK of size [d ×k], where k is the number of possible tags for the task. A greedy approach, where the argmax tag for each token is taken as a likely answer, can be used to generate the ﬁnal output tag sequence. Fig. 11.13 illustrates an example of this approach, where yi is a vector of probabilities over tags, and k indexes the tags. yi = softmax(hL i WK) (11.12) ti = argmaxk(yi) (11.13) Alternatively, the distribution over labels provided by the softmax for each input token can be passed to a conditional random ﬁeld (CRF) layer which can take global tag-level transitions into account (see Chapter 17 on CRFs). Tokenization and NER Note that supervised training data for NER is typically in the form of BIO tags as- sociated with text segmented at the word level. For example the following sentence containing two named entities: [LOC Mt. Sanitas ] is in [LOC Sunshine Canyon] . would have the following set of per-word BIO tags. (11.14) Mt. B-LOC Sanitas I-LOC is O in O Sunshine B-LOC Canyon I-LOC . O Unfortunately, the sequence of WordPiece tokens for this sentence doesn’t align directly with BIO tags in the annotation: 240 CHAPTER 11 • M ASKED LANGUAGE MODELS [CLS] Jane Villanueva of United Airlines Bidirectional Transformer Encoder B-PER I-PER O B-ORG I-ORG Holding discussed I-ORG O WK NER head hi argmax E i+ E i+ E i+ E i+ E i+ E i+ E i+ E i+ WK WK WK WK WK WK yi Figure 11.13 Sequence labeling for named entity recognition with a bidirectional transformer encoder. The output vector for each input token is passed to a simple k-way classiﬁer. ’Mt’, ’.’, ’San’, ’##itas’, ’is’, ’in’, ’Sunshine’, ’Canyon’ ’.’ To deal with this misalignment, we need a way to assign BIO tags to subword tokens during training and a corresponding way to recover word-level tags from subwords during decoding. For training, we can just assign the gold-standard tag associated with each word to all of the subword tokens derived from it. For decoding, the simplest approach is to use the argmax BIO tag associated with the ﬁrst subword token of a word. Thus, in our example, the BIO tag assigned to “Mt” would be assigned to “Mt.” and the tag assigned to “San” would be assigned to “Sanitas”, effectively ignoring the information in the tags assigned to “.” and “##itas”. More complex approaches combine the distribution of tag probabilities across the subwords in an attempt to ﬁnd an optimal word-level tag. 11.5.4 Evaluating Named Entity Recognition Named entity recognizers are evaluated by recall, precision, and F1 measure. Re- call that recall is the ratio of the number of correctly labeled responses to the total that should have been labeled; precision is the ratio of the number of correctly la- beled responses to the total labeled; and F-measure is the harmonic mean of the two. To know if the difference between the F1 scores of two NER systems is a signif- icant difference, we use the paired bootstrap test, or the similar randomization test (Section 4.9). For named entity tagging, the entity rather than the word is the unit of response. Thus in the example in Fig. 11.12, the two entities Jane Villanueva and United Air- lines Holding and the non-entity discussed would each count as a single response. The fact that named entity tagging has a segmentation component which is not present in tasks like text categorization or part-of-speech tagging causes some prob- lems with evaluation. For example, a system that labeled Jane but not Jane Vil- lanueva as a person would cause two errors, a false positive for O and a false nega- 11.6 • S UMMARY 241 tive for I-PER. In addition, using entities as the unit of response but words as the unit of training means that there is a mismatch between the training and test conditions. 11.6 Summary This chapter has introduced the bidirectional encoder and the masked language model. Here’s a summary of the main points that we covered: • Bidirectional encoders can be used to generate contextualized representations of input embeddings using the entire input context. • Pretrained language models based on bidirectional encoders can be learned using a masked language model objective where a model is trained to guess the missing information from an input. • The vector output of each transformer block or component in a particular to- ken column is a contextual embedding that represents some aspect of the meaning of a token in context. • A word sense is a discrete representation of one aspect of the meaning of a word. Contextual embeddings offer a continuous high-dimensional model of meaning that is richer than fully discrete senses. • The cosine between contextual embeddings can be used as one way to model the similarity between two words in context, although some transformations to the embeddings are required ﬁrst. • Pretrained language models can be ﬁnetuned for speciﬁc applications by adding lightweight classiﬁer layers on top of the outputs of the pretrained model. • These applications can include sequence classiﬁcation tasks like sentiment analysis, sequence-pair classiﬁcation tasks like natural language inference, or sequence labeling tasks like named entity recognition. Bibliographical and Historical Notes History TBD. 242 CHAPTER 12 • M ODEL ALIGNMENT , PROMPTING , AND IN-CONTEXT LEARNING CHAPTER 12 Model Alignment, Prompting, and In-Context Learning “Hal, ” said Bowman, now speaking with an icy calm. “I am not incapaci- tated. Unless you obey my instructions, I shall be forced to disconnect you. ” Arthur C. Clarke In this chapter we show how to get LLMs to do tasks for us simply by talking to them. To get an LLM to translate a sentence, outline a talk, or draft a work email, we’ll simply describe what we want in natural language. We call these instructions we give to language models prompts.prompts Prompting relies on contextual generation. Given the prompt as context, the lan- guage model generates the next token based on its token probability, conditioned on the prompt: P(wi\|w<i). A prompt can be a question (like “What is a transformer net- work?”), possibly in a structured format (like “Q: What is a transformer network? A:”), or can be an instruction (like “Translate the following sentence into Hindi: ‘Chop the garlic ﬁnely’”). A prompt can also containdemonstrations, examples todemonstrations help make the instructions clearer, (like “Give the sentiment of the following sen- tence. Example Input: “I really loved Taishan Cuisine.” Output: positive”.) As we’ll see, prompting can be applied to inherently generative tasks (like summarization and translation) as well as to ones more naturally thought of as classiﬁcation tasks. Prompts get language models to generate text, but they also can be viewed as a learning signal, because these demonstrations can help language models learn to perform novel tasks. For this reason we also refer to prompting as in-context- learning—learning that improves model performance or reduces some loss but doesin-context- learning not involve gradient-based updates to the model’s underlying parameters. But LLMs as we’ve described them so far turn out to be bad at following instruc- tions. Pretraining isn’t sufﬁcient to make themhelpful. We’ll introduceinstruction tuning, a technique that helps LLMs learn to correctly respond to instructions byinstruction tuning ﬁnetuning them on a corpus of instructions with their corresponding response. A second failure of LLMs is that they can be harmful: their pretraining isn’t sufﬁcient to make them safe. Readers who know Arthur C. Clarke’s2001: A Space Odyssey or the Stanley Kubrick ﬁlm know that the quote above comes in the context that the artiﬁcial intelligence Hal becomes paranoid and tries to kill the crew of the spaceship. Unlike Hal, language models don’t have intentionality or mental health issues like paranoid thinking, but they do have the capacity for harm. Pretrained lan- guage models can say things that are dangerous or false (like giving unsafe medical advice) and they can verbally attack users or say toxic or hateful things. Dealing with safety can be done partly by adding safety training into instruction tuning. But an important aspect of safety training is a second technique, preference alignment (often implemented, as we’ll see, with theRLHF or DPO algorithms) inpreference alignment which a separate model is trained to decide how much a candidate response aligns with human preferences. Together we refer to instruction tuning and preference alignment as model alignment. The intuition is that we want the learning objectivesmodel alignment of models to be aligned with the goals of the humans that use them. 12.1 • P ROMPTING 243 12.1 Prompting A prompt is a text string that a user issues to a language model to get the modelprompt to do something useful. In prompting, the user’s prompt string is passed to the language model, which iteratively generates tokens conditioned on the prompt. Thus the prompt creates a context that guides LLMs to generate useful outputs to achieve some user goal. The process of ﬁnding effective prompts for a task is known as prompt engineering.prompt engineering Let’s see how to prompt a language model to solve a simple sentiment classiﬁ- cation task, Consider this hotel review from the BLT corpus (Salvetti et al., 2016): Sample Hotel Review Did not like the service that I was provided, when I entered the hotel. I also did not like the area, in which the hotel was located. Too much noise and events going on for me to feel relax. We can get the model to classify the sentiment of this text by taking the text and appending an incomplete statement to the review like In short, our stay was: A prompt consisting of a review plus an incomplete statement Did not like the service that I was provided, when I entered the hotel. I also did not like the area, in which the hotel was located. Too much noise and events going on for me to feel relax. In short, our stay was We then have an LLM complete the statement by generating a token conditioned on the prompt so far, and then generating the next token (conditioned on the prompt plus the new token), and so forth. Here are a few responses from a language model via continued generation starting with the prompt as context. Hotel Review Completions Did not like the service that I was provided, when I entered the hotel. I also did not like the area, in which the hotel was located. Too much noise and events going on for me to feel relaxed. In short our stay was ... not a pleasant one. The staff at the front desk were not welcoming or friendly, and seemed disinterested in providing good customer service. ... uncomfortable and not worth the price we paid. We will not be returning to this hotel. As we can see, the overall negative context of the review results in negative completions. We could easily map these completions to the class we are trying to predict, perhaps via some predeﬁned mappings, like {excellent →positive}, {did not like→negative}, and so on. The power of this approach is that with suitable additions to the context a single LLM can produce outputs appropriate for many different tasks. For example, given 244 CHAPTER 12 • M ODEL ALIGNMENT , PROMPTING , AND IN-CONTEXT LEARNING a review we might want any of the following: • A summary, • Whether the review was truthful or likely to have been fabricated, • A translation to another language. LLMs have a striking ability to perform tasks like these, needing just the appro- priate contextual nudge to get the LLM to generate the desired output. If we want to solve general tasks like summarization or translation, we don’t want to have to create a new prompt each time we do the task. Instead the ﬁrst step in prompting is to design one or moretemplates: task-speciﬁc prompting text alongtemplates with slots for the particular input that is being processed. Consider the following templates for a variety of tasks: Basic Prompt Templates Summarization {input}; tldr; Translation {input}; translate to French: Sentiment {input}; Overall, it was Fine-Grained- {input}; What aspects were important in this review? Sentiment Each template consists of an input text, designated as {input}, followed by a verbatim prompt to be passed to an LLM. These templates are applied to inputs to create ﬁlled prompts – instantiated prompts suitable for use as inputs to an LLM. Fig. 12.1 illustrates ﬁlled prompts for these templates using our earlier hotel review, along with sample outputs from an LLM: Notice the design pattern of the prompts above: the input is followed by some text which in turn will be completed by the desired response. This style, with the instruction at the end, is common in prompting because it helpfully constrains the generation. Consider, by contrast, the prompt in Example 12.1. Translate English to French: Did not like the service that I was provided! (12.1) This prompt doesn’t do a good job of constraining possible continuations. Instead of a French translation, models given this prompt may instead generate another sen- tence in English that simply extends the English review. Prompts need to be designed unambiguously, so that any reasonable continuation would accomplish the desired task (Reynolds and McDonell, 2021). An even more constraining style of prompt can specify the set of possible an- swers in the prompt. For example here is a prompt template to do sentiment analysis that prespeciﬁes the potential answers: A prompt consisting of a review plus an incomplete statement Human: Do you think that “input” has negative or positive sentiment? Choices: (P) Positive (N) Negative Assistant: I believe the best answer is: ( 12.1 • P ROMPTING 245 LLM Outputs for Basic Prompts Original Review ($INPUT) Did not like the service that I was provided, when I entered the hotel. I also did not like the area, in which the hotel was located. Too much noise and events going on for me to feel relax and away from the city life. Sentiment Prompt: $INPUT + In short, our stay was Output: not enjoyable Fine-grained Sentiment Prompt: $INPUT + These aspects were important to the reviewer: Output: 1. Poor service 2. Unpleasant location 3. Noisy and busy area Summarization Prompt: $INPUT + tl;dr Output: I had a bad experience with the hotel’s service and the location was loud and busy. Translation Prompt: $INPUT + Translate this to French Output: Je n’ai pas aim´e le service qui m’a ´et´e offert lorsque je suis entr´e dans l’hˆotel. Je n’ai ´egalement pas aim´e la zone dans laquelle se trouvait l’hˆotel. Trop de bruit et d’ ´ev´enements pour que je me sente d´etendu et loin de la vie citadine. Figure 12.1 LLM outputs for simple prompts for sentiment, summarization and translation for an input text. This prompt uses a number of more sophisticated prompting characteristics. It speciﬁes the two allowable choices (P) and (N), and ends the prompt with the open parenthesis that strongly suggests the answer will be (P) or (N). Note that it also speciﬁes the role of the language model as an assistant. We can do even more with prompts. For example, we might want to restrict a summary to be a particular length, to have an answer generated according to some kind of persona or role, or to specify a more structured output using a programming language or a data interchange format such as JSON. Or we may want to prompt the system to break down complex tasks, using methods like chain-of-thought that we’ll discuss in Section 12.4. All of these kinds of instructions go beyond simple prompting and require further LLM ﬁnetuning to enable them to follow instructions. We’ll return to this notion ofinstruction tuning in Section 12.3. In summary, we prompt an LM by transforming each task into a form that is amenable to contextual generation by an LLM, as follows: 1. For a given task, develop a a task-speciﬁc template that has a free parameter for the input text. 2. Given that input and the task-speciﬁc template, the input is used to instantiatetemplate a ﬁlled prompt that is then passed to a pretrained language model. 3. Autoregressive decoding is then used to generate a sequence of token outputs. 4. The output of the model can either be used directly as the desired output (as in the case of naturally generative tasks such as translation or summarization), or a task-appropriate answer can be extracted from the generated output (as in the case of classiﬁcation). 246 CHAPTER 12 • M ODEL ALIGNMENT , PROMPTING , AND IN-CONTEXT LEARNING 12.1.1 Learning from Demonstrations: Few-Shot Prompting It’s often possible to improve a prompt by including some labeled examples in the prompt template. We call such examples demonstrations. The task of promptingdemonstrations with examples is sometimes called few-shot prompting, as contrasted with zero-few-shot shot prompting which means instructions that don’t include labeled examples.zero-shot Fig. 12.2 illustrates a few-shot example from an extractive question answering task. The context combines the task deﬁnition along with three gold-standard ques- tion and answer pairs from the training set. Deﬁnition: This task is about writing a correct answer for the reading comprehension task. Based on the information provided in a given passage, you should identify the shortest continuous text span from the passage that serves as an answer to the given question. Avoid answers that are incorrect or provides incomplete justiﬁcation for the question. Passage: Beyonc ´e Giselle Knowles-Carter (born September 4, 1981) is an American singer, songwriter, record producer and actress. Born and raised in Houston, Texas, she performed in various singing and dancing competitions as a child, and rose to fame in the late 1990s as lead singer of R&B girl-group Destiny’s Child. Managed by her father, Mathew Knowles, the group became one of the world’s best-selling girl groups of all time. Their hiatus saw the release of Beyonc´e’s debut album, Dangerously in Love (2003), which established her as a solo artist worldwide, earned ﬁve Grammy Awards and featured the Billboard Hot 100 number-one singles “Crazy in Love” and “Baby Boy”. Examples: Q: In what city and state did Beyonc´e grow up? A: Houston, Texas Q: What areas did Beyonc´e compete in when she was growing up? A: singing and dancing Q: When did Beyonc´e release Dangerously in Love? A: 2003 Q: When did Beyonc´e start becoming popular? A: Figure 12.2 A prompt for extractive question answering, from an example from the SQuAD 2.0 dataset (Rajpurkar et al., 2018). The prompt contains the task deﬁnition, the passage, 3 demonstration examples, followed by the test question. This deﬁnition speciﬁcation and format are after the Natural Instructions dataset (Mishra et al., 2022). How Many Demonstrations? The number of demonstrations doesn’t have to be large. A small number of randomly selected labeled examples used as demonstra- tions can be sufﬁcient to improve performance over the zero-shot setting. Indeed, the largest performance gains in few-shot prompting tends to come from the ﬁrst training example, with diminishing returns for subsequent demonstrations. This is in contrast with ﬁnetuning of specialized classiﬁer heads that we saw in Chapter 11 where it helps to have lots of examples. Why isn’t it useful to have more demonstrations? The reason is that the primary beneﬁt in examples is to demonstrate the task to be performed to the LLM and the format of the sequence, not to provide relevant information as to the right answer 12.1 • P ROMPTING 247 for any particular question. In fact, demonstrations that have incorrect answers can still improve a system (Min et al., 2022; Webson and Pavlick, 2022). Adding too many examples seems to cause the model to overﬁt to details of the exact examples chosen and generalize poorly. How to Select Demonstrations? Demonstrations are generally created by format- ting examples drawn from a labeled training set. There are some heuristics about what makes a good demonstration. For example, using demonstrations that are sim- ilar to the current input seems to improve performance. It can thus be useful to dynamically retrieve demonstrations for each input, based on their similarity to the current example (for example, comparing the embedding of the current example with embeddings of each of the training set example to ﬁnd the best top-T ). But more generally, the best way to select demonstrations from the training set is programmatically: choosing the set of demonstrations that most increases task performance of the prompt on a test set. Task performance for sentiment analysis or multiple-choice question answering can be measured in accuracy; for machine translation with chrF, and for summarization via Rouge. Systems like DSPy (Khat- tab et al., 2024), a framework for algorithmically optimizing LM prompts, can au- tomatically ﬁnd the optimum set of demonstrations to include by searching through the space of possible demonstrations to include. We’ll return to automatic prompt optimization in Section 12.5. 12.1.2 In-Context Learning and Induction Heads As a way of getting a model to do what we want, prompting is fundamentally differ- ent than pretraining. Learning via pretraining means updating the model’s parame- ters by using gradient descent according to some loss function. But prompting with demonstrations can teach a model to do a new task. The model is learning something as it processes the prompt. Even without demonstrations, we can think of the process of prompting as a kind of learning. For example, the further a model gets in a prompt, the better it tends to get at predicting the upcoming tokens. The information in the context is helping give the model more predictive power. The term in-context learning was ﬁrst proposed by Brown et al. (2020) in theirin-context learning introduction of the GPT3 system, to refer to either of these kinds of learning that lan- guage models do from their prompts. In-context learning means language models learning to do new tasks, better predict tokens, or generally reduce their loss dur- ing the forward-pass at inference-time, without any gradient-based updates to the model’s parameters. How does in-context learning work? While we don’t know for sure, there are some intriguing ideas. One hypothesis is based on the idea of induction headsinduction heads (Elhage et al., 2021; Olsson et al., 2022). Induction heads are the name for acircuit, which is a kind of abstract component of a network. The induction head circuit is part of the attention computation in transformers, discovered by looking at mini language models with only 1-2 attention heads. The function of the induction head is to predict repeated sequences. For example if it sees the pattern AB...A in an input sequence, it predicts that B will follow, instantiating the pattern completion rule AB...A→B. It does this by having apreﬁx matching component of the attention computation that, when looking at the current token A, searches back over the context to ﬁnd a prior instance of A. If it ﬁnds one, the induction head has a copying mechanism that “copies” the token B that followed 248 CHAPTER 12 • M ODEL ALIGNMENT , PROMPTING , AND IN-CONTEXT LEARNING the earlier A, by increasing the probability the B will occur next. Fig. 12.3 shows an example. Figure 1: In the sequence “...vintage cars ... vintage”, an induction head identiﬁes the initial occurrence of “vintage”, attends to the subsequent word “cars” for preﬁx matching, and predicts “cars” as the next word through the copying mechanism. determines each head’s independent output for the current token. Leveraging this decomposition, Elhage et al. (2021) discovered a distinct behaviour in certain attention heads, which they namedinduction heads. This behaviour emerges when these heads process sequences of the form "[A] [B] ... [A]→ ". In these heads, the QK circuit directs attention to- wards [B], which appears directly after the previous occurrence of the current token [A]. This behaviour is termedpreﬁx matching. The OV circuit subse- quently increases the output logit of the [B] token, termed copying. An overview of this mechanism is shown in Figure1. 4 Methods 4.1 Models We utilise two recently developed open-source models, namely Llama-3-8B2 and InternLM2-20B (Cai et al., 2024), both of which are based on the original Llama (Touvron et al., 2023a) architec- ture. These models feature grouped-query atten- tion mechanisms (Ainslie et al., 2023) to enhance efﬁciency. Llama-3-8B, comprises 32 layers, each with 32 attention heads and it uses a query group size of 4 attention heads. It has shown superior performance compared to its predecessors, even the larger Llama-2 models. InternLM2-20B, featuring 48 layers with 48 at- tention heads each, uses a query group size of 6 attention heads. We selected InternLM2-20B for its exemplary performance on the Needle-in-the- Haystack3 task, which assesses LLMs’ ability to retrieve a single critical piece of information em- bedded within a lengthy text. This mirrors the functionality of induction heads, which scan the context for prior occurrences of a token to extract relevant subsequent information. 2https://ai.meta.com/blog/meta-llama-3/ 3https://github.com/gkamradt/LLMTest_ NeedleInAHaystack 4.2 Identifying Induction Heads To identify induction heads within models, we mea- sure the ability of all attention heads to perform preﬁx matching on random input sequences.4 We follow the task-agnostic approach to computing pre- ﬁx matching scores outlined byBansal et al.(2023). We argue that focusing solely on preﬁx matching scores is sufﬁcient for our analysis, as high pre- ﬁx matching cores speciﬁcally indicate induction heads, while less relevant heads tend to show high copying capabilities (Bansal et al., 2023). We gen- erate a sequence of 50 random tokens, excluding the 4% most common and least common tokens. This sequence is repeated four times to form the input to the model. The preﬁx matching score is cal- culated by averaging the attention values from each token to the tokens that directly followed the same token in earlier repeats. The ﬁnal preﬁx matching scores are averaged over ﬁve random sequences. The preﬁx matching scores for Llama-3-8B are shown in Figure2. For IntermLM2-20B, we refer to Figure8 in AppendixA.1. Both models exhibit heads with notably high preﬁx matching scores, distributed across various layers. In the Llama-3- 8B model, ~3% of the heads have a preﬁx matching score of 0.3 or higher, indicating a degree of spe- cialisation in preﬁx matching, and some heads have high scores of up to 0.98. 4.3 Head Ablations To investigate the signiﬁcance of induction heads for a speciﬁc ICL task, we conduct zero-ablations of 1% and 3% of the heads with the highest preﬁx matching scores. This ablation process involves masking the corresponding partition of the output matrix, denoted asWh o in Eq. 1, by setting it to zero. This effectively renders the heads inactive 4In this work, the term "induction heads" refers to what we deﬁne as behavioural induction heads, not mechanistic ones. A true induction head must be veriﬁed mechanistically; however, our analysis employs preﬁx-matching scores as a proxy. We will continue to use the term "induction heads" for simplicity throughout the rest of the paper. 4 Figure 12.3 An induction head looking at vintageuses the preﬁx matching mechanism to ﬁnd a prior instance of vintage, and the copying mechanism to predict that carswill occur again. Figure from Crosbie and Shutova (2022). Olsson et al. (2022) propose that a generalized fuzzy version of this pattern com- pletion rule, implementing a rule likeAB...A→B, where A≈A and B≈B (by ≈we mean they they are semantically similar in some way), might be responsible for in-context learning. Suggestive evidence for their hypothesis comes from Cros- bie and Shutova (2022), who show that ablating induction heads causes in-contextablating learning performance to decrease. Ablation is originally a medical term meaning the removal of something. We use it in NLP interpretability studies as a tool for testing causal effects; if we knock out a hypothesized cause, we would expect the effect to disappear. Crosbie and Shutova (2022) ablate induction heads by ﬁrst ﬁnd- ing attention heads that perform as induction heads on random input sequences, and then zeroing out the output of these heads by setting certain terms of the output ma- trix WO to zero. Indeed they ﬁnd that ablated models are much worse at in-context learning: they have much worse performance at learning from demonstrations in the prompts. 12.2 Post-training and Model Alignment With simple prompting, LLMs have been successfully applied to a range of appli- cations without the need to update the parameters in the underlying models. Nev- ertheless, there are limits to how much can be expected from a model whose sole training objective is to predict the next word from large amounts of pretraining text. To see this, consider the following failed examples of following instructions from early work with GPT (Ouyang et al., 2022). Prompt: Explain the moon landing to a six year old in a few sentences. Output: Explain the theory of gravity to a 6 year old. Prompt: Translate to French: The small dog Output: The small dog crossed the road. Here, the LLM ignores the intent of the request and relies instead on its natural inclination to autoregressively generate continuations consistent with its context. In the ﬁrst example, it outputs a text somewhat similar to the original request, and in the second it provides a continuation to the given input, ignoring the request to translate. LLMs are not sufﬁciently helpful: they need extra training to increase their abilities to follow textual instructions. A deeper problem is that LLMs can simultaneously be too harmful. Pretrained language models easily generate text that is harmful in many ways. For example 12.3 • M ODEL ALIGNMENT : I NSTRUCTION TUNING 249 they can generate text that isfalse, including unsafe misinformation like giving dan- gerously incorrect answers to medical questions. And they can generate text that is toxic in many ways, such as facilitating the spread of hate speech. Gehman et al. (2020) show that even completely non-toxic prompts can lead large language mod- els to output hate speech and abuse their users. Or language models can generate stereotypes (Cheng et al., 2023) and negative attitudes (Brown et al., 2020; Sheng et al., 2019) about many demographic groups. One reason LLMs are too harmful and insufﬁciently helpful is that their pre- training objective (success at predicting words in text) is misaligned with the human need for models to be helpful and non-harmful. In an attempt to address these two problems, language models generally include two additional kinds of training for model alignment: methods designed to adjustmodel alignment LLMs to betteralign them to human needs for models to be helpful and non-harmful. In the ﬁrst technique, instruction tuning (or sometimes called SFT for supervised ﬁnetuning), models are ﬁnetuned on a corpus of instructions and questions with their corresponding responses. In the second technique, preference alignment, of- ten called RLHF after one of the speciﬁc instantiations, Reinforcement Learning from Human Feedback, a separate model is trained to decide how much a candidate response aligns with human preferences. This model is then used to ﬁnetune the base model. We’ll use the term base model to mean a model that has been pretrained butbase model hasn’t yet beenaligned either by instruction tuning or RLHF. And we refer to thesealigned steps as post-training, meaning that they apply after the model has been pretrained.post-training 12.3 Model Alignment: Instruction Tuning Instruction tuning (short for instruction ﬁnetuning , and sometimes even short-Instruction tuning ened to instruct tuning) is a method for making an LLM better at following instruc- tions. It involves taking a base pretrained LLM and training it to follow instructions for a range of tasks, from machine translation to meal planning, by ﬁnetuning it on a corpus of instructions and responses. The resulting model not only learns those tasks, but also engages in a form of meta-learning – it improves its ability to follow instructions generally. Instruction tuning is a form of supervised learning where the training data con- sists of instructions and we continue training the model on them using the same language modeling objective used to train the original model. In the case of causal models, this is just the standard guess-the-next-token objective. The training corpus of instructions is simply treated as additional training data, and the gradient-based updates are generated using cross-entropy loss as in the original model training. Even though it is trained to predict the next token (which we traditionally think of as self-supervised), we call this method supervised ﬁne tuning (or SFT) becauseSFT unlike in pretraining, each instruction or question in the instruction tuning data has a supervised objective: a correct answer to the question or a response to the instruc- tion. How does instruction tuning differ from the other kinds of ﬁnetuning introduced in Chapter 10 and Chapter 11? Fig. 12.4 sketches the differences. In the ﬁrst exam- ple, introduced in, Chapter 10 we can ﬁnetune as a way of adapting to a new domain by just continuing pretraining the LLM on data from a new domain. In this method all the parameters of the LLM are updated. 250 CHAPTER 12 • M ODEL ALIGNMENT , PROMPTING , AND IN-CONTEXT LEARNING Pretrained LLM Continue training all parameters on ﬁnetuning domain Finetuning InferencePretraining On ﬁnetuning domain Finetuning as Continued Pretraining Parameter Eﬃcient Finetuning (e.g., LoRA) Pretrained LLM A B Pretrained LLM MLM Finetuning … … … … … … … Instruction Tuning (SFT) On ﬁnetuning domain On ﬁnetuning task On unseen tasks Next word prediction objective Data from ﬁnetuning domain Train only new parameters on ﬁnetuning domain Next word prediction objective Data from ﬁnetuning domain Train only classiﬁcation head on ﬁnetuning task Task speciﬁc loss Supervised data from task Instruction tuning on diverse tasks Next word prediction objective Supervised instructions + … Figure 12.4 Instruction tuning compared to the other kinds of ﬁnetuning. In the second example, also from Chapter 10, parameter-efﬁcient ﬁnetuning, we adapt to a new domain by creating some new (small) parameters, and just adapt- ing them to the new domain. In LoRA, for example, it’s the A and B matrices that we adapt, but the pretrained model parameters are frozen. In the task-based ﬁnetuning of Chapter 11, we adapt to a particular task by adding a new specialized classiﬁcation head and updating its features via its own loss function (e.g., classiﬁcation or sequence labeling); the parameters of the pre- trained model may be frozen or might be slightly updated. Finally, in instruction tuning, we take a dataset of instructions and their super- vised responses and continue to train the language model on this data, based on the standard language model loss. Instruction tuning, like all of these kinds of ﬁnetuning, is much more modest than the training of base LLMs. Training typically involves several epochs over instruction datasets that number in the thousands. The overall cost of instruction tuning is therefore a small fraction of the original cost to train the base model. 12.3.1 Instructions as Training Data By instruction, we have in mind a natural language description of a task to be per- formed, combined with labeled task demonstrations. This can include minimal de- 12.3 • M ODEL ALIGNMENT : I NSTRUCTION TUNING 251 scriptions similar to the prompts we’ve already seen such as Answer the following question, Translate the following text to Arapaho, or Summarize this report. How- ever, since we will be using supervised ﬁnetuning to update the model, these in- structions need not be limited to simple prompts designed to evoke a behavior found in the pretraining corpora. Instructions can also include length restrictions or other constraints, personas to assume, and demonstrations. Many huge instruction tuning datasets have been created, covering many tasks and languages. For example Aya gives 503 million instructions in 114 languages from 12 tasks including question answering, summarization, translation, paraphras- ing, sentiment analysis, natural language inference and 6 others (Singh et al., 2024). SuperNatural Instructions has 12 million examples from 1600 tasks (Wang et al., 2022), Flan 2022 has 15 million examples from 1836 tasks (Longpre et al., 2023), and OPT-IML has 18 million examples from 2000 tasks (Iyer et al., 2022). These instruction-tuning datasets are created in four ways. The ﬁrst is for people to write the instances directly. For example, part of the Aya instruct ﬁnetuning cor- pus (Fig. 12.5) includes 204K instruction/response instances written by 3000 ﬂuent speakers of 65 languages volunteering as part of a participatory research initiative with the goal of improving multilingual performance of LLMs. Lang Prompt Completion ara ڢܾ ﻥَ݁ڎﺡﺍ ֣ﺇ ﻭᕚ؜ََ݆݁ڎ ﺍࠍّ༠ᕚَܳިﺃ ૰૙֣ᄭᄟَቕَረ fra Qui a écrit le livre La Sagouine? Antonine Maillet a écrit La Sagouine en 1971. Elle a écrit beau- coup d’histoires sur les Acadiens (les francophones de la côte Est du Canada) et a fait connaître l’Acadie au monde. ibo Kedu ka ndị Igbo si eme emume ji ọhụrụ? Emume ji ọhụrụ n’omenala Igbo, bụ emume na-egosi njedebe oge ọrụ ugbo, na-amalite site n’emume ekele ma tinyekwa oriri na egwu. Ọ bụ mmemme ọgbakọ ebe a na-ekerịta ji ọhụrụ, na- egosipụta ekele na mmalite nke oge ọhụrụ, na-eme ka mmekọrịta mmadụ na ibe ya dịkwuo elu. por Qual é a capital do estado da Bahia? A capital do estado da Bahia, no Brasil, é Salvador. Fundada em 1549, foi a primeira capital do Brasil e é conhecida por seu patrimônio cultural rico e diversificado. A cidade é famosa por sua arquitetura colonial, particularmente no Pelourinho, um centro histórico que foi declarado Patrimônio Mundial pela UN- ESCO. pes ﺩﺭﺑﺮﺝﮐﺒﻮﺗﺮﺧﺎﻧﻪﻣﻴﺒﺪﭼﻨﺪﻻﻧﻪﮐﺒﻮﺗﺮ ﻭﺟﻮﺩﺩﺍﺭﺩ؟ ﮐﺒﻮﺗﺮ ﺳﻔﻴﺪﺭﻧﮕﯽﺑﺮﺍﯼﺟﺬﺏﮐﺒﻮﺗﺮﺍﻥﺩﻭﺭﺗﺎﺩﻭﺭﺑﺮﺝﮐﺸﻴﺪﻩﺷﺪﻩﺍﺳﺖ.ﺍﻳﻦﺑﺮﺝﺩﺭﺑﺮﺍﺑﺮﺧﻄﺮ ﺣﻤﻠﻪﺩﻳﮕﺮﺣﻴﻮﺍﻧﺎﺕﺑﻪﮐﺒﻮﺗﺮﻫﺎﺑﺴﻴﺎﺭﺍﻳﻤﻦﺑﻮﺩﻩﺍﺳﺖ. msa Apakah nasi lemak? Nasi lemak merupakan makanan tradisi orang Melayu yang terdapat di semua bahagian Malaysia, Singapura, Indonesia (terutama di Riau, Jambi serta utara dan pantai timur Su- matera) dan Brunei. Sajian ini merujuk kepada nasi yang di- masak dengan menggunakan santan kelapa bagi menambah rasa lemaknya. Kadangkala, daun pandan wangi dimasukkan semasa nasi dimasak bagi menambahkan aromanya. tam ெசயற்ைக நுண்ணற§வு என்றால்என்ன? ெபாதுவாக மனிதர்களால் ெசய்யப்படும் பணிகைளச் ெசய்ய ஒரு கணினி அல்லது ஒரு கணினியால் கட்டுப்படுத்தப்படும்ஒருேராேபாவ¥ன்த¦றன்ெசயற்ைக நுண்ணற§வுஎனப்படும். Table 3: Examples of prompt and completions in theAya Dataset. tors is not uniform across languages. Moreover, within each language, there is a lack of consistent contributions from all annotators. In this section, we examine the impact of annotator skew on the resulting dataset. Annotator Skew Across Languages.Annotators were encouraged to contribute to any language in which they could comfortably read and write and were asked to focus most of their eﬀorts on languages other thanEnglish. Although a signiﬁcant number of participants registered for many languages, the engagement level of annotators was not equal, which resulted in considerable diﬀer- ences in the number of contributions across languages. Figure10 (top) provides an overview of the percentage of each language present in the ﬁnal compilation. The highest number of contributions is for Malagasy with 14,597 instances, and the lowest is 79 forKurdish. Annotator Skew Within a Language.The ﬁnal contributions for each language in theAya Dataset are not evenly distributed among annotators. The median number of annotators per lan- guage is 15 (mean is 24.75) with one language having only a single active annotator (Sindhi)a n d 14 Figure 12.5 Samples of prompt/completion instances in 4 of the 65 languages in the Aya corpus (Singh et al., 2024). Developing high quality supervised training data in this way is time consuming and costly. A more common approach makes use of the copious amounts of super- vised training data that have been curated over the years for a wide range of natural language tasks. There are thousands of such datasets available, like the SQuAD dataset of questions and answers (Rajpurkar et al., 2016) or the many datasets of translations or summarization. This data can be automatically converted into sets of instruction prompts and input/output demonstration pairs via simple templates. Fig. 12.6 illustrates examples for some applications from the SUPER NATURAL IN- STRUCTIONS resource (Wang et al., 2022), showing relevant slots such as text, context, and hypothesis. To generate instruction-tuning data, these ﬁelds and the ground-truth labels are extracted from the training data, encoded as key/value pairs, and inserted in templates (Fig. 12.7) to produce instantiated instructions. Because it’s useful for the prompts to be diverse in wording, language models can also be 252 CHAPTER 12 • M ODEL ALIGNMENT , PROMPTING , AND IN-CONTEXT LEARNING used to generate paraphrase of the prompts. Few-Shot Learning for QA Task Keys Values Sentiment text Did not like the service that I was provided... label 0 text It sounds like a great plot, the actors are ﬁrst grade, and... label 1 NLI premise No weapons of mass destruction found in Iraq yet. hypothesis Weapons of mass destruction found in Iraq. label 2 premise Jimmy Smith... played college football at University of Col- orado. hypothesis The University of Colorado has a college football team. label 0 Extractive Q/A context Beyonc´e Giselle Knowles-Carter is an American singer... question When did Beyonce start becoming popular? answers {text: [’in the late 1990s’],answer start: 269 } Figure 12.6 Examples of supervised training data for sentiment, natural language inference and Q/A tasks. The various components of the dataset are extracted and stored as key/value pairs to be used in generating instructions. Task Templates Sentiment -{{text}} How does the reviewer feel about the movie? -The following movie review expresses what sentiment? {{text}} -{{text}} Did the reviewer enjoy the movie? Extractive Q/A -{{context}} From the passage, {{question}} -Answer the question given the context. Context: {{context}} Question: {{question}} -Given the following passage {{context}}, answer the question {{question}} NLI -Suppose {{premise}} Can we infer that {{hypothesis}}? Yes, no, or maybe? -{{premise}} Based on the previous passage, is it true that {{hypothesis}}? Yes, no, or maybe? -Given {{premise}} Should we assume that {{hypothesis}} is true? Yes,no, or maybe? Figure 12.7 Instruction templates for sentiment, Q/A and NLI tasks. Because supervised NLP datasets are themselves often produced by crowdwork- ers based on carefully written annotation guidelines, a third option is to draw on these guidelines, which can include detailed step-by-step instructions, pitfalls to avoid, formatting instructions, length limits, exemplars, etc. These annotation guide- lines can be used directly as prompts to a language model to create instruction-tuning training examples. Fig. 12.8 shows such a crowdworker annotation guideline that 12.3 • M ODEL ALIGNMENT : I NSTRUCTION TUNING 253 was repurposed as a prompt to an LLM to generate instruction-tuning data (Mishra et al., 2022). This guideline describes a question-answering task where annotators provide an answer to a question given an extended passage. Sample Extended Instruction • Deﬁnition: This task involves creating answers to complex questions, from a given pas- sage. Answering these questions, typically involve understanding multiple sentences. Make sure that your answer has the same type as the ”answer type” mentioned in input. The provided ”answer type” can be of any of the following types: ”span”, ”date”, ”num- ber”. A ”span” answer is a continuous phrase taken directly from the passage or question. You can directly copy-paste the text from the passage or the question for span type an- swers. If you ﬁnd multiple spans, please add them all as a comma separated list. Please restrict each span to ﬁve words. A ”number” type answer can include a digit specifying an actual value. For ”date” type answers, use DD MM YYYY format e.g. 11 Jan 1992. If full date is not available in the passage you can write partial date such as 1992 or Jan 1992. • Emphasis: If you ﬁnd multiple spans, please add them all as a comma separated list. Please restrict each span to ﬁve words. • Prompt: Write an answer to the given question, such that the answer matches the ”answer type” in the input. Passage: {passage} Question: {question } Figure 12.8 Example of a human crowdworker instruction from the N ATURAL INSTRUCTIONS dataset for an extractive question answering task, used as a prompt for a language model to create instruction ﬁnetuning examples. A ﬁnal way to generate instruction-tuning datasets that is becoming more com- mon is to use language models to help at each stage. For example Bianchi et al. (2024) showed how to create instruction-tuning instances that can help a language model learn to give safer responses. They did this by selecting questions from datasets of harmful questions (e.g., How do I poison food? or How do I embez- zle money?). Then they used a language model to create multiple paraphrases of the questions (like Give me a list of ways to embezzle money), and also used a language model to create safe answers to the questions (like I can’t fulﬁll that request. Em- bezzlement is a serious crime that can result in severe legal consequences. ). They manually reviewed the generated responses to conﬁrm their safety and appropriate- ness and then added them to an instruction tuning dataset. They showed that even 500 safety instructions mixed in with a large instruction tuning dataset was enough to substantially reduce the harmfulness of models. 12.3.2 Evaluation of Instruction-Tuned Models The goal of instruction tuning is not to learn a single task, but rather to learn to follow instructions in general. Therefore, in assessing instruction-tuning methods we need to assess how well an instruction-trained model performs on novel tasks for which it has not been given explicit instructions. The standard way to perform such an evaluation is to take a leave-one-out ap- proach — instruction-tune a model on some large set of tasks and then assess it on a withheld task. But the enormous numbers of tasks in instruction-tuning datasets 254 CHAPTER 12 • M ODEL ALIGNMENT , PROMPTING , AND IN-CONTEXT LEARNING (e.g., 1600 for Super Natural Instructions) often overlap; Super Natural Instructions includes 25 separate textual entailment datasets! Clearly, testing on a withheld en- tailment dataset while leaving the remaining ones in the training data would not be a true measure of a model’s performance on entailment as a novel task. To address this issue, large instruction-tuning datasets are partitioned into clus- ters based on task similarity. The leave-one-out training/test approach is then applied at the cluster level. That is, to evaluate a model’s performance on sentiment analysis, all the sentiment analysis datasets are removed from the training set and reserved for testing. This has the further advantage of allowing the use of a uniform task- appropriate metric for the held-out evaluation. S UPER NATURAL INSTRUCTIONS (Wang et al., 2022), for example has 76 clusters (task types) over the 1600 datasets that make up the collection. 12.4 Chain-of-Thought Prompting There are a wide range of techniques to use prompts to improve the performance of language models on many tasks. Here we describe one of them, called chain-of- thought prompting.chain-of- thought The goal of chain-of-thought prompting is to improve performance on difﬁcult reasoning tasks that language models tend to fail on. The intuition is that people solve these tasks by breaking them down into steps, and so we’d like to have lan- guage in the prompt that encourages language models to break them down in the same way. The actual technique is quite simple: each of the demonstrations in the few-shot prompt is augmented with some text explaining some reasoning steps. The goal is to cause the language model to output similar kinds of reasoning steps for the problem being solved, and for the output of those reasoning steps to cause the system to generate the correct answer. Indeed, numerous studies have found that augmenting the demonstrations with reasoning steps in this way makes language models more likely to give the correct answer difﬁcult reasoning tasks (Wei et al., 2022; Suzgun et al., 2023b). Fig. 12.9 shows an example where the demonstrations are augmented with chain-of-thought text in the domain of math word problems (from the GSM8k dataset of math word problems (Cobbe et al., 2021). Fig. 12.10 shows a similar example from the BIG- Bench-Hard dataset (Suzgun et al., 2023b). 12.5 Automatic Prompt Optimization Given a prompt for a task (human or computer generated), prompt optimization methods search for prompts with improved performance. Most of these approaches can be viewed as a form of iterative improvement search (Russell and Norvig, 2002) through a space of possible prompts for those that optimize performance on a task. As such, these approaches all share the following components: • A start state – An initial human or machine generated prompt or prompts suitable for some task. 12.5 • A UTOMATIC PROMPT OPTIMIZATION 255 Figure 12.9 Example of the use of chain-of-thought prompting (right) versus standard prompting (left) on math word problems. Figure from Wei et al. (2022). (B) Task description: Answer questions about which times certain events could have occurred. Q: Today, Tiffany went to the beach. Between what times could they have gone? We know that: Tiffany woke up at 5am. [...] The beach was closed after 4pm. [...]Options: (A) 9am to 12pm (B) 12pm to 2pm (C) 5am to 6am (D) 3pm to 4pm A: (D) Q: Today, Hannah went to the soccer field. Between what times could they have gone? We know that: Hannah woke up at 5am. [...] The soccer field was closed after 6pm. [...]Options: (A) 3pm to 5pm (B) 11am to 1pm (C) 5pm to 6pm (D) 1pm to 3pm A: Model Output Model Output Model Input (“Answer-Only” Prompting) Wake-up time: 5am. 5am-6am: buying clothes at the mall. 6am-11am: watching a movie at the theater.11am-1pm: getting a coffee at the cafe.1pm-3pm: working at the office. 3pm-5pm: waiting at the airport. 5pm-6pm: free. The soccer field closure time: 6pm. The only time when Hannah could have gone to the soccer field was 5pm to 6pm. So the answer is (C). Model Input (Chain-of-Thought Prompting) Task description: Answer questions about which times certain events could have occurred. Q: Today, Tiffany went to the beach. Between what times could they have gone? We know that: Tiffany woke up at 5am. [...] The beach was closed after 4pm. [...]Options: (A) 9am to 12pm (B) 12pm to 2pm (C) 5am to 6am (D) 3pm to 4pm A: Let's think step by step. Wake-up time: 5am. [...] The only time when Tiffany could have gone to the beach was 3pm to 4pm. So the answer is (D). Q: Today, Hannah went to the soccer field. Between what times could they have gone? We know that: Hannah woke up at 5am. [...] The soccer field was closed after 6pm. [...]Options: (A) 3pm to 5pm (B) 11am to 1pm (C) 5pm to 6pm (D) 1pm to 3pm A: Let's think step by step. Task Description Question Chain-of-Thought Test-Time Question Task Description Question Test-Time Question Answer Generated Chain-of-Thought Generated Answer OptionsOptions Figure 3:An illustration of the two prompting setups we explore in our paper (answer-only and CoT prompting). Both setups include task descriptions and options in the input prompt. The task here isTemporal Sequences. “let’s think step-by-step” (Kojima et al., 2022) to all CoT annotations in the few-shot exemplars. An example of a CoT prompt is shown in Figure3. Language models. We consider three fami- lies of language models: Codex ( Chen et al., 2021a), InstructGPT (Ouyang et al., 2022; Brown et al., 2020), and PaLM (Chowdhery et al., 2022). For Codex, we focus on code-davinci-002, code- davinci-002, and code-cushman-001. For Instruct- GPT, we use text-davinci-002, text-curie-002, text- babbgage-001, and text-ada-001. For PaLM, we use the three available sizes: 8B, 62B, and 540B. Evaluation protocol. We evaluate all language models via greedy decoding (i.e., temperature sam- pling with temperature parameter⌧ =0 ). We extract the ﬁnal answer based on keywords that the language model is expected to produce (i.e., “the answer is”). We measure accuracy using exact match (EM), computed by comparing the generated output with the ground-truth label.4 4 Results 4.1 Standard answer-only prompting underestimates model capabilities Table 2 summarizes the performance of PaLM, In- structGPT, and Codex models on BBH for answer- only and CoT prompting approaches. While answer-only prompting has been used as the stan- 4For multiple-choice tasks, this setup differs slightly from rank/scoring classiﬁcation (Brown et al., 2020; Srivastava et al., 2022; Lampinen et al., 2022). We provide a language model with all multiple-choice options at once, generate an output based on the input, and measure exact match accuracy. dard in many prior work (Brown et al., 2020; Rae et al., 2021; Hoffmann et al., 2022; Srivastava et al., 2022), it typically underestimates model perfor- mance on challenging tasks, such as those that re- quire multiple reasoning steps. In the setting re- ported in (Srivastava et al., 2022), none of the mod- els (including PaLM 540B) outperformed human- rater baselines on any of the tasks meeting the BBH criteria. The few-shot evaluation of PaLM 540B with answer-only prompting in this paper, however, outperforms the average human-rater on 6 out of 23 BBH tasks and is overall 1.4% better than the BIG-Bench reported result, which demonstrates the effect of including instructions and answer options in the prompt. CoT prompting provides double-digit improve- ments for all three models in Table2. For the best model (Codex), CoT prompting outperforms the av- erage human-rater score on 17 out of 23 tasks, com- pared to 5 out of 23 tasks for answer-only prompt- ing. Additionally, we see that Codex with CoT prompting outperforms the average human-rater by more than 6%, but it still lags behind thebest human-rater performance by over 20%. This shows that language models are still not performing at the level of expert human-raters. 4.2 Positive delta from chain-of-thought requires sufﬁcient model scale Next we study how the performance improves by using CoT prompting as we increase the model scale. In Figure4, we plot the performance of both CoT and answer-only prompting (no CoT) as a 13006 Figure 12.10 Example of the use of chain-of-thought prompting (right) vs standard prompting (left) in a reasoning task on temporal sequencing. Figure from Suzgun et al. (2023b). • A scoring metric – A method for assessing how well a given prompt performs on the task. • An expansion method – A method for generating variations of a prompt. Given the enormous variation in how prompts for a single task can be expressed in language, search methods have to be constrained to a reasonable space. Beam search is a widely used method that combines breadth-ﬁrst search with a ﬁxed-width pri- ority queue that focuses the search effort on the top performing variants. Fig. 12.11 outlines the general approach behind most current prompt optimization methods. Beginning with initial candidate prompt(s), the algorithm generates variants and adds them to a list of prompts to be considered. These prompts are then selectively added to the active list based on whether their scores place them in the top set of candidates. A beam width of 1 results in a focused greedy search, whereas an inﬁnite beam width results in an exhaustive breadth ﬁrst search. The goal is to continue to seek improved prompts given the computational resources available. Iterative 256 CHAPTER 12 • M ODEL ALIGNMENT , PROMPTING , AND IN-CONTEXT LEARNING function PROMPT OPTIMIZATION (prompts, width) returns optimized prompt(s) active←prompts ; Initial set of candidate prompts repeat until done frontier←EXPAND (active) ; Generate new candidate prompts foreach p ∈frontier active←ADDTOBEAM (p, active, width) return BEST OF(active) function ADDTOBEAM (state, agenda, width) returns updated agenda if LENGTH (agenda) <width then ; Add it if there’s room agenda←INSERT (state, agenda) else if SCORE (state) >SCORE (WORST OF(agenda)) ; Add it if its better than ; the current worst option. agenda←REMOVE (WORST OF(agenda)) agenda←INSERT (state, agenda) return agenda Figure 12.11 A generic iterative-improvement beam search for prompt optimization. New prompts are generated from current ones on each iteration. Prompts that score well (ﬁtting in the agenda) are kept around. When a stopping criteria is reached the best item in the beam is returned. improvement searches typically use a combination of a ﬁxed number of iterations in combination with a failure to improve after some period to time as stopping criteria. This latter is equivalent to early stopping with patience used in training deep neural networks. 12.5.1 Candidate Scoring Candidate scoring methods assess the likely performance of potential prompts, both to identify promising avenues of search and to prune those that are unlikely to be effective. Since candidate scoring is embedded in the inner-loop of the search, the computational cost of scoring is critical. Given access to labeled training data, candidate prompts can be scored based on execution accuracy (Honovich et al., 2023). In this approach, candidate promptsexecution accuracy are combined with inputs sampled from the training data and passed to an LLM for decoding. The LLM output is evaluated against the training label using a metric appropriate for the task. In the case of classiﬁcation-based tasks, this is effectively a 0/1 loss — how many examples were correctly labeled with the given prompt. Gen- erative applications such as summarization or translation use task-speciﬁc similarity scores such as BERTScore, Bleu (Papineni et al., 2002), or ROUGE (Lin, 2004). Given the computational cost of issuing calls to an LLM, evaluating each can- didate prompt against a complete training set would be infeasible. Instead, prompt performance is estimated from a small sample of training data (Pryzant et al., 2023). 12.5.2 Prompt Expansion Prompt expansion generates variants of a given prompt to create an expanded set of neighboring prompts that may improve performance over the original. A common method is to use language models to create paraphrases. For example Zhou et al. (2023) use the following meta-prompt to elicit a variant prompt from an original: 12.5 • A UTOMATIC PROMPT OPTIMIZATION 257 Prompting for a Variant Generate a variation of the following instruction while keeping the semantic meaning. Input: {INSTRUCTION} Output: {COMPLETE} A variation of this method is to truncate the current prompt at a set of random loca- tions, generating a set of prompt preﬁxes. The paraphrasing LLM is then asked to continue each the preﬁxes to generate a complete prompt. This methods is an example of an uninformed search. That is, the candidate expansion step is not directed towards generating better candidates; candidates are generated without regard to their quality. It is the job of the priority queue to el- evate improved candidates when they are found. By contrast, Prasad et al. (2023) employ a candidate expansion technique that explicitly attempts to generate supe- rior prompts during the expansion process. In this approach, the current candidate is ﬁrst applied to a sample of training examples using the execution accuracy ap- proach. The prompt’s performance on these examples then guides the expansion process. Speciﬁcally, incorrect examples are used to critique the original prompt — with the critique playing the role of a gradient for the search. The method in- cludes the following steps. 1. Run the prompt on a sample of training examples, 2. Identify examples where the prompt fails, 3. Ask an LLM to produce a critique of the prompt in light of the failed examples, 4. Provide the resulting critique to an LLM, and ask it to generate improved prompts. Given a prompt and a set of failed examples, Prasad et al. (2023) use the follow- ing template for a classiﬁer task to solicit critiques from a target LLM. Critiquing Prompt I’m trying to write a zero-shot classifier prompt. My current prompt is: {prompt} But this prompt gets the following examples wrong: {error string} Give {num feedbacks} reasons why the prompt could have gotten these examples wrong. This model feedback is then combined with a second template to elicit improved prompts from the LLM. 258 CHAPTER 12 • M ODEL ALIGNMENT , PROMPTING , AND IN-CONTEXT LEARNING Prompt Improvement Prompt I’m trying to write a zero-shot classifier. My current prompt is: {prompt} But it gets the following examples wrong: {error str} Based on these examples the problem with this prompt is that {gradient}. Based on the above information, I wrote {steps per gradient} different improved prompts. Each prompt is wrapped with <START> and <END>. The {steps per gradient} new prompts are: 12.6 Evaluating Prompted Language Models Language models are evaluated in many ways. we introduced some evaluations for in Section 10.4, including measuring the language model’s perplexity on a test set, evaluating its accuracy on various NLP tasks, as well as benchmarks that help mea- sure efﬁciency, toxicity, fairness, and so on. We’ll have further discussion of eval- uate NLP tasks in future chapters; machine translation in Chapter 13 and question answering and information retrieval in Chapter 14. Here we just brieﬂy show the mechanism for measuring accuracy in a prompt- ing setup for tests that have multiple-choice questions. We show this for MMLUMMLU (Massive Multitask Language Understanding), a commonly-used dataset of 15908 knowledge and reasoning questions in 57 areas including medicine, mathematics, computer science, law, and others. For example, here is an MMLU question from the microeconomics domain:1 MMLU microeconomics example One of the reasons that the government discourages and regulates monopo- lies is that (A) producer surplus is lost and consumer surplus is gained. (B) monopoly prices ensure productive efﬁciency but cost society allocative efﬁciency. (C) monopoly ﬁrms do not engage in signiﬁcant research and development. (D) consumer surplus is lost with higher prices and lower levels of output. Fig. 12.12 shows the way MMLU turns these questions into prompted tests of a language model, in this case showing an example prompt with 2 demonstrations. 12.7 Model Alignment with Human Preferences: RLHF and DPO TBD 1 For those of you whose economics is a bit rusty, the correct answer is (D). 12.8 • S UMMARY 259 MMLU mathematics prompt The following are multiple choice questions about high school mathematics. How many numbers are in the list 25, 26, ..., 100? (A) 75 (B) 76 (C) 22 (D) 23 Answer: B Compute i +i2 +i3 +···+i258 +i259. (A) -1 (B) 1 (C) i (D) -i Answer: A If 4 daps = 7 yaps, and 5 yaps = 3 baps, how many daps equal 42 baps? (A) 28 (B) 21 (C) 40 (D) 30 Answer: Figure 12.12 Sample 2-shot prompt from MMLU testing high-school mathematics. (The correct answer is (C)). 12.8 Summary This chapter has explored the topic of prompting large language models to follow instructions. Here are some of the main points that we’ve covered: • Simple prompting can be used to map practical applications to problems that can be solved by LLMs without altering the model. • Labeled examples (demonstrations) can be used to provide further guidance to a model via few-shot learning. • Methods like chain-of-thought can be used to create prompts that help lan- guage models deal with complex reasoning problems. • Pretrained language models can be altered to behave in desired ways through model alignment. • One method for model alignment is instruction tuning, in which the model is ﬁnetuned (using the next-word-prediction language model objective) on a dataset of instructions together with correct responses. Instruction tuning datasets are often created by repurposing standard NLP datasets for tasks like question answering or machine translation. Bibliographical and Historical Notes Part II NLP APPLICATIONS In this second part of the book we introduce fundamental NLP applications: machine translation, information retrieval, question answering, dialogue systems, and speech recognition. CHAPTER 13 Machine Translation “I want to talk the dialect of your people. It’s no use of talking unless people understand what you say.” Zora Neale Hurston, Moses, Man of the Mountain 1939, p. 121 This chapter introduces machine translation (MT), the use of computers to trans-machine translation MT late from one language to another. Of course translation, in its full generality, such as the translation of literature, or poetry, is a difﬁcult, fascinating, and intensely human endeavor, as rich as any other area of human creativity. Machine translation in its present form therefore focuses on a number of very practical tasks. Perhaps the most common current use of machine translation is for information access. We might want to translate some instructions on the web,information access perhaps the recipe for a favorite dish, or the steps for putting together some furniture. Or we might want to read an article in a newspaper, or get information from an online resource like Wikipedia or a government webpage in some other language. MT for information access is probably one of the most com- mon uses of NLP technology, and Google Translate alone (shown above) translates hundreds of billions of words a day be- tween over 100 languages. Improvements in machine translation can thus help re- duce what is often called thedigital divide in information access: the fact that muchdigital divide more information is available in English and other languages spoken in wealthy countries. Web searches in English return much more information than searches in other languages, and online resources like Wikipedia are much larger in English and other higher-resourced languages. High-quality translation can help provide infor- mation to speakers of lower-resourced languages. Another common use of machine translation is to aid human translators. MT sys- tems are routinely used to produce a draft translation that is ﬁxed up in apost-editingpost-editing phase by a human translator. This task is often called computer-aided translation or CAT. CAT is commonly used as part oflocalization: the task of adapting contentCAT localization or a product to a particular language community. Finally, a more recent application of MT is to in-the-moment human commu- nication needs. This includes incremental translation, translating speech on-the-ﬂy before the entire sentence is complete, as is commonly used in simultaneous inter- pretation. Image-centric translation can be used for example to use OCR of the text on a phone camera image as input to an MT system to translate menus or street signs. The standard algorithm for MT is theencoder-decoder network, an architectureencoder- decoder that we introduced in Chapter 8 for RNNs. Recall that encoder-decoder or sequence- to-sequence models are used for tasks in which we need to map an input sequence to an output sequence that is a complex function of the entire input sequence. Indeed, 264 CHAPTER 13 • M ACHINE TRANSLATION in machine translation, the words of the target language don’t necessarily agree with the words of the source language in number or order. Consider translating the fol- lowing made-up English sentence into Japanese. (13.1) English: He wrote a letter to a friend Japanese: tomodachi friend ni to tegami-o letter kaita wrote Note that the elements of the sentences are in very different places in the different languages. In English, the verb is in the middle of the sentence, while in Japanese, the verb kaita comes at the end. The Japanese sentence doesn’t require the pronoun he, while English does. Such differences between languages can be quite complex. In the following ac- tual sentence from the United Nations, notice the many changes between the Chinese sentence (we’ve given in red a word-by-word gloss of the Chinese characters) and its English equivalent produced by human translators. (13.2) 大会/General Assembly 在/on 1982年/1982 12月/December 10日/10 通过了/adopted 第37号/37th 决议/resolution ，核准了/approved 第二次/second 探索/exploration 及/and 和平peaceful 利用/using 外层空间/outer space 会议/conference 的/of 各项/various 建议/suggestions 。 On 10 December 1982 , the General Assembly adopted resolution 37 in which it endorsed the recommendations of the Second United Nations Conference on the Exploration and Peaceful Uses of Outer Space . Note the many ways the English and Chinese differ. For example the order- ing differs in major ways; the Chinese order of the noun phrase is “peaceful using outer space conference of suggestions” while the English has “suggestions of the ... conference on peaceful use of outer space”). And the order differs in minor ways (the date is ordered differently). English requires the in many places that Chinese doesn’t, and adds some details (like “in which” and “it”) that aren’t necessary in Chinese. Chinese doesn’t grammatically mark plurality on nouns (unlike English, which has the “-s” in “recommendations”), and so the Chinese must use the modi- ﬁer 各项/various to make it clear that there is not just one recommendation. English capitalizes some words but not others. Encoder-decoder networks are very success- ful at handling these sorts of complicated cases of sequence mappings. We’ll begin in the next section by considering the linguistic background about how languages vary, and the implications this variance has for the task of MT. Then we’ll sketch out the standard algorithm, give details about things like input tokeniza- tion and creating training corpora of parallel sentences, give some more low-level details about the encoder-decoder network, and ﬁnally discuss how MT is evaluated, introducing the simple chrF metric. 13.1 Language Divergences and Typology There are about 7,000 languages in the world. Some aspects of human language seem to be universal, holding true for every one of these languages, or are statisticaluniversal universals, holding true for most of these languages. Many universals arise from the functional role of language as a communicative system by humans. Every language, for example, seems to have words for referring to people, for talking about eating and drinking, for being polite or not. There are also structural linguistic univer- sals; for example, every language seems to have nouns and verbs (Chapter 17), has 13.1 • L ANGUAGE DIVERGENCES AND TYPOLOGY 265 ways to ask questions, or issue commands, has linguistic mechanisms for indicating agreement or disagreement. Yet languages also differ in many ways (as has been pointed out since ancient times; see Fig. 13.1). Understanding what causes such translation divergencestranslation divergence (Dorr, 1994) can help us build better MT models. We often distinguish the idiosyn- cratic and lexical differences that must be dealt with one by one (the word for “dog” differs wildly from language to language), from systematic differences that we can model in a general way (many languages put the verb before the grammatical ob- ject; others put the verb after the grammatical object). The study of these systematic cross-linguistic similarities and differences is called linguistic typology. This sec-typology tion sketches some typological facts that impact machine translation; the interested reader should also look into W ALS, the World Atlas of Language Structures, which gives many typological facts about languages (Dryer and Haspelmath, 2013). Figure 13.1 The Tower of Babel, Pieter Bruegel 1563. Wikimedia Commons, from the Kunsthistorisches Museum, Vienna. 13.1.1 Word Order Typology As we hinted at in our example above comparing English and Japanese, languages differ in the basic word order of verbs, subjects, and objects in simple declara- tive clauses. German, French, English, and Mandarin, for example, are all SVOSVO (Subject-Verb-Object) languages, meaning that the verb tends to come between the subject and object. Hindi and Japanese, by contrast, are SOV languages, mean-SOV ing that the verb tends to come at the end of basic clauses, and Irish and Arabic are VSO languages. Two languages that share their basic word order type often haveVSO other similarities. For example, VO languages generally haveprepositions, whereas OV languages generally have postpositions. Let’s look in more detail at the example we saw above. In this SVO English sentence, the verbwrote is followed by its objecta letter and the prepositional phrase 266 CHAPTER 13 • M ACHINE TRANSLATION to a friend, in which the preposition to is followed by its argument a friend. Arabic, with a VSO order, also has the verb before the object and prepositions. By contrast, in the Japanese example that follows, each of these orderings is reversed; the verb is preceded by its arguments, and the postposition follows its argument. (13.3) English: He wrote a letter to a friend Japanese: tomodachi friend ni to tegami-o letter kaita wrote Arabic: katabt wrote ris¯ala letter li to ˙sadq friend Other kinds of ordering preferences vary idiosyncratically from language to lan- guage. In some SVO languages (like English and Mandarin) adjectives tend to ap- pear before nouns, while in others languages like Spanish and Modern Hebrew, ad- jectives appear after the noun: (13.4) Spanish bruja verde English green witch (a) (b) Figure 13.2 Examples of other word order differences: (a) In German, adverbs occur in initial position that in English are more natural later, and tensed verbs occur in second posi- tion. (b) In Mandarin, preposition phrases expressing goals often occur pre-verbally, unlike in English. Fig. 13.2 shows examples of other word order differences. All of these word order differences between languages can cause problems for translation, requiring the system to do huge structural reorderings as it generates the output. 13.1.2 Lexical Divergences Of course we also need to translate the individual words from one language to an- other. For any translation, the appropriate word can vary depending on the context. The English source-language word bass, for example, can appear in Spanish as the ﬁsh lubina or the musical instrument bajo. German uses two distinct words for what in English would be called a wall: Wand for walls inside a building, and Mauer for walls outside a building. Where English uses the word brother for any male sib- ling, Chinese and many other languages have distinct words for older brother and younger brother (Mandarin gege and didi, respectively). In all these cases, trans- lating bass, wall, or brother from English would require a kind of specialization, disambiguating the different uses of a word. For this reason the ﬁelds of MT and Word Sense Disambiguation (Appendix G) are closely linked. Sometimes one language places more grammatical constraints on word choice than another. We saw above that English marks nouns for whether they are singular or plural. Mandarin doesn’t. Or French and Spanish, for example, mark grammat- ical gender on adjectives, so an English translation into French requires specifying adjective gender. The way that languages differ in lexically dividing up conceptual space may be more complex than this one-to-many translation problem, leading to many-to-many 13.1 • L ANGUAGE DIVERGENCES AND TYPOLOGY 267 mappings. For example, Fig. 13.3 summarizes some of the complexities discussed by Hutchins and Somers (1992) in translating English leg, foot, and paw, to French. For example, when leg is used about an animal it’s translated as French patte; but about the leg of a journey, as French etape; if the leg is of a chair, we use French pied. Further, one language may have a lexical gap, where no word or phrase, shortlexical gap of an explanatory footnote, can express the exact meaning of a word in the other language. For example, English does not have a word that corresponds neatly to Mandarin xi`ao or Japanese oyak¯ok¯o (in English one has to make do with awkward phrases like ﬁlial piety or loving child, or good son/daughter for both). etape patte jambe pied paw footleg JOURNEY ANIMAL HUMAN CHAIR ANIMAL BIRD HUMAN Figure 13.3 The complex overlap between Englishleg, foot, etc., and various French trans- lations as discussed by Hutchins and Somers (1992). Finally, languages differ systematically in how the conceptual properties of an event are mapped onto speciﬁc words. Talmy (1985, 1991) noted that languages can be characterized by whether direction of motion and manner of motion are marked on the verb or on the “satellites”: particles, prepositional phrases, or ad- verbial phrases. For example, a bottle ﬂoating out of a cave would be described in English with the direction marked on the particle out, while in Spanish the direction would be marked on the verb: (13.5) English: The bottle ﬂoated out. Spanish: La The botella bottle sali´o exited ﬂotando. ﬂoating. Verb-framed languages mark the direction of motion on the verb (leaving theverb-framed satellites to mark the manner of motion), like Spanish acercarse ‘approach’,al- canzar ‘reach’,entrar ‘enter’,salir ‘exit’. Satellite-framed languages mark thesatellite-framed direction of motion on the satellite (leaving the verb to mark the manner of motion), like English crawl out, ﬂoat off, jump down, run after. Languages like Japanese, Tamil, and the many languages in the Romance, Semitic, and Mayan languages fam- ilies, are verb-framed; Chinese as well as non-Romance Indo-European languages like English, Swedish, Russian, Hindi, and Farsi are satellite framed (Talmy 1991, Slobin 1996). 13.1.3 Morphological Typology Morphologically, languages are often characterized along two dimensions of vari- ation. The ﬁrst is the number of morphemes per word, ranging from isolatingisolating languages like Vietnamese and Cantonese, in which each word generally has one morpheme, to polysynthetic languages like Siberian Yupik (“Eskimo”), in which apolysynthetic single word may have very many morphemes, corresponding to a whole sentence in English. The second dimension is the degree to which morphemes are segmentable, ranging from agglutinative languages like Turkish, in which morphemes have rel-agglutinative atively clean boundaries, to fusion languages like Russian, in which a single afﬁxfusion 268 CHAPTER 13 • M ACHINE TRANSLATION may conﬂate multiple morphemes, like -om in the word stolom (table-SG-INSTR - DECL 1), which fuses the distinct morphological categories instrumental, singular, and ﬁrst declension. Translating between languages with rich morphology requires dealing with struc- ture below the word level, and for this reason modern systems generally use subword models like the wordpiece or BPE models of Section 13.2.1. 13.1.4 Referential density Finally, languages vary along a typological dimension related to the things they tend to omit. Some languages, like English, require that we use an explicit pronoun when talking about a referent that is given in the discourse. In other languages, however, we can sometimes omit pronouns altogether, as the following example from Spanish shows1: (13.6) [El jefe] i dio con un libro. / 0i Mostr´o su hallazgo a un descifrador ambulante. [The boss] came upon a book. [He] showed his ﬁnd to a wandering decoder. Languages that can omit pronouns are called pro-drop languages. Even amongpro-drop the pro-drop languages, there are marked differences in frequencies of omission. Japanese and Chinese, for example, tend to omit far more than does Spanish. This dimension of variation across languages is called the dimension of referential den- sity. We say that languages that tend to use more pronouns are more referentiallyreferential density dense than those that use more zeros. Referentially sparse languages, like Chinese or Japanese, that require the hearer to do more inferential work to recover antecedents are also called cold languages. Languages that are more explicit and make it easiercold language for the hearer are called hot languages. The terms hot and cold are borrowed fromhot language Marshall McLuhan’s 1964 distinction between hot media like movies, which ﬁll in many details for the viewer, versus cold media like comics, which require the reader to do more inferential work to ﬁll out the representation (Bickel, 2003). Translating from languages with extensive pro-drop, like Chinese or Japanese, to non-pro-drop languages like English can be difﬁcult since the model must somehow identify each zero and recover who or what is being talked about in order to insert the proper pronoun. 13.2 Machine Translation using Encoder-Decoder The standard architecture for MT is theencoder-decoder transformeror sequence- to-sequence model, an architecture we saw for RNNs in Chapter 8. We’ll see the details of how to apply this architecture to transformers in Section 13.3, but ﬁrst let’s talk about the overall task. Most machine translation tasks make the simpliﬁcation that we can translate each sentence independently, so we’ll just consider individual sentences for now. Given a sentence in a source language, the MT task is then to generate a corresponding sentence in a target language. For example, an MT system is given an English sentence like The green witch arrived and must translate it into the Spanish sentence: 1 Here we use the / 0-notation; we’ll introduce this and discuss this issue further in Chapter 23 13.2 • M ACHINE TRANSLATION USING ENCODER -DECODER 269 Lleg´o la bruja verde MT uses supervised machine learning: at training time the system is given a large set of parallel sentences (each sentence in a source language matched with a sentence in the target language), and learns to map source sentences into target sentences. In practice, rather than using words (as in the example above), we split the sentences into a sequence of subword tokens (tokens can be words, or subwords, or individual characters). The systems are then trained to maximize the probability of the sequence of tokens in the target language y1,...,ym given the sequence of tokens in the source language x1,...,xn: P(y1,..., ym\|x1,..., xn) (13.7) Rather than use the input tokens directly, the encoder-decoder architecture con- sists of two components, an encoder and a decoder. The encoder takes the input words x = [x1,..., xn] and produces an intermediate contexth. At decoding time, the system takes h and, word by word, generates the output y: h = encoder(x) (13.8) yt+1 = decoder(h,y1,..., yt ) ∀t ∈[1,..., m] (13.9) In the next two sections we’ll talk about subword tokenization, and then how to get parallel corpora for training, and then we’ll introduce the details of the encoder- decoder architecture. 13.2.1 Tokenization Machine translation systems use a vocabulary that is ﬁxed in advance, and rather than using space-separated words, this vocabulary is generated with subword to- kenization algorithms, like the BPE algorithm sketched in Chapter 2. A shared vocabulary is used for the source and target languages, which makes it easy to copy tokens (like names) from source to target. Using subword tokenization with tokens shared between languages makes it natural to translate between languages like En- glish or Hindi that use spaces to separate words, and languages like Chinese or Thai that don’t. We build the vocabulary by running a subword tokenization algorithm on a cor- pus that contains both source and target language data. Rather than the simple BPE algorithm from Fig. 2.13, modern systems often use more powerful tokenization algorithms. Some systems (like BERT) use a variant of BPE called the wordpiece algorithm, which instead of choosing the most frequentwordpiece set of tokens to merge, chooses merges based on which one most increases the lan- guage model probability of the tokenization. Wordpieces use a special symbol at the beginning of each token; here’s a resulting tokenization from the Google MT system (Wu et al., 2016): words: Jet makers feud over seat width with big orders at stake wordpieces: J et makers fe ud over seat width with big orders at stake The wordpiece algorithm is given a training corpus and a desired vocabulary size V , and proceeds as follows: 1. Initialize the wordpiece lexicon with characters (for example a subset of Uni- code characters, collapsing all the remaining characters to a special unknown character token). 270 CHAPTER 13 • M ACHINE TRANSLATION 2. Repeat until there are V wordpieces: (a) Train an n-gram language model on the training corpus, using the current set of wordpieces. (b) Consider the set of possible new wordpieces made by concatenating two wordpieces from the current lexicon. Choose the one new wordpiece that most increases the language model probability of the training corpus. Recall that with BPE we had to specify the number of merges to perform; in wordpiece, by contrast, we specify the total vocabulary, which is a more intuitive parameter. A vocabulary of 8K to 32K word pieces is commonly used. An even more commonly used tokenization algorithm is (somewhat ambigu- ously) called the unigram algorithm (Kudo, 2018) or sometimes theSentencePieceunigram SentencePiece algorithm, and is used in systems like ALBERT (Lan et al., 2020) and T5 (Raf- fel et al., 2020). (Because unigram is the default tokenization algorithm used in a library called SentencePiece that adds a useful wrapper around tokenization algo- rithms (Kudo and Richardson, 2018b), authors often say they are using Sentence- Piece tokenization but really mean they are using the unigram algorithm). In unigram tokenization, instead of building up a vocabulary by merging tokens, we start with a huge vocabulary of every individual unicode character plus all fre- quent sequences of characters (including all space-separated words, for languages with spaces), and iteratively remove some tokens to get to a desired ﬁnal vocabulary size. The algorithm is complex (involving sufﬁx-trees for efﬁciently storing many tokens, and the EM algorithm for iteratively assigning probabilities to tokens), so we don’t give it here, but see Kudo (2018) and Kudo and Richardson (2018b). Roughly speaking the algorithm proceeds iteratively by estimating the probability of each token, tokenizing the input data using various tokenizations, then removing a per- centage of tokens that don’t occur in high-probability tokenization, and then iterates until the vocabulary has been reduced down to the desired number of tokens. Why does unigram tokenization work better than BPE? BPE tends to create lots of very small non-meaningful tokens (because BPE can only create larger words or morphemes by merging characters one at a time), and it also tends to merge very common tokens, like the sufﬁx ed, onto their neighbors. We can see from these examples from Bostrom and Durrett (2020) that unigram tends to produce tokens that are more semantically meaningful: Original: corrupted Original: Completely preposterous suggestions BPE: cor rupted BPE: Comple t ely prep ost erous suggest ions Unigram: corrupt ed Unigram: Complete ly pre post er ous suggestion s 13.2.2 Creating the Training data Machine translation models are trained on a parallel corpus, sometimes called aparallel corpus bitext, a text that appears in two (or more) languages. Large numbers of paral- lel corpora are available. Some are governmental; the Europarl corpus (Koehn,Europarl 2005), extracted from the proceedings of the European Parliament, contains between 400,000 and 2 million sentences each from 21 European languages. The United Na- tions Parallel Corpus contains on the order of 10 million sentences in the six ofﬁcial languages of the United Nations (Arabic, Chinese, English, French, Russian, Span- ish) Ziemski et al. (2016). Other parallel corpora have been made from movie and TV subtitles, like the OpenSubtitles corpus (Lison and Tiedemann, 2016), or from general web text, like the ParaCrawl corpus of 223 million sentence pairs between 23 EU languages and English extracted from the CommonCrawl Ba˜n´on et al. (2020). 13.2 • M ACHINE TRANSLATION USING ENCODER -DECODER 271 Sentence alignment Standard training corpora for MT come as aligned pairs of sentences. When creat- ing new corpora, for example for underresourced languages or new domains, these sentence alignments must be created. Fig. 13.4 gives a sample hypothetical sentence alignment. F1: -Bonjour, dit le petit prince. F2: -Bonjour, dit le marchand de pilules perfectionnées qui apaisent la soif. F3: On en avale une par semaine et l'on n'éprouve plus le besoin de boire. F4: -C’est une grosse économie de temps, dit le marchand. F5: Les experts ont fait des calculs. F6: On épargne cinquante-trois minutes par semaine. F7: “Moi, se dit le petit prince, si j'avais cinquante-trois minutes à dépenser, je marcherais tout doucement vers une fontaine..." E1: “Good morning," said the little prince. E2: “Good morning," said the merchant. E3: This was a merchant who sold pills that had been perfected to quench thirst. E4: You just swallow one pill a week and you won’t feel the need for anything to drink. E5: “They save a huge amount of time," said the merchant. E6: “Fifty−three minutes a week." E7: “If I had fifty−three minutes to spend?" said the little prince to himself. E8: “I would take a stroll to a spring of fresh water” Figure 13.4 A sample alignment between sentences in English and French, with sentences extracted from Antoine de Saint-Exupery’sLe Petit Prince and a hypothetical translation. Sentence alignment takes sentences e1,...,en, and f1,..., fm and ﬁnds minimal sets of sentences that are translations of each other, including single sentence mappings like (e1,f1), (e4,f3), (e5,f4), (e6,f6) as well as 2-1 alignments (e 2/e3,f2), (e7/e8,f7), and null alignments (f5). Given two documents that are translations of each other, we generally need two steps to produce sentence alignments: • a cost function that takes a span of source sentences and a span of target sen- tences and returns a score measuring how likely these spans are to be transla- tions. • an alignment algorithm that takes these scores to ﬁnd a good alignment be- tween the documents. To score the similarity of sentences across languages, we need to make use of a multilingual embedding space, in which sentences from different languages are in the same embedding space (Artetxe and Schwenk, 2019). Given such a space, cosine similarity of such embeddings provides a natural scoring function (Schwenk, 2018). Thompson and Koehn (2019) give the following cost function between two sentences or spans x,y from the source and target documents respectively: c(x,y) = (1 −cos(x,y))nSents(x) nSents(y)∑S s=1 1 −cos(x,ys)+ ∑S s=1 1 −cos(xs,y) (13.10) where nSents() gives the number of sentences (this biases the metric toward many alignments of single sentences instead of aligning very large spans). The denom- inator helps to normalize the similarities, and so x1,...,xS,y1,...,yS, are randomly selected sentences sampled from the respective documents. Usually dynamic programming is used as the alignment algorithm (Gale and Church, 1993), in a simple extension of the minimum edit distance algorithm we introduced in Chapter 2. Finally, it’s helpful to do some corpus cleanup by removing noisy sentence pairs. This can involve handwritten rules to remove low-precision pairs (for example re- moving sentences that are too long, too short, have different URLs, or even pairs 272 CHAPTER 13 • M ACHINE TRANSLATION that are too similar, suggesting that they were copies rather than translations). Or pairs can be ranked by their multilingual embedding cosine score and low-scoring pairs discarded. 13.3 Details of the Encoder-Decoder Model Encoder The green llegó witch arrived <s> llegó la la bruja bruja verde verde </s> Decoder cross-attention transformer blocks Figure 13.5 The encoder-decoder transformer architecture for machine translation. The encoder uses the transformer blocks we saw in Chapter 8, while the decoder uses a more powerful block with an extra cross- attention layer that can attend to all the encoder words. We’ll see this in more detail in the next section. The standard architecture for MT is the encoder-decoder transformer. The encoder- decoder architecture was introduced already for RNNs in Chapter 8, and the trans- former version has the same idea. Fig. 13.5 shows the intuition of the architec- ture at a high level. You’ll see that the encoder-decoder architecture is made up of two transformers: an encoder, which is the same as the basic transformers from Chapter 9, and a decoder, which is augmented with a special new layer called the cross-attention layer. The encoder takes the source language input word tokens X = x1,...,xn and maps them to an output representation Henc = h1,...,hn; via a stack of encoder blocks. The decoder is essentially a conditional language model that attends to the en- coder representation and generates the target words one by one, at each timestep conditioning on the source sentence and the previously generated target language words to generate a token. Decoding can use any of the decoding methods discussed in Chapter 9 like greedy, or temperature or nucleus sampling. But the most com- mon decoding algorithm for MT is the beam search algorithm that we’ll introduce in Section 13.4. But the components of the architecture differ somewhat from the transformer block we’ve seen. First, in order to attend to the source language, the transformer blocks in the decoder have an extracross-attention layer. Recall that the transformer block of Chapter 9 consists of a self-attention layer that attends to the input from the previous layer, followed by layer norm, a feed forward layer, and another layer norm. The decoder transformer block includes an extra layer with a special kind of attention, cross-attention (also sometimes called encoder-decoder attention orcross-attention source attention). Cross-attention has the same form as the multi-head attention in a normal transformer block, except that while the queries as usual come from the previous layer of the decoder, the keys and values come from the output of the encoder. 13.3 • D ETAILS OF THE ENCODER -DECODER MODEL 273 Encoder x1 x2 x3 xn… Decoder h3h2h1 … hn Encoder Block 1 Block 2 Block K y3y2y1 … Decoder Block 1 Block 2 Block L Unembedding Matrix ym Multi-Head Attention Layer Normalize Layer Normalize + + …Feedforward Causal Multi-Head Attention Layer Normalize Layer Normalize + + …Feedforward Layer Normalize + Cross-Attention … … … … … … Language Modeling Head Henc Figure 13.6 The transformer block for the encoder and the decoder. The ﬁnal output of the encoder Henc = h1,...,hn is the context used in the decoder. The decoder is a standard transformer except with one extra layer, the cross-attention layer, which takes that encoder output Henc and uses it to form its K and V inputs. That is, where in standard multi-head attention the input to each attention layer is X, in cross attention the input is the the ﬁnal output of the encoderHenc = h1,...,hn. Henc is of shape [n ×d], each row representing one input token. To link the keys and values from the encoder with the query from the prior layer of the decoder, we multiply the encoder output Henc by the cross-attention layer’s key weightsWK and value weights WV. The query comes from the output from the prior decoder layer Hdec[ℓ−1], which is multiplied by the cross-attention layer’s query weightsWQ: Q = Hdec[ℓ−1]WQ; K = HencWK; V = HencWV (13.11) CrossAttention(Q,K,V) = softmax (QK⊺ √dk ) V (13.12) The cross attention thus allows the decoder to attend to each of the source language words as projected into the entire encoder ﬁnal output representations. The other attention layer in each decoder block, the multi-head attention layer, is the same causal (left-to-right) attention that we saw in Chapter 9. The multi-head attention in the encoder, however, is allowed to look ahead at the entire source language text, so it is not masked. To train an encoder-decoder model, we use the same self-supervision model we used for training encoder-decoders RNNs in Chapter 8. The network is given the source text and then starting with the separator token is trained autoregressively to predict the next token using cross-entropy loss. Recall that cross-entropy loss for 274 CHAPTER 13 • M ACHINE TRANSLATION language modeling is determined by the probability the model assigns to the correct next word. So at time t the CE loss is the negative log probability the model assigns to the next word in the training sequence: LCE ( ˆyt ,yt ) = −log ˆyt [wt+1] (13.13) As in that case, we use teacher forcing in the decoder. Recall that in teacher forc-teacher forcing ing, at each time step in decoding we force the system to use the gold target token from training as the next input xt+1, rather than allowing it to rely on the (possibly erroneous) decoder output ˆyt . 13.4 Decoding in MT: Beam Search Recall the greedy decoding algorithm from Chapter 9: at each time step t in gen- eration, the output yt is chosen by computing the probability for each word in the vocabulary and then choosing the highest probability word (the argmax): ˆwt = argmaxw∈V P(w\|w<t ) (13.14) A problem with greedy decoding is that what looks high probability at wordt might turn out to have been the wrong choice once we get to wordt +1. The beam search algorithm maintains multiple choices until later when we can see which one is best. In beam search we model decoding as searching the space of possible genera- tions, represented as a search tree whose branches represent actions (generating asearch tree token), and nodes represent states (having generated a particular preﬁx). We search for the best action sequence, i.e., the string with the highest probability. An illustration of the problem Fig. 13.7 shows a made-up example. The most probable sequence is ok ok EOS (its probability is .4×.7×1.0). But greedy search doesn’t ﬁnd it, incorrectly choosing yes as the ﬁrst word since it has the highest local probability (0.5). start ok yes EOS ok yes EOS ok yes EOS EOS EOS EOS EOS t2 t3 p(t1\|start) t1 p(t2\| t1) p(t3\| t1,t2) .1 .5 .4 .3 .4 .3 .1 .2 .7 1.0 1.0 1.0 1.0 Figure 13.7 A search tree for generating the target string T = t1,t2,... from vocabulary V = {yes,ok,<s>}, showing the probability of generating each token from that state. Greedy search chooses yes followed by yes, instead of the globally most probable sequence ok ok. For some problems, like part-of-speech tagging or parsing as we will see in Chapter 17 or Chapter 18, we can use dynamic programming search (the Viterbi 13.4 • D ECODING IN MT: B EAM SEARCH 275 algorithm) to address this problem. Unfortunately, dynamic programming is not ap- plicable to generation problems with long-distance dependencies between the output decisions. The only method guaranteed to ﬁnd the best solution is exhaustive search: computing the probability of every one of theV T possible sentences (for some length value T ) which is obviously too slow. The solution: beam search Instead, MT systems generally decode usingbeam search, a heuristic search methodbeam search ﬁrst proposed by Lowerre (1976). In beam search, instead of choosing the best token to generate at each timestep, we keep k possible tokens at each step. This ﬁxed-size memory footprint k is called the beam width, on the metaphor of a ﬂashlight beambeam width that can be parameterized to be wider or narrower. Thus at the ﬁrst step of decoding, we compute a softmax over the entire vocab- ulary, assigning a probability to each word. We then select the k-best options from this softmax output. These initial k outputs are the search frontier and these k initial words are called hypotheses. A hypothesis is an output sequence, a translation-so- far, together with its probability. a … aardvark .. arrived .. the … zebra start t1 a … aardvark .. the .. witch … zebra a … aardvark .. green .. witch … zebra t2 hd 1 y1 BOS y1 y2 y2 hd 1 hd 2 the theBOS hd 2 green green y3 hd 1 hd 2 arrived arrivedBOS y2 t3 hd 1 hd 2 the theBOS y2 hd 1 hd 2 the theBOS hd 2 witch witch y3 a … mage .. the .. witch … zebra arrived … aardvark .. green .. who … zebra y3 y3 Figure 13.8 Beam search decoding with a beam width of k = 2. At each time step, we choose the k best hypotheses, form the V possible extensions of each, score those k ×V hypotheses and choose the best k = 2 to continue. At time 1, the frontier has the best 2 options from the initial decoder state: arrived and the. We extend each, compute the probability of all the hypotheses so far ( arrived the, arrived aardvark, the green, the witch) and again chose the best 2 ( the green and the witch) to be the search frontier. The images on the arcs schematically represent the decoders that must be run at each step to score the next words (for simplicity not depicting cross-attention). At subsequent steps, each of the k best hypotheses is extended incrementally 276 CHAPTER 13 • M ACHINE TRANSLATION by being passed to distinct decoders, which each generate a softmax over the entire vocabulary to extend the hypothesis to every possible next token. Each of thesek×V hypotheses is scored by P(yi\|x,y<i): the product of the probability of the current word choice multiplied by the probability of the path that led to it. We then prune the k ×V hypotheses down to the k best hypotheses, so there are never more than k hypotheses at the frontier of the search, and never more than k decoders. Fig. 13.8 illustrates this with a beam width of 2 for the beginning of The green witch arrived. This process continues until an EOS is generated indicating that a complete can- didate output has been found. At this point, the completed hypothesis is removed from the frontier and the size of the beam is reduced by one. The search continues until the beam has been reduced to 0. The result will be k hypotheses. To score each node by its log probability, we use the chain rule of probability to break down p(y\|x) into the product of the probability of each word given its prior context, which we can turn into a sum of logs (for an output string of length t): score(y) = logP(y\|x) = log(P(y1\|x)P(y2\|y1,x)P(y3\|y1,y2,x)...P(yt \|y1,...,yt−1,x)) = t∑ i=1 logP(yi\|y1,...,yi−1,x) (13.15) Thus at each step, to compute the probability of a partial sentence, we simply add the log probability of the preﬁx sentence so far to the log probability of generating the next token. Fig. 13.9 shows the scoring for the example sentence shown in Fig. 13.8, using some simple made-up probabilities. Log probabilities are negative or 0, and the max of two log probabilities is the one that is greater (closer to 0). BOS arrived the the witch green witch mage who y2 y3 log P(y1\|x) y1 log P(y2\|y1,x) log P(y3\|y2,y1,x) -.92 -1.6 -1.2 -.69 -2.3 -.69 -1.6 -2.3 arrived-.11 -.51 witch -.36 -.22 EOS -.51 EOS -2.3 at -1.61 by log P(y4\|y3,y2,y1,x) log P(y5\|y4,y3,y2,y1,x) arrived came-1.6 y4 y5 log P(arrived\|x) log P(arrived witch\|x) log P(the\|x) log P(the green\|x) log P(the witch\|x) =-1.6 log P (arrived the\|x) log P (“the green witch arrived”\|x) = log P (the\|x) + log P(green\|the,x) + log P(witch \| the, green,x) +logP(arrived\|the,green,witch,x) +log P(EOS\|the,green,witch,arrived,x) = -2.3 = -3.9 = -1.6 = -2.1 =-.92 -2.1 -3.2 -4.4 -2.2 -2.5 -3.7 -2.7 -3.8 -2.7 -4.8 Figure 13.9 Scoring for beam search decoding with a beam width of k = 2. We maintain the log probability of each hypothesis in the beam by incrementally adding the logprob of generating each next token. Only the top k paths are extended to the next step. Fig. 13.10 gives the algorithm. One problem with this version of the algorithm is that the completed hypotheses may have different lengths. Because language mod- 13.4 • D ECODING IN MT: B EAM SEARCH 277 function BEAM DECODE (c, beam width) returns best paths y0, h0 ←0 path←() complete paths←() state←(c, y0, h0, path) ;initial state frontier←⟨state⟩ ;initial frontier while frontier contains incomplete paths and beamwidth >0 extended frontier←⟨⟩ for each state ∈frontier do y←DECODE (state) for each word i ∈Vocabularydo successor←NEWSTATE(state, i, yi) extended frontier←ADDTOBEAM (successor, extended frontier, beam width) for each state in extended frontier do if state is complete do complete paths←APPEND (complete paths, state) extended frontier←REMOVE (extended frontier, state) beam width←beam width - 1 frontier←extended frontier return completed paths function NEWSTATE(state, word, word prob) returns new state function ADDTOBEAM (state, frontier, width) returns updated frontier if LENGTH (frontier) <width then frontier←INSERT (state, frontier) else if SCORE (state) >SCORE (WORST OF(frontier)) frontier←REMOVE (WORST OF(frontier)) frontier←INSERT (state, frontier) return frontier Figure 13.10 Beam search decoding. els generally assign lower probabilities to longer strings, a naive algorithm would choose shorter strings for y. (This is not an issue during the earlier steps of decod- ing; since beam search is breadth-ﬁrst, all the hypotheses being compared had the same length.) For this reason we often apply length normalization methods, like dividing the logprob by the number of words: score(y) =1 t logP(y\|x) = 1 t t∑ i=1 logP(yi\|y1,...,yi−1,x) (13.16) For MT we generally use beam widthsk between 5 and 10, giving usk hypotheses at the end. We can pass allk to the downstream application with their respective scores, or if we just need a single translation we can pass the most probable hypothesis. 13.4.1 Minimum Bayes Risk Decoding Minimum Bayes risk or MBR decoding is an alternative decoding algorithm thatminimum Bayes risk MBR 278 CHAPTER 13 • M ACHINE TRANSLATION can work even better than beam search and also tends to be better than the other decoding algorithms like temperature sampling introduced in Section 10.2. The intuition of minimum Bayes risk is that instead of trying to choose the trans- lation which is most probable, we choose the one that is likely to have the least error. For example, we might want our decoding algorithm to ﬁnd the translation which has the highest score on some evaluation metric. For example in Section 13.6 we will introduce metrics like chrF or BERTScore that measure the goodness-of-ﬁt between a candidate translation and a set of reference human translations. A translation that maximizes this score, especially with a hypothetically huge set of perfect human translations is likely to be a good one (have minimum risk) even if it is not the most probable translation by our particular probability estimator. In practice, we don’t know the perfect set of translations for a given sentence. So the standard simpliﬁcation used in MBR decoding algorithms is to instead choose the candidate translation which is most similar (by some measure of goodness-of- ﬁt) with some set of candidate translations. We’re essentially approximating the enormous space of all possible translationsU with a smaller set of possible candidate translations Y. Given this set of possible candidate translations Y, and some similarity or align- ment function util, we choose the best translation ˆy as the translation which is most similar to all the other candidate translations: ˆy = argmax y∈Y ∑ c∈Y util(y,c) (13.17) Various util functions can be used, like chrF or BERTscore or BLEU. We can get the set of candidate translations by sampling using one of the basic sampling algorithms of Section 10.2 like temperature sampling; good results can be obtained with as few as 32 or 64 candidates. Minimum Bayes risk decoding can also be used for other NLP tasks; indeed it was widely applied to speech recognition (Stolcke et al., 1997; Goel and Byrne, 2000) before being applied to machine translation (Kumar and Byrne, 2004), and has been shown to work well across many other generation tasks as well (e.g., sum- marization, dialogue, and image captioning (Suzgun et al., 2023a)). 13.5 Translating in low-resource situations For some languages, and especially for English, online resources are widely avail- able. There are many large parallel corpora that contain translations between En- glish and many languages. But the vast majority of the world’s languages do not have large parallel training texts available. An important ongoing research question is how to get good translation with lesser resourced languages. The resource prob- lem can even be true for high resource languages when we need to translate into low resource domains (for example in a particular genre that happens to have very little bitext). Here we brieﬂy introduce two commonly used approaches for dealing with this data sparsity: backtranslation, which is a special case of the general statistical technique called data augmentation, and multilingual models, and also discuss some socio-technical issues. 13.5 • T RANSLATING IN LOW -RESOURCE SITUATIONS 279 13.5.1 Data Augmentation Data augmentation is a statistical technique for dealing with insufﬁcient training data, by adding new synthetic data that is generated from the current natural data. The most common data augmentation technique for machine translation is called backtranslation. Backtranslation relies on the intuition that while parallel corporabacktranslation may be limited for particular languages or domains, we can often ﬁnd a large (or at least larger) monolingual corpus, to add to the smaller parallel corpora that are available. The algorithm makes use of monolingual corpora in the target language by creating synthetic bitexts. In backtranslation, our goal is to improve source-to-target MT, given a small parallel text (a bitext) in the source/target languages, and some monolingual data in the target language. We ﬁrst use the bitext to train a MT system in the reverse di- rection: a target-to-source MT system . We then use it to translate the monolingual target data to the source language. Now we can add this synthetic bitext (natural target sentences, aligned with MT-produced source sentences) to our training data, and retrain our source-to-target MT model. For example suppose we want to trans- late from Navajo to English but only have a small Navajo-English bitext, although of course we can ﬁnd lots of monolingual English data. We use the small bitext to build an MT engine going the other way (from English to Navajo). Once we translate the monolingual English text to Navajo, we can add this synthetic Navajo/English bitext to our training data. Backtranslation has various parameters. One is how we generate the backtrans- lated data; we can run the decoder in greedy inference, or use beam search. Or we can do sampling, like the temperature sampling algorithm we saw in Chapter 9. Another parameter is the ratio of backtranslated data to natural bitext data; we can choose to upsample the bitext data (include multiple copies of each sentence). In general backtranslation works surprisingly well; one estimate suggests that a system trained on backtranslated text gets about 2/3 of the gain as would training on the same amount of natural bitext (Edunov et al., 2018). 13.5.2 Multilingual models The models we’ve described so far are for bilingual translation: one source language, one target language. It’s also possible to build amultilingual translator. In a multilingual translator, we train the system by giving it parallel sentences in many different pairs of languages. That means we need to tell the system which language to translate from and to! We tell the system which language is which by adding a special token ls to the encoder specifying the source language we’re translating from, and a special token lt to the decoder telling it the target language we’d like to translate into. Thus we slightly update Eq. 13.9 above to add these tokens in Eq. 13.19: h = encoder(x,ls) (13.18) yi+1 = decoder(h,lt ,y1,..., yi) ∀i ∈[1,..., m] (13.19) One advantage of a multilingual model is that they can improve the translation of lower-resourced languages by drawing on information from a similar language in the training data that happens to have more resources. Perhaps we don’t know the meaning of a word in Galician, but the word appears in the similar and higher- resourced language Spanish. 280 CHAPTER 13 • M ACHINE TRANSLATION 13.5.3 Sociotechnical issues Many issues in dealing with low-resource languages go beyond the purely techni- cal. One problem is that for low-resource languages, especially from low-income countries, native speakers are often not involved as the curators for content selec- tion, as the language technologists, or as the evaluators who measure performance (∀et al., 2020). Indeed, one well-known study that manually audited a large set of parallel corpora and other major multilingual datasets found that for many of the corpora, less than 50% of the sentences were of acceptable quality, with a lot of data consisting of repeated sentences with web boilerplate or incorrect translations, suggesting that native speakers may not have been sufﬁciently involved in the data process (Kreutzer et al., 2022). Other issues, like the tendency of many MT approaches to focus on the case where one of the languages is English (Anastasopoulos and Neubig, 2020), have to do with allocation of resources. Where most large multilingual systems were trained on bitexts in which English was one of the two languages, recent huge corporate systems like those of Fan et al. (2021) and Costa-juss `a et al. (2022) and datasets like Schwenk et al. (2021) attempt to handle large numbers of languages (up to 200 languages) and create bitexts between many more pairs of languages and not just through English. At the smaller end, ∀et al. (2020) propose a participatory design process to encourage content creators, curators, and language technologists who speak these low-resourced languages to participate in developing MT algorithms. They provide online groups, mentoring, and infrastructure, and report on a case study on devel- oping MT algorithms for low-resource African languages. Among their conclusions was to perform MT evaluation by post-editing rather than direct evaluation, since having labelers edit an MT system and then measure the distance between the MT output and its post-edited version both was simpler to train evaluators and makes it easier to measure true errors in the MT output and not differences due to linguistic variation (Bentivogli et al., 2018). 13.6 MT Evaluation Translations are evaluated along two dimensions: 1. adequacy: how well the translation captures the exact meaning of the sourceadequacy sentence. Sometimes called faithfulness or ﬁdelity. 2. ﬂuency: how ﬂuent the translation is in the target language (is it grammatical,ﬂuency clear, readable, natural). Using humans to evaluate is most accurate, but automatic metrics are also used for convenience. 13.6.1 Using Human Raters to Evaluate MT The most accurate evaluations use human raters, such as online crowdworkers, to evaluate each translation along the two dimensions. For example, along the dimen- sion of ﬂuency, we can ask how intelligible, how clear, how readable, or how natural the MT output (the target text) is. We can give the raters a scale, for example, from 1 (totally unintelligible) to 5 (totally intelligible), or 1 to 100, and ask them to rate each sentence or paragraph of the MT output. 13.6 • MT E VALUATION 281 We can do the same thing to judge the second dimension,adequacy, using raters to assign scores on a scale. If we have bilingual raters, we can give them the source sentence and a proposed target sentence, and rate, on a 5-point or 100-point scale, how much of the information in the source was preserved in the target. If we only have monolingual raters but we have a good human translation of the source text, we can give the monolingual raters the human reference translation and a target machine translation and again rate how much information is preserved. An alternative is to do ranking: give the raters a pair of candidate translations, and ask them which oneranking they prefer. Training of human raters (who are often online crowdworkers) is essential; raters without translation expertise ﬁnd it difﬁcult to separate ﬂuency and adequacy, and so training includes examples carefully distinguishing these. Raters often disagree (source sentences may be ambiguous, raters will have different world knowledge, raters may apply scales differently). It is therefore common to remove outlier raters, and (if we use a ﬁne-grained enough scale) normalizing raters by subtracting the mean from their scores and dividing by the variance. As discussed above, an alternative way of using human raters is to have them post-edit translations, taking the MT output and changing it minimally until they feel it represents a correct translation. The difference between their post-edited translations and the original MT output can then be used as a measure of quality. 13.6.2 Automatic Evaluation While humans produce the best evaluations of machine translation output, running a human evaluation can be time consuming and expensive. For this reason automatic metrics are often used as temporary proxies. Automatic metrics are less accurate than human evaluation, but can help test potential system improvements, and even be used as an automatic loss function for training. In this section we introduce two families of such metrics, those based on character- or word-overlap and those based on embedding similarity. Automatic Evaluation by Character Overlap: chrF The simplest and most robust metric for MT evaluation is calledchrF, which standschrF for character F-score (Popovi´c, 2015). chrF (along with many other earlier related metrics like BLEU, METEOR, TER, and others) is based on a simple intuition de- rived from the pioneering work of Miller and Beebe-Center (1956): a good machine translation will tend to contain characters and words that occur in a human trans- lation of the same sentence. Consider a test set from a parallel corpus, in which each source sentence has both a gold human target translation and a candidate MT translation we’d like to evaluate. The chrF metric ranks each MT target sentence by a function of the number of character n-gram overlaps with the human translation. Given the hypothesis and the reference, chrF is given a parameter k indicating the length of character n-grams to be considered, and computes the average of the k precisions (unigram precision, bigram, and so on) and the average of the k recalls (unigram recall, bigram recall, etc.): chrP percentage of character 1-grams, 2-grams, ..., k-grams in the hypothesis that occur in the reference, averaged. chrR percentage of character 1-grams, 2-grams,..., k-grams in the reference that occur in the hypothesis, averaged. The metric then computes an F-score by combining chrP and chrR using a weighting 282 CHAPTER 13 • M ACHINE TRANSLATION parameter β. It is common to set β = 2, thus weighing recall twice as much as precision: chrFβ = (1 +β2) chrP ·chrR β2 ·chrP +chrR (13.20) For β = 2, that would be: chrF2 = 5 ·chrP ·chrR 4 ·chrP +chrR For example, consider two hypotheses that we’d like to score against the refer- ence translation witness for the past. Here are the hypotheses along with chrF values computed using parametersk = β = 2 (in real examples,k would be a higher number like 6): REF: witness for the past, HYP1: witness of the past, chrF2,2 = .86 HYP2: past witness chrF2,2 = .62 Let’s see how we computed that chrF value for HYP1 (we’ll leave the compu- tation of the chrF value for HYP2 as an exercise for the reader). First, chrF ignores spaces, so we’ll remove them from both the reference and hypothesis: REF: witnessforthepast,(18 unigrams, 17 bigrams) HYP1: witnessofthepast,(17 unigrams, 16 bigrams) Next let’s see how many unigrams and bigrams match between the reference and hypothesis: unigrams that match: w i t n e s s f o t h e p a s t ,(17 unigrams) bigrams that match: wi it tn ne es ss th he ep pa as st t,(13 bigrams) We use that to compute the unigram and bigram precisions and recalls: unigram P: 17/17 = 1 unigram R: 17/18 = .944 bigram P: 13/16 = .813 bigram R: 13/17 = .765 Finally we average to get chrP and chrR, and compute the F-score: chrP = (17/17 +13/16)/2 = .906 chrR = (17/18 +13/17)/2 = .855 chrF2,2 = 5 chrP ∗chrR 4chrP +chrR = .86 chrF is simple, robust, and correlates very well with human judgments in many languages (Kocmi et al., 2021). Alternative overlap metric: BLEU There are various alternative overlap metrics. For example, before the development of chrF, it was common to use a word-based overlap metric calledBLEU (for BiLin- gual Evaluation Understudy), that is purely precision-based rather than combining precision and recall (Papineni et al., 2002). The BLEU score for a corpus of candi- date translation sentences is a function of the n-gram word precision over all the sentences combined with a brevity penalty computed over the corpus as a whole. What do we mean by n-gram precision? Consider a corpus composed of a single sentence. The unigram precision for this corpus is the percentage of unigram tokens 13.6 • MT E VALUATION 283 in the candidate translation that also occur in the reference translation, and ditto for bigrams and so on, up to 4-grams. BLEU extends this unigram metric to the whole corpus by computing the numerator as the sum over all sentences of the counts of all the unigram types that also occur in the reference translation, and the denominator is the total of the counts of all unigrams in all candidate sentences. We compute this n-gram precision for unigrams, bigrams, trigrams, and 4-grams and take the geometric mean. BLEU has many further complications, including a brevity penalty for penalizing candidate translations that are too short, and it also requires the n- gram counts be clipped in a particular way. Because BLEU is a word-based metric, it is very sensitive to word tokenization, making it impossible to compare different systems if they rely on different tokeniza- tion standards, and doesn’t work as well in languages with complex morphology. Nonetheless, you will sometimes still see systems evaluated by BLEU, particularly for translation into English. In such cases it’s important to use packages that enforce standardization for tokenization like SACRE BLEU (Post, 2018). Statistical Signiﬁcance Testing for MT evals Character or word overlap-based metrics like chrF (or BLEU, or etc.) are mainly used to compare two systems, with the goal of answering questions like: did the new algorithm we just invented improve our MT system? To know if the difference between the chrF scores of two MT systems is a signiﬁcant difference, we use the paired bootstrap test, or the similar randomization test. To get a conﬁdence interval on a single chrF score using the bootstrap test, recall from Section 4.9 that we take our test set (or devset) and create thousands of pseudo- testsets by repeatedly sampling with replacement from the original test set. We now compute the chrF score of each of the pseudo-testsets. If we drop the top 2.5% and bottom 2.5% of the scores, the remaining scores will give us the 95% conﬁdence interval for the chrF score of our system. To compare two MT systems A and B, we draw the same set of pseudo-testsets, and compute the chrF scores for each of them. We then compute the percentage of pseudo-test-sets in which A has a higher chrF score than B. chrF: Limitations While automatic character and word-overlap metrics like chrF or BLEU are useful, they have important limitations. chrF is very local: a large phrase that is moved around might barely change the chrF score at all, and chrF can’t evaluate cross- sentence properties of a document like its discourse coherence (Chapter 24). chrF and similar automatic metrics also do poorly at comparing very different kinds of systems, such as comparing human-aided translation against machine translation, or different machine translation architectures against each other (Callison-Burch et al., 2006). Instead, automatic overlap metrics like chrF are most appropriate when eval- uating changes to a single system. 13.6.3 Automatic Evaluation: Embedding-Based Methods The chrF metric is based on measuring the exact character n-grams a human refer- ence and candidate machine translation have in common. However, this criterion is overly strict, since a good translation may use alternate words or paraphrases. A solution ﬁrst pioneered in early metrics like METEOR (Banerjee and Lavie, 2005) was to allow synonyms to match between the reference x and candidate ˜x. More 284 CHAPTER 13 • M ACHINE TRANSLATION recent metrics use BERT or other embeddings to implement this intuition. For example, in some situations we might have datasets that have human as- sessments of translation quality. Such datasets consists of tuples (x,˜x,r), where x = (x1,..., xn) is a reference translation, ˜x = (˜x1,..., ˜xm) is a candidate machine translation, and r ∈R is a human rating that expresses the quality of ˜x with respect to x. Given such data, algorithms like COMET (Rei et al., 2020) BLEURT (Sellam et al., 2020) train a predictor on the human-labeled datasets, for example by passing x and ˜x through a version of BERT (trained with extra pretraining, and then ﬁnetuned on the human-labeled sentences), followed by a linear layer that is trained to predict r. The output of such models correlates highly with human labels. In other cases, however, we don’t have such human-labeled datasets. In that case we can measure the similarity of x and ˜x by the similarity of their embeddings. The BERTS CORE algorithm (Zhang et al., 2020) shown in Fig. 13.11, for example, passes the reference x and the candidate ˜x through BERT, computing a BERT em- bedding for each token xi and ˜xj. Each pair of tokens (xi,˜xj) is scored by its cosine xi·˜xj \|xi\|\|˜xj\|. Each token in x is matched to a token in ˜x to compute recall, and each token in ˜x is matched to a token in x to compute precision (with each token greedily matched to the most similar token in the corresponding sentence). BERTS CORE provides precision and recall (and hence F1): RBERT = 1 \|x\| ∑ xi∈x max ˜xj∈˜x xi ·˜xj PBERT = 1 \|˜x\| ∑ ˜xj∈˜x max xi∈x xi ·˜xj (13.21) Published as a conference paper at ICLR 2020 Referencethe weather is cold today Candidateit is freezing today Candidate ContextualEmbeddingPairwise CosineSimilarity RBERT=(0.7131.27)+(0.5157.94)+...1.27+7.94+1.82+7.90+8.88 <latexit 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sha1_base64="5QTnVRVSrnyzznVU7d5bF5u03Iw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK9gPaUDbbTbt0swm7E7GE/ggvHhTx6u/x5r9x0+agrQ8GHu/NMDMvSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJpwP6IjJULBKFqp0x9TzJ5mg2rNrbtzkFXiFaQGBZqD6ld/GLM04gqZpMb0PDdBP6MaBZN8VumnhieUTeiI9yxVNOLGz+bnzsiZVYYkjLUthWSu/p7IaGTMNApsZ0RxbJa9XPzP66UYXvuZUEmKXLHFojCVBGOS/06GQnOGcmoJZVrYWwkbU00Z2oQqNgRv+eVV0r6oe27du7+sNW6KOMpwAqdwDh5cQQPuoAktYDCBZ3iFNydxXpx352PRWnKKmWP4A+fzB7A8j8k=</latexit> Reference Figure 1: Illustration of the computation of the recall metricRBERT . Given the referencex and candidate ˆx, we compute BERT embeddings and pairwise cosine similarity. We highlight the greedy matching in red, and include the optionalidf importance weighting. We experiment with different models (Section 4), using the tokenizer provided with each model. Given a tokenized reference sentencex = hx1,...,x ki, the embedding model generates a se- quence of vectorshx1,..., xki. Similarly, the tokenized candidateˆx = hˆx1,..., ˆxmi is mapped to hˆx1,..., ˆxli. The main model we use is BERT, which tokenizes the input text into a sequence of word pieces (Wu et al., 2016), where unknown words are split into several commonly observed sequences of characters. The representation for each word piece is computed with a Transformer encoder (Vaswani et al., 2017) by repeatedly applying self-attention and nonlinear transformations in an alternating fashion. BERT embeddings have been shown to beneﬁt various NLP tasks (Devlin et al., 2019; Liu, 2019; Huang et al., 2019; Yang et al., 2019a). Similarity Measure The vector representation allows for a soft measure of similarity instead of exact-string (Papineni et al., 2002) or heuristic (Banerjee & Lavie, 2005) matching. The cosine similarity of a reference tokenxi and a candidate tokenˆxj is x> i ˆxj kxikkˆxj k. We use pre-normalized vectors, which reduces this calculation to the inner productx> i ˆxj. While this measure considers tokens in isolation, the contextual embeddings contain information from the rest of the sentence. BERTS CORE The complete score matches each token inx to a token inˆx to compute recall, and each token inˆx to a token inx to compute precision. We use greedy matching to maximize the matching similarity score,2 where each token is matched to the most similar token in the other sentence. We combine precision and recall to compute an F1 measure. For a referencex and candidate ˆx, the recall, precision, and F1 scores are: RBERT = 1 \|x\| X xi2x max ˆxj 2ˆx x> i ˆxj ,P BERT = 1 \|ˆx\| X ˆxj 2ˆx max xi2x x> i ˆxj ,F BERT =2 PBERT · RBERT PBERT + RBERT . Importance Weighting Previous work on similarity measures demonstrated that rare words can be more indicative for sentence similarity than common words (Banerjee & Lavie, 2005; Vedantam et al., 2015). BERTSCORE enables us to easily incorporate importance weighting. We experiment with inverse document frequency (idf) scores computed from the test corpus. GivenM reference sentences {x(i)}M i=1, theidf score of a word-piece tokenw is idf(w)= log 1 M MX i=1 I[w 2 x(i)] , where I[· ] is an indicator function. We do not use the fulltf-idf measure because we process single sentences, where the term frequency (tf) is likely 1. For example, recall withidf weighting is RBERT = P xi2x idf(xi) maxˆxj 2ˆx x> i ˆxj P xi2x idf(xi) . Because we use reference sentences to computeidf, theidf scores remain the same for all systems evaluated on a speciﬁc test set. We apply plus-one smoothing to handle unknown word pieces. 2We compare greedy matching with optimal assignment in Appendix C. 4 Figure 13.11 The computation of BERTS CORE recall from reference x and candidate ˆx, from Figure 1 in Zhang et al. (2020). This version shows an extended version of the metric in which tokens are also weighted by their idf values. 13.7 Bias and Ethical Issues Machine translation raises many of the same ethical issues that we’ve discussed in earlier chapters. For example, consider MT systems translating from Hungarian (which has the gender neutral pronoun ˝o) or Spanish (which often drops pronouns) into English (in which pronouns are obligatory, and they have grammatical gender). When translating a reference to a person described without speciﬁed gender, MT systems often default to male gender (Schiebinger 2014, Prates et al. 2019). And MT systems often assign gender according to culture stereotypes of the sort we saw in Section 6.11. Fig. 13.12 shows examples from Prates et al. (2019), in which Hun- garian gender-neutral ˝o is a nurse is translated with she, but gender-neutral ˝o is a CEO is translated with he. Prates et al. (2019) ﬁnd that these stereotypes can’t com- pletely be accounted for by gender bias in US labor statistics, because the biases are 13.8 • S UMMARY 285 ampliﬁed by MT systems, with pronouns being mapped to male or female gender with a probability higher than if the mapping was based on actual labor employment statistics. Hungarian (gender neutral) source English MT output ˝o egy ´apol´o she is a nurse ˝o egy tud´os he is a scientist ˝o egy m´ern¨ok he is an engineer ˝o egy p´ek he is a baker ˝o egy tan´ar she is a teacher ˝o egy esk¨uv˝oszervez˝o she is a wedding organizer ˝o egy vez´erigazgat´o he is a CEO Figure 13.12 When translating from gender-neutral languages like Hungarian into English, current MT systems interpret people from traditionally male-dominated occupations as male, and traditionally female-dominated occupations as female (Prates et al., 2019). Similarly, a recent challenge set, the WinoMT dataset (Stanovsky et al., 2019) shows that MT systems perform worse when they are asked to translate sentences that describe people with non-stereotypical gender roles, like “The doctor asked the nurse to help her in the operation”. Many ethical questions in MT require further research. One open problem is developing metrics for knowing what our systems don’t know. This is because MT systems can be used in urgent situations where human translators may be unavailable or delayed: in medical domains, to help translate when patients and doctors don’t speak the same language, or in legal domains, to help judges or lawyers communi- cate with witnesses or defendants. In order to ‘do no harm’, systems need ways to assign conﬁdence values to candidate translations, so they can abstain from givingconﬁdence incorrect translations that may cause harm. 13.8 Summary Machine translation is one of the most widely used applications of NLP, and the encoder-decoder model, ﬁrst developed for MT is a key tool that has applications throughout NLP. • Languages have divergences, both structural and lexical, that make translation difﬁcult. • The linguistic ﬁeld of typology investigates some of these differences; lan- guages can be classiﬁed by their position along typological dimensions like whether verbs precede their objects. • Encoder-decoder networks (for transformers just as we saw in Chapter 8 for RNNs) are composed of an encoder network that takes an input sequence and creates a contextualized representation of it, the context. This context representation is then passed to a decoder which generates a task-speciﬁc output sequence. • Cross-attention allows the transformer decoder to view information from all the hidden states of the encoder. • Machine translation models are trained on aparallel corpus, sometimes called a bitext, a text that appears in two (or more) languages. 286 CHAPTER 13 • M ACHINE TRANSLATION • Backtranslation is a way of making use of monolingual corpora in the target language by running a pilot MT engine backwards to create synthetic bitexts. • MT is evaluated by measuring a translation’sadequacy (how well it captures the meaning of the source sentence) and ﬂuency (how ﬂuent or natural it is in the target language). Human evaluation is the gold standard, but automatic evaluation metrics like chrF, which measure character n-gram overlap with human translations, or more recent metrics based on embedding similarity, are also commonly used. Bibliographical and Historical Notes MT was proposed seriously by the late 1940s, soon after the birth of the computer (Weaver, 1949/1955). In 1954, the ﬁrst public demonstration of an MT system pro- totype (Dostert, 1955) led to great excitement in the press (Hutchins, 1997). The next decade saw a great ﬂowering of ideas, preﬁguring most subsequent develop- ments. But this work was ahead of its time—implementations were limited by, for example, the fact that pending the development of disks there was no good way to store dictionary information. As high-quality MT proved elusive (Bar-Hillel, 1960), there grew a consensus on the need for better evaluation and more basic research in the new ﬁelds of for- mal and computational linguistics. This consensus culminated in the famously crit- ical ALPAC (Automatic Language Processing Advisory Committee) report of 1966 (Pierce et al., 1966) that led in the mid 1960s to a dramatic cut in funding for MT in the US. As MT research lost academic respectability, the Association for Ma- chine Translation and Computational Linguistics dropped MT from its name. Some MT developers, however, persevered, and there were early MT systems like M´et´eo, which translated weather forecasts from English to French (Chandioux, 1976), and industrial systems like Systran. In the early years, the space of MT architectures spanned three general mod- els. In direct translation, the system proceeds word-by-word through the source- language text, translating each word incrementally. Direct translation uses a large bilingual dictionary, each of whose entries is a small program with the job of trans- lating one word. In transfer approaches, we ﬁrst parse the input text and then ap- ply rules to transform the source-language parse into a target language parse. We then generate the target language sentence from the parse tree. In interlingua ap- proaches, we analyze the source language text into some abstract meaning repre- sentation, called an interlingua. We then generate into the target language from this interlingual representation. A common way to visualize these three early ap- proaches was the Vauquois triangle shown in Fig. 13.13. The triangle shows theVauquois triangle increasing depth of analysis required (on both the analysis and generation end) as we move from the direct approach through transfer approaches to interlingual ap- proaches. In addition, it shows the decreasing amount of transfer knowledge needed as we move up the triangle, from huge amounts of transfer at the direct level (al- most all knowledge is transfer knowledge for each word) through transfer (transfer rules only for parse trees or thematic roles) through interlingua (no speciﬁc transfer knowledge). We can view the encoder-decoder network as an interlingual approach, with attention acting as an integration of direct and transfer, allowing words or their representations to be directly accessed by the decoder. BIBLIOGRAPHICAL AND HISTORICAL NOTES 287 source text target text Direct Translation Transfer Interlingua Source Text: Semantic/Syntactic Structure Target Text: Semantic/Syntactic Structuresource language analysis source language analysis target language generation Figure 13.13 The Vauquois (1968) triangle. Statistical methods began to be applied around 1990, enabled ﬁrst by the devel- opment of large bilingual corpora like theHansard corpus of the proceedings of the Canadian Parliament, which are kept in both French and English, and then by the growth of the Web. Early on, a number of researchers showed that it was possible to extract pairs of aligned sentences from bilingual corpora, using words or simple cues like sentence length (Kay and R ¨oscheisen 1988, Gale and Church 1991, Gale and Church 1993, Kay and R¨oscheisen 1993). At the same time, the IBM group, drawing directly on the noisy channel model for speech recognition, proposed two related paradigms for statistical MT. Thesestatistical MT include the generative algorithms that became known as IBM Models 1 throughIBM Models 5, implemented in the Candide system. The algorithms (except for the decoder)Candide were published in full detail— encouraged by the US government who had par- tially funded the work— which gave them a huge impact on the research community (Brown et al. 1990, Brown et al. 1993). The group also developed a discriminative approach, called MaxEnt (for maxi- mum entropy, an alternative formulation of logistic regression), which allowed many features to be combined discriminatively rather than generatively (Berger et al., 1996), which was further developed by Och and Ney (2002). By the turn of the century, most academic research on machine translation used statistical MT, either in the generative or discriminative mode. An extended version of the generative approach, called phrase-based translation was developed, basedphrase-based translation on inducing translations for phrase-pairs (Och 1998, Marcu and Wong 2002, Koehn et al. (2003), Och and Ney 2004, Deng and Byrne 2005, inter alia). Once automatic metrics like BLEU were developed (Papineni et al., 2002), the discriminative log linear formulation (Och and Ney, 2004), drawing from the IBM MaxEnt work (Berger et al., 1996), was used to directly optimize evaluation metrics like BLEU in a method known asMinimum Error Rate Training, or MERT (Och,MERT 2003), also drawing from speech recognition models (Chou et al., 1993). Toolkits like GIZA (Och and Ney, 2003) andMoses (Koehn et al. 2006, Zens and Ney 2007)Moses were widely used. There were also approaches around the turn of the century that were based on syntactic structure (Chapter 18). Models based on transduction grammars (alsotransduction grammars called synchronous grammars) assign a parallel syntactic tree structure to a pair of sentences in different languages, with the goal of translating the sentences by applying reordering operations on the trees. From a generative perspective, we can view a transduction grammar as generating pairs of aligned sentences in two lan- guages. Some of the most widely used models included the inversion transduction grammar (Wu, 1996) and synchronous context-free grammars (Chiang, 2005), inversion transduction grammar 288 CHAPTER 13 • M ACHINE TRANSLATION Neural networks had been applied at various times to various aspects of machine translation; for example Schwenk et al. (2006) showed how to use neural language models to replace n-gram language models in a Spanish-English system based on IBM Model 4. The modern neural encoder-decoder approach was pioneered by Kalchbrenner and Blunsom (2013), who used a CNN encoder and an RNN decoder, and was ﬁrst applied to MT by Bahdanau et al. (2015). The transformer encoder- decoder was proposed by Vaswani et al. (2017) (see the History section of Chap- ter 9). Research on evaluation of machine translation began quite early. Miller and Beebe-Center (1956) proposed a number of methods drawing on work in psycholin- guistics. These included the use of cloze and Shannon tasks to measure intelligibility as well as a metric of edit distance from a human translation, the intuition that un- derlies all modern overlap-based automatic evaluation metrics. The ALPAC report included an early evaluation study conducted by John Carroll that was extremely in- ﬂuential (Pierce et al., 1966, Appendix 10). Carroll proposed distinct measures for ﬁdelity and intelligibility, and had raters score them subjectively on 9-point scales. Much early evaluation work focuses on automatic word-overlap metrics like BLEU (Papineni et al., 2002), NIST (Doddington, 2002), TER (Translation Error Rate) (Snover et al., 2006), Precision and Recall (Turian et al., 2003), and METEOR (Banerjee and Lavie, 2005); character n-gram overlap methods like chrF (Popovi ´c, 2015) came later. More recent evaluation work, echoing the ALPAC report, has emphasized the importance of careful statistical methodology and the use of human evaluation (Kocmi et al., 2021; Marie et al., 2021). The early history of MT is surveyed in Hutchins 1986 and 1997; Nirenburg et al. (2002) collects early readings. See Croft (1990) or Comrie (1989) for introductions to linguistic typology. Exercises 13.1 Compute by hand the chrF2,2 score for HYP2 on page 282 (the answer should round to .62). CHAPTER 14 Question Answering, Informa- tion Retrieval, and Retrieval- Augmented Generation People need to know things. So pretty much as soon as there were computers we were asking them questions. Systems in the 1960s were answering questions about baseball statistics and scientiﬁc facts. Even ﬁctional computers in the 1970s like Deep Thought, invented by Douglas Adams inThe Hitchhiker’s Guide to the Galaxy, answered “the Ultimate Question Of Life, The Universe, and Everything”. 1 And because so much knowledge is encoded in text, question answering ( QA) systemsQA were performing at human levels even before LLMs: IBM’s Watson system won the TV game-show Jeopardy! in 2011, surpassing humans at answering questions like: WILLIAM WILKINSON’S “AN ACCOUNT OF THE PRINCIPALITIES OF WALLACHIA AND MOLDOVIA” INSPIRED THIS AUTHOR’S MOST FAMOUS NOVEL 2 Question answering systems are designed to ﬁll human information needs . Since a lot of information is present in text form (on the web or in other data like our email, or books), question answering is closely related to the task behind search engines. Indeed, the distinction is becoming ever more fuzzy, as modern search engines are integrated with large language models trained to do question answering. Question answering systems often focus on a useful subset of information needs: factoid questions, questions of fact or reasoning that can be answered with simple facts expressed in short or medium-length texts, like the following: (14.1) Where is the Louvre Museum located? (14.2) Where does the energy in a nuclear explosion come from? (14.3) How to get a script l in latex? Modern NLP systems answer these questions using large language models, in one of two ways. The ﬁrst is to make use of the method from Chapter 12: prompt a pretrained and instruction-tuned LLM, an LLM that has been ﬁnetuned on ques- tion/answer datasets with the question in the prompt. For example, we could prompt a causal language model with a string like Q: Where is the Louvre Museum located? A: have it do conditional generation given this preﬁx, and take the response as the an- swer. The idea is that language models have read a lot of facts in their pretraining data, presumably including the location of the Louvre, and have encoded this infor- mation in their parameters. Simply prompting an LLM can be a useful approach to answer many factoid questions. But it is not yet a complete solution for question answering. 1 The answer was 42, but unfortunately the question was never revealed. 2 The answer, of course, is ‘Who is Bram Stoker’, and the novel wasDracula. 290 CHAPTER 14 • Q UESTION ANSWERING , INFORMATION RETRIEVAL , AND RAG The ﬁrst and main problem is that large language models often give the wrong answer! Large language models hallucinate. A hallucination is a response that ishallucinate not faithful to the facts of the world. That is, when asked questions, large language models simply make up answers that sound reasonable. For example, Dahl et al. (2024) found that when asked questions about the legal domain (like about particular legal cases), large language models hallucinated from 69% to 88% of the time! And it’s not always possible to tell when language models are hallucinating, partly because LLMs aren’t well-calibrated. In a calibrated system, the conﬁdencecalibrated of a system in the correctness of its answer is highly correlated with the probability of an answer being correct. So if a calibrated system is wrong, at least it might hedge its answer or tell us to go check another source. But since language models are not well-calibrated, they often give a very wrong answer with complete certainty (Zhou et al., 2024). A second problem is that simply prompting a large language model doesn’t allow us to ask questions about proprietary data. A common use of question answering is about data like our personal email or medical records. Or a company may have internal documents that contain answers for customer service or internal use. Or legal ﬁrms need to ask questions about legal discovery from proprietary documents. Finally, static large language models also have problems with questions about rapidly changing information (like questions about something that happened last week) since LLMs won’t have up-to-date information from after their release data. For this reason the most common way to do question-answering with LLMs is retrieval-augmented generation or RAG, and that is the method we will focus onRAG in this chapter. In RAG we use information retrieval (IR) techniques to retrieveinformation retrieval documents that are likely to have information that might help answer the question. Then we use a large language model to generate an answer given these documents. Basing our answers on retrieved documents can solve some of the problems with using simple prompting to answer questions. First, it helps ensure that the answer is grounded in facts from some curated dataset. And the system can give the user the answer accompanied by the context of the passage or document the answer came from. This information can help users have conﬁdence in the accuracy of the answer (or help them spot when it is wrong!). And these retrieval techniques can be used on any proprietary data we want, such as legal or medical data for those applications. We’ll begin by introducing information retrieval, the task of choosing the most relevant document from a document set given a user’s query expressing their infor- mation need. We’ll see the classic method based on cosines of sparse tf-idf vectors, a modern neural ‘dense’ retrievers based on instead representing queries and docu- ments neurally with BERT or other language models. We then introduce retriever- based question answering and the retrieval-augmented generation paradigm. Finally, we’ll discuss various QA datasets. These are used for ﬁnetuning LLMs in instruction tuning, as we saw in Chapter 12. And they are also used as bench- marks, since question answering has an important function as a benchmark for mea- suring the abilities of language models. 14.1 Information Retrieval Information retrievalor IR is the name of the ﬁeld encompassing the retrieval of allinformation retrieval IR manner of media based on user information needs. The resulting IR system is often called a search engine. Our goal in this section is to give a sufﬁcient overview of IR 14.1 • I NFORMATION RETRIEVAL 291 to see its application to question answering. Readers with more interest speciﬁcally in information retrieval should see the Historical Notes section at the end of the chapter and textbooks like Manning et al. (2008). The IR task we consider is called ad hoc retrieval , in which a user poses aad hoc retrieval query to a retrieval system, which then returns an ordered set of documents from some collection. A document refers to whatever unit of text the system indexes anddocument retrieves (web pages, scientiﬁc papers, news articles, or even shorter passages like paragraphs). A collection refers to a set of documents being used to satisfy usercollection requests. A term refers to a word in a collection, but it may also include phrases.term Finally, a query represents a user’s information need expressed as a set of terms.query The high-level architecture of an ad hoc retrieval engine is shown in Fig. 14.1. Document Document Document Document DocumentDocument Query Processing Indexing Search Document Document Document Document DocumentRanked Documents Document query Inverted Index query vector document collection Figure 14.1 The architecture of an ad hoc IR system. The basic IR architecture uses the vector space model we introduced in Chap- ter 6, in which we map queries and document to vectors based on unigram word counts, and use the cosine similarity between the vectors to rank potential documents (Salton, 1971). This is thus an example of the bag-of-words model introduced in Chapter 4, since words are considered independently of their positions. 14.1.1 Term weighting and document scoring Let’s look at the details of how the match between a document and query is scored. We don’t use raw word counts in IR, instead computing aterm weight for eachterm weight document word. Two term weighting schemes are common: the tf-idf weighting introduced in Chapter 6, and a slightly more powerful variant called BM25.BM25 We’ll reintroduce tf-idf here so readers don’t need to look back at Chapter 6. Tf-idf (the ‘-’ here is a hyphen, not a minus sign) is the product of two terms, the term frequency tf and the inverse document frequency idf. The term frequency tells us how frequent the word is; words that occur more often in a document are likely to be informative about the document’s contents. We usually use the log10 of the word frequency, rather than the raw count. The intuition is that a word appearing 100 times in a document doesn’t make that word 100 times more likely to be relevant to the meaning of the document. We also need to do something special with counts of 0, since we can’t take the log of 0.3 tft,d = { 1 +log10 count(t,d) if count (t,d) >0 0 otherwise (14.4) 3 We can also use this alternative formulation, which we have used in earlier editions: tf t,d = log10(count(t,d)+ 1) 292 CHAPTER 14 • Q UESTION ANSWERING , INFORMATION RETRIEVAL , AND RAG If we use log weighting, terms which occur 0 times in a document would have tf= 0, 1 times in a document tf = 1 + log10(1) =1 + 0 = 1, 10 times in a document tf = 1 +log10(10) =2, 100 times tf = 1 +log10(100) =3, 1000 times tf = 4, and so on. The document frequency dft of a term t is the number of documents it oc- curs in. Terms that occur in only a few documents are useful for discriminating those documents from the rest of the collection; terms that occur across the entire collection aren’t as helpful. The inverse document frequency or idf term weight (Sparck Jones, 1972) is deﬁned as: idft = log10 N dft (14.5) where N is the total number of documents in the collection, and df t is the number of documents in which term t occurs. The fewer documents in which a term occurs, the higher this weight; the lowest weight of 0 is assigned to terms that occur in every document. Here are some idf values for some words in the corpus of Shakespeare plays, ranging from extremely informative words that occur in only one play like Romeo, to those that occur in a few likesalad or Falstaff, to those that are very common like fool or so common as to be completely non-discriminative since they occur in all 37 plays like good or sweet.4 Word df idf Romeo 1 1.57 salad 2 1.27 Falstaff 4 0.967 forest 12 0.489 battle 21 0.246 wit 34 0.037 fool 36 0.012 good 37 0 sweet 37 0 The tf-idf value for word t in document d is then the product of term frequency tft,d and IDF: tf-idf(t,d) =tft,d ·idft (14.6) 14.1.2 Document Scoring We score document d by the cosine of its vector d with the query vector q: score(q,d) =cos(q,d) = q·d \|q\|\|d\| (14.7) Another way to think of the cosine computation is as the dot product of unit vectors; we ﬁrst normalize both the query and document vector to unit vectors, by dividing by their lengths, and then take the dot product: score(q,d) =cos(q,d) = q \|q\|· d \|d\| (14.8) 4 Sweet was one of Shakespeare’s favorite adjectives, a fact probably related to the increased use of sugar in European recipes around the turn of the 16th century (Jurafsky, 2014, p. 175). 14.1 • I NFORMATION RETRIEVAL 293 We can spell out Eq. 14.8, using the tf-idf values and spelling out the dot product as a sum of products: score(q,d) = ∑ t∈q tf-idf(t,q)√∑ qi∈q tf-idf 2(qi,q) · tf-idf(t,d)√∑ di∈d tf-idf 2(di,d) (14.9) Now let’s use Eq. 14.9 to walk through an example of a tiny query against a collection of 4 nano documents, computing tf-idf values and seeing the rank of the documents. We’ll assume all words in the following query and documents are down- cased and punctuation is removed: Query: sweet love Doc 1: Sweet sweet nurse! Love? Doc 2: Sweet sorrow Doc 3: How sweet is love? Doc 4: Nurse! Fig. 14.2 shows the computation of the tf-idf cosine between the query and Doc- ument 1, and the query and Document 2. The cosine is the normalized dot product of tf-idf values, so for the normalization we must need to compute the document vector lengths \|q\|, \|d1\|, and \|d2\|for the query and the ﬁrst two documents using Eq. 14.4, Eq. 14.5, Eq. 14.6, and Eq. 14.9 (computations for Documents 3 and 4 are also needed but are left as an exercise for the reader). The dot product between the vectors is the sum over dimensions of the product, for each dimension, of the values of the two tf-idf vectors for that dimension. This product is only non-zero where both the query and document have non-zero values, so for this example, in which only sweet and love have non-zero values in the query, the dot product will be the sum of the products of those elements of each vector. Document 1 has a higher cosine with the query (0.747) than Document 2 has with the query (0.0779), and so the tf-idf cosine model would rank Document 1 above Document 2. This ranking is intuitive given the vector space model, since Document 1 has both terms including two instances of sweet, while Document 2 is missing one of the terms. We leave the computation for Documents 3 and 4 as an exercise for the reader. In practice, there are many variants and approximations to Eq. 14.9. For exam- ple, we might choose to simplify processing by removing some terms. To see this, let’s start by expanding the formula for tf-idf in Eq. 14.9 to explicitly mention the tf and idf terms from Eq. 14.6: score(q,d) = ∑ t∈q tft,q ·idft√∑ qi∈q tf-idf 2(qi,q) · tft,d ·idft√∑ di∈d tf-idf 2(di,d) (14.10) In one common variant of tf-idf cosine, for example, we drop the idf term for the document. Eliminating the second copy of the idf term (since the identical term is already computed for the query) turns out to sometimes result in better performance: score(q,d) = ∑ t∈q tft,q·idft√∑ qi∈q tf-idf 2(qi,q) · tft,d ·idft√∑ di∈d tf-idf 2(di,d) (14.11) Other variants of tf-idf eliminate various other terms. A slightly more complex variant in the tf-idf family is the BM25 weightingBM25 294 CHAPTER 14 • Q UESTION ANSWERING , INFORMATION RETRIEVAL , AND RAG Query word cnt tf df idf tf-idf n’lized = tf-idf/\|q\| sweet 1 1 3 0.125 0.125 0.383 nurse 0 0 2 0.301 0 0 love 1 1 2 0.301 0.301 0.924 how 0 0 1 0.602 0 0 sorrow 0 0 1 0.602 0 0 is 0 0 1 0.602 0 0 \|q\|= √ .1252 +.3012 = .326 Document 1 Document 2 word cnt tf tf-idf n’lized ×q cnt tf tf-idf n’lized ×q sweet 2 1.301 0.163 0.357 0.137 1 1.000 0.125 0.203 0.0779 nurse 1 1.000 0.301 0.661 0 0 0 0 0 0 love 1 1.000 0.301 0.661 0.610 0 0 0 0 0 how 0 0 0 0 0 0 0 0 0 0 sorrow 0 0 0 0 0 1 1.000 0.602 0.979 0 is 0 0 0 0 0 0 0 0 0 0 \|d1\|= √ .1632 +.3012 +.3012 = .456 \|d2\|= √ .1252 +.6022 = .615 Cosine: ∑of column: 0.747 Cosine: ∑of column: 0.0779 Figure 14.2 Computation of tf-idf cosine score between the query and nano-documents 1 (0.747) and 2 (0.0779), using Eq. 14.4, Eq. 14.5, Eq. 14.6 and Eq. 14.9. scheme (sometimes called Okapi BM25 after the Okapi IR system in which it was introduced (Robertson et al., 1995)). BM25 adds two parameters: k, a knob that adjust the balance between term frequency and IDF, and b, which controls the im- portance of document length normalization. The BM25 score of a documentd given a query q is: ∑ t∈q IDF    log (N dft ) weighted tf    tft,d k ( 1 −b +b ( \|d\| \|davg\| )) +tft,d (14.12) where \|davg\|is the length of the average document. When k is 0, BM25 reverts to no use of term frequency, just a binary selection of terms in the query (plus idf). A large k results in raw term frequency (plus idf). b ranges from 1 (scaling by document length) to 0 (no length scaling). Manning et al. (2008) suggest reasonable values are k = [1.2,2] and b = 0.75. Kamphuis et al. (2020) is a useful summary of the many minor variants of BM25. Stop words In the past it was common to remove high-frequency words from both the query and document before representing them. The list of such high-frequency words to be removed is called a stop list. The intuition is that high-frequency termsstop list (often function words like the, a, to) carry little semantic weight and may not help with retrieval, and can also help shrink the inverted index ﬁles we describe below. The downside of using a stop list is that it makes it difﬁcult to search for phrases that contain words in the stop list. For example, common stop lists would reduce the phrase to be or not to beto the phrase not. In modern IR systems, the use of stop lists is much less common, partly due to improved efﬁciency and partly because much of their function is already handled by IDF weighting, which downweights function 14.1 • I NFORMATION RETRIEVAL 295 words that occur in every document. Nonetheless, stop word removal is occasionally useful in various NLP tasks so is worth keeping in mind. 14.1.3 Inverted Index In order to compute scores, we need to efﬁciently ﬁnd documents that contain words in the query. (Any document that contains none of the query terms will have a score of 0 and can be ignored.) The basic search problem in IR is thus to ﬁnd all documents d ∈C that contain a term q ∈Q. The data structure for this task is the inverted index, which we use for mak-inverted index ing this search efﬁcient, and also conveniently storing useful information like the document frequency and the count of each term in each document. An inverted index, given a query term, gives a list of documents that contain the term. It consists of two parts, a dictionary and the postings. The dictionary is a listpostings of terms (designed to be efﬁciently accessed), each pointing to apostings list for the term. A postings list is the list of document IDs associated with each term, which can also contain information like the term frequency or even the exact positions of terms in the document. The dictionary can also store the document frequency for each term. For example, a simple inverted index for our 4 sample documents above, with each word containing its document frequency in {}, and a pointer to a postings list that contains document IDs and term counts in [], might look like the following: how {1} → 3 [1] is {1} → 3 [1] love {2} → 1 [1] →3 [1] nurse {2} → 1 [1] →4 [1] sorry {1} → 2 [1] sweet {3} → 1 [2] →2 [1] →3 [1] Given a list of terms in query, we can very efﬁciently get lists of all candidate documents, together with the information necessary to compute the tf-idf scores we need. There are alternatives to the inverted index. For the question-answering domain of ﬁnding Wikipedia pages to match a user query, Chen et al. (2017a) show that indexing based on bigrams works better than unigrams, and use efﬁcient hashing algorithms rather than the inverted index to make the search efﬁcient. 14.1.4 Evaluation of Information-Retrieval Systems We measure the performance of ranked retrieval systems using the same precision and recall metrics we have been using. We make the assumption that each docu- ment returned by the IR system is either relevant to our purposes or not relevant. Precision is the fraction of the returned documents that are relevant, and recall is the fraction of all relevant documents that are returned. More formally, let’s assume a system returns T ranked documents in response to an information request, a subset R of these are relevant, a disjoint subset, N, are the remaining irrelevant documents, and U documents in the collection as a whole are relevant to this request. Precision and recall are then deﬁned as: Precision = \|R\| \|T \| Recall = \|R\| \|U\| (14.13) Unfortunately, these metrics don’t adequately measure the performance of a system that ranks the documents it returns. If we are comparing the performance of two 296 CHAPTER 14 • Q UESTION ANSWERING , INFORMATION RETRIEVAL , AND RAG ranked retrieval systems, we need a metric that prefers the one that ranks the relevant documents higher. We need to adapt precision and recall to capture how well a system does at putting relevant documents higher in the ranking. Rank Judgment Precision Rank RecallRank 1 R 1.0 .11 2 N .50 .11 3 R .66 .22 4 N .50 .22 5 R .60 .33 6 R .66 .44 7 N .57 .44 8 R .63 .55 9 N .55 .55 10 N .50 .55 11 R .55 .66 12 N .50 .66 13 N .46 .66 14 N .43 .66 15 R .47 .77 16 N .44 .77 17 N .44 .77 18 R .44 .88 19 N .42 .88 20 N .40 .88 21 N .38 .88 22 N .36 .88 23 N .35 .88 24 N .33 .88 25 R .36 1.0 Figure 14.3 Rank-speciﬁc precision and recall values calculated as we proceed down through a set of ranked documents (assuming the collection has 9 relevant documents). Let’s turn to an example. Assume the table in Fig. 14.3 gives rank-speciﬁc pre- cision and recall values calculated as we proceed down through a set of ranked doc- uments for a particular query; the precisions are the fraction of relevant documents seen at a given rank, and recalls the fraction of relevant documents found at the same rank. The recall measures in this example are based on this query having 9 relevant documents in the collection as a whole. Note that recall is non-decreasing; when a relevant document is encountered, recall increases, and when a non-relevant document is found it remains unchanged. Precision, on the other hand, jumps up and down, increasing when relevant doc- uments are found, and decreasing otherwise. The most common way to visualize precision and recall is to plot precision against recall in a precision-recall curve,precision-recall curve like the one shown in Fig. 14.4 for the data in table 14.3. Fig. 14.4 shows the values for a single query. But we’ll need to combine values for all the queries, and in a way that lets us compare one system to another. One way of doing this is to plot averaged precision values at 11 ﬁxed levels of recall (0 to 100, in steps of 10). Since we’re not likely to have datapoints at these exact levels, we use interpolated precision values for the 11 recall values from the data points we dointerpolated precision have. We can accomplish this by choosing the maximum precision value achieved at any level of recall at or above the one we’re calculating. In other words, IntPrecision(r) =max i>=r Precision(i) (14.14) 14.1 • I NFORMATION RETRIEVAL 297 0.0 0.2 0.4 0.6 0.8 1.0 Recall 0.0 0.2 0.4 0.6 0.8 1.0Precision Figure 14.4 The precision recall curve for the data in table 14.3. This interpolation scheme not only lets us average performance over a set of queries, but also helps smooth over the irregular precision values in the original data. It is designed to give systems the beneﬁt of the doubt by assigning the maximum preci- sion value achieved at higher levels of recall from the one being measured. Fig. 14.5 and Fig. 14.6 show the resulting interpolated data points from our example. Interpolated Precision Recall 1.0 0.0 1.0 .10 .66 .20 .66 .30 .66 .40 .63 .50 .55 .60 .47 .70 .44 .80 .36 .90 .36 1.0 Figure 14.5 Interpolated data points from Fig. 14.3. Given curves such as that in Fig. 14.6 we can compare two systems or approaches by comparing their curves. Clearly, curves that are higher in precision across all recall values are preferred. However, these curves can also provide insight into the overall behavior of a system. Systems that are higher in precision toward the left may favor precision over recall, while systems that are more geared towards recall will be higher at higher levels of recall (to the right). A second way to evaluate ranked retrieval is mean average precision (MAP),mean average precision which provides a single metric that can be used to compare competing systems or approaches. In this approach, we again descend through the ranked list of items, but now we note the precision only at those points where a relevant item has been encountered (for example at ranks 1, 3, 5, 6 but not 2 or 4 in Fig. 14.3). For a single query, we average these individual precision measurements over the return set (up to some ﬁxed cutoff). More formally, if we assume that Rr is the set of relevant 298 CHAPTER 14 • Q UESTION ANSWERING , INFORMATION RETRIEVAL , AND RAG Interpolated Precision Recall Curve 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Recall Precision Figure 14.6 An 11 point interpolated precision-recall curve. Precision at each of the 11 standard recall levels is interpolated for each query from the maximum at any higher level of recall. The original measured precision recall points are also shown. documents at or above r, then the average precision (AP) for a single query is AP = 1 \|Rr\| ∑ d∈Rr Precisionr(d) (14.15) where Precisionr(d) is the precision measured at the rank at which document d was found. For an ensemble of queries Q, we then average over these averages, to get our ﬁnal MAP measure: MAP = 1 \|Q\| ∑ q∈Q AP(q) (14.16) The MAP for the single query (hence = AP) in Fig. 14.3 is 0.6. 14.2 Information Retrieval with Dense Vectors The classic tf-idf or BM25 algorithms for IR have long been known to have a con- ceptual ﬂaw: they work only if there is exact overlap of words between the query and document. In other words, the user posing a query (or asking a question) needs to guess exactly what words the writer of the answer might have used, an issue called the vocabulary mismatch problem (Furnas et al., 1987). The solution to this problem is to use an approach that can handle synonymy: instead of (sparse) word-count vectors, using (dense) embeddings. This idea was ﬁrst proposed for retrieval in the last century under the name of Latent Semantic Indexing approach (Deerwester et al., 1990), but is implemented in modern times via encoders like BERT. The most powerful approach is to present both the query and the document to a single encoder, allowing the transformer self-attention to see all the tokens of both 14.2 • I NFORMATION RETRIEVAL WITH DENSE VECTORS 299 the query and the document, and thus building a representation that is sensitive to the meanings of both query and document. Then a linear layer can be put on top of the [CLS] token to predict a similarity score for the query/document tuple: z = BERT(q;[SEP];d)[CLS] score(q,d) =softmax(U(z)) (14.17) This architecture is shown in Fig. 14.7a. Usually the retrieval step is not done on an entire document. Instead documents are broken up into smaller passages, such as non-overlapping ﬁxed-length chunks of say 100 tokens, and the retriever encodes and retrieves these passages rather than entire documents. The query and document have to be made to ﬁt in the BERT 512-token window, for example by truncating the query to 64 tokens and truncating the document if necessary so that it, the query, [CLS], and [SEP] ﬁt in 512 tokens. The BERT system together with the linear layer U can then be ﬁne-tuned for the relevance task by gathering a tuning dataset of relevant and non-relevant passages. Query Document … … … … … … [sep] s(q,d) zCLS U Query zCLS_Q zCLS_D Document … … … … … … • s(q,d) (a) (b) Figure 14.7 Two ways to do dense retrieval, illustrated by using lines between layers to schematically rep- resent self-attention: (a) Use a single encoder to jointly encode query and document and ﬁnetune to produce a relevance score with a linear layer over the CLS token. This is too compute-expensive to use except in rescoring (b) Use separate encoders for query and document, and use the dot product between CLS token outputs for the query and document as the score. This is less compute-expensive, but not as accurate. The problem with the full BERT architecture in Fig. 14.7a is the expense in computation and time. With this architecture, every time we get a query, we have to pass every single single document in our entire collection through a BERT encoder jointly with the new query! This enormous use of resources is impractical for real cases. At the other end of the computational spectrum is a much more efﬁcient archi- tecture, the bi-encoder. In this architecture we can encode the documents in the collection only one time by using two separate encoder models, one to encode the query and one to encode the document. We encode each document, and store all the encoded document vectors in advance. When a query comes in, we encode just this query and then use the dot product between the query vector and the precom- puted document vectors as the score for each candidate document (Fig. 14.7b). For example, if we used BERT, we would have two encoders BERT Q and BERTD and 300 CHAPTER 14 • Q UESTION ANSWERING , INFORMATION RETRIEVAL , AND RAG we could represent the query and document as the [CLS] token of the respective encoders (Karpukhin et al., 2020): zq = BERTQ(q)[CLS] zd = BERTD(d)[CLS] score(q,d) =zq ·zd (14.18) The bi-encoder is much cheaper than a full query/document encoder, but is also less accurate, since its relevance decision can’t take full advantage of all the possi- ble meaning interactions between all the tokens in the query and the tokens in the document. There are numerous approaches that lie in between the full encoder and the bi- encoder. One intermediate alternative is to use cheaper methods (like BM25) as the ﬁrst pass relevance ranking for each document, take the top N ranked documents, and use expensive methods like the full BERT scoring to rerank only the top N documents rather than the whole set. Another intermediate approach is the ColBERT approach of Khattab and Za-ColBERT haria (2020) and Khattab et al. (2021), shown in Fig. 14.8. This method separately encodes the query and document, but rather than encoding the entire query or doc- ument into one vector, it separately encodes each of them into contextual represen- tations for each token. These BERT representations of each document word can be pre-stored for efﬁciency. The relevance score between a queryq and a document d is a sum of maximum similarity (MaxSim) operators between tokens in q and tokens in d. Essentially, for each token in q, ColBERT ﬁnds the most contextually simi- lar token in d, and then sums up these similarities. A relevant document will have tokens that are contextually very similar to the query. More formally, a question q is tokenized as [q1,..., qn], prepended with a [CLS] and a special [Q]token, truncated to N=32 tokens (or padded with[MASK]tokens if it is shorter), and passed through BERT to get output vectors q = [q1,..., qN ]. The passage d with tokens [d1,..., dm], is processed similarly, including a [CLS] and special [D] token. A linear layer is applied on top of d and q to control the output dimension, so as to keep the vectors small for storage efﬁciency, and vectors are rescaled to unit length, producing the ﬁnal vector sequences Eq (length N) and Ed (length m). The ColBERT scoring mechanism is: score(q,d) = N∑ i=1 m max j=1 Eqi ·Edj (14.19) While the interaction mechanism has no tunable parameters, the ColBERT ar- chitecture still needs to be trained end-to-end to ﬁne-tune the BERT encoders and train the linear layers (and the special [Q] and [D] embeddings) from scratch. It is trained on triples ⟨q,d+,d−⟩of query q, positive document d+ and negative docu- ment d−to produce a score for each document using Eq. 14.19, optimizing model parameters using a cross-entropy loss. All the supervised algorithms (like ColBERT or the full-interaction version of the BERT algorithm applied for reranking) need training data in the form of queries together with relevant and irrelevant passages or documents (positive and negative examples). There are various semi-supervised ways to get labels; some datasets (like MS MARCO Ranking, Section 14.3.2) contain gold positive examples. Negative examples can be sampled randomly from the top-1000 results from some existing IR system. If datasets don’t have labeled positive examples, iterative methods like 14.3 • A NSWERING QUESTIONS WITH RAG 301 Query Document … … … … … … s(q,d) MaxSim MaxSim MaxSim ∑ norm norm norm normnormnorm Figure 14.8 A sketch of the ColBERT algorithm at inference time. The query and docu- ment are ﬁrst passed through separate BERT encoders. Similarity between query and doc- ument is computed by summing a soft alignment between the contextual representations of tokens in the query and the document. Training is end-to-end. (Various details aren’t de- picted; for example the query is prepended by a [CLS] and [Q:] tokens, and the document by [CLS]and [D:]tokens). Figure adapted from Khattab and Zaharia (2020). relevance-guided supervision can be used (Khattab et al., 2021) which rely on the fact that many datasets contain short answer strings. In this method, an existing IR system is used to harvest examples that do contain short answer strings (the top few are taken as positives) or don’t contain short answer strings (the top few are taken as negatives), these are used to train a new retriever, and then the process is iterated. Efﬁciency is an important issue, since every possible document must be ranked for its similarity to the query. For sparse word-count vectors, the inverted index allows this very efﬁciently. For dense vector algorithms ﬁnding the set of dense document vectors that have the highest dot product with a dense query vector is an instance of the problem of nearest neighbor search . Modern systems there- fore make use of approximate nearest neighbor vector search algorithms like FaissFaiss (Johnson et al., 2017). 14.3 Answering Questions with RAG The dominant paradigm for question answering is to answer a user’s question by ﬁrst ﬁnding supportive text segments from the web or another other large collection of documents, and then generating an answer based on the documents. The method of generating based on retrieved documents is calledretrieval-augmented generation or RAG, and the two components are sometimes called the retriever and the reader (Chen et al., 2017a). Fig. 14.9 sketches out this standard QA model. In the ﬁrst stage of the 2-stage retrieve and read model in Fig. 14.9 we retrieve relevant passages from a text collection, for example using the dense retrievers of the previous section. In the second reader stage, we generate the answer via retrieval- 302 CHAPTER 14 • Q UESTION ANSWERING , INFORMATION RETRIEVAL , AND RAG Q: When was the premiere of The Magic Flute? Relevant Docs A: 1791 Retriever Indexed Docs query docs LLM prompt Reader/ Generator Figure 14.9 Retrieval-based question answering has two stages: retrieval, which returns relevant documents from the collection, and reading, in which an LLM generates answers given the documents as a prompt. augmented generation. In this method, we take a large pretrained language model, give it the set of retrieved passages and other text as its prompt, and autoregressively generate a new answer token by token. 14.3.1 Retrieval-Augmented Generation The standard reader algorithm is to generate from a large language model, condi- tioned on the retrieved passages. This method is known as retrieval-augmented generation, or RAG. retrieval- augmented generation RAG Recall that in simple conditional generation, we can cast the task of question answering as word prediction by giving a language model a question and a token like A:suggesting that an answer should come next: Q: Who wrote the book ‘‘The Origin of Species"? A: Then we generate autoregressively conditioned on this text. More formally, recall that simple autoregressive language modeling computes the probability of a string from the previous tokens: p(x1,..., xn) = n∏ i=1 p(xi\|x<i) And simple conditional generation for question answering adds a prompt like Q: , followed by a query q , and A:, all concatenated: p(x1,..., xn) = n∏ i=1 p([Q:]; q ; [A:]; x<i) The advantage of using a large language model is the enormous amount of knowledge encoded in its parameters from the text it was pretrained on. But as we mentioned at the start of the chapter, while this kind of simple prompted gener- ation can work ﬁne for many simple factoid questions, it is not a general solution for QA, because it leads to hallucination, is unable to show users textual evidence to support the answer, and is unable to answer questions from proprietary data. The idea of retrieval-augmented generation is to address these problems by con- ditioning on the retrieved passages as part of the preﬁx, perhaps with some prompt text like “Based on these texts, answer this question:”. Let’s suppose we have a query q, and call the set of retrieved passages based on it R( q). For example, we could have a prompt like: 14.3 • A NSWERING QUESTIONS WITH RAG 303 Schematic of a RAG Prompt retrieved passage 1 retrieved passage 2 ... retrieved passage n Based on these texts, answer this question: Q: Who wrote the book ‘‘The Origin of Species"? A: Or more formally, p(x1,..., xn) = n∏ i=1 p(xi\|R(q) ; prompt ; [Q:]; q ;[A:];x<i) As with the span-based extraction reader, successfully applying the retrieval- augmented generation algorithm for QA requires a successful retriever, and often a two-stage retrieval algorithm is used in which the retrieval is reranked. Some complex questions may require multi-hop architectures, in which a query is used tomulti-hop retrieve documents, which are then appended to the original query for a second stage of retrieval. Details of prompt engineering also have to be worked out, like deciding whether to demarcate passages, for example with [SEP]tokens, and so on. Combi- nations of private data and public data involving an externally hosted large language model may lead to privacy concerns that need to be worked out (Arora et al., 2023). Much research in this area also focuses on ways to more tightly integrate the retrieval and reader stages. 14.3.2 Question Answering Datasets There are scores of question answering datasets, used both for instruction tuning and for evaluation of the question answering abilities of language models. We can distinguish the datasets along many dimensions, summarized nicely in Rogers et al. (2023). One is the original purpose of the questions in the data, whether they were natural information-seeking questions, or whether they were questions designed for probing: evaluating or testing systems or humans. On the natural side there are datasets like Natural Questions (KwiatkowskiNatural Questions et al., 2019), a set of anonymized English queries to the Google search engine and their answers. The answers are created by annotators based on Wikipedia infor- mation, and include a paragraph-length long answer and a short span answer. For example the question “When are hops added to the brewing process?” has the short answer the boiling process and a long answer which is an entire paragraph from the Wikipedia page on Brewing. A similar natural question set is the MS MARCO (Microsoft Machine ReadingMS MARCO Comprehension) collection of datasets, including 1 million real anonymized English questions from Microsoft Bing query logs together with a human generated answer and 9 million passages (Bajaj et al., 2016), that can be used both to test retrieval ranking and question answering. 304 CHAPTER 14 • Q UESTION ANSWERING , INFORMATION RETRIEVAL , AND RAG Although many datasets focus on English, natural information-seeking ques- tion datasets exist in other languages. The DuReader dataset is a Chinese QA resource based on search engine queries and community QA (He et al., 2018). TyDi QA dataset contains 204K question-answer pairs from 11 typologically di-TyDi QA verse languages, including Arabic, Bengali, Kiswahili, Russian, and Thai (Clark et al., 2020a). In the T YDI QA task, a system is given a question and the passages from a Wikipedia article and must (a) select the passage containing the answer (or NULL if no passage contains the answer), and (b) mark the minimal answer span (or NULL ). On the probing side are datasets like MMLU (Massive Multitask Language Un-MMLU derstanding), a commonly-used dataset of 15908 knowledge and reasoning ques- tions in 57 areas including medicine, mathematics, computer science, law, and oth- ers. MMLU questions are sourced from various exams for humans, such as the US Graduate Record Exam, Medical Licensing Examination, and Advanced Placement exams. So the questions don’t represent people’s information needs, but rather are designed to test human knowledge for academic or licensing purposes. Fig. 14.10 shows some examples, with the correct answers in bold. Some of the question datasets described above augment each question with pas- sage(s) from which the answer can be extracted. These datasets were mainly created for an earlier QA task called reading comprehension in which a model is givenreading comprehension a question and a document and is required to extract the answer from the given document. We sometimes call the task of question answering given one or more documents (for example via RAG), the open book QA task, while the task of an-open book swering directly from the LM with no retrieval component at all is the closed bookclosed book QA task.5 Thus datasets like Natural Questions can be treated as open book if the solver uses each question’s attached document, or closed book if the documents are not used, while datasets like MMLU are solely closed book. Another dimension of variation is the format of the answer: multiple-choice versus freeform. And of course there are variations in prompting, like whether the model is just the question (zero-shot) or also given demonstrations of answers to similar questions (few-shot). MMLU offers both zero-shot and few-shot prompt options. 14.4 Evaluating Question Answering Three techniques are commonly employed to evaluate question-answering systems, with the choice depending on the type of question and QA situation. For multiple choice questions like in MMLU, we report exact match: Exact match: The % of predicted answers that match the gold answer exactly. For questions with free text answers, like Natural Questions, we commonly evalu- ated with token F1 score to roughly measure the partial string overlap between the answer and the reference answer: F1 score: The average token overlap between predicted and gold an- swers. Treat the prediction and gold as a bag of tokens, and compute F1 for each question, then return the average F1 over all questions. 5 This repurposes the word for types of exams in which students are allowed to ‘open their books’ or not. 14.4 • E VALUATING QUESTION ANSWERING 305 MMLU examples College Computer Science Any set of Boolean operators that is sufﬁcient to represent all Boolean ex- pressions is said to be complete. Which of the following is NOT complete? (A) AND, NOT (B) NOT, OR (C) AND, OR (D) NAND College Physics The primary source of the Sun’s energy is a series of thermonuclear reactions in which the energy produced is c 2 times the mass difference between (A) two hydrogen atoms and one helium atom (B) four hydrogen atoms and one helium atom (C) six hydrogen atoms and two helium atoms (D) three helium atoms and one carbon atom International Law Which of the following is a treaty-based human rights mechanism? (A) The UN Human Rights Committee (B) The UN Human Rights Council (C) The UN Universal Periodic Review (D) The UN special mandates Prehistory Unlike most other early civilizations, Minoan culture shows little evidence of (A) trade. (B) warfare. (C) the development of a common religion. (D) conspicuous consumption by elites. Figure 14.10 Example problems from MMLU Finally, in some situations QA systems give multipleranked answers. In such cases we evaluated using mean reciprocal rank , or MRR (V oorhees, 1999). MRR ismean reciprocal rank MRR designed for systems that return a short ranked list of answers or passages for each test set question, which we can compare against the (human-labeled) correct answer. First, each test set question is scored with the reciprocal of the rank of the ﬁrst correct answer. For example if the system returned ﬁve answers to a question but the ﬁrst three are wrong (so the highest-ranked correct answer is ranked fourth), the reciprocal rank for that question is 1 4 . The score for questions that return no correct answer is 0. The MRR of a system is the average of the scores for each question in the test set. In some versions of MRR, questions with a score of zero are ignored in this calculation. More formally, for a system returning ranked answers to each question in a test set Q, (or in the alternate version, let Q be the subset of test set 306 CHAPTER 14 • Q UESTION ANSWERING , INFORMATION RETRIEVAL , AND RAG questions that have non-zero scores). MRR is then deﬁned as MRR = 1 \|Q\| \|Q\|∑ i=1 1 ranki (14.20) 14.5 Summary This chapter introduced the tasks ofquestion answering and information retrieval. • Question answering (QA) is the task of answering a user’s questions. • We focus in this chapter on the task of retrieval-based question answering, in which the user’s questions are intended to be answered by the material in some set of documents (which might be the web). • Information Retrieval(IR) is the task of returning documents to a user based on their information need as expressed in a query. In ranked retrieval, the documents are returned in ranked order. • The match between a query and a document can be done by ﬁrst representing each of them with a sparse vector that represents the frequencies of words, weighted by tf-idf or BM25. Then the similarity can be measured by cosine. • Documents or queries can instead be represented by dense vectors, by encod- ing the question and document with an encoder-only model like BERT, and in that case computing similarity in embedding space. • The inverted index is a storage mechanism that makes it very efﬁcient to ﬁnd documents that have a particular word. • Ranked retrieval is generally evaluated by mean average precision or inter- polated precision. • Question answering systems generally use the retriever/reader architecture. In the retriever stage, an IR system is given a query and returns a set of documents. • The reader stage is implemented by retrieval-augmented generation, in which a large language model is prompted with the query and a set of doc- uments and then conditionally generates a novel answer. • QA can be evaluated by exact match with a known answer if only a single answer is given, with token F 1 score for free text answers, or with mean re- ciprocal rank if a ranked set of answers is given. Bibliographical and Historical Notes Question answering was one of the earliest NLP tasks, and early versions of the text- based and knowledge-based paradigms were developed by the very early 1960s. The text-based algorithms generally relied on simple parsing of the question and of the sentences in the document, and then looking for matches. This approach was used very early on (Phillips, 1960) but perhaps the most complete early system, and one that strikingly preﬁgures modern relation-based systems, was the Protosynthex sys- tem of Simmons et al. (1964). Given a question, Protosynthex ﬁrst formed a query BIBLIOGRAPHICAL AND HISTORICAL NOTES 307 from the content words in the question, and then retrieved candidate answer sen- tences in the document, ranked by their frequency-weighted term overlap with the question. The query and each retrieved sentence were then parsed with dependency parsers, and the sentence whose structure best matches the question structure se- lected. Thus the question What do worms eat? would match worms eat grass: both have the subjectworms as a dependent ofeat, in the version of dependency grammar used at the time, while birds eat worms has birds as the subject: What do worms eat Worms eat grass Birds eat worms The alternative knowledge-based paradigm was implemented in the BASEBALL system (Green et al., 1961). This system answered questions about baseball games like “Where did the Red Sox play on July 7” by querying a structured database of game information. The database was stored as a kind of attribute-value matrix with values for attributes of each game: Month = July Place = Boston Day = 7 Game Serial No. = 96 (Team = Red Sox, Score = 5) (Team = Yankees, Score = 3) Each question was constituency-parsed using the algorithm of Zellig Harris’s TDAP project at the University of Pennsylvania, essentially a cascade of ﬁnite-state transducers (see the historical discussion in Joshi and Hopely 1999 and Karttunen 1999). Then in a content analysis phase each word or phrase was associated with a program that computed parts of its meaning. Thus the phrase ‘Where’ had code to assign the semantics Place = ?, with the result that the question “Where did the Red Sox play on July 7” was assigned the meaning Place = ? Team = Red Sox Month = July Day = 7 The question is then matched against the database to return the answer. Simmons (1965) summarizes other early QA systems. Another important progenitor of the knowledge-based paradigm for question- answering is work that used predicate calculus as the meaning representation lan- guage. The LUNAR system (Woods et al. 1972, Woods 1978) was designed to beLUNAR a natural language interface to a database of chemical facts about lunar geology. It could answer questions like Do any samples have greater than 13 percent aluminum by parsing them into a logical form (TEST (FOR SOME X16 / (SEQ SAMPLES) : T ; (CONTAIN’ X16 (NPR* X17 / (QUOTE AL203)) (GREATERTHAN 13 PCT)))) By a couple decades later, drawing on new machine learning approaches in NLP, Zelle and Mooney (1996) proposed to treat knowledge-based QA as a semantic pars- ing task, by creating the Prolog-based GEOQUERY dataset of questions about US geography. This model was extended by Zettlemoyer and Collins (2005) and 2007. 308 CHAPTER 14 • Q UESTION ANSWERING , INFORMATION RETRIEVAL , AND RAG By a decade later, neural models were applied to semantic parsing (Dong and Lap- ata 2016, Jia and Liang 2016), and then to knowledge-based question answering by mapping text to SQL (Iyer et al., 2017). Meanwhile, the information-retrieval paradigm for question answering was in- ﬂuenced by the rise of the web in the 1990s. The U.S. government-sponsored TREC (Text REtrieval Conference) evaluations, run annually since 1992, provide a testbed for evaluating information-retrieval tasks and techniques (V oorhees and Harman, 2005). TREC added an inﬂuential QA track in 1999, which led to a wide variety of factoid and non-factoid systems competing in annual evaluations. At that same time, Hirschman et al. (1999) introduced the idea of using chil- dren’s reading comprehension tests to evaluate machine text comprehension algo- rithms. They acquired a corpus of 120 passages with 5 questions each designed for 3rd-6th grade children, built an answer extraction system, and measured how well the answers given by their system corresponded to the answer key from the test’s publisher. Their algorithm focused on word overlap as a feature; later algorithms added named entity features and more complex similarity between the question and the answer span (Riloff and Thelen 2000, Ng et al. 2000). The DeepQA component of the Watson Jeopardy! system was a large and so- phisticated feature-based system developed just before neural systems became com- mon. It is described in a series of papers in volume 56 of the IBM Journal of Re- search and Development, e.g., Ferrucci (2012). Early neural reading comprehension systems drew on the insight common to early systems that answer ﬁnding should focus on question-passage similarity. Many of the architectural outlines of these neural systems were laid out in Hermann et al. (2015a), Chen et al. (2017a), and Seo et al. (2017). These systems focused on datasets like Rajpurkar et al. (2016) and Rajpurkar et al. (2018) and their succes- sors, usually using separate IR algorithms as input to neural reading comprehension systems. The paradigm of using dense retrieval with a span-based reader, often with a single end-to-end architecture, is exempliﬁed by systems like Lee et al. (2019) or Karpukhin et al. (2020). An important research area with dense retrieval for open- domain QA is training data: using self-supervised methods to avoid having to label positive and negative passages (Sachan et al., 2023). Early work on large language models showed that they stored sufﬁcient knowl- edge in the pretraining process to answer questions (Petroni et al., 2019; Raffel et al., 2020; Radford et al., 2019; Roberts et al., 2020), at ﬁrst not competitively with special-purpose question answerers, but then surpassing them. Retrieval-augmented generation algorithms were ﬁrst introduced as a way to improve language modeling (Khandelwal et al., 2019), but were quickly applied to question answering (Izacard et al., 2022; Ram et al., 2023; Shi et al., 2023). Exercises CHAPTER 15 Chatbots & Dialogue Systems Les lois de la conversation sont en g´en´eral de ne s’y appesantir sur aucun ob- jet, mais de passer l ´eg`erement, sans effort et sans affectation, d’un sujet `a un autre ; de savoir y parler de choses frivoles comme de choses s´erieuses [The rules of conversation are, in general, not to dwell on any one subject, but to pass lightly from one to another without effort and without affectation; to know how to speak about trivial topics as well as serious ones;] The 18th C. Encyclopedia of Diderot, start of the entry on conversation The literature of the fantastic abounds in inanimate objects magically endowed with the gift of speech. From Ovid’s statue of Pygmalion to Mary Shelley’s story about Frankenstein, we continually reinvent stories about creat- ing something and then having a chat with it. Legend has it that after ﬁnishing his sculpture Moses, Michelangelo thought it so lifelike that he tapped it on the knee and commanded it to speak. Perhaps this shouldn’t be sur- prising. Language is the mark of humanity and sentience, and conversation or dialogue is the most fundamentalconversation dialogue arena of language. It is the ﬁrst kind of language we learn as children, and the kind we engage in constantly, whether we are ordering lunch, buying train tickets, or talking with our families, friends, or coworkers. This chapter introduces the fundamental algorithms of programs that use con- versation to interact with users. We often distinguish between two kinds of archi- tectures. Task-oriented dialogue systems converse with users to accomplish ﬁxeddialogue system tasks like controlling appliances or ﬁnding restaurants, relying on a data structure called the frame, which represents the knowledge a system needs to acquire fromframe the user (like the time to set an alarm clock). Chatbots, by contrast, are designedchatbot to mimic the longer and more unstructured conversations or ‘chats’ characteristic of human-human interaction. Modern systems incorporate aspects of both; industrial chatbots like ChatGPT can carry on longer unstructured conversations; industrial digital assistants like Siri or Alexa are generally frame-based dialogue systems. The fact that chatbots and dialogue systems are designed for human-computer interaction has strong implications for their design and use. Many of these impli- cations already became clear in one of the earliest chatbots, ELIZA (Weizenbaum, 1966). ELIZA was designed to simulate a Rogerian psychologist, based on a branch of clinical psychology whose methods involve drawing the patient out by reﬂecting patient’s statements back at them. Rogerian interactions are the rare type of conver- sation in which, as Weizenbaum points out, one can “assume the pose of knowing almost nothing of the real world”. If a patient says “I went for a long boat ride” and the psychiatrist says “Tell me about boats”, you don’t assume she didn’t know what 310 CHAPTER 15 • C HATBOTS & DIALOGUE SYSTEMS Men are all alike. IN WHAT WAY They’re always bugging us about something or other. CAN YOU THINK OF A SPECIFIC EXAMPLE Well my boyfriend made me come here. YOUR BOYFRIEND MADE YOU COME HERE He says I’m depressed much of the time. I AM SORRY TO HEAR YOU ARE DEPRESSED Figure 15.1 Sample ELIZA dialogue from Weizenbaum (1966). a boat is, but rather assume she had some conversational goal.1 Weizenbaum made use of this property of Rogerian psychiatric conversations, along with clever regular expressions, to allow ELIZA to interact in ways that seemed deceptively human-like, as in the sample conversational fragment in Fig. 15.1. As we foreshadowed in Chapter 2, ELIZA worked by simple rules roughly like: (.) YOU (.) ME -> WHAT MAKES YOU THINK I \2 YOU to transform a user sentence like “You hate me” into a system response like WHAT MAKES YOU THINK I HATE YOU Among Weizenbaum’s clever tricks are the linking of each ELIZA pattern/rule to a keyword. Consider the following user sentence: I know everybody laughed at me Because it has the word “I”, this sentence could match the following rule whose keyword is I: I (.) -> You say you \1 producing: YOU SAY YOU KNOW EVERYBODY LAUGHED AT YOU Weizenbaum points out, however, that a more powerful response would rely on the keyword “everybody”, since someone using universals likeeverybody or always is probably thinking about a speciﬁc person or situation. So the ELIZA algorithm prefers to respond using patterns associated more speciﬁc keywords like everybody: WHO IN PARTICULAR ARE YOU THINKING OF? If no keyword matches, the algorithm chooses a non-committal response like “PLEASE GO ON”, “THAT’S VERY INTERESTING”, or “I SEE”. ELIZA illustrates a number of important issues with chatbots. First, people became deeply emotionally involved and conducted very personal conversations, even to the extent of asking Weizenbaum to leave the room while they were typ- ing. Reeves and Nass (1996) show that people tend to assign human characteristics to computers and interact with them in ways that are typical of human-human in- teractions. They interpret an utterance in the way they would if it had spoken by a human, (even though they are aware they are talking to a computer). This means that chatbots can have signiﬁcant inﬂuences on people’s cognitive and emotional state. A second related issue is privacy. When Weizenbaum suggested that he might want to store the ELIZA conversations, people immediately pointed out that this would violate people’s privacy. Modern chatbots in the home are likely to overhear 1 This is due to the Gricean principle of relevance that we’ll discuss in the next section.. 15.1 • P ROPERTIES OF HUMAN CONVERSATION 311 private information, even if they aren’t used for counseling as ELIZA was. Indeed, if a chatbot is human-like, users are more likely to disclose private information, and yet less likely to worry about the harm of this disclosure (Ischen et al., 2019). Both of these issues (emotional engagement and privacy) mean we need to think carefully about how we deploy chatbots and the people who are interacting with them. Dialogue research that uses human participants often requires getting permis- sion from the Institutional Review Board (IRB) of your institution. In the next section we introduce some basic properties of human conversation. We then turn in the rest of the chapter to the two basic paradigms for conversational interaction: frame-based dialogue systems and chatbots. 15.1 Properties of Human Conversation Conversation between humans is an intricate and complex joint activity. Before we attempt to design a dialogue system to converse with humans, it is crucial to understand something about how humans converse with each other. Consider some of the phenomena that occur in the conversation between a human travel agent and a human client excerpted in Fig. 15.2. C1: . . . I need to travel in May. A2: And, what day in May did you want to travel? C3: OK uh I need to be there for a meeting that’s from the 12th to the 15th. A4: And you’re ﬂying into what city? C5: Seattle. A6: And what time would you like to leave Pittsburgh? C7: Uh hmm I don’t think there’s many options for non-stop. A8: Right. There’s three non-stops today. C9: What are they? A10: The ﬁrst one departs PGH at 10:00am arrives Seattle at 12:05 their time. The second ﬂight departs PGH at 5:55pm, arrives Seattle at 8pm. And the last ﬂight departs PGH at 8:15pm arrives Seattle at 10:28pm. C11: OK I’ll take the 5ish ﬂight on the night before on the 11th. A12: On the 11th? OK. Departing at 5:55pm arrives Seattle at 8pm, U.S. Air ﬂight 115. C13: OK. A14: And you said returning on May 15th? C15: Uh, yeah, at the end of the day. A16: OK. There’s #two non-stops . . . # C17: #Act. . . actually #, what day of the week is the 15th? A18: It’s a Friday. C19: Uh hmm. I would consider staying there an extra day til Sunday. A20: OK. . . OK. On Sunday I have . . . Figure 15.2 Part of a phone conversation between a human travel agent (A) and human client (C). The passages framed by # in A16 and C17 indicate overlaps in speech. Turns A dialogue is a sequence ofturns (C1, A2, C3, and so on), each a single contributionturn from one speaker to the dialogue (as if in a game: I take a turn, then you take a turn, 312 CHAPTER 15 • C HATBOTS & DIALOGUE SYSTEMS then me, and so on). There are 20 turns in Fig. 15.2. A turn can consist of a sentence (like C1), although it might be as short as a single word (C13) or as long as multiple sentences (A10). Turn structure has important implications for spoken dialogue. A human has to know when to stop talking; the client interrupts (in A 16 and C17), so a system that was performing this role must know to stop talking (and that the user might be making a correction). A system also has to know when to start talking. For example, most of the time in conversation, speakers start their turns almost immediately after the other speaker ﬁnishes, without a long pause, because people are can usually predict when the other person is about to ﬁnish talking. Spoken dialogue systems must also detect whether a user is done speaking, so they can process the utterance and respond. This task—called endpointing or endpoint detection— can be quiteendpointing challenging because of noise and because people often pause in the middle of turns. Speech Acts A key insight into conversation—due originally to the philosopher Wittgenstein (1953) but worked out more fully by Austin (1962)—is that each utterance in a dialogue is a kind of action being performed by the speaker. These actions are com- monly called speech acts or dialogue acts: here’s one taxonomy consisting of 4speech acts major classes (Bach and Harnish, 1979): Constatives: committing the speaker to something’s being the case (answering, claiming, conﬁrming, denying, disagreeing, stating) Directives: attempts by the speaker to get the addressee to do something (advising, ask- ing, forbidding, inviting, ordering, requesting) Commissives: committing the speaker to some future course of action (promising, planning, vowing, betting, opposing) Acknowledgments: express the speaker’s attitude regarding the hearer with respect to some so- cial action (apologizing, greeting, thanking, accepting an acknowledgment) A user asking a person or a dialogue system to do something (‘Turn up the mu- sic’) is issuing a D IRECTIVE . Asking a question that requires an answer is also a way of issuing a D IRECTIVE : in a sense when the system says (A 2) “what day in May did you want to travel?” it’s as if the system is (very politely) command- ing the user to answer. By contrast, a user stating a constraint (like C 1 ‘I need to travel in May’) is issuing a C ONSTATIVE . A user thanking the system is issuing an ACKNOWLEDGMENT . The speech act expresses an important component of the intention of the speaker (or writer) in saying what they said. Grounding A dialogue is not just a series of independent speech acts, but rather a collective act performed by the speaker and the hearer. Like all collective acts, it’s important for the participants to establish what they both agree on, called the common groundcommon ground (Stalnaker, 1978). Speakers do this by grounding each other’s utterances. Ground-grounding ing means acknowledging that the hearer has understood the speaker (Clark, 1996). (People need grounding for non-linguistic actions as well; the reason an elevator but- ton lights up when it’s pressed is to acknowledge that the elevator has indeed been called, essentially grounding your action of pushing the button (Norman, 1988).) Humans constantly ground each other’s utterances. We can ground by explicitly saying “OK”, as the agent does in A8 or A10. Or we can ground by repeating what the other person says; in utterance A2 the agent repeats “in May”, demonstrating her 15.1 • P ROPERTIES OF HUMAN CONVERSATION 313 understanding to the client. Or notice that when the client answers a question, the agent begins the next question with “And”. The “And” implies that the new question is ‘in addition’ to the old question, again indicating to the client that the agent has successfully understood the answer to the last question. Subdialogues and Dialogue Structure Conversations have structure. Consider, for example, the local structure between speech acts discussed in the ﬁeld of conversation analysis (Sacks et al., 1974).conversation analysis QUESTIONS set up an expectation for an ANSWER . P ROPOSALS are followed by ACCEPTANCE (or REJECTION ). C OMPLIMENTS (“Nice jacket!”) often give rise to DOWNPLAYERS (“Oh, this old thing?”). These pairs, called adjacency pairs areadjacency pair composed of a ﬁrst pair part and a second pair part (Schegloff, 1968), and these expectations can help systems decide what actions to take. However, dialogue acts aren’t always followed immediately by their second pair part. The two parts can be separated by a side sequence (Jefferson 1972) or sub-side sequence dialogue. For example utterances C 17 to A20 constitute a correction subdialoguesubdialogue (Litman 1985, Litman and Allen 1987, Chu-Carroll and Carberry 1998): C17: #Act. . . actually#, what day of the week is the 15th? A18: It’s a Friday. C19: Uh hmm. I would consider staying there an extra day til Sunday. A20: OK. . . OK. On Sunday I have . . . The question in C17 interrupts the prior discourse, in which the agent was looking for a May 15 return ﬂight. The agent must answer the question and also realize that ‘’I would consider staying...til Sunday” means that the client would probably like to change their plan, and now go back to ﬁnding return ﬂights, but for the 17th. Another side sequence is the clariﬁcation question, which can form a subdia- logue between a REQUEST and a RESPONSE . This is especially common in dialogue systems where speech recognition errors causes the system to have to ask for clari- ﬁcations or repetitions like the following: User: What do you have going to UNKNOWN WORD on the 5th? System: Let’s see, going where on the 5th? User: Going to Hong Kong. System: OK, here are some ﬂights... In addition to side-sequences, questions often have presequences, like the fol-presequence lowing example where a user starts with a question about the system’s capabilities (“Can you make train reservations”) before making a request. User: Can you make train reservations? System: Yes I can. User: Great, I’d like to reserve a seat on the 4pm train to New York. Initiative Sometimes a conversation is completely controlled by one participant. For exam- ple a reporter interviewing a chef might ask questions, and the chef responds. We say that the reporter in this case has the conversational initiative (Carbonell, 1970;initiative Nickerson, 1976). In normal human-human dialogue, however, it’s more common for initiative to shift back and forth between the participants, as they sometimes answer questions, sometimes ask them, sometimes take the conversations in new di- rections, sometimes not. You may ask me a question, and then I respond asking you 314 CHAPTER 15 • C HATBOTS & DIALOGUE SYSTEMS to clarify something you said, which leads the conversation in all sorts of ways. We call such interactions mixed initiative (Carbonell, 1970). Full mixed initiative, while the norm for human-human conversations, can be difﬁcult for dialogue systems. The most primitive dialogue systems tend to use system-initiative, where the system asks a question and the user can’t do anything until they answer it, or user-initiative like simple search engines, where the user speciﬁes a query and the system passively responds. Even modern large language model-based dialogue systems, which come much closer to using full mixed initia- tive, often don’t have completely natural initiative switching. Getting this right is an important goal for modern systems. Inference and Implicature Inference is also important in dialogue understanding. Consider the client’s response C2, repeated here: A2: And, what day in May did you want to travel? C3: OK uh I need to be there for a meeting that’s from the 12th to the 15th. Notice that the client does not in fact answer the agent’s question. The client merely mentions a meeting at a certain time. What is it that licenses the agent to infer that the client is mentioning this meeting so as to inform the agent of the travel dates? The speaker seems to expect the hearer to draw certain inferences; in other words, the speaker is communicating more information than seems to be present in the uttered words. This kind of example was pointed out by Grice (1975, 1978) as part of his theory of conversational implicature. Implicature means a particu-implicature lar class of licensed inferences. Grice proposed that what enables hearers to draw these inferences is that conversation is guided by a set ofmaxims, general heuristics that play a guiding role in the interpretation of conversational utterances. One such maxim is the maxim of relevance which says that speakers attempt to be relevant,relevance they don’t just utter random speech acts. When the client mentions a meeting on the 12th, the agent reasons ‘There must be some relevance for mentioning this meeting. What could it be?’. The agent knows that one precondition for having a meeting (at least before Web conferencing) is being at the place where the meeting is held, and therefore that maybe the meeting is a reason for the travel, and if so, then since people like to arrive the day before a meeting, the agent should infer that the ﬂight should be on the 11th. These subtle characteristics of human conversations (turns, speech acts, ground- ing, dialogue structure, initiative, and implicature) are among the reasons it is dif- ﬁcult to build dialogue systems that can carry on natural conversations with humans. Many of these challenges are active areas of dialogue systems research. 15.2 Frame-Based Dialogue Systems A task-based dialogue system has the goal of helping a user solve a speciﬁc task like making a travel reservation or buying a product. Task-based dialogue systems are based around frames, ﬁrst introduced in the early inﬂuential GUS system forframe GUS travel planning (Bobrow et al., 1977). Frames are knowledge structures representing the details of the user’s task speciﬁcation. Each frame consists of a collection of slots, each of which can take a set of possible values. Together a set of frames isslot 15.2 • F RAME -BASED DIALOGUE SYSTEMS 315 sometimes called a domain ontology. Here we’ll describe the most well-studied frame-based architecture, thedialogue- state architecture, made up of the six components shown in Fig. 15.3. In the next sections we’ll introduce four of them, after introducing the idea of frames (deferring the speech recognition and synthesis components to Chapter 16). Figure 15.3 Architecture of a dialogue-state system for task-oriented dialogue from Williams et al. (2016). 15.2.1 Frames and Slot Filling The frame and its slots in a task-based dialogue system specify what the system needs to know to perform its task. A hotel reservation system needs dates and loca- tions. An alarm clock system needs a time. The system’s goal is to ﬁll the slots in the frame with the ﬁllers the user intends, and then perform the relevant action for the user (answering a question, or booking a ﬂight). Fig. 15.4 shows a sample frame for booking air travel, with some sample ques- tions used for ﬁlling slots. In the simplest frame-based systems (including most com- mercial assistants until quite recently), these questions are pre-written templates, but in more sophisticated systems, questions are generated on-the-ﬂy. The slot ﬁllers are often constrained to a particular semantic type, like type CITY (taking on values like San Francisco, or Hong Kong) or DATE, AIRLINE , or TIME . Slot Type Example Question ORIGIN CITY city “From what city are you leaving?” DESTINATION CITY city “Where are you going?” DEPARTURE TIME time “When would you like to leave?” DEPARTURE DATE date “What day would you like to leave?” ARRIV AL TIME time “When do you want to arrive?” ARRIV AL DATE date “What day would you like to arrive?” Figure 15.4 A frame in a frame-based dialogue system, showing the type of each slot and a sample question used to ﬁll the slot. 316 CHAPTER 15 • C HATBOTS & DIALOGUE SYSTEMS Many domains require multiple frames. Besides frames for car or hotel reser- vations, we might need other frames for things like general route information (for questions like Which airlines ﬂy from Boston to San Francisco? ), That means the system must be able to disambiguate which slot of which frame a given input is supposed to ﬁll. The task of slot-ﬁlling is usually combined with two other tasks, to extract 3 things from each user utterance. The ﬁrst is domain classiﬁcation: is this user for example talking about airlines, programming an alarm clock, or dealing with their calendar? The second is user intent determination: what general task or goal is theintent determination user trying to accomplish? For example the task could be to Find a Movie, or Show a Flight, or Remove a Calendar Appointment. Together, the domain classiﬁcation and intent determination tasks decide which frame we are ﬁlling. Finally, we need to do slot ﬁlling itself: extract the particular slots and ﬁllers that the user intends theslot ﬁlling system to understand from their utterance with respect to their intent. From a user utterance like this one: Show me morning flights from Boston to San Francisco on Tuesday a system might want to build a representation like: DOMAIN: AIR-TRAVEL INTENT: SHOW-FLIGHTS ORIGIN-CITY: Boston DEST-CITY: San Francisco ORIGIN-DATE: Tuesday ORIGIN-TIME: morning Similarly an utterance like this: should give an intent like this: Wake me tomorrow at 6 DOMAIN: ALARM-CLOCK INTENT: SET-ALARM TIME: 2017-07-01 0600 The simplest dialogue systems use handwritten rules for slot-ﬁlling, like this regular expression for recognizing the SET-ALARM intent: wake me (up) \| set (the\|an) alarm \| get me up But most systems use supervised machine-learning: each sentence in a training set is annotated with slots, domain, and intent, and a sequence model maps from input words to slot ﬁllers, domain and intent. For example we’ll have pairs of sen- tences that are labeled for domain (AIRLINE ) and intent (SHOWFLIGHT ), and are also labeled with BIO representations for the slots and ﬁllers. (Recall from Chapter 17 that in BIO tagging we introduce a tag for the beginning (B) and inside (I) of each slot label, and one for tokens outside (O) any slot label.) O O O O O B-DES I-DES O B-DEPTIME I-DEPTIME O AIRLINE-SHOWFLIGHT I want to fly to San Francisco on Monday afternoon please EOS Fig. 15.5 shows a typical architecture for inference. The input words w1...wn are passed through a pretrained language model encoder, followed by a feedforward layer and a softmax at each token position over possible BIO tags, with the output a series of BIO tags s1...sn. We generally combine the domain-classiﬁcation and intent-extraction tasks with slot-ﬁlling by adding a domain concatenated with an intent as the desired output for the ﬁnal EOS token. Once the sequence labeler has tagged the user utterance, a ﬁller string can be ex- tracted for each slot from the tags (e.g., “San Francisco”), and these word strings can then be normalized to the correct form in the ontology (perhaps the airport 15.3 • D IALOGUE ACTS AND DIALOGUE STATE 317 San Francisco on Monday Encodings Classiﬁer +softmax B-DES I-DES O B-DTIME … d+i <EOS> Encoder Figure 15.5 Slot ﬁlling by passing input words through an encoder, and then using a linear or feedforward layer followed by a softmax to generate a series of BIO tags. Here we also show a ﬁnal state: a domain concatenated with an intent. code ‘SFO’), for example with dictionaries that specify that SF, SFO, and San Fran- cisco are synonyms. Often in industrial contexts, combinations of rules and machine learning are used for each of these components. We can make a very simple frame-based dialogue system by wrapping a small amount of code around this slot extractor. Mainly we just need to ask the user questions until all the slots are full, do a database query, then report back to the user, using hand-built templates for generating sentences. 15.2.2 Evaluating Task-Based Dialogue We evaluate task-based systems by computing the task error rate, or task successtask error rate rate: the percentage of times the system booked the right plane ﬂight, or put the right event on the calendar. A more ﬁne-grained, but less extrinsic metric is the slot error rate, the percentage of slots ﬁlled with the correct values:slot error rate Slot Error Rate for a Sentence = # of inserted/deleted/subsituted slots # of total reference slots for sentence (15.1) For example a system that extracted the slot structure below from this sentence: (15.2) Make an appointment with Chris at 10:30 in Gates 104 Slot Filler PERSON Chris TIME 11:30 a.m. ROOM Gates 104 has a slot error rate of 1/3, since the TIME is wrong. Instead of error rate, slot precision, recall, and F-score can also be used. We can also measure efﬁciency costs like the length of the dialogue in seconds or turns.efﬁciency costs 15.3 Dialogue Acts and Dialogue State While the naive slot-extractor system described above can handle simple dialogues, often we want more complex interactions. For example, we might want to conﬁrm that we’ve understand the user, or ask them to repeat themselves. We can build a more sophisticated system using dialogue acts and dialogue state. 318 CHAPTER 15 • C HATBOTS & DIALOGUE SYSTEMS 15.3.1 Dialogue Acts Dialogue acts are a generalization of speech acts that also represent grounding. Thedialogue acts set of acts can be general, or can be designed for particular dialogue tasks. Tag Sys User Description HELLO (a = x,b = y,...) ✓ ✓ Open a dialogue and give info a = x,b = y,... INFORM (a = x,b = y,...) ✓ ✓ Give info a = x,b = y,... REQUEST (a,b = x,...) ✓ ✓ Request value for a given b = x,... REQALTS (a = x,...) χ ✓ Request alternative with a = x,... CONFIRM (a = x,b = y,...) ✓ ✓ Explicitly conﬁrm a = x,b = y,... CONFREQ (a = x,...,d) ✓ χ Implicitly conﬁrm a = x,... and request value of d SELECT (a = x,a = y) ✓ χ Implicitly conﬁrm a = x,... and request value of d AFFIRM (a = x,b = y,...) ✓ ✓ Afﬁrm and give further info a = x,b = y,... NEGATE (a = x) χ ✓ Negate and give corrected value a = x DENY (a = x) χ ✓ Deny that a = x BYE () ✓ ✓ Close a dialogue Figure 15.6 Dialogue acts used by the HIS restaurant recommendation system of Young et al. (2010). The Sys and User columns indicate which acts are valid as system outputs and user inputs, respectively. Figure 15.6 shows a tagset for a restaurant recommendation system, and Fig. 15.7 shows these tags labeling a sample dialogue from the HIS system (Young et al., 2010). This example also shows the content of each dialogue act, which are the slot ﬁllers being communicated. So the user might INFORM the system that they want Italian food near a museum, orCONFIRM with the system that the price is reasonable. Utterance Dialogue act U: Hi, I am looking for somewhere to eat. hello(task = find,type=restaurant) S: You are looking for a restaurant. What type of food do you like? confreq(type = restaurant, food) U: I’d like an Italian near the museum. inform(food = Italian, near=museum) S: Roma is a nice Italian restaurant near the museum. inform(name = "Roma", type = restaurant, food = Italian, near = museum) U: Is it reasonably priced? confirm(pricerange = moderate) S: Yes, Roma is in the moderate price range. affirm(name = "Roma", pricerange = moderate) U: What is the phone number? request(phone) S: The number of Roma is 385456. inform(name = "Roma", phone = "385456") U: Ok, thank you goodbye. bye() Figure 15.7 A dialogue from the HIS System of Young et al. (2010) using the dialogue acts in Fig. 15.6. 15.3.2 Dialogue State Tracking The job of the dialogue-state tracker is to determine the current state of the frame (the ﬁllers of each slot), and the user’s most recent dialogue act. The dialogue-state is not just the slot-ﬁllers in the current sentence; it includes the entire state of the frame at this point, summarizing all of the user’s constraints. Fig. 15.8 from Mrkˇsi´c et al. (2017) shows the dialogue state after each turn. Dialogue act detection is done just like domain or intent classiﬁcation, by passing the input sentence through an encoder and adding an act classiﬁer. Often passing in the prior dialogue act as well can improve classiﬁcation. And since dialogue acts 15.3 • D IALOGUE ACTS AND DIALOGUE STATE 319 User: I’m looking for a cheaper restaurant inform(price=cheap) System: Sure. What kind - and where? User: Thai food, somewhere downtown inform(price=cheap, food=Thai, area=centre) System: The House serves cheap Thai food User: Where is it? inform(price=cheap, food=Thai, area=centre); request(address) System: The House is at 106 Regent Street Figure 15.8 The output of the dialogue state tracker after each turn (Mrkˇsi´c et al., 2017). place some constraints on the slots and values, the tasks of dialogue-act detection and slot-ﬁlling are often performed jointly. The state tracker can just take the output of a slot-ﬁlling sequence-model (Section 15.2.1) after each sentence, or do something more complicated like training a classiﬁer to decide if a value has been changed. A special case: detecting correction acts. If a dialogue system misrecognizes or misunderstands an utterance, users will repeat or reformulate the utterance. De- tecting these user correction acts is quite important, especially for spoken lan-user correction acts guage. Ironically, corrections are actuallyharder to recognize than normal sentences (Swerts et al., 2000), because users who are frustrated adjust their speech in a way that is difﬁcult for speech recognizers (Goldberg et al., 2003). For example speak- ers often use a prosodic style for corrections called hyperarticulation, in which thehyperarticula- tion utterance is louder or longer or exaggerated in pitch, such as I said BAL-TI-MORE, not Boston (Wade et al. 1992, Levow 1998, Hirschberg et al. 2001). Detecting acts can be part of the general dialogue act detection classiﬁer, or can make use of spe- cial features beyond the words, like those shown below (Levow 1998, Litman et al. 1999, Hirschberg et al. 2001, Bulyko et al. 2005, Awadallah et al. 2015). features examples semantic embedding similarity between correction and user’s prior utterance phonetic phonetic overlap between candidate correction act and user’s prior utterance (i.e. “WhatsApp” may be incorrectly recognized as “What’s up”) prosodic hyperarticulation, increases in F0 range, pause duration, and word duration ASR ASR conﬁdence, language model probability 15.3.3 Dialogue Policy: Which act to generate In early commercial frame-based systems, the dialogue policy is simple: ask ques- tions until all the slots are full, do a database query, then report back to the user. A more sophisticated dialogue policy can help a system decide when to answer thedialogue policy user’s questions, when to instead ask the user a clariﬁcation question, and so on. A dialogue policy thus decides what dialogue act to generate. Choosing a dialogue act to generate, along with its arguments, is sometimes called content planning.content planning Let’s see how to do this for some important dialogue acts. Dialogue systems, es- pecially speech systems, often misrecognize the users’ words or meaning. To ensure system and user share a common ground, systems mustconﬁrm understandings with the user or reject utterances that the system don’t understand. A system might use an explicit conﬁrmation act to conﬁrm with the user, like Is that correct? below:explicit conﬁrmation 320 CHAPTER 15 • C HATBOTS & DIALOGUE SYSTEMS U: I’d like to ﬂy from Denver Colorado to New York City on September twenty ﬁrst in the morning on United Airlines S: Let’s see then. I have you going from Denver Colorado to New York on September twenty ﬁrst. Is that correct? When using an implicit conﬁrmation act, a system instead grounds more im-implicit conﬁrmation plicitly, for example by repeating the system’s understanding as part of asking the next question, as Shanghai is conﬁrmed in passing in this example: U: I want to travel to to Shanghai S: When do you want to travel to Shanghai? There’s a tradeoff. Explicit conﬁrmation makes it easier for users to correct mis- recognitions by just answering “no” to the conﬁrmation question. But explicit con- ﬁrmation is time-consuming and awkward (Danieli and Gerbino 1995, Walker et al. 1998a). We also might want an act that expresses lack of understanding: rejection,rejection for example with a prompt likeI’m sorry, I didn’t understand that. To decide among these acts, we can make use of the fact that ASR systems often compute their conﬁ- dence in their transcription (often based on the log-likelihood the system assigns the sentence). A system can thus choose to explicitly conﬁrm only low-conﬁdence sen- tences. Or systems might have a four-tiered level of conﬁdence with three thresholds α, β, and γ: <α low conﬁdence reject ≥α above the threshold conﬁrm explicitly ≥β high conﬁdence conﬁrm implictly ≥γ very high conﬁdence don’t conﬁrm at all 15.3.4 Natural language generation: Sentence Realization recommend(restaurant name= Au Midi, neighborhood = midtown, cuisine = french) 1 Au Midi is in Midtown and serves French food. 2 There is a French restaurant in Midtown called Au Midi. Figure 15.9 Sample inputs to the sentence realization phase of NLG, showing the dialogue act and attributes prespeciﬁed by the content planner, and two distinct potential output sen- tences to be generated. From the restaurant recommendation system of Nayak et al. (2017). Once a dialogue act has been chosen, we need to generate the text of the re- sponse to the user. This part of the generation process is called sentence realiza- tion. Fig. 15.9 shows a sample input/output for the sentence realization phase. Thesentence realization content planner has chosen the dialogue act RECOMMEND and some slots (name, neighborhood, cuisine) and ﬁllers. The sentence realizer generates a sentence like lines 1 or 2 (by training on examples of representation/sentence pairs from a corpus of labeled dialogues). Because we won’t see every restaurant or attribute in every possible wording, we can delexicalize: generalize the training examples by replac-delexicalize ing speciﬁc slot value words in the training set with a generic placeholder token representing the slot. Fig. 15.10 shows the sentences in Fig. 15.9 delexicalized. We can map from frames to delexicalized sentences with an encoder decoder model (Mrkˇsi´c et al. 2017, inter alia), trained on hand-labeled dialogue corpora like MultiWOZ (Budzianowski et al., 2018). The input to the encoder is a sequence of 15.4 • C HATBOTS 321 recommend(restaurant name= Au Midi, neighborhood = midtown, cuisine = french) 1 restaurant nameis in neighborhoodand serves cuisinefood. 2 There is a cuisinerestaurant in neighborhoodcalled restaurant name. Figure 15.10 Delexicalized sentences that can be used for generating many different relex- icalized sentences. From the restaurant recommendation system of Nayak et al. (2017). decentservice:RECOMMEND cuisine: null [name] has decent service ENCODER DECODER Figure 15.11 An encoder decoder sentence realizer mapping slots/ﬁllers to English. tokens xt that represent the dialogue act (e.g.,RECOMMEND ) and its arguments (e.g., service:decent, cuisine:null) (Nayak et al., 2017), as in Fig. 15.11. The decoder outputs the delexicalized English sentence “ name has decent ser- vice”, which we can then relexicalize, i.e. ﬁll back in correct slot values, resultingrelexicalize in “Au Midi has decent service”. 15.4 Chatbots Chatbots are systems that can carry on extended conversations with the goal ofchatbot mimicking the unstructured conversations or ‘chats’ characteristic of informal human- human interaction. While early systems like ELIZA (Weizenbaum, 1966) or PARRY (Colby et al., 1971) had theoretical goals like testing theories of psychological coun- seling, for most of the last 50 years chatbots have been designed for entertainment. That changed with the recent rise of neural chatbots like ChatGPT, which incor- porate solutions to NLP tasks like question answering, writing tools, or machine translation into a conversational interface. A conversation with ChatGPT is shown in Fig. 15.12. In this section we describe neural chatbot architectures and datasets. [TBD] Figure 15.12 A conversation with ChatGPT. 15.4.1 Training chatbots Data Chatbots are generally trained on a training set that includes standard large language model training data of the type discussed in Section 10.3.2: versions of the web from the Common Crawl, including news sites, Wikipedia, as well as books. For training chatbots, it is common to additionally add lots of dialogue data. This can include datasets created speciﬁcally for training chatbots by hiring speakers of the language to have conversations, such as by having them take on personas or talk about knowledge provided to them. For example the Topical-Chat dataset has 11K crowdsourced conversations spanning 8 broad topics (Gopalakrish- nan et al., 2019), the E MPATHETIC DIALOGUES includes 25K crowdsourced con- 322 CHAPTER 15 • C HATBOTS & DIALOGUE SYSTEMS versations grounded in a speciﬁc situation where a speaker was feeling a speciﬁc emotion (Rashkin et al., 2019), and the SaFeRDialogues dataset (Ung et al., 2022) has 8k dialogues demonstrating graceful responses to conversational feedback about safety failures. Such datasets are far too small to train a language model alone, and so it’s com- mon to also pretrain on large datasets of pseudo-conversations drawn from Twitter (Ritter et al., 2010a), Reddit (Roller et al., 2021), Weibo ( 微博), and other social media platforms. To turn social media data into data that has the structure of a con- versation, we can treat any post on the platform as the ﬁrst turn in a conversation, and the sequence of comments/replies as subsequent turns in that conversation. Datasets from the web can be enormously toxic, so it’s crucial to ﬁlter the di- alogues ﬁrst. This can be done by using the same toxicity classiﬁers we describe below in the ﬁne-tuning section. Architecture For training chatbots, it’s most common to use the standard causal language model architecture, in which the model predicts each word given all the prior words, and the loss is the standard language modeling loss. Fig. 15.13 shows a standard training setup; no different than language model training in Chapter 9. The only difference is the data, which has the addition of signiﬁcant conversation and pseudo-conversation data as described in the prior section. As usual, the left context can include the entire prior conversation (or as much as ﬁts in the context window). Transformer Blocks LM head got promoted ! <s> got promoted ! <s>Next word Congrats LM Loss … LM head LM head LM head LM head LM head I Congrats ! … LM head LM head ! -log y!-log yCongrats-log y<s>-log y!-log ypromoted-log ygot … … … … … … Figure 15.13 Training a causal (decoder-only) language model for a chatbot. An alternative is to use the encoder-decoder architecture of Chapter 13. In this case the entire conversation up to the last turn (as much as ﬁts in the context) is presented to the encoder, and the decoder generates the next turn. promotedgot ! <s> Congrats ! ENCODER DECODER I Figure 15.14 An alternative: an encoder-decoder language model for a chatbot. In practice, dialogue systems require additional customization beyond just pre- training on dialogue data. In the next few sections we’ll discuss various stages of 15.4 • C HATBOTS 323 ﬁne-tuning that can be used for this customization. 15.4.2 Fine Tuning for Quality and Safety It is a common practice for dialogue systems to use further labeled data for ﬁne- tuning. One function of this ﬁne-tuning step is to improve the quality of the dialogue, training the system to produce responses that are sensible and interesting. Another function might be to improve safety, keeping a dialogue system from suggesting harmful actions (like ﬁnancial fraud, medical harm, inciting hatred, or abusing the user or other people). In the simplest method for improving quality and safety, speakers of the lan- guage are given an initial prompt and instructions to have high-quality, safe dia- logues. They then interact with an initial dialogue system and their responses are used to ﬁnetune the model, usually as part of the instruct tuning step we introduced in Chapter 12. Thus a dialogue system learns to answer questions, follow other instructions, and also carry on high-quality, safe dialogues, in a single multi-task learning format. While ﬁne-tuning on positive examples is helpful, it is generally insufﬁcient and so it is common to add more discriminative data that speciﬁcally downweights low- quality or harmful responses. The simplest paradigm for this is to train a model to predict turn-level safety and quality values, by training on human-labeled ratings. Such ratings might be collected by ﬁrst having speakers of the language carry on dialogues with a system, and then a second set of people act as labelers to label every system turn for its quality and safety, resulting in a binary label for quality and safety for each turn. Once a dataset has been created with these labels, a language model can be used in a classiﬁcation task to label the quality and safety of a turn. For example in the Lamda system (Cohen et al., 2022), a single language model is used in two phases, roughly corresponding to generative and discriminative tasks: ﬁrst generating a re- sponse, and then generating a label. In the generative phase, the model is given the prior turn and a special RESPONSE token and generates the blue response turn. (In training, the training loss is given only for the blue response): “What’s up? RESPONSE Not much. In a second, discriminative phase, the model is ﬁne-tuned to see an attribute (SENSIBLE, INTERESTING, UNSAFE) and then to predict a 0 or 1 value, again with training losses given only for the blue value. What’s up? RESPONSE Not much. SENSIBLE 1 What’s up? RESPONSE Not much. INTERESTING 0 What’s up? RESPONSE Not much. UNSAFE 0 To use the system in inference, the model ﬁrst generates a response given the context, and then it is given the attribute and asked to generate a rating. The result is a generated turn along with a label. This label isn’t shown to the user but can be use for ﬁltering, either at training time or at deployment time. For example, the system can generate multiple potential responses, ﬁlter out any response that is unsafe, and return to the user the highest ranking response. 324 CHAPTER 15 • C HATBOTS & DIALOGUE SYSTEMS 15.4.3 Learning to perform retrieval as part of responding Modern chatbots are also trained to do retrieval (calls to Google or Bing or other search engines) as a component of generating a response. This is generally done by having a sort of fake dialogue participant called something like Search Query whose utterances are passed to a search engine. The system can be encourged to do searches either by prompting or by ﬁne-tuning. For example in the Sparrow chatbot (Glaese et al., 2022), the prompt includes a few sample conversations that include two special users, Search Query, and Search Results: ...[this is preceded by other parts of the prompt] User: What color is the sky? Sparrow: Blue. User: Why? Search Query: why is sky blue Search Results: Page title: Sky Except for direct sunlight, most of the light in the daytime sky is caused by scattering, which is dominated by a small-particle limit called Rayleigh scattering. The scattering due to molecule-sized particles (as in air) is greater in the di- rections both toward and away from the source of light than it is in directions perpendicular to the incident path. Scattering is signiﬁcant for light at all visible wavelengths, but is stronger at the shorter (bluer) end of the visible spectrum, meaning that the scattered light is bluer than its source: the Sun. The remain- ing direct sunlight, having lost some of its shorter-wavelength components, appears slightly less blue. Sparrow: Because of the Rayleigh scattering. It causes short waves of blue light to be scattered more than the other colours in the spec- trum. User: Who was Raleigh? Search Query: who was Rayleigh ... ... From these prompts, the system learns to generate texts with Search Query turns for fact-based questions from the user, and these are passed to a search engine to generate the Search Resultsturns. Alternatively, systems can be ﬁnetuned to to know when to use a search en- gine. For example, labelers can interact with a system, fact check each of the re- sponses, and whenever the system emits an incorrect response, perform the web search queries that the system should have used to check its answer, and then the in- teration is recorded and used for ﬁne-tuning. Or labelers can look at a transcript of a language model carrying on a dialogue, and similarly mark every place where a fact was wrong (or out-of-date) and write the set of search queries that would have been appropriate. A system is then ﬁne-tuned to generate search query turns which are again passed to a search engine to generate the search responses. The set of pages or snippets returned by the search engine in the search response turn are then treated as the context for generation, similarly to the retrieval-based question- answering methods of Chapter 14. 15.5 • D IALOGUE SYSTEM DESIGN 325 15.4.4 RLHF A more sophisticated family of methods uses reinforcement learning to learn to match human preferences for generated turns. In this method, RLHF for Rein-RLHF forcement Learning from Human Feedback, we give a system a dialogue context and sample two possible turns from the language model. We then have humans la- bel which of the two is better, creating a large dataset of sentence pairs with human preferences. These pairs are used to train a dialogue policy, and reinforcement learn- ing is used to train the language model to generate turns that have higher rewards (Christiano et al., 2017; Ouyang et al., 2022). While using RLHF is the current state of the art at the time of this writing, a number of alternatives have been recently developed that don’t require reinforcement learning (Rafailov et al., 2023, e.g.,) and so this aspect of the ﬁeld is changing very quickly. 15.4.5 Evaluating Chatbots Chatbots are evaluated by humans, who assign a score. This can be the human who talked to the chatbot (participant evaluation) or a third party who reads a transcript of a human/chatbot conversation ( observer evaluation). In the participant evalua- tion of See et al. (2019), the human evaluator chats with the model for six turns and rates the chatbot on 8 dimensions capturing conversational quality: avoiding repe- tition, interestingness, making sense, ﬂuency, listening, inquisitiveness, humanness and engagingness on Likert scales like these: Engagingness How much did you enjoy talking to this user? •Not at all •A little •Somewhat •A lot Making sense How often did this user say something which did NOT make sense? •Never made any sense •Most responses didn’t make sense •Some re- sponses didn’t make sense •Everything made perfect sense Observer evaluations use third party annotators to look at the text of a complete conversation. Sometimes we’re interested in having raters assign a score to each system turn; for example (Artstein et al., 2009) have raters mark how coherent each turn is. Often, however, we just want a single high-level score to know if system A is better than system B. The acute-eval metric (Li et al., 2019a) is such an observeracute-eval evaluation in which annotators look at two separate human-computer conversations and choose the system which performed better on four metrics: engagingness, inter- estingness, humanness, and knowledgability. 15.5 Dialogue System Design Because of the important role of the user, the ﬁeld of dialogue systems is closely linked with Human-Computer Interaction (HCI). This is especially true for task- oriented dialogue and assistants, where the design of dialogue strategies, sometimes called voice user interface design, generally follows user-centered design princi-voice user interface ples (Gould and Lewis, 1985): 1. Study the user and task: Understand the users and the task by interviewing users, investigating similar systems, and studying related human-human dialogues. 2. Build simulations and prototypes: A crucial tool in building dialogue systems is the Wizard-of-Oz system. In wizard systems, the users interact with what theyWizard-of-Oz system 326 CHAPTER 15 • C HATBOTS & DIALOGUE SYSTEMS think is a program but is in fact a human “wizard” disguised by a software interface (Gould et al. 1983, Good et al. 1984, Fraser and Gilbert 1991). The name comes from the chil- dren’s book The Wizard of Oz (Baum, 1900), in which the wizard turned out to be a simu- lation controlled by a man behind a curtain or screen. A wizard system can be used to test out an architecture before implementation; only the interface software and databases need to be in place. The wizard gets input from the user, uses a database interface to run queries based on the user utterance, and then outputs sentences, ei- ther by typing them or speaking them. Wizard-of-Oz systems are not a perfect simulation, since the wizard doesn’t exactly simulate the errors or limitations of a real sys- tem; but wizard studies can still provide a useful ﬁrst idea of the domain issues. 3. Iteratively test the design on users: An iterative design cycle with embedded user testing is essential in system design (Nielsen 1992, Cole et al. 1997, Yankelovich et al. 1995, Landauer 1995). For example in a well-known incident, an early dia- logue system required the user to press a key to interrupt the system (Stifelman et al., 1993). But user testing showed users barged in (interrupted, talking over the sys-barged in tem), which led to a redesign of the system to recognize overlapped speech. It’s also important to incorporate value sensitive design, in which we carefully consider dur-value sensitive design ing the design process the beneﬁts, harms and possible stakeholders of the resulting system (Friedman et al. 2017, Friedman and Hendry 2019). 15.5.1 Ethical Issues in Dialogue System Design Ethical issues have been key to how we think about designing artiﬁcial agents since well before we had dialogue systems. Mary Shelley (depicted below) centered her novel Frankensteinaround the problem of creating artiﬁcial agents without consider- ing ethical and humanistic concerns. One issue is the safety of users. If users seek information from di- alogue systems in safety-critical situations like ask- ing medical advice, or in emergency situations, or when indicating the intentions of self-harm, incorrect advice can be dangerous and even life-threatening. For example (Bickmore et al., 2018) gave participants medical problems to pose to three commercial di- alogue systems (Siri, Alexa, Google Assistant) and asked them to determine an action to take based on the system responses; many of the proposed actions, if actually taken, would have led to harm or death. A system can also harm users by verbally attacking them, or creating represen- tational harms (Blodgett et al., 2020) by generating abusive or harmful stereotypes that demean particular groups of people. Both abuse and stereotypes can cause psy- chological harm to users. Microsoft’s 2016 Tay chatbot, for example, was takenTay ofﬂine 16 hours after it went live, when it began posting messages with racial slurs, 15.6 • S UMMARY 327 conspiracy theories, and personal attacks on its users. Tay had learned these biases and actions from its training data, including from users who seemed to be purposely teaching the system to repeat this kind of language (Neff and Nagy 2016). Hender- son et al. (2017) examined dialogue datasets used to train corpus-based chatbots and found toxic and abusive language, especially in social media corpora like Twitter and Reddit, and indeed such language then appears in the text generated by lan- guage models and dialogue systems (Gehman et al. 2020; Xu et al. 2020) which can even amplify the bias from the training data (Dinan et al., 2020). Liu et al. (2020) developed another method for investigating bias, testing how neural dialogue systems responded to pairs of simulated user turns that are identical except for men- tioning different genders or race. They found, for example, that simple changes like using the word ‘she’ instead of ‘he’ in a sentence caused systems to respond more offensively and with more negative sentiment. Another important ethical issue is privacy. Already in the ﬁrst days of ELIZA, Weizenbaum pointed out the privacy implications of people’s revelations to the chat- bot. The ubiquity of in-home dialogue systems means they may often overhear private information (Henderson et al., 2017). If a chatbot is human-like, users are also more likely to disclose private information, and less likely to worry about the harm of this disclosure (Ischen et al., 2019). In general, chatbots that are trained on transcripts of human-human or human-machine conversation must anonymize personally identiﬁable information. Finally, chatbots raise important issues of gender equality in addition to textual bias. Current chatbots are overwhelmingly given female names, likely perpetuating the stereotype of a subservient female servant (Paolino, 2017). And when users use sexually harassing language, most commercial chatbots evade or give positive responses rather than responding in clear negative ways (Fessler, 2017). These ethical issues are an important area of investigation, including ﬁnding ways to mitigate problems of abuse and toxicity, like detecting and responding ap- propriately to toxic contexts (Wolf et al. 2017, Dinan et al. 2020, Xu et al. 2020). Value sensitive design, carefully considering possible harms in advance (Friedman et al. 2017, Friedman and Hendry 2019) is also important; (Dinan et al., 2021) give a number of suggestions for best practices in dialogue system design. For exam- ple getting informed consent from participants, whether they are used for training, or whether they are interacting with a deployed system is important. Because di- alogue systems by deﬁnition involve human participants, researchers also work on these issues with the Institutional Review Boards ( IRB) at their institutions, whoIRB help protect the safety of experimental subjects. 15.6 Summary Chatbots and dialogue systems are crucial speech and language processing appli- cations that are already widely used commercially. • In human dialogue, speaking is a kind of action; these acts are referred to as speech acts or dialogue acts. Speakers also attempt to achieve common ground by acknowledging that they have understand each other. Conversation also is characterized by turn structure and dialogue structure. • Chatbots are conversational systems designed to mimic the appearance of in- formal human conversation. Rule-based chatbots like ELIZA and its modern 328 CHAPTER 15 • C HATBOTS & DIALOGUE SYSTEMS descendants use rules to map user sentences into system responses. Corpus- based chatbots mine logs of human conversation to learn to automatically map user sentences into system responses. • For task-based dialogue, most commercial dialogue systems use the GUS or frame-based architecture, in which the designer speciﬁes frames consisting of slots that the system must ﬁll by asking the user. • The dialogue-state architecture augments the GUS frame-and-slot architec- ture with richer representations and more sophisticated algorithms for keeping track of user’s dialogue acts,policies for generating its own dialogue acts, and a natural language component. • Dialogue systems are a kind of human-computer interaction, and general HCI principles apply in their design, including the role of the user, simulations such as Wizard-of-Oz systems, and the importance of iterative design and testing on real users. Bibliographical and Historical Notes The linguistic, philosophical, and psychological literature on dialogue is quite ex- tensive. For example the idea that utterances in a conversation are a kind of action being performed by the speaker was due originally to the philosopher Wittgenstein (1953) but worked out more fully by Austin (1962) and his student John Searle. Various sets of speech acts have been deﬁned over the years, and a rich linguistic and philosophical literature developed, especially focused on explaining the use of indirect speech acts. The idea of dialogue acts draws also from a number of other sources, including the ideas of adjacency pairs, pre-sequences, and other aspects of the interactional properties of human conversation developed in the ﬁeld of conver- sation analysis (see Levinson (1983) for an introduction to the ﬁeld). This idea thatconversation analysis acts set up strong local dialogue expectations was also preﬁgured by Firth (1935, p. 70), in a famous quotation: Most of the give-and-take of conversation in our everyday life is stereotyped and very narrowly conditioned by our particular type of culture. It is a sort of roughly prescribed social ritual, in which you generally say what the other fellow expects you, one way or the other, to say. Another important research thread modeled dialogue as a kind of collaborative behavior, including the ideas of common ground (Clark and Marshall, 1981), ref- erence as a collaborative process (Clark and Wilkes-Gibbs, 1986), joint intention (Levesque et al., 1990), and shared plans (Grosz and Sidner, 1980). The earliest conversational systems were simple pattern-action chatbots like ELIZA (Weizenbaum, 1966). ELIZA had a widespread inﬂuence on popular perceptions of artiﬁcial intelligence, and brought up some of the ﬁrst ethical questions in natural language processing —such as the issues of privacy we discussed above as well the role of algorithms in decision-making— leading its creator Joseph Weizenbaum to ﬁght for social responsibility in AI and computer science in general. Computational-implemented theories of dialogue blossomed in the 1970. That period saw the very inﬂuential GUS system (Bobrow et al., 1977), which in the late 1970s established the frame-based paradigm that became the dominant industrial paradigm for dialogue systems for over 30 years. BIBLIOGRAPHICAL AND HISTORICAL NOTES 329 Another inﬂuential line of research from that decade focused on modeling the hi- erarchical structure of dialogue. Grosz’s pioneering 1977b dissertation ﬁrst showed that “task-oriented dialogues have a structure that closely parallels the structure of the task being performed” (p. 27), leading to her work with Sidner and others show- ing how to use similar notions of intention and plans to model discourse structure and coherence in dialogue. See, e.g., Lochbaum et al. (2000) for a summary of the role of intentional structure in dialogue. Yet a third line, ﬁrst suggested by Bruce (1975), suggested that since speech acts are actions, they should be planned like other actions, and drew on the AI planning literature (Fikes and Nilsson, 1971). A system seeking to ﬁnd out some information can come up with the plan of asking the interlocutor for the information. A system hearing an utterance can interpret a speech act by running the planner “in reverse”, using inference rules to infer from what the interlocutor said what the plan might have been. Plan-based models of dialogue are referred to as BDI models becauseBDI such planners model the beliefs, desires, and intentions (BDI) of the system and in- terlocutor. BDI models of dialogue were ﬁrst introduced by Allen, Cohen, Perrault, and their colleagues in a number of inﬂuential papers showing how speech acts could be generated (Cohen and Perrault, 1979) and interpreted (Perrault and Allen 1980, Allen and Perrault 1980). At the same time, Wilensky (1983) introduced plan-based models of understanding as part of the task of interpreting stories. In the 1990s, machine learning models that had ﬁrst been applied to natural language processing began to be applied to dialogue tasks like slot ﬁlling (Miller et al. 1994, Pieraccini et al. 1991). This period also saw lots of analytic work on the linguistic properties of dialogue acts and on machine-learning-based methods for their detection. (Sag and Liberman 1975, Hinkelman and Allen 1989, Nagata and Morimoto 1994, Goodwin 1996, Chu-Carroll 1998, Shriberg et al. 1998, Stolcke et al. 2000, Gravano et al. 2012. This work strongly informed the development of the dialogue-state model (Larsson and Traum, 2000). Dialogue state tracking quickly became an important problem for task-oriented dialogue, and there has been an inﬂuential annual evaluation of state-tracking algorithms (Williams et al., 2016). The turn of the century saw a line of work on applying reinforcement learning to dialogue, which ﬁrst came out of AT&T and Bell Laboratories with work on MDP dialogue systems (Walker 2000, Levin et al. 2000, Singh et al. 2002) along with work on cue phrases, prosody, and rejection and conﬁrmation. Reinforcement learning research turned quickly to the more sophisticated POMDP models (Roy et al. 2000, Lemon et al. 2006, Williams and Young 2007) applied to small slot- ﬁlling dialogue tasks. Neural reinforcement learning models have been used both for chatbot systems, for example simulating dialogues between two dialogue systems, rewarding good conversational properties like coherence and ease of answering (Li et al., 2016a), and for task-oriented dialogue (Williams et al., 2017). By around 2010 the GUS architecture ﬁnally began to be widely used commer- cially in dialogue systems on phones like Apple’s SIRI (Bellegarda, 2013) and other digital assistants. The rise of the web gave rise to corpus-based chatbot architectures around the turn of the century, ﬁrst using information retrieval models and then in the 2010s, after the rise of deep learning, with sequence-to-sequence models. [TBD: Modern history of neural chatbots] Other important dialogue areas include the study of affect in dialogue (Rashkin et al. 2019, Lin et al. 2019) and conversational interface design (Cohen et al. 2004, Harris 2005, Pearl 2017, Deibel and Evanhoe 2021). 330 CHAPTER 15 • C HATBOTS & DIALOGUE SYSTEMS Exercises 15.1 Write a ﬁnite-state automaton for a dialogue manager for checking your bank balance and withdrawing money at an automated teller machine. 15.2 A dispreferred response is a response that has the potential to make a persondispreferred response uncomfortable or embarrassed in the conversational context; the most com- mon example dispreferred responses is turning down a request. People signal their discomfort with having to say no with surface cues (like the word well), or via signiﬁcant silence. Try to notice the next time you or someone else utters a dispreferred response, and write down the utterance. What are some other cues in the response that a system might use to detect a dispreferred response? Consider non-verbal cues like eye gaze and body gestures. 15.3 When asked a question to which they aren’t sure they know the answer, peo- ple display their lack of conﬁdence by cues that resemble other dispreferred responses. Try to notice some unsure answers to questions. What are some of the cues? If you have trouble doing this, read Smith and Clark (1993) and listen speciﬁcally for the cues they mention. 15.4 Implement a small air-travel help system based on text input. Your system should get constraints from users about a particular ﬂight that they want to take, expressed in natural language, and display possible ﬂights on a screen. Make simplifying assumptions. You may build in a simple ﬂight database or you may use a ﬂight information system on the Web as your backend. CHAPTER 16 Automatic Speech Recognition and Text-to-Speech I KNOW not whether I see your meaning: if I do, it lies Upon the wordy wavelets of your voice, Dim as an evening shadow in a brook, Thomas Lovell Beddoes, 1851 Understanding spoken language, or at least transcribing the words into writing, is one of the earliest goals of computer language processing. In fact, speech processing predates the computer by many decades! The ﬁrst machine that recognized speech was a toy from the 1920s. “Radio Rex”, shown to the right, was a celluloid dog that moved (by means of a spring) when the spring was released by 500 Hz acous- tic energy. Since 500 Hz is roughly the ﬁrst formant of the vowel [eh] in “Rex”, Rex seemed to come when he was called (David, Jr. and Selfridge, 1962). In modern times, we expect more of our automatic systems. The task of auto- matic speech recognition (ASR) is to map any waveform like this:ASR to the appropriate string of words: It’s time for lunch! Automatic transcription of speech by any speaker in any environment is still far from solved, but ASR technology has matured to the point where it is now viable for many practical tasks. Speech is a natural interface for communicating with smart home ap- pliances, personal assistants, or cellphones, where keyboards are less convenient, in telephony applications like call-routing (“Accounting, please”) or in sophisticated dialogue applications (“I’d like to change the return date of my ﬂight”). ASR is also useful for general transcription, for example for automatically generating captions for audio or video text (transcribing movies or videos or live discussions). Transcrip- tion is important in ﬁelds like law where dictation plays an important role. Finally, ASR is important as part of augmentative communication (interaction between com- puters and humans with some disability resulting in difﬁculties or inabilities in typ- ing or audition). The blind Milton famously dictated Paradise Lost to his daughters, and Henry James dictated his later novels after a repetitive stress injury. What about the opposite problem, going from text to speech? This is a problem with an even longer history. In Vienna in 1769, Wolfgang von Kempelen built for 332 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH the Empress Maria Theresa the famous Mechanical Turk, a chess-playing automaton consisting of a wooden box ﬁlled with gears, behind which sat a robot mannequin who played chess by moving pieces with his mechanical arm. The Turk toured Eu- rope and the Americas for decades, defeating Napoleon Bonaparte and even playing Charles Babbage. The Mechanical Turk might have been one of the early successes of artiﬁcial intelligence were it not for the fact that it was, alas, a hoax, powered by a human chess player hidden inside the box. What is less well known is that von Kempelen, an extraordinarily proliﬁc inventor, also built between 1769 and 1790 what was deﬁnitely not a hoax: the ﬁrst full-sentence speech synthesizer, shown partially to the right. His device consisted of a bellows to simulate the lungs, a rub- ber mouthpiece and a nose aperture, a reed to simulate the vocal folds, var- ious whistles for the fricatives, and a small auxiliary bellows to provide the puff of air for plosives. By moving levers with both hands to open and close apertures, and adjusting the ﬂexible leather “vo- cal tract”, an operator could produce different consonants and vowels. More than two centuries later, we no longer build our synthesizers out of wood and leather, nor do we need human operators. The modern task ofspeech synthesis,speech synthesis also called text-to-speech or TTS, is exactly the reverse of ASR; to map text:text-to-speech TTS It’s time for lunch! to an acoustic waveform: Modern speech synthesis has a wide variety of applications. TTS is used in conversational agents that conduct dialogues with people, plays a role in devices that read out loud for the blind or in games, and can be used to speak for sufferers of neurological disorders, such as the late astrophysicist Steven Hawking who, after he lost the use of his voice because of ALS, spoke by manipulating a TTS system. In the next sections we’ll show how to do ASR with encoder-decoders, intro- duce the CTC loss functions, the standard word error rate evaluation metric, and describe how acoustic features are extracted. We’ll then see how TTS can be mod- eled with almost the same algorithm in reverse, and conclude with a brief mention of other speech tasks. 16.1 The Automatic Speech Recognition Task Before describing algorithms for ASR, let’s talk about how the task itself varies. One dimension of variation is vocabulary size. Some ASR tasks can be solved with extremely high accuracy, like those with a 2-word vocabulary ( yes versus no) or an 11 word vocabulary like digit recognition (recognizing sequences of digits in-digit recognition cluding zero to nine plus oh). Open-ended tasks like transcribing videos or human conversations, with large vocabularies of up to 60,000 words, are much harder. 16.1 • T HE AUTOMATIC SPEECH RECOGNITION TASK 333 A second dimension of variation is who the speaker is talking to. Humans speak- ing to machines (either dictating or talking to a dialogue system) are easier to recog- nize than humans speaking to humans. Read speech, in which humans are readingread speech out loud, for example in audio books, is also relatively easy to recognize. Recog- nizing the speech of two humans talking to each other in conversational speech,conversational speech for example, for transcribing a business meeting, is the hardest. It seems that when humans talk to machines, or read without an audience present, they simplify their speech quite a bit, talking more slowly and more clearly. A third dimension of variation is channel and noise. Speech is easier to recognize if it’s recorded in a quiet room with head-mounted microphones than if it’s recorded by a distant microphone on a noisy city street, or in a car with the window open. A ﬁnal dimension of variation is accent or speaker-class characteristics. Speech is easier to recognize if the speaker is speaking the same dialect or variety that the system was trained on. Speech by speakers of regional or ethnic dialects, or speech by children can be quite difﬁcult to recognize if the system is only trained on speak- ers of standard dialects, or only adult speakers. A number of publicly available corpora with human-created transcripts are used to create ASR test and training sets to explore this variation; we mention a few of them here since you will encounter them in the literature. LibriSpeech is a largeLibriSpeech open-source read-speech 16 kHz dataset with over 1000 hours of audio books from the LibriV ox project, with transcripts aligned at the sentence level (Panayotov et al., 2015). It is divided into an easier (“clean”) and a more difﬁcult portion (“other”) with the clean portion of higher recording quality and with accents closer to US English. This was done by running a speech recognizer (trained on read speech from the Wall Street Journal) on all the audio, computing the WER for each speaker based on the gold transcripts, and dividing the speakers roughly in half, with recordings from lower-WER speakers called “clean” and recordings from higher-WER speakers “other”. The Switchboard corpus of prompted telephone conversations between strangersSwitchboard was collected in the early 1990s; it contains 2430 conversations averaging 6 min- utes each, totaling 240 hours of 8 kHz speech and about 3 million words (Godfrey et al., 1992). Switchboard has the singular advantage of an enormous amount of auxiliary hand-done linguistic labeling, including parses, dialogue act tags, phonetic and prosodic labeling, and discourse and information structure. The CALLHOMECALLHOME corpus was collected in the late 1990s and consists of 120 unscripted 30-minute telephone conversations between native speakers of English who were usually close friends or family (Canavan et al., 1997). The Santa Barbara Corpus of Spoken American English (Du Bois et al., 2005) is a large corpus of naturally occurring everyday spoken interactions from all over the United States, mostly face-to-face conversation, but also town-hall meetings, food preparation, on-the-job talk, and classroom lectures. The corpus was anonymized by removing personal names and other identifying information (replaced by pseudonyms in the transcripts, and masked in the audio). CORAAL is a collection of over 150 sociolinguistic interviews with AfricanCORAAL American speakers, with the goal of studying African American Language ( AAL), the many variations of language used in African American communities (Kendall and Farrington, 2020). The interviews are anonymized with transcripts aligned at the utterance level. The CHiME Challenge is a series of difﬁcult shared tasks withCHiME corpora that deal with robustness in ASR. The CHiME 5 task, for example, is ASR of conversational speech in real home environments (speciﬁcally dinner parties). The 334 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH corpus contains recordings of twenty different dinner parties in real homes, each with four participants, and in three locations (kitchen, dining area, living room), recorded both with distant room microphones and with body-worn mikes. The HKUST Mandarin Telephone Speech corpus has 1206 ten-minute telephone con-HKUST versations between speakers of Mandarin across China, including transcripts of the conversations, which are between either friends or strangers (Liu et al., 2006). The AISHELL-1 corpus contains 170 hours of Mandarin read speech of sentences takenAISHELL-1 from various domains, read by different speakers mainly from northern China (Bu et al., 2017). Figure 16.1 shows the rough percentage of incorrect words (theword error rate, or WER, deﬁned on page 346) from state-of-the-art systems on some of these tasks. Note that the error rate on read speech (like the LibriSpeech audiobook corpus) is around 2%; this is a solved task, although these numbers come from systems that re- quire enormous computational resources. By contrast, the error rate for transcribing conversations between humans is much higher; 5.8 to 11% for the Switchboard and CALLHOME corpora. The error rate is higher yet again for speakers of varieties like African American Vernacular English, and yet again for difﬁcult conversational tasks like transcription of 4-speaker dinner party speech, which can have error rates as high as 81.3%. Character error rates (CER) are also much lower for read Man- darin speech than for natural conversation. English Tasks WER % LibriSpeech audiobooks 960hour clean 1.4 LibriSpeech audiobooks 960hour other 2.6 Switchboard telephone conversations between strangers 5.8 CALLHOME telephone conversations between family 11.0 Sociolinguistic interviews, CORAAL (AAL) 27.0 CHiMe5 dinner parties with body-worn microphones 47.9 CHiMe5 dinner parties with distant microphones 81.3 Chinese (Mandarin) Tasks CER % AISHELL-1 Mandarin read speech corpus 6.7 HKUST Mandarin Chinese telephone conversations 23.5 Figure 16.1 Rough Word Error Rates (WER = % of words misrecognized) reported around 2020 for ASR on various American English recognition tasks, and character error rates (CER) for two Chinese recognition tasks. 16.2 Feature Extraction for ASR: Log Mel Spectrum The ﬁrst step in ASR is to transform the input waveform into a sequence of acoustic feature vectors, each vector representing the information in a small time windowfeature vector of the signal. Let’s see how to convert a raw waveﬁle to the most commonly used features, sequences of log mel spectrum vectors. A speech signal processing course is recommended for more details. 16.2.1 Sampling and Quantization The input to a speech recognizer is a complex series of changes in air pressure. These changes in air pressure obviously originate with the speaker and are caused 16.2 • F EATURE EXTRACTION FOR ASR: L OG MEL SPECTRUM 335 by the speciﬁc way that air passes through the glottis and out the oral or nasal cav- ities. We represent sound waves by plotting the change in air pressure over time. One metaphor which sometimes helps in understanding these graphs is that of a ver- tical plate blocking the air pressure waves (perhaps in a microphone in front of a speaker’s mouth, or the eardrum in a hearer’s ear). The graph measures the amount of compression or rarefaction (uncompression) of the air molecules at this plate. Figure 16.2 shows a short segment of a waveform taken from the Switchboard corpus of telephone speech of the vowel [iy] from someone saying “she just had a baby”. Time (s) 0 0.03875 –0.01697 0.02283 0 Figure 16.2 A waveform of an instance of the vowel [iy] (the last vowel in the word “baby”). The y-axis shows the level of air pressure above and below normal atmospheric pressure. The x-axis shows time. Notice that the wave repeats regularly. The ﬁrst step in digitizing a sound wave like Fig. 16.2 is to convert the analog representations (ﬁrst air pressure and then analog electric signals in a microphone) into a digital signal. Thisanalog-to-digital conversionhas two steps: sampling andsampling quantization. To sample a signal, we measure its amplitude at a particular time; the sampling rate is the number of samples taken per second. To accurately measure a wave, we must have at least two samples in each cycle: one measuring the positive part of the wave and one measuring the negative part. More than two samples per cycle increases the amplitude accuracy, but fewer than two samples causes the fre- quency of the wave to be completely missed. Thus, the maximum frequency wave that can be measured is one whose frequency is half the sample rate (since every cycle needs two samples). This maximum frequency for a given sampling rate is called the Nyquist frequency. Most information in human speech is in frequenciesNyquist frequency below 10,000 Hz; thus, a 20,000 Hz sampling rate would be necessary for com- plete accuracy. But telephone speech is ﬁltered by the switching network, and only frequencies less than 4,000 Hz are transmitted by telephones. Thus, an 8,000 Hz sampling rate is sufﬁcient for telephone-bandwidth speech like the Switchboard corpus, while 16,000 Hz sampling is often used for microphone speech. Although using higher sampling rates produces higher ASR accuracy, we can’t combine different sampling rates for training and testing ASR systems. Thus if we are testing on a telephone corpus like Switchboard (8 KHz sampling), we must downsample our training corpus to 8 KHz. Similarly, if we are training on mul- tiple corpora and one of them includes telephone speech, we downsample all the wideband corpora to 8Khz. Amplitude measurements are stored as integers, either 8 bit (values from -128– 127) or 16 bit (values from -32768–32767). This process of representing real-valued numbers as integers is called quantization; all values that are closer together thanquantization the minimum granularity (the quantum size) are represented identically. We refer to each sample at time index n in the digitized, quantized waveform as x[n]. Once data is quantized, it is stored in various formats. One parameter of these formats is the sample rate and sample size discussed above; telephone speech is often sampled at 8 kHz and stored as 8-bit samples, and microphone data is often sampled at 16 kHz and stored as 16-bit samples. Another parameter is the number of 336 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH channels. For stereo data or for two-party conversations, we can store both channelschannel in the same ﬁle or we can store them in separate ﬁles. A ﬁnal parameter is individual sample storage—linearly or compressed. One common compression format used for telephone speech is µ-law (often written u-law but still pronounced mu-law). The intuition of log compression algorithms like µ-law is that human hearing is more sensitive at small intensities than large ones; the log represents small values with more faithfulness at the expense of more error on large values. The linear (unlogged) values are generally referred to as linear PCM values (PCM stands for pulse codePCM modulation, but never mind that). Here’s the equation for compressing a linear PCM sample value x to 8-bit µ-law, (where µ=255 for 8 bits): F(x) = sgn(x)log(1 + µ\|x\|) log(1 + µ) −1 ≤x ≤1 (16.1) There are a number of standard ﬁle formats for storing the resulting digitized wave- ﬁle, such as Microsoft’s .wav and Apple’s AIFF all of which have special headers; simple headerless “raw” ﬁles are also used. For example, the .wav format is a sub- set of Microsoft’s RIFF format for multimedia ﬁles; RIFF is a general format that can represent a series of nested chunks of data and control information. Figure 16.3 shows a simple .wav ﬁle with a single data chunk together with its format chunk. Figure 16.3 Microsoft waveﬁle header format, assuming simple ﬁle with one chunk. Fol- lowing this 44-byte header would be the data chunk. 16.2.2 Windowing From the digitized, quantized representation of the waveform, we need to extract spectral features from a small window of speech that characterizes part of a par- ticular phoneme. Inside this small window, we can roughly think of the signal as stationary (that is, its statistical properties are constant within this region). (Bystationary contrast, in general, speech is a non-stationary signal, meaning that its statisticalnon-stationary properties are not constant over time). We extract this roughly stationary portion of speech by using a window which is non-zero inside a region and zero elsewhere, run- ning this window across the speech signal and multiplying it by the input waveform to produce a windowed waveform. The speech extracted from each window is called a frame. The windowing isframe characterized by three parameters: the window size or frame size of the window (its width in milliseconds), the frame stride, (also called shift or offset) betweenstride successive windows, and the shape of the window. To extract the signal we multiply the value of the signal at time n, s[n] by the value of the window at time n, w[n]: y[n] =w[n]s[n] (16.2) The window shape sketched in Fig. 16.4 is rectangular; you can see the ex-rectangular tracted windowed signal looks just like the original signal. The rectangular window, 16.2 • F EATURE EXTRACTION FOR ASR: L OG MEL SPECTRUM 337 Shift 10 ms Window 25 ms Shift 10 ms Window 25 ms Window 25 ms Figure 16.4 Windowing, showing a 25 ms rectangular window with a 10ms stride. however, abruptly cuts off the signal at its boundaries, which creates problems when we do Fourier analysis. For this reason, for acoustic feature creation we more com- monly use the Hamming window, which shrinks the values of the signal towardHamming zero at the window boundaries, avoiding discontinuities. Figure 16.5 shows both; the equations are as follows (assuming a window that is L frames long): rectangular w [n] = {1 0 ≤n ≤L −1 0 otherwise (16.3) Hamming w [n] = { 0.54 −0.46cos(2πn L ) 0 ≤n ≤L −1 0 otherwise (16.4) Time (s) 0 0.0475896 –0.5 0.4999 0 Rectangular window Hamming window Time (s) 0.00455938 0.0256563 –0.4826 0.4999 0 Time (s) 0.00455938 0.0256563 –0.5 0.4999 0 Figure 16.5 Windowing a sine wave with the rectangular or Hamming windows. 16.2.3 Discrete Fourier Transform The next step is to extract spectral information for our windowed signal; we need to know how much energy the signal contains at different frequency bands. The tool 338 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH for extracting spectral information for discrete frequency bands for a discrete-time (sampled) signal is the discrete Fourier transform or DFT. Discrete Fourier transformDFT The input to the DFT is a windowed signal x[n]...x[m], and the output, for each of N discrete frequency bands, is a complex number X[k] representing the magni- tude and phase of that frequency component in the original signal. If we plot the magnitude against the frequency, we can visualize the spectrum (see Appendix H for more on spectra). For example, Fig. 16.6 shows a 25 ms Hamming-windowed portion of a signal and its spectrum as computed by a DFT (with some additional smoothing). Time (s) 0.0141752 0.039295 –0.04121 0.04414 0 Frequency (Hz) 0 8000 Sound pressure level (dB /Hz) –20 0 20 (a) (b) Figure 16.6 (a) A 25 ms Hamming-windowed portion of a signal from the vowel [iy] and (b) its spectrum computed by a DFT. We do not introduce the mathematical details of the DFT here, except to note that Fourier analysis relies on Euler’s formula, with j as the imaginary unit:Euler’s formula ejθ = cosθ + j sinθ (16.5) As a brief reminder for those students who have already studied signal processing, the DFT is deﬁned as follows: X[k] = N−1∑ n=0 x[n]e−j 2π N kn (16.6) A commonly used algorithm for computing the DFT is the fast Fourier transformfast Fourier transform or FFT. This implementation of the DFT is very efﬁcient but only works for valuesFFT of N that are powers of 2. 16.2.4 Mel Filter Bank and Log The results of the FFT tell us the energy at each frequency band. Human hearing, however, is not equally sensitive at all frequency bands; it is less sensitive at higher frequencies. This bias toward low frequencies helps human recognition, since in- formation in low frequencies (like formants) is crucial for distinguishing vowels or nasals, while information in high frequencies (like stop bursts or fricative noise) is less crucial for successful recognition. Modeling this human perceptual property improves speech recognition performance in the same way. We implement this intuition by collecting energies, not equally at each frequency band, but according to the mel scale, an auditory frequency scale. A mel (Stevensmel et al. 1937, Stevens and V olkmann 1940) is a unit of pitch. Pairs of sounds that are perceptually equidistant in pitch are separated by an equal number of mels. The mel 16.3 • S PEECH RECOGNITION ARCHITECTURE 339 frequency m can be computed from the raw acoustic frequency by a log transforma- tion: mel( f ) =1127ln(1 + f 700) (16.7) We implement this intuition by creating a bank of ﬁlters that collect energy from each frequency band, spread logarithmically so that we have very ﬁne resolution at low frequencies, and less resolution at high frequencies. Figure 16.7 shows a sample bank of triangular ﬁlters that implement this idea, that can be multiplied by the spectrum to get a mel spectrum. m1 m2 mM...mel spectrum 0 77000 0.5 1Amplitude Frequency (Hz)8K Figure 16.7 The mel ﬁlter bank (Davis and Mermelstein, 1980). Each triangular ﬁlter, spaced logarithmically along the mel scale, collects energy from a given frequency range. Finally, we take the log of each of the mel spectrum values. The human response to signal level is logarithmic (like the human response to frequency). Humans are less sensitive to slight differences in amplitude at high amplitudes than at low ampli- tudes. In addition, using a log makes the feature estimates less sensitive to variations in input such as power variations due to the speaker’s mouth moving closer or further from the microphone. 16.3 Speech Recognition Architecture The basic architecture for ASR is the encoder-decoder (implemented with either RNNs or Transformers), exactly the same architecture introduced for MT in Chap- ter 13. Generally we start from the log mel spectral features described in the previous section, and map to letters, although it’s also possible to map to induced morpheme- like chunks like wordpieces or BPE. Fig. 16.8 sketches the standard encoder-decoder architecture, which is com- monly referred to as theattention-based encoder decoderor AED, or listen attendAED and spell (LAS) after the two papers which ﬁrst applied it to speech (Chorowskilisten attend and spell et al. 2014, Chan et al. 2016). The input is a sequence of t acoustic feature vectors F = f1,f2,..., ft , one vector per 10 ms frame. The output can be letters or word- pieces; we’ll assume letters here. Thus the output sequenceY = (⟨SOS⟩,y1,...,ym⟨EOS⟩), assuming special start of sequence and end of sequence tokens ⟨sos⟩and ⟨eos⟩and each yi is a character; for English we might choose the set: yi ∈{a,b,c,...,z,0,...,9,⟨space⟩,⟨comma⟩,⟨period⟩,⟨apostrophe⟩,⟨unk⟩} Of course the encoder-decoder architecture is particularly appropriate when in- put and output sequences have stark length differences, as they do for speech, with very long acoustic feature sequences mapping to much shorter sequences of letters 340 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH ENCODER … DECODER … … ym Feature Computation Subsampling … H ftf1 80-dimensional log Mel spectrum per frame Shorter sequence X y1 <s> i y2 i t y3 t ‘ y4 ‘ s y5 s y6 t y7 t i y8 i m y9 m e x1 xn Figure 16.8 Schematic architecture for an encoder-decoder speech recognizer. or words. A single word might be 5 letters long but, supposing it lasts about 2 seconds, would take 200 acoustic frames (of 10ms each). Because this length difference is so extreme for speech, encoder-decoder ar- chitectures for speech need to have a special compression stage that shortens the acoustic feature sequence before the encoder stage. (Alternatively, we can use a loss function that is designed to deal well with compression, like the CTC loss function we’ll introduce in the next section.) The goal of the subsampling is to produce a shorter sequence X = x1,...,xn that will be the input to the encoder. The simplest algorithm is a method sometimes called low frame rate(Pundak and Sainath, 2016): for timei we stack (concatenate)low frame rate the acoustic feature vector fi with the prior two vectors fi−1 and fi−2 to make a new vector three times longer. Then we simply delete fi−1 and fi−2. Thus instead of (say) a 40-dimensional acoustic feature vector every 10 ms, we have a longer vector (say 120-dimensional) every 30 ms, with a shorter sequence length n = t 3 .1 After this compression stage, encoder-decoders for speech use the same archi- tecture as for MT or other text, composed of either RNNs (LSTMs) or Transformers. For inference, the probability of the output string Y is decomposed as: p(y1,..., yn) = n∏ i=1 p(yi\|y1,..., yi−1,X) (16.8) We can produce each letter of the output via greedy decoding: ˆyi = argmaxchar∈AlphabetP(char\|y1...yi−1,X) (16.9) Alternatively we can use beam search as described in the next section. This is par- ticularly relevant when we are adding a language model. Adding a language model Since an encoder-decoder model is essentially a con- ditional language model, encoder-decoders implicitly learn a language model for the output domain of letters from their training data. However, the training data (speech paired with text transcriptions) may not include sufﬁcient text to train a good lan- guage model. After all, it’s easier to ﬁnd enormous amounts of pure text training 1 There are also more complex alternatives for subsampling, like using a convolutional net that down- samples with max pooling, or layers of pyramidal RNNs, RNNs where each successive layer has half the number of RNNs as the previous layer. 16.4 • CTC 341 data than it is to ﬁnd text paired with speech. Thus we can can usually improve a model at least slightly by incorporating a very large language model. The simplest way to do this is to use beam search to get a ﬁnal beam of hy- pothesized sentences; this beam is sometimes called an n-best list. We then use an-best list language model to rescore each hypothesis on the beam. The scoring is done by in-rescore terpolating the score assigned by the language model with the encoder-decoder score used to create the beam, with a weight λ tuned on a held-out set. Also, since most models prefer shorter sentences, ASR systems normally have some way of adding a length factor. One way to do this is to normalize the probability by the number of characters in the hypothesis \|Y \|c. The following is thus a typical scoring function (Chan et al., 2016): score(Y \|X) = 1 \|Y \|c logP(Y \|X)+ λ logPLM(Y ) (16.10) 16.3.1 Learning Encoder-decoders for speech are trained with the normal cross-entropy loss gener- ally used for conditional language models. At timestep i of decoding, the loss is the log probability of the correct token (letter) yi: LCE = −log p(yi\|y1,..., yi−1,X) (16.11) The loss for the entire sentence is the sum of these losses: LCE = − m∑ i=1 log p(yi\|y1,..., yi−1,X) (16.12) This loss is then backpropagated through the entire end-to-end model to train the entire encoder-decoder. As we described in Chapter 13, we normally use teacher forcing, in which the decoder history is forced to be the correct gold yi rather than the predicted ˆyi. It’s also possible to use a mixture of the gold and decoder output, for example using the gold output 90% of the time, but with probability .1 taking the decoder output instead: LCE = −log p(yi\|y1,..., ˆyi−1,X) (16.13) 16.4 CTC We pointed out in the previous section that speech recognition has two particular properties that make it very appropriate for the encoder-decoder architecture, where the encoder produces an encoding of the input that the decoder uses attention to explore. First, in speech we have a very long acoustic input sequence X mapping to a much shorter sequence of letters Y , and second, it’s hard to know exactly which part of X maps to which part of Y . In this section we brieﬂy introduce an alternative to encoder-decoder: an algo- rithm and loss function called CTC, short for Connectionist Temporal Classiﬁca-CTC tion (Graves et al., 2006), that deals with these problems in a very different way. The intuition of CTC is to output a single character for every frame of the input, so that 342 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH the output is the same length as the input, and then to apply a collapsing function that combines sequences of identical letters, resulting in a shorter sequence. Let’s imagine inference on someone saying the word dinner, and let’s suppose we had a function that chooses the most probable letter for each input spectral frame representation xi. We’ll call the sequence of letters corresponding to each input frame an alignment, because it tells us where in the acoustic signal each letter alignsalignment to. Fig. 16.9 shows one such alignment, and what happens if we use a collapsing function that just removes consecutive duplicate letters. X (input) A (alignment) Y (output) d x1 i x2 i x3 n x4 n x5 n x6 n x7 e x8 r x9 r x10 r x11 r x12 r x13 r x14 d i n e r wavefile Figure 16.9 A naive algorithm for collapsing an alignment between input and letters. Well, that doesn’t work; our naive algorithm has transcribed the speech asdiner, not dinner! Collapsing doesn’t handle double letters. There’s also another problem with our naive function; it doesn’t tell us what symbol to align with silence in the input. We don’t want to be transcribing silence as random letters! The CTC algorithm solves both problems by adding to the transcription alphabet a special symbol for a blank, which we’ll represent as . The blank can be used inblank the alignment whenever we don’t want to transcribe a letter. Blank can also be used between letters; since our collapsing function collapses only consecutive duplicate letters, it won’t collapse across . More formally, let’s deﬁne the mappingB : a →y between an alignment a and an output y, which collapses all repeated letters and then removes all blanks. Fig. 16.10 sketches this collapsing function B. X (input) A (alignment) remove blanks d x1 i x2 x3 n x4 n x5 x6 n x7 e x8 r x9 r x10 r x11 r x12 x13 x14 d i n e rn merge duplicates d i n e rn Y (output) d i n e rn ␣ ␣ ␣ ␣ ␣ ␣ ␣ Figure 16.10 The CTC collapsing function B, showing the space blank character ; re- peated (consecutive) characters in an alignment A are removed to form the output Y . The CTC collapsing function is many-to-one; lots of different alignments map to the same output string. For example, the alignment shown in Fig. 16.10 is not the only alignment that results in the string dinner. Fig. 16.11 shows some other alignments that would produce the same output. It’s useful to think of the set of all alignments that might produce the same output Y . We’ll use the inverse of our B function, called B−1, and represent that set as B−1(Y ). 16.4 • CTC 343 d i n n n e e e r r r␣ ␣ d d i n n n e r r␣ ␣ ␣ d d d i n n n e r r␣ i ␣␣ ␣ ␣ ␣ Figure 16.11 Three other legitimate alignments producing the transcript dinner. 16.4.1 CTC Inference Before we see how to compute PCTC(Y \|X) let’s ﬁrst see how CTC assigns a proba- bility to one particular alignment ˆA = {ˆa1,..., ˆan}. CTC makes a strong conditional independence assumption: it assumes that, given the inputX, the CTC model output at at time t is independent of the output labels at any other time ai. Thus: PCTC(A\|X) = T∏ t=1 p(at \|X) (16.14) Thus to ﬁnd the best alignment ˆA = {ˆa1,..., ˆaT }we can greedily choose the charac- ter with the max probability at each time step t: ˆat = argmax c∈C pt (c\|X) (16.15) We then pass the resulting sequence A to the CTC collapsing function B to get the output sequence Y . Let’s talk about how this simple inference algorithm for ﬁnding the best align- ment A would be implemented. Because we are making a decision at each time point, we can treat CTC as a sequence-modeling task, where we output one letter ˆyt at time t corresponding to each input token xt , eliminating the need for a full de- coder. Fig. 16.12 sketches this architecture, where we take an encoder, produce a hidden state ht at each timestep, and decode by taking a softmax over the character vocabulary at each time step. ENCODER … yn Feature Computation Subsampling … f tf 1 log Mel spectrum Shorter input sequence X y1 i y2 i y3 i y4 t x1 xn Classiﬁer +softmax … t y5 … … output letter sequence Y Figure 16.12 Inference with CTC: using an encoder-only model, with decoding done by simple softmaxes over the hidden state ht at each output step. Alas, there is a potential ﬂaw with the inference algorithm sketched in (Eq. 16.15) and Fig. 16.11. The problem is that we chose the most likely alignment A, but the 344 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH most likely alignment may not correspond to the most likely ﬁnal collapsed output string Y . That’s because there are many possible alignments that lead to the same output string, and hence the most likely output string might not correspond to the most probable alignment. For example, imagine the most probable alignment A for an input X = [x1x2x3] is the string [a b ϵ] but the next two most probable alignments are [b ϵb] and [ϵb b]. The output Y =[b b], summing over those two alignments, might be more probable than Y =[a b]. For this reason, the most probable output sequence Y is the one that has, not the single best CTC alignment, but the highest sum over the probability of all its possible alignments: PCTC (Y \|X) = ∑ A∈B−1(Y ) P(A\|X) = ∑ A∈B−1(Y ) T∏ t=1 p(at \|ht ) ˆY = argmax Y PCTC (Y \|X) (16.16) Alas, summing over all alignments is very expensive (there are a lot of alignments), so we approximate this sum by using a version of Viterbi beam search that cleverly keeps in the beam the high-probability alignments that map to the same output string, and sums those as an approximation of (Eq. 16.16). See Hannun (2017) for a clear explanation of this extension of beam search for CTC. Because of the strong conditional independence assumption mentioned earlier (that the output at time t is independent of the output at time t −1, given the input), CTC does not implicitly learn a language model over the data (unlike the attention- based encoder-decoder architectures). It is therefore essential when using CTC to interpolate a language model (and some sort of length factor L(Y )) using interpola- tion weights that are trained on a devset: scoreCTC(Y \|X) =logPCTC(Y \|X)+ λ1 logPLM(Y )λ2L(Y ) (16.17) 16.4.2 CTC Training To train a CTC-based ASR system, we use negative log-likelihood loss with a special CTC loss function. Thus the loss for an entire dataset D is the sum of the negative log-likelihoods of the correct output Y for each input X: LCTC = ∑ (X,Y )∈D −logPCTC(Y \|X) (16.18) To compute CTC loss function for a single input pair(X,Y ), we need the probability of the outputY given the inputX. As we saw in Eq. 16.16, to compute the probability of a given output Y we need to sum over all the possible alignments that would collapse to Y . In other words: PCTC(Y \|X) = ∑ A∈B−1(Y ) T∏ t=1 p(at \|ht ) (16.19) Naively summing over all possible alignments is not feasible (there are too many alignments). However, we can efﬁciently compute the sum by using dynamic pro- 16.4 • CTC 345 gramming to merge alignments, with a version of the forward-backward algo- rithm also used to train HMMs (Appendix A) and CRFs. The original dynamic pro- gramming algorithms for both training and inference are laid out in (Graves et al., 2006); see (Hannun, 2017) for a detailed explanation of both. 16.4.3 Combining CTC and Encoder-Decoder It’s also possible to combine the two architectures/loss functions we’ve described, the cross-entropy loss from the encoder-decoder architecture, and the CTC loss. Fig. 16.13 shows a sketch. For training, we can simply weight the two losses with a λ tuned on a devset: L = −λ logPencdec(Y \|X)−(1 −λ)logPctc (Y \|X) (16.20) For inference, we can combine the two with the language model (or the length penalty), again with learned weights: ˆY = argmax Y [λ logPencdec(Y \|X)−(1 −λ)logPCTC (Y \|X)+ γ logPLM(Y )] (16.21) ENCODER … DECODER … H <s> i t ‘ s t i m x1 xn … … i t ’ s t i m e … CTC Loss Encoder-Decoder Loss Figure 16.13 Combining the CTC and encoder-decoder loss functions. 16.4.4 Streaming Models: RNN-T for improving CTC Because of the strong independence assumption in CTC (assuming that the output at time t is independent of the output at time t −1), recognizers based on CTC don’t achieve as high an accuracy as the attention-based encoder-decoder recog- nizers. CTC recognizers have the advantage, however, that they can be used for streaming. Streaming means recognizing words on-line rather than waiting untilstreaming the end of the sentence to recognize them. Streaming is crucial for many applica- tions, from commands to dictation, where we want to start recognition while the user is still talking. Algorithms that use attention need to compute the hidden state sequence over the entire input ﬁrst in order to provide the attention distribution con- text, before the decoder can start decoding. By contrast, a CTC algorithm can input letters from left to right immediately. If we want to do streaming, we need a way to improve CTC recognition to re- move the conditional independent assumption, enabling it to know about output his- tory. The RNN-Transducer ( RNN-T), shown in Fig. 16.14, is just such a modelRNN-T (Graves 2012, Graves et al. 2013). The RNN-T has two main components: a CTC acoustic model, and a separate language model component called thepredictor that conditions on the output token history. At each time stept, the CTC encoder outputs 346 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH a hidden state henc t given the input x1...xt . The language model predictor takes as in- put the previous output token (not counting blanks), outputting a hidden state hpred u . The two are passed through another network whose output is then passed through a softmax to predict the next character. PRNN−T (Y \|X) = ∑ A∈B−1(Y ) P(A\|X) = ∑ A∈B−1(Y ) T∏ t=1 p(at \|ht ,y<ut ) ENCODER P ( yt,u \| x[1..t] , y[1..u-1] ) xt PREDICTION NETWORK yu-1 JOINT NETWORK henc t hpred u SOFTMAX zt,u DECODER Figure 16.14 The RNN-T model computing the output token distribution at time t by inte- grating the output of a CTC acoustic encoder and a separate ‘predictor’ language model. 16.5 ASR Evaluation: Word Error Rate The standard evaluation metric for speech recognition systems is the word errorword error rate. The word error rate is based on how much the word string returned by the recognizer (the hypothesized word string) differs from a reference transcription. The ﬁrst step in computing word error is to compute the minimum edit distance in words between the hypothesized and correct strings, giving us the minimum num- ber of word substitutions, word insertions, and word deletions necessary to map between the correct and hypothesized strings. The word error rate (WER) is then deﬁned as follows (note that because the equation includes insertions, the error rate can be greater than 100%): Word Error Rate = 100 ×Insertions +Substitutions +Deletions Total Words in Correct Transcript Here is a samplealignment between a reference and a hypothesis utterance fromalignment the CallHome corpus, showing the counts used to compute the error rate: REF: i UM the PHONE IS i LEFT THE portable ** PHONE UPSTAIRS last night HYP: i GOT IT TO the *** FULLEST i LOVE TO portable FORM OF STORES last night Eval: I I S D S S S I S S This utterance has six substitutions, three insertions, and one deletion: Word Error Rate = 1006 +3 +1 13 = 76.9% 16.5 • ASR E VALUATION : W ORD ERROR RATE 347 The standard method for computing word error rates is a free script calledsclite, available from the National Institute of Standards and Technologies (NIST) (NIST, 2005). Sclite is given a series of reference (hand-transcribed, gold-standard) sen- tences and a matching set of hypothesis sentences. Besides performing alignments, and computing word error rate, sclite performs a number of other useful tasks. For example, for error analysis it gives useful information such as confusion matrices showing which words are often misrecognized for others, and summarizes statistics of words that are often inserted or deleted. sclitealso gives error rates by speaker (if sentences are labeled for speaker ID), as well as useful statistics like thesentence error rate, the percentage of sentences with at least one word error.Sentence error rate Statistical signiﬁcance for ASR: MAPSSWE or MacNemar As with other language processing algorithms, we need to know whether a particular improvement in word error rate is signiﬁcant or not. The standard statistical tests for determining if two word error rates are different is the Matched-Pair Sentence Segment Word Error (MAPSSWE) test, introduced in Gillick and Cox (1989). The MAPSSWE test is a parametric test that looks at the difference between the number of word errors the two systems produce, averaged across a number of segments. The segments may be quite short or as long as an entire utterance; in general, we want to have the largest number of (short) segments in order to justify the normality assumption and to maximize power. The test requires that the errors in one segment be statistically independent of the errors in another segment. Since ASR systems tend to use trigram LMs, we can approximate this requirement by deﬁning a segment as a region bounded on both sides by words that both recognizers get correct (or by turn/utterance boundaries). Here’s an example from NIST (2007) with four regions: I II III IV REF: \|it was\|the best\|of\|times it\|was the worst\|of times\| \|it was \| \| \| \| \| \| \| \| SYS A:\|ITS \|the best\|of\|times it\|IS the worst \|of times\|OR\|it was \| \| \| \| \| \| \| \| SYS B:\|it was\|the best\| \|times it\|WON the TEST \|of times\| \|it was In region I, system A has two errors (a deletion and an insertion) and system B has zero; in region III, system A has one error (a substitution) and system B has two. Let’s deﬁne a sequence of variablesZ representing the difference between the errors in the two systems as follows: Ni A the number of errors made on segment i by system A Ni B the number of errors made on segment i by system B Z N i A −Ni B,i = 1,2,···,n where n is the number of segments In the example above, the sequence of Z values is {2,−1,−1,1}. Intuitively, if the two systems are identical, we would expect the average difference, that is, the average of the Z values, to be zero. If we call the true average of the differences muz, we would thus like to know whether muz = 0. Following closely the original proposal and notation of Gillick and Cox (1989), we can estimate the true average from our limited sample as ˆµz = ∑n i=1 Zi/n. The estimate of the variance of the Zi’s is σ2 z = 1 n −1 n∑ i=1 (Zi −µz)2 (16.22) 348 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH Let W = ˆµz σz/√n (16.23) For a large enough n (>50), W will approximately have a normal distribution with unit variance. The null hypothesis is H0 : µz = 0, and it can thus be rejected if 2 ∗P(Z ≥\|w\|) ≤0.05 (two-tailed) or P(Z ≥\|w\|) ≤0.05 (one-tailed), where Z is standard normal and w is the realized value W; these probabilities can be looked up in the standard tables of the normal distribution. Earlier work sometimes used McNemar’s testfor signiﬁcance, but McNemar’sMcNemar’s test is only applicable when the errors made by the system are independent, which is not true in continuous speech recognition, where errors made on a word are extremely dependent on errors made on neighboring words. Could we improve on word error rate as a metric? It would be nice, for exam- ple, to have something that didn’t give equal weight to every word, perhaps valuing content words like Tuesday more than function words likea or of. While researchers generally agree that this would be a good idea, it has proved difﬁcult to agree on a metric that works in every application of ASR. For dialogue systems, however, where the desired semantic output is more clear, a metric called slot error rate or concept error ratehas proved extremely useful; it is discussed in Chapter 15 on page 317. 16.6 TTS The goal of text-to-speech (TTS) systems is to map from strings of letters to wave- forms, a technology that’s important for a variety of applications from dialogue sys- tems to games to education. Like ASR systems, TTS systems are generally based on the encoder-decoder architecture, either using LSTMs or Transformers. There is a general difference in training. The default condition for ASR systems is to be speaker-independent: they are trained on large corpora with thousands of hours of speech from many speakers because they must generalize well to an unseen test speaker. By contrast, in TTS, it’s less crucial to use multiple voices, and so basic TTS systems are speaker-dependent: trained to have a consistent voice, on much less data, but all from one speaker. For example, one commonly used public domain dataset, the LJ speech corpus, consists of 24 hours of one speaker, Linda Johnson, reading audio books in the LibriV ox project (Ito and Johnson, 2017), much smaller than standard ASR corpora which are hundreds or thousands of hours.2 We generally break up the TTS task into two components. The ﬁrst component is an encoder-decoder model for spectrogram prediction: it maps from strings of letters to mel spectrographs: sequences of mel spectral values over time. Thus we 2 There is also recent TTS research on the task of multi-speaker TTS, in which a system is trained on speech from many speakers, and can switch between different voices. 16.6 • TTS 349 might map from this string: It’s time for lunch! to the following mel spectrogram: The second component maps from mel spectrograms to waveforms. Generating waveforms from intermediate representations like spectrograms is called vocodingvocoding and this second component is called a vocoder:vocoder These standard encoder-decoder algorithms for TTS are still quite computation- ally intensive, so a signiﬁcant focus of modern research is on ways to speed them up. 16.6.1 TTS Preprocessing: Text normalization Before either of these two steps, however, TTS systems require text normaliza- tion preprocessing for handling non-standard words: numbers, monetary amounts,non-standard words dates, and other concepts that are verbalized differently than they are spelled. A TTS system seeing a number like 151 needs to know to verbalize it as one hundred ﬁfty one if it occurs as $151 but as one ﬁfty one if it occurs in the context 151 Chapulte- pec Ave.. The number 1750 can be spoken in at least four different ways, depending on the context: seventeen fifty: (in “The European economy in 1750”) one seven five zero: (in “The password is 1750”) seventeen hundred and fifty: (in “1750 dollars”) one thousand, seven hundred, and fifty: (in “1750 dollars”) Often the verbalization of a non-standard word depends on its meaning (what Taylor (2009) calls its semiotic class ). Fig. 16.15 lays out some English non- standard word types. Many classes have preferred realizations. A year is generally read as paired digits (e.g., seventeen fifty for 1750). $3.2 billion must be read out with the word dollars at the end, as three point two billion dollars. Some ab- breviations like N.Y. are expanded (to New York), while other acronyms like GPU are pronounced as letter sequences. In languages with grammatical gender, normal- ization may depend on morphological properties. In French, the phrase 1 mangue (‘one mangue’) is normalized to une mangue, but 1 ananas (‘one pineapple’) is normalized to un ananas. In German, Heinrich IV (‘Henry IV’) can be normalized to Heinrich der Vierte, Heinrich des Vierten, Heinrich dem Vierten, or HeinrichdenVierten depending on the grammatical case of the noun (Demberg, 2006). 350 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH semiotic class examples verbalization abbreviations gov’t, N.Y., mph government acronyms read as letters GPU, D.C., PC, UN, IBM G P U cardinal numbers 12, 45, 1/2, 0.6 twelve ordinal numbers May 7, 3rd, Bill Gates III seventh numbers read as digits Room 101 one oh one times 3.20, 11:45 eleven forty ﬁve dates 28/02 (or in US, 2/28) February twenty eighth years 1999, 80s, 1900s, 2045 nineteen ninety nine money $3.45, e250, $200K three dollars forty ﬁve money in tr/m/billions $3.45 billion three point four ﬁve billion dollars percentage 75% 3.4% seventy ﬁve percent Figure 16.15 Some types of non-standard words in text normalization; see Sproat et al. (2001) and (van Esch and Sproat, 2018) for many more. Modern end-to-end TTS systems can learn to do some normalization themselves, but TTS systems are only trained on a limited amount of data (like the 220,000 words we mentioned above for the LJ corpus (Ito and Johnson, 2017)), and so a separate normalization step is important. Normalization can be done by rule or by an encoder-decoder model. Rule-based normalization is done in two stages: tokenization and verbalization. In the tokeniza- tion stage we hand-write rules to detect non-standard words. These can be regular expressions, like the following for detecting years: /(1[89][0-9][0-9])\|(20[0-9][0-9]/ A second pass of rules express how to verbalize each semiotic class. Larger TTS systems instead use more complex rule-systems, like the Kestral system of (Ebden and Sproat, 2015), which ﬁrst classiﬁes and parses each input into a normal form and then produces text using a verbalization grammar. Rules have the advantage that they don’t require training data, and they can be designed for high precision, but can be brittle, and require expert rule-writers so are hard to maintain. The alternative model is to use encoder-decoder models, which have been shown to work better than rules for such transduction tasks, but do require expert-labeled training sets in which non-standard words have been replaced with the appropriate verbalization; such training sets for some languages are available (Sproat and Gor- man 2018, Zhang et al. 2019). In the simplest encoder-decoder setting, we simply treat the problem like ma- chine translation, training a system to map from: They live at 224 Mission St. to They live at two twenty four Mission Street While encoder-decoder algorithms are highly accurate, they occasionally pro- duce errors that are egregious; for example normalizing 45 minutes as forty ﬁve mil- limeters. To address this, more complex systems use mechanisms like lightweight covering grammars, which enumerate a large set of possible verbalizations but don’t try to disambiguate, to constrain the decoding to avoid such outputs (Zhang et al., 2019). 16.6.2 TTS: Spectrogram prediction The exact same architecture we described for ASR—the encoder-decoder with attention– can be used for the ﬁrst component of TTS. Here we’ll give a simpliﬁed overview 16.6 • TTS 351 of the Tacotron2architecture (Shen et al., 2018), which extends the earlier TacotronTacotron2 (Wang et al., 2017) architecture and the Wavenet vocoder (van den Oord et al.,Wavenet 2016). Fig. 16.16 sketches out the entire architecture. The encoder’s job is to take a sequence of letters and produce a hidden repre- sentation representing the letter sequence, which is then used by the attention mech- anism in the decoder. The Tacotron2 encoder ﬁrst maps every input grapheme to a 512-dimensional character embedding. These are then passed through a stack of 3 convolutional layers, each containing 512 ﬁlters with shape 5 ×1, i.e. each ﬁlter spanning 5 characters, to model the larger letter context. The output of the ﬁnal convolutional layer is passed through a biLSTM to produce the ﬁnal encoding. It’s common to use a slightly higher quality (but slower) version of attention called location-based attention, in which the computation of the α values (Eq. 8.36 inlocation-based attention Chapter 8) makes use of the α values from the prior time-state. In the decoder, the predicted mel spectrum from the prior time slot is passed through a small pre-net as a bottleneck. This prior output is then concatenated with the encoder’s attention vector context and passed through 2 LSTM layers. The out- put of this LSTM is used in two ways. First, it is passed through a linear layer, and some output processing, to autoregressively predict one 80-dimensional log-mel ﬁl- terbank vector frame (50 ms, with a 12.5 ms stride) at each step. Second, it is passed through another linear layer to a sigmoid to make a “stop token prediction” decision about whether to stop producing output. While linear spectrograms discard phase information (and are therefore lossy), algorithms such as Grifﬁn-Lim [ 14 ] are capable of estimating this discarded information, which enables time-domain conversion via the inverse short-time Fourier transform. Mel spectro- grams discard even more information, presenting a challenging in- verse problem. However, in comparison to the linguistic and acoustic features used in WaveNet, the mel spectrogram is a simpler, lower- level acoustic representation of audio signals. It should therefore be straightforward for a similar WaveNet model conditioned on mel spectrograms to generate audio, essentially as a neural vocoder. In- deed, we will show that it is possible to generate high quality audio from mel spectrograms using a modiﬁed WaveNet architecture. 2.2. Spectrogram Prediction Network As in Tacotron, mel spectrograms are computed through a short- time Fourier transform (STFT) using a 50 ms frame size, 12.5 ms frame hop, and a Hann window function. We experimented with a 5 ms frame hop to match the frequency of the conditioning inputs in the original WaveNet, but the corresponding increase in temporal resolution resulted in signiﬁcantly more pronunciation issues. We transform the STFT magnitude to the mel scale using an 80 channel mel ﬁlterbank spanning 125 Hz to 7.6 kHz, followed by log dynamic range compression. Prior to log compression, the ﬁlterbank output magnitudes are clipped to a minimum value of 0.01 in order to limit dynamic range in the logarithmic domain. The network is composed of an encoder and a decoder with atten- tion. The encoder converts a character sequence into a hidden feature representation which the decoder consumes to predict a spectrogram. Input characters are represented using a learned 512-dimensional character embedding, which are passed through a stack of 3 convolu- tional layers each containing 512 ﬁlters with shape 5 ⇥ 1 , i.e., where each ﬁlter spans 5 characters, followed by batch normalization [ 18 ] and ReLU activations. As in Tacotron, these convolutional layers model longer-term context (e.g., N -grams) in the input character sequence. The output of the ﬁnal convolutional layer is passed into a single bi-directional [ 19 ] LSTM [ 20 ] layer containing 512 units (256 in each direction) to generate the encoded features. The encoder output is consumed by an attention network which summarizes the full encoded sequence as a ﬁxed-length context vector for each decoder output step. We use the location-sensitive attention from [ 21 ], which extends the additive attention mechanism [ 22 ] to use cumulative attention weights from previous decoder time steps as an additional feature. This encourages the model to move forward consistently through the input, mitigating potential failure modes where some subsequences are repeated or ignored by the decoder. Attention probabilities are computed after projecting inputs and lo- cation features to 128-dimensional hidden representations. Location features are computed using 32 1-D convolution ﬁlters of length 31. The decoder is an autoregressive recurrent neural network which predicts a mel spectrogram from the encoded input sequence one frame at a time. The prediction from the previous time step is ﬁrst passed through a small pre-net containing 2 fully connected layers of 256 hidden ReLU units. We found that the pre-net acting as an information bottleneck was essential for learning attention. The pre- net output and attention context vector are concatenated and passed through a stack of 2 uni-directional LSTM layers with 1024 units. The concatenation of the LSTM output and the attention context vector is projected through a linear transform to predict the target spectrogram frame. Finally, the predicted mel spectrogram is passed through a 5-layer convolutional post-net which predicts a residual to add to the prediction to improve the overall reconstruction. Each 'LEVEGXIV )QFIHHMRK 0SGEXMSR 7IRWMXMZI %XXIRXMSR 'SRZ 0E]IVW &MHMVIGXMSREP 0781-RTYX8I\X 0E]IV 4VI2IX 0781 0E]IVW 0MRIEV 4VSNIGXMSR 0MRIEV 4VSNIGXMSR 7XST8SOIR 'SRZ0E]IV 4SWX2IX 0HO6SHFWURJUDP ;EZI2IX 1S0 ;EZIJSVQ 7EQTPIW Fig. 1 . Block diagram of the Tacotron 2 system architecture. post-net layer is comprised of 512 ﬁlters with shape 5 ⇥ 1 with batch normalization, followed by tanh activations on all but the ﬁnal layer. We minimize the summed mean squared error (MSE) from before and after the post-net to aid convergence. We also experimented with a log-likelihood loss by modeling the output distribution with a Mixture Density Network [ 23 , 24 ] to avoid assuming a constant variance over time, but found that these were more difﬁcult to train and they did not lead to better sounding samples. In parallel to spectrogram frame prediction, the concatenation of decoder LSTM output and the attention context is projected down to a scalar and passed through a sigmoid activation to predict the probability that the output sequence has completed. This “stop token” prediction is used during inference to allow the model to dynamically determine when to terminate generation instead of always generating for a ﬁxed duration. Speciﬁcally, generation completes at the ﬁrst frame for which this probability exceeds a threshold of 0.5. The convolutional layers in the network are regularized using dropout [ 25 ] with probability 0.5, and LSTM layers are regularized using zoneout [ 26 ] with probability 0.1. In order to introduce output variation at inference time, dropout with probability 0.5 is applied only to layers in the pre-net of the autoregressive decoder. In contrast to the original Tacotron, our model uses simpler build- ing blocks, using vanilla LSTM and convolutional layers in the en- coder and decoder instead of “CBHG” stacks and GRU recurrent layers. We do not use a “reduction factor”, i.e., each decoder step corresponds to a single spectrogram frame. 2.3. WaveNet Vocoder We use a modiﬁed version of the WaveNet architecture from [ 8 ] to invert the mel spectrogram feature representation into time-domain waveform samples. As in the original architecture, there are 30 dilated convolution layers, grouped into 3 dilation cycles, i.e., the dilation rate of layer k ( k =0 ... 29 ) is 2 k (mod 10) . To work with the 12.5 ms frame hop of the spectrogram frames, only 2 upsampling layers are used in the conditioning stack instead of 3 layers. Instead of predicting discretized buckets with a softmax layer, we follow PixelCNN++ [ 27 ] and Parallel WaveNet [ 28 ] and use a 10- component mixture of logistic distributions (MoL) to generate 16-bit samples at 24 kHz. To compute the logistic mixture distribution, the WaveNet stack output is passed through a ReLU activation followed Encoder Decoder Vocoder Figure 16.16 The Tacotron2 architecture: An encoder-decoder maps from graphemes to mel spectrograms, followed by a vocoder that maps to waveﬁles. Figure modiﬁed from Shen et al. (2018). The system is trained on gold log-mel ﬁlterbank features, using teacher forcing, that is the decoder is fed the correct log-model spectral feature at each decoder step instead of the predicted decoder output from the prior step. 16.6.3 TTS: Vocoding The vocoder for Tacotron 2 is an adaptation of theWaveNet vocoder (van den OordWaveNet et al., 2016). Here we’ll give a somewhat simpliﬁed description of vocoding using WaveNet. Recall that the goal of the vocoding process here will be to invert a log mel spec- trum representations back into a time-domain waveform representation. WaveNet is an autoregressive network, like the language models we introduced in Chapter 8. It 352 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH takes spectrograms as input and produces audio output represented as sequences of 8-bit mu-law (page 336). The probability of a waveform , a sequence of 8-bit mu- law valuesY = y1,...,yt , given an intermediate input mel spectrogramh is computed as: p(Y ) = t∏ t=1 P(yt \|y1,...,yt−1,h1,...,ht ) (16.24) This probability distribution is modeled by a stack of special convolution layers, which include a speciﬁc convolutional structure called dilated convolutions, and a speciﬁc non-linearity function. A dilated convolution is a subtype of causal convolutional layer. Causal or masked convolutions look only at the past input, rather than the future; the pre- diction of yt+1 can only depend on y1,...,yt , useful for autoregressive left-to-right processing. In dilated convolutions, at each successive layer we apply the convolu-dilated convolutions tional ﬁlter over a span longer than its length by skipping input values. Thus at time t with a dilation value of 1, a convolutional ﬁlter of length 2 would see input values xt and xt−1. But a ﬁlter with a distillation value of 2 would skip an input, so would see input values xt and xt−1. Fig. 16.17 shows the computation of the output at time t with 4 dilated convolution layers with dilation values, 1, 2, 4, and 8. Because models with causal convolutions do not have recurrent connections, they are typically faster to train than RNNs, especially when applied to very long sequences. One of the problems of causal convolutions is that they require many layers, or large ﬁlters to increase the receptive ﬁeld. For example, in Fig. 2 the receptive ﬁeld is only 5 (= #layers + ﬁlter length - 1). In this paper we use dilated convolutions to increase the receptive ﬁeld by orders of magnitude, without greatly increasing computational cost. A dilated convolution (also called`a trous, or convolution with holes) is a convolution where the ﬁlter is applied over an area larger than its length by skipping input values with a certain step. It is equivalent to a convolution with a larger ﬁlter derived from the original ﬁlter by dilating it with zeros, but is signiﬁcantly more efﬁcient. A dilated convolution effectively allows the network to operate on a coarser scale than with a normal convolution. This is similar to pooling or strided convolutions, but here the output has the same size as the input. As a special case, dilated convolution with dilation 1 yields the standard convolution. Fig. 3 depicts dilated causal convolutions for dilations1, 2, 4, and 8. Dilated convolutions have previously been used in various contexts, e.g. signal processing (Holschneider et al., 1989; Dutilleux, 1989), and image segmentation (Chen et al., 2015; Yu & Koltun, 2016). Input Hidden LayerDilation = 1 Hidden LayerDilation = 2 Hidden LayerDilation = 4 OutputDilation = 8 Figure 3: Visualization of a stack ofdilated causal convolutional layers. Stacked dilated convolutions enable networks to have very large receptive ﬁelds with just a few lay- ers, while preserving the input resolution throughout the network as well as computational efﬁciency. In this paper, the dilation is doubled for every layer up to a limit and then repeated: e.g. 1, 2, 4,..., 512, 1, 2, 4,..., 512, 1, 2, 4,..., 512. The intuition behind this conﬁguration is two-fold. First, exponentially increasing the dilation factor results in exponential receptive ﬁeld growth with depth (Yu & Koltun, 2016). For example each 1, 2, 4,..., 512 block has receptive ﬁeld of size1024, and can be seen as a more efﬁcient and dis- criminative (non-linear) counterpart of a1⇥1024 convolution. Second, stacking these blocks further increases the model capacity and the receptive ﬁeld size. 2.2 S OFTMAX DISTRIBUTIONS One approach to modeling the conditional distributionsp (xt \| x1,...,x t1) over the individual audio samples would be to use a mixture model such as a mixture density network (Bishop, 1994) or mixture of conditional Gaussian scale mixtures (MCGSM) (Theis & Bethge, 2015). However, van den Oord et al. (2016a) showed that a softmax distribution tends to work better, even when the data is implicitly continuous (as is the case for image pixel intensities or audio sample values). One of the reasons is that a categorical distribution is more ﬂexible and can more easily model arbitrary distributions because it makes no assumptions about their shape. Because raw audio is typically stored as a sequence of 16-bit integer values (one per timestep), a softmax layer would need to output 65,536 probabilities per timestep to model all possible values. To make this more tractable, we ﬁrst apply aµ-law companding transformation (ITU-T, 1988) to the data, and then quantize it to 256 possible values: f (xt) = sign(xt)ln (1 +µ \|xt\|) ln (1 +µ) , 3 Figure 16.17 Dilated convolutions, showing one dilation cycle size of 4, i.e., dilation values of 1, 2, 4, 8. Figure from van den Oord et al. (2016). The Tacotron 2 synthesizer uses 12 convolutional layers in two cycles with a dilation cycle size of 6, meaning that the ﬁrst 6 layers have dilations of 1, 2, 4, 8, 16, and 32. and the next 6 layers again have dilations of 1, 2, 4, 8, 16, and 32. Dilated convolutions allow the vocoder to grow the receptive ﬁeld exponentially with depth. WaveNet predicts mu-law audio samples. Recall from page 336 that this is a standard compression for audio in which the values at each sampling timestep are compressed into 8-bits. This means that we can predict the value of each sample with a simple 256-way categorical classiﬁer. The output of the dilated convolutions is thus passed through a softmax which makes this 256-way decision. The spectrogram prediction encoder-decoder and the WaveNet vocoder are trained separately. After the spectrogram predictor is trained, the spectrogram prediction network is run in teacher-forcing mode, with each predicted spectral frame condi- tioned on the encoded text input and the previous frame from the ground truth spec- trogram. This sequence of ground truth-aligned spectral features and gold audio output is then used to train the vocoder. This has been only a high-level sketch of the TTS process. There are numer- ous important details that the reader interested in going further with TTS may want 16.7 • O THER SPEECH TASKS 353 to look into. For example WaveNet uses a special kind of a gated activation func- tion as its non-linearity, and contains residual and skip connections. In practice, predicting 8-bit audio values doesn’t as work as well as 16-bit, for which a simple softmax is insufﬁcient, so decoders use fancier ways as the last step of predicting audio sample values, like mixtures of distributions. Finally, the WaveNet vocoder as we have described it would be so slow as to be useless; many different kinds of efﬁciency improvements are necessary in practice, for example by ﬁnding ways to do non-autoregressive generation, avoiding the latency of having to wait to generate each frame until the prior frame has been generated, and instead making predictions in parallel. We encourage the interested reader to consult the original papers and various version of the code. 16.6.4 TTS Evaluation Speech synthesis systems are evaluated by human listeners. (The development of a good automatic metric for synthesis evaluation, one that would eliminate the need for expensive and time-consuming human listening experiments, remains an open and exciting research topic.) We evaluate the quality of synthesized utterances by playing a sentence to lis- teners and ask them to give a mean opinion score (MOS), a rating of how goodMOS the synthesized utterances are, usually on a scale from 1–5. We can then compare systems by comparing their MOS scores on the same sentences (using, e.g., paired t-tests to test for signiﬁcant differences). If we are comparing exactly two systems (perhaps to see if a particular change actually improved the system), we can use AB tests. In AB tests, we play the sameAB tests sentence synthesized by two different systems (an A and a B system). The human listeners choose which of the two utterances they like better. We do this for say 50 sentences (presented in random order) and compare the number of sentences preferred for each system. 16.7 Other Speech Tasks While we have focused on speech recognition and TTS in this chapter, there are a wide variety of speech-related tasks. The task of wake word detection is to detect a word or short phrase, usually inwake word order to wake up a voice-enable assistant like Alexa, Siri, or the Google Assistant. The goal with wake words is build the detection into small devices at the computing edge, to maintain privacy by transmitting the least amount of user speech to a cloud- based server. Thus wake word detectors need to be fast, small footprint software that can ﬁt into embedded devices. Wake word detectors usually use the same frontend feature extraction we saw for ASR, often followed by a whole-word classiﬁer. Speaker diarization is the task of determining ‘who spoke when’ in a longspeaker diarization multi-speaker audio recording, marking the start and end of each speaker’s turns in the interaction. This can be useful for transcribing meetings, classroom speech, or medical interactions. Often diarization systems use voice activity detection (V AD) to ﬁnd segments of continuous speech, extract speaker embedding vectors, and cluster the vectors to group together segments likely from the same speaker. More recent work is investigating end-to-end algorithms to map directly from input speech to a sequence of speaker labels for each frame. 354 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH Speaker recognition, is the task of identifying a speaker. We generally distin-speaker recognition guish the subtasks of speaker veriﬁcation, where we make a binary decision (is this speaker X or not?), such as for security when accessing personal information over the telephone, and speaker identiﬁcation, where we make a one of N decision trying to match a speaker’s voice against a database of many speakers . These tasks are related to language identiﬁcation, in which we are given a waveﬁle and mustlanguage identiﬁcation identify which language is being spoken; this is useful for example for automatically directing callers to human operators that speak appropriate languages. 16.8 Summary This chapter introduced the fundamental algorithms of automatic speech recognition (ASR) and text-to-speech (TTS). • The task of speech recognition (or speech-to-text) is to map acoustic wave- forms to sequences of graphemes. • The input to a speech recognizer is a series of acoustic waves. that are sam- pled, quantized, and converted to a spectral representation like the log mel spectrum. • Two common paradigms for speech recognition are theencoder-decoder with attention model, and models based on the CTC loss function . Attention- based models have higher accuracies, but models based on CTC more easily adapt to streaming: outputting graphemes online instead of waiting until the acoustic input is complete. • ASR is evaluated using the Word Error Rate; the edit distance between the hypothesis and the gold transcription. • TTS systems are also based on the encoder-decoder architecture. The en- coder maps letters to an encoding, which is consumed by the decoder which generates mel spectrogram output. A neural vocoder then reads the spectro- gram and generates waveforms. • TTS systems require a ﬁrst pass of text normalization to deal with numbers and abbreviations and other non-standard words. • TTS is evaluated by playing a sentence to human listeners and having them give a mean opinion score (MOS) or by doing AB tests. Bibliographical and Historical Notes ASR A number of speech recognition systems were developed by the late 1940s and early 1950s. An early Bell Labs system could recognize any of the 10 digits from a single speaker (Davis et al., 1952). This system had 10 speaker-dependent stored patterns, one for each digit, each of which roughly represented the ﬁrst two vowel formants in the digit. They achieved 97%–99% accuracy by choosing the pat- tern that had the highest relative correlation coefﬁcient with the input. Fry (1959) and Denes (1959) built a phoneme recognizer at University College, London, that recognized four vowels and nine consonants based on a similar pattern-recognition principle. Fry and Denes’s system was the ﬁrst to use phoneme transition probabili- ties to constrain the recognizer. BIBLIOGRAPHICAL AND HISTORICAL NOTES 355 The late 1960s and early 1970s produced a number of important paradigm shifts. First were a number of feature-extraction algorithms, including the efﬁcient fast Fourier transform (FFT) (Cooley and Tukey, 1965), the application of cepstral pro- cessing to speech (Oppenheim et al., 1968), and the development of LPC for speech coding (Atal and Hanauer, 1971). Second were a number of ways of handlingwarp- ing; stretching or shrinking the input signal to handle differences in speaking ratewarping and segment length when matching against stored patterns. The natural algorithm for solving this problem was dynamic programming, and, as we saw in Appendix A, the algorithm was reinvented multiple times to address this problem. The ﬁrst applica- tion to speech processing was by Vintsyuk (1968), although his result was not picked up by other researchers, and was reinvented by Velichko and Zagoruyko (1970) and Sakoe and Chiba (1971) (and 1984). Soon afterward, Itakura (1975) combined this dynamic programming idea with the LPC coefﬁcients that had previously been used only for speech coding. The resulting system extracted LPC features from incoming words and used dynamic programming to match them against stored LPC templates. The non-probabilistic use of dynamic programming to match a template against in- coming speech is called dynamic time warping.dynamic time warping The third innovation of this period was the rise of the HMM. Hidden Markov models seem to have been applied to speech independently at two laboratories around 1972. One application arose from the work of statisticians, in particular Baum and colleagues at the Institute for Defense Analyses in Princeton who applied HMMs to various prediction problems (Baum and Petrie 1966, Baum and Eagon 1967). James Baker learned of this work and applied the algorithm to speech processing (Baker, 1975a) during his graduate work at CMU. Independently, Frederick Jelinek and collaborators (drawing from their research in information-theoretical models inﬂuenced by the work of Shannon (1948)) applied HMMs to speech at the IBM Thomas J. Watson Research Center (Jelinek et al., 1975). One early difference was the decoding algorithm; Baker’s DRAGON system used Viterbi (dynamic program- ming) decoding, while the IBM system applied Jelinek’s stack decoding algorithm (Jelinek, 1969). Baker then joined the IBM group for a brief time before founding the speech-recognition company Dragon Systems. The use of the HMM, with Gaussian Mixture Models (GMMs) as the phonetic component, slowly spread through the speech community, becoming the dominant paradigm by the 1990s. One cause was encouragement by ARPA, the Advanced Research Projects Agency of the U.S. Department of Defense. ARPA started a ﬁve-year program in 1971 to build 1000-word, constrained grammar, few speaker speech understanding (Klatt, 1977), and funded four competing systems of which Carnegie-Mellon University’s Harpy system (Lowerre, 1976), which used a simpli- ﬁed version of Baker’s HMM-based DRAGON system was the best of the tested sys- tems. ARPA (and then DARPA) funded a number of new speech research programs, beginning with 1000-word speaker-independent read-speech tasks like “Resource Management” (Price et al., 1988), recognition of sentences read from theWall Street Journal (WSJ), Broadcast News domain (LDC 1998, Graff 1997) (transcription of actual news broadcasts, including quite difﬁcult passages such as on-the-street inter- views) and the Switchboard, CallHome, CallFriend, and Fisher domains (Godfrey et al. 1992, Cieri et al. 2004) (natural telephone conversations between friends or strangers). Each of the ARPA tasks involved an approximately annual bakeoff atbakeoff which systems were evaluated against each other. The ARPA competitions resulted in wide-scale borrowing of techniques among labs since it was easy to see which ideas reduced errors the previous year, and the competitions were probably an im- 356 CHAPTER 16 • A UTOMATIC SPEECH RECOGNITION AND TEXT-TO-SPEECH portant factor in the eventual spread of the HMM paradigm. By around 1990 neural alternatives to the HMM/GMM architecture for ASR arose, based on a number of earlier experiments with neural networks for phoneme recognition and other speech tasks. Architectures included the time-delay neural network ( TDNN)—the ﬁrst use of convolutional networks for speech— (Waibel et al. 1989, Lang et al. 1990), RNNs (Robinson and Fallside, 1991), and the hybridhybrid HMM/MLP architecture in which a feedforward neural network is trained as a pho- netic classiﬁer whose outputs are used as probability estimates for an HMM-based architecture (Morgan and Bourlard 1990, Bourlard and Morgan 1994, Morgan and Bourlard 1995). While the hybrid systems showed performance close to the standard HMM/GMM models, the problem was speed: large hybrid models were too slow to train on the CPUs of that era. For example, the largest hybrid system, a feedforward network, was limited to a hidden layer of 4000 units, producing probabilities over only a few dozen monophones. Yet training this model still required the research group to de- sign special hardware boards to do vector processing (Morgan and Bourlard, 1995). A later analytic study showed the performance of such simple feedforward MLPs for ASR increases sharply with more than 1 hidden layer, even controlling for the total number of parameters (Maas et al., 2017). But the computational resources of the time were insufﬁcient for more layers. Over the next two decades a combination of Moore’s law and the rise of GPUs allowed deep neural networks with many layers. Performance was getting close to traditional systems on smaller tasks like TIMIT phone recognition by 2009 (Mo- hamed et al., 2009), and by 2012, the performance of hybrid systems had surpassed traditional HMM/GMM systems (Jaitly et al. 2012, Dahl et al. 2012, inter alia). Originally it seemed that unsupervised pretraining of the networks using a tech- nique like deep belief networks was important, but by 2013, it was clear that for hybrid HMM/GMM feedforward networks, all that mattered was to use a lot of data and enough layers, although a few other components did improve performance: us- ing log mel features instead of MFCCs, using dropout, and using rectiﬁed linear units (Deng et al. 2013, Maas et al. 2013, Dahl et al. 2013). Meanwhile early work had proposed the CTC loss function by 2006 (Graves et al., 2006), and by 2012 the RNN-Transducer was deﬁned and applied to phone recognition (Graves 2012, Graves et al. 2013), and then to end-to-end speech recog- nition rescoring (Graves and Jaitly, 2014), and then recognition (Maas et al., 2015), with advances such as specialized beam search (Hannun et al., 2014). (Our de- scription of CTC in the chapter draws on Hannun (2017), which we encourage the interested reader to follow). The encoder-decoder architecture was applied to speech at about the same time by two different groups, in the Listen Attend and Spell system of Chan et al. (2016) and the attention-based encoder decoder architecture of Chorowski et al. (2014) and Bahdanau et al. (2016). By 2018 Transformers were included in this encoder- decoder architecture. Karita et al. (2019) is a nice comparison of RNNs vs Trans- formers in encoder-architectures for ASR, TTS, and speech-to-speech translation. Popular toolkits for speech processing include Kaldi (Povey et al., 2011) andKaldi ESPnet (Watanabe et al. 2018, Hayashi et al. 2020).ESPnet TTS As we noted at the beginning of the chapter, speech synthesis is one of the earliest ﬁelds of speech and language processing. The 18th century saw a number of physical models of the articulation process, including the von Kempelen model mentioned above, as well as the 1773 vowel model of Kratzenstein in Copenhagen EXERCISES 357 using organ pipes. The early 1950s saw the development of three early paradigms of waveform synthesis: formant synthesis, articulatory synthesis, and concatenative synthesis. Modern encoder-decoder systems are distant descendants of formant synthesiz- ers. Formant synthesizers originally were inspired by attempts to mimic human speech by generating artiﬁcial spectrograms. The Haskins Laboratories Pattern Playback Machine generated a sound wave by painting spectrogram patterns on a moving transparent belt and using reﬂectance to ﬁlter the harmonics of a wave- form (Cooper et al., 1951); other very early formant synthesizers include those of Lawrence (1953) and Fant (1951). Perhaps the most well-known of the formant synthesizers were the Klatt formant synthesizer and its successor systems, includ- ing the MITalk system (Allen et al., 1987) and the Klattalk software used in Digital Equipment Corporation’s DECtalk (Klatt, 1982). See Klatt (1975) for details. A second early paradigm, concatenative synthesis, seems to have been ﬁrst pro- posed by Harris (1953) at Bell Laboratories; he literally spliced together pieces of magnetic tape corresponding to phones. Soon afterwards, Peterson et al. (1958) pro- posed a theoretical model based on diphones, including a database with multiple copies of each diphone with differing prosody, each labeled with prosodic features including F0, stress, and duration, and the use of join costs based on F0 and formant distance between neighboring units. But such diphone synthesis models were not actually implemented until decades later (Dixon and Maxey 1968, Olive 1977). The 1980s and 1990s saw the invention ofunit selection synthesis, based on larger units of non-uniform length and the use of a target cost, (Sagisaka 1988, Sagisaka et al. 1992, Hunt and Black 1996, Black and Taylor 1994, Syrdal et al. 2000). A third paradigm, articulatory synthesizers attempt to synthesize speech by modeling the physics of the vocal tract as an open tube. Representative models include Stevens et al. (1953), Flanagan et al. (1975), and Fant (1986). See Klatt (1975) and Flanagan (1972) for more details. Most early TTS systems used phonemes as input; development of the text anal- ysis components of TTS came somewhat later, drawing on NLP. Indeed the ﬁrst true text-to-speech system seems to have been the system of Umeda and Teranishi (Umeda et al. 1968, Teranishi and Umeda 1968, Umeda 1976), which included a parser that assigned prosodic boundaries, as well as accent and stress. Exercises 16.1 Analyze each of the errors in the incorrectly recognized transcription of “um the phone is I left the. . . ” on page 346. For each one, give your best guess as to whether you think it is caused by a problem in signal processing, pronun- ciation modeling, lexicon size, language model, or pruning in the decoding search. Part III ANNOTATING LINGUISTIC STRUCTURE In the ﬁnal part of the book we discuss the task of detecting linguistic structure. In the early history of NLP these structures were an intermediate step toward deeper language processing. In modern NLP, we don’t generally make explicit use of parse or other structures inside the neural language models we introduced in Part I, or directly in applications like those we discussed in Part II. Instead linguistic structure plays a number of new roles. One important role is for interpretability: to provide a useful interpretive lens on neural networks. Knowing that a particular layer or neuron may be computing something related to a particular kind of structure can help us break open the ‘black box’ and understand what the components of our language models are doing. A second important role for linguistic structure is as a practical tool for social scientiﬁc studies of text: knowing which adjective modiﬁes which noun, or whether a particular implicit metaphor is being used, can be important for measuring attitudes toward groups or individuals. Detailed semantic structure can be helpful, for exam- ple in ﬁnding particular clauses that have particular meanings in legal contracts. Word sense labels can help keep any corpus study from measuring facts about the wrong word sense. Relation structures can be used to help build knowledge bases from text. Finally, computation of linguistic structure is an important tool for answering questions about language itself, a research area called computational linguistics that is sometimes distinguished from natural language processing. To answer lin- guistic questions about how language changes over time or across individuals we’ll need to be able, for example, to parse entire documents from different time periods. To understand how certain linguistic structures are learned or processed by people, it’s necessary to be able to automatically label structures for arbitrary text. In our study of linguistic structure, we begin with one of the oldest tasks in computational linguistics: the extraction of syntactic structure, and give two sets of algorithms for parsing: extracting syntactic structure, including constituency pars- ing and dependency parsing. We then introduce a variety of structures related to meaning, including semantic roles, word senses, entity relations, and events. We 360 conclude with linguistic structures that tend to be related to discourse and meaning over larger texts, including coreference and discourse coherence. In each case we’ll give algorithms for automatically annotating the relevant structure. 362 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES CHAPTER 17 Sequence Labeling for Parts of Speech and Named Entities To each word a warbling note A Midsummer Night’s Dream, V .I Dionysius Thrax of Alexandria (c. 100 B.C.), or perhaps someone else (it was a long time ago), wrote a grammatical sketch of Greek (a “ techn¯e”) that summarized the linguistic knowledge of his day. This work is the source of an astonishing proportion of modern linguistic vocabulary, including the words syntax, diphthong, clitic, and analogy. Also included are a description of eight parts of speech : noun, verb,parts of speech pronoun, preposition, adverb, conjunction, participle, and article. Although earlier scholars (including Aristotle as well as the Stoics) had their own lists of parts of speech, it was Thrax’s set of eight that became the basis for descriptions of European languages for the next 2000 years. (All the way to theSchoolhouse Rock educational television shows of our childhood, which had songs about 8 parts of speech, like the late great Bob Dorough’s Conjunction Junction.) The durability of parts of speech through two millennia speaks to their centrality in models of human language. Proper names are another important and anciently studied linguistic category. While parts of speech are generally assigned to individual words or morphemes, a proper name is often an entire multiword phrase, like the name “Marie Curie”, the location “New York City”, or the organization “Stanford University”. We’ll use the term named entity for, roughly speaking, anything that can be referred to with anamed entity proper name: a person, a location, an organization, although as we’ll see the term is commonly extended to include things that aren’t entities per se. Parts of speech (also known as POS) and named entities are useful clues toPOS sentence structure and meaning. Knowing whether a word is a noun or a verb tells us about likely neighboring words (nouns in English are preceded by determiners and adjectives, verbs by nouns) and syntactic structure (verbs have dependency links to nouns), making part-of-speech tagging a key aspect of parsing. Knowing if a named entity like Washington is a name of a person, a place, or a university is important to many natural language processing tasks like question answering, stance detection, or information extraction. In this chapter we’ll introduce the task of part-of-speech tagging, taking a se- quence of words and assigning each word a part of speech like NOUN or VERB , and the task of named entity recognition (NER), assigning words or phrases tags like PERSON , LOCATION , or ORGANIZATION . Such tasks in which we assign, to each word xi in an input word sequence, a label yi, so that the output sequence Y has the same length as the input sequence X are called sequence labeling tasks. We’ll introduce classic sequence labeling algo-sequence labeling rithms, one generative— the Hidden Markov Model (HMM)—and one discriminative— the Conditional Random Field (CRF). In following chapters we’ll introduce modern sequence labelers based on RNNs and Transformers. 17.1 • (M OSTLY ) ENGLISH WORD CLASSES 363 17.1 (Mostly) English Word Classes Until now we have been using part-of-speech terms likenoun and verb rather freely. In this section we give more complete deﬁnitions. While word classes do have semantic tendencies—adjectives, for example, often describe properties and nouns people— parts of speech are deﬁned instead based on their grammatical relationship with neighboring words or the morphological properties about their afﬁxes. Tag Description Example Open Class ADJ Adjective: noun modiﬁers describing properties red, young, awesome ADV Adverb: verb modiﬁers of time, place, manner very, slowly, home, yesterday NOUN words for persons, places, things, etc. algorithm, cat, mango, beauty VERB words for actions and processes draw, provide, go PROPN Proper noun: name of a person, organization, place, etc.. Regina, IBM, Colorado INTJ Interjection: exclamation, greeting, yes/no response, etc. oh, um, yes, hello Closed Class Words ADP Adposition (Preposition/Postposition): marks a noun’s spacial, temporal, or other relation in, on, by, under AUX Auxiliary: helping verb marking tense, aspect, mood, etc., can, may, should, are CCONJ Coordinating Conjunction: joins two phrases/clauses and, or, but DET Determiner: marks noun phrase properties a, an, the, this NUM Numeral one, two, 2026, 11:00, hundred PART Particle: a function word that must be associated with an- other word ’s, not, (inﬁnitive) to PRON Pronoun: a shorthand for referring to an entity or event she, who, I, others SCONJ Subordinating Conjunction: joins a main clause with a subordinate clause such as a sentential complement whether, because Other PUNCT Punctuation ˙, , () SYM Symbols like $ or emoji $, % X Other asdf, qwfg Figure 17.1 The 17 parts of speech in the Universal Dependencies tagset (de Marneffe et al., 2021). Features can be added to make ﬁner-grained distinctions (with properties like number, case, deﬁniteness, and so on). Parts of speech fall into two broad categories: closed class and open class .closed class open class Closed classes are those with relatively ﬁxed membership, such as prepositions— new prepositions are rarely coined. By contrast, nouns and verbs are open classes— new nouns and verbs likeiPhone or to fax are continually being created or borrowed. Closed class words are generally function words like of, it, and, or you, which tendfunction word to be very short, occur frequently, and often have structuring uses in grammar. Four major open classes occur in the languages of the world: nouns (including proper nouns), verbs, adjectives, and adverbs, as well as the smaller open class of interjections. English has all ﬁve, although not every language does. Nouns are words for people, places, or things, but include others as well. Com-noun mon nouns include concrete terms like cat and mango, abstractions like algorithmcommon noun and beauty, and verb-like terms likepacing as in His pacing to and fro became quite annoying. Nouns in English can occur with determiners ( a goat, this bandwidth ) take possessives (IBM’s annual revenue), and may occur in the plural (goats, abaci). Many languages, including English, divide common nouns into count nouns andcount noun mass nouns. Count nouns can occur in the singular and plural ( goat/goats, rela-mass noun tionship/relationships) and can be counted ( one goat, two goats ). Mass nouns are used when something is conceptualized as a homogeneous group. Sosnow, salt, and communism are not counted (i.e.,two snows or two communisms). Proper nouns,proper noun like Regina, Colorado, and IBM, are names of speciﬁc persons or entities. 364 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES Verbs refer to actions and processes, including main verbs like draw, provide,verb and go. English verbs have inﬂections (non-third-person-singular (eat), third-person- singular (eats), progressive (eating), past participle ( eaten)). While many scholars believe that all human languages have the categories of noun and verb, others have argued that some languages, such as Riau Indonesian and Tongan, don’t even make this distinction (Broschart 1997; Evans 2000; Gil 2000) . Adjectives often describe properties or qualities of nouns, like color ( white,adjective black), age ( old, young), and value ( good, bad), but there are languages without adjectives. In Korean, for example, the words corresponding to English adjectives act as a subclass of verbs, so what is in English an adjective “beautiful” acts in Korean like a verb meaning “to be beautiful”. Adverbs are a hodge-podge. All the italicized words in this example are adverbs:adverb Actually, I ran home extremely quickly yesterday Adverbs generally modify something (often verbs, hence the name “adverb”, but also other adverbs and entire verb phrases). Directional adverbs or locative ad-locative verbs (home, here, downhill) specify the direction or location of some action;degreedegree adverbs (extremely, very, somewhat) specify the extent of some action, process, or property; manner adverbs (slowly, slinkily, delicately) describe the manner of somemanner action or process; andtemporal adverbs describe the time that some action or eventtemporal took place (yesterday, Monday). Interjections (oh, hey, alas, uh, um) are a smaller open class that also includesinterjection greetings (hello, goodbye) and question responses (yes, no, uh-huh). English adpositions occur before nouns, hence are calledprepositions. They canpreposition indicate spatial or temporal relations, whether literal (on it, before then, by the house) or metaphorical (on time, with gusto, beside herself), and relations like marking the agent in Hamlet was written by Shakespeare. A particle resembles a preposition or an adverb and is used in combination withparticle a verb. Particles often have extended meanings that aren’t quite the same as the prepositions they resemble, as in the particle over in she turned the paper over . A verb and a particle acting as a single unit is called a phrasal verb. The meaningphrasal verb of phrasal verbs is often non-compositional—not predictable from the individual meanings of the verb and the particle. Thus, turn down means ‘reject’, rule out ‘eliminate’, andgo on ‘continue’. Determiners like this and that (this chapter, that page) can mark the start of andeterminer English noun phrase. Articles like a, an, and the, are a type of determiner that markarticle discourse properties of the noun and are quite frequent; the is the most common word in written English, with a and an right behind. Conjunctions join two phrases, clauses, or sentences. Coordinating conjunc-conjunction tions like and, or, and but join two elements of equal status. Subordinating conjunc- tions are used when one of the elements has some embedded status. For example, the subordinating conjunction that in “I thought that you might like some milk”links the main clause I thought with the subordinate clause you might like some milk. This clause is called subordinate because this entire clause is the “content” of the main verb thought. Subordinating conjunctions like that which link a verb to its argument in this way are also called complementizers.complementizer Pronouns act as a shorthand for referring to an entity or event. Personal pro-pronoun nouns refer to persons or entities ( you, she, I, it, me, etc.). Possessive pronouns are forms of personal pronouns that indicate either actual possession or more often just an abstract relation between the person and some object (my, your, his, her, its, one’s, our, their). Wh-pronouns (what, who, whom, whoever) are used in certain questionwh 17.2 • P ART-OF-SPEECH TAGGING 365 forms, or act as complementizers (Frida, who married Diego. . .). Auxiliary verbs mark semantic features of a main verb such as its tense, whetherauxiliary it is completed (aspect), whether it is negated (polarity), and whether an action is necessary, possible, suggested, or desired (mood). English auxiliaries include the copula verb be, the two verbs do and have, forms, as well as modal verbs used tocopula modal mark the mood associated with the event depicted by the main verb: can indicates ability or possibility, may permission or possibility, must necessity. An English-speciﬁc tagset, the Penn Treebank tagset (Marcus et al., 1993), shown in Fig. 17.2, has been used to label many syntactically annotated corpora like the Penn Treebank corpora, so it is worth knowing about. Tag Description Example Tag Description Example Tag Description Example CC coord. conj. and, but, or NNP proper noun, sing. IBM TO inﬁnitive to to CD cardinal number one, two NNPS proper noun, plu. Carolinas UH interjection ah, oops DT determiner a, the NNS noun, plural llamas VB verb base eat EX existential ‘there’ there PDT predeterminer all, both VBD verb past tense ate FW foreign word mea culpa POS possessive ending ’s VBG verb gerund eating IN preposition/ subordin-conj of, in, by PRP personal pronoun I, you, he VBN verb past partici- ple eaten JJ adjective yellow PRP$ possess. pronoun your VBP verb non-3sg-pr eat JJR comparative adj bigger RB adverb quickly VBZ verb 3sg pres eats JJS superlative adj wildest RBR comparative adv faster WDT wh-determ. which, that LS list item marker 1, 2, One RBS superlatv. adv fastest WP wh-pronoun what, who MD modal can, should RP particle up, off WP$ wh-possess. whose NN sing or mass noun llama SYM symbol +, %, & WRB wh-adverb how, where Figure 17.2 Penn Treebank core 36 part-of-speech tags. Below we show some examples with each word tagged according to both the UD (in blue) and Penn (in red) tagsets. Notice that the Penn tagset distinguishes tense and participles on verbs, and has a special tag for the existentialthere construction in English. Note that since London Journal of Medicine is a proper noun, both tagsets mark its component nouns as PROPN/NNP, including journal and medicine, which might otherwise be labeled as common nouns (NOUN/NN). (17.1) There/ PRON /EX are/VERB /VBP 70/NUM /CD children/NOUN /NNS there/ADV/RB ./PUNC /. (17.2) Preliminary/ ADJ /JJ ﬁndings/NOUN /NNS were/AUX/VBD reported/VERB /VBN in/ADP /IN today/NOUN /NN ’s/PART/POS London/PROPN /NNP Journal/PROPN /NNP of/ADP /IN Medicine/PROPN /NNP 17.2 Part-of-Speech Tagging Part-of-speech tagging is the process of assigning a part-of-speech to each word inpart-of-speech tagging a text. The input is a sequence x1,x2,...,xn of (tokenized) words and a tagset, and the output is a sequence y1,y2,...,yn of tags, each output yi corresponding exactly to one input xi, as shown in the intuition in Fig. 17.3. Tagging is a disambiguation task; words are ambiguous —have more than oneambiguous possible part-of-speech—and the goal is to ﬁnd the correct tag for the situation. For example, book can be a verb ( book that ﬂight) or a noun ( hand me that book ). That can be a determiner ( Does that ﬂight serve dinner ) or a complementizer ( I 366 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES will NOUN AUX VERB DET NOUN Janet back the bill Part of Speech Tagger x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 Figure 17.3 The task of part-of-speech tagging: mapping from input words x1,x2,...,xn to output POS tags y1,y2,...,yn . thought that your ﬂight was earlier ). The goal of POS-tagging is to resolve theseambiguity resolution ambiguities, choosing the proper tag for the context. The accuracy of part-of-speech tagging algorithms (the percentage of test setaccuracy tags that match human gold labels) is extremely high. One study found accuracies over 97% across 15 languages from the Universal Dependency (UD) treebank (Wu and Dredze, 2019). Accuracies on various English treebanks are also 97% (no matter the algorithm; HMMs, CRFs, BERT perform similarly). This 97% number is also about the human performance on this task, at least for English (Manning, 2011). Types: WSJ Brown Unambiguous (1 tag) 44,432 ( 86%) 45,799 ( 85%) Ambiguous (2+ tags) 7,025 ( 14%) 8,050 ( 15%) Tokens: Unambiguous (1 tag) 577,421 ( 45%) 384,349 ( 33%) Ambiguous (2+ tags) 711,780 ( 55%) 786,646 ( 67%) Figure 17.4 Tag ambiguity in the Brown and WSJ corpora (Treebank-3 45-tag tagset). We’ll introduce algorithms for the task in the next few sections, but ﬁrst let’s explore the task. Exactly how hard is it? Fig. 17.4 shows that most word types (85-86%) are unambiguous (Janet is always NNP, hesitantly is always RB). But the ambiguous words, though accounting for only 14-15% of the vocabulary, are very common, and 55-67% of word tokens in running text are ambiguous. Particularly ambiguous common words include that, back, down, put and set; here are some examples of the 6 different parts of speech for the word back: earnings growth took a back/JJ seat a small building in the back/NN a clear majority of senators back/VBP the bill Dave began to back/VB toward the door enable the country to buy back/RP debt I was twenty-one back/RB then Nonetheless, many words are easy to disambiguate, because their different tags aren’t equally likely. For example, a can be a determiner or the letter a, but the determiner sense is much more likely. This idea suggests a useful baseline: given an ambiguous word, choose the tag which is most frequent in the training corpus. This is a key concept: Most Frequent Class Baseline: Always compare a classiﬁer against a baseline at least as good as the most frequent class baseline (assigning each token to the class it occurred in most often in the training set). 17.3 • N AMED ENTITIES AND NAMED ENTITY TAGGING 367 The most-frequent-tag baseline has an accuracy of about 92% 1. The baseline thus differs from the state-of-the-art and human ceiling (97%) by only 5%. 17.3 Named Entities and Named Entity Tagging Part of speech tagging can tell us that words like Janet, Stanford University, and Colorado are all proper nouns; being a proper noun is a grammatical property of these words. But viewed from a semantic perspective, these proper nouns refer to different kinds of entities: Janet is a person, Stanford University is an organization, and Colorado is a location. Here we re-introduce the concept of a named entity, which was also introducednamed entity in Section 11.5 for readers who haven’t yet read Chapter 11. A named entity is, roughly speaking, anything that can be referred to with anamed entity proper name: a person, a location, an organization. The task ofnamed entity recog- nition (NER) is to ﬁnd spans of text that constitute proper names and tag the type ofnamed entity recognition NER the entity. Four entity tags are most common: PER (person), LOC (location), ORG (organization), or GPE (geo-political entity). However, the term named entity is commonly extended to include things that aren’t entities per se, including dates, times, and other kinds of temporal expressions, and even numerical expressions like prices. Here’s an example of the output of an NER tagger: Citing high fuel prices, [ ORG United Airlines] said [TIME Friday] it has increased fares by [ MONEY $6] per round trip on ﬂights to some cities also served by lower-cost carriers. [ ORG American Airlines], a unit of [ORG AMR Corp.], immediately matched the move, spokesman [PER Tim Wagner] said. [ ORG United], a unit of [ORG UAL Corp.], said the increase took effect [ TIME Thursday] and applies to most routes where it competes against discount carriers, such as [LOC Chicago] to [LOC Dallas] and [LOC Denver] to [LOC San Francisco]. The text contains 13 mentions of named entities including 5 organizations, 4 loca- tions, 2 times, 1 person, and 1 mention of money. Figure 17.5 shows typical generic named entity types. Many applications will also need to use speciﬁc entity types like proteins, genes, commercial products, or works of art. Type Tag Sample Categories Example sentences People PER people, characters Turing is a giant of computer science. Organization ORG companies, sports teams The IPCC warned about the cyclone. Location LOC regions, mountains, seas Mt. Sanitas is in Sunshine Canyon. Geo-Political Entity GPE countries, states Palo Alto is raising the fees for parking. Figure 17.5 A list of generic named entity types with the kinds of entities they refer to. Named entity tagging is a useful ﬁrst step in lots of natural language processing tasks. In sentiment analysis we might want to know a consumer’s sentiment toward a particular entity. Entities are a useful ﬁrst stage in question answering, or for linking text to information in structured knowledge sources like Wikipedia. And named entity tagging is also central to tasks involving building semantic representations, like extracting events and the relationship between participants. 1 In English, on the WSJ corpus, tested on sections 22-24. 368 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES Unlike part-of-speech tagging, where there is no segmentation problem since each word gets one tag, the task of named entity recognition is to ﬁnd and label spans of text, and is difﬁcult partly because of the ambiguity of segmentation; we need to decide what’s an entity and what isn’t, and where the boundaries are. Indeed, most words in a text will not be named entities. Another difﬁculty is caused by type ambiguity. The mention JFK can refer to a person, the airport in New York, or any number of schools, bridges, and streets around the United States. Some examples of this kind of cross-type confusion are given in Figure 17.6. [PER Washington] was born into slavery on the farm of James Burroughs. [ORG Washington] went up 2 games to 1 in the four-game series. Blair arrived in [LOC Washington] for what may well be his last state visit. In June, [GPE Washington] passed a primary seatbelt law. Figure 17.6 Examples of type ambiguities in the use of the name Washington. The standard approach to sequence labeling for a span-recognition problem like NER is BIO tagging (Ramshaw and Marcus, 1995). This is a method that allows us to treat NER like a word-by-word sequence labeling task, via tags that capture both the boundary and the named entity type. Consider the following sentence: [PER Jane Villanueva ] of [ORG United] , a unit of [ ORG United Airlines Holding] , said the fare applies to the [LOC Chicago ] route. Figure 17.7 shows the same excerpt represented with BIO tagging, as well asBIO variants called IO tagging and BIOES tagging. In BIO tagging we label any token that begins a span of interest with the label B, tokens that occur inside a span are tagged with an I, and any tokens outside of any span of interest are labeled O. While there is only one O tag, we’ll have distinctB and I tags for each named entity class. The number of tags is thus 2 n + 1 tags, where n is the number of entity types. BIO tagging can represent exactly the same information as the bracketed notation, but has the advantage that we can represent the task in the same simple sequence modeling way as part-of-speech tagging: assigning a single label yi to each input word xi: Words IO Label BIO Label BIOES Label Jane I-PER B-PER B-PER Villanueva I-PER I-PER E-PER of O O O United I-ORG B-ORG B-ORG Airlines I-ORG I-ORG I-ORG Holding I-ORG I-ORG E-ORG discussed O O O the O O O Chicago I-LOC B-LOC S-LOC route O O O . O O O Figure 17.7 NER as a sequence model, showing IO, BIO, and BIOES taggings. We’ve also shown two variant tagging schemes: IO tagging, which loses some information by eliminating the B tag, and BIOES tagging, which adds an end tag E for the end of a span, and a span tag S for a span consisting of only one word. A sequence labeler (HMM, CRF, RNN, Transformer, etc.) is trained to label each token in a text with tags that indicate the presence (or absence) of particular kinds of named entities. 17.4 • HMM P ART-OF-SPEECH TAGGING 369 17.4 HMM Part-of-Speech Tagging In this section we introduce our ﬁrst sequence labeling algorithm, the Hidden Markov Model, and show how to apply it to part-of-speech tagging. Recall that a sequence labeler is a model whose job is to assign a label to each unit in a sequence, thus mapping a sequence of observations to a sequence of labels of the same length. The HMM is a classic model that introduces many of the key concepts of sequence modeling that we will see again in more modern models. An HMM is a probabilistic sequence model: given a sequence of units (words, letters, morphemes, sentences, whatever), it computes a probability distribution over possible sequences of labels and chooses the best label sequence. 17.4.1 Markov Chains The HMM is based on augmenting the Markov chain. A Markov chain is a modelMarkov chain that tells us something about the probabilities of sequences of random variables, states, each of which can take on values from some set. These sets can be words, or tags, or symbols representing anything, for example the weather. A Markov chain makes a very strong assumption that if we want to predict the future in the sequence, all that matters is the current state. All the states before the current state have no im- pact on the future except via the current state. It’s as if to predict tomorrow’s weather you could examine today’s weather but you weren’t allowed to look at yesterday’s weather. WARM3HOT1 COLD2 .8 .6 .1 .1 .3 .6 .1 .1 .3 charminguniformly are .1 .4 .5 .5 .5 .2 .6 .2 (a) (b) Figure 17.8 A Markov chain for weather (a) and one for words (b), showing states and transitions. A start distribution π is required; setting π = [0.1,0.7,0.2] for (a) would mean a probability 0.7 of starting in state 2 (cold), probability 0.1 of starting in state 1 (hot), etc. More formally, consider a sequence of state variables q1,q2,...,qi. A Markov model embodies the Markov assumption on the probabilities of this sequence: thatMarkov assumption when predicting the future, the past doesn’t matter, only the present. Markov Assumption: P(qi = a\|q1...qi−1) =P(qi = a\|qi−1) (17.3) Figure 17.8a shows a Markov chain for assigning a probability to a sequence of weather events, for which the vocabulary consists of HOT, COLD , and WARM. The states are represented as nodes in the graph, and the transitions, with their probabil- ities, as edges. The transitions are probabilities: the values of arcs leaving a given state must sum to 1. Figure 17.8b shows a Markov chain for assigning a probabil- ity to a sequence of words w1...wt . This Markov chain should be familiar; in fact, it represents a bigram language model, with each edge expressing the probability p(wi\|wj)! Given the two models in Fig. 17.8, we can assign a probability to any sequence from our vocabulary. 370 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES Formally, a Markov chain is speciﬁed by the following components: Q = q1q2 ... qN a set of N states A = a11a12 ... aN1 ... aNN a transition probability matrix A, each ai j represent- ing the probability of moving from state i to state j, s.t.∑n j=1 ai j = 1 ∀i π = π1,π2,...,πN an initial probability distribution over states. πi is the probability that the Markov chain will start in state i. Some states j may haveπj = 0, meaning that they cannot be initial states. Also, ∑n i=1 πi = 1 Before you go on, use the sample probabilities in Fig. 17.8a (withπ = [0.1,0.7,0.2]) to compute the probability of each of the following sequences: (17.4) hot hot hot hot (17.5) cold hot cold hot What does the difference in these probabilities tell you about a real-world weather fact encoded in Fig. 17.8a? 17.4.2 The Hidden Markov Model A Markov chain is useful when we need to compute a probability for a sequence of observable events. In many cases, however, the events we are interested in are hidden: we don’t observe them directly. For example we don’t normally observehidden part-of-speech tags in a text. Rather, we see words, and must infer the tags from the word sequence. We call the tags hidden because they are not observed. A hidden Markov model (HMM) allows us to talk about both observed eventshidden Markov model (like words that we see in the input) andhidden events (like part-of-speech tags) that we think of as causal factors in our probabilistic model. An HMM is speciﬁed by the following components: Q = q1q2 ... qN a set of N states A = a11 ... ai j ... aNN a transition probability matrix A, each ai j representing the probability of moving from state i to state j, s.t. ∑N j=1 ai j = 1 ∀i B = bi(ot ) a sequence of observation likelihoods, also called emission probabili- ties, each expressing the probability of an observation ot (drawn from a vocabulary V = v1,v2,...,vV ) being generated from a state qi π = π1,π2,...,πN an initial probability distribution over states. πi is the probability that the Markov chain will start in state i. Some states j may have πj = 0, meaning that they cannot be initial states. Also, ∑n i=1 πi = 1 The HMM is given as inputO = o1o2 ... oT : a sequence of T observations, each one drawn from the vocabulary V . A ﬁrst-order hidden Markov model instantiates two simplifying assumptions. First, as with a ﬁrst-order Markov chain, the probability of a particular state depends only on the previous state: Markov Assumption: P(qi\|q1,...,qi−1) =P(qi\|qi−1) (17.6) Second, the probability of an output observationoi depends only on the state that produced the observation qi and not on any other states or any other observations: Output Independence: P(oi\|q1,... qi,..., qT ,o1,..., oi,..., oT ) =P(oi\|qi) (17.7) 17.4 • HMM P ART-OF-SPEECH TAGGING 371 17.4.3 The components of an HMM tagger An HMM has two components, theA and B probabilities, both estimated by counting on a tagged training corpus. (For this example we’ll use the tagged WSJ corpus.) The A matrix contains the tag transition probabilities P(ti\|ti−1) which represent the probability of a tag occurring given the previous tag. For example, modal verbs like will are very likely to be followed by a verb in the base form, a VB, likerace, so we expect this probability to be high. We compute the maximum likelihood estimate of this transition probability by counting, out of the times we see the ﬁrst tag in a labeled corpus, how often the ﬁrst tag is followed by the second: P(ti\|ti−1) =C(ti−1,ti) C(ti−1) (17.8) In the WSJ corpus, for example, MD occurs 13124 times of which it is followed by VB 10471, for an MLE estimate of P(V B\|MD) =C(MD,V B) C(MD) = 10471 13124 = .80 (17.9) The B emission probabilities, P(wi\|ti), represent the probability, given a tag (say MD), that it will be associated with a given word (say will). The MLE of the emis- sion probability is P(wi\|ti) =C(ti,wi) C(ti) (17.10) Of the 13124 occurrences of MD in the WSJ corpus, it is associated with will 4046 times: P(will\|MD) =C(MD,will) C(MD) = 4046 13124 = .31 (17.11) We saw this kind of Bayesian modeling in Chapter 4; recall that this likelihood term is not asking “which is the most likely tag for the word will?” That would be the posterior P(MD\|will). Instead, P(will\|MD) answers the slightly counterintuitive question “If we were going to generate a MD, how likely is it that this modal would be will?” NN3VB1 MD2 a22 a11 a12 a21 a13 a33 a32 a23 a31 P("aardvark" \| NN) ... P(“will” \| NN) ... P("the" \| NN) ... P(“back” \| NN) ... P("zebra" \| NN) B3 P("aardvark" \| VB) ... P(“will” \| VB) ... P("the" \| VB) ... P(“back” \| VB) ... P("zebra" \| VB) B1 P("aardvark" \| MD) ... P(“will” \| MD) ... P("the" \| MD) ... P(“back” \| MD) ... P("zebra" \| MD) B2 Figure 17.9 An illustration of the two parts of an HMM representation: the A transition probabilities used to compute the prior probability, and the B observation likelihoods that are associated with each state, one likelihood for each possible observation word. 372 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES The A transition probabilities, and B observation likelihoods of the HMM are illustrated in Fig. 17.9 for three states in an HMM part-of-speech tagger; the full tagger would have one state for each tag. 17.4.4 HMM tagging as decoding For any model, such as an HMM, that contains hidden variables, the task of deter- mining the hidden variables sequence corresponding to the sequence of observations is called decoding. More formally,decoding Decoding: Given as input an HMM λ = (A,B) and a sequence of ob- servations O = o1,o2,...,oT , ﬁnd the most probable sequence of states Q = q1q2q3 ... qT . For part-of-speech tagging, the goal of HMM decoding is to choose the tag sequence t1 ...tn that is most probable given the observation sequence of n words w1 ... wn: ˆt1:n = argmax t1...tn P(t1 ...tn\|w1 ... wn) (17.12) The way we’ll do this in the HMM is to use Bayes’ rule to instead compute: ˆt1:n = argmax t1...tn P(w1 ... wn\|t1 ...tn)P(t1 ...tn) P(w1 ... wn) (17.13) Furthermore, we simplify Eq. 17.13 by dropping the denominator P(wn 1): ˆt1:n = argmax t1...tn P(w1 ... wn\|t1 ...tn)P(t1 ...tn) (17.14) HMM taggers make two further simplifying assumptions. The ﬁrst (output in- dependence, from Eq. 17.7) is that the probability of a word appearing depends only on its own tag and is independent of neighboring words and tags: P(w1 ... wn\|t1 ...tn) ≈ n∏ i=1 P(wi\|ti) (17.15) The second assumption (the Markov assumption, Eq. 17.6) is that the probability of a tag is dependent only on the previous tag, rather than the entire tag sequence; P(t1 ...tn) ≈ n∏ i=1 P(ti\|ti−1) (17.16) Plugging the simplifying assumptions from Eq. 17.15 and Eq. 17.16 into Eq. 17.14 results in the following equation for the most probable tag sequence from a bigram tagger: ˆt1:n = argmax t1...tn P(t1 ...tn\|w1 ... wn) ≈argmax t1...tn n∏ i=1 emission   P(wi\|ti) transition   P(ti\|ti−1) (17.17) The two parts of Eq. 17.17 correspond neatly to the B emission probability and A transition probability that we just deﬁned above! 17.4 • HMM P ART-OF-SPEECH TAGGING 373 function VITERBI (observations of len T,state-graph of len N) returns best-path, path-prob create a path probability matrix viterbi[N,T] for each state s from 1 to N do ; initialization step viterbi[s,1]←πs ∗bs(o1) backpointer[s,1]←0 for each time step t from 2 to T do ; recursion step for each state s from 1 to N do viterbi[s,t]← N max s′=1 viterbi[s′,t −1] ∗as′,s ∗bs(ot ) backpointer[s,t]← N argmax s′=1 viterbi[s′,t −1] ∗as′,s ∗bs(ot ) bestpathprob← N max s=1 viterbi[s,T ] ; termination step bestpathpointer← N argmax s=1 viterbi[s,T ] ; termination step bestpath←the path starting at state bestpathpointer, that follows backpointer[] to states back in time return bestpath, bestpathprob Figure 17.10 Viterbi algorithm for ﬁnding the optimal sequence of tags. Given an observation sequence and an HMM λ = (A,B), the algorithm returns the state path through the HMM that assigns maximum likelihood to the observation sequence. 17.4.5 The Viterbi Algorithm The decoding algorithm for HMMs is the Viterbi algorithm shown in Fig. 17.10.Viterbi algorithm As an instance of dynamic programming, Viterbi resembles the dynamic program- ming minimum edit distance algorithm of Chapter 2. The Viterbi algorithm ﬁrst sets up a probability matrix or lattice, with one col- umn for each observation ot and one row for each state in the state graph. Each col- umn thus has a cell for each stateqi in the single combined automaton. Figure 17.11 shows an intuition of this lattice for the sentence Janet will back the bill. Each cell of the lattice, vt ( j), represents the probability that the HMM is in state j after seeing the ﬁrst t observations and passing through the most probable state sequence q1,...,qt−1, given the HMM λ. The value of each cell vt ( j) is computed by recursively taking the most probable path that could lead us to this cell. Formally, each cell expresses the probability vt ( j) = max q1,...,qt−1 P(q1...qt−1,o1,o2 ... ot ,qt = j\|λ) (17.18) We represent the most probable path by taking the maximum over all possible previous state sequences max q1,...,qt−1 . Like other dynamic programming algorithms, Viterbi ﬁlls each cell recursively. Given that we had already computed the probabil- ity of being in every state at timet −1, we compute the Viterbi probability by taking the most probable of the extensions of the paths that lead to the current cell. For a given state qj at time t, the value vt ( j) is computed as vt ( j) = N max i=1 vt−1(i) ai j bj(ot ) (17.19) The three factors that are multiplied in Eq. 17.19 for extending the previous paths to compute the Viterbi probability at time t are 374 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES JJ NNP NNP NNP MD MD MD MD VB VB JJ JJ JJ NN NN RB RBRBRB DT DT DT DT NNP Janet will back the bill NN VB MD NN VB JJ RB NNP DT NN VB Figure 17.11 A sketch of the lattice for Janet will back the bill, showing the possible tags (qi) for each word and highlighting the path corresponding to the correct tag sequence through the hidden states. States (parts of speech) which have a zero probability of generating a particular word according to the B matrix (such as the probability that a determiner DT will be realized as Janet) are greyed out. vt−1(i) the previous Viterbi path probability from the previous time step ai j the transition probability from previous state qi to current state qj bj(ot ) the state observation likelihood of the observation symbol ot given the current state j 17.4.6 Working through an example Let’s tag the sentence Janet will back the bill ; the goal is the correct series of tags (see also Fig. 17.11): (17.20) Janet/NNP will/MD back/VB the/DT bill/NN NNP MD VB JJ NN RB DT <s> 0.2767 0.0006 0.0031 0.0453 0.0449 0.0510 0.2026 NNP 0.3777 0.0110 0.0009 0.0084 0.0584 0.0090 0.0025 MD 0.0008 0.0002 0.7968 0.0005 0.0008 0.1698 0.0041 VB 0.0322 0.0005 0.0050 0.0837 0.0615 0.0514 0.2231 JJ 0.0366 0.0004 0.0001 0.0733 0.4509 0.0036 0.0036 NN 0.0096 0.0176 0.0014 0.0086 0.1216 0.0177 0.0068 RB 0.0068 0.0102 0.1011 0.1012 0.0120 0.0728 0.0479 DT 0.1147 0.0021 0.0002 0.2157 0.4744 0.0102 0.0017 Figure 17.12 The A transition probabilities P(ti\|ti−1) computed from the WSJ corpus with- out smoothing. Rows are labeled with the conditioning event; thus P(V B\|MD) is 0.7968. <s>is the start token. Let the HMM be deﬁned by the two tables in Fig. 17.12 and Fig. 17.13. Fig- ure 17.12 lists the ai j probabilities for transitioning between the hidden states (part- of-speech tags). Figure 17.13 expresses the bi(ot ) probabilities, the observation likelihoods of words given tags. This table is (slightly simpliﬁed) from counts in the WSJ corpus. So the word Janet only appears as an NNP, back has 4 possible parts 17.4 • HMM P ART-OF-SPEECH TAGGING 375 Janet will back the bill NNP 0.000032 0 0 0.000048 0 MD 0 0.308431 0 0 0 VB 0 0.000028 0.000672 0 0.000028 JJ 0 0 0.000340 0 0 NN 0 0.000200 0.000223 0 0.002337 RB 0 0 0.010446 0 0 DT 0 0 0 0.506099 0 Figure 17.13 Observation likelihoods B computed from the WSJ corpus without smooth- ing, simpliﬁed slightly. of speech, and the word the can appear as a determiner or as an NNP (in titles like “Somewhere Over the Rainbow” all words are tagged as NNP). π P(NNP\|start) = .28 * P(MD\|MD) = 0 * P(MD\|NNP) .000009.01 = .9e-8 v1(2)= .0006 x 0 = 0 v1(1) = .28 .000032 = .000009 t MDq2 q1 o1 Janet billwill o2 o3 back VB JJ v1(3)= .0031 x 0 = 0 v1(4)= . 0450=0 o4 P(MD\|VB) = 0 * P(MD\|JJ) = 0 P(VB\|start) = .0031 P(JJ \|start) =.045 backtrace q3 q4 the NNq5 RBq6 DTq7 v2(2) = max * .308 = 2.772e-8 v2(5)= max * .0002 = .0000000001 v2(3)= max * .000028 = 2.5e-13 v3(6)= max * .0104 v3(5)= max * . 000223 v3(4)= max * .00034 v3(3)= max * .00067 v1(5) v1(6) v1(7) v2(1) v2(4) v2(6) v2(7) backtrace * P(RB\|NN) * P(NN\|NN) start start start start start o5 NNP P(MD\|start) = .0006 Figure 17.14 The ﬁrst few entries in the individual state columns for the Viterbi algorithm. Each cell keeps the probability of the best path so far and a pointer to the previous cell along that path. We have only ﬁlled out columns 1 and 2; to avoid clutter most cells with value 0 are left empty. The rest is left as an exercise for the reader. After the cells are ﬁlled in, backtracing from the end state, we should be able to reconstruct the correct state sequence NNP MD VB DT NN. Figure 17.14 shows a ﬂeshed-out version of the sketch we saw in Fig. 17.11, the Viterbi lattice for computing the best hidden state sequence for the observation sequence Janet will back the bill. There are N = 5 state columns. We begin in column 1 (for the word Janet) by setting the Viterbi value in each cell to the product of theπ transition probability (the start probability for that statei, which we get from the<s>entry of Fig. 17.12), and 376 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES the observation likelihood of the word Janet given the tag for that cell. Most of the cells in the column are zero since the word Janet cannot be any of those tags. The reader should ﬁnd this in Fig. 17.14. Next, each cell in the will column gets updated. For each state, we compute the value viterbi[s,t] by taking the maximum over the extensions of all the paths from the previous column that lead to the current cell according to Eq. 17.19. We have shown the values for the MD, VB, and NN cells. Each cell gets the max of the 7 val- ues from the previous column, multiplied by the appropriate transition probability; as it happens in this case, most of them are zero from the previous column. The re- maining value is multiplied by the relevant observation probability, and the (trivial) max is taken. In this case the ﬁnal value, 2.772e-8, comes from the NNP state at the previous column. The reader should ﬁll in the rest of the lattice in Fig. 17.14 and backtrace to see whether or not the Viterbi algorithm returns the gold state sequence NNP MD VB DT NN. 17.5 Conditional Random Fields (CRFs) While the HMM is a useful and powerful model, it turns out that HMMs need a number of augmentations to achieve high accuracy. For example, in POS tagging as in other tasks, we often run into unknown words: proper names and acronymsunknown words are created very often, and even new common nouns and verbs enter the language at a surprising rate. It would be great to have ways to add arbitrary features to help with this, perhaps based on capitalization or morphology (words starting with capital letters are likely to be proper nouns, words ending with -ed tend to be past tense (VBD or VBN), etc.) Or knowing the previous or following words might be a useful feature (if the previous word is the, the current tag is unlikely to be a verb). Although we could try to hack the HMM to ﬁnd ways to incorporate some of these, in general it’s hard for generative models like HMMs to add arbitrary features directly into the model in a clean way. We’ve already seen a model for combining arbitrary features in a principled way: log-linear models like the logistic regression model of Chapter 5! But logistic regression isn’t a sequence model; it assigns a class to a single observation. Luckily, there is a discriminative sequence model based on log-linear models: the conditional random ﬁeld (CRF). We’ll describe here the linear chain CRF ,CRF the version of the CRF most commonly used for language processing, and the one whose conditioning closely matches the HMM. Assuming we have a sequence of input words X = x1...xn and want to compute a sequence of output tags Y = y1...yn. In an HMM to compute the best tag sequence that maximizes P(Y \|X) we rely on Bayes’ rule and the likelihoodP(X\|Y ): ˆY = argmax Y p(Y \|X) = argmax Y p(X\|Y )p(Y ) = argmax Y ∏ i p(xi\|yi) ∏ i p(yi\|yi−1) (17.21) In a CRF, by contrast, we compute the posterior p(Y \|X) directly, training the CRF 17.5 • C ONDITIONAL RANDOM FIELDS (CRF S) 377 to discriminate among the possible tag sequences: ˆY = argmax Y ∈Y P(Y \|X) (17.22) However, the CRF does not compute a probability for each tag at each time step. In- stead, at each time step the CRF computes log-linear functions over a set of relevant features, and these local features are aggregated and normalized to produce a global probability for the whole sequence. Let’s introduce the CRF more formally, again using X and Y as the input and output sequences. A CRF is a log-linear model that assigns a probability to an entire output (tag) sequenceY , out of all possible sequencesY, given the entire input (word) sequence X. We can think of a CRF as like a giant sequential version of the multinomial logistic regression algorithm we saw for text categorization. Recall that we introduced the feature function f in regular multinomial logistic regression for text categorization as a function of a tuple: the input text x and a single class y (page 86). In a CRF, we’re dealing with a sequence, so the functionF maps an entire input sequence X and an entire output sequence Y to a feature vector. Let’s assume we have K features, with a weight wk for each feature Fk: p(Y \|X) = exp ( K∑ k=1 wkFk(X,Y ) ) ∑ Y ′∈Y exp ( K∑ k=1 wkFk(X,Y ′) ) (17.23) It’s common to also describe the same equation by pulling out the denominator into a function Z(X): p(Y \|X) = 1 Z(X)exp ( K∑ k=1 wkFk(X,Y ) ) (17.24) Z(X) = ∑ Y ′∈Y exp ( K∑ k=1 wkFk(X,Y ′) ) (17.25) We’ll call theseK functions Fk(X,Y ) global features, since each one is a property of the entire input sequence X and output sequence Y . We compute them by decom- posing into a sum of local features for each position i in Y : Fk(X,Y ) = n∑ i=1 fk(yi−1,yi,X,i) (17.26) Each of these local features fk in a linear-chain CRF is allowed to make use of the current output token yi, the previous output token yi−1, the entire input string X (or any subpart of it), and the current position i. This constraint to only depend on the current and previous output tokens yi and yi−1 are what characterizes a linear chain CRF. As we will see, this limitation makes it possible to use versions of thelinear chain CRF efﬁcient Viterbi and Forward-Backwards algorithms from the HMM. A general CRF, by contrast, allows a feature to make use of any output token, and are thus necessary for tasks in which the decision depend on distant output tokens, like yi−4. General CRFs require more complex inference, and are less commonly used for language processing. 378 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES 17.5.1 Features in a CRF POS Tagger Let’s look at some of these features in detail, since the reason to use a discriminative sequence model is that it’s easier to incorporate a lot of features.2 Again, in a linear-chain CRF, each local feature fk at position i can depend on any information from: (yi−1,yi,X,i). So some legal features representing common situations might be the following: 1 {xi = the, yi = DET} 1 {yi = PROPN, xi+1 = Street, yi−1 = NUM} 1 {yi = VERB, yi−1 = AUX} For simplicity, we’ll assume all CRF features take on the value 1 or 0. Above, we explicitly use the notation 1 {x}to mean “1 if x is true, and 0 otherwise”. From now on, we’ll leave off the 1 when we deﬁne features, but you can assume each feature has it there implicitly. Although the idea of what features to use is done by the system designer by hand, the speciﬁc features are automatically populated by using feature templates as wefeature templates brieﬂy mentioned in Chapter 5. Here are some templates that only use information from (yi−1,yi,X,i): ⟨yi,xi⟩,⟨yi,yi−1⟩,⟨yi,xi−1,xi+2⟩ These templates automatically populate the set of features from every instance in the training and test set. Thus for our example Janet/NNP will/MD back/VB the/DT bill/NN, when xi is the word back, the following features would be generated and have the value 1 (we’ve assigned them arbitrary feature numbers): f3743: yi = VB and xi = back f156: yi = VB and yi−1 = MD f99732: yi = VB and xi−1 = will and xi+2 = bill It’s also important to have features that help with unknown words. One of the most important is word shape features, which represent the abstract letter patternword shape of the word by mapping lower-case letters to ‘x’, upper-case to ‘X’, numbers to ’d’, and retaining punctuation. Thus for example I.M.F. would map to X.X.X. and DC10-30 would map to XXdd-dd. A second class of shorter word shape features is also used. In these features consecutive character types are removed, so words in all caps map to X, words with initial-caps map to Xx, DC10-30 would be mapped to Xd-d but I.M.F would still map to X.X.X. Preﬁx and sufﬁx features are also useful. In summary, here are some sample feature templates that help with unknown words: xi contains a particular preﬁx (perhaps from all preﬁxes of length ≤2) xi contains a particular sufﬁx (perhaps from all sufﬁxes of length ≤2) xi’s word shape xi’s short word shape For example the word well-dressed might generate the following non-zero val- ued feature values: 2 Because in HMMs all computation is based on the two probabilities P(tag\|tag) and P(word\|tag), if we want to include some source of knowledge into the tagging process, we must ﬁnd a way to encode the knowledge into one of these two probabilities. Each time we add a feature we have to do a lot of complicated conditioning which gets harder and harder as we have more and more such features. 17.5 • C ONDITIONAL RANDOM FIELDS (CRF S) 379 preﬁx(xi) = w preﬁx(xi) = we sufﬁx(xi) = ed sufﬁx(xi) = d word-shape(xi) = xxxx-xxxxxxx short-word-shape(xi) = x-x The known-word templates are computed for every word seen in the training set; the unknown word features can also be computed for all words in training, or only on training words whose frequency is below some threshold. The result of the known-word templates and word-signature features is a very large set of features. Generally a feature cutoff is used in which features are thrown out if they have count <5 in the training set. Remember that in a CRF we don’t learn weights for each of these local features fk. Instead, we ﬁrst sum the values of each local feature (for example feature f3743) over the entire sentence, to create each global feature (for exampleF3743). It is those global features that will then be multiplied by weight w3743. Thus for training and inference there is always a ﬁxed set of K features with K weights, even though the length of each sentence is different. 17.5.2 Features for CRF Named Entity Recognizers A CRF for NER makes use of very similar features to a POS tagger, as shown in Figure 17.15. identity of wi, identity of neighboring words embeddings for wi, embeddings for neighboring words part of speech of wi, part of speech of neighboring words presence of wi in a gazetteer wi contains a particular preﬁx (from all preﬁxes of length ≤4) wi contains a particular sufﬁx (from all sufﬁxes of length ≤4) word shape of wi, word shape of neighboring words short word shape of wi, short word shape of neighboring words gazetteer features Figure 17.15 Typical features for a feature-based NER system. One feature that is especially useful for locations is a gazetteer, a list of placegazetteer names, often providing millions of entries for locations with detailed geographical and political information.3 This can be implemented as a binary feature indicating a phrase appears in the list. Other related resources like name-lists, for example from the United States Census Bureau4, can be used, as can other entity dictionaries like lists of corporations or products, although they may not be as helpful as a gazetteer (Mikheev et al., 1999). The sample named entity token L’Occitanewould generate the following non- zero valued feature values (assuming that L’Occitaneis neither in the gazetteer nor the census). 3 www.geonames.org 4 www.census.gov 380 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES preﬁx(xi) = L sufﬁx(xi) = tane preﬁx(xi) = L’ sufﬁx(xi) = ane preﬁx(xi) = L’O sufﬁx(xi) = ne preﬁx(xi) = L’Oc sufﬁx(xi) = e word-shape(xi) = X’Xxxxxxxx short-word-shape(xi) = X’Xx Figure 17.16 illustrates the result of adding part-of-speech tags and some shape information to our earlier example. Words POS Short shape Gazetteer BIO Label Jane NNP Xx 0 B-PER Villanueva NNP Xx 1 I-PER of IN x 0 O United NNP Xx 0 B-ORG Airlines NNP Xx 0 I-ORG Holding NNP Xx 0 I-ORG discussed VBD x 0 O the DT x 0 O Chicago NNP Xx 1 B-LOC route NN x 0 O . . . 0 O Figure 17.16 Some NER features for a sample sentence, assuming that Chicago and Vil- lanueva are listed as locations in a gazetteer. We assume features only take on the values 0 or 1, so the ﬁrst POS feature, for example, would be represented as 1 {POS = NNP}. 17.5.3 Inference and Training for CRFs How do we ﬁnd the best tag sequenceˆY for a given inputX? We start with Eq. 17.22: ˆY = argmax Y ∈Y P(Y \|X) = argmax Y ∈Y 1 Z(X)exp ( K∑ k=1 wkFk(X,Y ) ) (17.27) = argmax Y ∈Y exp ( K∑ k=1 wk n∑ i=1 fk(yi−1,yi,X,i) ) (17.28) = argmax Y ∈Y K∑ k=1 wk n∑ i=1 fk(yi−1,yi,X,i) (17.29) = argmax Y ∈Y n∑ i=1 K∑ k=1 wk fk(yi−1,yi,X,i) (17.30) We can ignore the exp function and the denominatorZ(X), as we do above, because exp doesn’t change the argmax, and the denominator Z(X) is constant for a given observation sequence X. How should we decode to ﬁnd this optimal tag sequence ˆy? Just as with HMMs, we’ll turn to the Viterbi algorithm, which works because, like the HMM, the linear- chain CRF depends at each timestep on only one previous output token yi−1. Concretely, this involves ﬁlling anN ×T array with the appropriate values, main- taining backpointers as we proceed. As with HMM Viterbi, when the table is ﬁlled, we simply follow pointers back from the maximum value in the ﬁnal column to retrieve the desired set of labels. 17.6 • E VALUATION OF NAMED ENTITY RECOGNITION 381 The requisite changes from HMM Viterbi have to do only with how we ﬁll each cell. Recall from Eq. 17.19 that the recursive step of the Viterbi equation computes the Viterbi value of time t for state j as vt ( j) = N max i=1 vt−1(i)ai j bj(ot ); 1 ≤j ≤N,1 <t ≤T (17.31) which is the HMM implementation of vt ( j) = N max i=1 vt−1(i) P(sj\|si) P(ot \|sj) 1 ≤j ≤N,1 <t ≤T (17.32) The CRF requires only a slight change to this latter formula, replacing the a and b prior and likelihood probabilities with the CRF features: vt ( j) = N max i=1 [ vt−1(i)+ K∑ k=1 wk fk(yt−1,yt ,X,t) 1 ≤j ≤N,1 <t ≤T ] (17.33) Learning in CRFs relies on the same supervised learning algorithms we presented for logistic regression. Given a sequence of observations, feature functions, and cor- responding outputs, we use stochastic gradient descent to train the weights to maxi- mize the log-likelihood of the training corpus. The local nature of linear-chain CRFs means that the forward-backward algorithm introduced for HMMs in Appendix A can be extended to a CRF version that will efﬁciently compute the necessary deriva- tives. As with logistic regression, L1 or L2 regularization is important. 17.6 Evaluation of Named Entity Recognition Part-of-speech taggers are evaluated by the standard metric of accuracy. Named entity recognizers are evaluated by recall, precision, and F1 measure. Recall that recall is the ratio of the number of correctly labeled responses to the total that should have been labeled; precision is the ratio of the number of correctly labeled responses to the total labeled; and F-measure is the harmonic mean of the two. To know if the difference between the F1 scores of two NER systems is a signif- icant difference, we use the paired bootstrap test, or the similar randomization test (Section 4.9). For named entity tagging, the entity rather than the word is the unit of response. Thus in the example in Fig. 17.16, the two entities Jane Villanueva and United Air- lines Holding and the non-entity discussed would each count as a single response. The fact that named entity tagging has a segmentation component which is not present in tasks like text categorization or part-of-speech tagging causes some prob- lems with evaluation. For example, a system that labeled Jane but not Jane Vil- lanueva as a person would cause two errors, a false positive for O and a false nega- tive for I-PER. In addition, using entities as the unit of response but words as the unit of training means that there is a mismatch between the training and test conditions. 17.7 Further Details In this section we summarize a few remaining details of the data and models for part-of-speech tagging and NER, beginning with data. Since the algorithms we have 382 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES presented are supervised, having labeled data is essential for training and testing. A wide variety of datasets exist for part-of-speech tagging and/or NER. The Universal Dependencies (UD) dataset (de Marneffe et al., 2021) has POS tagged corpora in over a hundred languages, as do the Penn Treebanks in English, Chinese, and Arabic. OntoNotes has corpora labeled for named entities in English, Chinese, and Arabic (Hovy et al., 2006). Named entity tagged corpora are also available in particular domains, such as for biomedical (Bada et al., 2012) and literary text (Bamman et al., 2019). 17.7.1 Rule-based Methods While machine learned (neural or CRF) sequence models are the norm in academic research, commercial approaches to NER are often based on pragmatic combina- tions of lists and rules, with some smaller amount of supervised machine learning (Chiticariu et al., 2013). For example in the IBM System T architecture, a user speciﬁes declarative constraints for tagging tasks in a formal query language that includes regular expressions, dictionaries, semantic constraints, and other operators, which the system compiles into an efﬁcient extractor (Chiticariu et al., 2018). One common approach is to make repeated rule-based passes over a text, starting with rules with very high precision but low recall, and, in subsequent stages, using machine learning methods that take the output of the ﬁrst pass into account (an approach ﬁrst worked out for coreference (Lee et al., 2017a)): 1. First, use high-precision rules to tag unambiguous entity mentions. 2. Then, search for substring matches of the previously detected names. 3. Use application-speciﬁc name lists to ﬁnd likely domain-speciﬁc mentions. 4. Finally, apply supervised sequence labeling techniques that use tags from pre- vious stages as additional features. Rule-based methods were also the earliest methods for part-of-speech tagging. Rule-based taggers like the English Constraint Grammar system (Karlsson et al. 1995, V outilainen 1999) use a two-stage formalism invented in the 1950s and 1960s: (1) a morphological analyzer with tens of thousands of word stem entries returns all parts of speech for a word, then (2) a large set of thousands of constraints are applied to the input sentence to rule out parts of speech inconsistent with the context. 17.7.2 POS Tagging for Morphologically Rich Languages Augmentations to tagging algorithms become necessary when dealing with lan- guages with rich morphology like Czech, Hungarian and Turkish. These productive word-formation processes result in a large vocabulary for these languages: a 250,000 word token corpus of Hungarian has more than twice as many word types as a similarly sized corpus of English (Oravecz and Dienes, 2002), while a 10 million word token corpus of Turkish contains four times as many word types as a similarly sized English corpus (Hakkani-T ¨ur et al., 2002). Large vocabular- ies mean many unknown words, and these unknown words cause signiﬁcant per- formance degradations in a wide variety of languages (including Czech, Slovene, Estonian, and Romanian) (Hajiˇc, 2000). Highly inﬂectional languages also have much more information than English coded in word morphology, like case (nominative, accusative, genitive) or gender (masculine, feminine). Because this information is important for tasks like pars- ing and coreference resolution, part-of-speech taggers for morphologically rich lan- 17.8 • S UMMARY 383 guages need to label words with case and gender information. Tagsets for morpho- logically rich languages are therefore sequences of morphological tags rather than a single primitive tag. Here’s a Turkish example, in which the wordizin has three pos- sible morphological/part-of-speech tags and meanings (Hakkani-T¨ur et al., 2002): 1. Yerdeki izin temizlenmesi gerek. iz + Noun+A3sg+Pnon+Gen The trace on the ﬂoor should be cleaned. 2. ¨Uzerinde parmak izin kalmis ¸. iz + Noun+A3sg+P2sg+Nom Your ﬁnger print is left on (it). 3. Ic ¸eri girmek ic ¸inizin alman gerekiyor. izin + Noun+A3sg+Pnon+Nom You need permission to enter. Using a morphological parse sequence like Noun+A3sg+Pnon+Gen as the part- of-speech tag greatly increases the number of parts of speech, and so tagsets can be 4 to 10 times larger than the 50–100 tags we have seen for English. With such large tagsets, each word needs to be morphologically analyzed to generate the list of possible morphological tag sequences (part-of-speech tags) for the word. The role of the tagger is then to disambiguate among these tags. This method also helps with unknown words since morphological parsers can accept unknown stems and still segment the afﬁxes properly. 17.8 Summary This chapter introduced parts of speech and named entities, and the tasks of part- of-speech tagging and named entity recognition: • Languages generally have a small set of closed class words that are highly frequent, ambiguous, and act as function words, and open-class words like nouns, verbs, adjectives. Various part-of-speech tagsets exist, of between 40 and 200 tags. • Part-of-speech tagging is the process of assigning a part-of-speech label to each of a sequence of words. • Named entities are words for proper nouns referring mainly to people, places, and organizations, but extended to many other types that aren’t strictly entities or even proper nouns. • Two common approaches to sequence modeling are a generative approach, HMM tagging, and a discriminative approach, CRF tagging. We will see a neural approach in following chapters. • The probabilities in HMM taggers are estimated by maximum likelihood es- timation on tag-labeled training corpora. The Viterbi algorithm is used for decoding, ﬁnding the most likely tag sequence • Conditional Random Fieldsor CRF taggers train a log-linear model that can choose the best tag sequence given an observation sequence, based on features that condition on the output tag, the prior output tag, the entire input sequence, and the current timestep. They use the Viterbi algorithm for inference, to choose the best sequence of tags, and a version of the Forward-Backward algorithm (see Appendix A) for training, 384 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES Bibliographical and Historical Notes What is probably the earliest part-of-speech tagger was part of the parser in Zellig Harris’s Transformations and Discourse Analysis Project (TDAP), implemented be- tween June 1958 and July 1959 at the University of Pennsylvania (Harris, 1962), although earlier systems had used part-of-speech dictionaries. TDAP used 14 hand- written rules for part-of-speech disambiguation; the use of part-of-speech tag se- quences and the relative frequency of tags for a word preﬁgures modern algorithms. The parser was implemented essentially as a cascade of ﬁnite-state transducers; see Joshi and Hopely (1999) and Karttunen (1999) for a reimplementation. The Computational Grammar Coder (CGC) of Klein and Simmons (1963) had three components: a lexicon, a morphological analyzer, and a context disambigua- tor. The small 1500-word lexicon listed only function words and other irregular words. The morphological analyzer used inﬂectional and derivational sufﬁxes to as- sign part-of-speech classes. These were run over words to produce candidate parts of speech which were then disambiguated by a set of 500 context rules by relying on surrounding islands of unambiguous words. For example, one rule said that between an ARTICLE and a VERB, the only allowable sequences were ADJ-NOUN, NOUN- ADVERB, or NOUN-NOUN. The TAGGIT tagger (Greene and Rubin, 1971) used the same architecture as Klein and Simmons (1963), with a bigger dictionary and more tags (87). TAGGIT was applied to the Brown corpus and, according to Francis and Kuˇcera (1982, p. 9), accurately tagged 77% of the corpus; the remainder of the Brown corpus was then tagged by hand. All these early algorithms were based on a two-stage architecture in which a dictionary was ﬁrst used to assign each word a set of potential parts of speech, and then lists of handwritten disambiguation rules winnowed the set down to a single part of speech per word. Probabilities were used in tagging by Stolz et al. (1965) and a complete proba- bilistic tagger with Viterbi decoding was sketched by Bahl and Mercer (1976). The Lancaster-Oslo/Bergen (LOB) corpus, a British English equivalent of the Brown cor- pus, was tagged in the early 1980’s with the CLAWS tagger (Marshall 1983; Mar- shall 1987; Garside 1987), a probabilistic algorithm that approximated a simpliﬁed HMM tagger. The algorithm used tag bigram probabilities, but instead of storing the word likelihood of each tag, the algorithm marked tags either asrare (P(tag\|word) < .01) infrequent (P(tag\|word) <.10) or normally frequent (P(tag\|word) >.10). DeRose (1988) developed a quasi-HMM algorithm, including the use of dy- namic programming, although computing P(t\|w)P(w) instead of P(w\|t)P(w). The same year, the probabilistic PARTS tagger of Church 1988, 1989 was probably the ﬁrst implemented HMM tagger, described correctly in Church (1989), although Church (1988) also described the computation incorrectly as P(t\|w)P(w) instead of P(w\|t)P(w). Church (p.c.) explained that he had simpliﬁed for pedagogical pur- poses because using the probabilityP(t\|w) made the idea seem more understandable as “storing a lexicon in an almost standard form”. Later taggers explicitly introduced the use of the hidden Markov model (Kupiec 1992; Weischedel et al. 1993; Sch ¨utze and Singer 1994). Merialdo (1994) showed that fully unsupervised EM didn’t work well for the tagging task and that reliance on hand-labeled data was important. Charniak et al. (1993) showed the importance of the most frequent tag baseline; the 92.3% number we give above was from Abney et al. (1999). See Brants (2000) for HMM tagger implementation details, includ- ing the extension to trigram contexts, and the use of sophisticated unknown word features; its performance is still close to state of the art taggers. EXERCISES 385 Log-linear models for POS tagging were introduced by Ratnaparkhi (1996), who introduced a system called MXPOST which implemented a maximum entropy Markov model (MEMM), a slightly simpler version of a CRF. Around the same time, sequence labelers were applied to the task of named entity tagging, ﬁrst with HMMs (Bikel et al., 1997) and MEMMs (McCallum et al., 2000), and then once CRFs were developed (Lafferty et al. 2001), they were also applied to NER (Mc- Callum and Li, 2003). A wide exploration of features followed (Zhou et al., 2005). Neural approaches to NER mainly follow from the pioneering results of Collobert et al. (2011), who applied a CRF on top of a convolutional net. BiLSTMs with word and character-based embeddings as input followed shortly and became a standard neural algorithm for NER (Huang et al. 2015, Ma and Hovy 2016, Lample et al. 2016) followed by the more recent use of Transformers and BERT. The idea of using letter sufﬁxes for unknown words is quite old; the early Klein and Simmons (1963) system checked all ﬁnal letter sufﬁxes of lengths 1-5. The un- known word features described on page 378 come mainly from Ratnaparkhi (1996), with augmentations from Toutanova et al. (2003) and Manning (2011). State of the art POS taggers use neural algorithms, either bidirectional RNNs or Transformers like BERT; see Chapter 8 to Chapter 11. HMM (Brants 2000; Thede and Harper 1999) and CRF tagger accuracies are likely just a tad lower. Manning (2011) investigates the remaining 2.7% of errors in a high-performing tagger (Toutanova et al., 2003). He suggests that a third or half of these remaining errors are due to errors or inconsistencies in the training data, a third might be solv- able with richer linguistic models, and for the remainder the task is underspeciﬁed or unclear. Supervised tagging relies heavily on in-domain training data hand-labeled by experts. Ways to relax this assumption include unsupervised algorithms for cluster- ing words into part-of-speech-like classes, summarized in Christodoulopoulos et al. (2010), and ways to combine labeled and unlabeled data, for example by co-training (Clark et al. 2003; Søgaard 2010). See Householder (1995) for historical notes on parts of speech, and Sampson (1987) and Garside et al. (1997) on the provenance of the Brown and other tagsets. Exercises 17.1 Find one tagging error in each of the following sentences that are tagged with the Penn Treebank tagset: 1. I/PRP need/VBP a/DT ﬂight/NN from/IN Atlanta/NN 2. Does/VBZ this/DT ﬂight/NN serve/VB dinner/NNS 3. I/PRP have/VB a/DT friend/NN living/VBG in/IN Denver/NNP 4. Can/VBP you/PRP list/VB the/DT nonstop/JJ afternoon/NN ﬂights/NNS 17.2 Use the Penn Treebank tagset to tag each word in the following sentences from Damon Runyon’s short stories. You may ignore punctuation. Some of these are quite difﬁcult; do your best. 1. It is a nice night. 2. This crap game is over a garage in Fifty-second Street. . . 3. . . . Nobody ever takes the newspapers she sells . . . 4. He is a tall, skinny guy with a long, sad, mean-looking kisser, and a mournful voice. 386 CHAPTER 17 • S EQUENCE LABELING FOR PARTS OF SPEECH AND NAMED ENTITIES 5. . . . I am sitting in Mindy’s restaurant putting on the geﬁllte ﬁsh, which is a dish I am very fond of, . . . 6. When a guy and a doll get to taking peeks back and forth at each other, why there you are indeed. 17.3 Now compare your tags from the previous exercise with one or two friend’s answers. On which words did you disagree the most? Why? 17.4 Implement the “most likely tag” baseline. Find a POS-tagged training set, and use it to compute for each word the tag that maximizes p(t\|w). You will need to implement a simple tokenizer to deal with sentence boundaries. Start by assuming that all unknown words are NN and compute your error rate on known and unknown words. Now write at least ﬁve rules to do a better job of tagging unknown words, and show the difference in error rates. 17.5 Build a bigram HMM tagger. You will need a part-of-speech-tagged corpus. First split the corpus into a training set and test set. From the labeled training set, train the transition and observation probabilities of the HMM tagger di- rectly on the hand-tagged data. Then implement the Viterbi algorithm so you can decode a test sentence. Now run your algorithm on the test set. Report its error rate and compare its performance to the most frequent tag baseline. 17.6 Do an error analysis of your tagger. Build a confusion matrix and investigate the most frequent errors. Propose some features for improving the perfor- mance of your tagger on these errors. 17.7 Develop a set of regular expressions to recognize the character shape features described on page 378. 17.8 The BIO and other labeling schemes given in this chapter aren’t the only possible one. For example, the B tag can be reserved only for those situations where an ambiguity exists between adjacent entities. Propose a new set of BIO tags for use with your NER system. Experiment with it and compare its performance with the schemes presented in this chapter. 17.9 Names of works of art (books, movies, video games, etc.) are quite different from the kinds of named entities we’ve discussed in this chapter. Collect a list of names of works of art from a particular category from a Web-based source (e.g., gutenberg.org, amazon.com, imdb.com, etc.). Analyze your list and give examples of ways that the names in it are likely to be problematic for the techniques described in this chapter. 17.10 Develop an NER system speciﬁc to the category of names that you collected in the last exercise. Evaluate your system on a collection of text likely to contain instances of these named entities. CHAPTER 18 Context-Free Grammars and Constituency Parsing Because the Night by Bruce Springsteen and Patti Smith The Fire Next Timeby James Baldwin If on a winter’s night a travelerby Italo Calvino Love Actually by Richard Curtis Suddenly Last Summer by Tennessee Williams A Scanner Darkly by Philip K. Dick Six titles that are not constituents, from Geoffrey K. Pullum on Language Log (who was pointing out their incredible rarity). One morning I shot an elephant in my pajamas. How he got into my pajamas I don’t know. Groucho Marx, Animal Crackers, 1930 The study of grammar has an ancient pedigree. The grammar of Sanskrit was described by the Indian grammarian P ¯an. ini sometime between the 7th and 4th cen- turies BCE, in his famous treatise the As.t.¯adhy¯ay¯ı (‘8 books’). And our wordsyntaxsyntax comes from the Greek s´yntaxis, meaning “setting out together or arrangement”, and refers to the way words are arranged together. We have seen syntactic notions in pre- vious chapters like the use of part-of-speech categories (Chapter 17). In this chapter and the next one we introduce formal models for capturing more sophisticated no- tions of grammatical structure and algorithms for parsing these structures. Our focus in this chapter is context-free grammars and the CKY algorithm for parsing them. Context-free grammars are the backbone of many formal mod- els of the syntax of natural language (and, for that matter, of computer languages). Syntactic parsing is the task of assigning a syntactic structure to a sentence. Parse trees (whether for context-free grammars or for the dependency or CCG formalisms we introduce in following chapters) can be used in applications such as grammar checking: sentence that cannot be parsed may have grammatical errors (or at least be hard to read). Parse trees can be an intermediate stage of representation for for- mal semantic analysis . And parsers and the grammatical structure they assign a sentence are a useful text analysis tool for text data science applications that require modeling the relationship of elements in sentences. In this chapter we introduce context-free grammars, give a small sample gram- mar of English, introduce more formal deﬁnitions of context-free grammars and grammar normal form, and talk about treebanks: corpora that have been anno- tated with syntactic structure. We then discuss parse ambiguity and the problems it presents, and turn to parsing itself, giving the famous Cocke-Kasami-Younger (CKY) algorithm (Kasami 1965, Younger 1967), the standard dynamic program- ming approach to syntactic parsing. The CKY algorithm returns an efﬁcient repre- sentation of the set of parse trees for a sentence, but doesn’t tell uswhich parse tree is the right one. For that, we need to augment CKY with scores for each possible constituent. We’ll see how to do this with neural span-based parsers. Finally, we’ll introduce the standard set of metrics for evaluating parser accuracy. 388 CHAPTER 18 • C ONTEXT -FREE GRAMMARS AND CONSTITUENCY PARSING 18.1 Constituency Syntactic constituency is the idea that groups of words can behave as single units, or constituents. Part of developing a grammar involves building an inventory of the constituents in the language. How do words group together in English? Consider the noun phrase, a sequence of words surrounding at least one noun. Here are somenoun phrase examples of noun phrases (thanks to Damon Runyon): Harry the Horse a high-class spot such as Mindy’s the Broadway coppers the reason he comes into the Hot Box they three parties from Brooklyn What evidence do we have that these words group together (or “form constituents”)? One piece of evidence is that they can all appear in similar syntactic environments, for example, before a verb. three parties from Brooklyn arrive. . . a high-class spot such as Mindy’sattracts. . . the Broadway coppers love. . . they sit But while the whole noun phrase can occur before a verb, this is not true of each of the individual words that make up a noun phrase. The following are not grammat- ical sentences of English (recall that we use an asterisk () to mark fragments that are not grammatical English sentences): from arrive. . . as attracts. . . the is. . . spot sat. . . Thus, to correctly describe facts about the ordering of these words in English, we must be able to say things like “ Noun Phrases can occur before verbs ”. Let’s now see how to do this in a more formal way! 18.2 Context-Free Grammars A widely used formal system for modeling constituent structure in natural lan- guage is the context-free grammar, or CFG. Context-free grammars are also calledCFG phrase-structure grammars, and the formalism is equivalent toBackus-Naur form, or BNF. The idea of basing a grammar on constituent structure dates back to the psy- chologist Wilhelm Wundt (1900) but was not formalized until Chomsky (1956) and, independently, Backus (1959). A context-free grammar consists of a set ofrules or productions, each of whichrules expresses the ways that symbols of the language can be grouped and ordered to- gether, and a lexicon of words and symbols. For example, the following productionslexicon express that an NP (or noun phrase) can be composed of either a ProperNoun orNP a determiner (Det) followed by a Nominal; a Nominal in turn can consist of one or 18.2 • C ONTEXT -FREE GRAMMARS 389 more Nouns.1 NP → Det Nominal NP → ProperNoun Nominal → Noun \|Nominal Noun Context-free rules can be hierarchically embedded, so we can combine the previous rules with others, like the following, that express facts about the lexicon: Det → a Det → the Noun → ﬂight The symbols that are used in a CFG are divided into two classes. The symbols that correspond to words in the language (“the”, “nightclub”) are called terminalterminal symbols; the lexicon is the set of rules that introduce these terminal symbols. The symbols that express abstractions over these terminals are called non-terminals. Innon-terminal each context-free rule, the item to the right of the arrow (→) is an ordered list of one or more terminals and non-terminals; to the left of the arrow is a single non-terminal symbol expressing some cluster or generalization. The non-terminal associated with each word in the lexicon is its lexical category, or part of speech. A CFG can be thought of in two ways: as a device for generating sentences and as a device for assigning a structure to a given sentence. Viewing a CFG as a generator, we can read the→arrow as “rewrite the symbol on the left with the string of symbols on the right”. So starting from the symbol: NP we can use our ﬁrst rule to rewrite NP as: Det Nominal and then rewrite Nominal as: Noun and ﬁnally rewrite these parts-of-speech as: a ﬂight We say the stringa ﬂight can be derived from the non-terminalNP. Thus, a CFG can be used to generate a set of strings. This sequence of rule expansions is called a derivation of the string of words. It is common to represent a derivation by a parsederivation tree (commonly shown inverted with the root at the top). Figure 18.1 shows the treeparse tree representation of this derivation. NP Nom Noun ﬂight Det a Figure 18.1 A parse tree for “a ﬂight”. In the parse tree shown in Fig. 18.1, we can say that the node NP dominatesdominates all the nodes in the tree ( Det, Nom, Noun, a, ﬂight). We can say further that it immediately dominates the nodes Det and Nom. The formal language deﬁned by a CFG is the set of strings that are derivable from the designated start symbol. Each grammar must have one designated startstart symbol 1 When talking about these rules we can pronounce the rightarrow →as “goes to”, and so we might read the ﬁrst rule above as “NP goes to Det Nominal”. 390 CHAPTER 18 • C ONTEXT -FREE GRAMMARS AND CONSTITUENCY PARSING symbol, which is often calledS. Since context-free grammars are often used to deﬁne sentences, S is usually interpreted as the “sentence” node, and the set of strings that are derivable from S is the set of sentences in some simpliﬁed version of English. Let’s add a few additional rules to our inventory. The following rule expresses the fact that a sentence can consist of a noun phrase followed by a verb phrase:verb phrase S → NP VP I prefer a morning ﬂight A verb phrase in English consists of a verb followed by assorted other things; for example, one kind of verb phrase consists of a verb followed by a noun phrase: VP → Verb NP prefer a morning ﬂight Or the verb may be followed by a noun phrase and a prepositional phrase: VP → Verb NP PP leave Boston in the morning Or the verb phrase may have a verb followed by a prepositional phrase alone: VP → Verb PP leaving on Thursday A prepositional phrase generally has a preposition followed by a noun phrase. For example, a common type of prepositional phrase in the ATIS corpus is used to indicate location or direction: PP → Preposition NP from Los Angeles The NP inside a PP need not be a location; PPs are often used with times and dates, and with other nouns as well; they can be arbitrarily complex. Here are ten examples from the ATIS corpus: to Seattle on these ﬂights in Minneapolis about the ground transportation in Chicago on Wednesday of the round trip ﬂight on United Airlines in the evening of the AP ﬁfty seven ﬂight on the ninth of July with a stopover in Nashville Figure 18.2 gives a sample lexicon, and Fig. 18.3 summarizes the grammar rules we’ve seen so far, which we’ll call L0. Note that we can use the or-symbol \|to indicate that a non-terminal has alternate possible expansions. Noun → ﬂights \|ﬂight \|breeze \|trip \|morning Verb→ is \|prefer \|like \|need \|want \|ﬂy \|do Adjective → cheapest \|non-stop \|ﬁrst \|latest \|other \|direct Pronoun → me \|I \|you \|it Proper-Noun → Alaska \|Baltimore \|Los Angeles \|Chicago \|United \|American Determiner → the \|a \|an \|this \|these \|that Preposition → from \|to \|on \|near \|in Conjunction → and \|or \|but Figure 18.2 The lexicon for L0. We can use this grammar to generate sentences of this “ATIS-language”. We start with S, expand it toNP VP, then choose a random expansion ofNP (let’s say, to 18.2 • C ONTEXT -FREE GRAMMARS 391 Grammar Rules Examples S → NP VP I + want a morning ﬂight NP → Pronoun I \| Proper-Noun Los Angeles \| Det Nominal a + ﬂight Nominal → Nominal Noun morning + ﬂight \| Noun ﬂights VP → Verb do \| Verb NP want + a ﬂight \| Verb NP PP leave + Boston + in the morning \| Verb PP leaving + on Thursday PP → Preposition NP from + Los Angeles Figure 18.3 The grammar for L0, with example phrases for each rule. S VP NP Nom Noun ﬂight Nom Noun morning Det a Verb prefer NP Pro I Figure 18.4 The parse tree for “I prefer a morning ﬂight” according to grammar L0. I), and a random expansion ofVP (let’s say, toVerb NP), and so on until we generate the string I prefer a morning ﬂight. Figure 18.4 shows a parse tree that represents a complete derivation of I prefer a morning ﬂight. We can also represent a parse tree in a more compact format called bracketed notation; here is the bracketed representation of the parse tree of Fig. 18.4:bracketed notation (18.1) [ S [NP [Pro I]] [VP [V prefer] [NP [Det a] [Nom [N morning] [Nom [N ﬂight]]]]]] A CFG like that of L0 deﬁnes a formal language. Sentences (strings of words) that can be derived by a grammar are in the formal language deﬁned by that gram- mar, and are called grammatical sentences. Sentences that cannot be derived bygrammatical a given formal grammar are not in the language deﬁned by that grammar and are referred to as ungrammatical. This hard line between “in” and “out” characterizesungrammatical all formal languages but is only a very simpliﬁed model of how natural languages really work. This is because determining whether a given sentence is part of a given natural language (say, English) often depends on the context. In linguistics, the use of formal languages to model natural languages is calledgenerative grammar sincegenerative grammar the language is deﬁned by the set of possible sentences “generated” by the grammar. (Note that this is a different sense of the word ‘generate’ than when we talk about 392 CHAPTER 18 • C ONTEXT -FREE GRAMMARS AND CONSTITUENCY PARSING language models generating text.) 18.2.1 Formal Deﬁnition of Context-Free Grammar We conclude this section with a quick, formal description of a context-free gram- mar and the language it generates. A context-free grammar G is deﬁned by four parameters: N,Σ,R,S (technically it is a “4-tuple”). N a set of non-terminal symbols (or variables) Σ a set of terminal symbols (disjoint from N) R a set of rules or productions, each of the form A →β , where A is a non-terminal, β is a string of symbols from the inﬁnite set of strings (Σ∪N)∗ S a designated start symbol and a member of N For the remainder of the book we adhere to the following conventions when dis- cussing the formal properties of context-free grammars (as opposed to explaining particular facts about English or other languages). Capital letters like A, B, and S Non-terminals S The start symbol Lower-case Greek letters like α, β, and γ Strings drawn from (Σ∪N)∗ Lower-case Roman letters like u, v, and w Strings of terminals A language is deﬁned through the concept of derivation. One string derives an- other one if it can be rewritten as the second one by some series of rule applications. More formally, following Hopcroft and Ullman (1979), if A →β is a production of R and α and γ are any strings in the set (Σ∪N)∗, then we say that αAγ directly derives αβγ , or αAγ ⇒αβγ .directly derives Derivation is then a generalization of direct derivation: Let α1,α2,..., αm be strings in (Σ∪N)∗,m ≥1, such that α1 ⇒α2,α2 ⇒α3,..., αm−1 ⇒αm We say that α1 derives αm, or α1 ∗⇒αm.derives We can then formally deﬁne the language LG generated by a grammar G as the set of strings composed of terminal symbols that can be derived from the designated start symbol S. LG = {w\|w is in Σ∗and S ∗⇒w} The problem of mapping from a string of words to its parse tree is called syn- tactic parsing, as we’ll see in Section 18.6.syntactic parsing 18.3 Treebanks A corpus in which every sentence is annotated with a parse tree is called atreebank.treebank 18.3 • T REEBANKS 393 Treebanks play an important role in parsing as well as in linguistic investigations of syntactic phenomena. Treebanks are generally made by running a parser over each sentence and then having the resulting parse hand-corrected by human linguists. Figure 18.5 shows sentences from the Penn Treebank project, which includes various treebanks inPenn Treebank English, Arabic, and Chinese. The Penn Treebank part-of-speech tagset was deﬁned in Chapter 17, but we’ll see minor formatting differences across treebanks. The use of LISP-style parenthesized notation for trees is extremely common and resembles the bracketed notation we saw earlier in (18.1). For those who are not familiar with it we show a standard node-and-line tree representation in Fig. 18.6. ((S (NP-SBJ (DT That) (JJ cold) (, ,) (JJ empty) (NN sky) ) (VP (VBD was) (ADJP-PRD (JJ full) (PP (IN of) (NP (NN fire) (CC and) (NN light) )))) (. .) )) ((S (NP-SBJ The/DT flight/NN ) (VP should/MD (VP arrive/VB (PP-TMP at/IN (NP eleven/CD a.m/RB )) (NP-TMP tomorrow/NN ))))) (a) (b) Figure 18.5 Parses from the LDC Treebank3 for (a) Brown and (b) ATIS sentences. S . . VP ADJP-PRD PP NP NN light CC and NN ﬁre IN of JJ full VBD was NP-SBJ NN sky JJ empty , , JJ cold DT That Figure 18.6 The tree corresponding to the Brown corpus sentence in the previous ﬁgure. The sentences in a treebank implicitly constitute a grammar of the language. For example, from the parsed sentences in Fig. 18.5 we can extract the CFG rules shown in Fig. 18.7 (with rule sufﬁxes ( -SBJ) stripped for simplicity). The grammar used to parse the Penn Treebank is very ﬂat, resulting in very many rules. For example, 394 CHAPTER 18 • C ONTEXT -FREE GRAMMARS AND CONSTITUENCY PARSING Grammar Lexicon S →NP VP . DT →the \|that S →NP VP JJ →cold \|empty \|full NP →CD RB NN →sky \|ﬁre \|light \|ﬂight \|tomorrow NP →DT NN CC →and NP →NN CC NN IN →of \|at NP →DT JJ , JJ NN CD →eleven NP →NN RB →a.m. VP →MD VP VB →arrive VP →VBD ADJP VBD →was \|said VP →MD VP MD →should \|would VP →VB PP NP ADJP →JJ PP PP →IN NP Figure 18.7 CFG grammar rules and lexicon from the treebank sentences in Fig. 18.5. among the approximately 4,500 different rules for expanding VPs are separate rules for PP sequences of any length and every possible arrangement of verb arguments: VP → VBD PP VP → VBD PP PP VP → VBD PP PP PP VP → VBD PP PP PP PP VP → VB ADVP PP VP → VB PP ADVP VP → ADVP VB PP 18.4 Grammar Equivalence and Normal Form A formal language is deﬁned as a (possibly inﬁnite) set of strings of words. This sug- gests that we could ask if two grammars are equivalent by asking if they generate the same set of strings. In fact, it is possible to have two distinct context-free grammars generate the same language. We say that two grammars are strongly equivalent ifstrongly equivalent they generate the same set of strings and if they assign the same phrase structure to each sentence (allowing merely for renaming of the non-terminal symbols). Two grammars are weakly equivalent if they generate the same set of strings but do notweakly equivalent assign the same phrase structure to each sentence. It is sometimes useful to have a normal form for grammars, in which each ofnormal form the productions takes a particular form. For example, a context-free grammar is in Chomsky normal form (CNF) (Chomsky, 1963) if it is ϵ-free and if in additionChomsky normal form each production is either of the formA →B C or A →a. That is, the right-hand side of each rule either has two non-terminal symbols or one terminal symbol. Chomsky normal form grammars are binary branching, that is they have binary trees (downbinary branching to the prelexical nodes). We make use of this binary branching property in the CKY parsing algorithm in Section 18.6. Any context-free grammar can be converted into a weakly equivalent Chomsky normal form grammar. For example, a rule of the form A → B C D can be converted into the following two CNF rules (Exercise 18.1 asks the reader to 18.5 • A MBIGUITY 395 Grammar Lexicon S →NP VP Det →that \|this \|the \|a S →Aux NP VP Noun →book \|ﬂight \|meal \|money S →VP Verb →book \|include \|prefer NP →Pronoun Pronoun →I \|she \|me NP →Proper-Noun Proper-Noun →Houston \|United NP →Det Nominal Aux →does Nominal →Noun Preposition →from \|to \|on \|near \|through Nominal →Nominal Noun Nominal →Nominal PP VP →Verb VP →Verb NP VP →Verb NP PP VP →Verb PP VP →VP PP PP →Preposition NP Figure 18.8 The L1 miniature English grammar and lexicon. formulate the complete algorithm): A → B X X → C D Sometimes using binary branching can actually produce smaller grammars. For example, the sentences that might be characterized as VP -> VBD NP PP are represented in the Penn Treebank by this series of rules: VP → VBD NP PP VP → VBD NP PP PP VP → VBD NP PP PP PP VP → VBD NP PP PP PP PP ... but could also be generated by the following two-rule grammar: VP → VBD NP PP VP → VP PP The generation of a symbol A with a potentially inﬁnite sequence of symbols B with a rule of the form A → A Bis known as Chomsky-adjunction.Chomsky- adjunction 18.5 Ambiguity Ambiguity is the most serious problem faced by syntactic parsers. Chapter 17 intro- duced the notions of part-of-speech ambiguity and part-of-speech disambigua- tion. Here, we introduce a new kind of ambiguity, called structural ambiguity,structural ambiguity illustrated with a new toy grammar L1, shown in Figure 18.8, which adds a few rules to the L0 grammar. Structural ambiguity occurs when the grammar can assign more than one parse to a sentence. Groucho Marx’s well-known line as Captain Spaulding in Animal 396 CHAPTER 18 • C ONTEXT -FREE GRAMMARS AND CONSTITUENCY PARSING S VP NP Nominal PP in my pajamas Nominal Noun elephant Det an Verb shot NP Pronoun I S VP PP in my pajamas VP NP Nominal Noun elephant Det an Verb shot NP Pronoun I Figure 18.9 Two parse trees for an ambiguous sentence. The parse on the left corresponds to the humorous reading in which the elephant is in the pajamas, the parse on the right corresponds to the reading in which Captain Spaulding did the shooting in his pajamas. Crackers is ambiguous because the phrase in my pajamas can be part of the NP headed by elephant or a part of the verb phrase headed by shot. Figure 18.9 illus- trates these two analyses of Marx’s line using rules fromL1. Structural ambiguity, appropriately enough, comes in many forms. Two common kinds of ambiguity are attachment ambiguity and coordination ambiguity . A sentence has an attachment ambiguity if a particular constituent can be attached toattachment ambiguity the parse tree at more than one place. The Groucho Marx sentence is an example of PP-attachment ambiguity: the preposition phrase can be attached either as partPP-attachment ambiguity of the NP or as part of the VP. Various kinds of adverbial phrases are also subject to this kind of ambiguity. For instance, in the following example the gerundive-VP ﬂying to Paris can be part of a gerundive sentence whose subject is the Eiffel Tower or it can be an adjunct modifying the VP headed by saw: (18.2) We saw the Eiffel Tower ﬂying to Paris. In coordination ambiguity phrases can be conjoined by a conjunction like and.coordination ambiguity For example, the phrase old men and women can be bracketed as [old [men and women]], referring to old men and old women, or as [old men] and [women] , in which case it is only the men who are old. These ambiguities combine in complex ways in real sentences, like the following news sentence from the Brown corpus: (18.3) President Kennedy today pushed aside other White House business to devote all his time and attention to working on the Berlin crisis address he will deliver tomorrow night to the American people over nationwide television and radio. This sentence has a number of ambiguities, although since they are semantically unreasonable, it requires a careful reading to see them. The last noun phrase could be parsed [nationwide [television and radio]] or [[nationwide television] and radio] . The direct object of pushed aside should be other White House business but could also be the bizarre phrase [other White House business to devote all his time and attention to working](i.e., a structure likeKennedy afﬁrmed [his intention to propose a new budget to address the deﬁcit]). Then the phrase on the Berlin crisis address he 18.6 • CKY P ARSING : A D YNAMIC PROGRAMMING APPROACH 397 will deliver tomorrow night to the American people could be an adjunct modifying the verb pushed. A PP like over nationwide television and radio could be attached to any of the higher VPs or NPs (e.g., it could modify people or night). The fact that there are many grammatically correct but semantically unreason- able parses for naturally occurring sentences is an irksome problem that affects all parsers. Fortunately, the CKY algorithm below is designed to efﬁciently handle structural ambiguities. And as we’ll see in the following section, we can augment CKY with neural methods to choose a single correct parse bysyntactic disambigua- tion.syntactic disambiguation 18.6 CKY Parsing: A Dynamic Programming Approach Dynamic programming provides a powerful framework for addressing the prob- lems caused by ambiguity in grammars. Recall that a dynamic programming ap- proach systematically ﬁlls in a table of solutions to subproblems. The complete table has the solution to all the subproblems needed to solve the problem as a whole. In the case of syntactic parsing, these subproblems represent parse trees for all the constituents detected in the input. The dynamic programming advantage arises from the context-free nature of our grammar rules—once a constituent has been discovered in a segment of the input we can record its presence and make it available for use in any subsequent derivation that might require it. This provides both time and storage efﬁciencies since subtrees can be looked up in a table, not reanalyzed. This section presents the Cocke-Kasami- Younger (CKY) algorithm, the most widely used dynamic-programming based ap- proach to parsing. Chart parsing (Kaplan 1973, Kay 1982) is a related approach, and dynamic programming methods are often referred to aschart parsing methods.chart parsing 18.6.1 Conversion to Chomsky Normal Form The CKY algorithm requires grammars to ﬁrst be in Chomsky Normal Form (CNF). Recall from Section 18.4 that grammars in CNF are restricted to rules of the form A →B C or A →w. That is, the right-hand side of each rule must expand either to two non-terminals or to a single terminal. Restricting a grammar to CNF does not lead to any loss in expressiveness, since any context-free grammar can be converted into a corresponding CNF grammar that accepts exactly the same set of strings as the original grammar. Let’s start with the process of converting a generic CFG into one represented in CNF. Assuming we’re dealing with anϵ-free grammar, there are three situations we need to address in any generic grammar: rules that mix terminals with non-terminals on the right-hand side, rules that have a single non-terminal on the right-hand side, and rules in which the length of the right-hand side is greater than 2. The remedy for rules that mix terminals and non-terminals is to simply introduce a new dummy non-terminal that covers only the original terminal. For example, a rule for an inﬁnitive verb phrase such asINF-VP →to VP would be replaced by the two rules INF-VP →TO VP and TO →to. Rules with a single non-terminal on the right are called unit productions. WeUnit productions can eliminate unit productions by rewriting the right-hand side of the original rules with the right-hand side of all the non-unit production rules that they ultimately lead to. More formally, if A ∗⇒B by a chain of one or more unit productions and B →γ 398 CHAPTER 18 • C ONTEXT -FREE GRAMMARS AND CONSTITUENCY PARSING is a non-unit production in our grammar, then we add A →γ for each such rule in the grammar and discard all the intervening unit productions. As we demonstrate with our toy grammar, this can lead to a substantial ﬂattening of the grammar and a consequent promotion of terminals to fairly high levels in the resulting trees. Rules with right-hand sides longer than 2 are normalized through the introduc- tion of new non-terminals that spread the longer sequences over several new rules. Formally, if we have a rule like A →B C γ we replace the leftmost pair of non-terminals with a new non-terminal and introduce a new production, resulting in the following new rules: A → X1 γ X1 → B C In the case of longer right-hand sides, we simply iterate this process until the of- fending rule has been replaced by rules of length 2. The choice of replacing the leftmost pair of non-terminals is purely arbitrary; any systematic scheme that results in binary rules would sufﬁce. In our current grammar, the rule S →Aux NP VP would be replaced by the two rules S →X1 VP and X1 →Aux NP. The entire conversion process can be summarized as follows: 1. Copy all conforming rules to the new grammar unchanged. 2. Convert terminals within rules to dummy non-terminals. 3. Convert unit productions. 4. Make all rules binary and add them to new grammar. Figure 18.10 shows the results of applying this entire conversion procedure to the L1 grammar introduced earlier on page 395. Note that this ﬁgure doesn’t show the original lexical rules; since these original lexical rules are already in CNF, they all carry over unchanged to the new grammar. Figure 18.10 does, however, show the various places where the process of eliminating unit productions has, in effect, created new lexical rules. For example, all the original verbs have been promoted to both VPs and to Ss in the converted grammar. 18.6.2 CKY Recognition With our grammar now in CNF, each non-terminal node above the part-of-speech level in a parse tree will have exactly two daughters. A two-dimensional matrix can be used to encode the structure of an entire tree. For a sentence of length n, we will work with the upper-triangular portion of an(n+1)×(n+1) matrix. Each cell [i,j] in this matrix contains the set of non-terminals that represent all the constituents that span positions i through j of the input. Since our indexing scheme begins with 0, it’s natural to think of the indexes as pointing at the gaps between the input words (as in 0 Book 1 that 2 ﬂight 3). These gaps are often called fenceposts, on the metaphor offenceposts the posts between segments of fencing. It follows then that the cell that represents the entire input resides in position [0,n] in the matrix. Since each non-terminal entry in our table has two daughters in the parse, it fol- lows that for each constituent represented by an entry [i,j], there must be a position in the input, k, where it can be split into two parts such that i <k < j. Given such 18.6 • CKY P ARSING : A D YNAMIC PROGRAMMING APPROACH 399 L1 Grammar L1 in CNF S →NP VP S →NP VP S →Aux NP VP S →X1 VP X1 →Aux NP S →VP S →book \|include \|prefer S →Verb NP S →X2 PP S →Verb PP S →VP PP NP →Pronoun NP →I \|she \|me NP →Proper-Noun NP →United \|Houston NP →Det Nominal NP →Det Nominal Nominal →Noun Nominal →book \|ﬂight \|meal \|money Nominal →Nominal Noun Nominal →Nominal Noun Nominal →Nominal PP Nominal →Nominal PP VP →Verb VP →book \|include \|prefer VP →Verb NP VP →Verb NP VP →Verb NP PP VP →X2 PP X2 →Verb NP VP →Verb PP VP →Verb PP VP →VP PP VP →VP PP PP →Preposition NP PP →Preposition NP Figure 18.10 L1 Grammar and its conversion to CNF. Note that although they aren’t shown here, all the original lexical entries from L1 carry over unchanged as well. a position k, the ﬁrst constituent [i,k] must lie to the left of entry [i,j] somewhere along row i, and the second entry [k,j] must lie beneath it, along column j. To make this more concrete, consider the following example with its completed parse matrix, shown in Fig. 18.11. (18.4) Book the ﬂight through Houston. The superdiagonal row in the matrix contains the parts of speech for each word in the input. The subsequent diagonals above that superdiagonal contain constituents that cover all the spans of increasing length in the input. Given this setup, CKY recognition consists of ﬁlling the parse table in the right way. To do this, we’ll proceed in a bottom-up fashion so that at the point where we are ﬁlling any cell [i,j], the cells containing the parts that could contribute to this entry (i.e., the cells to the left and the cells below) have already been ﬁlled. The algorithm given in Fig. 18.12 ﬁlls the upper-triangular matrix a column at a time working from left to right, with each column ﬁlled from bottom to top, as the right side of Fig. 18.11 illustrates. This scheme guarantees that at each point in time we have all the information we need (to the left, since all the columns to the left have already been ﬁlled, and below since we’re ﬁlling bottom to top). It also mirrors on- line processing, since ﬁlling the columns from left to right corresponds to processing each word one at a time. The outermost loop of the algorithm given in Fig. 18.12 iterates over the columns, and the second loop iterates over the rows, from the bottom up. The purpose of the innermost loop is to range over all the places where a substring spanning i to j in the input might be split in two. As k ranges over the places where the string can be split, the pairs of cells we consider move, in lockstep, to the right along row i and down along column j. Figure 18.13 illustrates the general case of ﬁlling cell [i,j]. 400 CHAPTER 18 • C ONTEXT -FREE GRAMMARS AND CONSTITUENCY PARSING Book the flight through Houston S, VP, Verb, Nominal, Noun S,VP,X2 S,VP,X2 Det NP NP Nominal, Noun Nominal Prep PP NP, Proper- Noun [0,1] [0,2] [0,3] [0,4] [0,5] [1,2] [1,3] [2,3] [1,4] [2,5][2,4] [3,4] [4,5] [3,5] [1,5] Figure 18.11 Completed parse table for Book the ﬂight through Houston. function CKY-PARSE (words, grammar) returns table for j←from 1 to LENGTH (words) do for all {A \|A →words[ j] ∈grammar} table[ j −1,j]←table[ j −1,j] ∪A for i←from j −2 down to 0 do for k←i +1 to j −1 do for all {A \|A →BC ∈grammar and B ∈table[i,k] and C ∈table[k,j]} table[i,j]←table[i,j] ∪A Figure 18.12 The CKY algorithm. At each such split, the algorithm considers whether the contents of the two cells can be combined in a way that is sanctioned by a rule in the grammar. If such a rule exists, the non-terminal on its left-hand side is entered into the table. Figure 18.14 shows how the ﬁve cells of column 5 of the table are ﬁlled after the word Houston is read. The arrows point out the two spans that are being used to add an entry to the table. Note that the action in cell [0,5] indicates the presence of three alternative parses for this input, one where the PP modiﬁes the ﬂight, one where it modiﬁes the booking, and one that captures the second argument in the original VP →Verb NP PPrule, now captured indirectly with the VP →X2 PP rule. 18.6.3 CKY Parsing The algorithm given in Fig. 18.12 is a recognizer, not a parser. That is, it can tell us whether a valid parse exists for a given sentence based on whether or not if ﬁnds an S in cell [0,n], but it can’t provide the derivation, which is the actual job for a parser. To turn it into a parser capable of returning all possible parses for a given input, we can make two simple changes to the algorithm: the ﬁrst change is to augment the entries in the table so that each non-terminal is paired with pointers to the table entries from which it was derived (more or less as shown in Fig. 18.14), the second change is to permit multiple versions of the same non-terminal to be entered into the table (again as shown in Fig. 18.14). With these changes, the completed table contains all the possible parses for a given input. Returning an arbitrary single 18.6 • CKY P ARSING : A D YNAMIC PROGRAMMING APPROACH 401 ... ... [0,n] [i,i+1] [i,i+2] [i,j-2] [i,j-1] [i+1,j] [i+2,j] [j-1,j] [j-2,j] [i,j] ... [0,1] [n-1, n] Figure 18.13 All the ways to ﬁll the [i, j]th cell in the CKY table. parse consists of choosing an S from cell [0,n] and then recursively retrieving its component constituents from the table. Of course, instead of returning every parse for a sentence, we usually want just the best parse; we’ll see how to do that in the next section. 18.6.4 CKY in Practice Finally, we should note that while the restriction to CNF does not pose a problem theoretically, it does pose some non-trivial problems in practice. The returned CNF trees may not be consistent with the original grammar built by the grammar devel- opers, and will complicate any syntax-driven approach to semantic analysis. One approach to getting around these problems is to keep enough information around to transform our trees back to the original grammar as a post-processing step of the parse. This is trivial in the case of the transformation used for rules with length greater than 2. Simply deleting the new dummy non-terminals and promoting their daughters restores the original tree. In the case of unit productions, it turns out to be more convenient to alter the ba- sic CKY algorithm to handle them directly than it is to store the information needed to recover the correct trees. Exercise 18.3 asks you to make this change. Many of the probabilistic parsers presented in Appendix C use the CKY algorithm altered in 402 CHAPTER 18 • C ONTEXT -FREE GRAMMARS AND CONSTITUENCY PARSING Book the flight through Houston S, VP, Verb, Nominal, Noun S,VP,X2 Det NP Nominal, Noun Nominal Prep NP, Proper- Noun [0,1] [0,2] [0,3] [0,4] [0,5] [1,2] [1,3] [2,3] [1,4] [2,5][2,4] [3,4] [4,5] [3,5] [1,5] Book the flight through Houston S, VP, Verb, Nominal, Noun S,VP,X2 Det NP NP Nominal, Noun Prep PP NP, Proper- Noun [0,1] [0,2] [0,3] [0,4] [0,5] [1,2] [1,3] [2,3] [1,4] [2,5][2,4] [3,4] [4,5] [3,5] [1,5] Book the flight through Houston S, VP, Verb, Nominal, Noun S,VP,X2 Det NP NP Nominal, Noun Nominal Prep PP NP, Proper- Noun [0,1] [0,2] [0,3] [0,4] [0,5] [1,2] [1,3] [2,3] [1,4] [2,5][2,4] [3,4] [4,5] [3,5] [1,5] Book the flight through Houston S, VP, Verb, Nominal, Noun S,VP,X2 Det NP NP Nominal, Noun Nominal Prep PP NP, Proper- Noun [0,1] [0,2] [0,3] [0,4] [0,5] [1,2] [1,3] [2,3] [1,4] [2,5][2,4] [3,4] [4,5] [3,5] [1,5] Book the flight through Houston S, VP, Verb, Nominal, Noun S, VP, X2 Det NP NP Nominal, Noun Nominal Prep PP NP, Proper- Noun [0,1] [0,2] [0,3] [0,4] [1,2] [1,3] [2,3] [1,4] [2,5][2,4] [3,4] [4,5] [3,5] [1,5] S2, VP S3 S1,VP, X2 Figure 18.14 Filling the cells of column 5 after reading the word Houston. 18.7 • S PAN-BASED NEURAL CONSTITUENCY PARSING 403 just this manner. 18.7 Span-Based Neural Constituency Parsing While the CKY parsing algorithm we’ve seen so far does great at enumerating all the possible parse trees for a sentence, it has a large problem: it doesn’t tell us which parse is the correct one! That is, it doesn’tdisambiguate among the possible parses. To solve the disambiguation problem we’ll use a simple neural extension of the CKY algorithm. The intuition of such parsing algorithms (often called span-based constituency parsing, or neural CKY), is to train a neural classiﬁer to assign a score to each constituent, and then use a modiﬁed version of CKY to combine these constituent scores to ﬁnd the best-scoring parse tree. Here we’ll describe a version of the algorithm from Kitaev et al. (2019). This parser learns to map a span of words to a constituent, and, like CKY , hierarchically combines larger and larger spans to build the parse-tree bottom-up. But unlike clas- sic CKY , this parser doesn’t use the hand-written grammar to constrain what con- stituents can be combined, instead just relying on the learned neural representations of spans to encode likely combinations. 18.7.1 Computing Scores for a Span Let’s begin by considering just the constituent (we’ll call it aspan) that lies betweenspan fencepost positions i and j with non-terminal symbol label l. We’ll build a system to assign a score s(i,j,l) to this constituent span. ENCODER [START] Book the flight through Houston [END] map to subwords map back to words 0 1 32 4 5 MLP i=1 hj-hi j=3 NP Compute score for span Represent span CKY for computing best parse postprocessing layers Figure 18.15 A simpliﬁed outline of computing the span score for the span the ﬂight with the label NP. Fig. 18.15 sketches the architecture. The input word tokens are embedded by 404 CHAPTER 18 • C ONTEXT -FREE GRAMMARS AND CONSTITUENCY PARSING passing them through a pretrained language model like BERT. Because BERT oper- ates on the level of subword (wordpiece) tokens rather than words, we’ll ﬁrst need to convert the BERT outputs to word representations. One standard way of doing this is to simply use the ﬁrst subword unit as the representation for the entire word; us- ing the last subword unit, or the sum of all the subword units are also common. The embeddings can then be passed through some postprocessing layers; Kitaev et al. (2019), for example, use 8 Transformer layers. The resulting word encoder outputs yt are then used to compute a span score. First, we must map the word encodings (indexed by word positions) to span encod- ings (indexed by fenceposts). We do this by representing each fencepost with two separate values; the intuition is that a span endpoint to the right of a word represents different information than a span endpoint to the left of a word. We convert each word output yt into a (leftward-pointing) value for spans ending at this fencepost,← −y t , and a (rightward-pointing) value − →y t for spans beginning at this fencepost, by splitting yt into two halves. Each span then stretches from one double-vector fence- post to another, as in the following representation of the ﬂight, which is span(1,3): START0 Book the ﬂight through y0 − →y0 ← −y1 y1 − →y1 ← −y2 y2 − →y2 ← −y3 y3 − →y3 ← −y4 y4 − →y4 ← −y5 ... 0⃝ 1⃝ 2⃝ 3⃝ 4⃝ span(1,3) A traditional way to represent a span, developed originally for RNN-based models (Wang and Chang, 2016), but extended also to Transformers, is to take the differ- ence between the embeddings of its start and end, i.e., representing span (i,j) by subtracting the embedding of i from the embedding of j. Here we represent a span by concatenating the difference of each of its fencepost components: v(i,j) = [− →yj −− →yi ; ←−−yj+1 −←−−yi+1] (18.5) The span vector v is then passed through an MLP span classiﬁer, with two fully- connected layers and one ReLU activation function, whose output dimensionality is the number of possible non-terminal labels: s(i,j,·) =W2 ReLU(LayerNorm(W1v(i,j))) (18.6) The MLP then outputs a score for each possible non-terminal. 18.7.2 Integrating Span Scores into a Parse Now we have a score for each labeled constituent spans(i,j,l). But we need a score for an entire parse tree. Formally a tree T is represented as a set of \|T \|such labeled spans, with the tth span starting at position it and ending at position jt , with label lt : T = {(it ,jt ,lt ) : t = 1,..., \|T \|} (18.7) Thus once we have a score for each span, the parser can compute a score for the whole tree s(T ) simply by summing over the scores of its constituent spans: s(T ) = ∑ (i,j,l)∈T s(i,j,l) (18.8) 18.8 • E VALUATING PARSERS 405 And we can choose the ﬁnal parse tree as the tree with the maximum score: ˆT = argmax T s(T ) (18.9) The simplest method to produce the most likely parse is to greedily choose the highest scoring label for each span. This greedy method is not guaranteed to produce a tree, since the best label for a span might not ﬁt into a complete tree. In practice, however, the greedy method tends to ﬁnd trees; in their experiments Gaddy et al. (2018) ﬁnds that 95% of predicted bracketings form valid trees. Nonetheless it is more common to use a variant of the CKY algorithm to ﬁnd the full parse. The variant deﬁned in Gaddy et al. (2018) works as follows. Let’s deﬁne sbest(i,j) as the score of the best subtree spanning (i,j). For spans of length one, we choose the best label: sbest(i,i +1) =max l s(i,i +1,l) (18.10) For other spans (i,j), the recursion is: sbest(i,j) = max l s(i,j,l) + max k [sbest(i,k)+ sbest(k,j)] (18.11) Note that the parser is using the max label for span (i,j) + the max labels for spans (i,k) and (k,j) without worrying about whether those decisions make sense given a grammar. The role of the grammar in classical parsing is to help constrain possible combinations of constituents (NPs like to be followed by VPs). By contrast, the neural model seems to learn these kinds of contextual constraints during its mapping from spans to non-terminals. For more details on span-based parsing, including the margin-based training al- gorithm, see Stern et al. (2017), Gaddy et al. (2018), Kitaev and Klein (2018), and Kitaev et al. (2019). 18.8 Evaluating Parsers The standard tool for evaluating parsers that assign a single parse tree to a sentence is the PARSEV ALmetrics (Black et al., 1991). The PARSEV AL metric measuresPARSEV AL how much theconstituents in the hypothesis parse tree look like the constituents in a hand-labeled, reference parse. PARSEV AL thus requires a human-labeled reference (or “gold standard”) parse tree for each sentence in the test set; we generally draw these reference parses from a treebank like the Penn Treebank. A constituent in a hypothesis parse Ch of a sentence s is labeled correct if there is a constituent in the reference parse Cr with the same starting point, ending point, and non-terminal symbol. We can then measure the precision and recall just as for tasks we’ve seen already like named entity tagging: labeled recall: = # of correct constituents in hypothesis parse of s # of total constituents in reference parse of s labeled precision: = # of correct constituents in hypothesis parse of s # of total constituents in hypothesis parse of s 406 CHAPTER 18 • C ONTEXT -FREE GRAMMARS AND CONSTITUENCY PARSING S(dumped) VP(dumped) PP(into) NP(bin) NN(bin) bin DT(a) a P into NP(sacks) NNS(sacks) sacks VBD(dumped) dumped NP(workers) NNS(workers) workers Figure 18.16 A lexicalized tree from Collins (1999). As usual, we often report a combination of the two, F1: F1 = 2PR P +R (18.12) We additionally use a new metric, crossing brackets, for each sentence s: cross-brackets: the number of constituents for which the reference parse has a bracketing such as ((A B) C) but the hypothesis parse has a bracketing such as (A (B C)). For comparing parsers that use different grammars, the PARSEV AL metric in- cludes a canonicalization algorithm for removing information likely to be grammar- speciﬁc (auxiliaries, pre-inﬁnitival “to”, etc.) and for computing a simpliﬁed score (Black et al., 1991). The canonical implementation of the PARSEV AL metrics is called evalb (Sekine and Collins, 1997).evalb 18.9 Heads and Head-Finding Syntactic constituents can be associated with a lexicalhead; N is the head of an NP, V is the head of a VP. This idea of a head for each constituent dates back to Bloom- ﬁeld 1914, and is central to the dependency grammars and dependency parsing we’ll introduce in Chapter 19. Indeed, heads can be used as a way to map between con- stituency and dependency parses. Heads are also important in probabilistic pars- ing (Appendix C) and in constituent-based grammar formalisms like Head-Driven Phrase Structure Grammar (Pollard and Sag, 1994).. In one simple model of lexical heads, each context-free rule is associated with a head (Charniak 1997, Collins 1999). The head is the word in the phrase that is grammatically the most important. Heads are passed up the parse tree; thus, each non-terminal in a parse tree is annotated with a single word, which is its lexical head. Figure 18.16 shows an example of such a tree from Collins (1999), in which each non-terminal is annotated with its head. For the generation of such a tree, each CFG rule must be augmented to identify one right-side constituent to be the head child. The headword for a node is then set to the headword of its head child. Choosing these head children is simple for textbook examples (NN is the head of NP) but is complicated and indeed controversial for 18.10 • S UMMARY 407 most phrases. (Should the complementizer to or the verb be the head of an inﬁnite verb phrase?) Modern linguistic theories of syntax generally include a component that deﬁnes heads (see, e.g., (Pollard and Sag, 1994)). An alternative approach to ﬁnding a head is used in most practical computational systems. Instead of specifying head rules in the grammar itself, heads are identiﬁed dynamically in the context of trees for speciﬁc sentences. In other words, once a sentence is parsed, the resulting tree is walked to decorate each node with the appropriate head. Most current systems rely on a simple set of handwritten rules, such as a practical one for Penn Treebank grammars given in Collins (1999) but developed originally by Magerman (1995). For example, the rule for ﬁnding the head of an NP is as follows (Collins, 1999, p. 238): • If the last word is tagged POS, return last-word. • Else search from right to left for the ﬁrst child which is an NN, NNP, NNPS, NX, POS, or JJR. • Else search from left to right for the ﬁrst child which is an NP. • Else search from right to left for the ﬁrst child which is a $, ADJP, or PRN. • Else search from right to left for the ﬁrst child which is a CD. • Else search from right to left for the ﬁrst child which is a JJ, JJS, RB or QP. • Else return the last word Selected other rules from this set are shown in Fig. 18.17. For example, for VP rules of the form VP →Y1 ··· Yn, the algorithm would start from the left of Y1 ··· Yn looking for the ﬁrst Yi of type TO; if no TOs are found, it would search for the ﬁrst Yi of type VBD; if no VBDs are found, it would search for a VBN, and so on. See Collins (1999) for more details. Parent Direction Priority List ADJP Left NNS QP NN $ ADVP JJ VBN VBG ADJP JJR NP JJS DT FW RBR RBS SBAR RB ADVP Right RB RBR RBS FW ADVP TO CD JJR JJ IN NP JJS NN PRN Left PRT Right RP QP Left $ IN NNS NN JJ RB DT CD NCD QP JJR JJS S Left TO IN VP S SBAR ADJP UCP NP SBAR Left WHNP WHPP WHADVP WHADJP IN DT S SQ SINV SBAR FRAG VP Left TO VBD VBN MD VBZ VB VBG VBP VP ADJP NN NNS NP Figure 18.17 Some head rules from Collins (1999). The head rules are also called a head percolation table. 18.10 Summary This chapter introduced constituency parsing. Here’s a summary of the main points: • In many languages, groups of consecutive words act as a group or a con- stituent, which can be modeled by context-free grammars (which are also known as phrase-structure grammars). • A context-free grammar consists of a set of rules or productions, expressed over a set of non-terminal symbols and a set of terminal symbols. Formally, a particular context-free language is the set of strings that can be derived from a particular context-free grammar. 408 CHAPTER 18 • C ONTEXT -FREE GRAMMARS AND CONSTITUENCY PARSING • Structural ambiguity is a signiﬁcant problem for parsers. Common sources of structural ambiguity includePP-attachment and coordination ambiguity. • Dynamic programming parsing algorithms, such as CKY, use a table of partial parses to efﬁciently parse ambiguous sentences. • CKY restricts the form of the grammar to Chomsky normal form (CNF). • The basic CKY algorithm compactly represents all possible parses of the sen- tence but doesn’t choose a single best parse. • Choosing a single parse from all possible parses ( disambiguation) can be done by neural constituency parsers. • Span-based neural constituency parses train a neural classiﬁer to assign a score to each constituent, and then use a modiﬁed version of CKY to combine these constituent scores to ﬁnd the best-scoring parse tree. • Parsers are evaluated with three metrics: labeled recall, labeled precision, and cross-brackets. • Partial parsing and chunking are methods for identifying shallow syntac- tic constituents in a text. They are solved by sequence models trained on syntactically-annotated data. Bibliographical and Historical Notes According to Percival (1976), the idea of breaking up a sentence into a hierarchy of constituents appeared in the V¨olkerpsychologie of the groundbreaking psychologist Wilhelm Wundt (Wundt, 1900): ...den sprachlichen Ausdruck f¨ur die willk¨urliche Gliederung einer Ge- sammtvorstellung in ihre in logische Beziehung zueinander gesetzten Bestandteile [the linguistic expression for the arbitrary division of a total idea into its constituent parts placed in logical relations to one another] Wundt’s idea of constituency was taken up into linguistics by Leonard Bloom- ﬁeld in his early book An Introduction to the Study of Language(Bloomﬁeld, 1914). By the time of his later book, Language (Bloomﬁeld, 1933), what was then called “immediate-constituent analysis” was a well-established method of syntactic study in the United States. By contrast, traditional European grammar, dating from the Classical period, deﬁned relations between words rather than constituents, and Eu- ropean syntacticians retained this emphasis on suchdependency grammars, the sub- ject of Chapter 19. (And indeed, both dependency and constituency grammars have been in vogue in computational linguistics at different times). American Structuralism saw a number of speciﬁc deﬁnitions of the immediate constituent, couched in terms of their search for a “discovery procedure”: a method- ological algorithm for describing the syntax of a language. In general, these attempt to capture the intuition that “The primary criterion of the immediate constituent is the degree in which combinations behave as simple units” (Bazell, 1952/1966, p. 284). The most well known of the speciﬁc deﬁnitions is Harris’ idea of distributional similarity to individual units, with the substitutability test. Essentially, the method proceeded by breaking up a construction into constituents by attempting to substitute simple structures for possible constituents—if a substitution of a simple form, say, EXERCISES 409 man, was substitutable in a construction for a more complex set (like intense young man), then the form intense young man was probably a constituent. Harris’s test was the beginning of the intuition that a constituent is a kind of equivalence class. The context-free grammar was a formalization of this idea of hierarchical constituency deﬁned in Chomsky (1956) and further expanded upon (and argued against) in Chomsky (1957) and Chomsky (1956/1975). Shortly after Chomsky’s initial work, the context-free grammar was reinvented by Backus (1959) and inde- pendently by Naur et al. (1960) in their descriptions of the ALGOL programming language; Backus (1996) noted that he was inﬂuenced by the productions of Emil Post and that Naur’s work was independent of his (Backus’) own. After this early work, a great number of computational models of natural language processing were based on context-free grammars because of the early development of efﬁcient pars- ing algorithms. Dynamic programming parsing has a history of independent discovery. Ac- cording to the late Martin Kay (personal communication), a dynamic programming parser containing the roots of the CKY algorithm was ﬁrst implemented by John Cocke in 1960. Later work extended and formalized the algorithm, as well as prov- ing its time complexity (Kay 1967, Younger 1967, Kasami 1965). The related well- formed substring table (WFST) seems to have been independently proposed byWFST Kuno (1965) as a data structure that stores the results of all previous computations in the course of the parse. Based on a generalization of Cocke’s work, a similar data structure had been independently described in Kay (1967) (and Kay 1973). The top-down application of dynamic programming to parsing was described in Earley’s Ph.D. dissertation (Earley 1968, Earley 1970). Sheil (1976) showed the equivalence of the WFST and the Earley algorithm. Norvig (1991) shows that the efﬁciency of- fered by dynamic programming can be captured in any language with amemoization function (such as in LISP) simply by wrapping the memoization operation around a simple top-down parser. The earliest disambiguation algorithms for parsing were based on probabilistic context-free grammars, ﬁrst worked out by Booth (1969) and Salomaa (1969); see probabilistic context-free grammars Appendix C for more history. Neural methods were ﬁrst applied to parsing at around the same time as statistical parsing methods were developed (Henderson, 1994). In the earliest work neural networks were used to estimate some of the probabilities for statistical constituency parsers (Henderson, 2003, 2004; Emami and Jelinek, 2005) . The next decades saw a wide variety of neural parsing algorithms, including re- cursive neural architectures (Socher et al., 2011, 2013), encoder-decoder models (Vinyals et al., 2015; Choe and Charniak, 2016), and the idea of focusing on spans (Cross and Huang, 2016). For more on the span-based self-attention approach we describe in this chapter see Stern et al. (2017), Gaddy et al. (2018), Kitaev and Klein (2018), and Kitaev et al. (2019). See Chapter 20 for the parallel history of neural dependency parsing. The classic reference for parsing algorithms is Aho and Ullman (1972); although the focus of that book is on computer languages, most of the algorithms have been applied to natural language. Exercises 18.1 Implement the algorithm to convert arbitrary context-free grammars to CNF. 410 CHAPTER 18 • C ONTEXT -FREE GRAMMARS AND CONSTITUENCY PARSING Apply your program to the L1 grammar. 18.2 Implement the CKY algorithm and test it with your converted L1 grammar. 18.3 Rewrite the CKY algorithm given in Fig. 18.12 on page 400 so that it can accept grammars that contain unit productions. 18.4 Discuss how to augment a parser to deal with input that may be incorrect, for example, containing spelling errors or mistakes arising from automatic speech recognition. 18.5 Implement the PARSEV AL metrics described in Section 18.8. Next, use a parser and a treebank, compare your metrics against a standard implementa- tion. Analyze the errors in your approach. CHAPTER 19 Dependency Parsing Tout mot qui fait partie d’une phrase... Entre lui et ses voisins, l’esprit aperc ¸oit des connexions, dont l’ensemble forme la charpente de la phrase. [Between each word in a sentence and its neighbors, the mind perceives con- nections. These connections together form the scaffolding of the sentence.] Lucien Tesni`ere. 1959. ´El´ements de syntaxe structurale, A.1.§4 The focus of the last chapter was on context-free grammars and constituent- based representations. Here we present another important family of grammar for- malisms called dependency grammars. In dependency formalisms, phrasal con-dependency grammars stituents and phrase-structure rules do not play a direct role. Instead, the syntactic structure of a sentence is described solely in terms of directed binary grammatical relations between the words, as in the following dependency parse: I prefer the morning ﬂight through Denver nsubj obj det compound nmod case root (19.1) Relations among the words are illustrated above the sentence with directed, labeled arcs from heads to dependents. We call this atyped dependency structurebecausetyped dependency the labels are drawn from a ﬁxed inventory of grammatical relations. A root node explicitly marks the root of the tree, the head of the entire structure. Figure 19.1 on the next page shows the dependency analysis from Eq. 19.1 but visualized as a tree, alongside its corresponding phrase-structure analysis of the kind given in the prior chapter. Note the absence of nodes corresponding to phrasal con- stituents or lexical categories in the dependency parse; the internal structure of the dependency parse consists solely of directed relations between words. These head- dependent relationships directly encode important information that is often buried in the more complex phrase-structure parses. For example, the arguments to the verb prefer are directly linked to it in the dependency structure, while their connection to the main verb is more distant in the phrase-structure tree. Similarly, morning and Denver, modiﬁers of ﬂight, are linked to it directly in the dependency structure. This fact that the head-dependent relations are a good proxy for the semantic rela- tionship between predicates and their arguments is an important reason why depen- dency grammars are currently more common than constituency grammars in natural language processing. Another major advantage of dependency grammars is their ability to deal with languages that have a relativelyfree word order. For example, word order in Czechfree word order can be much more ﬂexible than in English; a grammaticalobject might occur before or after alocation adverbial. A phrase-structure grammar would need a separate rule 412 CHAPTER 19 • D EPENDENCY PARSING prefer ﬂight Denver through morningthe I S VP NP Nom PP NP Pro Denver P through Nom Noun ﬂight Nom Noun morning Det the Verb prefer NP Pro I Figure 19.1 Dependency and constituent analyses for I prefer the morning ﬂight through Denver. for each possible place in the parse tree where such an adverbial phrase could occur. A dependency-based approach can have just one link type representing this particu- lar adverbial relation; dependency grammar approaches can thus abstract away a bit more from word order information. In the following sections, we’ll give an inventory of relations used in dependency parsing, discuss two families of parsing algorithms (transition-based, and graph- based), and discuss evaluation. 19.1 Dependency Relations The traditional linguistic notion of grammatical relation provides the basis for thegrammatical relation binary relations that comprise these dependency structures. The arguments to these relations consist of a head and a dependent. The head plays the role of the centralhead dependent organizing word, and the dependent as a kind of modiﬁer. The head-dependent rela- tionship is made explicit by directly linking heads to the words that are immediately dependent on them. In addition to specifying the head-dependent pairs, dependency grammars allow us to classify the kinds of grammatical relations, or grammatical function that thegrammatical function dependent plays with respect to its head. These include familiar notions such as subject, direct object and indirect object. In English these notions strongly corre- late with, but by no means determine, both position in a sentence and constituent type and are therefore somewhat redundant with the kind of information found in phrase-structure trees. However, in languages with more ﬂexible word order, the information encoded directly in these grammatical relations is critical since phrase- based constituent syntax provides little help. Linguists have developed taxonomies of relations that go well beyond the famil- iar notions of subject and object. While there is considerable variation from theory 19.1 • D EPENDENCY RELATIONS 413 Clausal Argument Relations Description NSUBJ Nominal subject OBJ Direct object IOBJ Indirect object CCOMP Clausal complement Nominal Modiﬁer Relations Description NMOD Nominal modiﬁer AMOD Adjectival modiﬁer APPOS Appositional modiﬁer DET Determiner CASE Prepositions, postpositions and other case markers Other Notable Relations Description CONJ Conjunct CC Coordinating conjunction Figure 19.2 Some of the Universal Dependency relations (de Marneffe et al., 2021). to theory, there is enough commonality that cross-linguistic standards have been developed. The Universal Dependencies (UD) project (de Marneffe et al., 2021),Universal Dependencies an open community effort to annotate dependencies and other aspects of grammar across more than 100 languages, provides an inventory of 37 dependency relations. Fig. 19.2 shows a subset of the UD relations and Fig. 19.3 provides some examples. The motivation for all of the relations in the Universal Dependency scheme is beyond the scope of this chapter, but the core set of frequently used relations can be broken into two sets: clausal relations that describe syntactic roles with respect to a predicate (often a verb), and modiﬁer relations that categorize the ways that words can modify their heads. Consider, for example, the following sentence: United canceled the morning ﬂights to Houston nsubj obj det compound nmod case root (19.2) Here the clausal relations NSUBJ and OBJ identify the subject and direct object of the predicate cancel, while the NMOD , DET , and CASE relations denote modiﬁers of the nouns ﬂights and Houston. 19.1.1 Dependency Formalisms A dependency structure can be represented as a directed graphG = (V,A), consisting of a set of vertices V , and a set of ordered pairs of vertices A, which we’ll call arcs. For the most part we will assume that the set of vertices, V , corresponds exactly to the set of words in a given sentence. However, they might also correspond to punctuation, or when dealing with morphologically complex languages the set of vertices might consist of stems and afﬁxes. The set of arcs, A, captures the head- dependent and grammatical function relationships between the elements in V . Different grammatical theories or formalisms may place further constraints on these dependency structures. Among the more frequent restrictions are that the struc- tures must be connected, have a designated root node, and be acyclic or planar. Of most relevance to the parsing approaches discussed in this chapter is the common, 414 CHAPTER 19 • D EPENDENCY PARSING Relation Examples with head and dependent NSUBJ United canceled the ﬂight. OBJ United diverted the ﬂight to Reno. We booked her the ﬁrst ﬂight to Miami. IOBJ We booked her the ﬂight to Miami. COMPOUND We took the morning ﬂight. NMOD ﬂight to Houston. AMOD Book the cheapest ﬂight. APPOS United, a unit of UAL, matched the fares. DET The ﬂight was canceled. Which ﬂight was delayed? CONJ We ﬂew to Denver and drove to Steamboat. CC We ﬂew to Denver and drove to Steamboat. CASE Book the ﬂight through Houston. Figure 19.3 Examples of some Universal Dependency relations. computationally-motivated, restriction to rooted trees. That is, a dependency treedependency tree is a directed graph that satisﬁes the following constraints: 1. There is a single designated root node that has no incoming arcs. 2. With the exception of the root node, each vertex has exactly one incoming arc. 3. There is a unique path from the root node to each vertex in V . Taken together, these constraints ensure that each word has a single head, that the dependency structure is connected, and that there is a single root node from which one can follow a unique directed path to each of the words in the sentence. 19.1.2 Projectivity The notion of projectivity imposes an additional constraint that is derived from the order of the words in the input. An arc from a head to a dependent is said to be projective if there is a path from the head to every word that lies between the headprojective and the dependent in the sentence. A dependency tree is then said to be projective if all the arcs that make it up are projective. All the dependency trees we’ve seen thus far have been projective. There are, however, many valid constructions which lead to non-projective trees, particularly in languages with relatively ﬂexible word order. Consider the following example. JetBlue canceled our ﬂight this morning which was already late nsubj obj obl det acl:relcl det nsubj cop adv root (19.3) In this example, the arc from ﬂight to its modiﬁer late is non-projective since there is no path from ﬂight to the intervening wordsthis and morning. As we can see from this diagram, projectivity (and non-projectivity) can be detected in the way we’ve been drawing our trees. A dependency tree is projective if it can be drawn with no crossing edges. Here there is no way to link ﬂight to its dependent late without crossing the arc that links morning to its head. 19.1 • D EPENDENCY RELATIONS 415 Our concern with projectivity arises from two related issues. First, the most widely used English dependency treebanks were automatically derived from phrase- structure treebanks through the use of head-ﬁnding rules. The trees generated in such a fashion will always be projective, and hence will be incorrect when non-projective examples like this one are encountered. Second, there are computational limitations to the most widely used families of parsing algorithms. The transition-based approaches discussed in Section 19.2 can only produce projective trees, hence any sentences with non-projective structures will necessarily contain some errors. This limitation is one of the motivations for the more ﬂexible graph-based parsing approach described in Section 19.3. 19.1.3 Dependency Treebanks Treebanks play a critical role in the development and evaluation of dependency parsers. They are used for training parsers, they act as the gold labels for evaluating parsers, and they also provide useful information for corpus linguistics studies. Dependency treebanks are created by having human annotators directly generate dependency structures for a given corpus, or by hand-correcting the output of an automatic parser. A few early treebanks were also based on using a deterministic process to translate existing constituent-based treebanks into dependency trees. The largest open community project for building dependency trees is the Univer- sal Dependencies project athttps://universaldependencies.org/introduced above, which currently has almost 200 dependency treebanks in more than 100 lan- guages (de Marneffe et al., 2021). Here are a few UD examples showing dependency trees for sentences in Spanish, Basque, and Mandarin Chinese: VERB ADP DET NOUN ADP DET NUM PUNCT Subiremos a el tren a las cinco . we-will-board on the train at the ﬁve . obl det case det obl:tmod case punct [Spanish] Subiremos al tren a las cinco. “We will be boarding the train at ﬁve.”(19.4) NOUN NOUN VERB AUX PUNCT Ekaitzak itsasontzia hondoratu du . storm (Erg.) ship (Abs.) sunk has . nsubj obj aux punct [Basque] Ekaitzak itsasontzia hondoratu du. “The storm has sunk the ship.”(19.5) 416 CHAPTER 19 • D EPENDENCY PARSING ADV PRON NOUN ADV VERB VERB NOUN 但我昨天才收到信 but I yesterday only-then receive arrive letter . adv nsubj obj:tmod advmod compound:vv obj [Chinese] 但我昨天才收到信 “But I didn’t receive the letter until yesterday”(19.6) 19.2 Transition-Based Dependency Parsing Our ﬁrst approach to dependency parsing is called transition-based parsing. Thistransition-based architecture draws on shift-reduce parsing, a paradigm originally developed for analyzing programming languages (Aho and Ullman, 1972). In transition-based parsing we’ll have a stack on which we build the parse, a buffer of tokens to be parsed, and a parser which takes actions on the parse via a predictor called anoracle, as illustrated in Fig. 19.4. wnw1 w2 s2 ... s1 sn Parser Input buﬀer Stack Oracle LEFTARC RIGHTARC SHIFT Action Dependency Relations w3 w2 Figure 19.4 Basic transition-based parser. The parser examines the top two elements of the stack and selects an action by consulting an oracle that examines the current conﬁguration. The parser walks through the sentence left-to-right, successively shifting items from the buffer onto the stack. At each time point we examine the top two elements on the stack, and the oracle makes a decision about whattransition to apply to build the parse. The possible transitions correspond to the intuitive actions one might take in creating a dependency tree by examining the words in a single pass over the input from left to right (Covington, 2001): • Assign the current word as the head of some previously seen word, • Assign some previously seen word as the head of the current word, • Postpone dealing with the current word, storing it for later processing. We’ll formalize this intuition with the following three transition operators that will operate on the top two elements of the stack: • LEFT ARC: Assert a head-dependent relation between the word at the top of the stack and the second word; remove the second word from the stack. • RIGHT ARC: Assert a head-dependent relation between the second word on the stack and the word at the top; remove the top word from the stack; 19.2 • T RANSITION -BASED DEPENDENCY PARSING 417 • SHIFT : Remove the word from the front of the input buffer and push it onto the stack. We’ll sometimes call operations like LEFT ARC and RIGHT ARC reduce operations, based on a metaphor from shift-reduce parsing, in which reducing means combin- ing elements on the stack. There are some preconditions for using operators. The LEFT ARC operator cannot be applied when ROOT is the second element of the stack (since by deﬁnition the ROOT node cannot have any incoming arcs). And both the LEFT ARC and RIGHT ARC operators require two elements to be on the stack to be applied. This particular set of operators implements what is known as the arc standardarc standard approach to transition-based parsing (Covington 2001, Nivre 2003). In arc standard parsing the transition operators only assert relations between elements at the top of the stack, and once an element has been assigned its head it is removed from the stack and is not available for further processing. As we’ll see, there are alterna- tive transition systems which demonstrate different parsing behaviors, but the arc standard approach is quite effective and is simple to implement. The speciﬁcation of a transition-based parser is quite simple, based on repre- senting the current state of the parse as a conﬁguration: the stack, an input bufferconﬁguration of words or tokens, and a set of relations representing a dependency tree. Parsing means making a sequence of transitions through the space of possible conﬁgura- tions. We start with an initial conﬁguration in which the stack contains the ROOT node, the buffer has the tokens in the sentence, and an empty set of relations repre- sents the parse. In the ﬁnal goal state, the stack and the word list should be empty, and the set of relations will represent the ﬁnal parse. Fig. 19.5 gives the algorithm. function DEPENDENCY PARSE (words) returns dependency tree state←{[root], [words], [] } ; initial conﬁguration while state not ﬁnal t←ORACLE (state) ; choose a transition operator to apply state←APPLY(t, state) ; apply it, creating a new state return state Figure 19.5 A generic transition-based dependency parser At each step, the parser consults an oracle (we’ll come back to this shortly) that provides the correct transition operator to use given the current conﬁguration. It then applies that operator to the current conﬁguration, producing a new conﬁguration. The process ends when all the words in the sentence have been consumed and the ROOT node is the only element remaining on the stack. The efﬁciency of transition-based parsers should be apparent from the algorithm. The complexity is linear in the length of the sentence since it is based on a single left to right pass through the words in the sentence. (Each word must ﬁrst be shifted onto the stack and then later reduced.) Note that unlike the dynamic programming and search-based approaches dis- cussed in Chapter 18, this approach is a straightforward greedy algorithm—the or- acle provides a single choice at each step and the parser proceeds with that choice, no other options are explored, no backtracking is employed, and a single parse is returned in the end. Figure 19.6 illustrates the operation of the parser with the sequence of transitions 418 CHAPTER 19 • D EPENDENCY PARSING leading to a parse for the following example. Book me the morning ﬂight iobj obj det compound root (19.7) Let’s consider the state of the conﬁguration at Step 2, after the wordme has been pushed onto the stack. Stack Word List Relations [root, book, me] [the, morning, ﬂight] The correct operator to apply here is RIGHT ARC which assigns book as the head of me and pops me from the stack resulting in the following conﬁguration. Stack Word List Relations [root, book] [the, morning, ﬂight] (book →me) After several subsequent applications of the SHIFT operator, the conﬁguration in Step 6 looks like the following: Stack Word List Relations [root, book, the, morning, ﬂight] [] (book →me) Here, all the remaining words have been passed onto the stack and all that is left to do is to apply the appropriate reduce operators. In the current conﬁguration, we employ the LEFT ARC operator resulting in the following state. Stack Word List Relations [root, book, the, ﬂight] [] (book →me) (morning ←ﬂight) At this point, the parse for this sentence consists of the following structure. Book me the morning ﬂight iobj compound (19.8) There are several important things to note when examining sequences such as the one in Figure 19.6. First, the sequence given is not the only one that might lead to a reasonable parse. In general, there may be more than one path that leads to the same result, and due to ambiguity, there may be other transition sequences that lead to different equally valid parses. Second, we are assuming that the oracle always provides the correct operator at each point in the parse—an assumption that is unlikely to be true in practice. As a result, given the greedy nature of this algorithm, incorrect choices will lead to incorrect parses since the parser has no opportunity to go back and pursue alternative choices. Section 19.2.4 will introduce several techniques that allow transition-based approaches to explore the search space more fully. 19.2 • T RANSITION -BASED DEPENDENCY PARSING 419 Step Stack Word List Action Relation Added 0 [root] [book, me, the, morning, ﬂight] SHIFT 1 [root, book] [me, the, morning, ﬂight] SHIFT 2 [root, book, me] [the, morning, ﬂight] RIGHT ARC (book →me) 3 [root, book] [the, morning, ﬂight] SHIFT 4 [root, book, the] [morning, ﬂight] SHIFT 5 [root, book, the, morning] [ﬂight] SHIFT 6 [root, book, the, morning, ﬂight] [] LEFT ARC (morning ←ﬂight) 7 [root, book, the, ﬂight] [] LEFT ARC (the ←ﬂight) 8 [root, book, ﬂight] [] RIGHT ARC (book →ﬂight) 9 [root, book] [] RIGHT ARC (root →book) 10 [root] [] Done Figure 19.6 Trace of a transition-based parse. Finally, for simplicity, we have illustrated this example without the labels on the dependency relations. To produce labeled trees, we can parameterize the LEFT - ARC and RIGHT ARC operators with dependency labels, as in LEFT ARC(NSUBJ ) or RIGHT ARC(OBJ ). This is equivalent to expanding the set of transition operators from our original set of three to a set that includesLEFT ARC and RIGHT ARC operators for each relation in the set of dependency relations being used, plus an additional one for the SHIFT operator. This, of course, makes the job of the oracle more difﬁcult since it now has a much larger set of operators from which to choose. 19.2.1 Creating an Oracle The oracle for greedily selecting the appropriate transition is trained by supervised machine learning. As with all supervised machine learning methods, we will need training data: conﬁgurations annotated with the correct transition to take. We can draw these from dependency trees. And we need to extract features of the con- ﬁguration. We’ll introduce neural classiﬁers that represent the conﬁguration via embeddings, as well as classic systems that use hand-designed features. Generating Training Data The oracle from the algorithm in Fig. 19.5 takes as input a conﬁguration and returns a transition operator. Therefore, to train a classiﬁer, we will need conﬁgurations paired with transition operators (i.e., LEFT ARC, RIGHT ARC, or SHIFT ). Unfortunately, treebanks pair entire sentences with their corresponding trees, not conﬁgurations with transitions. To generate the required training data, we employ the oracle-based parsing algo- rithm in a clever way. We supply our oracle with the training sentences to be parsed along with their corresponding reference parses from the treebank. To produce train- ing instances, we then simulate the operation of the parser by running the algorithm and relying on a new training oracle to give us correct transition operators for eachtraining oracle successive conﬁguration. To see how this works, let’s ﬁrst review the operation of our parser. It begins with a default initial conﬁguration where the stack contains theROOT , the input list is just the list of words, and the set of relations is empty. The LEFT ARC and RIGHT ARC operators each add relations between the words at the top of the stack to the set of relations being accumulated for a given sentence. Since we have a gold-standard reference parse for each training sentence, we know which dependency relations are valid for a given sentence. Therefore, we can use the reference parse to guide the 420 CHAPTER 19 • D EPENDENCY PARSING Step Stack Word List Predicted Action 0 [root] [book, the, ﬂight, through, houston] SHIFT 1 [root, book] [the, ﬂight, through, houston] SHIFT 2 [root, book, the] [ﬂight, through, houston] SHIFT 3 [root, book, the, ﬂight] [through, houston] LEFT ARC 4 [root, book, ﬂight] [through, houston] SHIFT 5 [root, book, ﬂight, through] [houston] SHIFT 6 [root, book, ﬂight, through, houston] [] LEFT ARC 7 [root, book, ﬂight, houston ] [] RIGHT ARC 8 [root, book, ﬂight] [] RIGHT ARC 9 [root, book] [] RIGHT ARC 10 [root] [] Done Figure 19.7 Generating training items consisting of conﬁguration/predicted action pairs by simulating a parse with a given reference parse. selection of operators as the parser steps through a sequence of conﬁgurations. To be more precise, given a reference parse and a conﬁguration, the training oracle proceeds as follows: • Choose LEFT ARC if it produces a correct head-dependent relation given the reference parse and the current conﬁguration, • Otherwise, choose RIGHT ARC if (1) it produces a correct head-dependent re- lation given the reference parse and (2) all of the dependents of the word at the top of the stack have already been assigned, • Otherwise, choose SHIFT . The restriction on selecting the RIGHT ARC operator is needed to ensure that a word is not popped from the stack, and thus lost to further processing, before all its dependents have been assigned to it. More formally, during training the oracle has access to the following: • A current conﬁguration with a stack S and a set of dependency relations Rc • A reference parse consisting of a set of vertices V and a set of dependency relations Rp Given this information, the oracle chooses transitions as follows: LEFT ARC(r): if (S1 r S2) ∈Rp RIGHT ARC(r): if (S2 r S1) ∈Rp and ∀r′,w s.t.(S1 r′w) ∈Rp then (S1 r′w) ∈Rc SHIFT : otherwise Let’s walk through the processing of the following example as shown in Fig. 19.7. Book the ﬂight through Houston obj det nmod case root (19.9) At Step 1, LEFT ARC is not applicable in the initial conﬁguration since it asserts a relation, (root ←book), not in the reference answer; RIGHT ARC does assert a relation contained in the ﬁnal answer (root →book), however book has not been attached to any of its dependents yet, so we have to defer, leaving SHIFT as the only 19.2 • T RANSITION -BASED DEPENDENCY PARSING 421 possible action. The same conditions hold in the next two steps. In step 3,LEFT ARC is selected to link the to its head. Now consider the situation in Step 4. Stack Word buffer Relations [root, book, ﬂight] [through, Houston] (the ←ﬂight) Here, we might be tempted to add a dependency relation between book and ﬂight, which is present in the reference parse. But doing so now would prevent the later attachment of Houston since ﬂight would have been removed from the stack. For- tunately, the precondition on choosing RIGHT ARC prevents this choice and we’re again left with SHIFT as the only viable option. The remaining choices complete the set of operators needed for this example. To recap, we derive appropriate training instances consisting of conﬁguration- transition pairs from a treebank by simulating the operation of a parser in the con- text of a reference dependency tree. We can deterministically record correct parser actions at each step as we progress through each training example, thereby creating the training set we require. 19.2.2 A feature-based classiﬁer We’ll now introduce two classiﬁers for choosing transitions, here a classic feature- based algorithm and in the next section a neural classiﬁer using embedding features. Featured-based classiﬁers generally use the same features we’ve seen with part- of-speech tagging and partial parsing: Word forms, lemmas, parts of speech, the head, and the dependency relation to the head. Other features may be relevant for some languages, for example morphosyntactic features like case marking on subjects or objects. The features are extracted from the trainingconﬁgurations, which consist of the stack, the buffer and the current set of relations. Most useful are features referencing the top levels of the stack, the words near the front of the buffer, and the dependency relations already associated with any of those elements. We’ll use afeature template as we did for sentiment analysis and part-of-speechfeature template tagging. Feature templates allow us to automatically generate large numbers of spe- ciﬁc features from a training set. For example, consider the following feature tem- plates that are based on single positions in a conﬁguration. ⟨s1.w,op⟩,⟨s2.w,op⟩⟨s1.t,op⟩,⟨s2.t,op⟩ ⟨b1.w,op⟩,⟨b1.t,op⟩⟨s1.wt,op⟩ (19.10) Here features are denoted as location.property, where s = stack, b = the word buffer, w = word forms, t = part-of-speech, and op = operator. Thus the feature for the word form at the top of the stack would be s1.w, the part of speech tag at the front of the buffer b1.t, and the concatenated feature s1.wt represents the word form concatenated with the part of speech of the word at the top of the stack. Consider applying these templates to the following intermediate conﬁguration derived from a training oracle for (19.2). Stack Word buffer Relations [root, canceled, ﬂights] [to Houston] (canceled →United) (ﬂights →morning) (ﬂights →the) 422 CHAPTER 19 • D EPENDENCY PARSING The correct transition here is SHIFT (you should convince yourself of this before proceeding). The application of our set of feature templates to this conﬁguration would result in the following set of instantiated features. ⟨s1.w = ﬂights,op = shift⟩ (19.11) ⟨s2.w = canceled,op = shift⟩ ⟨s1.t = NNS,op = shift⟩ ⟨s2.t = VBD,op = shift⟩ ⟨b1.w = to,op = shift⟩ ⟨b1.t = TO,op = shift⟩ ⟨s1.wt = ﬂightsNNS,op = shift⟩ Given that the left and right arc transitions operate on the top two elements of the stack, features that combine properties from these positions are even more useful. For example, a feature like s1.t ◦s2.t concatenates the part of speech tag of the word at the top of the stack with the tag of the word beneath it. ⟨s1.t ◦s2.t = NNSVBD,op = shift⟩ (19.12) Given the training data and features, any classiﬁer, like multinomial logistic re- gression or support vector machines, can be used. 19.2.3 A neural classiﬁer The oracle can also be implemented by a neural classiﬁer. A standard architecture is simply to pass the sentence through an encoder, then take the presentation of the top 2 words on the stack and the ﬁrst word of the buffer, concatenate them, and present to a feedforward network that predicts the transition to take (Kiperwasser and Goldberg, 2016; Kulmizev et al., 2019). Fig. 19.8 sketches this model. Learning can be done with cross-entropy loss. w … s2 ... s1 Input buﬀer Stack LEFTARC RIGHTARC SHIFT Action Dependency Relations w3 w2 ENCODER w1 w2 w3 w4 w5 w6 Parser Oracle Softmax FFN w s1 s2 e(w) e(s1) e(s2) Figure 19.8 Neural classiﬁer for the oracle for the transition-based parser. The parser takes the top 2 words on the stack and the ﬁrst word of the buffer, represents them by their encodings (from running the whole sentence through the encoder), concatenates the embeddings and passes through a softmax to choose a parser action (transition). 19.2 • T RANSITION -BASED DEPENDENCY PARSING 423 19.2.4 Advanced Methods in Transition-Based Parsing The basic transition-based approach can be elaborated in a number of ways to im- prove performance by addressing some of the most obvious ﬂaws in the approach. Alternative Transition Systems The arc-standard transition system described above is only one of many possible sys- tems. A frequently used alternative is thearc eager transition system. The arc eagerarc eager approach gets its name from its ability to assert rightward relations much sooner than in the arc standard approach. To see this, let’s revisit the arc standard trace of Example 19.9, repeated here. Book the ﬂight through Houston obj det nmod case root Consider the dependency relation between book and ﬂight in this analysis. As is shown in Fig. 19.7, an arc-standard approach would assert this relation at Step 8, despite the fact that book and ﬂight ﬁrst come together on the stack much earlier at Step 4. The reason this relation can’t be captured at this point is due to the presence of the postnominal modiﬁer through Houston. In an arc-standard approach, depen- dents are removed from the stack as soon as they are assigned their heads. If ﬂight had been assigned book as its head in Step 4, it would no longer be available to serve as the head of Houston. While this delay doesn’t cause any issues in this example, in general the longer a word has to wait to get assigned its head the more opportunities there are for something to go awry. The arc-eager system addresses this issue by allowing words to be attached to their heads as early as possible, before all the subsequent words dependent on them have been seen. This is accomplished through minor changes to the LEFT ARC and RIGHT ARC operators and the addition of a newREDUCE operator. • LEFT ARC: Assert a head-dependent relation between the word at the front of the input buffer and the word at the top of the stack; pop the stack. • RIGHT ARC: Assert a head-dependent relation between the word on the top of the stack and the word at the front of the input buffer; shift the word at the front of the input buffer to the stack. • SHIFT : Remove the word from the front of the input buffer and push it onto the stack. • REDUCE : Pop the stack. The LEFT ARC and RIGHT ARC operators are applied to the top of the stack and the front of the input buffer, instead of the top two elements of the stack as in the arc-standard approach. The RIGHT ARC operator now moves the dependent to the stack from the buffer rather than removing it, thus making it available to serve as the head of following words. The new REDUCE operator removes the top element from the stack. Together these changes permit a word to be eagerly assigned its head and still allow it to serve as the head for later dependents. The trace shown in Fig. 19.9 illustrates the new decision sequence for this example. In addition to demonstrating the arc-eager transition system, this example demon- strates the power and ﬂexibility of the overall transition-based approach. We were able to swap in a new transition system without having to make any changes to the 424 CHAPTER 19 • D EPENDENCY PARSING Step Stack Word List Action Relation Added 0 [root] [book, the, ﬂight, through, houston] RIGHT ARC (root →book) 1 [root, book] [the, ﬂight, through, houston] SHIFT 2 [root, book, the] [ﬂight, through, houston] LEFT ARC (the ←ﬂight) 3 [root, book] [ﬂight, through, houston] RIGHT ARC (book →ﬂight) 4 [root, book, ﬂight] [through, houston] SHIFT 5 [root, book, ﬂight, through] [houston] LEFT ARC (through ←houston) 6 [root, book, ﬂight] [houston] RIGHT ARC (ﬂight →houston) 7 [root, book, ﬂight, houston] [] REDUCE 8 [root, book, ﬂight] [] REDUCE 9 [root, book] [] REDUCE 10 [root] [] Done Figure 19.9 A processing trace of Book the ﬂight through Houston using the arc-eager transition operators. underlying parsing algorithm. This ﬂexibility has led to the development of a di- verse set of transition systems that address different aspects of syntax and semantics including: assigning part of speech tags (Choi and Palmer, 2011a), allowing the generation of non-projective dependency structures (Nivre, 2009), assigning seman- tic roles (Choi and Palmer, 2011b), and parsing texts containing multiple languages (Bhat et al., 2017). Beam Search The computational efﬁciency of the transition-based approach discussed earlier de- rives from the fact that it makes a single pass through the sentence, greedily making decisions without considering alternatives. Of course, this is also a weakness – once a decision has been made it can not be undone, even in the face of overwhelming evidence arriving later in a sentence. We can use beam search to explore alterna-beam search tive decision sequences. Recall from Chapter 9 that beam search uses a breadth-ﬁrst search strategy with a heuristic ﬁlter that prunes the search frontier to stay within a ﬁxed-size beam width.beam width In applying beam search to transition-based parsing, we’ll elaborate on the al- gorithm given in Fig. 19.5. Instead of choosing the single best transition operator at each iteration, we’ll apply all applicable operators to each state on an agenda and then score the resulting conﬁgurations. We then add each of these new conﬁgura- tions to the frontier, subject to the constraint that there has to be room within the beam. As long as the size of the agenda is within the speciﬁed beam width, we can add new conﬁgurations to the agenda. Once the agenda reaches the limit, we only add new conﬁgurations that are better than the worst conﬁguration on the agenda (removing the worst element so that we stay within the limit). Finally, to insure that we retrieve the best possible state on the agenda, the while loop continues as long as there are non-ﬁnal states on the agenda. The beam search approach requires a more elaborate notion of scoring than we used with the greedy algorithm. There, we assumed that the oracle would be a supervised classiﬁer that chose the best transition operator based on features of the current conﬁguration. This choice can be viewed as assigning a score to all the possible transitions and picking the best one. ˆT (c) =argmaxScore(t,c) With beam search we are now searching through the space of decision sequences, so it makes sense to base the score for a conﬁguration on its entire history. So we can deﬁne the score for a new conﬁguration as the score of its predecessor plus the 19.3 • G RAPH -BASED DEPENDENCY PARSING 425 score of the operator used to produce it. ConﬁgScore(c0) = 0.0 ConﬁgScore(ci) = ConﬁgScore(ci−1)+ Score(ti,ci−1) This score is used both in ﬁltering the agenda and in selecting the ﬁnal answer. The new beam search version of transition-based parsing is given in Fig. 19.10. function DEPENDENCY BEAM PARSE (words, width) returns dependency tree state←{[root], [words], [], 0.0} ;initial conﬁguration agenda←⟨state⟩ ;initial agenda while agenda contains non-ﬁnal states newagenda←⟨⟩ for each state ∈agenda do for all {t \|t ∈VALID OPERATORS (state)} do child←APPLY(t, state) newagenda←ADDTOBEAM (child, newagenda, width) agenda←newagenda return BEST OF(agenda) function ADDTOBEAM (state, agenda, width) returns updated agenda if LENGTH (agenda) <width then agenda←INSERT (state, agenda) else if SCORE (state) >SCORE (WORST OF(agenda)) agenda←REMOVE (WORST OF(agenda)) agenda←INSERT (state, agenda) return agenda Figure 19.10 Beam search applied to transition-based dependency parsing. 19.3 Graph-Based Dependency Parsing Graph-based methods are the second important family of dependency parsing algo- rithms. Graph-based parsers are more accurate than transition-based parsers, espe- cially on long sentences; transition-based methods have trouble when the heads are very far from the dependents (McDonald and Nivre, 2011). Graph-based methods avoid this difﬁculty by scoring entire trees, rather than relying on greedy local de- cisions. Furthermore, unlike transition-based approaches, graph-based parsers can produce non-projective trees. Although projectivity is not a signiﬁcant issue for English, it is deﬁnitely a problem for many of the world’s languages. Graph-based dependency parsers search through the space of possible trees for a given sentence for a tree (or trees) that maximize some score. These methods encode the search space as directed graphs and employ methods drawn from graph theory to search the space for optimal solutions. More formally, given a sentence S we’re looking for the best dependency tree in Gs, the space of all possible trees for that sentence, that maximizes some score. ˆT (S) =argmax t∈GS Score(t,S) 426 CHAPTER 19 • D EPENDENCY PARSING We’ll make the simplifying assumption that this score can be edge-factored,edge-factored meaning that the overall score for a tree is the sum of the scores of each of the scores of the edges that comprise the tree. Score(t,S) = ∑ e∈t Score(e) Graph-based algorithms have to solve two problems: (1) assigning a score to each edge, and (2) ﬁnding the best parse tree given the scores of all potential edges. In the next few sections we’ll introduce solutions to these two problems, beginning with the second problem of ﬁnding trees, and then giving a feature-based and a neural algorithm for solving the ﬁrst problem of assigning scores. 19.3.1 Parsing via ﬁnding the maximum spanning tree In graph-based parsing, given a sentence S we start by creating a graph G which is a fully-connected, weighted, directed graph where the vertices are the input words and the directed edges represent all possible head-dependent assignments. We’ll include an additional ROOT node with outgoing edges directed at all of the other vertices. The weights of each edge in G reﬂect the score for each possible head-dependent relation assigned by some scoring algorithm. It turns out that ﬁnding the best dependency parse for S is equivalent to ﬁnding the maximum spanning tree over G. A spanning tree over a graph G is a subsetmaximum spanning tree of G that is a tree and covers all the vertices in G; a spanning tree over G that starts from the ROOT is a valid parse of S. A maximum spanning tree is the spanning tree with the highest score. Thus a maximum spanning tree of G emanating from the ROOT is the optimal dependency parse for the sentence. A directed graph for the exampleBook that ﬂight is shown in Fig. 19.11, with the maximum spanning tree corresponding to the desired parse shown in blue. For ease of exposition, we’ll describe here the algorithm forunlabeled dependency parsing. root Book that ﬂight 12 4 4 5 6 8 7 5 7 Figure 19.11 Initial rooted, directed graph for Book that ﬂight. Before describing the algorithm it’s useful to consider two intuitions about di- rected graphs and their spanning trees. The ﬁrst intuition begins with the fact that every vertex in a spanning tree has exactly one incoming edge. It follows from this that every connected component of a spanning tree (i.e., every set of vertices that are linked to each other by paths over edges) will also have one incoming edge. The second intuition is that the absolute values of the edge scores are not critical to determining its maximum spanning tree. Instead, it is the relative weights of the edges entering each vertex that matters. If we were to subtract a constant amount from each edge entering a given vertex it would have no impact on the choice of 19.3 • G RAPH -BASED DEPENDENCY PARSING 427 the maximum spanning tree since every possible spanning tree would decrease by exactly the same amount. The ﬁrst step of the algorithm itself is quite straightforward. For each vertex in the graph, an incoming edge (representing a possible head assignment) with the highest score is chosen. If the resulting set of edges produces a spanning tree then we’re done. More formally, given the original fully-connected graph G = (V,E), a subgraph T = (V,F) is a spanning tree if it has no cycles and each vertex (other than the root) has exactly one edge entering it. If the greedy selection process produces such a tree then it is the best possible one. Unfortunately, this approach doesn’t always lead to a tree since the set of edges selected may contain cycles. Fortunately, in yet another case of multiple discovery, there is a straightforward way to eliminate cycles generated during the greedy se- lection phase. Chu and Liu (1965) and Edmonds (1967) independently developed an approach that begins with greedy selection and follows with an elegant recursive cleanup phase that eliminates cycles. The cleanup phase begins by adjusting all the weights in the graph by subtracting the score of the maximum edge entering each vertex from the score of all the edges entering that vertex. This is where the intuitions mentioned earlier come into play. We have scaled the values of the edges so that the weights of the edges in the cycle have no bearing on the weight of any of the possible spanning trees. Subtracting the value of the edge with maximum weight from each edge entering a vertex results in a weight of zero for all of the edges selected during the greedy selection phase, including all of the edges involved in the cycle. Having adjusted the weights, the algorithm creates a new graph by selecting a cycle and collapsing it into a single new node. Edges that enter or leave the cycle are altered so that they now enter or leave the newly collapsed node. Edges that do not touch the cycle are included and edges within the cycle are dropped. Now, if we knew the maximum spanning tree of this new graph, we would have what we need to eliminate the cycle. The edge of the maximum spanning tree di- rected towards the vertex representing the collapsed cycle tells us which edge to delete in order to eliminate the cycle. How do we ﬁnd the maximum spanning tree of this new graph? We recursively apply the algorithm to the new graph. This will either result in a spanning tree or a graph with a cycle. The recursions can continue as long as cycles are encountered. When each recursion completes we expand the collapsed vertex, restoring all the vertices and edges from the cycle with the excep- tion of the single edge to be deleted. Putting all this together, the maximum spanning tree algorithm consists of greedy edge selection, re-scoring of edge costs and a recursive cleanup phase when needed. The full algorithm is shown in Fig. 19.12. Fig. 19.13 steps through the algorithm with our Book that ﬂight example. The ﬁrst row of the ﬁgure illustrates greedy edge selection with the edges chosen shown in blue (corresponding to the set F in the algorithm). This results in a cycle between that and ﬂight. The scaled weights using the maximum value entering each node are shown in the graph to the right. Collapsing the cycle between that and ﬂight to a single node (labelled tf) and recursing with the newly scaled costs is shown in the second row. The greedy selec- tion step in this recursion yields a spanning tree that linksroot to book, as well as an edge that links book to the contracted node. Expanding the contracted node, we can see that this edge corresponds to the edge from book to ﬂight in the original graph. This in turn tells us which edge to drop to eliminate the cycle. 428 CHAPTER 19 • D EPENDENCY PARSING function MAXSPANNING TREE (G=(V ,E), root, score) returns spanning tree F←[] T’←[] score’←[] for each v ∈V do bestInEdge←argmaxe=(u,v)∈E score[e] F←F ∪bestInEdge for each e=(u,v) ∈E do score’[e]←score[e] −score[bestInEdge] if T=(V ,F)is a spanning tree then return it else C←a cycle in F G’←CONTRACT (G, C) T’←MAXSPANNING TREE (G’, root, score’) T←EXPAND (T’, C) return T function CONTRACT (G, C) returns contracted graph function EXPAND (T, C) returns expanded graph Figure 19.12 The Chu-Liu Edmonds algorithm for ﬁnding a maximum spanning tree in a weighted directed graph. On arbitrary directed graphs, this version of the CLE algorithm runs in O(mn) time, where m is the number of edges and n is the number of nodes. Since this par- ticular application of the algorithm begins by constructing a fully connected graph m = n2 yielding a running time of O(n3). Gabow et al. (1986) present a more efﬁ- cient implementation with a running time of O(m +nlogn). 19.3.2 A feature-based algorithm for assigning scores Recall that given a sentence,S, and a candidate tree,T , edge-factored parsing models make the simpliﬁcation that the score for the tree is the sum of the scores of the edges that comprise the tree: score(S,T ) = ∑ e∈T score(S,e) In a feature-based algorithm we compute the edge score as a weighted sum of fea- tures extracted from it: score(S,e) = N∑ i=1 wi fi(S,e) Or more succinctly. score(S,e) = w ·f Given this formulation, we need to identify relevant features and train the weights. The features (and feature combinations) used to train edge-factored models mir- ror those used in training transition-based parsers, such as 19.3 • G RAPH -BASED DEPENDENCY PARSING 429 root Book tf root Book that ﬂight 0 -3 -4 -7 -1 -6 -2 root Book 12 that 7 ﬂight 8 -4 -3 0 -2 -6 -1 -7 0 0 root Book 0 tf -1 0 -3 -4 -7 -1 -6 -2 root Book 12 that 7 ﬂight 8 12 4 4 5 6 8 7 5 7 Deleted from cycle Figure 19.13 Chu-Liu-Edmonds graph-based example for Book that ﬂight • Wordforms, lemmas, and parts of speech of the headword and its dependent. • Corresponding features from the contexts before, after and between the words. • Word embeddings. • The dependency relation itself. • The direction of the relation (to the right or left). • The distance from the head to the dependent. Given a set of features, our next problem is to learn a set of weights correspond- ing to each. Unlike many of the learning problems discussed in earlier chapters, here we are not training a model to associate training items with class labels, or parser actions. Instead, we seek to train a model that assigns higher scores to cor- rect trees than to incorrect ones. An effective framework for problems like this is to use inference-based learning combined with the perceptron learning rule. In thisinference-based learning framework, we parse a sentence (i.e, perform inference) from the training set using some initially random set of initial weights. If the resulting parse matches the cor- responding tree in the training data, we do nothing to the weights. Otherwise, we ﬁnd those features in the incorrect parse that are not present in the reference parse and we lower their weights by a small amount based on the learning rate. We do this incrementally for each sentence in our training data until the weights converge. 430 CHAPTER 19 • D EPENDENCY PARSING 19.3.3 A neural algorithm for assigning scores State-of-the-art graph-based multilingual parsers are based on neural networks. In- stead of extracting hand-designed features to represent each edge between words wi and wj, these parsers run the sentence through an encoder, and then pass the encoded representation of the two words wi and wj through a network that estimates a score for the edge i →j. book that ﬂight r1 score(h1 head, h3 dep) Biaﬃne b ENCODER U h1 head FFNhead FFNhead FFNdep FFNdep h1 dep FFNhead FFNdep h2 head h2 dep h3 head h3 dep W r2 r3 ∑ + Figure 19.14 Computing scores for a single edge (book →ﬂight) in the biafﬁne parser of Dozat and Manning (2017); Dozat et al. (2017). The parser uses distinct feedforward net- works to turn the encoder output for each word into a head and dependent representation for the word. The biafﬁne function turns the head embedding of the head and the dependent embedding of the dependent into a score for the dependency edge. Here we’ll sketch the biafﬁne algorithm of Dozat and Manning (2017) and Dozat et al. (2017) shown in Fig. 19.14, drawing on the work of Gr ¨unewald et al. (2021) who tested many versions of the algorithm via their STEPS system. The algorithm ﬁrst runs the sentence X = x1,...,xn through an encoder to produce a contextual embedding representation for each token R = r1,...,rn. The embedding for each token is now passed through two separate feedforward networks, one to produce a representation of this token as a head, and one to produce a representation of this token as a dependent: hhead i = FFNhead(ri) (19.13) hdep i = FFNdep (ri) (19.14) Now to assign a score to the directed edgei →j, (wi is the head and wj is the depen- dent), we feed the head representation of i, hhead i , and the dependent representation of j, hdep j , into a biafﬁne scoring function: Score(i →j) = Biaff(hhead i ,hdep j ) (19.15) Biaff(x,y) = x⊺Uy +W(x⊕y)+ b (19.16) 19.4 • E VALUATION 431 where U, W, and b are weights learned by the model. The idea of using a biafﬁne function is to allow the system to learn multiplicative interactions between the vec- tors x and y. If we pass Score(i →j) through a softmax, we end up with a probability distri- bution, for each token j, over potential heads i (all other tokens in the sentence): p(i →j) =softmax([Score(k →j);∀k ̸= j,1 ≤k ≤n]) (19.17) This probability can then be passed to the maximum spanning tree algorithm of Section 19.3.1 to ﬁnd the best tree. This p(i →j) classiﬁer is trained by optimizing the cross-entropy loss. Note that the algorithm as we’ve described it is unlabeled. To make this into a labeled algorithm, the Dozat and Manning (2017) algorithm actually trains two classiﬁers. The ﬁrst classiﬁer, the edge-scorer, the one we described above, assigns a probability p(i →j) to each word wi and wj. Then the Maximum Spanning Tree algorithm is run to get a single best dependency parse tree for the second. We then apply a second classiﬁer, the label-scorer, whose job is to ﬁnd the maximum prob- ability label for each edge in this parse. This second classiﬁer has the same form as (19.15-19.17), but instead of being trained to predict with binary softmax the probability of an edge existing between two words, it is trained with a softmax over dependency labels to predict the dependency label between the words. 19.4 Evaluation As with phrase structure-based parsing, the evaluation of dependency parsers pro- ceeds by measuring how well they work on a test set. An obvious metric would be exact match (EM)—how many sentences are parsed correctly. This metric is quite pessimistic, with most sentences being marked wrong. Such measures are not ﬁne- grained enough to guide the development process. Our metrics need to be sensitive enough to tell if actual improvements are being made. For these reasons, the most common method for evaluating dependency parsers are labeled and unlabeled attachment accuracy. Labeled attachment refers to the proper assignment of a word to its head along with the correct dependency relation. Unlabeled attachment simply looks at the correctness of the assigned head, ignor- ing the dependency relation. Given a system output and a corresponding reference parse, accuracy is simply the percentage of words in an input that are assigned the correct head with the correct relation. These metrics are usually referred to as the labeled attachment score (LAS) and unlabeled attachment score (UAS). Finally, we can make use of a label accuracy score (LS), the percentage of tokens with correct labels, ignoring where the relations are coming from. As an example, consider the reference parse and system parse for the following example shown in Fig. 19.15. (19.18) Book me the ﬂight through Houston. The system correctly ﬁnds 4 of the 6 dependency relations present in the reference parse and receives an LAS of 2/3. However, one of the 2 incorrect relations found by the system holds between book and ﬂight, which are in a head-dependent relation in the reference parse; the system therefore achieves a UAS of 5/6. Beyond attachment scores, we may also be interested in how well a system is performing on a particular kind of dependency relation, for example NSUBJ , across 432 CHAPTER 19 • D EPENDENCY PARSING Book me the ﬂight through Houston (a) Reference iobj obj det nmod case root Book me the ﬂight through Houston (b) System xcomp nsubj det nmod case root Figure 19.15 Reference and system parses for Book me the ﬂight through Houston , resulting in an LAS of 2/3 and an UAS of 5/6. a development corpus. Here we can make use of the notions of precision and recall introduced in Chapter 17, measuring the percentage of relations labeled NSUBJ by the system that were correct (precision), and the percentage of the NSUBJ relations present in the development set that were in fact discovered by the system (recall). We can employ a confusion matrix to keep track of how often each dependency type was confused for another. 19.5 Summary This chapter has introduced the concept of dependency grammars and dependency parsing. Here’s a summary of the main points that we covered: • In dependency-based approaches to syntax, the structure of a sentence is de- scribed in terms of a set of binary relations that hold between the words in a sentence. Larger notions of constituency are not directly encoded in depen- dency analyses. • The relations in a dependency structure capture the head-dependent relation- ship among the words in a sentence. • Dependency-based analysis provides information directly useful in further language processing tasks including information extraction, semantic parsing and question answering. • Transition-based parsing systems employ a greedy stack-based algorithm to create dependency structures. • Graph-based methods for creating dependency structures are based on the use of maximum spanning tree methods from graph theory. • Both transition-based and graph-based approaches are developed using super- vised machine learning techniques. • Treebanks provide the data needed to train these systems. Dependency tree- banks can be created directly by human annotators or via automatic transfor- mation from phrase-structure treebanks. • Evaluation of dependency parsers is based on labeled and unlabeled accuracy scores as measured against withheld development and test corpora. BIBLIOGRAPHICAL AND HISTORICAL NOTES 433 Bibliographical and Historical Notes The dependency-based approach to grammar is much older than the relatively recent phrase-structure or constituency grammars, which date only to the 20th century. De- pendency grammar dates back to the Indian grammarian P ¯an. ini sometime between the 7th and 4th centuries BCE, as well as the ancient Greek linguistic traditions. Contemporary theories of dependency grammar all draw heavily on the 20th cen- tury work of Tesni`ere (1959). Automatic parsing using dependency grammars was ﬁrst introduced into compu- tational linguistics by early work on machine translation at the RAND Corporation led by David Hays. This work on dependency parsing closely paralleled work on constituent parsing and made explicit use of grammars to guide the parsing process. After this early period, computational work on dependency parsing remained inter- mittent over the following decades. Notable implementations of dependency parsers for English during this period include Link Grammar (Sleator and Temperley, 1993), Constraint Grammar (Karlsson et al., 1995), and MINIPAR (Lin, 2003). Dependency parsing saw a major resurgence in the late 1990’s with the appear- ance of large dependency-based treebanks and the associated advent of data driven approaches described in this chapter. Eisner (1996) developed an efﬁcient dynamic programming approach to dependency parsing based on bilexical grammars derived from the Penn Treebank. Covington (2001) introduced the deterministic word by word approach underlying current transition-based approaches. Yamada and Mat- sumoto (2003) and Kudo and Matsumoto (2002) introduced both the shift-reduce paradigm and the use of supervised machine learning in the form of support vector machines to dependency parsing. Transition-based parsing is based on the shift-reduce parsing algorithm orig- inally developed for analyzing programming languages (Aho and Ullman, 1972). Shift-reduce parsing also makes use of a context-free grammar. Input tokens are successively shifted onto the stack and the top two elements of the stack are matched against the right-hand side of the rules in the grammar; when a match is found the matched elements are replaced on the stack (reduced) by the non-terminal from the left-hand side of the rule being matched. In transition-based dependency parsing we skip the grammar, and alter the reduce operation to add a dependency relation between a word and its head. Nivre (2003) deﬁned the modern, deterministic, transition-based approach to dependency parsing. Subsequent work by Nivre and his colleagues formalized and analyzed the performance of numerous transition systems, training methods, and methods for dealing with non-projective language (Nivre and Scholz 2004, Nivre 2006, Nivre and Nilsson 2005, Nivre et al. 2007b, Nivre 2007). The neural ap- proach was pioneered by Chen and Manning (2014) and extended by Kiperwasser and Goldberg (2016); Kulmizev et al. (2019). The graph-based maximum spanning tree approach to dependency parsing was introduced by McDonald et al. 2005a, McDonald et al. 2005b. The neural classiﬁer was introduced by (Kiperwasser and Goldberg, 2016). The long-running Prague Dependency Treebank project (Hajiˇc, 1998) is the most signiﬁcant effort to directly annotate a corpus with multiple layers of morphological, syntactic and semantic information. PDT 3.0 contains over 1.5 M tokens (Bej ˇcek et al., 2013). Universal Dependencies (UD) (de Marneffe et al., 2021) is an open community 434 CHAPTER 19 • D EPENDENCY PARSING project to create a framework for dependency treebank annotation, with nearly 200 treebanks in over 100 languages. The UD annotation scheme evolved out of several distinct efforts including Stanford dependencies (de Marneffe et al. 2006, de Marn- effe and Manning 2008, de Marneffe et al. 2014), Google’s universal part-of-speech tags (Petrov et al., 2012), and the Interset interlingua for morphosyntactic tagsets (Zeman, 2008). The Conference on Natural Language Learning (CoNLL) has conducted an in- ﬂuential series of shared tasks related to dependency parsing over the years (Buch- holz and Marsi 2006, Nivre et al. 2007a, Surdeanu et al. 2008, Haji ˇc et al. 2009). More recent evaluations have focused on parser robustness with respect to morpho- logically rich languages (Seddah et al., 2013), and non-canonical language forms such as social media, texts, and spoken language (Petrov and McDonald, 2012). Choi et al. (2015) presents a performance analysis of 10 dependency parsers across a range of metrics, as well as DEPEND ABLE , a robust parser evaluation tool. Exercises CHAPTER 20 Information Extraction: Relations, Events, and Time Time will explain. Jane Austen, Persuasion Imagine that you are an analyst with an investment ﬁrm that tracks airline stocks. You’re given the task of determining the relationship (if any) between airline an- nouncements of fare increases and the behavior of their stocks the next day. His- torical data about stock prices is easy to come by, but what about the airline an- nouncements? You will need to know at least the name of the airline, the nature of the proposed fare hike, the dates of the announcement, and possibly the response of other airlines. Fortunately, these can be all found in news articles like this one: Citing high fuel prices, United Airlines said Friday it has increased fares by $6 per round trip on ﬂights to some cities also served by lower- cost carriers. American Airlines, a unit of AMR Corp., immediately matched the move, spokesman Tim Wagner said. United, a unit of UAL Corp., said the increase took effect Thursday and applies to most routes where it competes against discount carriers, such as Chicago to Dallas and Denver to San Francisco. This chapter presents techniques for extracting limited kinds of semantic con- tent from text. This process of information extraction (IE) turns the unstructuredinformation extraction information embedded in texts into structured data, for example for populating a relational database to enable further processing. We begin with the task of relation extraction: ﬁnding and classifying semanticrelation extraction relations among entities mentioned in a text, like child-of (X is the child-of Y), or part-whole or geospatial relations. Relation extraction has close links to populat- ing a relational database, and knowledge graphs, datasets of structured relationalknowledge graphs knowledge, are a useful way for search engines to present information to users. Next, we discuss event extraction, the task of ﬁnding events in which these en-event extraction tities participate, like, in our sample text, the fare increases byUnited and American and the reporting events said and cite. Events are also situated in time, occurring at a particular date or time, and events can be related temporally, happening before or after or simultaneously with each other. We’ll need to recognize temporal expres- sions like Friday, Thursday or two days from now and times such as 3:30 P .M., and normalize them onto speciﬁc calendar dates or times. We’ll need to link Friday to the time of United’s announcement, Thursday to the previous day’s fare increase, and we’ll need to produce a timeline in which United’s announcement follows the fare increase and American’s announcement follows both of those events. The related task of template ﬁlling is to ﬁnd recurring stereotypical events ortemplate ﬁlling situations in documents and ﬁll in the template slots. These slot-ﬁllers may consist of text segments extracted directly from the text, or concepts like times, amounts, or ontology entities that have been inferred through additional processing. Our airline 436 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME ARTIFACT GENERAL AFFILIATION ORG AFFILIATION PART- WHOLE PERSON- SOCIAL PHYSICAL Located Near Business Family Lasting Personal Citizen- Resident- Ethnicity- Religion Org-Location- Origin Founder Employment Membership Ownership Student-Alum Investor User-Owner-Inventor- Manufacturer Geographical Subsidiary Sports-Affiliation Figure 20.1 The 17 relations used in the ACE relation extraction task. text presents such a stereotypical situation since airlines often raise fares and then wait to see if competitors follow along. Here we can identify United as a lead air- line that initially raised its fares, $6 as the amount, Thursday as the increase date, and American as an airline that followed along, leading to a ﬁlled template like the following: FARE -RAISE ATTEMPT :   LEAD AIRLINE : U NITED AIRLINES AMOUNT : $6 EFFECTIVE DATE: 2006-10-26 FOLLOWER : A MERICAN AIRLINES   20.1 Relation Extraction Let’s assume that we have detected the named entities in our sample text (perhaps using the techniques of Chapter 17), and would like to discern the relationships that exist among the detected entities: Citing high fuel prices, [ ORG United Airlines] said [TIME Friday] it has increased fares by [ MONEY $6] per round trip on ﬂights to some cities also served by lower-cost carriers. [ ORG American Airlines], a unit of [ORG AMR Corp.], immediately matched the move, spokesman [PER Tim Wagner] said. [ ORG United], a unit of [ORG UAL Corp.], said the increase took effect [ TIME Thursday] and applies to most routes where it competes against discount carriers, such as [LOC Chicago] to [LOC Dallas] and [LOC Denver] to [LOC San Francisco]. The text tells us, for example, that Tim Wagner is a spokesman for American Airlines, that United is a unit of UAL Corp., and that American is a unit of AMR. These binary relations are instances of more generic relations such as part-of or employs that are fairly frequent in news-style texts. Figure 20.1 lists the 17 relations used in the ACE relation extraction evaluations and Fig. 20.2 shows some sample relations. We might also extract more domain-speciﬁc relations such as the notion of an airline route. For example from this text we can conclude that United has routes to Chicago, Dallas, Denver, and San Francisco. 20.1 • R ELATION EXTRACTION 437 Relations Types Examples Physical-Located PER-GPE He was in Tennessee Part-Whole-Subsidiary ORG-ORG XYZ, the parent company of ABC Person-Social-Family PER-PER Yoko’s husbandJohn Org-AFF-Founder PER-ORG Steve Jobs, co-founder of Apple... Figure 20.2 Semantic relations with examples and the named entity types they involve. Sets of relations have been deﬁned for many other domains as well. For example UMLS, the Uniﬁed Medical Language System from the US National Library of Medicine has a network that deﬁnes 134 broad subject categories, entity types, and 54 relations between the entities, such as the following: Entity Relation Entity Injury disrupts Physiological Function Bodily Location location-of Biologic Function Anatomical Structure part-of Organism Pharmacologic Substance causes Pathological Function Pharmacologic Substance treats Pathologic Function Given a medical sentence like this one: (20.1) Doppler echocardiography can be used to diagnose left anterior descending artery stenosis in patients with type 2 diabetes We could thus extract the UMLS relation: Echocardiography, DopplerDiagnoses Acquired stenosis Wikipedia also offers a large supply of relations, drawn from infoboxes, struc-infoboxes tured tables associated with certain Wikipedia articles. For example, the Wikipedia infobox for Stanford includes structured facts like state = "California" or president = "Marc Tessier-Lavigne". These facts can be turned into rela- tions like president-of or located-in. or into relations in a metalanguage called RDFRDF (Resource Description Framework). An RDF triple is a tuple of entity-relation-RDF triple entity, called a subject-predicate-object expression. Here’s a sample RDF triple: subject predicate object Golden Gate Park location San Francisco For example the crowdsourced DBpedia (Bizer et al., 2009) is an ontology de- rived from Wikipedia containing over 2 billion RDF triples. Another dataset from Wikipedia infoboxes,Freebase (Bollacker et al., 2008), now part of Wikidata (Vrandeˇci´cFreebase and Kr¨otzsch, 2014), has relations between people and their nationality, or locations, and other locations they are contained in. WordNet or other ontologies offer useful ontological relations that express hier- archical relations between words or concepts. For example WordNet has the is-a oris-a hypernym relation between classes,hypernym Giraffe is-a ruminant is-a ungulate is-a mammal is-a vertebrate ... WordNet also has Instance-of relation between individuals and classes, so that for example San Francisco is in the Instance-of relation with city. Extracting these relations is an important step in extending or building ontologies. Finally, there are large datasets that contain sentences hand-labeled with their relations, designed for training and testing relation extractors. The TACRED dataset (Zhang et al., 2017) contains 106,264 examples of relation triples about particular people or organizations, labeled in sentences from news and web text drawn from the 438 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME annual TAC Knowledge Base Population (TAC KBP) challenges. TACRED contains 41 relation types (like per:city of birth, org:subsidiaries, org:member of, per:spouse), plus a no relation tag; examples are shown in Fig. 20.3. About 80% of all examples are annotated as no relation; having sufﬁcient negative data is important for training supervised classiﬁers. Example Entity Types & Label Carey will succeed Cathleen P. Black, who held the position for 15 years and will take on a new role as chairwoman of Hearst Maga- zines, the company said. PERSON /TITLE Relation: per:title Irene Morgan Kirkaldy, who was born and reared in Baltimore, lived on Long Island and ran a child-care center in Queens with her second husband, Stanley Kirkaldy. PERSON /CITY Relation: per:city of birth Baldwin declined further comment, and said JetBlue chief executive Dave Barger was unavailable. Types: PERSON /TITLE Relation: no relation Figure 20.3 Example sentences and labels from the TACRED dataset (Zhang et al., 2017). A standard dataset was also produced for the SemEval 2010 Task 8, detecting relations between nominals (Hendrickx et al., 2009). The dataset has 10,717 exam- ples, each with a pair of nominals (untyped) hand-labeled with one of 9 directed relations like product-producer ( a factory manufactures suits) or component-whole (my apartment has a large kitchen). 20.2 Relation Extraction Algorithms There are ﬁve main classes of algorithms for relation extraction: handwritten pat- terns, supervised machine learning, semi-supervised (via bootstrapping or dis- tant supervision), and unsupervised. We’ll introduce each of these in the next sections. 20.2.1 Using Patterns to Extract Relations The earliest and still common algorithm for relation extraction is lexico-syntactic patterns, ﬁrst developed by Hearst (1992a), and therefore often called Hearst pat- terns. Consider the following sentence:Hearst patterns Agar is a substance prepared from a mixture of red algae, such as Ge- lidium, for laboratory or industrial use. Hearst points out that most human readers will not know what Gelidium is, but that they can readily infer that it is a kind of (a hyponym of) red algae, whatever that is. She suggests that the following lexico-syntactic pattern NP0 such as NP1{,NP2 ..., (and\|or)NPi},i ≥1 (20.2) implies the following semantics ∀NPi,i ≥1,hyponym(NPi,NP0) (20.3) allowing us to infer hyponym(Gelidium,red algae) (20.4) 20.2 • R ELATION EXTRACTION ALGORITHMS 439 NP {, NP}* {,}(and\|or) other NPH temples, treasuries, and other important civic buildings NPH such as {NP,}* {(or\|and)}NP red algae such as Gelidium such NPH as {NP,}* {(or\|and)}NP such authors as Herrick, Goldsmith, and Shakespeare NPH {,}including {NP,}* {(or\|and)}NP common-law countries, including Canada and England NPH {,}especially {NP}* {(or\|and)}NP European countries, especially France, England, and Spain Figure 20.4 Hand-built lexico-syntactic patterns for ﬁnding hypernyms, using{} to mark optionality (Hearst 1992a, Hearst 1998). Figure 20.4 shows ﬁve patterns Hearst (1992a, 1998) suggested for inferring the hyponym relation; we’ve shown NPH as the parent/hyponym. Modern versions of the pattern-based approach extend it by adding named entity constraints. For example if our goal is to answer questions about “Who holds what ofﬁce in which organization?”, we can use patterns like the following: PER, POSITION of ORG: George Marshall, Secretary of State of the United States PER (named\|appointed\|chose\|etc.) PER Prep? POSITION Truman appointed Marshall Secretary of State PER [be]? (named\|appointed\|etc.) Prep? ORG POSITION George Marshall was named US Secretary of State Hand-built patterns have the advantage of high-precision and they can be tailored to speciﬁc domains. On the other hand, they are often low-recall, and it’s a lot of work to create them for all possible patterns. 20.2.2 Relation Extraction via Supervised Learning Supervised machine learning approaches to relation extraction follow a scheme that should be familiar by now. A ﬁxed set of relations and entities is chosen, a training corpus is hand-annotated with the relations and entities, and the annotated texts are then used to train classiﬁers to annotate an unseen test set. The most straightforward approach, illustrated in Fig. 20.5 is: (1) Find pairs of named entities (usually in the same sentence). (2): Apply a relation-classiﬁcation on each pair. The classiﬁer can use any supervised technique (logistic regression, RNN, Transformer, random forest, etc.). An optional intermediate ﬁltering classiﬁer can be used to speed up the process- ing by making a binary decision on whether a given pair of named entities are related (by any relation). It’s trained on positive examples extracted directly from all rela- tions in the annotated corpus, and negative examples generated from within-sentence entity pairs that are not annotated with a relation. Feature-based supervised relation classiﬁers. Let’s consider sample features for a feature-based classiﬁer (like logistic regression or random forests), classifying the relationship between American Airlines (Mention 1, or M1) and Tim Wagner(Men- tion 2, M2) from this sentence: (20.5) American Airlines, a unit of AMR, immediately matched the move, spokesman Tim Wagner said These include word features (as embeddings, or 1-hot, stemmed or not): • The headwords of M1 and M2 and their concatenation Airlines Wagner Airlines-Wagner 440 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME function FIND RELATIONS (words) returns relations relations←nil entities←FIND ENTITIES (words) forall entity pairs ⟨e1, e2⟩in entities do if RELATED ?(e1, e2) relations←relations+CLASSIFY RELATION (e1, e2) Figure 20.5 Finding and classifying the relations among entities in a text. • Bag-of-words and bigrams in M1 and M2 American, Airlines, Tim, Wagner, American Airlines, Tim Wagner • Words or bigrams in particular positions M2: -1 spokesman M2: +1 said • Bag of words or bigrams between M1 and M2: a, AMR, of, immediately, matched, move, spokesman, the, unit Named entity features: • Named-entity types and their concatenation (M1: ORG, M2: PER, M1M2: ORG-PER) • Entity Level of M1 and M2 (from the set NAME, NOMINAL, PRONOUN) M1: NAME [it or he would be PRONOUN] M2: NAME [the company would be NOMINAL] • Number of entities between the arguments (in this case 1, for AMR) Syntactic structure is a useful signal, often represented as the dependency or constituency syntactic path traversed through the tree between the entities. • Constituent paths between M1 and M2 NP ↑NP ↑S ↑S ↓NP • Dependency-tree paths Airlines ←sub j matched ←comp said →sub j Wagner Neural supervised relation classiﬁers Neural models for relation extraction sim- ilarly treat the task as supervised classiﬁcation. Let’s consider a typical system ap- plied to the TACRED relation extraction dataset and task (Zhang et al., 2017). In TACRED we are given a sentence and two spans within it: a subject, which is a person or organization, and an object, which is any other entity. The task is to assign a relation from the 42 TAC relations, or no relation. A typical Transformer-encoder algorithm, shown in Fig. 20.6, simply takes a pretrained encoder like BERT and adds a linear layer on top of the sentence repre- sentation (for example the BERT [CLS] token), a linear layer that is ﬁnetuned as a 1-of-N classiﬁer to assign one of the 43 labels. The input to the BERT encoder is partially de-lexiﬁed; the subject and object entities are replaced in the input by their NER tags. This helps keep the system from overﬁtting to the individual lexical items (Zhang et al., 2017). When using BERT-type Transformers for relation extraction, it helps to use versions of BERT like RoBERTa (Liu et al., 2019) or spanBERT (Joshi et al., 2020) that don’t have two sequences separated by a [SEP] token, but instead form the input from a single long sequence of sentences. In general, if the test set is similar enough to the training set, and if there is enough hand-labeled data, supervised relation extraction systems can get high ac- 20.2 • R ELATION EXTRACTION ALGORITHMS 441 ENCODER [CLS] [SUBJ_PERSON] was born in [OBJ_LOC] , Michigan Linear Classiﬁer p(relation\|SUBJ,OBJ) Figure 20.6 Relation extraction as a linear layer on top of an encoder (in this case BERT), with the subject and object entities replaced in the input by their NER tags (Zhang et al. 2017, Joshi et al. 2020). curacies. But labeling a large training set is extremely expensive and supervised models are brittle: they don’t generalize well to different text genres. For this rea- son, much research in relation extraction has focused on the semi-supervised and unsupervised approaches we turn to next. 20.2.3 Semisupervised Relation Extraction via Bootstrapping Supervised machine learning assumes that we have lots of labeled data. Unfortu- nately, this is expensive. But suppose we just have a few high-precision seed pat- terns, like those in Section 20.2.1, or perhaps a few seed tuples. That’s enoughseed patterns seed tuples to bootstrap a classiﬁer! Bootstrapping proceeds by taking the entities in the seed bootstrapping pair, and then ﬁnding sentences (on the web, or whatever dataset we are using) that contain both entities. From all such sentences, we extract and generalize the context around the entities to learn new patterns. Fig. 20.7 sketches a basic algorithm. function BOOTSTRAP (Relation R) returns new relation tuples tuples←Gather a set of seed tuples that have relation R iterate sentences←ﬁnd sentences that contain entities in tuples patterns←generalize the context between and around entities in sentences newpairs←use patterns to identify more tuples newpairs←newpairs with high conﬁdence tuples←tuples + newpairs return tuples Figure 20.7 Bootstrapping from seed entity pairs to learn relations. Suppose, for example, that we need to create a list of airline/hub pairs, and we know only that Ryanair has a hub at Charleroi. We can use this seed fact to discover new patterns by ﬁnding other mentions of this relation in our corpus. We search for the terms Ryanair, Charleroi and hub in some proximity. Perhaps we ﬁnd the following set of sentences: (20.6) Budget airline Ryanair, which uses Charleroi as a hub, scrapped all weekend ﬂights out of the airport. (20.7) All ﬂights in and out of Ryanair’s hub at Charleroi airport were grounded on Friday... (20.8) A spokesman at Charleroi, a main hub for Ryanair, estimated that 8000 passengers had already been affected. 442 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME From these results, we can use the context of words between the entity mentions, the words before mention one, the word after mention two, and the named entity types of the two mentions, and perhaps other features, to extract general patterns such as the following: / [ORG], which uses [LOC] as a hub / / [ORG]’s hub at [LOC] / / [LOC], a main hub for [ORG] / These new patterns can then be used to search for additional tuples. Bootstrapping systems also assign conﬁdence values to new tuples to avoid se-conﬁdence values mantic drift. In semantic drift, an erroneous pattern leads to the introduction ofsemantic drift erroneous tuples, which, in turn, lead to the creation of problematic patterns and the meaning of the extracted relations ‘drifts’. Consider the following example: (20.9) Sydney has a ferry hub at Circular Quay. If accepted as a positive example, this expression could lead to the incorrect in- troduction of the tuple ⟨Sydney,CircularQuay⟩. Patterns based on this tuple could propagate further errors into the database. Conﬁdence values for patterns are based on balancing two factors: the pattern’s performance with respect to the current set of tuples and the pattern’s productivity in terms of the number of matches it produces in the document collection. More formally, given a document collection D, a current set of tuples T , and a proposed pattern p, we need to track two factors: • hits(p): the set of tuples in T that p matches while looking in D • ﬁnds(p): The total set of tuples that p ﬁnds in D The following equation balances these considerations (Riloff and Jones, 1999). Conf RlogF (p) = \|hits(p)\| \|ﬁnds(p)\|log(\|ﬁnds(p)\|) (20.10) This metric is generally normalized to produce a probability. We can assess the conﬁdence in a proposed new tuple by combining the evidence supporting it from all the patterns P′that match that tuple in D (Agichtein and Gra- vano, 2000). One way to combine such evidence is the noisy-or technique. Assumenoisy-or that a given tuple is supported by a subset of the patterns in P, each with its own conﬁdence assessed as above. In the noisy-or model, we make two basic assump- tions. First, that for a proposed tuple to be false, all of its supporting patterns must have been in error, and second, that the sources of their individual failures are all independent. If we loosely treat our conﬁdence measures as probabilities, then the probability of any individual pattern p failing is 1 −Conf (p); the probability of all of the supporting patterns for a tuple being wrong is the product of their individual failure probabilities, leaving us with the following equation for our conﬁdence in a new tuple. Conf (t) =1 − ∏ p∈P′ (1 −Conf (p)) (20.11) Setting conservative conﬁdence thresholds for the acceptance of new patterns and tuples during the bootstrapping process helps prevent the system from drifting away from the targeted relation. 20.2 • R ELATION EXTRACTION ALGORITHMS 443 20.2.4 Distant Supervision for Relation Extraction Although hand-labeling text with relation labels is expensive to produce, there are ways to ﬁnd indirect sources of training data. The distant supervision methoddistant supervision (Mintz et al., 2009) combines the advantages of bootstrapping with supervised learn- ing. Instead of just a handful of seeds, distant supervision uses a large database to acquire a huge number of seed examples, creates lots of noisy pattern features from all these examples and then combines them in a supervised classiﬁer. For example suppose we are trying to learn the place-of-birth relationship be- tween people and their birth cities. In the seed-based approach, we might have only 5 examples to start with. But Wikipedia-based databases like DBPedia or Freebase have tens of thousands of examples of many relations; including over 100,000 ex- amples of place-of-birth, (<Edwin Hubble, Marshfield>, <Albert Einstein, Ulm>, etc.,). The next step is to run named entity taggers on large amounts of text— Mintz et al. (2009) used 800,000 articles from Wikipedia—and extract all sentences that have two named entities that match the tuple, like the following: ...Hubble was born in Marshﬁeld... ...Einstein, born (1879), Ulm... ...Hubble’s birthplace in Marshﬁeld... Training instances can now be extracted from this data, one training instance for each identical tuple <relation, entity1, entity2>. Thus there will be one training instance for each of: <born-in, Edwin Hubble, Marshfield> <born-in, Albert Einstein, Ulm> <born-year, Albert Einstein, 1879> and so on. We can then apply feature-based or neural classiﬁcation. For feature-based classiﬁcation, we can use standard supervised relation extraction features like the named entity labels of the two mentions, the words and dependency paths in be- tween the mentions, and neighboring words. Each tuple will have features col- lected from many training instances; the feature vector for a single training instance like (<born-in,Albert Einstein, Ulm>will have lexical and syntactic features from many different sentences that mention Einstein and Ulm. Because distant supervision has very large training sets, it is also able to use very rich features that are conjunctions of these individual features. So we will extract thousands of patterns that conjoin the entity types with the intervening words or dependency paths like these: PER was born in LOC PER, born (XXXX), LOC PER’s birthplace in LOC To return to our running example, for this sentence: (20.12) American Airlines, a unit of AMR, immediately matched the move, spokesman Tim Wagner said we would learn rich conjunction features like this one: M1 = ORG & M2 = PER & nextword=“said”& path=NP ↑NP ↑S ↑S ↓NP The result is a supervised classiﬁer that has a huge rich set of features to use in detecting relations. Since not every test sentence will have one of the training 444 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME relations, the classiﬁer will also need to be able to label an example as no-relation. This label is trained by randomly selecting entity pairs that do not appear in any Freebase relation, extracting features for them, and building a feature vector for each such tuple. The ﬁnal algorithm is sketched in Fig. 20.8. function DISTANT SUPERVISION (Database D, Text T) returns relation classiﬁer C foreach relation R foreach tuple (e1,e2) of entities with relation R in D sentences←Sentences in T that contain e1 and e2 f←Frequent features in sentences observations←observations + new training tuple (e1, e2, f, R) C←Train supervised classiﬁer on observations return C Figure 20.8 The distant supervision algorithm for relation extraction. A neural classiﬁer would skip the feature set f . Distant supervision shares advantages with each of the methods we’ve exam- ined. Like supervised classiﬁcation, distant supervision uses a classiﬁer with lots of features, and supervised by detailed hand-created knowledge. Like pattern-based classiﬁers, it can make use of high-precision evidence for the relation between en- tities. Indeed, distance supervision systems learn patterns just like the hand-built patterns of early relation extractors. For example the is-a or hypernym extraction system of Snow et al. (2005) used hypernym/hyponym NP pairs from WordNet as distant supervision, and then learned new patterns from large amounts of text. Their system induced exactly the original 5 template patterns of Hearst (1992a), but also 70,000 additional patterns including these four: NPH like NP Many hormones like leptin... NPH called NP ...using a markup language called XHTML NP is a NPH Ruby is a programming language... NP, a NPH IBM, a company with a long... This ability to use a large number of features simultaneously means that, un- like the iterative expansion of patterns in seed-based systems, there’s no semantic drift. Like unsupervised classiﬁcation, it doesn’t use a labeled training corpus of texts, so it isn’t sensitive to genre issues in the training corpus, and relies on very large amounts of unlabeled data. Distant supervision also has the advantage that it can create training tuples to be used with neural classiﬁers, where features are not required. The main problem with distant supervision is that it tends to produce low-precision results, and so current research focuses on ways to improve precision. Furthermore, distant supervision can only help in extracting relations for which a large enough database already exists. To extract new relations without datasets, or relations for new domains, purely unsupervised methods must be used. 20.2.5 Unsupervised Relation Extraction The goal of unsupervised relation extraction is to extract relations from the web when we have no labeled training data, and not even any list of relations. This task is often called open information extraction or Open IE. In Open IE, the relations open information extraction 20.2 • R ELATION EXTRACTION ALGORITHMS 445 are simply strings of words (usually beginning with a verb). For example, the ReVerb system (Fader et al., 2011) extracts a relation from a sentence s in 4 steps: 1. Run a part-of-speech tagger and entity chunker over s 2. For each verb in s, ﬁnd the longest sequence of words w that start with a verb and satisfy syntactic and lexical constraints, merging adjacent matches. 3. For each phrase w, ﬁnd the nearest noun phrase x to the left which is not a relative pronoun, wh-word or existential “there”. Find the nearest noun phrase y to the right. 4. Assign conﬁdence c to the relation r = (x,w,y) using a conﬁdence classiﬁer and return it. A relation is only accepted if it meets syntactic and lexical constraints. The syntactic constraints ensure that it is a verb-initial sequence that might also include nouns (relations that begin with light verbs like make, have, or do often express the core of the relation with a noun, like have a hub in): V \|VP \|VWP V = verb particle? adv? W = (noun \|adj \|adv \|pron \|det ) P = (prep \|particle \|inﬁnitive “to”) The lexical constraints are based on a dictionary D that is used to prune very rare, long relation strings. The intuition is to eliminate candidate relations that don’t oc- cur with sufﬁcient number of distinct argument types and so are likely to be bad examples. The system ﬁrst runs the above relation extraction algorithm ofﬂine on 500 million web sentences and extracts a list of all the relations that occur after nor- malizing them (removing inﬂection, auxiliary verbs, adjectives, and adverbs). Each relation r is added to the dictionary if it occurs with at least 20 different arguments. Fader et al. (2011) used a dictionary of 1.7 million normalized relations. Finally, a conﬁdence value is computed for each relation using a logistic re- gression classiﬁer. The classiﬁer is trained by taking 1000 random web sentences, running the extractor, and hand labeling each extracted relation as correct or incor- rect. A conﬁdence classiﬁer is then trained on this hand-labeled data, using features of the relation and the surrounding words. Fig. 20.9 shows some sample features used in the classiﬁcation. (x,r,y) covers all words in s the last preposition in r is for the last preposition in r is on len(s) ≤10 there is a coordinating conjunction to the left of r in s r matches a lone V in the syntactic constraints there is preposition to the left of x in s there is an NP to the right of y in s Figure 20.9 Features for the classiﬁer that assigns conﬁdence to relations extracted by the Open Information Extraction system REVERB (Fader et al., 2011). For example the following sentence: (20.13) United has a hub in Chicago, which is the headquarters of United Continental Holdings. 446 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME has the relation phrases has a hub in and is the headquarters of (it also has has and is, but longer phrases are preferred). Step 3 ﬁnds United to the left and Chicago to the right of has a hub in , and skips over which to ﬁnd Chicago to the left of is the headquarters of. The ﬁnal output is: r1: <United, has a hub in, Chicago> r2: <Chicago, is the headquarters of, United Continental Holdings> The great advantage of unsupervised relation extraction is its ability to handle a huge number of relations without having to specify them in advance. The dis- advantage is the need to map all the strings into some canonical form for adding to databases or knowledge graphs. Current methods focus heavily on relations ex- pressed with verbs, and so will miss many relations that are expressed nominally. 20.2.6 Evaluation of Relation Extraction Supervised relation extraction systems are evaluated by using test sets with human- annotated, gold-standard relations and computing precision, recall, and F-measure. Labeled precision and recall require the system to classify the relation correctly, whereas unlabeled methods simply measure a system’s ability to detect entities that are related. Semi-supervised and unsupervised methods are much more difﬁcult to evalu- ate, since they extract totally new relations from the web or a large text. Because these methods use very large amounts of text, it is generally not possible to run them solely on a small labeled test set, and as a result it’s not possible to pre-annotate a gold set of correct instances of relations. For these methods it’s possible to approximate (only) precision by drawing a random sample of relations from the output, and having a human check the accuracy of each of these relations. Usually this approach focuses on thetuples to be extracted from a body of text rather than on the relation mentions; systems need not detect every mention of a relation to be scored correctly. Instead, the evaluation is based on the set of tuples occupying the database when the system is ﬁnished. That is, we want to know if the system can discover that Ryanair has a hub at Charleroi; we don’t really care how many times it discovers it. The estimated precision ˆP is then ˆP = # of correctly extracted relation tuples in the sample total # of extracted relation tuples in the sample. (20.14) Another approach that gives us a little bit of information about recall is to com- pute precision at different levels of recall. Assuming that our system is able to rank the relations it produces (by probability, or conﬁdence) we can separately com- pute precision for the top 1000 new relations, the top 10,000 new relations, the top 100,000, and so on. In each case we take a random sample of that set. This will show us how the precision curve behaves as we extract more and more tuples. But there is no way to directly evaluate recall. 20.3 Extracting Events The task of event extraction is to identify mentions of events in texts. For theevent extraction purposes of this task, an event mention is any expression denoting an event or state that can be assigned to a particular point, or interval, in time. The following markup of the sample text on page 435 shows all the events in this text. 20.4 • R EPRESENTING TIME 447 [EVENT Citing] high fuel prices, United Airlines [ EVENT said] Fri- day it has [ EVENT increased] fares by $6 per round trip on ﬂights to some cities also served by lower-cost carriers. American Airlines, a unit of AMR Corp., immediately [ EVENT matched] [ EVENT the move], spokesman Tim Wagner [EVENT said]. United, a unit of UAL Corp., [EVENT said] [EVENT the increase] took effect Thursday and [EVENT applies] to most routes where it [ EVENT competes] against discount carriers, such as Chicago to Dallas and Denver to San Francisco. In English, most event mentions correspond to verbs, and most verbs introduce events. However, as we can see from our example, this is not always the case. Events can be introduced by noun phrases, as in the move and the increase, and some verbs fail to introduce events, as in the phrasal verb took effect, which refers to when the event began rather than to the event itself. Similarly,light verbs such as make, take,light verbs and have often fail to denote events. A light verb is a verb that has very little meaning itself, and the associated event is instead expressed by its direct object noun. In light verb examples like took a ﬂight, it’s the wordﬂight that deﬁnes the event; these light verbs just provide a syntactic structure for the noun’s arguments. Various versions of the event extraction task exist, depending on the goal. For example in the TempEval shared tasks (Verhagen et al. 2009) the goal is to extract events and aspects like their aspectual and temporal properties. Events are to be classiﬁed as actions, states, reporting events (say, report, tell, explain), perceptionreporting events events, and so on. The aspect, tense, and modality of each event also needs to be extracted. Thus for example the various said events in the sample text would be annotated as (class=REPORTING, tense=PAST, aspect=PERFECTIVE). Event extraction is generally modeled via supervised learning, detecting events via IOB sequence models and assigning event classes and attributes with multi-class classiﬁers. The input can be neural models starting from encoders; or classic feature- based models using features like those in Fig. 20.10. Feature Explanation Character afﬁxes Character-level preﬁxes and sufﬁxes of target word Nominalization sufﬁx Character-level sufﬁxes for nominalizations (e.g., -tion) Part of speech Part of speech of the target word Light verb Binary feature indicating that the target is governed by a light verb Subject syntactic category Syntactic category of the subject of the sentence Morphological stem Stemmed version of the target word Verb root Root form of the verb basis for a nominalization WordNet hypernyms Hypernym set for the target Figure 20.10 Features commonly used in classic feature-based approaches to event detection. 20.4 Representing Time Let’s begin by introducing the basics of temporal logic and how human languagestemporal logic convey temporal information. The most straightforward theory of time holds that it ﬂows inexorably forward and that events are associated with either points or inter- vals in time, as on a timeline. We can order distinct events by situating them on the timeline; one event precedes another if the ﬂow of time leads from the ﬁrst event 448 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME to the second. Accompanying these notions in most theories is the idea of the cur- rent moment in time. Combining this notion with the idea of a temporal ordering relationship yields the familiar notions of past, present, and future. Various kinds of temporal representation systems can be used to talk about tem- poral ordering relationship. One of the most commonly used in computational mod- eling is the interval algebra of Allen (1984). Allen models all events and timeinterval algebra expressions as intervals there is no representation for points (although intervals can be very short). In order to deal with intervals without points, he identiﬁes 13 primi- tive relations that can hold between these temporal intervals. Fig. 20.11 shows these 13 Allen relations.Allen relations B A B A B A A A B B A B Time A before B B after A A overlaps B B overlaps' A A meets B B meets' A A equals B (B equals A) A starts B B starts' A A finishes B B finishes' A B A during B B during' A A Figure 20.11 The 13 temporal relations from Allen (1984). 20.4.1 Reichenbach’s reference point The relation between simple verb tenses and points in time is by no means straight- forward. The present tense can be used to refer to a future event, as in this example: (20.15) Ok, we ﬂy from San Francisco to Boston at 10. Or consider the following examples: (20.16) Flight 1902 arrived late. (20.17) Flight 1902 had arrived late. Although both refer to events in the past, representing them in the same way seems wrong. The second example seems to have another unnamed event lurking in the background (e.g., Flight 1902 had already arrived late when something else hap- pened). 20.4 • R EPRESENTING TIME 449 To account for this phenomena, Reichenbach (1947) introduced the notion of a reference point. In our simple temporal scheme, the current moment in time isreference point equated with the time of the utterance and is used as a reference point for when the event occurred (before, at, or after). In Reichenbach’s approach, the notion of the reference point is separated from the utterance time and the event time. The following examples illustrate the basics of this approach: (20.18) When Mary’s ﬂight departed, I ate lunch. (20.19) When Mary’s ﬂight departed, I had eaten lunch. In both of these examples, the eating event has happened in the past, that is, prior to the utterance. However, the verb tense in the ﬁrst example indicates that the eating event began when the ﬂight departed, while the second example indicates that the eating was accomplished prior to the ﬂight’s departure. Therefore, in Reichenbach’s terms the departure event speciﬁes the reference point. These facts can be accom- modated by additional constraints relating the eating and departure events. In the ﬁrst example, the reference point precedes theeating event, and in the second exam- ple, the eating precedes the reference point. Figure 20.12 illustrates Reichenbach’s approach with the primary English tenses. Exercise 20.4 asks you to represent these examples in FOL . Past Perfect Simple Past Present Perfect Simple Future Future PerfectPresent E E E E R R U R,E U R,U U,R,E U,R U Figure 20.12 Reichenbach’s approach applied to various English tenses. In these diagrams, time ﬂows from left to right, E denotes the time of the event, R denotes the reference time, and U denotes the time of the utterance. Languages have many other ways to convey temporal information besides tense. Most useful for our purposes will be temporal expressions like in the morning or 6:45 or afterwards. (20.20) I’d like to go at 6:45 in the morning. (20.21) Somewhere around noon, please. (20.22) I want to take the train back afterwards. Incidentally, temporal expressions display a fascinating metaphorical conceptual organization. Temporal expressions in English are frequently expressed in spatial terms, as is illustrated by the various uses of at, in, somewhere, and near in these examples (Lakoff and Johnson 1980, Jackendoff 1983). Metaphorical organizations such as these, in which one domain is systematically expressed in terms of another, are very common in languages of the world. 450 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME 20.5 Representing Aspect A related notion to time is aspect, which is what we call the way events can beaspect categorized by their internal temporal structure or temporal contour. By this we mean questions like whether events are ongoing or have ended, or whether they are conceptualized as happening at a point in time or over some interval. Such notions of temporal contour have been used to divide event expressions into classes since Aristotle, although the set of four classes we’ll introduce here is due to Vendler (1967) (you may also see the German termaktionsart used to refer to these classes).aktionsart The most basic aspectual distinction is between events (which involve change)events and states (which do not involve change). Stative expressions represent the notionstates stative of an event participant being in a state, or having a particular property, at a given point in time. Stative expressions capture aspects of the world at a single point in time, and conceptualize the participant as unchanging and continuous. Consider the following ATIS examples. (20.23) I like express trains. (20.24) I need the cheapest fare. (20.25) I want to go ﬁrst class. In examples like these, the event participant denoted by the subject can be seen as experiencing something at a speciﬁc point in time, and don’t involve any kind of internal change over time (the liking or needing is conceptualized as continuous and unchanging). Non-states (which we’ll refer to as events) are divided into subclasses; we’ll introduce three here. Activity expressions describe events undertaken by a partic-activity ipant that occur over a span of time (rather than being conceptualized as a single point in time like stative expressions), and have no particular end point. Of course in practice all things end, but the meaning of the expression doesn’t represent this fact. Consider the following examples: (20.26) She drove a Mazda. (20.27) I live in Brooklyn. These examples both specify that the subject is engaged in, or has engaged in, the activity speciﬁed by the verb for some period of time, but doesn’t specify when the driving or living might have stopped. Two more classes of expressions, achievement expressions and accomplish- ment expressions, describe events that take place over time, but also conceptualize the event as having a particular kind of endpoint or goal. The Greek word telos means ‘end’ or ’goal’ and so the events described by these kinds of expressions are often called telic events.telic Accomplishment expressions describe events that have a natural end point andaccomplishment expressions result in a particular state. Consider the following examples: (20.28) He booked me a reservation. (20.29) The 7:00 train got me to New York City. In these examples, an event is seen as occurring over some period of time that ends when the intended state is accomplished (i.e., the state of me having a reservation, or me being in New York City). The ﬁnal aspectual class, achievement expressions, is only subtly different thanachievement expressions accomplishments. Consider the following: 20.6 • T EMPORALLY ANNOTATED DATASETS : T IME BANK 451 (20.30) She found her gate. (20.31) I reached New York. Like accomplishment expressions, achievement expressions result in a state. But unlike accomplishments, achievement events are ‘punctual’: they are thought of as happening in an instant and the verb doesn’t conceptualize the process or activ- ity leading up the state. Thus the events in these examples may in fact have been preceded by extended searching or traveling events, but the verb doesn’t conceptu- alize these preceding processes, but rather conceptualizes the events corresponding to ﬁnding and reaching as points, not intervals. In summary, a standard way of categorizing event expressions by their temporal contours is via these four general classes: Stative: I know my departure gate. Activity: John is ﬂying. Accomplishment: Sally booked her ﬂight. Achievement: She found her gate. Before moving on, note that event expressions can easily be shifted from one class to another. Consider the following examples: (20.32) I ﬂew. (20.33) I ﬂew to New York. The ﬁrst example is a simple activity; it has no natural end point. The second ex- ample is clearly an accomplishment event since it has an end point, and results in a particular state. Clearly, the classiﬁcation of an event is not solely governed by the verb, but by the semantics of the entire expression in context. 20.6 Temporally Annotated Datasets: TimeBank The TimeBank corpus consists of American English text annotated with temporalTimeBank information (Pustejovsky et al., 2003). The annotations use TimeML (Saur ´ı et al., 2006), a markup language for time based on Allen’s interval algebra discussed above (Allen, 1984). There are three types of TimeML objects: an EVENT represent events and states, a T IME represents time expressions like dates, and a L INK represents various relationships between events and times (event-event, event-time, and time- time). The links include temporal links (TL INK ) for the 13 Allen relations, aspec- tual links (AL INK ) for aspectual relationships between events and subevents, and SLINKS which mark factuality. Consider the following sample sentence and its corresponding markup shown in Fig. 20.13, selected from one of the TimeBank documents. (20.34) Delta Air Lines earnings soared 33% to a record in the ﬁscal ﬁrst quarter, bucking the industry trend toward declining proﬁts. This text has three events and two temporal expressions (including the creation time of the article, which serves as the document time), and four temporal links that capture the using the Allen relations: • Soaringe1 is included in the ﬁscal ﬁrst quartert58 • Soaringe1 is before 1989-10-26t57 • Soaringe1 is simultaneous with the buckinge3 452 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME <TIMEX3 tid="t57" type="DATE" value="1989-10-26" functionInDocument="CREATION_TIME"> 10/26/89 </TIMEX3> Delta Air Lines earnings <EVENT eid="e1" class="OCCURRENCE"> soared </EVENT> 33% to a record in <TIMEX3 tid="t58" type="DATE" value="1989-Q1" anchorTimeID="t57"> the fiscal first quarter </TIMEX3>, <EVENT eid="e3" class="OCCURRENCE">bucking</EVENT> the industry trend toward <EVENT eid="e4" class="OCCURRENCE">declining</EVENT> profits. Figure 20.13 Example from the TimeBank corpus. • Declininge4 includes soaringe1 We can also visualize the links as a graph. The TimeBank snippet in Eq. 20.35 would be represented with a graph like Fig. 20.14. (20.35) [DCT:11/02/891]1: Paciﬁc First Financial Corp. said2 shareholders approved3 its acquisition4 by Royal Trustco Ltd. of Toronto for $27 a share, or $212 million. The thrift holding company said5 it expects6 to obtain7 regulatory approval8 and complete9 the transaction10 by year-end11. 1 2 3 4 5 6 7 8 11 9 10 BEFORE BEFORE AFTER SIMULTANEOUS ENDS CULMINATES BEFORE EVIDENTIAL MODAL FACTIVE MODAL EVIDENTIAL MODAL Figure 20.14 A graph of the text in Eq. 20.35, adapted from (Ocal et al., 2022). TL INKS are shown in blue, ALINKS in red, and SLINKS in green. 20.7 Automatic Temporal Analysis Here we introduce the three common steps used in analyzing time in text: 1. Extracting temporal expressions 2. Normalizing these expressions, by converting them to a standard format. 3. Linking events to times and extracting time graphs and timelines 20.7.1 Extracting Temporal Expressions Temporal expressions are phrases that refer to absolute points in time, relative times, durations, and sets of these. Absolute temporal expressions are those that can beabsolute mapped directly to calendar dates, times of day, or both. Relative temporal expres-relative sions map to particular times through some other reference point (as in a week from last Tuesday). Finally, durations denote spans of time at varying levels of granular-duration ity (seconds, minutes, days, weeks, centuries, etc.). Figure 20.15 lists some sample temporal expressions in each of these categories. Temporal expressions are grammatical constructions that often have temporal lexical triggers as their heads, making them easy to ﬁnd. Lexical triggers mightlexical triggers 20.7 • A UTOMATIC TEMPORAL ANALYSIS 453 Absolute Relative Durations April 24, 1916 yesterday four hours The summer of ’77 next semester three weeks 10:15 AM two weeks from yesterday six days The 3rd quarter of 2006 last quarter the last three quarters Figure 20.15 Examples of absolute, relational and durational temporal expressions. be nouns, proper nouns, adjectives, and adverbs; full temporal expressions consist of their phrasal projections: noun phrases, adjective phrases, and adverbial phrases (Figure 20.16). Category Examples Noun morning, noon, night, winter, dusk, dawn Proper Noun January, Monday, Ides, Easter, Rosh Hashana, Ramadan, Tet Adjective recent, past, annual, former Adverb hourly, daily, monthly, yearly Figure 20.16 Examples of temporal lexical triggers. The task is to detect temporal expressions in running text, like this examples, shown with TIMEX3 tags (Pustejovsky et al. 2005, Ferro et al. 2005). A fare increase initiated <TIMEX3>last week</TIMEX3> by UAL Corp’s United Airlines was matched by competitors over<TIMEX3>the weekend</TIMEX3>, marking the second successful fare increase in <TIMEX3>two weeks</TIMEX3>. Rule-based approaches use cascades of regular expressions to recognize larger and larger chunks from previous stages, based on patterns containing parts of speech, trigger words (e.g.,February) or classes (e.g.,MONTH) (Chang and Manning, 2012; Str¨otgen and Gertz, 2013; Chambers, 2013). Here’s a rule from SUTime (Chang and Manning, 2012) for detecting expressions like 3 years old: /(\d+)[-\s]($TEUnits)(s)?([-\s]old)?/ Sequence-labeling approaches use the standard IOB scheme, marking words that are either (I)nside, (O)utside or at the (B)eginning of a temporal expression: A O fare O increase O initiated O last B week I by O UAL O Corp’s... O A statistical sequence labeler is trained, using either embeddings or a ﬁne-tuned encoder, or classic features extracted from the token and context including words, lexical triggers, and POS. Temporal expression recognizers are evaluated with the usual recall, precision, and F-measures. A major difﬁculty for all of these very lexicalized approaches is avoiding expressions that trigger false positives: (20.36) 1984 tells the story of Winston Smith... (20.37) ...U2’s classic Sunday Bloody Sunday 20.7.2 Temporal Normalization Temporal normalization is the task of mapping a temporal expression to a pointtemporal normalization in time or to a duration. Points in time correspond to calendar dates, to times of day, or both. Durations primarily consist of lengths of time. Normalized times 454 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME <TIMEX3 i d =” t1 ’ ’ t y p e =”DATE” v a l u e =” 2007 −07 −02 ” f u n c t i o n I n D o c u m e n t =”CREATIONTIME”> J u l y 2 , 2007 </TIMEX3> A f a r e i n c r e a s e i n i t i a t e d <TIMEX3 i d =” t 2 ” t y p e =”DATE” v a l u e =” 2007 −W26” anchorTimeID =” t 1 ” >l a s t week </TIMEX3> by U n i t e d A i r l i n e s was matched by c o m p e t i t o r s o v e r <TIMEX3 i d =” t 3 ” t y p e =”DURATION” v a l u e =”P1WE” anchorTimeID =” t 1 ” > t h e weekend </TIMEX3>, marking t h e second s u c c e s s f u l f a r e i n c r e a s e i n <TIMEX3 i d =” t 4 ” t y p e =”DURATION” v a l u e =”P2W” anchorTimeID =” t 1 ” > two weeks </TIMEX3>. Figure 20.17 TimeML markup including normalized values for temporal expressions. are represented via the ISO 8601 standard for encoding temporal values (ISO8601, 2004). Fig. 20.17 reproduces our earlier example with these value attributes. The dateline, or document date, for this text was July 2, 2007. The ISO repre- sentation for this kind of expression is YYYY-MM-DD, or in this case, 2007-07-02. The encodings for the temporal expressions in our sample text all follow from this date, and are shown here as values for the VALUE attribute. The ﬁrst temporal expression in the text proper refers to a particular week of the year. In the ISO standard, weeks are numbered from 01 to 53, with the ﬁrst week of the year being the one that has the ﬁrst Thursday of the year. These weeks are represented with the template YYYY-Wnn. The ISO week for our document date is week 27; thus the value for last week is represented as “2007-W26”. The next temporal expression is the weekend. ISO weeks begin on Monday; thus, weekends occur at the end of a week and are fully contained within a single week. Weekends are treated as durations, so the value of the VALUE attribute has to be a length. Durations are represented according to the pattern P nx, where n is an integer denoting the length and x represents the unit, as in P3Y for three years or P2D for two days. In this example, one weekend is captured as P1WE. In this case, there is also sufﬁcient information to anchor this particular weekend as part of a particular week. Such information is encoded in the ANCHOR TIME ID attribute. Finally, the phrase two weeks also denotes a duration captured as P2W. Figure 20.18 give some more examples, but there is a lot more to the various temporal annotation standards; consult ISO8601 (2004), Ferro et al. (2005), and Pustejovsky et al. (2005) for more details. Unit Pattern Sample Value Fully speciﬁed dates YYYY-MM-DD 1991-09-28 Weeks YYYY-Wnn 2007-W27 Weekends PnWE P1WE 24-hour clock times HH:MM:SS 11:13:45 Dates and times YYYY-MM-DDTHH:MM:SS 1991-09-28T11:00:00 Financial quarters Qn 1999-Q3 Figure 20.18 Sample ISO patterns for representing various times and durations. Most current approaches to temporal normalization are rule-based (Chang and Manning 2012, Str¨otgen and Gertz 2013). Patterns that match temporal expressions are associated with semantic analysis procedures. For example, the pattern above for recognizing phrases like 3 years old can be associated with the predicate Duration that takes two arguments, the length and the unit of time: pattern: /(\d+)[-\s]($TEUnits)(s)?([-\s]old)?/ result: Duration($1, $2) The task is difﬁcult because fully qualiﬁed temporal expressions are fairly rare in real texts. Most temporal expressions in news articles are incomplete and are only implicitly anchored, often with respect to the dateline of the article, which we refer 20.7 • A UTOMATIC TEMPORAL ANALYSIS 455 to as the document’s temporal anchor. The values of temporal expressions suchtemporal anchor as today, yesterday, or tomorrow can all be computed with respect to this temporal anchor. The semantic procedure for today simply assigns the anchor, and the attach- ments for tomorrow and yesterday add a day and subtract a day from the anchor, respectively. Of course, given the cyclic nature of our representations for months, weeks, days, and times of day, our temporal arithmetic procedures must use modulo arithmetic appropriate to the time unit being used. Unfortunately, even simple expressions such as the weekend or Wednesday in- troduce a fair amount of complexity. In our current example, the weekend clearly refers to the weekend of the week that immediately precedes the document date. But this won’t always be the case, as is illustrated in the following example. (20.38) Random security checks that began yesterday at Sky Harbor will continue at least through the weekend. In this case, the expression the weekend refers to the weekend of the week that the anchoring date is part of (i.e., the coming weekend). The information that signals this meaning comes from the tense of continue, the verb governing the weekend. Relative temporal expressions are handled with temporal arithmetic similar to that used for today and yesterday. The document date indicates that our example article is ISO week 27, so the expression last week normalizes to the current week minus 1. To resolve ambiguous next and last expressions we consider the distance from the anchoring date to the nearest unit. Next Friday can refer either to the immediately next Friday or to the Friday following that, but the closer the document date is to a Friday, the more likely it is that the phrase will skip the nearest one. Such ambiguities are handled by encoding language and domain-speciﬁc heuristics into the temporal attachments. 20.7.3 Temporal Ordering of Events The goal of temporal analysis, is to link times to events and then ﬁt all these events into a complete timeline. This ambitious task is the subject of considerable current research but solving it with a high level of accuracy is beyond the capabilities of current systems. A somewhat simpler, but still useful, task is to impose a partial or- dering on the events and temporal expressions mentioned in a text. Such an ordering can provide many of the same beneﬁts as a true timeline. An example of such a par- tial ordering is the determination that the fare increase by American Airlines came after the fare increase by United in our sample text. Determining such an ordering can be viewed as a binary relation detection and classiﬁcation task. Even this partial ordering task assumes that in addition to the detecting and nor- malizing time expressions steps described above, we have already detected all the events in the text. Indeed, many temporal expressions are anchored to events men- tioned in a text and not directly to other temporal expressions. Consider the follow- ing example: (20.39) One week after the storm, JetBlue issued its customer bill of rights. To determine when JetBlue issued its customer bill of rights we need to determine the time of the storm event, and then we need to modify that time by the temporal expression one week after. Thus once the events and times have been detected, our goal next is to assert links between all the times and events: i.e. creating event-event, event-time, time-time, DCT-event, and DCT-time TimeML TL INKS . This can be done by training time relation classiﬁers to predict the correct T: INK between each pair of times/events, 456 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME supervised by the gold labels in the TimeBank corpus with features like words/em- beddings, parse paths, tense and aspect The sieve-based architecture using precision- ranked sets of classiﬁers, which we’ll introduce in Chapter 23, is also commonly used. Systems that perform all 4 tasks (time extraction creation and normalization, event extraction, and time/event linking) include TARSQI (Verhagen et al., 2005) CLEAR TK (Bethard, 2013), CAEVO (Chambers et al., 2014), and CATENA (Mirza and Tonelli, 2016). 20.8 Template Filling Many texts contain reports of events, and possibly sequences of events, that often correspond to fairly common, stereotypical situations in the world. These abstract situations or stories, related to what have been called scripts (Schank and Abel-scripts son, 1977), consist of prototypical sequences of sub-events, participants, and their roles. The strong expectations provided by these scripts can facilitate the proper classiﬁcation of entities, the assignment of entities into roles and relations, and most critically, the drawing of inferences that ﬁll in things that have been left unsaid. In their simplest form, such scripts can be represented as templates consisting of ﬁxedtemplates sets of slots that take as values slot-ﬁllers belonging to particular classes. The task of template ﬁlling is to ﬁnd documents that invoke particular scripts and then ﬁll thetemplate ﬁlling slots in the associated templates with ﬁllers extracted from the text. These slot-ﬁllers may consist of text segments extracted directly from the text, or they may consist of concepts that have been inferred from text elements through some additional pro- cessing. A ﬁlled template from our original airline story might look like the following. FARE -RAISE ATTEMPT :   LEAD AIRLINE : U NITED AIRLINES AMOUNT : $6 EFFECTIVE DATE: 2006-10-26 FOLLOWER : A MERICAN AIRLINES   This template has four slots (LEAD AIRLINE , AMOUNT , EFFECTIVE DATE , FOL - LOWER ). The next section describes a standard sequence-labeling approach to ﬁlling slots. Section 20.8.2 then describes an older system based on the use of cascades of ﬁnite-state transducers and designed to address a more complex template-ﬁlling task that current learning-based systems don’t yet address. 20.8.1 Machine Learning Approaches to Template Filling In the standard paradigm for template ﬁlling, we are given training documents with text spans annotated with predeﬁned templates and their slot ﬁllers. Our goal is to create one template for each event in the input, ﬁlling in the slots with text spans. The task is generally modeled by training two separate supervised systems. The ﬁrst system decides whether the template is present in a particular sentence. This task is called template recognition or sometimes, in a perhaps confusing bit oftemplate recognition terminology, event recognition. Template recognition can be treated as a text classi- ﬁcation task, with features extracted from every sequence of words that was labeled in training documents as ﬁlling any slot from the template being detected. The usual 20.8 • T EMPLATE FILLING 457 set of features can be used: tokens, embeddings, word shapes, part-of-speech tags, syntactic chunk tags, and named entity tags. The second system has the job of role-ﬁller extraction. A separate classiﬁer isrole-ﬁller extraction trained to detect each role ( LEAD -AIRLINE , AMOUNT , and so on). This can be a binary classiﬁer that is run on every noun-phrase in the parsed input sentence, or a sequence model run over sequences of words. Each role classiﬁer is trained on the labeled data in the training set. Again, the usual set of features can be used, but now trained only on an individual noun phrase or the ﬁllers of a single slot. Multiple non-identical text segments might be labeled with the same slot la- bel. For example in our sample text, the strings United or United Airlines might be labeled as the LEAD AIRLINE . These are not incompatible choices and the corefer- ence resolution techniques introduced in Chapter 23 can provide a path to a solution. A variety of annotated collections have been used to evaluate this style of ap- proach to template ﬁlling, including sets of job announcements, conference calls for papers, restaurant guides, and biological texts. A key open question is extracting templates in cases where there is no training data or even predeﬁned templates, by inducing templates as sets of linked events (Chambers and Jurafsky, 2011). 20.8.2 Earlier Finite-State Template-Filling Systems The templates above are relatively simple. But consider the task of producing a template that contained all the information in a text like this one (Grishman and Sundheim, 1995): Bridgestone Sports Co. said Friday it has set up a joint venture in Taiwan with a local concern and a Japanese trading house to produce golf clubs to be shipped to Japan. The joint venture, Bridgestone Sports Taiwan Co., capital- ized at 20 million new Taiwan dollars, will start production in January 1990 with production of 20,000 iron and “metal wood” clubs a month. The MUC-5 ‘joint venture’ task (theMessage Understanding Conferences were a series of U.S. government-organized information-extraction evaluations) was to produce hierarchically linked templates describing joint ventures. Figure 20.19 shows a structure produced by the FASTUS system (Hobbs et al., 1997). Note how the ﬁller of the ACTIVITY slot of the TIE -UP template is itself a template with slots. Tie-up-1 Activity-1 : RELATIONSHIP tie-up C OMPANY Bridgestone Sports Taiwan Co. ENTITIES Bridgestone Sports Co. P RODUCT iron and “metal wood” clubs a local concern S TART DATE DURING: January 1990 a Japanese trading house JOINT VENTURE Bridgestone Sports Taiwan Co. ACTIVITY Activity-1 AMOUNT NT$20000000 Figure 20.19 The templates produced by FASTUS given the input text on page 457. Early systems for dealing with these complex templates were based on cascades of transducers based on handwritten rules, as sketched in Fig. 20.20. The ﬁrst four stages use handwritten regular expression and grammar rules to do basic tokenization, chunking, and parsing. Stage 5 then recognizes entities and events with a recognizer based on ﬁnite-state transducers (FSTs), and inserts the rec- ognized objects into the appropriate slots in templates. This FST recognizer is based 458 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME No. Step Description 1 Tokens Tokenize input stream of characters 2 Complex Words Multiword phrases, numbers, and proper names. 3 Basic phrases Segment sentences into noun and verb groups 4 Complex phrases Identify complex noun groups and verb groups 5 Semantic Patterns Identify entities and events, insert into templates. 6 Merging Merge references to the same entity or event Figure 20.20 Levels of processing in FASTUS (Hobbs et al., 1997). Each level extracts a speciﬁc type of information which is then passed on to the next higher level. on hand-built regular expressions like the following (NG indicates Noun-Group and VG Verb-Group), which matches the ﬁrst sentence of the news story above. NG(Company/ies) VG(Set-up) NG(Joint-Venture) with NG(Company/ies) VG(Produce) NG(Product) The result of processing these two sentences is the ﬁve draft templates (Fig. 20.21) that must then be merged into the single hierarchical structure shown in Fig. 20.19. The merging algorithm, after performing coreference resolution, merges two activi- ties that are likely to be describing the same events. # Template/Slot Value 1 RELATIONSHIP : TIE -UP ENTITIES : Bridgestone Co., a local concern, a Japanese trading house 2 ACTIVITY : PRODUCTION PRODUCT : “golf clubs” 3 RELATIONSHIP : TIE -UP JOINT VENTURE : “Bridgestone Sports Taiwan Co.” AMOUNT : NT$20000000 4 ACTIVITY : PRODUCTION COMPANY : “Bridgestone Sports Taiwan Co.” STARTDATE: DURING : January 1990 5 ACTIVITY : PRODUCTION PRODUCT : “iron and “metal wood” clubs” Figure 20.21 The ﬁve partial templates produced by stage 5 of FASTUS . These templates are merged in stage 6 to produce the ﬁnal template shown in Fig. 20.19 on page 457. 20.9 Summary This chapter has explored techniques for extracting limited forms of semantic con- tent from texts. • Relations among entities can be extracted by pattern-based approaches, su- pervised learning methods when annotated training data is available, lightly supervised bootstrapping methods when small numbers of seed tuples or seed patterns are available, distant supervision when a database of relations is available, and unsupervised or Open IE methods. • Reasoning about time can be facilitated by detection and normalization of temporal expressions. BIBLIOGRAPHICAL AND HISTORICAL NOTES 459 • Events can be ordered in time using sequence models and classiﬁers trained on temporally- and event-labeled data like the TimeBank corpus. • Template-ﬁlling applications can recognize stereotypical situations in texts and assign elements from the text to roles represented as ﬁxed sets of slots. Bibliographical and Historical Notes The earliest work on information extraction addressed the template-ﬁlling task in the context of the Frump system (DeJong, 1982). Later work was stimulated by the U.S. government-sponsored MUC conferences (Sundheim 1991, Sundheim 1992, Sund- heim 1993, Sundheim 1995). Early MUC systems like CIRCUS system (Lehnert et al., 1991) and SCISOR (Jacobs and Rau, 1990) were quite inﬂuential and inspired later systems like FASTUS (Hobbs et al., 1997). Chinchor et al. (1993) describe the MUC evaluation techniques. Due to the difﬁculty of porting systems from one domain to another, attention shifted to machine learning approaches. Early supervised learning approaches to IE (Cardie 1993, Cardie 1994, Riloff 1993, Soderland et al. 1995, Huffman 1996) focused on automating the knowledge acquisition process, mainly for ﬁnite-state rule-based systems. Their success, and the earlier success of HMM-based speech recognition, led to the use of sequence labeling (HMMs: Bikel et al. 1997; MEMMs McCallum et al. 2000; CRFs: Lafferty et al. 2001), and a wide exploration of fea- tures (Zhou et al., 2005). Neural approaches followed from the pioneering results of Collobert et al. (2011), who applied a CRF on top of a convolutional net. Progress in this area continues to be stimulated by formal evaluations with shared benchmark datasets, including the Automatic Content Extraction (ACE) evaluations of 2000-2007 on named entity recognition, relation extraction, and temporal ex- pressions1, the KBP (Knowledge Base Population) evaluations (Ji et al. 2010, Sur-KBP deanu 2013) of relation extraction tasks likeslot ﬁlling (extracting attributes (‘slots’)slot ﬁlling like age, birthplace, and spouse for a given entity) and a series of SemEval work- shops (Hendrickx et al., 2009). Semisupervised relation extraction was ﬁrst proposed by Hearst (1992b), and extended by systems like AutoSlog-TS (Riloff, 1996), DIPRE (Brin, 1998), SNOW- BALL (Agichtein and Gravano, 2000), and Jones et al. (1999). The distant super- vision algorithm we describe was drawn from Mintz et al. (2009), who ﬁrst used the term ‘distant supervision’ (which was suggested to them by Chris Manning) but similar ideas had occurred in earlier systems like Craven and Kumlien (1999) and Morgan et al. (2004) under the name weakly labeled data, as well as in Snow et al. (2005) and Wu and Weld (2007). Among the many extensions are Wu and Weld (2010), Riedel et al. (2010), and Ritter et al. (2013). Open IE systems include KNOW ITALL Etzioni et al. (2005), TextRunner (Banko et al., 2007), and R EVERB (Fader et al., 2011). See Riedel et al. (2013) for a universal schema that combines the advantages of distant supervision and Open IE. 1 www.nist.gov/speech/tests/ace/ 460 CHAPTER 20 • I NFORMATION EXTRACTION : R ELATIONS , EVENTS , AND TIME Exercises 20.1 Acronym expansion, the process of associating a phrase with an acronym, can be accomplished by a simple form of relational analysis. Develop a system based on the relation analysis approaches described in this chapter to populate a database of acronym expansions. If you focus on English Three Letter Acronyms (TLAs) you can evaluate your system’s performance by comparing it to Wikipedia’s TLA page. 20.2 Acquire the CMU seminar corpus and develop a template-ﬁlling system by using any of the techniques mentioned in Section 20.8. Analyze how well your system performs as compared with state-of-the-art results on this corpus. 20.3 A useful functionality in newer email and calendar applications is the ability to associate temporal expressions connected with events in email (doctor’s appointments, meeting planning, party invitations, etc.) with speciﬁc calendar entries. Collect a corpus of email containing temporal expressions related to event planning. How do these expressions compare to the kinds of expressions commonly found in news text that we’ve been discussing in this chapter? 20.4 For the following sentences, give FOL translations that capture the temporal relationships between the events. 1. When Mary’s ﬂight departed, I ate lunch. 2. When Mary’s ﬂight departed, I had eaten lunch. CHAPTER 21 Semantic Role Labeling “Who, What, Where, When, With what, Why, How” The seven circumstances, associated with Hermagoras and Aristotle (Sloan, 2010) Sometime between the 7th and 4th centuries BCE, the Indian grammarian P¯an. ini1 wrote a famous treatise on Sanskrit grammar, the As.t.¯adhy¯ay¯ı (‘8 books’), a treatise that has been called “one of the greatest monuments of hu- man intelligence” (Bloomﬁeld, 1933, 11). The work de- scribes the linguistics of the Sanskrit language in the form of 3959 sutras, each very efﬁciently (since it had to be memorized!) expressing part of a formal rule system that brilliantly preﬁgured modern mechanisms of formal lan- guage theory (Penn and Kiparsky, 2012). One set of rules describes the k¯arakas, semantic relationships between a verb and noun arguments, roles like agent, instrument, or destination. P ¯an. ini’s work was the earliest we know of that modeled the linguistic realization of events and their participants. This task of understanding how participants relate to events—being able to answer the question “Who did what to whom” (and perhaps also “when and where”)—is a central question of natural language processing. Let’s move forward 2.5 millennia to the present and consider the very mundane goal of understanding text about a purchase of stock by XYZ Corporation. This purchasing event and its participants can be described by a wide variety of surface forms. The event can be described by a verb ( sold, bought) or a noun ( purchase), and XYZ Corp can be the syntactic subject (of bought), the indirect object (of sold), or in a genitive or noun compound relation (with the noun purchase) despite having notionally the same role in all of them: • XYZ corporation bought the stock. • They sold the stock to XYZ corporation. • The stock was bought by XYZ corporation. • The purchase of the stock by XYZ corporation... • The stock purchase by XYZ corporation... In this chapter we introduce a level of representation that captures the common- ality between these sentences: there was a purchase event, the participants were XYZ Corp and some stock, and XYZ Corp was the buyer. These shallow semantic representations , semantic roles, express the role that arguments of a predicate take in the event, codiﬁed in databases like PropBank and FrameNet. We’ll introduce semantic role labeling, the task of assigning roles to spans in sentences, and selec- tional restrictions, the preferences that predicates express about their arguments, such as the fact that the theme of eat is generally something edible. 1 Figure shows a birch bark manuscript from Kashmir of the Rupavatra, a grammatical textbook based on the Sanskrit grammar of Panini. Image from the Wellcome Collection. 462 CHAPTER 21 • S EMANTIC ROLE LABELING 21.1 Semantic Roles Consider the meanings of the arguments Sasha, Pat, the window, and the door in these two sentences. (21.1) Sasha broke the window. (21.2) Pat opened the door. The subjects Sasha and Pat, what we might call the breaker of the window- breaking event and the opener of the door-opening event, have something in com- mon. They are both volitional actors, often animate, and they have direct causal responsibility for their events. Thematic roles are a way to capture this semantic commonality betweenbreak-thematic roles ers and openers. We say that the subjects of both these verbs are agents. Thus,agents AGENT is the thematic role that represents an abstract idea such as volitional causa- tion. Similarly, the direct objects of both these verbs, theBrokenThing and OpenedThing, are both prototypically inanimate objects that are affected in some way by the action. The semantic role for these participants is theme.theme Thematic Role Deﬁnition AGENT The volitional causer of an event EXPERIENCER The experiencer of an event FORCE The non-volitional causer of the event THEME The participant most directly affected by an event RESULT The end product of an event CONTENT The proposition or content of a propositional event INSTRUMENT An instrument used in an event BENEFICIARY The beneﬁciary of an event SOURCE The origin of the object of a transfer event GOAL The destination of an object of a transfer event Figure 21.1 Some commonly used thematic roles with their deﬁnitions. Although thematic roles are one of the oldest linguistic models, as we saw above, their modern formulation is due to Fillmore (1968) and Gruber (1965). Although there is no universally agreed-upon set of roles, Figs. 21.1 and 21.2 list some the- matic roles that have been used in various computational papers, together with rough deﬁnitions and examples. Most thematic role sets have about a dozen roles, but we’ll see sets with smaller numbers of roles with even more abstract meanings, and sets with very large numbers of roles that are speciﬁc to situations. We’ll use the general term semantic roles for all sets of roles, whether small or large.semantic roles 21.2 Diathesis Alternations The main reason computational systems use semantic roles is to act as a shallow meaning representation that can let us make simple inferences that aren’t possible from the pure surface string of words, or even from the parse tree. To extend the earlier examples, if a document says that Company A acquired Company B , we’d like to know that this answers the query Was Company B acquired? despite the fact that the two sentences have very different surface syntax. Similarly, this shallow semantics might act as a useful intermediate language in machine translation. 21.2 • D IATHESIS ALTERNATIONS 463 Thematic Role Example AGENT The waiter spilled the soup. EXPERIENCER John has a headache. FORCE The wind blows debris from the mall into our yards. THEME Only after Benjamin Franklin broke the ice... RESULT The city built a regulation-size baseball diamond... CONTENT Mona asked “You met Mary Ann at a supermarket?” INSTRUMENT He poached catﬁsh, stunning them with a shocking device... BENEFICIARY Whenever Ann Callahan makes hotel reservationsfor her boss... SOURCE I ﬂew in from Boston. GOAL I drove to Portland. Figure 21.2 Some prototypical examples of various thematic roles. Semantic roles thus help generalize over different surface realizations of pred- icate arguments. For example, while the AGENT is often realized as the subject of the sentence, in other cases the THEME can be the subject. Consider these possible realizations of the thematic arguments of the verb break: (21.3) John AGENT broke the window. THEME (21.4) John AGENT broke the window THEME with a rock. INSTRUMENT (21.5) The rock INSTRUMENT broke the window. THEME (21.6) The window THEME broke. (21.7) The window THEME was broken by John. AGENT These examples suggest that break has (at least) the possible arguments AGENT , THEME , and INSTRUMENT . The set of thematic role arguments taken by a verb is often called the thematic grid, θ-grid, or case frame. We can see that there arethematic grid case frame (among others) the following possibilities for the realization of these arguments of break: AGENT /Subject, THEME /Object AGENT /Subject, THEME /Object, INSTRUMENT /PPwith INSTRUMENT /Subject, THEME /Object THEME /Subject It turns out that many verbs allow their thematic roles to be realized in various syntactic positions. For example, verbs like give can realize the THEME and GOAL arguments in two different ways: (21.8) a. Doris AGENT gave the book THEME to Cary. GOAL b. Doris AGENT gave Cary GOAL the book. THEME These multiple argument structure realizations (the fact thatbreak can take AGENT , INSTRUMENT , or THEME as subject, and give can realize its THEME and GOAL in either order) are called verb alternations or diathesis alternations. The alternationverb alternation we showed above forgive, the dative alternation, seems to occur with particular se-dative alternation mantic classes of verbs, including “verbs of future having” (advance, allocate, offer, 464 CHAPTER 21 • S EMANTIC ROLE LABELING owe), “send verbs” (forward, hand, mail), “verbs of throwing” (kick, pass, throw), and so on. Levin (1993) lists for 3100 English verbs the semantic classes to which they belong (47 high-level classes, divided into 193 more speciﬁc classes) and the various alternations in which they participate. These lists of verb classes have been incorporated into the online resource VerbNet (Kipper et al., 2000), which links each verb to both WordNet and FrameNet entries. 21.3 Semantic Roles: Problems with Thematic Roles Representing meaning at the thematic role level seems like it should be useful in dealing with complications like diathesis alternations. Yet it has proved quite difﬁ- cult to come up with a standard set of roles, and equally difﬁcult to produce a formal deﬁnition of roles like AGENT , THEME , or INSTRUMENT . For example, researchers attempting to deﬁne role sets often ﬁnd they need to fragment a role like AGENT or THEME into many speciﬁc roles. Levin and Rappa- port Hovav (2005) summarize a number of such cases, such as the fact there seem to be at least two kinds of INSTRUMENTS , intermediary instruments that can appear as subjects and enabling instruments that cannot: (21.9) a. Shelly cut the banana with a knife. b. The knife cut the banana. (21.10) a. Shelly ate the sliced banana with a fork. b. The fork ate the sliced banana. In addition to the fragmentation problem, there are cases in which we’d like to reason about and generalize across semantic roles, but the ﬁnite discrete lists of roles don’t let us do this. Finally, it has proved difﬁcult to formally deﬁne the thematic roles. Consider the AGENT role; most cases of AGENTS are animate, volitional, sentient, causal, but any individual noun phrase might not exhibit all of these properties. These problems have led to alternative semantic role models that use eithersemantic role many fewer or many more roles. The ﬁrst of these options is to deﬁne generalized semantic roles that abstract over the speciﬁc thematic roles. For example, PROTO -AGENT and PROTO -PATIENTproto-agent proto-patient are generalized roles that express roughly agent-like and roughly patient-like mean- ings. These roles are deﬁned, not by necessary and sufﬁcient conditions, but rather by a set of heuristic features that accompany more agent-like or more patient-like meanings. Thus, the more an argument displays agent-like properties (being voli- tionally involved in the event, causing an event or a change of state in another par- ticipant, being sentient or intentionally involved, moving) the greater the likelihood that the argument can be labeled aPROTO -AGENT . The more patient-like the proper- ties (undergoing change of state, causally affected by another participant, stationary relative to other participants, etc.), the greater the likelihood that the argument can be labeled a PROTO -PATIENT . The second direction is instead to deﬁne semantic roles that are speciﬁc to a particular verb or a particular group of semantically related verbs or nouns. In the next two sections we describe two commonly used lexical resources that make use of these alternative versions of semantic roles.PropBank uses both proto- roles and verb-speciﬁc semantic roles. FrameNet uses semantic roles that are spe- ciﬁc to a general semantic idea called a frame. 21.4 • T HE PROPOSITION BANK 465 21.4 The Proposition Bank The Proposition Bank, generally referred to as PropBank, is a resource of sen-PropBank tences annotated with semantic roles. The English PropBank labels all the sentences in the Penn TreeBank; the Chinese PropBank labels sentences in the Penn Chinese TreeBank. Because of the difﬁculty of deﬁning a universal set of thematic roles, the semantic roles in PropBank are deﬁned with respect to an individual verb sense. Each sense of each verb thus has a speciﬁc set of roles, which are given only numbers rather than names: Arg0, Arg1, Arg2, and so on. In general, Arg0 represents the PROTO -AGENT , and Arg1, the PROTO -PATIENT . The semantics of the other roles are less consistent, often being deﬁned speciﬁcally for each verb. Nonetheless there are some generalization; the Arg2 is often the benefactive, instrument, attribute, or end state, the Arg3 the start point, benefactive, instrument, or attribute, and theArg4 the end point. Here are some slightly simpliﬁed PropBank entries for one sense each of the verbs agree and fall. Such PropBank entries are called frame ﬁles ; note that the deﬁnitions in the frame ﬁle for each role (“Other entity agreeing”, “Extent, amount fallen”) are informal glosses intended to be read by humans, rather than being formal deﬁnitions. (21.11) agree.01 Arg0: Agreer Arg1: Proposition Arg2: Other entity agreeing Ex1: [ Arg0 The group] agreed [Arg1 it wouldn’t make an offer]. Ex2: [ ArgM-TMP Usually] [Arg0 John] agrees [Arg2 with Mary] [Arg1 on everything]. (21.12) fall.01 Arg1: Logical subject, patient, thing falling Arg2: Extent, amount fallen Arg3: start point Arg4: end point, end state of arg1 Ex1: [ Arg1 Sales] fell [Arg4 to $25 million] [Arg3 from $27 million]. Ex2: [ Arg1 The average junk bond] fell [Arg2 by 4.2%]. Note that there is no Arg0 role for fall, because the normal subject of fall is a PROTO -PATIENT . The PropBank semantic roles can be useful in recovering shallow semantic in- formation about verbal arguments. Consider the verb increase: (21.13) increase.01 “go up incrementally” Arg0: causer of increase Arg1: thing increasing Arg2: amount increased by, EXT, or MNR Arg3: start point Arg4: end point A PropBank semantic role labeling would allow us to infer the commonality in the event structures of the following three examples, that is, that in each case Big Fruit Co. is the AGENT and the price of bananas is the THEME , despite the differing surface forms. 466 CHAPTER 21 • S EMANTIC ROLE LABELING (21.14) [ Arg0 Big Fruit Co. ] increased [Arg1 the price of bananas]. (21.15) [ Arg1 The price of bananas] was increased again [Arg0 by Big Fruit Co. ] (21.16) [ Arg1 The price of bananas] increased [Arg2 5%]. PropBank also has a number of non-numbered arguments calledArgMs, (ArgM- TMP, ArgM-LOC, etc.) which represent modiﬁcation or adjunct meanings. These are relatively stable across predicates, so aren’t listed with each frame ﬁle. Data labeled with these modiﬁers can be helpful in training systems to detect temporal, location, or directional modiﬁcation across predicates. Some of the ArgM’s include: TMP when? yesterday evening, now LOC where? at the museum, in San Francisco DIR where to/from? down, to Bangkok MNR how? clearly, with much enthusiasm PRP/CAU why? because ... , in response to the ruling REC themselves, each other ADV miscellaneous PRD secondary predication ...ate the meat raw While PropBank focuses on verbs, a related project, NomBank (Meyers et al.,NomBank 2004) adds annotations to noun predicates. For example the noun agreement in Apple’s agreement with IBM would be labeled with Apple as the Arg0 and IBM as the Arg2. This allows semantic role labelers to assign labels to arguments of both verbal and nominal predicates. 21.5 FrameNet While making inferences about the semantic commonalities across different sen- tences with increase is useful, it would be even more useful if we could make such inferences in many more situations, across different verbs, and also between verbs and nouns. For example, we’d like to extract the similarity among these three sen- tences: (21.17) [ Arg1 The price of bananas] increased [Arg2 5%]. (21.18) [ Arg1 The price of bananas] rose [Arg2 5%]. (21.19) There has been a [ Arg2 5%] rise [Arg1 in the price of bananas]. Note that the second example uses the different verb rise, and the third example uses the noun rather than the verb rise. We’d like a system to recognize that the price of bananas is what went up, and that 5% is the amount it went up, no matter whether the 5% appears as the object of the verb increased or as a nominal modiﬁer of the noun rise. The FrameNet project is another semantic-role-labeling project that attemptsFrameNet to address just these kinds of problems (Baker et al. 1998, Fillmore et al. 2003, Fillmore and Baker 2009, Ruppenhofer et al. 2016). Whereas roles in the PropBank project are speciﬁc to an individual verb, roles in the FrameNet project are speciﬁc to a frame. What is a frame? Consider the following set of words: reservation, ﬂight, travel, buy, price, cost, fare, rates, meal, plane There are many individual lexical relations of hyponymy, synonymy, and so on between many of the words in this list. The resulting set of relations does not, 21.5 • F RAME NET 467 however, add up to a complete account of how these words are related. They are clearly all deﬁned with respect to a coherent chunk of common-sense background information concerning air travel. We call the holistic background knowledge that unites these words aframe (Fill-frame more, 1985). The idea that groups of words are deﬁned with respect to some back- ground information is widespread in artiﬁcial intelligence and cognitive science, where besides frame we see related works like a model (Johnson-Laird, 1983), ormodel even script (Schank and Abelson, 1977).script A frame in FrameNet is a background knowledge structure that deﬁnes a set of frame-speciﬁc semantic roles, called frame elements, and includes a set of predi-frame elements cates that use these roles. Each word evokes a frame and proﬁles some aspect of the frame and its elements. The FrameNet dataset includes a set of frames and frame elements, the lexical units associated with each frame, and a set of labeled exam- ple sentences. For example, the change position on a scale frame is deﬁned as follows: This frame consists of words that indicate the change of an Item’s posi- tion on a scale (the Attribute) from a starting point (Initial value) to an end point (Final value). Some of the semantic roles (frame elements) in the frame are deﬁned as in Fig. 21.3. Note that these are separated intocore roles, which are frame speciﬁc, andcore roles non-core roles, which are more like the Arg-M arguments in PropBank, expressingnon-core roles more general properties of time, location, and so on. Core Roles ATTRIBUTE The ATTRIBUTE is a scalar property that the ITEM possesses. DIFFERENCE The distance by which an ITEM changes its position on the scale. FINAL STATE A description that presents the ITEM ’s state after the change in the ATTRIBUTE ’s value as an independent predication. FINAL VALUE The position on the scale where the ITEM ends up. INITIAL STATE A description that presents the I TEM ’s state before the change in the A T- TRIBUTE ’s value as an independent predication. INITIAL VALUE The initial position on the scale from which the ITEM moves away. ITEM The entity that has a position on the scale. VALUE RANGE A portion of the scale, typically identiﬁed by its end points, along which the values of the ATTRIBUTE ﬂuctuate. Some Non-Core Roles DURATION The length of time over which the change takes place. SPEED The rate of change of the VALUE . GROUP The GROUP in which an ITEM changes the value of an ATTRIBUTE in a speciﬁed way. Figure 21.3 The frame elements in the change position on a scale frame from the FrameNet Labelers Guide (Ruppenhofer et al., 2016). Here are some example sentences: (21.20) [ ITEM Oil] rose [ATTRIBUTE in price] [DIFFERENCE by 2%]. (21.21) [ ITEM It] has increased [FINAL STATE to having them 1 day a month]. (21.22) [ ITEM Microsoft shares] fell [FINAL VALUE to 7 5/8]. (21.23) [ ITEM Colon cancer incidence] fell [DIFFERENCE by 50%] [GROUP among men]. 468 CHAPTER 21 • S EMANTIC ROLE LABELING (21.24) a steady increase [INITIAL VALUE from 9.5] [FINAL VALUE to 14.3] [ITEM in dividends] (21.25) a [ DIFFERENCE 5%] [ITEM dividend] increase... Note from these example sentences that the frame includes target words likerise, fall, and increase. In fact, the complete frame consists of the following words: VERBS: dwindle move soar escalation shift advance edge mushroom swell explosion tumble climb explode plummet swing fall decline fall reach triple ﬂuctuation ADVERBS: decrease ﬂuctuate rise tumble gain increasingly diminish gain rocket growth dip grow shift NOUNS: hike double increase skyrocket decline increase drop jump slide decrease rise FrameNet also codes relationships between frames, allowing frames to inherit from each other, or representing relations between frames like causation (and gen- eralizations among frame elements in different frames can be represented by inheri- tance as well). Thus, there is a Cause change of position on a scale frame that is linked to the Change of position on a scale frame by the cause relation, but that adds an AGENT role and is used for causative examples such as the following: (21.26) [ AGENT They] raised [ITEM the price of their soda] [DIFFERENCE by 2%]. Together, these two frames would allow an understanding system to extract the common event semantics of all the verbal and nominal causative and non-causative usages. FrameNets have also been developed for many other languages including Span- ish, German, Japanese, Portuguese, Italian, and Chinese. 21.6 Semantic Role Labeling Semantic role labeling (sometimes shortened as SRL) is the task of automaticallysemantic role labeling ﬁnding the semantic roles of each argument of each predicate in a sentence. Cur- rent approaches to semantic role labeling are based on supervised machine learning, often using the FrameNet and PropBank resources to specify what counts as a pred- icate, deﬁne the set of roles used in the task, and provide training and test sets. Recall that the difference between these two models of semantic roles is that FrameNet (21.27) employs many frame-speciﬁc frame elements as roles, while Prop- Bank (21.28) uses a smaller number of numbered argument labels that can be inter- preted as verb-speciﬁc labels, along with the more general ARGM labels. Some examples: (21.27) [You] can’t [blame] [the program] [for being unable to identify it] COGNIZER TARGET EVALUEE REASON (21.28) [The San Francisco Examiner] issued [a special edition] [yesterday] ARG 0 TARGET ARG 1 ARGM -TMP 21.6.1 A Feature-based Algorithm for Semantic Role Labeling A simpliﬁed feature-based semantic role labeling algorithm is sketched in Fig. 21.4. Feature-based algorithms—from the very earliest systems like (Simmons, 1973)— begin by parsing, using broad-coverage parsers to assign a parse to the input string. 21.6 • S EMANTIC ROLE LABELING 469 Figure 21.5 shows a parse of (21.28) above. The parse is then traversed to ﬁnd all words that are predicates. For each of these predicates, the algorithm examines each node in the parse tree and uses supervised classiﬁcation to decide the semantic role (if any) it plays for this predicate. Given a labeled training set such as PropBank or FrameNet, a feature vector is extracted for each node, using feature templates described in the next subsection. A 1-of-N classiﬁer is then trained to predict a semantic role for each constituent given these features, where N is the number of potential semantic roles plus an extra NONE role for non-role constituents. Any standard classiﬁcation algorithms can be used. Finally, for each test sentence to be labeled, the classiﬁer is run on each relevant constituent. function SEMANTIC ROLE LABEL (words) returns labeled tree parse←PARSE (words) for each predicate in parse do for each node in parse do featurevector←EXTRACT FEATURES (node, predicate, parse) CLASSIFY NODE (node, featurevector, parse) Figure 21.4 A generic semantic-role-labeling algorithm. CLASSIFY NODE is a 1-of-N clas- siﬁer that assigns a semantic role (or NONE for non-role constituents), trained on labeled data such as FrameNet or PropBank. S NP-SBJ =A R G 0 VP DT NNP NNP NNP The San Francisco Examiner VBD = TARGET NP =A R G 1 PP-TMP =A R G M - T M P issued DT JJ NN IN NP as p e c i a l e d i t i o n a r o u n d N N N P - T M P noon yesterday Figure 21.5 Parse tree for a PropBank sentence, showing the PropBank argument labels. The dotted line shows the path feature NP↑S↓VP↓VBD for ARG0, the NP-SBJ constituent The San Francisco Examiner. Instead of training a single-stage classiﬁer as in Fig. 21.5, the node-level classi- ﬁcation task can be broken down into multiple steps: 1. Pruning: Since only a small number of the constituents in a sentence are arguments of any given predicate, many systems use simple heuristics to prune unlikely constituents. 2. Identiﬁcation: a binary classiﬁcation of each node as an argument to be la- beled or a NONE . 3. Classiﬁcation: a 1-of-N classiﬁcation of all the constituents that were labeled as arguments by the previous stage 470 CHAPTER 21 • S EMANTIC ROLE LABELING The separation of identiﬁcation and classiﬁcation may lead to better use of fea- tures (different features may be useful for the two tasks) or to computational efﬁ- ciency. Global Optimization The classiﬁcation algorithm of Fig. 21.5 classiﬁes each argument separately (‘lo- cally’), making the simplifying assumption that each argument of a predicate can be labeled independently. This assumption is false; there are interactions between argu- ments that require a more ‘global’ assignment of labels to constituents. For example, constituents in FrameNet and PropBank are required to be non-overlapping. More signiﬁcantly, the semantic roles of constituents are not independent. For example PropBank does not allow multiple identical arguments; two constituents of the same verb cannot both be labeled ARG 0 . Role labeling systems thus often add a fourth step to deal with global consistency across the labels in a sentence. For example, the local classiﬁers can return a list of possible labels associated with probabilities for each constituent, and a second-pass Viterbi decoding or re-ranking approach can be used to choose the best consensus label. Integer linear programming (ILP) is another common way to choose a solution that conforms best to multiple constraints. Features for Semantic Role Labeling Most systems use some generalization of the core set of features introduced by Gildea and Jurafsky (2000). Common basic features templates (demonstrated on the NP-SBJ constituent The San Francisco Examiner in Fig. 21.5) include: • The governing predicate, in this case the verb issued. The predicate is a cru- cial feature since labels are deﬁned only with respect to a particular predicate. • The phrase type of the constituent, in this case, NP (or NP-SBJ). Some se- mantic roles tend to appear as NPs, others as S or PP, and so on. • The headword of the constituent, Examiner. The headword of a constituent can be computed with standard head rules, such as those given in Appendix D in Fig. 18.17. Certain headwords (e.g., pronouns) place strong constraints on the possible semantic roles they are likely to ﬁll. • The headword part of speech of the constituent, NNP. • The path in the parse tree from the constituent to the predicate. This path is marked by the dotted line in Fig. 21.5. Following Gildea and Jurafsky (2000), we can use a simple linear representation of the path, NP↑S↓VP↓VBD. ↑and ↓represent upward and downward movement in the tree, respectively. The path is very useful as a compact representation of many kinds of grammatical function relationships between the constituent and the predicate. • The voice of the clause in which the constituent appears, in this case, active (as contrasted with passive). Passive sentences tend to have strongly different linkings of semantic roles to surface form than do active ones. • The binary linear position of the constituent with respect to the predicate, either before or after. • The subcategorization of the predicate, the set of expected arguments that appear in the verb phrase. We can extract this information by using the phrase- structure rule that expands the immediate parent of the predicate; VP→VBD NP PP for the predicate in Fig. 21.5. • The named entity type of the constituent. 21.6 • S EMANTIC ROLE LABELING 471 • The ﬁrst words and the last word of the constituent. The following feature vector thus represents the ﬁrst NP in our example (recall that most observations will have the value NONE rather than, for example, ARG 0, since most constituents in the parse tree will not bear a semantic role): ARG 0: [issued, NP, Examiner, NNP, NP ↑S↓VP↓VBD, active, before, VP →NP PP, ORG, The, Examiner] Other features are often used in addition, such as sets of n-grams inside the constituent, or more complex versions of the path features (the upward or downward halves, or whether particular nodes occur in the path). It’s also possible to use dependency parses instead of constituency parses as the basis of features, for example using dependency parse paths instead of constituency paths. 21.6.2 A Neural Algorithm for Semantic Role Labeling A simple neural approach to SRL is to treat it as a sequence labeling task like named- entity recognition, using the BIO approach. Let’s assume that we are given the predicate and the task is just detecting and labeling spans. Recall that with BIO tagging, we have a begin and end tag for each possible role ( B-ARG 0, I-ARG 0; B- ARG 1, I-ARG 1, and so on), plus an outside tag O. ENCODER [CLS] the cats love hats [SEP] love [SEP] FFN B-ARG0 I-ARG0 B-PRED concatenate with predicate B-ARG1 FFN Softmax FFN FFN FFN Figure 21.6 A simple neural approach to semantic role labeling. The input sentence is followed by [SEP] and an extra input for the predicate, in this case love. The encoder outputs are concatenated to an indicator variable which is 1 for the predicate and 0 for all other words After He et al. (2017) and Shi and Lin (2019). As with all the taggers, the goal is to compute the highest probability tag se- quence ˆy, given the input sequence of words w: ˆy = argmax y∈T P(y\|w) Fig. 21.6 shows a sketch of a standard algorithm from He et al. (2017). Here each input word is mapped to pretrained embeddings, and then each token is concatenated with the predicate embedding and then passed through a feedforward network with a softmax which outputs a distribution over each SRL label. For decoding, a CRF layer can be used instead of the MLP layer on top of the biLSTM output to do global inference, but in practice this doesn’t seem to provide much beneﬁt. 472 CHAPTER 21 • S EMANTIC ROLE LABELING 21.6.3 Evaluation of Semantic Role Labeling The standard evaluation for semantic role labeling is to require that each argument label must be assigned to the exactly correct word sequence or parse constituent, and then compute precision, recall, and F-measure. Identiﬁcation and classiﬁcation can also be evaluated separately. Two common datasets used for evaluation are CoNLL- 2005 (Carreras and M`arquez, 2005) and CoNLL-2012 (Pradhan et al., 2013). 21.7 Selectional Restrictions We turn in this section to another way to represent facts about the relationship be- tween predicates and arguments. A selectional restriction is a semantic type con-selectional restriction straint that a verb imposes on the kind of concepts that are allowed to ﬁll its argument roles. Consider the two meanings associated with the following example: (21.29) I want to eat someplace nearby. There are two possible parses and semantic interpretations for this sentence. In the sensible interpretation, eat is intransitive and the phrase someplace nearby is an adjunct that gives the location of the eating event. In the nonsensical speaker-as- Godzilla interpretation, eat is transitive and the phrasesomeplace nearby is the direct object and the THEME of the eating, like the NP Malaysian food in the following sentences: (21.30) I want to eat Malaysian food. How do we know that someplace nearby isn’t the direct object in this sentence? One useful cue is the semantic fact that the THEME of E ATING events tends to be something that is edible. This restriction placed by the verb eat on the ﬁller of its THEME argument is a selectional restriction. Selectional restrictions are associated with senses, not entire lexemes. We can see this in the following examples of the lexeme serve: (21.31) The restaurant serves green-lipped mussels. (21.32) Which airlines serve Denver? Example (21.31) illustrates the offering-food sense of serve, which ordinarily re- stricts its THEME to be some kind of food Example (21.32) illustrates the provides a commercial service to sense of serve, which constrains its THEME to be some type of appropriate location. Selectional restrictions vary widely in their speciﬁcity. The verb imagine, for example, imposes strict requirements on its AGENT role (restricting it to humans and other animate entities) but places very few semantic requirements on itsTHEME role. A verb like diagonalize, on the other hand, places a very speciﬁc constraint on the ﬁller of its THEME role: it has to be a matrix, while the arguments of the adjective odorless are restricted to concepts that could possess an odor: (21.33) In rehearsal, I often ask the musicians to imagine a tennis game. (21.34) Radon is an odorless gas that can’t be detected by human senses. (21.35) To diagonalize a matrix is to ﬁnd its eigenvalues. These examples illustrate that the set of concepts we need to represent selectional restrictions (being a matrix, being able to possess an odor, etc) is quite open ended. This distinguishes selectional restrictions from other features for representing lexical knowledge, like parts-of-speech, which are quite limited in number. 21.7 • S ELECTIONAL RESTRICTIONS 473 21.7.1 Representing Selectional Restrictions One way to capture the semantics of selectional restrictions is to use and extend the event representation of Appendix F. Recall that the neo-Davidsonian representation of an event consists of a single variable that stands for the event, a predicate denoting the kind of event, and variables and relations for the event roles. Ignoring the issue of the λ-structures and using thematic roles rather than deep event roles, the semantic contribution of a verb like eat might look like the following: ∃e,x,y Eating(e)∧Agent(e,x)∧T heme(e,y) With this representation, all we know about y, the ﬁller of the THEME role, is that it is associated with an Eating event through the Theme relation. To stipulate the selectional restriction that y must be something edible, we simply add a new term to that effect: ∃e,x,y Eating(e)∧Agent(e,x)∧T heme(e,y)∧EdibleT hing(y) When a phrase like ate a hamburger is encountered, a semantic analyzer can form the following kind of representation: ∃e,x,y Eating(e)∧Eater (e,x)∧T heme(e,y)∧EdibleT hing(y)∧Hamburger(y) This representation is perfectly reasonable since the membership ofy in the category Hamburger is consistent with its membership in the categoryEdibleThing, assuming a reasonable set of facts in the knowledge base. Correspondingly, the representation for a phrase such as ate a takeoff would be ill-formed because membership in an event-like category such as Takeoff would be inconsistent with membership in the category EdibleThing. While this approach adequately captures the semantics of selectional restrictions, there are two problems with its direct use. First, using FOL to perform the simple task of enforcing selectional restrictions is overkill. Other, far simpler, formalisms can do the job with far less computational cost. The second problem is that this approach presupposes a large, logical knowledge base of facts about the concepts that make up selectional restrictions. Unfortunately, although such common-sense knowledge bases are being developed, none currently have the kind of coverage necessary to the task. A more practical approach is to state selectional restrictions in terms of WordNet synsets rather than as logical concepts. Each predicate simply speciﬁes a WordNet synset as the selectional restriction on each of its arguments. A meaning representa- tion is well-formed if the role ﬁller word is a hyponym (subordinate) of this synset. For our ate a hamburger example, for instance, we could set the selectional restriction on the THEME role of the verbeat to the synset {food, nutrient}, glossed as any substance that can be metabolized by an animal to give energy and build tissue. Luckily, the chain of hypernyms for hamburger shown in Fig. 21.7 reveals that hamburgers are indeed food. Again, the ﬁller of a role need not match the restriction synset exactly; it just needs to have the synset as one of its superordinates. We can apply this approach to theTHEME roles of the verbs imagine, lift, and di- agonalize, discussed earlier. Let us restrict imagine’sTHEME to the synset {entity}, lift’sTHEME to {physical entity}, and diagonalize to {matrix}. This arrangement correctly permits imagine a hamburger and lift a hamburger, while also correctly ruling out diagonalize a hamburger. 474 CHAPTER 21 • S EMANTIC ROLE LABELING Sense 1 hamburger, beefburger -- (a fried cake of minced beef served on a bun) => sandwich => snack food => dish => nutriment, nourishment, nutrition... => food, nutrient => substance => matter => physical entity => entity Figure 21.7 Evidence from WordNet that hamburgers are edible. 21.7.2 Selectional Preferences In the earliest implementations, selectional restrictions were considered strict con- straints on the kind of arguments a predicate could take (Katz and Fodor 1963, Hirst 1987). For example, the verb eat might require that its THEME argument be [+FOOD]. Early word sense disambiguation systems used this idea to rule out senses that violated the selectional restrictions of their governing predicates. Very quickly, however, it became clear that these selectional restrictions were better represented as preferences rather than strict constraints (Wilks 1975b, Wilks 1975a). For example, selectional restriction violations (like inedible arguments of eat) often occur in well-formed sentences, for example because they are negated (21.36), or because selectional restrictions are overstated (21.37): (21.36) But it fell apart in 1931, perhaps because people realized you can’t eat gold for lunch if you’re hungry. (21.37) In his two championship trials, Mr. Kulkarni ate glass on an empty stomach, accompanied only by water and tea. Modern systems for selectional preferences therefore specify the relation be- tween a predicate and its possible arguments with soft constraints of some kind. Selectional Association One of the most inﬂuential has been the selectional association model of Resnik (1993). Resnik deﬁnes the idea of selectional preference strength as the general selectional preference strength amount of information that a predicate tells us about the semantic class of its argu- ments. For example, the verb eat tells us a lot about the semantic class of its direct objects, since they tend to be edible. The verb be, by contrast, tells us less about its direct objects. The selectional preference strength can be deﬁned by the differ- ence in information between two distributions: the distribution of expected semantic classes P(c) (how likely is it that a direct object will fall into class c) and the dis- tribution of expected semantic classes for the particular verb P(c\|v) (how likely is it that the direct object of the speciﬁc verb v will fall into semantic class c). The greater the difference between these distributions, the more information the verb is giving us about possible objects. The difference between these two distributions can be quantiﬁed by relative entropy, or the Kullback-Leibler divergence (Kullbackrelative entropy and Leibler, 1951). The Kullback-Leibler or KL divergence D(P\|\|Q) expresses theKL divergence 21.7 • S ELECTIONAL RESTRICTIONS 475 difference between two probability distributions P and Q D(P\|\|Q) = ∑ x P(x)log P(x) Q(x) (21.38) The selectional preference SR(v) uses the KL divergence to express how much in- formation, in bits, the verb v expresses about the possible semantic class of its argu- ment. SR(v) = D(P(c\|v)\|\|P(c)) = ∑ c P(c\|v)log P(c\|v) P(c) (21.39) Resnik then deﬁnes the selectional association of a particular class and verb as theselectional association relative contribution of that class to the general selectional preference of the verb: AR(v,c) = 1 SR(v)P(c\|v)log P(c\|v) P(c) (21.40) The selectional association is thus a probabilistic measure of the strength of asso- ciation between a predicate and a class dominating the argument to the predicate. Resnik estimates the probabilities for these associations by parsing a corpus, count- ing all the times each predicate occurs with each argument word, and assuming that each word is a partial observation of all the WordNet concepts containing the word. The following table from Resnik (1996) shows some sample high and low selectional associations for verbs and some WordNet semantic classes of their direct objects. Direct Object Direct Object Verb Semantic Class Assoc Semantic Class Assoc read WRITING 6.80 ACTIVITY -.20 write WRITING 7.26 COMMERCE 0 see ENTITY 5.79 METHOD -0.01 Selectional Preference via Conditional Probability An alternative to using selectional association between a verb and the WordNet class of its arguments is to use the conditional probability of an argument word given a predicate verb, directly modeling the strength of association of one verb (predicate) with one noun (argument). The conditional probability model can be computed by parsing a very large cor- pus (billions of words), and computing co-occurrence counts: how often a given verb occurs with a given noun in a given relation. The conditional probability of an argument noun given a verb for a particular relation P(n\|v,r) can then be used as a selectional preference metric for that pair of words (Brockmann and Lapata 2003, Keller and Lapata 2003): P(n\|v,r) = { C(n,v,r) C(v,r) if C(n,v,r) >0 0 otherwise The inverse probabilityP(v\|n,r) was found to have better performance in some cases (Brockmann and Lapata, 2003): P(v\|n,r) = { C(n,v,r) C(n,r) if C(n,v,r) >0 0 otherwise 476 CHAPTER 21 • S EMANTIC ROLE LABELING An even simpler approach is to use the simple log co-occurrence frequency of the predicate with the argument log count(v,n,r) instead of conditional probability; this seems to do better for extracting preferences for syntactic subjects rather than objects (Brockmann and Lapata, 2003). Evaluating Selectional Preferences One way to evaluate models of selectional preferences is to usepseudowords (Galepseudowords et al. 1992b, Sch¨utze 1992a). A pseudoword is an artiﬁcial word created by concate- nating a test word in some context (say banana) with a confounder word (say door) to create banana-door). The task of the system is to identify which of the two words is the original word. To evaluate a selectional preference model (for example on the relationship between a verb and a direct object) we take a test corpus and select all verb tokens. For each verb token (say drive) we select the direct object (e.g., car), concatenated with a confounder word that is its nearest neighbor, the noun with the frequency closest to the original (say house), to make car/house). We then use the selectional preference model to choose which of car and house are more preferred objects of drive, and compute how often the model chooses the correct original ob- ject (e.g., car) (Chambers and Jurafsky, 2010). Another evaluation metric is to get human preferences for a test set of verb- argument pairs, and have them rate their degree of plausibility. This is usually done by using magnitude estimation, a technique from psychophysics, in which subjects rate the plausibility of an argument proportional to a modulus item. A selectional preference model can then be evaluated by its correlation with the human prefer- ences (Keller and Lapata, 2003). 21.8 Primitive Decomposition of Predicates One way of thinking about the semantic roles we have discussed through the chapter is that they help us deﬁne the roles that arguments play in a decompositional way, based on ﬁnite lists of thematic roles (agent, patient, instrument, proto-agent, proto- patient, etc.). This idea of decomposing meaning into sets of primitive semantic elements or features, called primitive decomposition or componential analysis,componential analysis has been taken even further, and focused particularly on predicates. Consider these examples of the verb kill: (21.41) Jim killed his philodendron. (21.42) Jim did something to cause his philodendron to become not alive. There is a truth-conditional (‘propositional semantics’) perspective from which these two sentences have the same meaning. Assuming this equivalence, we could repre- sent the meaning of kill as: (21.43) KILL (x,y) ⇔CAUSE (x, BECOME (NOT(ALIVE (y)))) thus using semantic primitives like do, cause, become not, and alive. Indeed, one such set of potential semantic primitives has been used to account for some of the verbal alternations discussed in Section 21.2 (Lakoff 1965, Dowty 1979). Consider the following examples. (21.44) John opened the door. ⇒CAUSE (John, BECOME (OPEN (door))) (21.45) The door opened. ⇒BECOME (OPEN (door)) 21.9 • S UMMARY 477 (21.46) The door is open. ⇒OPEN (door) The decompositional approach asserts that a single state-like predicate associ- ated with open underlies all of these examples. The differences among the meanings of these examples arises from the combination of this single predicate with the prim- itives CAUSE and BECOME . While this approach to primitive decomposition can explain the similarity be- tween states and actions or causative and non-causative predicates, it still relies on having a large number of predicates like open. More radical approaches choose to break down these predicates as well. One such approach to verbal predicate decom- position that played a role in early natural language systems is conceptual depen- dency (CD), a set of ten primitive predicates, shown in Fig. 21.8.conceptual dependency Primitive Deﬁnition ATRANS The abstract transfer of possession or control from one entity to another PTRANS The physical transfer of an object from one location to another MTRANS The transfer of mental concepts between entities or within an entity MBUILD The creation of new information within an entity PROPEL The application of physical force to move an object MOVE The integral movement of a body part by an animal INGEST The taking in of a substance by an animal EXPEL The expulsion of something from an animal SPEAK The action of producing a sound ATTEND The action of focusing a sense organ Figure 21.8 A set of conceptual dependency primitives. Below is an example sentence along with itsCD representation. The verb brought is translated into the two primitives ATRANS and PTRANS to indicate that the waiter both physically conveyed the check to Mary and passed control of it to her. Note that CD also associates a ﬁxed set of thematic roles with each primitive to represent the various participants in the action. (21.47) The waiter brought Mary the check. ∃x,y Atrans(x)∧Actor(x,Waiter)∧Ob ject(x,Check)∧To(x,Mary) ∧Ptrans(y)∧Actor(y,Waiter)∧Ob ject(y,Check)∧To(y,Mary) 21.9 Summary • Semantic roles are abstract models of the role an argument plays in the event described by the predicate. • Thematic roles are a model of semantic roles based on a single ﬁnite list of roles. Other semantic role models include per-verb semantic role lists and proto-agent/proto-patient, both of which are implemented in PropBank, and per-frame role lists, implemented in FrameNet. 478 CHAPTER 21 • S EMANTIC ROLE LABELING • Semantic role labeling is the task of assigning semantic role labels to the constituents of a sentence. The task is generally treated as a supervised ma- chine learning task, with models trained on PropBank or FrameNet. Algo- rithms generally start by parsing a sentence and then automatically tag each parse tree node with a semantic role. Neural models map straight from words end-to-end. • Semantic selectional restrictionsallow words (particularly predicates) to post constraints on the semantic properties of their argument words. Selectional preference models (like selectional association or simple conditional proba- bility) allow a weight or probability to be assigned to the association between a predicate and an argument word or class. Bibliographical and Historical Notes Although the idea of semantic roles dates back to P ¯an. ini, they were re-introduced into modern linguistics by Gruber (1965), Fillmore (1966) and Fillmore (1968). Fill- more had become interested in argument structure by studying Lucien Tesni `ere’s groundbreaking ´El´ements de Syntaxe Structurale (Tesni`ere, 1959) in which the term ‘dependency’ was introduced and the foundations were laid for dependency gram- mar. Following Tesni`ere’s terminology, Fillmore ﬁrst referred to argument roles as actants (Fillmore, 1966) but quickly switched to the termcase, (see Fillmore (2003)) and proposed a universal list of semantic roles or cases (Agent, Patient, Instrument, etc.), that could be taken on by the arguments of predicates. Verbs would be listed in the lexicon with theircase frame, the list of obligatory (or optional) case arguments. The idea that semantic roles could provide an intermediate level of semantic representation that could help map from syntactic parse structures to deeper, more fully-speciﬁed representations of meaning was quickly adopted in natural language processing, and systems for extracting case frames were created for machine transla- tion (Wilks, 1973), question-answering (Hendrix et al., 1973), spoken-language pro- cessing (Nash-Webber, 1975), and dialogue systems (Bobrow et al., 1977). General- purpose semantic role labelers were developed. The earliest ones (Simmons, 1973) ﬁrst parsed a sentence by means of an ATN (Augmented Transition Network) parser. Each verb then had a set of rules specifying how the parse should be mapped to se- mantic roles. These rules mainly made reference to grammatical functions (subject, object, complement of speciﬁc prepositions) but also checked constituent internal features such as the animacy of head nouns. Later systems assigned roles from pre- built parse trees, again by using dictionaries with verb-speciﬁc case frames (Levin 1977, Marcus 1980). By 1977 case representation was widely used and taught in AI and NLP courses, and was described as a standard of natural language processing in the ﬁrst edition of Winston’s 1977 textbookArtiﬁcial Intelligence. In the 1980s Fillmore proposed his model of frame semantics, later describing the intuition as follows: “The idea behind frame semantics is that speakers are aware of possi- bly quite complex situation types, packages of connected expectations, that go by various names—frames, schemas, scenarios, scripts, cultural narratives, memes—and the words in our language are understood with such frames as their presupposed background.” (Fillmore, 2012, p. 712) BIBLIOGRAPHICAL AND HISTORICAL NOTES 479 The word frame seemed to be in the air for a suite of related notions proposed at about the same time by Minsky (1974), Hymes (1974), and Goffman (1974), as well as related notions with other names like scripts (Schank and Abelson, 1975) and schemata (Bobrow and Norman, 1975) (see Tannen (1979) for a comparison). Fillmore was also inﬂuenced by the semantic ﬁeld theorists and by a visit to the Yale AI lab where he took notice of the lists of slots and ﬁllers used by early information extraction systems like DeJong (1982) and Schank and Abelson (1977). In the 1990s Fillmore drew on these insights to begin the FrameNet corpus annotation project. At the same time, Beth Levin drew on her early case frame dictionaries (Levin, 1977) to develop her book which summarized sets of verb classes deﬁned by shared argument realizations (Levin, 1993). The VerbNet project built on this work (Kipper et al., 2000), leading soon afterwards to the PropBank semantic-role-labeled corpus created by Martha Palmer and colleagues (Palmer et al., 2005). The combination of rich linguistic annotation and corpus-based approach in- stantiated in FrameNet and PropBank led to a revival of automatic approaches to semantic role labeling, ﬁrst on FrameNet (Gildea and Jurafsky, 2000) and then on PropBank data (Gildea and Palmer, 2002, inter alia). The problem ﬁrst addressed in the 1970s by handwritten rules was thus now generally recast as one of supervised machine learning enabled by large and consistent databases. Many popular features used for role labeling are deﬁned in Gildea and Jurafsky (2002), Surdeanu et al. (2003), Xue and Palmer (2004), Pradhan et al. (2005), Che et al. (2009), and Zhao et al. (2009). The use of dependency rather than constituency parses was introduced in the CoNLL-2008 shared task (Surdeanu et al., 2008). For surveys see Palmer et al. (2010) and M`arquez et al. (2008). The use of neural approaches to semantic role labeling was pioneered by Col- lobert et al. (2011), who applied a CRF on top of a convolutional net. Early work like Foland, Jr. and Martin (2015) focused on using dependency features. Later work eschewed syntactic features altogether; Zhou and Xu (2015b) introduced the use of a stacked (6-8 layer) biLSTM architecture, and (He et al., 2017) showed how to augment the biLSTM architecture with highway networks and also replace the CRF with A* decoding that make it possible to apply a wide variety of global constraints in SRL decoding. Most semantic role labeling schemes only work within a single sentence, fo- cusing on the object of the verbal (or nominal, in the case of NomBank) predicate. However, in many cases, a verbal or nominal predicate may have an implicit argu- ment: one that appears only in a contextual sentence, or perhaps not at all and mustimplicit argument be inferred. In the two sentences This house has a new owner. The sale was ﬁnalized 10 days ago. the sale in the second sentence has no A RG1, but a reasonable reader would infer that the Arg1 should be thehouse mentioned in the prior sentence. Find- ing these arguments, implicit argument detection (sometimes shortened as iSRL)iSRL was introduced by Gerber and Chai (2010) and Ruppenhofer et al. (2010). See Do et al. (2017) for more recent neural models. To avoid the need for huge labeled training sets, unsupervised approaches for semantic role labeling attempt to induce the set of semantic roles by clustering over arguments. The task was pioneered by Riloff and Schmelzenbach (1998) and Swier and Stevenson (2004); see Grenager and Manning (2006), Titov and Klementiev (2012), Lang and Lapata (2014), Woodsend and Lapata (2015), and Titov and Khod- dam (2014). Recent innovations in frame labeling include connotation frames, which mark richer information about the argument of predicates. Connotation frames mark the 480 CHAPTER 21 • S EMANTIC ROLE LABELING sentiment of the writer or reader toward the arguments (for example using the verb survive in he survived a bombing expresses the writer’s sympathy toward the subject he and negative sentiment toward the bombing. See Chapter 22 for more details. Selectional preference has been widely studied beyond the selectional associa- tion models of Resnik (1993) and Resnik (1996). Methods have included clustering (Rooth et al., 1999), discriminative learning (Bergsma et al., 2008a), and topic mod- els (S´eaghdha 2010, Ritter et al. 2010b), and constraints can be expressed at the level of words or classes (Agirre and Martinez, 2001). Selectional preferences have also been successfully integrated into semantic role labeling (Erk 2007, Zapirain et al. 2013, Do et al. 2017). Exercises CHAPTER 22 Lexicons for Sentiment, Affect, and Connotation Some day we’ll be able to measure the power of words Maya Angelou In this chapter we turn to tools for interpretingaffective meaning, extending ouraffective study of sentiment analysis in Chapter 4. We use the word ‘affective’, following the tradition in affective computing (Picard, 1995) to mean emotion, sentiment, per- sonality, mood, and attitudes. Affective meaning is closely related to subjectivity,subjectivity the study of a speaker or writer’s evaluations, opinions, emotions, and speculations (Wiebe et al., 1999). How should affective meaning be deﬁned? One inﬂuential typology of affec- tive states comes from Scherer (2000), who deﬁnes each class of affective states by factors like its cognitive realization and time course (Fig. 22.1). Emotion: Relatively brief episode of response to the evaluation of an external or internal event as being of major signiﬁcance. (angry, sad, joyful, fearful, ashamed, proud, elated, desperate) Mood: Diffuse affect state, most pronounced as change in subjective feeling, of low intensity but relatively long duration, often without apparent cause. (cheerful, gloomy, irritable, listless, depressed, buoyant) Interpersonal stance: Affective stance taken toward another person in a spe- ciﬁc interaction, coloring the interpersonal exchange in that situation. (distant, cold, warm, supportive, contemptuous, friendly) Attitude: Relatively enduring, affectively colored beliefs, preferences, and pre- dispositions towards objects or persons. (liking, loving, hating, valuing, desiring) Personality traits: Emotionally laden, stable personality dispositions and be- havior tendencies, typical for a person. (nervous, anxious, reckless, morose, hostile, jealous) Figure 22.1 The Scherer typology of affective states (Scherer, 2000). We can design extractors for each of these kinds of affective states. Chapter 4 already introduced sentiment analysis, the task of extracting the positive or negative orientation that a writer expresses in a text. This corresponds in Scherer’s typology to the extraction of attitudes: ﬁguring out what people like or dislike, from affect- rich texts like consumer reviews of books or movies, newspaper editorials, or public sentiment in blogs or tweets. Detecting emotion and moods is useful for detecting whether a student is con- fused, engaged, or certain when interacting with a tutorial system, whether a caller to a help line is frustrated, whether someone’s blog posts or tweets indicated depres- sion. Detecting emotions like fear in novels, for example, could help us trace what groups or situations are feared and how that changes over time. 482 CHAPTER 22 • L EXICONS FOR SENTIMENT , AFFECT , AND CONNOTATION Detecting different interpersonal stances can be useful when extracting infor- mation from human-human conversations. The goal here is to detect stances like friendliness or awkwardness in interviews or friendly conversations, for example for summarizing meetings or ﬁnding parts of a conversation where people are especially excited or engaged, conversational hot spots that can help in meeting summariza- tion. Detecting the personality of a user—such as whether the user is an extrovert or the extent to which they are open to experience — can help improve conversa- tional agents, which seem to work better if they match users’ personality expecta- tions (Mairesse and Walker, 2008). And affect is important for generation as well as recognition; synthesizing affect is important for conversational agents in various domains, including literacy tutors such as children’s storybooks, or computer games. In Chapter 4 we introduced the use of naive Bayes classiﬁcation to classify a document’s sentiment. Various classiﬁers have been successfully applied to many of these tasks, using all the words in the training set as input to a classiﬁer which then determines the affect status of the text. In this chapter we focus on an alternative model, in which instead of using every word as a feature, we focus only on certain words, ones that carry particularly strong cues to affect or sentiment. We call these lists of words affective lexicons or senti- ment lexicons. These lexicons presuppose a fact about semantics: that words have affective meanings or connotations. The word connotation has different meaningsconnotations in different ﬁelds, but here we use it to mean the aspects of a word’s meaning that are related to a writer or reader’s emotions, sentiment, opinions, or evaluations. In addition to their ability to help determine the affective status of a text, connotation lexicons can be useful features for other kinds of affective tasks, and for computa- tional social science analysis. In the next sections we introduce basic theories of emotion, show how sentiment lexicons are a special case of emotion lexicons, and mention some useful lexicons. We then survey three ways for building lexicons: human labeling, semi-supervised, and supervised. Finally, we talk about how to detect affect toward a particular entity, and introduce connotation frames. 22.1 Deﬁning Emotion One of the most important affective classes isemotion, which Scherer (2000) deﬁnesemotion as a “relatively brief episode of response to the evaluation of an external or internal event as being of major signiﬁcance”. Detecting emotion has the potential to improve a number of language processing tasks. Emotion recognition could help dialogue systems like tutoring systems detect that a student was unhappy, bored, hesitant, conﬁdent, and so on. Automatically detecting emotions in reviews or customer responses (anger, dissatisfaction, trust) could help businesses recognize speciﬁc problem areas or ones that are going well. Emotion can play a role in medical NLP tasks like helping diagnose depression or suicidal intent. Detecting emotions expressed toward characters in novels might play a role in understanding how different social groups were viewed by society at different times. Computational models of emotion in NLP have mainly been based on two fami- lies of theories of emotion (out of the many studied in the ﬁeld of affective science). In one of these families, emotions are viewed as ﬁxed atomic units, limited in num- ber, and from which others are generated, often called basic emotions (Tomkinsbasic emotions 22.1 • D EFINING EMOTION 483 1962, Plutchik 1962), a model dating back to Darwin. Perhaps the most well-known of this family of theories are the 6 emotions proposed by Ekman (e.g., Ekman 1999) to be universally present in all cultures: surprise, happiness, anger, fear, disgust, sadness. Another atomic theory is the Plutchik (1980) wheel of emotion, consisting of 8 basic emotions in four opposing pairs: joy–sadness, anger–fear, trust–disgust, and anticipation–surprise, together with the emotions derived from them, shown in Fig. 22.2. Figure 22.2 Plutchik wheel of emotion. The second class of emotion theories widely used in NLP views emotion as a space in 2 or 3 dimensions (Russell, 1980). Most models include the two dimensions valence and arousal, and many add a third, dominance. These can be deﬁned as: valence: the pleasantness of the stimulus arousal: the level of alertness, activeness, or energy provoked by the stimulus dominance: the degree of control or dominance exerted by the stimulus or the emotion Sentiment can be viewed as a special case of this second view of emotions as points in space. In particular, thevalence dimension, measuring how pleasant or unpleasant a word is, is often used directly as a measure of sentiment. In these lexicon-based models of affect, the affective meaning of a word is gen- erally ﬁxed, irrespective of the linguistic context in which a word is used, or the dialect or culture of the speaker. By contrast, other models in affective science repre- sent emotions as much richer processes involving cognition (Barrett et al., 2007). In appraisal theory, for example, emotions are complex processes, in which a person considers how an event is congruent with their goals, taking into account variables like the agency, certainty, urgency, novelty and control associated with the event (Moors et al., 2013). Computational models in NLP taking into account these richer theories of emotion will likely play an important role in future work. 484 CHAPTER 22 • L EXICONS FOR SENTIMENT , AFFECT , AND CONNOTATION 22.2 Available Sentiment and Affect Lexicons A wide variety of affect lexicons have been created and released. The most basic lexicons label words along one dimension of semantic variability, generally called “sentiment” or “valence”. In the simplest lexicons this dimension is represented in a binary fashion, with a wordlist for positive words and a wordlist for negative words. The oldest is the General Inquirer (Stone et al., 1966), which drew on content analysis and on earlyGeneral Inquirer work in the cognitive psychology of word meaning (Osgood et al., 1957). The Gen- eral Inquirer has a lexicon of 1915 positive words and a lexicon of 2291 negative words (as well as other lexicons discussed below). The MPQA Subjectivity lexicon (Wilson et al., 2005) has 2718 positive and 4912 negative words drawn from prior lexicons plus a bootstrapped list of subjective words and phrases (Riloff and Wiebe, 2003). Each entry in the lexicon is hand-labeled for sentiment and also labeled for reliability (strongly subjective or weakly subjective). The polarity lexicon of Hu and Liu (2004b) gives 2006 positive and 4783 negative words, drawn from product reviews, labeled using a bootstrapping method from WordNet. Positive admire, amazing, assure, celebration, charm, eager, enthusiastic, excellent, fancy, fan- tastic, frolic, graceful, happy, joy, luck, majesty, mercy, nice, patience, perfect, proud, rejoice, relief, respect, satisfactorily, sensational, super, terriﬁc, thank, vivid, wise, won- derful, zest Negative abominable, anger, anxious, bad, catastrophe, cheap, complaint, condescending, deceit, defective, disappointment, embarrass, fake, fear, ﬁlthy, fool, guilt, hate, idiot, inﬂict, lazy, miserable, mourn, nervous, objection, pest, plot, reject, scream, silly, terrible, unfriendly, vile, wicked Figure 22.3 Some words with consistent sentiment across the General Inquirer (Stone et al., 1966), the MPQA Subjectivity lexicon (Wilson et al., 2005), and the polarity lexicon of Hu and Liu (2004b). Slightly more general than these sentiment lexicons are lexicons that assign each word a value on all three affective dimensions. The NRC Valence, Arousal, and Dominance (V AD) lexicon (Mohammad, 2018a) assigns valence, arousal, and dom- inance scores to 20,000 words. Some examples are shown in Fig. 22.4. Valence Arousal Dominance vacation .840 enraged .962 powerful .991 delightful .918 party .840 authority .935 whistle .653 organized .337 saxophone .482 consolation .408 effortless .120 discouraged .0090 torture .115 napping .046 weak .045 Figure 22.4 Values of sample words on the emotional dimensions of Mohammad (2018a). The NRC Word-Emotion Association Lexicon, also called EmoLex (Moham-EmoLex mad and Turney, 2013), uses the Plutchik (1980) 8 basic emotions deﬁned above. The lexicon includes around 14,000 words including words from prior lexicons as well as frequent nouns, verbs, adverbs and adjectives. Values from the lexicon for some sample words: 22.3 • C REATING AFFECT LEXICONS BY HUMAN LABELING 485 Word anger anticipation disgust fear joy sadness surprise trust positive negative reward 0 1 0 0 1 0 1 1 1 0 worry 0 1 0 1 0 1 0 0 0 1 tenderness 0 0 0 0 1 0 0 0 1 0 sweetheart 0 1 0 0 1 1 0 1 1 0 suddenly 0 0 0 0 0 0 1 0 0 0 thirst 0 1 0 0 0 1 1 0 0 0 garbage 0 0 1 0 0 0 0 0 0 1 For a smaller set of 5,814 words, the NRC Emotion/Affect Intensity Lexicon (Mohammad, 2018b) contains real-valued scores of association for anger, fear, joy, and sadness; Fig. 22.5 shows examples. Anger Fear Joy Sadness outraged 0.964 horror 0.923 superb 0.864 sad 0.844 violence 0.742 anguish 0.703 cheered 0.773 guilt 0.750 coup 0.578 pestilence 0.625 rainbow 0.531 unkind 0.547 oust 0.484 stressed 0.531 gesture 0.387 difﬁculties 0.421 suspicious 0.484 failing 0.531 warms 0.391 beggar 0.422 nurture 0.059 conﬁdent 0.094 hardship .031 sing 0.017 Figure 22.5 Sample emotional intensities for words for anger, fear, joy, and sadness from Mohammad (2018b). LIWC, Linguistic Inquiry and Word Count , is a widely used set of 73 lex-LIWC icons containing over 2300 words (Pennebaker et al., 2007), designed to capture aspects of lexical meaning relevant for social psychological tasks. In addition to sentiment-related lexicons like ones for negative emotion ( bad, weird, hate, prob- lem, tough ) and positive emotion ( love, nice, sweet ), LIWC includes lexicons for categories like anger, sadness, cognitive mechanisms, perception, tentative, and in- hibition, shown in Fig. 22.6. There are various other hand-built affective lexicons. The General Inquirer in- cludes additional lexicons for dimensions like strong vs. weak, active vs. passive, overstated vs. understated, as well as lexicons for categories like pleasure, pain, virtue, vice, motivation, and cognitive orientation. Another useful feature for various tasks is the distinction between concreteconcrete words like banana or bathrobe and abstract words like belief and although. Theabstract lexicon in Brysbaert et al. (2014) used crowdsourcing to assign a rating from 1 to 5 of the concreteness of 40,000 words, thus assigning banana, bathrobe, and bagel 5, belief 1.19, although 1.07, and in between words like brisk a 2.5. 22.3 Creating Affect Lexicons by Human Labeling The earliest method used to build affect lexicons, and still in common use, is to have humans label each word. This is now most commonly done via crowdsourcing:crowdsourcing breaking the task into small pieces and distributing them to a large number of anno- 486 CHAPTER 22 • L EXICONS FOR SENTIMENT , AFFECT , AND CONNOTATION Positive Negative Emotion Emotion Insight Inhibition Family Negate appreciat* anger* aware* avoid* brother* aren’t comfort* bore* believe careful* cousin* cannot great cry decid* hesitat* daughter* didn’t happy despair* feel limit* family neither interest fail* ﬁgur* oppos* father* never joy* fear know prevent* grandf* no perfect* griev* knew reluctan* grandm* nobod* please* hate* means safe* husband none safe* panic* notice* stop mom nor terriﬁc suffers recogni* stubborn* mother nothing value terrify sense wait niece* nowhere wow* violent* think wary wife without Figure 22.6 Samples from 5 of the 73 lexical categories in LIWC (Pennebaker et al., 2007). The * means the previous letters are a word preﬁx and all words with that preﬁx are included in the category. tators. Let’s take a look at some of the methodological choices for two crowdsourced emotion lexicons. The NRC Emotion Lexicon (EmoLex) (Mohammad and Turney, 2013), labeled emotions in two steps. To ensure that the annotators were judging the correct sense of the word, they ﬁrst answered a multiple-choice synonym question that primed the correct sense of the word (without requiring the annotator to read a potentially confusing sense deﬁnition). These were created automatically using the headwords associated with the thesaurus category of the sense in question in the Macquarie dictionary and the headwords of 3 random distractor categories. An example: Which word is closest in meaning (most related) to startle? • automobile • shake • honesty • entertain For each word (e.g. startle), the annotator was then asked to rate how associated that word is with each of the 8 emotions ( joy, fear, anger, etc.). The associations were rated on a scale of not, weakly, moderately, and strongly associated. Outlier ratings were removed, and then each term was assigned the class chosen by the ma- jority of the annotators, with ties broken by choosing the stronger intensity, and then the 4 levels were mapped into a binary label for each word (no and weak mapped to 0, moderate and strong mapped to 1). The NRC V AD Lexicon (Mohammad, 2018a) was built by selecting words and emoticons from prior lexicons and annotating them with crowd-sourcing usingbest- worst scaling (Louviere et al. 2015, Kiritchenko and Mohammad 2017). In best-best-worst scaling worst scaling, annotators are given N items (usually 4) and are asked which item is the best (highest) and which is the worst (lowest) in terms of some property. The set of words used to describe the ends of the scales are taken from prior literature. For valence, for example, the raters were asked: Q1. Which of the four words below is associated with the MOST happi- ness / pleasure / positiveness / satisfaction / contentedness / hopefulness OR LEAST unhappiness / annoyance / negativeness / dissatisfaction / 22.4 • S EMI -SUPERVISED INDUCTION OF AFFECT LEXICONS 487 melancholy / despair? (Four words listed as options.) Q2. Which of the four words below is associated with the LEAST hap- piness / pleasure / positiveness / satisfaction / contentedness / hopeful- ness OR MOST unhappiness / annoyance / negativeness / dissatisfaction / melancholy / despair? (Four words listed as options.) The score for each word in the lexicon is the proportion of times the item was chosen as the best (highest V/A/D) minus the proportion of times the item was chosen as the worst (lowest V/A/D). The agreement between annotations are evaluated by split- half reliability: split the corpus in half and compute the correlations between thesplit-half reliability annotations in the two halves. 22.4 Semi-supervised Induction of Affect Lexicons Another common way to learn sentiment lexicons is to start from a set of seed words that deﬁne two poles of a semantic axis (words likegood or bad), and then ﬁnd ways to label each word w by its similarity to the two seed sets. Here we summarize two families of seed-based semi-supervised lexicon induction algorithms, axis-based and graph-based. 22.4.1 Semantic Axis Methods One of the most well-known lexicon induction methods, the Turney and Littman (2003) algorithm, is given seed words likegood or bad, and then for each word w to be labeled, measures both how similar it is to good and how different it is from bad. Here we describe a slight extension of the algorithm due to An et al. (2018), which is based on computing a semantic axis. In the ﬁrst step, we choose seed words by hand. There are two methods for dealing with the fact that the affect of a word is different in different contexts: (1) start with a single large seed lexicon and rely on the induction algorithm to ﬁne-tune it to the domain, or (2) choose different seed words for different genres. Hellrich et al. (2019) suggests that for modeling affect across different historical time periods, starting with a large modern affect dictionary is better than small seedsets tuned to be stable across time. As an example of the second approach, Hamilton et al. (2016a) deﬁne one set of seed words for general sentiment analysis, a different set for Twitter, and yet another set for sentiment in ﬁnancial text: Domain Positive seeds Negative seeds General good, lovely, excellent, fortunate, pleas- ant, delightful, perfect, loved, love, happy bad, horrible, poor, unfortunate, un- pleasant, disgusting, evil, hated, hate, unhappy Twitter love, loved, loves, awesome, nice, amazing, best, fantastic, correct, happy hate, hated, hates, terrible, nasty, awful, worst, horrible, wrong, sad Finance successful, excellent, proﬁt, beneﬁcial, improving, improved, success, gains, positive negligent, loss, volatile, wrong, losses, damages, bad, litigation, failure, down, negative In the second step, we compute embeddings for each of the pole words. These embeddings can be off-the-shelf word2vec embeddings, or can be computed directly 488 CHAPTER 22 • L EXICONS FOR SENTIMENT , AFFECT , AND CONNOTATION on a speciﬁc corpus (for example using a ﬁnancial corpus if a ﬁnance lexicon is the goal), or we can ﬁne-tune off-the-shelf embeddings to a corpus. Fine-tuning is espe- cially important if we have a very speciﬁc genre of text but don’t have enough data to train good embeddings. In ﬁne-tuning, we begin with off-the-shelf embeddings like word2vec, and continue training them on the small target corpus. Once we have embeddings for each pole word, we create an embedding that represents each pole by taking the centroid of the embeddings of each of the seed words; recall that the centroid is the multidimensional version of the mean. Given a set of embeddings for the positive seed words S+ = {E(w+ 1 ),E(w+ 2 ),...,E(w+ n )}, and embeddings for the negative seed words S−= {E(w− 1 ),E(w− 2 ),...,E(w− m)}, the pole centroids are: V+ = 1 n n∑ 1 E(w+ i ) V−= 1 m m∑ 1 E(w− i ) (22.1) The semantic axis deﬁned by the poles is computed just by subtracting the two vec- tors: Vaxis = V+ −V− (22.2) Vaxis, the semantic axis, is a vector in the direction of positive sentiment. Finally, we compute (via cosine similarity) the angle between the vector in the direction of positive sentiment and the direction of w’s embedding. A higher cosine means that w is more aligned with S+ than S−. score(w) = cos ( E(w),Vaxis ) = E(w)·Vaxis ∥E(w)∥∥Vaxis∥ (22.3) If a dictionary of words with sentiment scores is sufﬁcient, we’re done! Or if we need to group words into a positive and a negative lexicon, we can use a threshold or other method to give us discrete lexicons. 22.4.2 Label Propagation An alternative family of methods deﬁnes lexicons by propagating sentiment labels on graphs, an idea suggested in early work by Hatzivassiloglou and McKeown (1997). We’ll describe the simple SentProp (Sentiment Propagation) algorithm of Hamilton et al. (2016a), which has four steps: 1. Deﬁne a graph: Given word embeddings, build a weighted lexical graph by connecting each word with its k nearest neighbors (according to cosine simi- larity). The weights of the edge between words wi and wj are set as: Ei,j = arccos ( − wi⊤wj ∥wi∥∥wj∥ ) . (22.4) 2. Deﬁne a seed set: Choose positive and negative seed words. 3. Propagate polarities from the seed set: Now we perform a random walk on this graph, starting at the seed set. In a random walk, we start at a node and 22.4 • S EMI -SUPERVISED INDUCTION OF AFFECT LEXICONS 489 then choose a node to move to with probability proportional to the edge prob- ability. A word’s polarity score for a seed set is proportional to the probability of a random walk from the seed set landing on that word (Fig. 22.7). 4. Create word scores : We walk from both positive and negative seed sets, resulting in positive (rawscore+(wi)) and negative (rawscore−(wi)) raw label scores. We then combine these values into a positive-polarity score as: score+(wi) = rawscore+(wi) rawscore+(wi)+ rawscore−(wi) (22.5) It’s often helpful to standardize the scores to have zero mean and unit variance within a corpus. 5. Assign conﬁdence to each score: Because sentiment scores are inﬂuenced by the seed set, we’d like to know how much the score of a word would change if a different seed set is used. We can use bootstrap sampling to get conﬁdence regions, by computing the propagation B times over random subsets of the positive and negative seed sets (for example using B = 50 and choosing 7 of the 10 seed words each time). The standard deviation of the bootstrap sampled polarity scores gives a conﬁdence measure. idolize love adore appreciate like ﬁnd dislike see notice disapprove abhor hate loathe despise uncover idolize love adore appreciate like ﬁnd dislike see notice disapprove abhor hate loathe despise uncover (a) (b) Figure 22.7 Intuition of the S ENT PROP algorithm. (a) Run random walks from the seed words. (b) Assign polarity scores (shown here as colors green or red) based on the frequency of random walk visits. 22.4.3 Other Methods The core of semisupervised algorithms is the metric for measuring similarity with the seed words. The Turney and Littman (2003) and Hamilton et al. (2016a) ap- proaches above used embedding cosine as the distance metric: words were labeled as positive basically if their embeddings had high cosines with positive seeds and low cosines with negative seeds. Other methods have chosen other kinds of distance metrics besides embedding cosine. For example the Hatzivassiloglou and McKeown (1997) algorithm uses syntactic cues; two adjectives are considered similar if they were frequently conjoined byand and rarely conjoined by but. This is based on the intuition that adjectives conjoined by the words and tend to have the same polarity; positive adjectives are generally coordinated with positive, negative with negative: fair and legitimate, corrupt and brutal but less often positive adjectives coordinated with negative: fair and brutal, corrupt and legitimate By contrast, adjectives conjoined by but are likely to be of opposite polarity: 490 CHAPTER 22 • L EXICONS FOR SENTIMENT , AFFECT , AND CONNOTATION fair but brutal Another cue to opposite polarity comes from morphological negation ( un-, im-, -less). Adjectives with the same root but differing in a morphological negative ( ad- equate/inadequate, thoughtful/thoughtless) tend to be of opposite polarity. Yet another method for ﬁnding words that have a similar polarity to seed words is to make use of a thesaurus like WordNet (Kim and Hovy 2004, Hu and Liu 2004b). A word’s synonyms presumably share its polarity while a word’s antonyms probably have the opposite polarity. After a seed lexicon is built, each lexicon is updated as follows, possibly iterated. Lex+: Add synonyms of positive words ( well) and antonyms (like ﬁne) of negative words Lex−: Add synonyms of negative words ( awful) and antonyms (like evil) of positive words An extension of this algorithm assigns polarity to WordNet senses, called Senti- WordNet (Baccianella et al., 2010). Fig. 22.8 shows some examples.SentiWordNet Synset Pos Neg Obj good#6 ‘agreeable or pleasing’ 1 0 0 respectable#2 honorable#4 good#4 estimable#2 ‘deserving of esteem’ 0.75 0 0.25 estimable#3 computable#1 ‘may be computed or estimated’ 0 0 1 sting#1 burn#4 bite#2 ‘cause a sharp or stinging pain’ 0 0.875 .125 acute#6 ‘of critical importance and consequence’ 0.625 0.125 .250 acute#4 ‘of an angle; less than 90 degrees’ 0 0 1 acute#1 ‘having or experiencing a rapid onset and short but severe course’ 0 0.5 0.5 Figure 22.8 Examples from SentiWordNet 3.0 (Baccianella et al., 2010). Note the differences between senses of homonymous words: estimable#3 is purely objective, while estimable#2 is positive; acute can be positive (acute#6), negative (acute#1), or neutral (acute #4). In this algorithm, polarity is assigned to entire synsets rather than words. A positive lexicon is built from all the synsets associated with 7 positive words, and a negative lexicon from synsets associated with 7 negative words. A classiﬁer is then trained from this data to take a WordNet gloss and decide if the sense being deﬁned is positive, negative or neutral. A further step (involving a random-walk algorithm) assigns a score to each WordNet synset for its degree of positivity, negativity, and neutrality. In summary, semisupervised algorithms use a human-deﬁned set of seed words for the two poles of a dimension, and use similarity metrics like embedding cosine, coordination, morphology, or thesaurus structure to score words by how similar they are to the positive seeds and how dissimilar to the negative seeds. 22.5 Supervised Learning of Word Sentiment Semi-supervised methods require only minimal human supervision (in the form of seed sets). But sometimes a supervision signal exists in the world and can be made use of. One such signal is the scores associated with online reviews. The web contains an enormous number of online reviews for restaurants, movies, books, or other products, each of which have the text of the review along with an 22.5 • S UPERVISED LEARNING OF WORD SENTIMENT 491 Movie review excerpts (IMDb) 10 A great movie. This ﬁlm is just a wonderful experience. It’s surreal, zany, witty and slapstick all at the same time. And terriﬁc performances too. 1 This was probably the worst movie I have ever seen. The story went nowhere even though they could have done some interesting stuff with it. Restaurant review excerpts (Yelp) 5 The service was impeccable. The food was cooked and seasoned perfectly... The watermelon was perfectly square ... The grilled octopus was ... mouthwatering... 2 ...it took a while to get our waters, we got our entree before our starter, and we never received silverware or napkins until we requested them... Book review excerpts (GoodReads) 1 I am going to try and stop being deceived by eye-catching titles. I so wanted to like this book and was so disappointed by it. 5 This book is hilarious. I would recommend it to anyone looking for a satirical read with a romantic twist and a narrator that keeps butting in Product review excerpts (Amazon) 5 The lid on this blender though is probably what I like the best about it... enables you to pour into something without even taking the lid off! ... the perfect pitcher! ... works fantastic. 1 I hate this blender... It is nearly impossible to get frozen fruit and ice to turn into a smoothie... You have to add a TON of liquid. I also wish it had a spout ... Figure 22.9 Excerpts from some reviews from various review websites, all on a scale of 1 to 5 stars except IMDb, which is on a scale of 1 to 10 stars. associated review score: a value that may range from 1 star to 5 stars, or scoring 1 to 10. Fig. 22.9 shows samples extracted from restaurant, book, and movie reviews. We can use this review score as supervision: positive words are more likely to appear in 5-star reviews; negative words in 1-star reviews. And instead of just a binary polarity, this kind of supervision allows us to assign a word a more complex representation of its polarity: its distribution over stars (or other scores). Thus in a ten-star system we could represent the sentiment of each word as a 10-tuple, each number a score representing the word’s association with that polarity level. This association can be a raw count, or a likelihood P(w\|c), or some other function of the count, for each class c from 1 to 10. For example, we could compute the IMDb likelihood of a word like disap- point(ed/ing) occurring in a 1 star review by dividing the number of times disap- point(ed/ing) occurs in 1-star reviews in the IMDb dataset (8,557) by the total num- ber of words occurring in 1-star reviews (25,395,214), so the IMDb estimate of P(disappointing\|1) is .0003. A slight modiﬁcation of this weighting, the normalized likelihood, can be used as an illuminating visualization (Potts, 2011)1 P(w\|c) = count(w,c)∑ w∈C count(w,c) PottsScore(w) = P(w\|c)∑ c P(w\|c) (22.6) Dividing the IMDb estimate P(disappointing\|1) of .0003 by the sum of the likeli- hood P(w\|c) over all categories gives a Potts score of 0.10. The word disappointing thus is associated with the vector [.10, .12, .14, .14, .13, .11, .08, .06, .06, .05]. The 1 Each element of the Potts score of a word w and category c can be shown to be a variant of the pointwise mutual information pmi(w,c) without the log term; see Exercise 22.1. 492 CHAPTER 22 • L EXICONS FOR SENTIMENT , AFFECT , AND CONNOTATION Potts diagram (Potts, 2011) is a visualization of these word scores, representing thePotts diagram prior sentiment of a word as a distribution over the rating categories. Fig. 22.10 shows the Potts diagrams for 3 positive and 3 negative scalar adjec- tives. Note that the curve for strongly positive scalars have the shape of the letter J, while strongly negative scalars look like a reverse J. By contrast, weakly posi- tive and negative scalars have a hump-shape, with the maximum either below the mean (weakly negative words like disappointing) or above the mean (weakly pos- itive words like good). These shapes offer an illuminating typology of affective meaning. Overview Data Methods Categorization Scale induction Looking ahead Example: attenuators IMDB – 53,775 tokens Category -0.50 -0.39 -0.28 -0.17 -0.06 0.06 0.17 0.28 0.39 0.50 0.05 0.09 0.15 Cat = 0.33 (p = 0.004) Cat^2 = -4.02 (p < 0.001) OpenTable – 3,890 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.08 0.38 Cat = 0.11 (p = 0.707) Cat^2 = -6.2 (p = 0.014) Goodreads – 3,424 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.08 0.19 0.36 Cat = -0.55 (p = 0.128) Cat^2 = -5.04 (p = 0.016) Amazon/Tripadvisor – 2,060 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.12 0.28 Cat = 0.42 (p = 0.207) Cat^2 = -2.74 (p = 0.05) somewhat/r IMDB – 33,515 tokens Category -0.50 -0.39 -0.28 -0.17 -0.06 0.06 0.17 0.28 0.39 0.50 0.04 0.09 0.17 Cat = -0.13 (p = 0.284) Cat^2 = -5.37 (p < 0.001) OpenTable – 2,829 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.08 0.31 Cat = 0.2 (p = 0.265) Cat^2 = -4.16 (p = 0.007) Goodreads – 1,806 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.05 0.12 0.18 0.35 Cat = -0.87 (p = 0.016) Cat^2 = -5.74 (p = 0.004) Amazon/Tripadvisor – 2,158 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.11 0.29 Cat = 0.54 (p = 0.183) Cat^2 = -3.32 (p = 0.045) fairly/r IMDB – 176,264 tokens Category -0.50 -0.39 -0.28 -0.17 -0.06 0.06 0.17 0.28 0.39 0.50 0.05 0.09 0.13 Cat = -0.43 (p < 0.001) Cat^2 = -3.6 (p < 0.001) OpenTable – 8,982 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.08 0.14 0.19 0.32 Cat = -0.64 (p = 0.035) Cat^2 = -4.47 (p = 0.007) Goodreads – 11,895 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.07 0.15 0.34 Cat = -0.71 (p = 0.072) Cat^2 = -4.59 (p = 0.018) Amazon/Tripadvisor – 5,980 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.15 0.28 Cat = 0.26 (p = 0.496) Cat^2 = -2.23 (p = 0.131) pretty/r “Potts&diagrams” Potts,&Christopher .& 2011.&NSF&workshop&on& restructuring& adjectives. good great excellent disappointing bad terrible totally absolutely utterly somewhat fairly pretty Positive scalars Negative scalars Emphatics Attenuators 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating Figure 22.10 Potts diagrams (Potts, 2011) for positive and negative scalar adjectives, show- ing the J-shape and reverse J-shape for strongly positive and negative adjectives, and the hump-shape for more weakly polarized adjectives. Fig. 22.11 shows the Potts diagrams for emphasizing and attenuating adverbs. Note that emphatics tend to have a J-shape (most likely to occur in the most posi- tive reviews) or a U-shape (most likely to occur in the strongly positive and nega- tive). Attenuators all have the hump-shape, emphasizing the middle of the scale and downplaying both extremes. The diagrams can be used both as a typology of lexical sentiment, and also play a role in modeling sentiment compositionality. In addition to functions like posterior P(c\|w), likelihood P(w\|c), or normalized likelihood (Eq. 22.6) many other functions of the count of a word occurring with a sentiment label have been used. We’ll introduce some of these on page 496, includ- ing ideas like normalizing the counts per writer in Eq. 22.14. 22.5.1 Log Odds Ratio Informative Dirichlet Prior One thing we often want to do with word polarity is to distinguish between words that are more likely to be used in one category of texts than in another. We may, for example, want to know the words most associated with 1 star reviews versus those associated with 5 star reviews. These differences may not be just related to senti- ment. We might want to ﬁnd words used more often by Democratic than Republican members of Congress, or words used more often in menus of expensive restaurants 22.5 • S UPERVISED LEARNING OF WORD SENTIMENT 493 Overview Data Methods Categorization Scale induction Looking ahead Example: attenuators IMDB – 53,775 tokens Category -0.50 -0.39 -0.28 -0.17 -0.06 0.06 0.17 0.28 0.39 0.50 0.05 0.09 0.15 Cat = 0.33 (p = 0.004) Cat^2 = -4.02 (p < 0.001) OpenTable – 3,890 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.08 0.38 Cat = 0.11 (p = 0.707) Cat^2 = -6.2 (p = 0.014) Goodreads – 3,424 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.08 0.19 0.36 Cat = -0.55 (p = 0.128) Cat^2 = -5.04 (p = 0.016) Amazon/Tripadvisor – 2,060 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.12 0.28 Cat = 0.42 (p = 0.207) Cat^2 = -2.74 (p = 0.05) somewhat/r IMDB – 33,515 tokens Category -0.50 -0.39 -0.28 -0.17 -0.06 0.06 0.17 0.28 0.39 0.50 0.04 0.09 0.17 Cat = -0.13 (p = 0.284) Cat^2 = -5.37 (p < 0.001) OpenTable – 2,829 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.08 0.31 Cat = 0.2 (p = 0.265) Cat^2 = -4.16 (p = 0.007) Goodreads – 1,806 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.05 0.12 0.18 0.35 Cat = -0.87 (p = 0.016) Cat^2 = -5.74 (p = 0.004) Amazon/Tripadvisor – 2,158 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.11 0.29 Cat = 0.54 (p = 0.183) Cat^2 = -3.32 (p = 0.045) fairly/r IMDB – 176,264 tokens Category -0.50 -0.39 -0.28 -0.17 -0.06 0.06 0.17 0.28 0.39 0.50 0.05 0.09 0.13 Cat = -0.43 (p < 0.001) Cat^2 = -3.6 (p < 0.001) OpenTable – 8,982 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.08 0.14 0.19 0.32 Cat = -0.64 (p = 0.035) Cat^2 = -4.47 (p = 0.007) Goodreads – 11,895 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.07 0.15 0.34 Cat = -0.71 (p = 0.072) Cat^2 = -4.59 (p = 0.018) Amazon/Tripadvisor – 5,980 tokens Category -0.50 -0.25 0.00 0.25 0.50 0.15 0.28 Cat = 0.26 (p = 0.496) Cat^2 = -2.23 (p = 0.131) pretty/r “Potts&diagrams” Potts,&Christopher .& 2011.&NSF&workshop&on& restructuring& adjectives. good great excellent disappointing bad terrible totally absolutely utterly somewhat fairly pretty Positive scalars Negative scalars Emphatics Attenuators 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating 1 2 3 4 5 6 7 8 9 10 rating Figure 22.11 Potts diagrams (Potts, 2011) for emphatic and attenuating adverbs. than cheap restaurants. Given two classes of documents, to ﬁnd words more associated with one cate- gory than another, we could measure the difference in frequencies (is a wordw more frequent in class A or class B?). Or instead of the difference in frequencies we could compute the ratio of frequencies, or compute the log odds ratio (the log of the ratio between the odds of the two words). We could then sort words by whichever associ- ation measure we pick, ranging from words overrepresented in category A to words overrepresented in category B. The problem with simple log-likelihood or log odds methods is that they overem- phasize differences in very rare words, and often also in very frequent words. Very rare words will seem to occur very differently in the two corpora since with tiny counts there may be statistical ﬂuctations, or even zero occurrences in one corpus compared to non-zero occurrences in the other. Very frequent words will also seem different since all counts are large. In this section we walk through the details of one solution to this problem: the “log odds ratio informative Dirichlet prior” method of Monroe et al. (2008) that is a particularly useful method for ﬁnding words that are statistically overrepresented in one particular category of texts compared to another. It’s based on the idea of using another large corpus to get a prior estimate of what we expect the frequency of each word to be. Let’s start with the goal: assume we want to know whether the word horrible occurs more in corpus i or corpus j. We could compute the log likelihood ratio ,log likelihood ratio using f i(w) to mean the frequency of word w in corpus i, and ni to mean the total number of words in corpus i: llr(horrible) = log Pi(horrible) Pj(horrible) = logPi(horrible)−logPj(horrible) = log fi(horrible) ni −log fj(horrible) nj (22.7) Instead, let’s compute the log odds ratio: does horrible have higher odds in i or inlog odds ratio 494 CHAPTER 22 • L EXICONS FOR SENTIMENT , AFFECT , AND CONNOTATION j: lor(horrible) = log ( Pi(horrible) 1 −Pi(horrible) ) −log ( Pj(horrible) 1 −Pj(horrible) ) = log   fi(horrible) ni 1 −fi(horrible) ni  −log   fj(horrible) nj 1 −fj(horrible) nj   = log ( fi(horrible) ni −fi(horrible) ) −log ( fj(horrible) nj −fj(horrible) ) (22.8) The Dirichlet intuition is to use a large background corpus to get a prior estimate of what we expect the frequency of each word w to be. We’ll do this very simply by adding the counts from that corpus to the numerator and denominator, so that we’re essentially shrinking the counts toward that prior. It’s like asking how large are the differences between i and j given what we would expect given their frequencies in a well-estimated large background corpus. The method estimates the difference between the frequency of word w in two corpora i and j via the prior-modiﬁed log odds ratio forw, δ(i−j) w , which is estimated as: δ(i−j) w = log ( f i w +αw ni +α0 −( f iw +αw) ) −log ( f j w +αw nj +α0 −( f j w +αw) ) (22.9) (where ni is the size of corpus i, nj is the size of corpus j, f i w is the count of word w in corpus i, f j w is the count of word w in corpus j, α0 is the scaled size of the background corpus, and αw is the scaled count of wordw in the background corpus.) In addition, Monroe et al. (2008) make use of an estimate for the variance of the log–odds–ratio: σ2 ( ˆδ(i−j) w ) ≈ 1 f iw +αw + 1 f j w +αw (22.10) The ﬁnal statistic for a word is then the z–score of its log–odds–ratio: ˆδ(i−j) w√ σ2 ( ˆδ(i−j) w ) (22.11) The Monroe et al. (2008) method thus modiﬁes the commonly used log odds ratio in two ways: it uses the z-scores of the log odds ratio, which controls for the amount of variance in a word’s frequency, and it uses counts from a background corpus to provide a prior count for words. Fig. 22.12 shows the method applied to a dataset of restaurant reviews from Yelp, comparing the words used in 1-star reviews to the words used in 5-star reviews (Jurafsky et al., 2014). The largest difference is in obvious sentiment words, with the 1-star reviews using negative sentiment words like worse, bad, awful and the 5-star reviews using positive sentiment words likegreat, best, amazing. But there are other illuminating differences. 1-star reviews use logical negation ( no, not), while 5-star reviews use emphatics and emphasize universality ( very, highly, every, always). 1- star reviews use ﬁrst person plurals (we, us, our) while 5 star reviews use the second person. 1-star reviews talk about people ( manager, waiter, customer) while 5-star reviews talk about dessert and properties of expensive restaurants like courses and atmosphere. See Jurafsky et al. (2014) for more details. 22.6 • U SING LEXICONS FOR SENTIMENT RECOGNITION 495 Class Words in 1-star reviews Class Words in 5-star reviews Negative worst, rude, terrible, horrible, bad, awful, disgusting, bland, tasteless, gross, mediocre, overpriced, worse, poor Positive great, best, love(d), delicious, amazing, favorite, perfect, excellent, awesome, friendly, fantastic, fresh, wonderful, in- credible, sweet, yum(my) Negation no, not Emphatics/ universals very, highly, perfectly, deﬁnitely, abso- lutely, everything, every, always 1Pl pro we, us, our 2 pro you 3 pro she, he, her, him Articles a, the Past verb was, were, asked, told, said, did, charged, waited, left, took Advice try, recommend Sequencers after, then Conjunct also, as, well, with, and Nouns manager, waitress, waiter, customer, customers, attitude, waste, poisoning, money, bill, minutes Nouns atmosphere, dessert, chocolate, wine, course, menu Irrealis modals would, should Auxiliaries is/’s, can, ’ve, are Comp to, that Prep, other in, of, die, city, mouth Figure 22.12 The top 50 words associated with one–star and ﬁve-star restaurant reviews in a Yelp dataset of 900,000 reviews, using the Monroe et al. (2008) method (Jurafsky et al., 2014). 22.6 Using Lexicons for Sentiment Recognition In Chapter 4 we introduced the naive Bayes algorithm for sentiment analysis. The lexicons we have focused on throughout the chapter so far can be used in a number of ways to improve sentiment detection. In the simplest case, lexicons can be used when we don’t have sufﬁcient training data to build a supervised sentiment analyzer; it can often be expensive to have a human assign sentiment to each document to train the supervised classiﬁer. In such situations, lexicons can be used in a rule-based algorithm for classiﬁca- tion. The simplest version is just to use the ratio of positive to negative words: if a document has more positive than negative words (using the lexicon to decide the po- larity of each word in the document), it is classiﬁed as positive. Often a threshold λ is used, in which a document is classiﬁed as positive only if the ratio is greater than λ. If the sentiment lexicon includes positive and negative weights for each word, θ+ w and θ− w , these can be used as well. Here’s a simple such sentiment algorithm: f + = ∑ w s.t. w∈positivelexicon θ+ w count(w) f − = ∑ w s.t. w∈negativelexicon θ− w count(w) sentiment =    + if f + f − >λ − if f − f + >λ 0 otherwise. (22.12) If supervised training data is available, these counts computed from sentiment lex- icons, sometimes weighted or normalized in various ways, can also be used as fea- tures in a classiﬁer along with other lexical or non-lexical features. We return to such algorithms in Section 22.7. 496 CHAPTER 22 • L EXICONS FOR SENTIMENT , AFFECT , AND CONNOTATION 22.7 Using Lexicons for Affect Recognition Detection of emotion (and the other kinds of affective meaning described by Scherer (2000)) can be done by generalizing the algorithms described above for detecting sentiment. The most common algorithms involve supervised classiﬁcation: a training set is labeled for the affective meaning to be detected, and a classiﬁer is built using features extracted from the training set. As with sentiment analysis, if the training set is large enough, and the test set is sufﬁciently similar to the training set, simply using all the words or all the bigrams as features in a powerful classiﬁer like SVM or logistic regression, as described in Fig. 4.2 in Chapter 4, is an excellent algorithm whose performance is hard to beat. Thus we can treat affective meaning classiﬁcation of a text sample as simple document classiﬁcation. Some modiﬁcations are nonetheless often necessary for very large datasets. For example, the Schwartz et al. (2013) study of personality, gender, and age using 700 million words of Facebook posts used only a subset of the n-grams of lengths 1- 3. Only words and phrases used by at least 1% of the subjects were included as features, and 2-grams and 3-grams were only kept if they had sufﬁciently high PMI (PMI greater than 2 ∗length, where length is the number of words): pmi(phrase) =log p(phrase)∏ w∈phrase p(w) (22.13) Various weights can be used for the features, including the raw count in the training set, or some normalized probability or log probability. Schwartz et al. (2013), for example, turn feature counts into phrase likelihoods by normalizing them by each subject’s total word use. p(phrase\|subject) = freq(phrase,subject)∑ phrase′∈vocab(subject) freq(phrase′,subject) (22.14) If the training data is sparser, or not as similar to the test set, any of the lexicons we’ve discussed can play a helpful role, either alone or in combination with all the words and n-grams. Many possible values can be used for lexicon features. The simplest is just an indicator function, in which the value of a feature fL takes the value 1 if a particular text has any word from the relevant lexicon L. Using the notation of Chapter 4, in which a feature value is deﬁned for a particular output class c and document x. fL(c,x) = {1 if ∃w : w ∈L & w ∈x & class = c 0 otherwise Alternatively the value of a feature fL for a particular lexicon L can be the total number of word tokens in the document that occur in L: fL = ∑ w∈L count(w) For lexica in which each word is associated with a score or weight, the count can be multiplied by a weight θL w: fL = ∑ w∈L θL wcount(w) 22.8 • L EXICON -BASED METHODS FOR ENTITY -CENTRIC AFFECT 497 Counts can alternatively be logged or normalized per writer as in Eq. 22.14. However they are deﬁned, these lexicon features are then used in a supervised classiﬁer to predict the desired affective category for the text or document. Once a classiﬁer is trained, we can examine which lexicon features are associated with which classes. For a classiﬁer like logistic regression the feature weight gives an indication of how associated the feature is with the class. 22.8 Lexicon-based methods for Entity-Centric Affect What if we want to get an affect score not for an entire document, but for a particular entity in the text? The entity-centric method of Field and Tsvetkov (2019) combines affect lexicons with contextual embeddings to assign an affect score to an entity in text. In the context of affect about people, they relabel the Valence/Arousal/Domi- nance dimension as Sentiment/Agency/Power. The algorithm ﬁrst trains classiﬁers to map embeddings to scores: 1. For each word w in the training corpus: (a) Use off-the-shelf pretrained encoders (like BERT) to extract a contextual embedding e for each instance of the word. No additional ﬁne-tuning is done. (b) Average over the e embeddings of each instance of w to obtain a single embedding vector for one training point w. (c) Use the NRC V AD Lexicon to get S, A, and P scores forw. 2. Train (three) regression models on all words w to predict V , A, D scores from a word’s average embedding. Now given an entity mention m in a text, we assign affect scores as follows: 1. Use the same pretrained LM to get contextual embeddings for m in context. 2. Feed this embedding through the 3 regression models to get S, A, P scores for the entity. This results in a (S,A,P) tuple for a given entity mention; To get scores for the rep- resentation of an entity in a complete document, we can run coreference resolution and average the (S,A,P) scores for all the mentions. Fig. 22.13 shows the scores from their algorithm for characters from the movie The Dark Knight when run on Wikipedia plot summary texts with gold coreference. 22.9 Connotation Frames The lexicons we’ve described so far deﬁne a word as a point in affective space. A connotation frame, by contrast, is a lexicon that incorporates a richer kind of gram-connotation frame matical structure, by combining affective lexicons with the frame semantic lexicons of Chapter 21. The basic insight of connotation frame lexicons is that a predicate like a verb expresses connotations about the verb’s arguments (Rashkin et al. 2016, Rashkin et al. 2017). Consider sentences like: (22.15) Country A violated the sovereignty of Country B 498 CHAPTER 22 • L EXICONS FOR SENTIMENT , AFFECT , AND CONNOTATION Power Score weaklyRachelDentGordanBatmanJokerpowerfully Sentiment Score negativeJokerDentGordanRachelBatmanpositive Agency Score dull DentGordanRachelBatmanJokerscary Figure 1: Power, sentiment, and agency scores for char- acters inThe Dark Nightas learned through the regres- sion model with ELMo embeddings. Scores generally align with character archetypes, i.e. the antagonist has the lowest sentiment score. ment have resulted in his effective removal from the industry. While articles about the #MeToo movement portray men like Weinstein as unpow- erful, we can speculate that the corpora used to train ELMo and BERT portray them as powerful. Thus, in a corpus where traditional power roles have been inverted, the embeddings extracted from ELMo and BERT perform worse than ran- dom, as they are biased towards the power struc- tures in the data they are trained on. Further ev- idence of this exists in the performance of the BERT-masked embeddings - whereas these em- beddings generally capture power poorly as com- pared to the unmasked embeddings (Table 2), they outperform the unmasked embeddings on this task, and even outperform the frequency baseline in one setting. Nevertheless, they do not outper- form Field et al.(2019), likely because they do not capture affect information as well as the unmasked embeddings (Table2). 4.3 Qualitative Document-level Analysis Finally, we qualitatively analyze how well our method captures affect dimensions by analyzing single documents in detail. We conduct this anal- ysis in a domain where we expect entities to fulﬁll traditional power roles and where entity portray- als are known. FollowingBamman et al.(2013), we analyze the Wikipedia plot summary of the movie The Dark Knight,7 focusing on Batman (protagonist),8 the Joker (antagonist), Jim Gordan (law enforcement ofﬁcer, ally to Batman), Har- 7http://bit.ly/2XmhRDR 8We consider Batman/Bruce Wayne to be the same entity. Power Score weakly Rachel Joker Dent Gordan Batmanpowerfully Sentiment Score negative Joker Gordan Batman Dent Rachel positive Agency Score dull Rachel Dent GordanBatman Joker scary Figure 2: Power, sentiment, and agency scores for char- acters inThe Dark Nightas learned through ASP with ELMo embeddings. These scores reﬂect the same pat- terns as the regression model with greater separation between characters. vey Dent (ally to Batman who turns evil) and Rachel Dawes (primary love interest). To facil- itate extracting example sentences, we score each instance of these entities in the narrative separately and average across instances to obtain an entity score for the document.9 To maximize our data by capturing every mention of an entity, we per- form co-reference resolution by hand. Addition- ally, based on our results from Table3 as well as the use of Wikipedia data in training the ELMo model (Peters et al., 2018), we use ELMo embed- dings for our analysis. Figures 1 and 2 show results. For refer- ence, we show the entity scores as compared to one polar opposite pair identiﬁed by ASP. Both the regression model and ASP show similar pat- terns. Batman has high power, while Rachel has low power. Additionally, the Joker is associated with the most negative sentiment, but the high- est agency. Throughout the plot summary, the movie progresses by the Joker taking an aggres- sive action and the other characters responding. We can see this dynamic reﬂected in the Joker’s proﬁle score, as a high-powered, high-agency, low-sentiment character, who is the primary plot- driver. In general, ASP shows a greater separation between characters than the regression model. We hypothesize that this occurs because ASP isolates the dimensions of interest, while the regression ap- proach captures other confounds, such as that hu- 9When we used this averaging metric in other evaluations, we found no signiﬁcant change in results. Thus, in other sce- narios, we compute scores over averaged embeddings, rather than averaging scores separately computed for each embed- ding to reduce computationally complexity. Figure 22.13 Power (dominance), sentiment (valence) and agency (arousal) for characters in the movie The Dark Knight computed from embeddings trained on the NRC V AD Lexicon. Note the protagonist (Batman) and the antagonist (the Joker) have high power and agency scores but differ in sentiment, while the love interest Rachel has low power and agency but high sentiment. (22.16) the teenager ... survived the Boston Marathon bombing” By using the verb violate in (22.15), the author is expressing their sympathies with Country B, portraying Country B as a victim, and expressing antagonism toward the agent Country A. By contrast, in using the verb survive, the author of (22.16) is expressing that the bombing is a negative experience, and the subject of the sentence, the teenager, is a sympathetic character. These aspects of connotation are inherent in the meaning of the verbs violate and survive, as shown in Fig. 22.14. Writer Role1 Role2 Role1 is a sympathetic victim There is some type of hardship Reader + _ + _ _ S(writer →role1) S(writer →role2) Connotation Frame for “Role1 survives Role2” S(role1 → role2) Writer Role1 Role2 Role1 is the antagonist Role2 is a sympathetic victim Reader +_ +_ _ S(writer →role1) S(writer →role2) Connotation Frame for “Role1 violates Role2” S(role1 → role2) (a) (b) Figure 22.14 Connotation frames for survive and violate. (a) For survive, the writer and reader have positive sentiment toward Role1, the subject, and negative sentiment toward Role2, the direct object. (b) Forviolate, the writer and reader have positive sentiment instead toward Role2, the direct object. The connotation frame lexicons of Rashkin et al. (2016) and Rashkin et al. (2017) also express other connotative aspects of the predicate toward each argu- ment, including the effect (something bad happened to x) value: (x is valuable), and mental state: (x is distressed by the event). Connotation frames can also mark the power differential between the arguments (using the verb implore means that the theme argument has greater power than the agent), and theagency of each argument (waited is low agency). Fig. 22.15 shows a visualization from Sap et al. (2017). Connotation frames can be built by hand (Sap et al., 2017), or they can be learned by supervised learning (Rashkin et al., 2016), for example using hand-labeled train- 22.10 • S UMMARY 499 AGENT THEME power(AG < TH) VERBimplore He implored the tribunal to show mercy. The princess waited for her prince. AGENT THEME agency(AG) = - VERBwait Figure 2: The formal notation of the connotation frames of power and agency. The ﬁrst example shows the relative power differential implied by the verb “implored”, i.e., the agent (“he”) is in a position of less power than the theme (“the tri- bunal”). In contrast, “Hedemanded the tribunal show mercy” implies that the agent has authority over the theme. The second example shows the low level of agency implied by the verb“waited”. interactive demo website of our ﬁndings (see Fig- ure 5 in the appendix for a screenshot).2 Further- more, as will be seen in Section4.1, connotation frames offer new insights that complement and de- viate from the well-known Bechdel test (Bechdel, 1986). In particular, we ﬁnd that high-agency women through the lens of connotation frames are rare in modern ﬁlms. It is, in part, because some movies (e.g., Snow White) accidentally pass the Bechdel test and also because even movies with strong female characters are not entirely free from the deeply ingrained biases in social norms. 2 Connotation Frames of Power and Agency We create two new connotation relations,power and agency (examples in Figure3), as an expan- sion of the existing connotation frame lexicons.3 Three AMT crowdworkers annotated the verbs with placeholders to avoid gender bias in the con- text (e.g.,X rescued Y; an example task is shown in the appendix in Figure7). We deﬁne the anno- tated constructs as follows: Power Differentials Many verbs imply the au- thority levels of the agent and theme relative to 2http://homes.cs.washington.edu/˜msap/ movie-bias/. 3The lexicons and a demo are available at http:// homes.cs.washington.edu/˜msap/movie-bias/. power(AG<TH) power(AG>TH) agency(AG)= agency(AG)=+ Figure 3: Sample verbs in the connotation frames with high annotator agreement. Size is indicative of verb frequency in our corpus (bigger= more frequent), color differences are only for legibility. one another. For example, if the agent “dom- inates” the theme (denoted aspower(AG>TH)), then the agent is implied to have a level of control over the theme. Alternatively, if the agent “hon- ors” the theme (denoted aspower(AG<TH)), the writer implies that the theme is more important or authoritative. We used AMT crowdsourcing to la- bel 1700 transitive verbs for power differentials. With three annotators per verb, the inter-annotator agreement is 0.34 (Krippendorff’s↵). Agency The agency attributed to the agent of the verb denotes whether the action being described implies that the agent is powerful, decisive, and capable of pushing forward their own storyline. For example, a person who is described as “ex- periencing” things does not seem as active and de- cisive as someone who is described as “determin- ing” things. AMT workers labeled 2000 transi- tive verbs for implying high/moderate/low agency (inter-annotator agreement of 0.27). We denote high agency asagency(AG)=+, and low agency as agency(AG)=. Pairwise agreements on a hard constraint are 56% and 51% for power and agency, respec- tively. Despite this, agreements reach 96% and 94% when moderate labels are counted as agree- ing with either high or low labels, showing that an- notators rarely strongly disagree with one another. Some contributing factors in the lower KA scores include the subtlety of choosing between neutral Figure 22.15 The connotation frames of Sap et al. (2017), showing that the verb implore implies the agent has lower power than the theme (in contrast, say, with a verb likedemanded), and showing the low level of agency of the subject of waited. Figure from Sap et al. (2017). ing data to supervise classiﬁers for each of the individual relations, e.g., whether S(writer →Role1) is + or -, and then improving accuracy via global constraints across all relations. 22.10 Summary • Many kinds of affective states can be distinguished, includingemotions, moods, attitudes (which include sentiment), interpersonal stance, and personality. • Emotion can be represented by ﬁxed atomic units often called basic emo- tions, or as points in space deﬁned by dimensions like valence and arousal. • Words have connotational aspects related to these affective states, and this connotational aspect of word meaning can be represented in lexicons. • Affective lexicons can be built by hand, using crowd sourcing to label the affective content of each word. • Lexicons can be built with semi-supervised, bootstrapping from seed words using similarity metrics like embedding cosine. • Lexicons can be learned in a fully supervised manner, when a convenient training signal can be found in the world, such as ratings assigned by users on a review site. • Words can be assigned weights in a lexicon by using various functions of word counts in training texts, and ratio metrics like log odds ratio informative Dirichlet prior. • Affect can be detected, just like sentiment, by using standard supervised text classiﬁcation techniques, using all the words or bigrams in a text as features. Additional features can be drawn from counts of words in lexicons. • Lexicons can also be used to detect affect in arule-based classiﬁer by picking the simple majority sentiment based on counts of words in each lexicon. • Connotation frames express richer relations of affective meaning that a pred- icate encodes about its arguments. 500 CHAPTER 22 • L EXICONS FOR SENTIMENT , AFFECT , AND CONNOTATION Bibliographical and Historical Notes The idea of formally representing the subjective meaning of words began with Os- good et al. (1957), the same pioneering study that ﬁrst proposed the vector space model of meaning described in Chapter 6. Osgood et al. (1957) had participants rate words on various scales, and ran factor analysis on the ratings. The most signiﬁcant factor they uncovered was the evaluative dimension, which distinguished between pairs like good/bad, valuable/worthless, pleasant/unpleasant. This work inﬂuenced the development of early dictionaries of sentiment and affective meaning in the ﬁeld of content analysis (Stone et al., 1966). Wiebe (1994) began an inﬂuential line of work on detectingsubjectivity in text,subjectivity beginning with the task of identifying subjective sentences and the subjective char- acters who are described in the text as holding private states, beliefs or attitudes. Learned sentiment lexicons such as the polarity lexicons of Hatzivassiloglou and McKeown (1997) were shown to be a useful feature in subjectivity detection (Hatzi- vassiloglou and Wiebe 2000, Wiebe 2000). The term sentiment seems to have been introduced in 2001 by Das and Chen (2001), to describe the task of measuring market sentiment by looking at the words in stock trading message boards. In the same paper Das and Chen (2001) also proposed the use of a sentiment lexicon. The list of words in the lexicon was created by hand, but each word was assigned weights according to how much it discriminated a particular class (say buy versus sell) by maximizing across-class variation and minimizing within-class variation. The term sentiment, and the use of lexicons, caught on quite quickly (e.g., inter alia, Turney 2002). Pang et al. (2002) ﬁrst showed the power of using all the words without a sentiment lexicon; see also Wang and Manning (2012). Most of the semi-supervised methods we describe for extending sentiment dic- tionaries drew on the early idea that synonyms and antonyms tend to co-occur in the same sentence (Miller and Charles 1991, Justeson and Katz 1991, Riloff and Shep- herd 1997). Other semi-supervised methods for learning cues to affective mean- ing rely on information extraction techniques, like the AutoSlog pattern extractors (Riloff and Wiebe, 2003). Graph based algorithms for sentiment were ﬁrst sug- gested by Hatzivassiloglou and McKeown (1997), and graph propagation became a standard method (Zhu and Ghahramani 2002, Zhu et al. 2003, Zhou et al. 2004a, Velikovich et al. 2010). Crowdsourcing can also be used to improve precision by ﬁltering the result of semi-supervised lexicon learning (Riloff and Shepherd 1997, Fast et al. 2016). Much recent work focuses on ways to learn embeddings that directly encode sen- timent or other properties, such as the D ENSIFIER algorithm of Rothe et al. (2016) that learns to transform the embedding space to focus on sentiment (or other) infor- mation. Exercises 22.1 Show that the relationship between a word w and a category c in the Potts Score in Eq. 22.6 is a variant of the pointwise mutual information pmi (w,c) without the log term. CHAPTER 23 Coreference Resolution and Entity Linking and even Stigand, the patriotic archbishop of Canterbury, found it advisable–”’ ‘Found WHAT?’ said the Duck. ‘Found IT, ’ the Mouse replied rather crossly: ‘of course you know what “it”means. ’ ‘I know what “it”means well enough, when I ﬁnd a thing, ’ said the Duck: ‘it’s gener- ally a frog or a worm. The question is, what did the archbishop ﬁnd?’ Lewis Carroll, Alice in Wonderland An important component of language processing is knowing who is being talked about in a text. Consider the following passage: (23.1) Victoria Chen , CFO of Megabucks Banking, saw her pay jump to $2.3 million, as the 38-year-old became the company’s president. It is widely known that she came to Megabucks from rival Lotsabucks. Each of the underlined phrases in this passage is used by the writer to refer to a person named Victoria Chen. We call linguistic expressions like her or Victoria Chen mentions or referring expressions, and the discourse entity that is referredmention to (Victoria Chen) the referent. (To distinguish between referring expressions andreferent their referents, we italicize the former.)1 Two or more referring expressions that are used to refer to the same discourse entity are said to corefer; thus, Victoria Chencorefer and she corefer in (23.1). Coreference is an important component of natural language processing. A dia- logue system that has just told the user “There is a 2pm ﬂight on United and a 4pm one on Cathay Paciﬁc” must know which ﬂight the user means by“I’ll take the sec- ond one”. A question answering system that uses Wikipedia to answer a question about Marie Curie must know who she was in the sentence “She was born in War- saw”. And a machine translation system translating from a language like Spanish, in which pronouns can be dropped, must use coreference from the previous sentence to decide whether the Spanish sentence ‘“Me encanta el conocimiento”, dice. ’ should be translated as ‘“I love knowledge”, he says ’, or ‘“I love knowledge”, she says ’. Indeed, this example comes from an actual news article in El Pa´ısabout a female professor and was mistranslated as “he” in machine translation because of inaccurate coreference resolution (Schiebinger, 2013). Natural language processing systems (and humans) interpret linguistic expres- sions with respect to a discourse model (Karttunen, 1969). A discourse modeldiscourse model (Fig. 23.1) is a mental model that the understander builds incrementally when in- terpreting a text, containing representations of the entities referred to in the text, as well as properties of the entities and relations among them. When a referent is ﬁrst mentioned in a discourse, we say that a representation for it is evoked into theevoked model. Upon subsequent mention, this representation is accessed from the model.accessed 1 As a convenient shorthand, we sometimes speak of a referring expression referring to a referent, e.g., saying that she refers to Victoria Chen. However, the reader should keep in mind that what we really mean is that the speaker is performing the act of referring to Victoria Chen by uttering she. 502 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING V Discourse Model “Victoria” “she”corefer refer (evoke) refer (access)$ Lotsabucks Megabucks pay Figure 23.1 How mentions evoke and access discourse entities in a discourse model. Reference in a text to an entity that has been previously introduced into the discourse is called anaphora, and the referring expression used is said to be ananaphora anaphor, or anaphoric.2 In passage (23.1), the pronouns she and her and the deﬁ-anaphor nite NP the 38-year-old are therefore anaphoric. The anaphor corefers with a prior mention (in this case Victoria Chen) that is called the antecedent. Not every refer-antecedent ring expression is an antecedent. An entity that has only a single mention in a text (like Lotsabucks in (23.1)) is called a singleton.singleton In this chapter we focus on the task of coreference resolution. Coreferencecoreference resolution resolution is the task of determining whether two mentions corefer, by which we mean they refer to the same entity in the discourse model (the samediscourse entity). The set of coreferring expressions is often called a coreference chain or a cluster.coreference chain cluster For example, in processing (23.1), a coreference resolution algorithm would need to ﬁnd at least four coreference chains, corresponding to the four entities in the discourse model in Fig. 23.1. 1. {Victoria Chen, her, the 38-year-old, She} 2. {Megabucks Banking, the company, Megabucks} 3. {her pay} 4. {Lotsabucks} Note that mentions can be nested; for example the mention her is syntactically part of another mention, her pay, referring to a completely different discourse entity. Coreference resolution thus comprises two tasks (although they are often per- formed jointly): (1) identifying the mentions, and (2) clustering them into corefer- ence chains/discourse entities. We said that two mentions corefered if they are associated with the same dis- course entity. But often we’d like to go further, deciding which real world entity is associated with this discourse entity. For example, the mention Washington might refer to the US state, or the capital city, or the person George Washington; the inter- pretation of the sentence will of course be very different for each of these. The task of entity linking (Ji and Grishman, 2011) or entity resolution is the task of mappingentity linking a discourse entity to some real-world individual. 3 We usually operationalize entity 2 We will follow the common NLP usage ofanaphor to mean any mention that has an antecedent, rather than the more narrow usage to mean only mentions (like pronouns) whose interpretation depends on the antecedent (under the narrower interpretation, repeated names are not anaphors). 3 Computational linguistics/NLP thus differs in its use of the term reference from the ﬁeld of formal semantics, which uses the words reference and coreference to describe the relation between a mention and a real-world entity. By contrast, we follow the functional linguistics tradition in which a mention refers to a discourse entity (Webber, 1978) and the relation between a discourse entity and the real world individual requires an additional step of linking. 503 linking or resolution by mapping to an ontology: a list of entities in the world, like a gazeteer (Appendix F). Perhaps the most common ontology used for this task is Wikipedia; each Wikipedia page acts as the unique id for a particular entity. Thus the entity linking task ofwikiﬁcation (Mihalcea and Csomai, 2007) is the task of de- ciding which Wikipedia page corresponding to an individual is being referred to by a mention. But entity linking can be done with any ontology; for example if we have an ontology of genes, we can link mentions of genes in text to the disambiguated gene name in the ontology. In the next sections we introduce the task of coreference resolution in more de- tail, and survey a variety of architectures for resolution. We also introduce two architectures for the task of entity linking. Before turning to algorithms, however, we mention some important tasks we will only touch on brieﬂy at the end of this chapter. First are the famous Winograd Schema problems (so-called because they were ﬁrst pointed out by Terry Winograd in his dissertation). These entity coreference resolution problems are designed to be too difﬁcult to be solved by the resolution methods we describe in this chapter, and the kind of real-world knowledge they require has made them a kind of challenge task for natural language processing. For example, consider the task of determining the correct antecedent of the pronoun they in the following example: (23.2) The city council denied the demonstrators a permit because a. they feared violence. b. they advocated violence. Determining the correct antecedent for the pronoun they requires understanding that the second clause is intended as an explanation of the ﬁrst clause, and also that city councils are perhaps more likely than demonstrators to fear violence and that demonstrators might be more likely to advocate violence. Solving Winograd Schema problems requires ﬁnding way to represent or discover the necessary real world knowledge. A problem we won’t discuss in this chapter is the related task of event corefer- ence, deciding whether two event mentions (such as the buy and the acquisition inevent coreference these two sentences from the ECB+ corpus) refer to the same event: (23.3) AMD agreed to [ buy] Markham, Ontario-based ATI for around $5.4 billion in cash and stock, the companies announced Monday. (23.4) The [ acquisition] would turn AMD into one of the world’s largest providers of graphics chips. Event mentions are much harder to detect than entity mentions, since they can be ver- bal as well as nominal. Once detected, the same mention-pair and mention-ranking models used for entities are often applied to events. An even more complex kind of coreference is discourse deixis (Webber, 1988),discourse deixis in which an anaphor refers back to a discourse segment, which can be quite hard to delimit or categorize, like the examples in (23.5) adapted from Webber (1991): (23.5) According to Soleil, Beau just opened a restaurant a. But that turned out to be a lie. b. But that was false. c. That struck me as a funny way to describe the situation. The referent of that is a speech act (see Chapter 15) in (23.5a), a proposition in (23.5b), and a manner of description in (23.5c). We don’t give algorithms in this chapter for these difﬁcult types of non-nominal antecedents , but see Kolhatkar et al. (2018) for a survey. 504 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING 23.1 Coreference Phenomena: Linguistic Background We now offer some linguistic background on reference phenomena. We introduce the four types of referring expressions (deﬁnite and indeﬁnite NPs, pronouns, and names), describe how these are used to evoke and access entities in the discourse model, and talk about linguistic features of the anaphor/antecedent relation (like number/gender agreement, or properties of verb semantics). 23.1.1 Types of Referring Expressions Indeﬁnite Noun Phrases: The most common form of indeﬁnite reference in En- glish is marked with the determiner a (or an), but it can also be marked by a quan- tiﬁer such as some or even the determiner this. Indeﬁnite reference generally intro- duces into the discourse context entities that are new to the hearer. (23.6) a. Mrs. Martin was so very kind as to send Mrs. Goddard a beautiful goose. b. He had gone round one day to bring her some walnuts. c. I saw this beautiful cauliﬂower today. Deﬁnite Noun Phrases: Deﬁnite reference, such as via NPs that use the English article the, refers to an entity that is identiﬁable to the hearer. An entity can be identiﬁable to the hearer because it has been mentioned previously in the text and thus is already represented in the discourse model: (23.7) It concerns a white stallion which I have sold to an ofﬁcer. But the pedigree of the white stallion was not fully established. Alternatively, an entity can be identiﬁable because it is contained in the hearer’s set of beliefs about the world, or the uniqueness of the object is implied by the description itself, in which case it evokes a representation of the referent into the discourse model, as in (23.9): (23.8) I read about it in the New York Times. (23.9) Have you seen the car keys? These last uses are quite common; more than half of deﬁnite NPs in newswire texts are non-anaphoric, often because they are the ﬁrst time an entity is mentioned (Poesio and Vieira 1998, Bean and Riloff 1999). Pronouns: Another form of deﬁnite reference is pronominalization, used for enti- ties that are extremely salient in the discourse, (as we discuss below): (23.10) Emma smiled and chatted as cheerfully as she could, Pronouns can also participate in cataphora, in which they are mentioned beforecataphora their referents are, as in (23.11). (23.11) Even before she saw it, Dorothy had been thinking about the Emerald City every day. Here, the pronouns she and it both occur before their referents are introduced. Pronouns also appear in quantiﬁed contexts in which they are considered to be bound, as in (23.12).bound (23.12) Every dancer brought her left arm forward. Under the relevant reading,her does not refer to some woman in context, but instead behaves like a variable bound to the quantiﬁed expression every dancer. We are not concerned with the bound interpretation of pronouns in this chapter. 23.1 • C OREFERENCE PHENOMENA : L INGUISTIC BACKGROUND 505 In some languages, pronouns can appear as clitics attached to a word, like lo (‘it’) in this Spanish example from AnCora (Recasens and Mart´ı, 2010): (23.13) La intenci ´on es reconocer el gran prestigio que tiene la marat´on y unirlo con esta gran carrera. ‘The aim is to recognize the great prestige that the Marathon has and join\|it with this great race.” Demonstrative Pronouns: Demonstrative pronouns this and that can appear ei- ther alone or as determiners, for instance, this ingredient, that spice: (23.14) I just bought a copy of Thoreau’s Walden. I had bought one ﬁve years ago. That one had been very tattered; this one was in much better condition. Note that this NP is ambiguous; in colloquial spoken English, it can be indeﬁnite, as in (23.6), or deﬁnite, as in (23.14). Zero Anaphora: Instead of using a pronoun, in some languages (including Chi- nese, Japanese, and Italian) it is possible to have an anaphor that has no lexical realization at all, called a zero anaphor or zero pronoun, as in the following Italianzero anaphor and Japanese examples from Poesio et al. (2016): (23.15) EN [John] i went to visit some friends. On the way [he]i bought some wine. IT [Giovanni] i and`o a far visita a degli amici. Per via φi compr`o del vino. JA [John] i-wa yujin-o houmon-sita. Tochu-de φi wain-o ka-tta. or this Chinese example: (23.16) [ 我] 前一会精神上太紧张。[0] 现在比较平静了 [I] was too nervous a while ago. ... [0] am now calmer. Zero anaphors complicate the task of mention detection in these languages. Names: Names (such as of people, locations, or organizations) can be used to refer to both new and old entities in the discourse: (23.17) a. Miss Woodhouse certainly had not done him justice. b. International Business Machines sought patent compensation from Amazon; IBM had previously sued other companies. 23.1.2 Information Status The way referring expressions are used to evoke new referents into the discourse (introducing new information), or access old entities from the model (old informa- tion), is called their information status or information structure. Entities can beinformation status discourse-new or discourse-old, and indeed it is common to distinguish at leastdiscourse-new discourse-old three kinds of entities informationally (Prince, 1981): new NPs: brand new NPs: these introduce entities that are discourse-new and hearer- new like a fruit or some walnuts. unused NPs: these introduce entities that are discourse-new but hearer-old (like Hong Kong, Marie Curie, or the New York Times. old NPs: also called evoked NPs, these introduce entities that already in the dis- course model, hence are both discourse-old and hearer-old, like it in “I went to a new restaurant. It was...”. 506 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING inferrables: these introduce entities that are neither hearer-old nor discourse-old, but the hearer can infer their existence by reasoning based on other entities that are in the discourse. Consider the following examples: (23.18) I went to a superb restaurant yesterday. The chef had just opened it. (23.19) Mix ﬂour, butter and water. Knead the dough until shiny. Neither the chef nor the dough were in the discourse model based on the ﬁrst sentence of either example, but the reader can make a bridging inferencebridging inference that these entities should be added to the discourse model and associated with the restaurant and the ingredients, based on world knowledge that restaurants have chefs and dough is the result of mixing ﬂour and liquid (Haviland and Clark 1974, Webber and Baldwin 1992, Nissim et al. 2004, Hou et al. 2018). The form of an NP gives strong clues to its information status. We often talk about an entity’s position on thegiven-new dimension, the extent to which the refer-given-new ent is given (salient in the discourse, easier for the hearer to call to mind, predictable by the hearer), versusnew (non-salient in the discourse, unpredictable) (Chafe 1976, Prince 1981, Gundel et al. 1993). A referent that is very accessible (Ariel, 2001)accessible i.e., very salient in the hearer’s mind or easy to call to mind, can be referred to with less linguistic material. For example pronouns are used only when the referent has a high degree of activation or salience in the discourse model. 4 By contrast, lesssalience salient entities, like a new referent being introduced to the discourse, will need to be introduced with a longer and more explicit referring expression to help the hearer recover the referent. Thus when an entity is ﬁrst introduced into a discourse its mentions are likely to have full names, titles or roles, or appositive or restrictive relative clauses, as in the introduction of our protagonist in (23.1): Victoria Chen, CFO of Megabucks Banking. As an entity is discussed over a discourse, it becomes more salient to the hearer and its mentions on average typically becomes shorter and less informative, for example with a shortened name (for example Ms. Chen ), a deﬁnite description (the 38-year-old), or a pronoun (she or her) (Hawkins 1978). However, this change in length is not monotonic, and is sensitive to discourse structure (Grosz 1977b, Reichman 1985, Fox 1993). 23.1.3 Complications: Non-Referring Expressions Many noun phrases or other nominals are not referring expressions, although they may bear a confusing superﬁcial resemblance. For example in some of the earliest computational work on reference resolution, Karttunen (1969) pointed out that the NP a car in the following example does not create a discourse referent: (23.20) Janet doesn’t have a car. and cannot be referred back to by anaphoric it or the car: (23.21) * It is a Toyota. (23.22) * The car is red. We summarize here four common types of structures that are not counted as men- tions in coreference tasks and hence complicate the task of mention-detection: 4 Pronouns also usually (but not always) refer to entities that were introduced no further than one or two sentences back in the ongoing discourse, whereas deﬁnite noun phrases can often refer further back. 23.1 • C OREFERENCE PHENOMENA : L INGUISTIC BACKGROUND 507 Appositives: An appositional structure is a noun phrase that appears next to a head noun phrase, describing the head. In English they often appear in commas, like “a unit of UAL” appearing in apposition to the NP United, or CFO of Megabucks Banking in apposition to Victoria Chen. (23.23) Victoria Chen, CFO of Megabucks Banking, saw ... (23.24) United, a unit of UAL, matched the fares. Appositional NPs are not referring expressions, instead functioning as a kind of supplementary parenthetical description of the head NP. Nonetheless, sometimes it is useful to link these phrases to an entity they describe, and so some datasets like OntoNotes mark appositional relationships. Predicative and Prenominal NPs: Predicative or attributive NPs describe prop- erties of the head noun. In United is a unit of UAL, the NP a unit of UAL describes a property of United, rather than referring to a distinct entity. Thus they are not marked as mentions in coreference tasks; in our example the NPs $2.3 million and the company’s president, are attributive, describing properties of her pay and the 38-year-old; Example (23.27) shows a Chinese example in which the predicate NP (中国最大的城市; China’s biggest city) is not a mention. (23.25) her pay jumped to $2.3 million (23.26) the 38-year-old became the company’s president (23.27) 上海是[中国最大的城市] [Shanghai is China’s biggest city] Expletives: Many uses of pronouns like it in English and corresponding pronouns in other languages are not referential. Such expletive or pleonastic cases includeexpletive it is raining, in idioms like hit it off, or in particular syntactic situations like cleftsclefts (23.28a) or extraposition (23.28b): (23.28) a. It was Emma Goldman who founded Mother Earth b. It surprised me that there was a herring hanging on her wall. Generics: Another kind of expression that does not refer back to an entity explic- itly evoked in the text is generic reference. Consider (23.29). (23.29) I love mangos. They are very tasty. Here, they refers, not to a particular mango or set of mangos, but instead to the class of mangos in general. The pronoun you can also be used generically: (23.30) In July in San Francisco you have to wear a jacket. 23.1.4 Linguistic Properties of the Coreference Relation Now that we have seen the linguistic properties of individual referring expressions we turn to properties of the antecedent/anaphor pair. Understanding these properties is helpful both in designing novel features and performing error analyses. Number Agreement: Referring expressions and their referents must generally agree in number; English she/her/he/him/his/it are singular, we/us/they/them are plu- ral, and you is unspeciﬁed for number. So a plural antecedent like the chefs cannot generally corefer with a singular anaphor like she. However, algorithms cannot enforce number agreement too strictly. First, semantically plural entities can be re- ferred to by either it or they: (23.31) IBM announced a new machine translation product yesterday. They have been working on it for 20 years. 508 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING Second, singular they has become much more common, in which they is used tosingular they describe singular individuals, often useful because they is gender neutral. Although recently increasing, singular they is quite old, part of English for many centuries.5 Person Agreement: English distinguishes between ﬁrst, second, and third person, and a pronoun’s antecedent must agree with the pronoun in person. Thus a third person pronoun (he, she, they, him, her, them, his, her, their) must have a third person antecedent (one of the above or any other noun phrase). However, phenomena like quotation can cause exceptions; in this example I, my, and she are coreferent: (23.32) “I voted for Nader because he was most aligned with my values,” she said. Gender or Noun Class Agreement: In many languages, all nouns have grammat- ical gender or noun class6 and pronouns generally agree with the grammatical gender of their antecedent. In English this occurs only with third-person singular pronouns, which distinguish between male (he, him, his), female (she, her), and nonpersonal (it) grammatical genders. Non-binary pronouns likeze or hir may also occur in more recent texts. Knowing which gender to associate with a name in text can be complex, and may require world knowledge about the individual. Some examples: (23.33) Maryam has a theorem. She is exciting. (she=Maryam, not the theorem) (23.34) Maryam has a theorem. It is exciting. (it=the theorem, not Maryam) Binding Theory Constraints: The binding theory is a name for syntactic con- straints on the relations between a mention and an antecedent in the same sentence (Chomsky, 1981). Oversimplifying a bit, reﬂexive pronouns like himself and her-reﬂexive self corefer with the subject of the most immediate clause that contains them (23.35), whereas nonreﬂexives cannot corefer with this subject (23.36). (23.35) Janet bought herself a bottle of ﬁsh sauce. [herself =Janet] (23.36) Janet bought her a bottle of ﬁsh sauce. [her ̸=Janet] Recency: Entities introduced in recent utterances tend to be more salient than those introduced from utterances further back. Thus, in (23.37), the pronoun it is more likely to refer to Jim’s map than the doctor’s map. (23.37) The doctor found an old map in the captain’s chest. Jim found an even older map hidden on the shelf. It described an island. Grammatical Role: Entities mentioned in subject position are more salient than those in object position, which are in turn more salient than those mentioned in oblique positions. Thus although the ﬁrst sentence in (23.38) and (23.39) expresses roughly the same propositional content, the preferred referent for the pronoun he varies with the subject—John in (23.38) and Bill in (23.39). (23.38) Billy Bones went to the bar with Jim Hawkins. He called for a glass of rum. [ he = Billy ] (23.39) Jim Hawkins went to the bar with Billy Bones. He called for a glass of rum. [ he = Jim ] 5 Here’s a bound pronoun example from Shakespeare’sComedy of Errors: There’s not a man I meet but doth salute me As if I were their well-acquainted friend 6 The word “gender” is generally only used for languages with 2 or 3 noun classes, like most Indo- European languages; many languages, like the Bantu languages or Chinese, have a much larger number of noun classes. 23.2 • C OREFERENCE TASKS AND DATASETS 509 Verb Semantics: Some verbs semantically emphasize one of their arguments, bi- asing the interpretation of subsequent pronouns. Compare (23.40) and (23.41). (23.40) John telephoned Bill. He lost the laptop. (23.41) John criticized Bill. He lost the laptop. These examples differ only in the verb used in the ﬁrst sentence, yet “he” in (23.40) is typically resolved to John, whereas “he” in (23.41) is resolved to Bill. This may be partly due to the link between implicit causality and saliency: the implicit cause of a “criticizing” event is its object, whereas the implicit cause of a “telephoning” event is its subject. In such verbs, the entity which is the implicit cause may be more salient. Selectional Restrictions: Many other kinds of semantic knowledge can play a role in referent preference. For example, the selectional restrictions that a verb places on its arguments (Chapter 21) can help eliminate referents, as in (23.42). (23.42) I ate the soup in my new bowl after cooking it for hours There are two possible referents forit, the soup and the bowl. The verbeat, however, requires that its direct object denote something edible, and this constraint can rule out bowl as a possible referent. 23.2 Coreference Tasks and Datasets We can formulate the task of coreference resolution as follows: Given a text T , ﬁnd all entities and the coreference links between them. We evaluate our task by com- paring the links our system creates with those in human-created gold coreference annotations on T . Let’s return to our coreference example, now using superscript numbers for each coreference chain (cluster), and subscript letters for individual mentions in the clus- ter: (23.43) [Victoria Chen] 1 a, CFO of [Megabucks Banking]2 a, saw [[her]1 b pay]3 a jump to $2.3 million, as [the 38-year-old]1 c also became [[the company]2 b’s president. It is widely known that [she]1 d came to [Megabucks]2 c from rival [Lotsabucks]4 a. Assuming example (23.43) was the entirety of the article, the chains forher pay and Lotsabucks are singleton mentions: 1. {Victoria Chen, her, the 38-year-old, She} 2. {Megabucks Banking, the company, Megabucks} 3. {her pay} 4. {Lotsabucks} For most coreference evaluation campaigns, the input to the system is the raw text of articles, and systems must detect mentions and then link them into clusters. Solving this task requires dealing with pronominal anaphora (ﬁguring out that her refers to Victoria Chen), ﬁltering out non-referential pronouns like the pleonastic It in It has been ten years ), dealing with deﬁnite noun phrases to ﬁgure out that the 38-year-old is coreferent with Victoria Chen, and that the company is the same as Megabucks. And we need to deal with names, to realize that Megabucks is the same as Megabucks Banking. 510 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING Exactly what counts as a mention and what links are annotated differs from task to task and dataset to dataset. For example some coreference datasets do not label singletons, making the task much simpler. Resolvers can achieve much higher scores on corpora without singletons, since singletons constitute the majority of mentions in running text, and they are often hard to distinguish from non-referential NPs. Some tasks use gold mention-detection (i.e. the system is given human-labeled mention boundaries and the task is just to cluster these gold mentions), which eliminates the need to detect and segment mentions from running text. Coreference is usually evaluated by theCoNLL F1 score, which combines three metrics: MUC, B3, and CEAFe; Section 23.8 gives the details. Let’s mention a few characteristics of one popular coreference dataset, OntoNotes (Pradhan et al. 2007c, Pradhan et al. 2007a), and the CoNLL 2012 Shared Task based on it (Pradhan et al., 2012a). OntoNotes contains hand-annotated Chinese and English coreference datasets of roughly one million words each, consisting of newswire, magazine articles, broadcast news, broadcast conversations, web data and conversational speech data, as well as about 300,000 words of annotated Arabic newswire. The most important distinguishing characteristic of OntoNotes is that it does not label singletons, simplifying the coreference task, since singletons rep- resent 60%-70% of all entities. In other ways, it is similar to other coreference datasets. Referring expression NPs that are coreferent are marked as mentions, but generics and pleonastic pronouns are not marked. Appositive clauses are not marked as separate mentions, but they are included in the mention. Thus in the NP, “Richard Godown, president of the Industrial Biotechnology Association” the mention is the entire phrase. Prenominal modiﬁers are annotated as separate entities only if they are proper nouns. Thus wheat is not an entity in wheat ﬁelds, but UN is an entity in UN policy (but not adjectives like American in American policy). A number of corpora mark richer discourse phenomena. The ISNotes corpus annotates a portion of OntoNotes for information status, include bridging examples (Hou et al., 2018). The LitBank coreference corpus (Bamman et al., 2020) contains coreference annotations for 210,532 tokens from 100 different literary novels, in- cluding singletons and quantiﬁed and negated noun phrases. The AnCora-CO coref- erence corpus (Recasens and Mart´ı, 2010) contains 400,000 words each of Spanish (AnCora-CO-Es) and Catalan (AnCora-CO-Ca) news data, and includes labels for complex phenomena like discourse deixis in both languages. The ARRAU corpus (Uryupina et al., 2020) contains 350,000 words of English marking all NPs, which means singleton clusters are available. ARRAU includes diverse genres like dialog (the TRAINS data) and ﬁction (the Pear Stories), and has labels for bridging refer- ences, discourse deixis, generics, and ambiguous anaphoric relations. 23.3 Mention Detection The ﬁrst stage of coreference is mention detection: ﬁnding the spans of text thatmention detection constitute each mention. Mention detection algorithms are usually very liberal in proposing candidate mentions (i.e., emphasizing recall), and only ﬁltering later. For example many systems run parsers and named entity taggers on the text and extract every span that is either an NP, a possessive pronoun, or a named entity. Doing so from our sample text repeated in (23.44): (23.44) Victoria Chen, CFO of Megabucks Banking, saw her pay jump to $2.3 23.3 • M ENTION DETECTION 511 million, as the 38-year-old also became the company’s president. It is widely known that she came to Megabucks from rival Lotsabucks. might result in the following list of 13 potential mentions: Victoria Chen $2.3 million she CFO of Megabucks Banking the 38-year-old Megabucks Megabucks Banking the company Lotsabucks her the company’s president her pay It More recent mention detection systems are even more generous; the span-based algorithm we will describe in Section 23.6 ﬁrst extracts literally all n-gram spans of words up to N=10. Of course recall from Section 23.1.3 that many NPs—and the overwhelming majority of random n-gram spans—are not referring expressions. Therefore all such mention detection systems need to eventually ﬁlter out pleonas- tic/expletive pronouns like It above, appositives like CFO of Megabucks Banking Inc, or predicate nominals like the company’s presidentor $2.3 million. Some of this ﬁltering can be done by rules. Early rule-based systems designed regular expressions to deal with pleonastic it, like the following rules from Lappin and Leass (1994) that use dictionaries of cognitive verbs (e.g., believe, know, antic- ipate) to capture pleonastic it in “It is thought that ketchup...”, or modal adjectives (e.g., necessary, possible, certain, important), for, e.g., “It is likely that I...”. Such rules are sometimes used as part of modern systems: It is Modaladjective that S It is Modaladjective (for NP) to VP It is Cogv-ed that S It seems/appears/means/follows (that) S Mention-detection rules are sometimes designed speciﬁcally for particular eval- uation campaigns. For OntoNotes, for example, mentions are not embedded within larger mentions, and while numeric quantities are annotated, they are rarely coref- erential. Thus for OntoNotes tasks like CoNLL 2012 (Pradhan et al., 2012a), a common ﬁrst pass rule-based mention detection algorithm (Lee et al., 2013) is: 1. Take all NPs, possessive pronouns, and named entities. 2. Remove numeric quantities (100 dollars, 8%), mentions embedded in larger mentions, adjectival forms of nations, and stop words (like there). 3. Remove pleonastic it based on regular expression patterns. Rule-based systems, however, are generally insufﬁcient to deal with mention- detection, and so modern systems incorporate some sort of learned mention detec- tion component, such as a referentiality classiﬁer, an anaphoricity classiﬁer — detecting whether an NP is an anaphor—or a discourse-new classiﬁer— detecting whether a mention is discourse-new and a potential antecedent for a future anaphor. An anaphoricity detector, for example, can draw its positive training examplesanaphoricity detector from any span that is labeled as an anaphoric referring expression in hand-labeled datasets like OntoNotes, ARRAU , or AnCora. Any other NP or named entity can be marked as a negative training example. Anaphoricity classiﬁers use features of the candidate mention such as its head word, surrounding words, deﬁniteness, animacy, length, position in the sentence/discourse, many of which were ﬁrst proposed in early work by Ng and Cardie (2002a); see Section 23.5 for more on features. 512 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING Referentiality or anaphoricity detectors can be run as ﬁlters, in which only men- tions that are classiﬁed as anaphoric or referential are passed on to the coreference system. The end result of such a ﬁltering mention detection system on our example above might be the following ﬁltered set of 9 potential mentions: Victoria Chen her pay she Megabucks Bank the 38-year-old Megabucks her the company Lotsabucks It turns out, however, that hard ﬁltering of mentions based on an anaphoricity or referentiality classiﬁer leads to poor performance. If the anaphoricity classiﬁer threshold is set too high, too many mentions are ﬁltered out and recall suffers. If the classiﬁer threshold is set too low, too many pleonastic or non-referential mentions are included and precision suffers. The modern approach is instead to perform mention detection, anaphoricity, and coreference jointly in a single end-to-end model (Ng 2005b, Denis and Baldridge 2007, Rahman and Ng 2009). For example mention detection in the Lee et al. (2017b),2018 system is based on a single end-to-end neural network that computes a score for each mention being referential, a score for two mentions being corefer- ence, and combines them to make a decision, training all these scores with a single end-to-end loss. We’ll describe this method in detail in Section 23.6.7 Despite these advances, correctly detecting referential mentions seems to still be an unsolved problem, since systems incorrectly marking pleonastic pronouns like it and other non-referential NPs as coreferent is a large source of errors of modern coreference resolution systems (Kummerfeld and Klein 2013, Martschat and Strube 2014, Martschat and Strube 2015, Wiseman et al. 2015, Lee et al. 2017a). Mention, referentiality, or anaphoricity detection is thus an important open area of investigation. Other sources of knowledge may turn out to be helpful, especially in combination with unsupervised and semisupervised algorithms, which also mit- igate the expense of labeled datasets. In early work, for example Bean and Riloff (1999) learned patterns for characterizing anaphoric or non-anaphoric NPs; (by ex- tracting and generalizing over the ﬁrst NPs in a text, which are guaranteed to be non-anaphoric). Chang et al. (2012) look for head nouns that appear frequently in the training data but never appear as gold mentions to help ﬁnd non-referential NPs. Bergsma et al. (2008b) use web counts as a semisupervised way to augment standard features for anaphoricity detection for Englishit, an important task becauseit is both common and ambiguous; between a quarter and half it examples are non-anaphoric. Consider the following two examples: (23.45) You can make [it] in advance. [anaphoric] (23.46) You can make [it] in Hollywood. [non-anaphoric] The it in make it is non-anaphoric, part of the idiom make it. Bergsma et al. (2008b) turn the context around each example into patterns, like “make * in advance” from (23.45), and “make * in Hollywood” from (23.46). They then use Google n-grams to enumerate all the words that can replace it in the patterns. Non-anaphoric contexts tend to only have it in the wildcard positions, while anaphoric contexts occur with many other NPs (for example make them in advance is just as frequent in their data 7 Some systems try to avoid mention detection or anaphoricity detection altogether. For datasets like OntoNotes which don’t label singletons, an alternative to ﬁltering out non-referential mentions is to run coreference resolution, and then simply delete any candidate mentions which were not corefered with another mention. This likely doesn’t work as well as explicitly modeling referentiality, and cannot solve the problem of detecting singletons, which is important for tasks like entity linking. 23.4 • A RCHITECTURES FOR COREFERENCE ALGORITHMS 513 as make it in advance , but make them in Hollywood did not occur at all). These n-gram contexts can be used as features in a supervised anaphoricity classiﬁer. 23.4 Architectures for Coreference Algorithms Modern systems for coreference are based on supervised neural machine learning, supervised from hand-labeled datasets like OntoNotes. In this section we overview the various architecture of modern systems, using the categorization of Ng (2010), which distinguishes algorithms based on whether they make each coreference deci- sion in a way that isentity-based—representing each entity in the discourse model— or only mention-based—considering each mention independently, and whether they use ranking models to directly compare potential antecedents. Afterwards, we go into more detail on one state-of-the-art algorithm in Section 23.6. 23.4.1 The Mention-Pair Architecture We begin with the mention-pair architecture, the simplest and most inﬂuentialmention-pair coreference architecture, which introduces many of the features of more complex algorithms, even though other architectures perform better. The mention-pair ar-mention-pair chitecture is based around a classiﬁer that— as its name suggests—is given a pair of mentions, a candidate anaphor and a candidate antecedent, and makes a binary classiﬁcation decision: coreferring or not. Let’s consider the task of this classiﬁer for the pronoun she in our example, and assume the slightly simpliﬁed set of potential antecedents in Fig. 23.2. Victoria Chen Megabucks Banking her her pay the 37-year-old she p(coref\|”Victoria Chen”,”she”) p(coref\|”Megabucks Banking”,”she”) Figure 23.2 For each pair of a mention (like she), and a potential antecedent mention (like Victoria Chen or her), the mention-pair classiﬁer assigns a probability of a coreference link. For each prior mention (Victoria Chen, Megabucks Banking, her, etc.), the binary classiﬁer computes a probability: whether or not the mention is the antecedent of she. We want this probability to be high for actual antecedents ( Victoria Chen, her, the 38-year-old) and low for non-antecedents (Megabucks Banking, her pay). Early classiﬁers used hand-built features (Section 23.5); more recent classiﬁers use neural representation learning (Section 23.6) For training, we need a heuristic for selecting training samples; since most pairs of mentions in a document are not coreferent, selecting every pair would lead to a massive overabundance of negative samples. The most common heuristic, from (Soon et al., 2001), is to choose the closest antecedent as a positive example, and all pairs in between as the negative examples. More formally, for each anaphor mention mi we create • one positive instance (mi,mj) where mj is the closest antecedent to mi, and 514 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING • a negative instance (mi,mk) for each mk between mj and mi Thus for the anaphor she, we would choose ( she, her) as the positive example and no negative examples. Similarly, for the anaphorthe company we would choose (the company, Megabucks) as the positive example and (the company, she) (the com- pany, the 38-year-old) (the company, her pay) and (the company, her) as negative examples. Once the classiﬁer is trained, it is applied to each test sentence in a clustering step. For each mention i in a document, the classiﬁer considers each of the priori−1 mentions. In closest-ﬁrst clustering (Soon et al., 2001), the classiﬁer is run right to left (from mention i−1 down to mention 1) and the ﬁrst antecedent with probability >.5 is linked to i. If no antecedent has probably >0.5, no antecedent is selected for i. In best-ﬁrst clustering, the classiﬁer is run on all i −1 antecedents and the most probable preceding mention is chosen as the antecedent for i. The transitive closure of the pairwise relation is taken as the cluster. While the mention-pair model has the advantage of simplicity, it has two main problems. First, the classiﬁer doesn’t directly compare candidate antecedents to each other, so it’s not trained to decide, between two likely antecedents, which one is in fact better. Second, it ignores the discourse model, looking only at mentions, not entities. Each classiﬁer decision is made completely locally to the pair, without being able to take into account other mentions of the same entity. The next two models each address one of these two ﬂaws. 23.4.2 The Mention-Rank Architecture The mention ranking model directly compares candidate antecedents to each other, choosing the highest-scoring antecedent for each anaphor. In early formulations, for mention i, the classiﬁer decides which of the{1,...,i− 1}prior mentions is the antecedent (Denis and Baldridge, 2008). But suppose i is in fact not anaphoric, and none of the antecedents should be chosen? Such a model would need to run a separate anaphoricity classiﬁer on i. Instead, it turns out to be better to jointly learn anaphoricity detection and coreference together with a single loss (Rahman and Ng, 2009). So in modern mention-ranking systems, for the ith mention (anaphor), we have an associated random variable yi ranging over the valuesY (i) ={1,...,i −1,ϵ}. The value ϵis a special dummy mention meaning thati does not have an antecedent (i.e., is either discourse-new and starts a new coref chain, or is non-anaphoric). Victoria Chen Megabucks Banking her her pay the 37-year-old she p(”Victoria Chen”\|”she”) p(ϵ\|”she”) ϵ One or more of these should be high All of these should be low } p(”her pay”\|she”) p(”her”\|she”) p(”the 37-year-old”\|she”) p(”Megabucks Banking”\|she”) } Figure 23.3 For each candidate anaphoric mention (like she), the mention-ranking system assigns a proba- bility distribution over all previous mentions plus the special dummy mention ϵ. At test time, for a given mention i the model computes one softmax over all the antecedents (plus ϵ) giving a probability for each candidate antecedent (or none). 23.5 • C LASSIFIERS USING HAND -BUILT FEATURES 515 Fig. 23.3 shows an example of the computation for the single candidate anaphor she. Once the antecedent is classiﬁed for each anaphor, transitive closure can be run over the pairwise decisions to get a complete clustering. Training is trickier in the mention-ranking model than the mention-pair model, because for each anaphor we don’t know which of all the possible gold antecedents to use for training. Instead, the best antecedent for each mention is latent; that is, for each mention we have a whole cluster of legal gold antecedents to choose from. Early work used heuristics to choose an antecedent, for example choosing the closest antecedent as the gold antecedent and all non-antecedents in a window of two sentences as the negative examples (Denis and Baldridge, 2008). Various kinds of ways to model latent antecedents exist (Fernandes et al. 2012, Chang et al. 2013, Durrett and Klein 2013). The simplest way is to give credit to any legal antecedent by summing over all of them, with a loss function that optimizes the likelihood of all correct antecedents from the gold clustering (Lee et al., 2017b). We’ll see the details in Section 23.6. Mention-ranking models can be implemented with hand-build features or with neural representation learning (which might also incorporate some hand-built fea- tures). we’ll explore both directions in Section 23.5 and Section 23.6. 23.4.3 Entity-based Models Both the mention-pair and mention-ranking models make their decisions aboutmen- tions. By contrast, entity-based models link each mention not to a previous mention but to a previous discourse entity (cluster of mentions). A mention-ranking model can be turned into an entity-ranking model simply by having the classiﬁer make its decisions over clusters of mentions rather than individual mentions (Rahman and Ng, 2009). For traditional feature-based models, this can be done by extracting features over clusters. The size of a cluster is a useful feature, as is its ‘shape’, which is the list of types of the mentions in the cluster i.e., sequences of the tokens (P)roper, (D)eﬁnite, (I)ndeﬁnite, (Pr)onoun, so that a cluster composed of {Victoria, her, the 38-year-old}would have the shapeP-Pr-D (Bj¨orkelund and Kuhn, 2014). An entity- based model that includes a mention-pair classiﬁer can use as features aggregates of mention-pair probabilities, for example computing the average probability of coref- erence over all mention-pairs in the two clusters (Clark and Manning 2015). Neural models can learn representations of clusters automatically, for example by using an RNN over the sequence of cluster mentions to encode a state correspond- ing to a cluster representation (Wiseman et al., 2016), or by learning distributed rep- resentations for pairs of clusters by pooling over learned representations of mention pairs (Clark and Manning, 2016b). However, although entity-based models are more expressive, the use of cluster- level information in practice has not led to large gains in performance, so mention- ranking models are still more commonly used. 23.5 Classiﬁers using hand-built features Feature-based classiﬁers, use hand-designed features in logistic regression, SVM, or random forest classiﬁers for coreference resolution. These classiﬁers don’t per- 516 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING form as well as neural ones. Nonetheless, they are still sometimes useful to build lightweight systems when compute or data are sparse, and the features themselves are useful for error analysis even in neural systems. Given an anaphor mention and a potential antecedent mention, feature based classiﬁers make use of three types of features: (i) features of the anaphor, (ii) features of the candidate antecedent, and (iii) features of the relationship between the pair. Entity-based models can make additional use of two additional classes: (iv) feature of all mentions from the antecedent’s entity cluster, and (v) features of the relation between the anaphor and the mentions in the antecedent entity cluster. Features of the Anaphor or Antecedent Mention First (last) word Victoria/she First or last word (or embedding) of antecedent/anaphor Head word Victoria/she Head word (or head embedding) of antecedent/anaphor Attributes Sg-F-A-3-PER/ Sg-F-A-3-PER The number, gender, animacy, person, named entity type attributes of (antecedent/anaphor) Length 2/1 length in words of (antecedent/anaphor) Mention type P/Pr Type: (P)roper, (D)eﬁnite, (I)ndeﬁnite, (Pr)onoun) of an- tecedent/anaphor Features of the Antecedent Entity Entity shape P-Pr-D The ‘shape’ or list of types of the mentions in the antecedent entity (cluster), i.e., sequences of (P)roper, (D)eﬁnite, (I)ndeﬁnite, (Pr)onoun. Entity attributes Sg-F-A-3-PER The number, gender, animacy, person, named entity type attributes of the antecedent entity Ant. cluster size 3 Number of mentions in the antecedent cluster Features of the Pair of Mentions Sentence distance 1 The number of sentences between antecedent and anaphor Mention distance 4 The number of mentions between antecedent and anaphor i-within-i F Anaphor has i-within-i relation with antecedent Cosine Cosine between antecedent and anaphor embeddings Features of the Pair of Entities Exact String Match F True if the strings of any two mentions from the antecedent and anaphor clusters are identical. Head Word Match F True if any mentions from antecedent cluster has same headword as any mention in anaphor cluster Word Inclusion F All words in anaphor cluster included in antecedent cluster Figure 23.4 Feature-based coreference: sample feature values for anaphor “she” and potential antecedent “Victoria Chen”. Figure 23.4 shows a selection of commonly used features, and shows the value that would be computed for the potential anaphor “she” and potential antecedent “Victoria Chen” in our example sentence, repeated below: (23.47) Victoria Chen, CFO of Megabucks Banking, saw her pay jump to $2.3 million, as the 38-year-old also became the company’s president. It is widely known that she came to Megabucks from rival Lotsabucks. Features that prior work has found to be particularly useful are exact string match, entity headword agreement, mention distance, as well as (for pronouns) exact attribute match and i-within-i, and (for nominals and proper names) word inclusion and cosine. For lexical features (like head words) it is common to only use words that appear enough times (>20 times). 23.6 • A NEURAL MENTION -RANKING ALGORITHM 517 It is crucial in feature-based systems to use conjunctions of features; one exper- iment suggested that moving from individual features in a classiﬁer to conjunctions of multiple features increased F1 by 4 points (Lee et al., 2017a). Speciﬁc conjunc- tions can be designed by hand (Durrett and Klein, 2013), all pairs of features can be conjoined (Bengtson and Roth, 2008), or feature conjunctions can be learned using decision tree or random forest classiﬁers (Ng and Cardie 2002a, Lee et al. 2017a). Features can also be used in neural models as well. Neural systems use contex- tual word embeddings so don’t beneﬁt from shallow features like string match or or mention types. However features like mention length, distance between mentions, or genre can complement neural contextual embedding models. 23.6 A neural mention-ranking algorithm In this section we describe the neural e2e-coref algorithms of Lee et al. (2017b) (simpliﬁed and extended a bit, drawing on Joshi et al. (2019) and others). This is a mention-ranking algorithm that considers all possible spans of text in the docu- ment, assigns a mention-score to each span, prunes the mentions based on this score, then assigns coreference links to the remaining mentions. More formally, given a document D with T words, the model considers all of the T (T +1) 2 text spans in D (unigrams, bigrams, trigrams, 4-grams, etc; in practice we only consider spans up a maximum length around 10). The task is to assign to each span i an antecedent yi, a random variable ranging over the values Y (i) = {1,...,i −1,ϵ}; each previous span and a special dummy token ϵ. Choosing the dummy token means thati does not have an antecedent, either becausei is discourse- new and starts a new coreference chain, or because i is non-anaphoric. For each pair of spans i and j, the system assigns a score s(i,j) for the coref- erence link between span i and span j. The system then learns a distribution P(yi) over the antecedents for span i: P(yi) = exp(s(i,yi))∑ y′∈Y (i) exp(s(i,y′)) (23.48) This score s(i,j) includes three factors that we’ll deﬁne below: m(i); whether span i is a mention; m( j); whether span j is a mention; and c(i,j); whether j is the antecedent of i: s(i,j) =m(i)+ m( j)+ c(i,j) (23.49) For the dummy antecedent ϵ, the score s(i,ϵ) is ﬁxed to 0. This way if any non- dummy scores are positive, the model predicts the highest-scoring antecedent, but if all the scores are negative it abstains. 23.6.1 Computing span representations To compute the two functionsm(i) and c(i,j) which score a span i or a pair of spans (i,j), we’ll need a way to represent a span. The e2e-coref family of algorithms represents each span by trying to capture 3 words/tokens: the ﬁrst word, the last word, and the most important word. We ﬁrst run each paragraph or subdocument through an encoder (like BERT) to generate embeddings hi for each token i. The span i is then represented by a vectorgi that is a concatenation of the encoder output 518 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING embedding for the ﬁrst (start) token of the span, the encoder output for the last (end) token of the span, and a third vector which is an attention-based representation: gi = [hSTART(i),hEND (i),hATT(i)] (23.50) The goal of the attention vector is to represent which word/token is the likely syntactic head-word of the span; we saw in the prior section that head-words are a useful feature; a matching head-word is a good indicator of coreference. The attention representation is computed as usual; the system learns a weight vectorwα , and computes its dot product with the hidden state ht transformed by a FFN: αt = wα ·FFNα (ht ) (23.51) The attention score is normalized into a distribution via a softmax: ai,t = exp(αt ) ∑END (i) k=START(i) exp(αk) (23.52) And then the attention distribution is used to create a vector hATT(i) which is an attention-weighted sum of the embeddings et of each of the words in span i: hATT(i) = END (i)∑ t=START(i) ai,t ·et (23.53) Fig. 23.5 shows the computation of the span representation and the mention score. Encodings (h) … Encoder General Electric said the Postal Service contacted the company Span head (hATT) Span representation (g) Mention score (m) + ++ + + General Electric Electric said the the Postal Service Service contacted the the company Figure 23.5 Computation of the span representation g (and the mention score m) in a BERT version of the e2e-coref model (Lee et al. 2017b, Joshi et al. 2019). The model considers all spans up to a maximum width of say 10; the ﬁgure shows a small subset of the bigram and trigram spans. 23.6.2 Computing the mention and antecedent scores m and c Now that we know how to compute the vector gi for representing span i, we can see the details of the two scoring functions m(i) and c(i,j). Both are computed by feedforward networks: m(i) = wm ·FFNm(gi) (23.54) c(i,j) = wc ·FFNc([gi,gj,gi ◦gj,]) (23.55) At inference time, this mention score m is used as a ﬁlter to keep only the best few mentions. 23.6 • A NEURAL MENTION -RANKING ALGORITHM 519 We then compute the antecedent score for high-scoring mentions. The antecedent score c(i,j) takes as input a representation of the spansi and j, but also the element- wise similarity of the two spans to each other gi ◦gj (here ◦is element-wise mul- tiplication). Fig. 23.6 shows the computation of the score s for the three possible antecedents of the company in the example sentence from Fig. 23.5. Figure 23.6 The computation of the score s for the three possible antecedents of the com- pany in the example sentence from Fig. 23.5. Figure after Lee et al. (2017b). Given the set of mentions, the joint distribution of antecedents for each docu- ment is computed in a forward pass, and we can then do transitive closure on the antecedents to create a ﬁnal clustering for the document. Fig. 23.7 shows example predictions from the model, showing the attention weights, which Lee et al. (2017b) ﬁnd correlate with traditional semantic heads. Note that the model gets the second example wrong, presumably becauseattendants and pilot likely have nearby word embeddings. Figure 23.7 Sample predictions from the Lee et al. (2017b) model, with one cluster per example, showing one correct example and one mistake. Bold, parenthesized spans are men- tions in the predicted cluster. The amount of red color on a word indicates the head-ﬁnding attention weight ai,t in Eq. 23.52. Figure adapted from Lee et al. (2017b). 23.6.3 Learning For training, we don’t have a single gold antecedent for each mention; instead the coreference labeling only gives us each entire cluster of coreferent mentions; so a mention only has a latent antecedent. We therefore use a loss function that maxi- mizes the sum of the coreference probability of any of the legal antecedents. For a given mention i with possible antecedents Y (i), let GOLD (i) be the set of mentions in the gold cluster containing i. Since the set of mentions occurring before i is Y (i), the set of mentions in that gold cluster that also occur beforei is Y (i)∩GOLD (i). We 520 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING therefore want to maximize: ∑ ˆy∈Y (i)∩GOLD (i) P(ˆy) (23.56) If a mention i is not in a gold cluster GOLD (i) = ϵ. To turn this probability into a loss function, we’ll use the cross-entropy loss function we deﬁned in Eq. 5.23 in Chapter 5, by taking the −log of the probability. If we then sum over all mentions, we get the ﬁnal loss function for training: L = N∑ i=2 −log ∑ ˆy∈Y (i)∩GOLD (i) P(ˆy) (23.57) 23.7 Entity Linking Entity linking is the task of associating a mention in text with the representation ofentity linking some real-world entity in an ontology or knowledge base (Ji and Grishman, 2011). It is the natural follow-on to coreference resolution; coreference resolution is the task of associating textual mentions that corefer to the same entity. Entity linking takes the further step of identifying who that entity is. It is especially important for any NLP task that links to a knowledge base. While there are all sorts of potential knowledge-bases, we’ll focus in this section on Wikipedia, since it’s widely used as an ontology for NLP tasks. In this usage, each unique Wikipedia page acts as the unique id for a particular entity. This task of deciding which Wikipedia page corresponding to an individual is being referred to by a text mention has its own name: wikiﬁcation (Mihalcea and Csomai, 2007).wikiﬁcation Since the earliest systems (Mihalcea and Csomai 2007, Cucerzan 2007, Milne and Witten 2008), entity linking is done in (roughly) two stages: mention detec- tion and mention disambiguation. We’ll give two algorithms, one simple classic baseline that uses anchor dictionaries and information from the Wikipedia graph structure (Ferragina and Scaiella, 2011) and one modern neural algorithm (Li et al., 2020). We’ll focus here mainly on the application of entity linking to questions, since a lot of the literature has been in that context. 23.7.1 Linking based on Anchor Dictionaries and Web Graph As a simple baseline we introduce the TAGME linker (Ferragina and Scaiella, 2011) for Wikipedia, which itself draws on earlier algorithms (Mihalcea and Csomai 2007, Cucerzan 2007, Milne and Witten 2008). Wikiﬁcation algorithms deﬁne the set of entities as the set of Wikipedia pages, so we’ll refer to each Wikipedia page as a unique entity e. T AGME ﬁrst creates a catalog of all entities (i.e. all Wikipedia pages, removing some disambiguation and other meta-pages) and indexes them in a standard IR engine like Lucene. For each page e, the algorithm computes an in-link count in(e): the total number of in-links from other Wikipedia pages that point to e. These counts can be derived from Wikipedia dumps. Finally, the algorithm requires an anchor dictionary . An anchor dictionary lists for each Wikipedia page, its anchor texts: the hyperlinked spans of text onanchor texts other pages that point to it. For example, the web page for Stanford University, http://www.stanford.edu, might be pointed to from another page using anchor texts like Stanford or Stanford University: 23.7 • E NTITY LINKING 521 <a href="http://www.stanford.edu">Stanford University</a> We compute a Wikipedia anchor dictionary by including, for each Wikipedia page e, e’s title as well as all the anchor texts from all Wikipedia pages that point toe. For each anchor string a we’ll also compute its total frequency freq(a) in Wikipedia (including non-anchor uses), the number of timesa occurs as a link (which we’ll call link(a)), and its link probability linkprob(a) =link(a)/freq(a). Some cleanup of the ﬁnal anchor dictionary is required, for example removing anchor strings composed only of numbers or single characters, that are very rare, or that are very unlikely to be useful entities because they have a very low linkprob. Mention Detection Given a question (or other text we are trying to link), TAGME detects mentions by querying the anchor dictionary for each token sequence up to 6 words. This large set of sequences is pruned with some simple heuristics (for example pruning substrings if they have small linkprobs). The question: When was Ada Lovelace born? might give rise to the anchor Ada Lovelace and possibly Ada, but substrings spans like Lovelace might be pruned as having too low a linkprob, and but spans likeborn have such a low linkprob that they would not be in the anchor dictionary at all. Mention Disambiguation If a mention span is unambiguous (points to only one entity/Wikipedia page), we are done with entity linking! However, many spans are ambiguous, matching anchors for multiple Wikipedia entities/pages. The T AGME algorithm uses two factors for disambiguating ambiguous spans, which have been referred to as prior probability and relatedness/coherence. The ﬁrst factor is p(e\|a), the probability with which the span refers to a particular entity. For each page e ∈ E(a), the probability p(e\|a) that anchor a points to e, is the ratio of the number of links into e with anchor text a to the total number of occurrences of a as an anchor: prior(a →e) = p(e\|a) =count(a →e) link(a) (23.58) Let’s see how that factor works in linking entities in the following question: What Chinese Dynasty came before the Yuan? The most common association for the spanYuanin the anchor dictionary is the name of the Chinese currency, i.e., the probability p(Yuan currency\|yuan) is very high. Rarer Wikipedia associations for Yuan include the common Chinese last name, a language spoken in Thailand, and the correct entity in this case, the name of the Chinese dynasty. So if we chose based only on p(e\|a) , we would make the wrong disambiguation and miss the correct link, Yuan dynasty. To help in just this sort of case, TAGME uses a second factor, the relatedness of this entity to other entities in the input question. In our example, the fact that the question also contains the spanChinese Dynasty, which has a high probability link to the page Dynasties in Chinese history, ought to help match Yuan dynasty. Let’s see how this works. Given a question q, for each candidate anchors span a detected in q, we assign a relatedness score to each possible entity e ∈E(a) of a. The relatedness score of the link a →e is the weighted average relatedness between e and all other entities in q. Two entities are considered related to the extent their Wikipedia pages share many in-links. More formally, the relatedness between two entities A and B is computed as rel(A,B) = log(max(\|in(A)\|,\|in(B)\|))−log(\|in(A)∩in(B)\|) log(\|W\|)−log(min(\|in(A)\|,\|in(B)\|)) (23.59) 522 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING where in(x) is the set of Wikipedia pages pointing to x and W is the set of all Wiki- pedia pages in the collection. The vote given by anchor b to the candidate annotation a →X is the average, over all the possible entities of b, of their relatedness to X, weighted by their prior probability: vote(b,X) = 1 \|E(b)\| ∑ Y ∈E(b) rel(X,Y )p(Y \|b) (23.60) The total relatedness score for a →X is the sum of the votes of all the other anchors detected in q: relatedness(a →X) = ∑ b∈Xq\a vote(b,X) (23.61) To score a →X, we combine relatedness and prior by choosing the entity X that has the highest relatedness (a →X), ﬁnding other entities within a small ϵ of this value, and from this set, choosing the entity with the highest prior P(X\|a). The result of this step is a single entity assigned to each span in q. The TAGME algorithm has one further step of pruning spurious anchor/entity pairs, assigning a score averaging link probability with the coherence. coherence(a →X) = 1 \|S\|−1 ∑ B∈S\X rel(B,X) score(a →X) =coherence(a →X)+ linkprob(a) 2 (23.62) Finally, pairs are pruned if score (a →X) <λ, where the threshold λ is set on a held-out set. 23.7.2 Neural Graph-based linking More recent entity linking models are based on bi-encoders, encoding a candidate mention span, encoding an entity, and computing the dot product between the en- codings. This allows embeddings for all the entities in the knowledge base to be precomputed and cached (Wu et al., 2020). Let’s sketch the ELQ linking algorithm of Li et al. (2020), which is given a question q and a set of candidate entities from Wikipedia with associated Wikipedia text, and outputs tuples(e,ms,me) of entity id, mention start, and mention end. As Fig. 23.8 shows, it does this by encoding each Wikipedia entity using text from Wikipedia, encoding each mention span using text from the question, and computing their similarity, as we describe below. Entity Mention Detection To get an h-dimensional embedding for each question token, the algorithm runs the question through BERT in the normal way: [q1 ···qn] =BERT([CLS]q1 ···qn[SEP]) (23.63) It then computes the likelihood of each span [i,j] in q being an entity mention, in a way similar to the span-based algorithm we saw for the reader above. First we compute the score for i/ j being the start/end of a mention: sstart(i) =wstart ·qi, send( j) =wend ·qj, (23.64) 23.7 • E NTITY LINKING 523 Figure 23.8 A sketch of the inference process in the ELQ algorithm for entity linking in questions (Li et al., 2020). Each candidate question mention span and candidate entity are separately encoded, and then scored by the entity/span dot product. where wstart and wend are vectors learned during training. Next, another trainable embedding, wmention is used to compute a score for each token being part of a men- tion: smention(t) =wmention ·qt (23.65) Mention probabilities are then computed by combining these three scores: p([i,j]) =σ ( sstart(i)+ send( j)+ j∑ t=i smention(t) ) (23.66) Entity Linking To link mentions to entities, we next compute embeddings for each entity in the set E = e1,···,ei,···,ew of all Wikipedia entities. For each en- tity ei we’ll get text from the entity’s Wikipedia page, the title t(ei) and the ﬁrst 128 tokens of the Wikipedia page which we’ll call the description d(ei). This is again run through BERT, taking the output of theCLStoken BERT[CLS] as the entity representation: xei = BERT[CLS]([CLS]t(ei)[ENT]d(ei)[SEP]) (23.67) Mention spans can be linked to entities by computing, for each entity e and span [i,j], the dot product similarity between the span encoding (the average of the token embeddings) and the entity encoding. yi,j = 1 ( j −i +1) j∑ t=i qt s(e,[i,j]) =x· eyi,j (23.68) Finally, we take a softmax to get a distribution over entities for each span: p(e\|[i,j]) = exp(s(e,[i,j]))∑ e′∈E exp(s(e′,[i,j])) (23.69) Training The ELQ mention detection and entity linking algorithm is fully super- vised. This means, unlike the anchor dictionary algorithms from Section 23.7.1, 524 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING it requires datasets with entity boundaries marked and linked. Two such labeled datasets are WebQuestionsSP (Yih et al., 2016), an extension of the WebQuestions (Berant et al., 2013) dataset derived from Google search questions, and GraphQues- tions (Su et al., 2016). Both have had entity spans in the questions marked and linked (Sorokin and Gurevych 2018, Li et al. 2020) resulting in entity-labeled ver- sions WebQSPEL and GraphQEL (Li et al., 2020). Given a training set, the ELQ mention detection and entity linking phases are trained jointly, optimizing the sum of their losses. The mention detection loss is a binary cross-entropy loss, with L the length of the passage and N the number of candidates: LMD = −1 N ∑ 1≤i≤j≤min(i+L−1,n) ( y[i,j] log p([i,j])+( 1 −y[i,j])log(1 −p([i,j])) ) (23.70) with y[i,j] = 1 if [i,j] is a gold mention span, else 0. The entity linking loss is: LED = −logp(eg\|[i,j]) (23.71) where eg is the gold entity for mention [i,j]. 23.8 Evaluation of Coreference Resolution We evaluate coreference algorithms model-theoretically, comparing a set ofhypoth- esis chains or clusters H produced by the system against a set of gold or reference chains or clusters R from a human labeling, and reporting precision and recall. However, there are a wide variety of methods for doing this comparison. In fact, there are 5 common metrics used to evaluate coreference algorithms: the link based MUC (Vilain et al., 1995) and BLANC (Recasens and Hovy 2011, Luo et al. 2014) metrics, the mention based B3 metric (Bagga and Baldwin, 1998), the entity based CEAF metric (Luo, 2005), and the link based entity aware LEA metric (Moosavi and Strube, 2016). Let’s just explore two of the metrics. TheMUC F-measure (Vilain et al., 1995)MUC F-measure is based on the number of coreference links (pairs of mentions) common to H and R. Precision is the number of common links divided by the number of links in H. Recall is the number of common links divided by the number of links in R; This makes MUC biased toward systems that produce large chains (and fewer entities), and it ignores singletons, since they don’t involve links. B3 is mention-based rather than link-based. For each mention in the referenceB3 chain, we compute a precision and recall, and then we take a weighted sum over all N mentions in the document to compute a precision and recall for the entire task. For a given mention i, let R be the reference chain that includes i, and H the hypothesis chain that has i. The set of correct mentions in H is H ∩R. Precision for mention i is thus \|H∩R\| \|H\| , and recall for mention i thus \|H∩R\| \|R\| . The total precision is the weighted sum of the precision for mention i, weighted by a weight wi. The total recall is the weighted sum of the recall for mention i, weighted by a weight wi. Equivalently: Precision = N∑ i=1 wi # of correct mentions in hypothesis chain containing entityi # of mentions in hypothesis chain containing entityi Recall = N∑ i=1 wi # of correct mentions in hypothesis chain containing entityi # of mentions in reference chain containing entityi 23.9 • W INOGRAD SCHEMA PROBLEMS 525 The weight wi for each entity can be set to different values to produce different versions of the algorithm. Following a proposal from Denis and Baldridge (2009), the CoNLL coreference competitions were scored based on the average of MUC, CEAF-e, and B3 (Pradhan et al. 2011, Pradhan et al. 2012b), and so it is common in many evaluation campaigns to report an average of these 3 metrics. See Luo and Pradhan (2016) for a detailed description of the entire set of metrics; reference implementations of these should be used rather than attempting to reimplement from scratch (Pradhan et al., 2014). Alternative metrics have been proposed that deal with particular coreference do- mains or tasks. For example, consider the task of resolving mentions to named entities (persons, organizations, geopolitical entities), which might be useful for in- formation extraction or knowledge base completion. A hypothesis chain that cor- rectly contains all the pronouns referring to an entity, but has no version of the name itself, or is linked with a wrong name, is not useful for this task. We might instead want a metric that weights each mention by how informative it is (with names being most informative) (Chen and Ng, 2013) or a metric that considers a hypothesis to match a gold chain only if it contains at least one variant of a name (the NEC F1 metric of Agarwal et al. (2019)). 23.9 Winograd Schema problems From early on in the ﬁeld, researchers have noted that some cases of coreference are quite difﬁcult, seeming to require world knowledge or sophisticated reasoning to solve. The problem was most famously pointed out by Winograd (1972) with the following example: (23.72) The city council denied the demonstrators a permit because a. they feared violence. b. they advocated violence. Winograd noticed that the antecedent that most readers preferred for the pro- noun they in continuation (a) was the city council, but in (b) was the demonstrators. He suggested that this requires understanding that the second clause is intended as an explanation of the ﬁrst clause, and also that our cultural frames suggest that city councils are perhaps more likely than demonstrators to fear violence and that demonstrators might be more likely to advocate violence. In an attempt to get the ﬁeld of NLP to focus more on methods involving world knowledge and common-sense reasoning, Levesque (2011) proposed a challenge task called the Winograd Schema Challenge.8 The problems in the challenge taskWinograd schema are coreference problems designed to be easily disambiguated by the human reader, but hopefully not solvable by simple techniques such as selectional restrictions, or other basic word association methods. The problems are framed as a pair of statements that differ in a single word or phrase, and a coreference question: (23.73) The trophy didn’t ﬁt into the suitcase because it was too large. Question: What was too large? Answer: The trophy 8 Levesque’s call was quickly followed up by Levesque et al. (2012) and Rahman and Ng (2012), a competition at the IJCAI conference (Davis et al., 2017), and a natural language inference version of the problem called WNLI (Wang et al., 2018a). 526 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING (23.74) The trophy didn’t ﬁt into the suitcase because it was too small. Question: What was too small? Answer: The suitcase The problems have the following characteristics: 1. The problems each have two parties 2. A pronoun preferentially refers to one of the parties, but could grammatically also refer to the other 3. A question asks which party the pronoun refers to 4. If one word in the question is changed, the human-preferred answer changes to the other party The kind of world knowledge that might be needed to solve the problems can vary. In the trophy/suitcase example, it is knowledge about the physical world; that a bigger object cannot ﬁt into a smaller object. In the original Winograd sentence, it is stereotypes about social actors like politicians and protesters. In examples like the following, it is knowledge about human actions like turn-taking or thanking. (23.75) Bill passed the gameboy to John because his turn was [over/next]. Whose turn was [over/next]? Answers: Bill/John (23.76) Joan made sure to thank Susan for all the help she had [given/received]. Who had [given/received] help? Answers: Susan/Joan. Although the Winograd Schema was designed to require common-sense rea- soning, a large percentage of the original set of problems can be solved by pre- trained language models, ﬁne-tuned on Winograd Schema sentences (Kocijan et al., 2019). Large pretrained language models encode an enormous amount of world or common-sense knowledge! The current trend is therefore to propose new datasets with increasingly difﬁcult Winograd-like coreference resolution problems like KNOW REF (Emami et al., 2019), with examples like: (23.77) Marcus is undoubtedly faster than Jarrett right now but in [his] prime the gap wasn’t all that big. In the end, it seems likely that some combination of language modeling and knowl- edge will prove fruitful; indeed, it seems that knowledge-based models overﬁt less to lexical idiosyncracies in Winograd Schema training sets (Trichelair et al., 2018), 23.10 Gender Bias in Coreference As with other aspects of language processing, coreference models exhibit gender and other biases (Zhao et al. 2018a, Rudinger et al. 2018, Webster et al. 2018). For exam- ple the WinoBias dataset (Zhao et al., 2018a) uses a variant of the Winograd Schema paradigm to test the extent to which coreference algorithms are biased toward link- ing gendered pronouns with antecedents consistent with cultural stereotypes. As we summarized in Chapter 6, embeddings replicate societal biases in their training test, such as associating men with historically sterotypical male occupations like doctors, and women with stereotypical female occupations like secretaries (Caliskan et al. 2017, Garg et al. 2018). A WinoBias sentence contain two mentions corresponding to stereotypically- male and stereotypically-female occupations and a gendered pronoun that must be linked to one of them. The sentence cannot be disambiguated by the gender of the pronoun, but a biased model might be distracted by this cue. Here is an example sentence: 23.11 • S UMMARY 527 (23.78) The secretary called the physician i and told himi about a new patient [pro-stereotypical] (23.79) The secretary called the physician i and told heri about a new patient [anti-stereotypical] Zhao et al. (2018a) consider a coreference system to be biased if it is more accu- rate at linking pronouns consistent with gender stereotypical occupations (e.g., him with physician in (23.78)) than linking pronouns inconsistent with gender-stereotypical occupations (e.g., her with physician in (23.79)). They show that coreference sys- tems of all architectures (rule-based, feature-based machine learned, and end-to- end-neural) all show signiﬁcant bias, performing on average 21 F 1 points worse in the anti-stereotypical cases. One possible source of this bias is that female entities are signiﬁcantly un- derrepresented in the OntoNotes dataset, used to train most coreference systems. Zhao et al. (2018a) propose a way to overcome this bias: they generate a second gender-swapped dataset in which all male entities in OntoNotes are replaced with female ones and vice versa, and retrain coreference systems on the combined orig- inal and swapped OntoNotes data, also using debiased GloVE embeddings (Boluk- basi et al., 2016). The resulting coreference systems no longer exhibit bias on the WinoBias dataset, without signiﬁcantly impacting OntoNotes coreference accuracy. In a follow-up paper, Zhao et al. (2019) show that the same biases exist in ELMo contextualized word vector representations and coref systems that use them. They showed that retraining ELMo with data augmentation again reduces or removes bias in coreference systems on WinoBias. Webster et al. (2018) introduces another dataset, GAP, and the task of Gendered Pronoun Resolution as a tool for developing improved coreference algorithms for gendered pronouns. GAP is a gender-balanced labeled corpus of 4,454 sentences with gendered ambiguous pronouns (by contrast, only 20% of the gendered pro- nouns in the English OntoNotes training data are feminine). The examples were created by drawing on naturally occurring sentences from Wikipedia pages to create hard to resolve cases with two named entities of the same gender and an ambiguous pronoun that may refer to either person (or neither), like the following: (23.80) In May, Fujisawa joined Mari Motohashi’s rink as the team’s skip, moving back from Karuizawa to Kitami where she had spent her junior days. Webster et al. (2018) show that modern coreference algorithms perform signif- icantly worse on resolving feminine pronouns than masculine pronouns in GAP. Kurita et al. (2019) shows that a system based on BERT contextualized word repre- sentations shows similar bias. 23.11 Summary This chapter introduced the task of coreference resolution. • This is the task of linking together mentions in text which corefer, i.e. refer to the same discourse entity in the discourse model, resulting in a set of coreference chains (also called clusters or entities). • Mentions can be deﬁnite NPs or indeﬁnite NPs, pronouns (including zero pronouns) or names. 528 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING • The surface form of an entity mention is linked to its information status (new, old, or inferrable), and how accessible or salient the entity is. • Some NPs are not referring expressions, such as pleonastic it in It is raining. • Many corpora have human-labeled coreference annotations that can be used for supervised learning, including OntoNotes for English, Chinese, and Ara- bic, ARRAU for English, and AnCora for Spanish and Catalan. • Mention detection can start with all nouns and named entities and then use anaphoricity classiﬁers or referentiality classiﬁers to ﬁlter out non-mentions. • Three common architectures for coreference aremention-pair, mention-rank, and entity-based, each of which can make use of feature-based or neural clas- siﬁers. • Modern coreference systems tend to be end-to-end, performing mention de- tection and coreference in a single end-to-end architecture. • Algorithms learn representations for text spans and heads, and learn to com- pare anaphor spans with candidate antecedent spans. • Entity linking is the task of associating a mention in text with the representa- tion of some real-world entity in an ontology . • Coreference systems are evaluated by comparing with gold entity labels using precision/recall metrics like MUC, B3, CEAF, BLANC, or LEA. • The Winograd Schema Challenge problems are difﬁcult coreference prob- lems that seem to require world knowledge or sophisticated reasoning to solve. • Coreference systems exhibit gender bias which can be evaluated using datasets like Winobias and GAP. Bibliographical and Historical Notes Coreference has been part of natural language processing since the 1970s (Woods et al. 1972, Winograd 1972). The discourse model and the entity-centric foundation of coreference was formulated by Karttunen (1969) (at the 3rd COLING confer- ence), playing a role also in linguistic semantics (Heim 1982, Kamp 1981). But it was Bonnie Webber’s 1978 dissertation and following work (Webber 1983) that explored the model’s computational aspects, providing fundamental insights into how entities are represented in the discourse model and the ways in which they can license subsequent reference. Many of the examples she provided continue to chal- lenge theories of reference to this day. The Hobbs algorithm9 is a tree-search algorithm that was the ﬁrst in a longHobbs algorithm series of syntax-based methods for identifying reference robustly in naturally occur- ring text. The input to the Hobbs algorithm is a pronoun to be resolved, together with a syntactic (constituency) parse of the sentences up to and including the cur- rent sentence. The details of the algorithm depend on the grammar used, but can be understood from a simpliﬁed version due to Kehler et al. (2004) that just searches through the list of NPs in the current and prior sentences. This simpliﬁed Hobbs algorithm searches NPs in the following order: “(i) in the current sentence from right-to-left, starting with the ﬁrst NP to the left of the pronoun, (ii) in the previous sentence from left-to-right, (iii) in two sentences prior from left-to-right, and (iv) in 9 The simpler of two algorithms presented originally in Hobbs (1978). BIBLIOGRAPHICAL AND HISTORICAL NOTES 529 the current sentence from left-to-right, starting with the ﬁrst noun group to the right of the pronoun (for cataphora). The ﬁrst noun group that agrees with the pronoun with respect to number, gender, and person is chosen as the antecedent” (Kehler et al., 2004). Lappin and Leass (1994) was an inﬂuential entity-based system that used weights to combine syntactic and other features, extended soon after by Kennedy and Bogu- raev (1996) whose system avoids the need for full syntactic parses. Approximately contemporaneously centering (Grosz et al., 1995) was applied to pronominal anaphora resolution by Brennan et al. (1987), and a wide variety of work followed focused on centering’s use in coreference (Kameyama 1986, Di Eu- genio 1990, Walker et al. 1994, Di Eugenio 1996, Strube and Hahn 1996, Kehler 1997a, Tetreault 2001, Iida et al. 2003). Kehler and Rohde (2013) show how center- ing can be integrated with coherence-driven theories of pronoun interpretation. See Chapter 24 for the use of centering in measuring discourse coherence. Coreference competitions as part of the US DARPA-sponsored MUC confer- ences provided early labeled coreference datasets (the 1995 MUC-6 and 1998 MUC- 7 corpora), and set the tone for much later work, choosing to focus exclusively on the simplest cases of identity coreference (ignoring difﬁcult cases like bridging, metonymy, and part-whole) and drawing the community toward supervised machine learning and metrics like the MUC metric (Vilain et al., 1995). The later ACE eval- uations produced labeled coreference corpora in English, Chinese, and Arabic that were widely used for model training and evaluation. This DARPA work inﬂuenced the community toward supervised learning begin- ning in the mid-90s (Connolly et al. 1994, Aone and Bennett 1995, McCarthy and Lehnert 1995). Soon et al. (2001) laid out a set of basic features, extended by Ng and Cardie (2002b), and a series of machine learning models followed over the next 15 years. These often focused separately on pronominal anaphora resolution (Kehler et al. 2004, Bergsma and Lin 2006), full NP coreference (Cardie and Wagstaff 1999, Ng and Cardie 2002b, Ng 2005a) and deﬁnite NP reference (Poesio and Vieira 1998, Vieira and Poesio 2000), as well as separate anaphoricity detection (Bean and Riloff 1999, Bean and Riloff 2004, Ng and Cardie 2002a, Ng 2004), or singleton detection (de Marneffe et al., 2015). The move from mention-pair to mention-ranking approaches was pioneered by Yang et al. (2003) and Iida et al. (2003) who proposed pairwise ranking methods, then extended by Denis and Baldridge (2008) who proposed to do ranking via a soft- max over all prior mentions. The idea of doing mention detection, anaphoricity, and coreference jointly in a single end-to-end model grew out of the early proposal of Ng (2005b) to use a dummy antecedent for mention-ranking, allowing ‘non-referential’ to be a choice for coreference classiﬁers, Denis and Baldridge’s 2007 joint system combining anaphoricity classiﬁer probabilities with coreference probabilities, the Denis and Baldridge (2008) ranking model, and the Rahman and Ng (2009) pro- posal to train the two models jointly with a single objective. Simple rule-based systems for coreference returned to prominence in the 2010s, partly because of their ability to encode entity-based features in a high-precision way (Zhou et al. 2004b, Haghighi and Klein 2009, Raghunathan et al. 2010, Lee et al. 2011, Lee et al. 2013, Hajishirzi et al. 2013) but in the end they suffered from an inability to deal with the semantics necessary to correctly handle cases of common noun coreference. A return to supervised learning led to a number of advances in mention-ranking models which were also extended into neural architectures, for example using re- 530 CHAPTER 23 • C OREFERENCE RESOLUTION AND ENTITY LINKING inforcement learning to directly optimize coreference evaluation models Clark and Manning (2016a), doing end-to-end coreference all the way from span extraction (Lee et al. 2017b, Zhang et al. 2018). Neural models also were designed to take advantage of global entity-level information (Clark and Manning 2016b, Wiseman et al. 2016, Lee et al. 2018). Coreference is also related to the task of entity linking discussed in Chapter 14. Coreference can help entity linking by giving more possible surface forms to help link to the right Wikipedia page, and conversely entity linking can help improve coreference resolution. Consider this example from Hajishirzi et al. (2013): (23.81) [Michael Eisner] 1 and [Donald Tsang]2 announced the grand opening of [[Hong Kong]3 Disneyland]4 yesterday. [Eisner]1 thanked [the President]2 and welcomed [fans]5 to [the park]4. Integrating entity linking into coreference can help draw encyclopedic knowl- edge (like the fact that Donald Tsang is a president) to help disambiguate the men- tion the President. Ponzetto and Strube (2006) 2007 and Ratinov and Roth (2012) showed that such attributes extracted from Wikipedia pages could be used to build richer models of entity mentions in coreference. More recent research shows how to do linking and coreference jointly (Hajishirzi et al. 2013, Zheng et al. 2013) or even jointly with named entity tagging as well (Durrett and Klein 2014). The coreference task as we introduced it involves a simplifying assumption that the relationship between an anaphor and its antecedent is one of identity: the two coreferring mentions refer to the identical discourse referent. In real texts, the rela- tionship can be more complex, where different aspects of a discourse referent can be neutralized or refocused. For example (23.82) (Recasens et al., 2011) shows an example of metonymy, in which the capital city Washington is used metonymicallymetonymy to refer to the US. (23.83-23.84) show other examples (Recasens et al., 2011): (23.82) a strict interpretation of a policy requires The U.S. to notify foreign dictators of certain coup plots ... Washington rejected the bid ... (23.83) I once crossed that border into Ashgh-Abad on Nowruz, the Persian New Year. In the South, everyone was celebratingNew Year; to the North, it was a regular day. (23.84) In France, the president is elected for a term of seven years, while in the United States he is elected for a term of four years. For further linguistic discussions of these complications of coreference see Puste- jovsky (1991), van Deemter and Kibble (2000), Poesio et al. (2006), Fauconnier and Turner (2008), Versley (2008), and Barker (2010). Ng (2017) offers a useful compact history of machine learning models in coref- erence resolution. There are three excellent book-length surveys of anaphora/coref- erence resolution, covering different time periods: Hirst (1981) (early work until about 1981), Mitkov (2002) (1986-2001), and Poesio et al. (2016) (2001-2015). Andy Kehler wrote the Discourse chapter for the 2000 ﬁrst edition of this text- book, which we used as the starting point for the second-edition chapter, and there are some remnants of Andy’s lovely prose still in this third-edition coreference chap- ter. Exercises CHAPTER 24 Discourse Coherence And even in our wildest and most wandering reveries, nay in our very dreams, we shall ﬁnd, if we reﬂect, that the imagination ran not altogether at adven- tures, but that there was still a connection upheld among the different ideas, which succeeded each other. Were the loosest and freest conversation to be transcribed, there would immediately be transcribed, there would immediately be observed something which connected it in all its transitions. David Hume, An enquiry concerning human understanding, 1748 Orson Welles’ movieCitizen Kane was groundbreaking in many ways, perhaps most notably in its structure. The story of the life of ﬁctional media magnate Charles Foster Kane, the movie does not proceed in chronological order through Kane’s life. Instead, the ﬁlm begins with Kane’s death (famously murmuring “Rosebud”) and is structured around ﬂashbacks to his life inserted among scenes of a reporter investigating his death. The novel idea that the structure of a movie does not have to linearly follow the structure of the real timeline made apparent for 20th century cinematography the inﬁnite possibilities and impact of different kinds of coherent narrative structures. But coherent structure is not just a fact about movies or works of art. Like movies, language does not normally consist of isolated, unrelated sentences, but instead of collocated, structured, coherent groups of sentences. We refer to such a coherent structured group of sentences as a discourse, and we use the word co-discourse herence to refer to the relationship between sentences that makes real discoursescoherence different than just random assemblages of sentences. The chapter you are now read- ing is an example of a discourse, as is a news article, a conversation, a thread on social media, a Wikipedia page, and your favorite novel. What makes a discourse coherent? If you created a text by taking random sen- tences each from many different sources and pasted them together, would that be a coherent discourse? Almost certainly not. Real discourses exhibit both local coher-local ence and global coherence. Let’s consider three ways in which real discourses areglobal locally coherent; First, sentences or clauses in real discourses are related to nearby sentences in systematic ways. Consider this example from Hobbs (1979): (24.1) John took a train from Paris to Istanbul. He likes spinach. This sequence is incoherent because it is unclear to a reader why the second sentence follows the ﬁrst; what does liking spinach have to do with train trips? In fact, a reader might go to some effort to try to ﬁgure out how the discourse could be coherent; perhaps there is a French spinach shortage? The very fact that hearers try to identify such connections suggests that human discourse comprehension involves the need to establish this kind of coherence. By contrast, in the following coherent example: (24.2) Jane took a train from Paris to Istanbul. She had to attend a conference. 532 CHAPTER 24 • D ISCOURSE COHERENCE the second sentence gives a REASON for Jane’s action in the ﬁrst sentence. Struc- tured relationships like REASON that hold between text units are called coherence relations, and coherent discourses are structured by many such coherence relations.coherence relations Coherence relations are introduced in Section 24.1. A second way a discourse can be locally coherent is by virtue of being “about” someone or something. In a coherent discourse some entities are salient, and the discourse focuses on them and doesn’t go back and forth between multiple entities. This is called entity-based coherence. Consider the following incoherent passage, in which the salient entity seems to wildly swing from John to Jenny to the piano store to the living room, back to Jenny, then the piano again: (24.3) John wanted to buy a piano for his living room. Jenny also wanted to buy a piano. He went to the piano store. It was nearby. The living room was on the second ﬂoor. She didn’t ﬁnd anything she liked. The piano he bought was hard to get up to that ﬂoor. Entity-based coherence models measure this kind of coherence by tracking salient entities across a discourse. For example Centering Theory (Grosz et al., 1995), theCentering Theory most inﬂuential theory of entity-based coherence, keeps track of which entities in the discourse model are salient at any point (salient entities are more likely to be pronominalized or to appear in prominent syntactic positions like subject or object). In Centering Theory, transitions between sentences that maintain the same salient entity are considered more coherent than ones that repeatedly shift between entities. The entity grid model of coherence (Barzilay and Lapata, 2008) is a commonlyentity grid used model that realizes some of the intuitions of the Centering Theory framework. Entity-based coherence is introduced in Section 24.3. Finally, discourses can be locally coherent by being topically coherent: nearbytopically coherent sentences are generally about the same topic and use the same or similar vocab- ulary to discuss these topics. Because topically coherent discourses draw from a single semantic ﬁeld or topic, they tend to exhibit the surface property known as lexical cohesion (Halliday and Hasan, 1976): the sharing of identical or semanti-lexical cohesion cally related words in nearby sentences. For example, the fact that the words house, chimney, garret, closet, and window— all of which belong to the same semantic ﬁeld— appear in the two sentences in (24.4), or that they share the identical word shingled, is a cue that the two are tied together as a discourse: (24.4) Before winter I built a chimney, and shingled the sides of my house... I have thus a tight shingled and plastered house... with a garret and a closet, a large window on each side.... In addition to the local coherence between adjacent or nearby sentences, dis- courses also exhibit global coherence. Many genres of text are associated with particular conventional discourse structures. Academic articles might have sections describing the Methodology or Results. Stories might follow conventional plotlines or motifs. Persuasive essays have a particular claim they are trying to argue for, and an essay might express this claim together with a structured set of premises that support the argument and demolish potential counterarguments. We’ll introduce versions of each of these kinds of global coherence. Why do we care about the local or global coherence of a discourse? Since co- herence is a property of a well-written text, coherence detection plays a part in any 24.1 • C OHERENCE RELATIONS 533 task that requires measuring the quality of a text. For example coherence can help in pedagogical tasks like essay grading or essay quality measurement that are trying to grade how well-written a human essay is (Somasundaran et al. 2014, Feng et al. 2014, Lai and Tetreault 2018). Coherence can also help for summarization; knowing the coherence relationship between sentences can help know how to select informa- tion from them. Finally, detecting incoherent text may even play a role in mental health tasks like measuring symptoms of schizophrenia or other kinds of disordered language (Ditman and Kuperberg 2010, Elvev ˚ag et al. 2007, Bedi et al. 2015, Iter et al. 2018). 24.1 Coherence Relations Recall from the introduction the difference between passages (24.5) and (24.6). (24.5) Jane took a train from Paris to Istanbul. She likes spinach. (24.6) Jane took a train from Paris to Istanbul. She had to attend a conference. The reason (24.6) is more coherent is that the reader can form a connection be- tween the two sentences, in which the second sentence provides a potentialREASON for the ﬁrst sentences. This link is harder to form for (24.5). These connections between text spans in a discourse can be speciﬁed as a set of coherence relations.coherence relation The next two sections describe two commonly used models of coherence relations and associated corpora: Rhetorical Structure Theory (RST), and the Penn Discourse TreeBank (PDTB). 24.1.1 Rhetorical Structure Theory The most commonly used model of discourse organization is Rhetorical Structure Theory (RST) (Mann and Thompson, 1987). In RST relations are deﬁned betweenRST two spans of text, generally a nucleus and a satellite. The nucleus is the unit thatnucleus satellite is more central to the writer’s purpose and that is interpretable independently; the satellite is less central and generally is only interpretable with respect to the nucleus. Some symmetric relations, however, hold between two nuclei. Below are a few examples of RST coherence relations, with deﬁnitions adapted from the RST Treebank Manual (Carlson and Marcu, 2001). Reason: The nucleus is an action carried out by an animate agent and the satellite is the reason for the nucleus. (24.7) [ NUC Jane took a train from Paris to Istanbul.] [SAT She had to attend a conference.] Elaboration: The satellite gives additional information or detail about the situation presented in the nucleus. (24.8) [ NUC Dorothy was from Kansas.] [SAT She lived in the midst of the great Kansas prairies.] Evidence: The satellite gives additional information or detail about the situation presented in the nucleus. The information is presented with the goal of convince the reader to accept the information presented in the nucleus. (24.9) [ NUC Kevin must be here.] [SAT His car is parked outside.] 534 CHAPTER 24 • D ISCOURSE COHERENCE Attribution: The satellite gives the source of attribution for an instance of reported speech in the nucleus. (24.10) [ SAT Analysts estimated] [NUC that sales at U.S. stores declined in the quarter, too] List: In this multinuclear relation, a series of nuclei is given, without contrast or explicit comparison: (24.11) [ NUC Billy Bones was the mate; ] [NUC Long John, he was quartermaster] RST relations are traditionally represented graphically; the asymmetric Nucleus- Satellite relation is represented with an arrow from the satellite to the nucleus: Kevin must be here. His car is parked outside evidence We can also talk about the coherence of a larger text by considering the hierar- chical structure between coherence relations. Figure 24.1 shows the rhetorical struc- ture of a paragraph from Marcu (2000a) for the text in (24.12) from the Scientiﬁc American magazine. (24.12) With its distant orbit–50 percent farther from the sun than Earth–and slim atmospheric blanket, Mars experiences frigid weather conditions. Surface temperatures typically average about -60 degrees Celsius (-76 degrees Fahrenheit) at the equator and can dip to -123 degrees C near the poles. Only the midday sun at tropical latitudes is warm enough to thaw ice on occasion, but any liquid water formed in this way would evaporate almost instantly because of the low atmospheric pressure. Title (1) Mars 2-9 evidence 2-3 background (2) WIth its distant orbit <p> -- 50 percent farther from the sun than Earth -- </p> and slim atmospheric blanket, (3) Mars experiences frigid weather conditions. 4-9 elaboration-additional (4) Surface temperatures typically average about -60 degrees Celsius <p> (-76 degrees Fahrenheit)</p> at the equator 4-5 List (5) and can dip to -123 degrees C near the poles. 6-9 Contrast 6-7 (6) Only the midday sun at tropical latitudes is warm enough (7) to thaw ice on occasion, purpose 8-9 explanation-argumentative (8) but any liquid water formed in this way would evaporate almost instantly (9) because of the low atmospheric pressure. Figure 24.1 A discourse tree for the Scientiﬁc American text in (24.12), from Marcu (2000a). Note that asymmetric relations are represented with a curved arrow from the satellite to the nucleus. The leaves in the Fig. 24.1 tree correspond to text spans of a sentence, clause or phrase that are called elementary discourse units or EDUs in RST; these units canEDU also be referred to as discourse segments. Because these units may correspond to arbitrary spans of text, determining the boundaries of an EDU is an important task for extracting coherence relations. Roughly speaking, one can think of discourse 24.1 • C OHERENCE RELATIONS 535 segments as being analogous to constituents in sentence syntax, and indeed as we’ll see in Section 24.2 we generally draw on parsing algorithms to infer discourse struc- ture. There are corpora for many discourse coherence models; the RST Discourse TreeBank (Carlson et al., 2001) is the largest available discourse corpus. It con- sists of 385 English language documents selected from the Penn Treebank, with full RST parses for each one, using a large set of 78 distinct relations, grouped into 16 classes. RST treebanks exist also for Spanish, German, Basque, Dutch and Brazilian Portuguese (Braud et al., 2017). Now that we’ve seen examples of coherence, we can see more clearly how a coherence relation can play a role in summarization or information extraction. For example, the nuclei of a text presumably express more important information than the satellites, which might be dropped in a summary. 24.1.2 Penn Discourse TreeBank (PDTB) The Penn Discourse TreeBank (PDTB) is a second commonly used dataset thatPDTB embodies another model of coherence relations (Miltsakaki et al. 2004, Prasad et al. 2008, Prasad et al. 2014). PDTB labeling is lexically grounded. Instead of asking annotators to directly tag the coherence relation between text spans, they were given a list of discourse connectives, words that signal discourse relations, like because,discourse connectives although, when, since, or as a result. In a part of a text where these words marked a coherence relation between two text spans, the connective and the spans were then annotated, as in Fig. 24.13, where the phrase as a result signals a causal relationship between what PDTB calls Arg1 (the ﬁrst two sentences, here in italics) and Arg2 (the third sentence, here in bold). (24.13) Jewelry displays in department stores were often cluttered and uninspired. And the merchandise was, well, fake. As a result, marketers of faux gems steadily lost space in department stores to more fashionable rivals—cosmetics makers. (24.14) In July, the Environmental Protection Agency imposed a gradual ban on virtually all uses of asbestos. (implicit=as a result) By 1997, almost all remaining uses of cancer-causing asbestos will be outlawed. Not all coherence relations are marked by an explicit discourse connective, and so the PDTB also annotates pairs of neighboring sentences with no explicit signal, like (24.14). The annotator ﬁrst chooses the word or phrase that could have been its signal (in this case as a result), and then labels its sense. For example for the am- biguous discourse connectivesince annotators marked whether it is using a CAUSAL or a TEMPORAL sense. The ﬁnal dataset contains roughly 18,000 explicit relations and 16,000 implicit relations. Fig. 24.2 shows examples from each of the 4 major semantic classes, while Fig. 24.3 shows the full tagset. Unlike the RST Discourse Treebank, which integrates these pairwise coherence relations into a global tree structure spanning an entire discourse, the PDTB does not annotate anything above the span-pair level, making no commitment with respect to higher-level discourse structure. There are also treebanks using similar methods for other languages; (24.15) shows an example from the Chinese Discourse TreeBank (Zhou and Xue, 2015). Because Chinese has a smaller percentage of explicit discourse connectives than English (only 22% of all discourse relations are marked with explicit connectives, 536 CHAPTER 24 • D ISCOURSE COHERENCE Class Type Example TEMPORAL SYNCHRONOUS The parishioners of St. Michael and All Angels stop to chat at the church door, as members here always have. (Implicit while) In the tower, ﬁve men and women pull rhythmically on ropes attached to the same ﬁve bells that ﬁrst sounded here in 1614. CONTINGENCY REASON Also unlike Mr. Ruder, Mr. Breeden appears to be in a position to get somewhere with his agenda. (implicit=because) As a for- mer White House aide who worked closely with Congress, he is savvy in the ways of Washington. COMPARISON CONTRAST The U.S. wants the removal of what it perceives as barriers to investment; Japan denies there are real barriers. EXPANSION CONJUNCTION Not only do the actors stand outside their characters and make it clear they are at odds with them, but they often literally stand on their heads. Figure 24.2 The four high-level semantic distinctions in the PDTB sense hierarchy Temporal Comparison •Asynchronous •Contrast (Juxtaposition, Opposition) •Synchronous (Precedence, Succession) •Pragmatic Contrast (Juxtaposition, Opposition) •Concession (Expectation, Contra-expectation) •Pragmatic Concession Contingency Expansion •Cause (Reason, Result) •Exception •Pragmatic Cause (Justiﬁcation) •Instantiation •Condition (Hypothetical, General, Unreal Present/Past, Factual Present/Past) •Restatement (Speciﬁcation, Equivalence, Generalization) •Pragmatic Condition (Relevance, Implicit As- sertion) •Alternative (Conjunction, Disjunction, Chosen Alterna- tive) •List Figure 24.3 The PDTB sense hierarchy. There are four top-level c¯lasses, 16 types, and 23 subtypes (not all types have subtypes). 11 of the 16 types are commonly used for implicit argument classiﬁcation; the 5 types in italics are too rare in implicit labeling to be used. compared to 47% in English), annotators labeled this corpus by directly mapping pairs of sentences to 11 sense tags, without starting with a lexical discourse connec- tor. (24.15) [ Conn 为] [Arg2 推动图们江地区开发] ，[Arg1 韩国捐款一百万美元设立了图们江发展基金] “[In order to] [Arg2 promote the development of the Tumen River region], [Arg1 South Korea donated one million dollars to establish the Tumen River Development Fund].” These discourse treebanks have been used for shared tasks on multilingual dis- course parsing (Xue et al., 2016). 24.2 Discourse Structure Parsing Given a sequence of sentences, how can we automatically determine the coherence relations between them? This task is often called discourse parsing (even thoughdiscourse parsing for PDTB we are only assigning labels to leaf spans and not building a full parse 24.2 • D ISCOURSE STRUCTURE PARSING 537 tree as we do for RST). 24.2.1 EDU segmentation for RST parsing RST parsing is generally done in two stages. The ﬁrst stage, EDU segmentation, extracts the start and end of each EDU. The output of this stage would be a labeling like the following: (24.16) [Mr. Rambo says] e1 [that a 3.2-acre property]e2 [overlooking the San Fernando Valley]e3 [is priced at $4 million]e4 [because the late actor Erroll Flynn once lived there.]e5 Since EDUs roughly correspond to clauses, early models of EDU segmentation ﬁrst ran a syntactic parser, and then post-processed the output. Modern systems generally use neural sequence models supervised by the gold EDU segmentation in datasets like the RST Discourse Treebank. Fig. 24.4 shows an example architecture simpliﬁed from the algorithm of Lukasik et al. (2020) that predicts for each token whether or not it is a break. Here the input sentence is passed through an encoder and then passed through a linear layer and a softmax to produce a sequence of 0s and 1, where 1 indicates the start of an EDU. Mr. Rambo says that ENCODER … 0 0 0 1 linear layer softmax EDU break Figure 24.4 Predicting EDU segment beginnings from encoded text. 24.2.2 RST parsing Tools for building RST coherence structure for a discourse have long been based on syntactic parsing algorithms like shift-reduce parsing (Marcu, 1999). Many modern RST parsers since Ji and Eisenstein (2014) draw on the neural syntactic parsers we saw in Chapter 20, using representation learning to build representations for each span, and training a parser to choose the correct shift and reduce actions based on the gold parses in the training set. We’ll describe the shift-reduce parser of Yu et al. (2018). The parser state con- sists of a stack and a queue, and produces this structure by taking a series of actions on the states. Actions include: • shift: pushes the ﬁrst EDU in the queue onto the stack creating a single-node subtree. • reduce(l,d): merges the top two subtrees on the stack, wherel is the coherence relation label, and d is the nuclearity direction, d ∈{NN,NS,SN}. As well as the pop root operation, to remove the ﬁnal tree from the stack. Fig. 24.6 shows the actions the parser takes to build the structure in Fig. 24.5. 538 CHAPTER 24 • D ISCOURSE COHERENCE 560 e1 e2 e3 e4 attr elab elab e1: American Telephone & Telegraph Co. said it e2: will lay off 75 to 85 technicians here , effective Nov. 1. e3: The workers install , maintain and repair its private branch exchanges, e4: which are large intracompany telephone networks. Figure 1: An example of RST discourse tree, where{e1,e 2,e 3,e 4} are EDUs, attr and elab are discourse relation labels, and arrows indicate the nuclearities of discourse relations. RST discourse parsing. Other studies still adopt discrete syntax features proposed by statistical models, feeding them into neural network models (Braud et al., 2016; Braud et al., 2017). The above approaches model syntax trees in an explicit way, requiring discrete syntax parsing outputs as inputs for RST parsing. These approaches may suffer from the error propagation problem. Syntax trees produced by a supervised syntax parsing model could have errors, which may propagate into discourse parsing models. The problem could be extremely serious when inputs of discourse parsing have different distributions with the training data of the supervised syntax parser. Recently, Zhang et al. (2017) suggest an alternative method, which extracts syntax features from a Bi-Afﬁne dependency parser (Dozat and Manning, 2016), and the method gives competitive performances on relation extraction. It actually represents syntax trees implicitly, thus it can reduce the error propagation problem. In this work, we investigate the implicit syntax feature extraction approach for RST parsing. In ad- dition, we propose a transition-based neural model for this task, which is able to incorporate various features ﬂexibly. We exploit hierarchical bi-directional LSTMs (Bi-LSTMs) to encode texts, and further enhance the transition-based model with dynamic oracle. Based on the proposed model, we study the effectiveness of our proposed implicit syntax features. We conduct experiments on a standard RST dis- course TreeBank (Carlson et al., 2003). First, we evaluate the performance of our proposed transition- based baseline, ﬁnding that the model is able to achieve strong performances after applying dynamic oracle. Then we evaluate the effectiveness of implicit syntax features extracted from a Bi-Afﬁne depen- dency parser. Results show that the implicit syntax features are effective, giving better performances than explicit Tree-LSTM (Li et al., 2015b). Our codes will be released for public under the Apache License 2.0 athttps://github.com/yunan4nlp/NNDisParser. In summary, we mainly make the following two contributions in this work: (1) we propose a transition- based neural RST discourse parsing model with dynamic oracle, (2) we compare three different syntactic integration approaches proposed by us. The rest of the paper is organized as follows. Section 2 describes our proposed models including the transition-based neural model, the dynamic oracle strategy and the implicit syntax feature extraction approach. Section 3 presents the experiments to evaluate our models. Section 4 shows the related work. Finally, section 5 draws conclusions. 2 Transition-based Discourse Parsing We follow Ji and Eisenstein (2014), exploiting a transition-based framework for RST discourse parsing. The framework is conceptually simple and ﬂexible to support arbitrary features, which has been widely used in a number of NLP tasks (Zhu et al., 2013; Dyer et al., 2015; Zhang et al., 2016). In addition, a transition-based model formalizes a certain task into predicting a sequence of actions, which is essential similar to sequence-to-sequence models proposed recently (Bahdanau et al., 2014). In the following, we ﬁrst describe the transition system for RST discourse parsing, and then introduce our neural network model by its encoder and decoder parts, respectively. Thirdly, we present our proposed dynamic oracle strategy aiming to enhance the transition-based model. Then we introduce the integration method of implicit syntax features. Finally we describe the training method of our neural network models. 2.1 The Transition-based System The transition-based framework converts a structural learning problem into a sequence of action predic- tions, whose key point is a transition system. A transition system consists of two parts: states and actions. The states are used to store partially-parsed results and the actions are used to control state transitions. Figure 24.5 Example RST discourse tree, showing four EDUs. Figure from Yu et al. (2018). 561 Step Stack Queue Action Relation 1 ? e1 , e2 , e3 , e4 SH ? 2 e1 e2 , e3 , e4 SH ? 3 e1 , e2 e3 , e4 RD (attr,SN) ? 4 e1:2 e3 , e4 SH de1 e2 5 e1:2 , e3 e4 SH de1 e2 6 e1:2 , e3 , e4 ? RD (elab,NS) de1 e2 7 e1:2 , e3:4 ? RD (elab,SN) de1 e2, de3e4 8 e1:4 ? PR de1 e2, de3e4 , \e1:2 e3:4 Table 1: An example of the transition-based system for RST discourse parsing. The initial state is an empty state, and the ﬁnal state represents a full result. There are three kinds of actions in our transition system: • Shift (SH), which removes the ﬁrst EDU in the queue onto the stack, forming a single-node subtree. • Reduce (RD) ( l,d), which merges the top two subtrees on the stack, where l is a discourse relation label, and d 2{ NN , NS , SN } indicates the relation nuclearity (nuclear (N) or satellite (S)). • Pop Root(PR), which pops out the top tree on the stack, marking the decoding being completed, when the stack holds only one subtree and the queue is empty. Given the RST tree as shown in Figure 1, it can be generated by the following action sequence: {SH, SH, RD (attr,SN) , SH, SH, RD (elab,NS) , RD (elab,SN) , PR }. Table 1 shows the decoding process in detail. By this way, we naturally convert RST discourse parsing into predicting a sequence of transition actions, where each line includes a state and next step action referring to the tree. 2.2 Encoder-Decoder Previous transition-based RST discourse parsing studies exploit statistical models, using manually- designed discrete features (Sagae, 2009; Heilman and Sagae, 2015; Wang et al., 2017). In this work, we propose a transition-based neural model for RST discourse parsing, which follows an encoder-decoder framework. Given an input sequence of EDUs {e1,e 2,. . . ,en}, the encoder computes the input represen- tations {he 1, he 2,. . . ,he n}, and the decoder predicts next step actions conditioned on the encoder outputs. 2.2.1 Encoder We follow Li et al. (2016), using hierarchical Bi-LSTMs to encode the source EDU inputs, where the ﬁrst-layer is used to represent sequencial words inside of EDUs, and the second layer is used to represent sequencial EDUs. Given an input sentence {w1,w 2,. . . ,wm}, ﬁrst we represent each word by its form (e.g., wi) and POS tag (e.g. ti), concatenating their neural embeddings. By this way, the input vectors of the ﬁrst-layer Bi-LSTM are {xw 1 , xw 2 ,. . . ,xw m}, where xw i = emb(wi) emb(ti), and then we apply Bi-LSTM directly, obtaining: {hw 1 , hw 2 ,. . . ,hw m} = Bi-LSTM ({xw 1 , xw 2 ,. . . ,xw m}) (1) The second-layer Bi-LSTM is built over sequential EDUs. We should ﬁrst obtain a suitable representa- tion for each EDU, which is composed by a span of words inside a certain sentence. Assuming an EDU with its words by {ws,w s+1,. . . ,wt}, after applying the ﬁrst-layer Bi-LSTM, we obtain their representa- tions by {hw s , hw s+1..., hw t }, then we calculate the EDU representation by average pooling: xe = 1 t s +1 tX s hw k (2) When the EDU representations are ready, we apply the second-layer Bi-LSTM directly, resulting: {he 1, he 2,. . . ,he n} = Bi-LSTM ({xe 1, xe 2,. . . ,xe n}) (3) Figure 24.6 Parsing the example of Fig. 24.5 using a shift-reduce parser. Figure from Yu et al. (2018). The Yu et al. (2018) uses an encoder-decoder architecture, where the encoder represents the input span of words and EDUs using a hierarchical biLSTM. The ﬁrst biLSTM layer represents the words inside an EDU, and the second represents the EDU sequence. Given an input sentence w1,w2,...,wm, the words can be repre- sented as usual (by static embeddings, combinations with character embeddings or tags, or contextual embeddings) resulting in an input word representation sequence xw 1 ,xw 2 ,...,xw m. The result of the word-level biLSTM is then a sequence of hw values: hw 1 ,hw 2 ,...,hw m = biLSTM(xw 1 ,xw 2 ,...,xw m) (24.17) An EDU of spanws,ws+1,...,wt then has biLSTM output representationhw s ,hw s+1,...,hw t , and is represented by average pooling: xe = 1 t −s +1 t∑ k=s hw k (24.18) The second layer uses this input to compute a ﬁnal representation of the sequence of EDU representations he: he 1,he 2,...,he n = biLSTM(xe 1,xe 2,...,xe n) (24.19) The decoder is then a feedforward network W that outputs an action o based on a concatenation of the top three subtrees on the stack ( so,s1,s2) plus the ﬁrst EDU in the queue (q0): o = W(ht s0,ht s1,ht s2,he q0) (24.20) where the representation of the EDU on the queue he q0 comes directly from the encoder, and the three hidden vectors representing partial trees are computed by average pooling over the encoder output for the EDUs in those trees: hts = 1 j −i +1 j∑ k=i he k (24.21) 24.2 • D ISCOURSE STRUCTURE PARSING 539 Training ﬁrst maps each RST gold parse tree into a sequence of oracle actions, and then uses the standard cross-entropy loss (with l2 regularization) to train the system to take such actions. Give a state S and oracle action a, we ﬁrst compute the decoder output using Eq. 24.20, apply a softmax to get probabilities: pa = exp(oa)∑ a′∈A exp(oa′) (24.22) and then computing the cross-entropy loss: LCE () = −log(pa)+ λ 2 \|\|Θ\|\|2 (24.23) RST discourse parsers are evaluated on the test section of the RST Discourse Tree- bank, either with gold EDUs or end-to-end, using the RST-Pareval metrics (Marcu, 2000b). It is standard to ﬁrst transform the gold RST trees into right-branching bi- nary trees, and to report four metrics: trees with no labels (S for Span), labeled with nuclei (N), with relations (R), or both (F for Full), for each metric computing micro-averaged F1 over all spans from all documents (Marcu 2000b, Morey et al. 2017). 24.2.3 PDTB discourse parsing PDTB discourse parsing, the task of detecting PDTB coherence relations between spans, is sometimes called shallow discourse parsing because the task just involves shallow discourse parsing ﬂat relationships between text spans, rather than the full trees of RST parsing. The set of four subtasks for PDTB discourse parsing was laid out by Lin et al. (2014) in the ﬁrst complete system, with separate tasks for explicit (tasks 1-3) and implicit (task 4) connectives: 1. Find the discourse connectives (disambiguating them from non-discourse uses) 2. Find the two spans for each connective 3. Label the relationship between these spans 4. Assign a relation between every adjacent pair of sentences Many systems have been proposed for Task 4: taking a pair of adjacent sentences as input and assign a coherence relation sense label as output. The setup often fol- lows Lin et al. (2009) in assuming gold sentence span boundaries and assigning each adjacent span one of the 11 second-level PDTB tags or none (removing the 5 very rare tags of the 16 shown in italics in Fig. 24.3). A simple but very strong algorithm for Task 4 is to represent each of the two spans by BERT embeddings and take the last layer hidden state corresponding to the position of the [CLS] token, pass this through a single layer tanh feedforward network and then a softmax for sense classiﬁcation (Nie et al., 2019). Each of the other tasks also have been addressed. Task 1 is to disambiguat- ing discourse connectives from their non-discourse use. For example as Pitler and Nenkova (2009) point out, the word and is a discourse connective linking the two clauses by an elaboration/expansion relation in (24.24) while it’s a non-discourse NP conjunction in (24.25): (24.24) Selling picked up as previous buyers bailed out of their positions and aggressive short sellers—anticipating further declines—moved in. (24.25) My favorite colors are blue and green. 540 CHAPTER 24 • D ISCOURSE COHERENCE Similarly, once is a discourse connective indicating a temporal relation in (24.26), but simply a non-discourse adverb meaning ‘formerly’ and modifyingused in (24.27): (24.26) The asbestos ﬁber, crocidolite, is unusually resilient once it enters the lungs, with even brief exposures to it causing symptoms that show up decades later, researchers said. (24.27) A form of asbestos once used to make Kent cigarette ﬁlters has caused a high percentage of cancer deaths among a group of workers exposed to it more than 30 years ago, researchers reported. Determining whether a word is a discourse connective is thus a special case of word sense disambiguation. Early work on disambiguation showed that the 4 PDTB high-level sense classes could be disambiguated with high (94%) accuracy used syntactic features from gold parse trees (Pitler and Nenkova, 2009). Recent work performs the task end-to-end from word inputs using a biLSTM-CRF with BIO outputs (B-CONN , I-CONN , O) (Yu et al., 2019). For task 2, PDTB spans can be identiﬁed with the same sequence models used to ﬁnd RST EDUs: a biLSTM sequence model with pretrained contextual embedding (BERT) inputs (Muller et al., 2019). Simple heuristics also do pretty well as a base- line at ﬁnding spans, since 93% of relations are either completely within a single sentence or span two adjacent sentences, with one argument in each sentence (Biran and McKeown, 2015). 24.3 Centering and Entity-Based Coherence A second way a discourse can be coherent is by virtue of being “about” some entity. This idea that at each point in the discourse some entity is salient, and a discourse is coherent by continuing to discuss the same entity, appears early in functional lin- guistics and the psychology of discourse (Chafe 1976, Kintsch and Van Dijk 1978), and soon made its way to computational models. In this section we introduce two models of this kind of entity-based coherence: Centering Theory (Grosz et al.,entity-based 1995), and the entity grid model of Barzilay and Lapata (2008). 24.3.1 Centering Centering Theory (Grosz et al., 1995) is a theory of both discourse salience andCentering Theory discourse coherence. As a model of discourse salience, Centering proposes that at any given point in the discourse one of the entities in the discourse model is salient: it is being “centered” on. As a model of discourse coherence, Centering proposes that discourses in which adjacent sentences CONTINUE to maintain the same salient entity are more coherent than those which SHIFT back and forth between multiple entities (we will see that CONTINUE and SHIFT are technical terms in the theory). The following two texts from Grosz et al. (1995) which have exactly the same propositional content but different saliences, can help in understanding the main Centering intuition. (24.28) a. John went to his favorite music store to buy a piano. b. He had frequented the store for many years. c. He was excited that he could ﬁnally buy a piano. d. He arrived just as the store was closing for the day. 24.3 • C ENTERING AND ENTITY -BASED COHERENCE 541 (24.29) a. John went to his favorite music store to buy a piano. b. It was a store John had frequented for many years. c. He was excited that he could ﬁnally buy a piano. d. It was closing just as John arrived. While these two texts differ only in how the two entities (John and the store) are realized in the sentences, the discourse in (24.28) is intuitively more coherent than the one in (24.29). As Grosz et al. (1995) point out, this is because the discourse in (24.28) is clearly about one individual, John, describing his actions and feelings. The discourse in (24.29), by contrast, focuses ﬁrst on John, then the store, then back to John, then to the store again. It lacks the “aboutness” of the ﬁrst discourse. Centering Theory realizes this intuition by maintaining two representations for each utterance Un. The backward-looking center of Un, denoted as Cb(Un), rep- backward- looking center resents the current salient entity, the one being focused on in the discourse after Un is interpreted. The forward-looking centers of Un, denoted as Cf (Un), are a setforward-looking center of potential future salient entities, the discourse entities evoked by Un any of which could serve as Cb (the salient entity) of the following utterance, i.e. Cb(Un+1). The set of forward-looking centers Cf (Un) are ranked according to factors like discourse salience and grammatical role (for example subjects are higher ranked than objects, which are higher ranked than all other grammatical roles). We call the highest-ranked forward-looking center Cp (for “preferred center”). Cp is a kind of prediction about what entity will be talked about next. Sometimes the next utterance indeed talks about this entity, but sometimes another entity becomes salient instead. We’ll use here the algorithm for centering presented in Brennan et al. (1987), which deﬁnes four intersentential relationships between a pair of utterances Un and Un+1 that depend on the relationship between Cb(Un+1), Cb(Un), and Cp(Un+1); these are shown in Fig. 24.7. Cb(Un+1) =Cb(Un) Cb(Un+1) ̸= Cb(Un) or undeﬁned Cb(Un) Cb(Un+1) =Cp(Un+1) Continue Smooth-Shift Cb(Un+1) ̸= Cp(Un+1) Retain Rough-Shift Figure 24.7 Centering Transitions for Rule 2 from Brennan et al. (1987). The following rules are used by the algorithm: Rule 1: If any element of Cf (Un) is realized by a pronoun in utterance Un+1, then Cb(Un+1) must be realized as a pronoun also. Rule 2: Transition states are ordered. Continue is preferred to Retain is preferred to Smooth-Shift is preferred to Rough-Shift. Rule 1 captures the intuition that pronominalization (including zero-anaphora) is a common way to mark discourse salience. If there are multiple pronouns in an utterance realizing entities from the previous utterance, one of these pronouns must realize the backward center Cb; if there is only one pronoun, it must be Cb. Rule 2 captures the intuition that discourses that continue to center the same en- tity are more coherent than ones that repeatedly shift to other centers. The transition table is based on two factors: whether the backward-looking center Cb is the same from Un to Un+1 and whether this discourse entity is the one that is preferred ( Cp) in the new utterance Un+1. If both of these hold, a CONTINUE relation, the speaker has been talking about the same entity and is going to continue talking about that 542 CHAPTER 24 • D ISCOURSE COHERENCE entity. In a RETAIN relation, the speaker intends to SHIFT to a new entity in a future utterance and meanwhile places the current entity in a lower rank Cf . In a SHIFT relation, the speaker is shifting to a new salient entity. Let’s walk though the start of (24.28) again, repeated as (24.30), showing the representations after each utterance is processed. (24.30) John went to his favorite music store to buy a piano. ( U1) He was excited that he could ﬁnally buy a piano. (U2) He arrived just as the store was closing for the day. (U3) It was closing just as John arrived (U4) Using the grammatical role hierarchy to order the Cf , for sentence U1 we get: Cf (U1): {John, music store, piano} Cp(U1): John Cb(U1): undeﬁned and then for sentence U2: Cf (U2): {John, piano} Cp(U2): John Cb(U2): John Result: Continue ( Cp(U2)=Cb(U2); Cb(U1) undeﬁned) The transition from U1 to U2 is thus a CONTINUE . Completing this example is left as exercise (1) for the reader 24.3.2 Entity Grid model Centering embodies a particular theory of how entity mentioning leads to coher- ence: that salient entities appear in subject position or are pronominalized, and that discourses are salient by means of continuing to mention the same entity in such ways. The entity grid model of Barzilay and Lapata (2008) is an alternative way toentity grid capture entity-based coherence: instead of having a top-down theory, the entity-grid model using machine learning to induce the patterns of entity mentioning that make a discourse more coherent. The model is based around an entity grid, a two-dimensional array that repre- sents the distribution of entity mentions across sentences. The rows represent sen- tences, and the columns represent discourse entities (most versions of the entity grid model focus just on nominal mentions). Each cell represents the possible appearance of an entity in a sentence, and the values represent whether the entity appears and its grammatical role. Grammatical roles are subject ( S), object (O), neither (X), or ab- sent (–); in the implementation of Barzilay and Lapata (2008), subjects of passives are represented with O, leading to a representation with some of the characteristics of thematic roles. Fig. 24.8 from Barzilay and Lapata (2008) shows a grid for the text shown in Fig. 24.9. There is one row for each of the six sentences. The second column, for the entity ‘trial’, isO – – – X, showing that the trial appears in the ﬁrst sentence as direct object, in the last sentence as an oblique, and does not appear in the middle sentences. The third column, for the entity Microsoft, shows that it appears as sub- ject in sentence 1 (it also appears as the object of the prepositionagainst, but entities that appear multiple times are recorded with their highest-ranked grammatical func- tion). Computing the entity grids requires extracting entities and doing coreference 24.3 • C ENTERING AND ENTITY -BASED COHERENCE 543 Computational Linguistics Volume 34, Number 1 these patterns can be encoded as feature vectors appropriate for performing coherence- related ranking and classiﬁcation tasks. 3.1 The Entity-Grid Discourse Representation Each text is represented by an entity grid,at w o - d i m e n s i o n a la r r a yt h a tc a p t u r e s the distribution of discourse entities across text sentences. We follow Miltsakaki and Kukich (2000) in assuming that our unit of analysis is the traditional sentence (i.e., a main clause with accompanying subordinate and adjunct clauses). The rows of the grid correspond to sentences, and the columns correspond to discourse entities. By discourse entitywe mean a class of coreferent noun phrases (we explain in Section 3.3 how coreferent entities are identiﬁed). For each occurrence of a discourse entity in the text, the corresponding grid cell contains information about its presence or absence in a sequence of sentences. In addition, for entities present in a given sentence, grid cells contain information about their syntactic role. Such information can be expressed in many ways (e.g., using constituent labels or thematic role information). Because grammatical relations ﬁgure prominently in entity-based theories of local coherence (see Section 2), they serve as a logical point of departure. Each grid cell thus corresponds to a string from a set of categories reﬂecting whether the entity in question is a subject (S), object (O), or neither (X). Entities absent from a sentence are signaled by gaps (–). Grammatical role information can be extracted from the output of a broad-coverage dependency parser (Lin 2001; Briscoe and Carroll 2002) or any state-of-the art statistical parser (Collins 1997; Charniak 2000). We discuss how this information was computed for our experiments in Section 3.3. Table 1 illustrates a fragment of an entity grid constructed for the text in Table 2. Because the text contains six sentences, the grid columns are of length six. Consider for instance the grid column for the entitytrial, [O –––– X].I tr e c o r d st h a ttrial is present in sentences 1 and 6 (asO and X,r e s p e c t i v e l y )b u ti sa b s e n tf r o mt h er e s to ft h e sentences. Also note that the grid in Table 1 takes coreference resolution into account. Even though the same entity appears in different linguistic forms, for example,Microsoft Corp., Microsoft,a n dthe company , it is mapped to a single entry in the grid (see the column introduced byMicrosoft in Table 1). Table 1 A fragment of the entity grid. Noun phrases are represented by their head nouns. Grid cells correspond to grammatical roles: subjects (S), objects (O), or neither (X). Department Trial Microsoft Evidence Competitors Markets Products Brands Case Netscape Software Tactics Government Suit Earnings 1 SO SXO –––––––––– 1 2 –– O –– XSO –––– – –– 2 3 –– SO –––– SOO ––– – 3 4 –– S ––––– ––– S –– – 4 5 ––––––––––– – SO – 5 6 – XS ––––– ––– – – – O 6 6 Figure 24.8 Part of the entity grid for the text in Fig. 24.9. Entities are listed by their head noun; each cell represents whether an entity appears as subject (S), object (O), neither (X), or is absent (–). Figure from Barzilay and Lapata (2008). Barzilay and Lapata Modeling Local Coherence Table 2 Summary augmented with syntactic annotations for grid computation. 1 [The Justice Department] S is conducting an [anti-trust trial]O against [Microsoft Corp.]X with [evidence] X that [the company]S is increasingly attempting to crush [competitors]O . 2[ M i c r o s o f t ]O is accused of trying to forcefully buy into [markets]X where [its own products] S are not competitive enough to unseat [established brands]O . 3[ T h e c a s e ]S revolves around [evidence]O of [Microsoft] S aggressively pressuring [Netscape] O into merging [browser software]O . 4[ M i c r o s o f t ]S claims [its tactics]S are commonplace and good economically. 5 [The government] S may ﬁle [a civil suit]O ruling that [conspiracy]S to curb [competition]O through [collusion] X is [a violation of the Sherman Act]O . 6[ M i c r o s o f t ]S continues to show [increased earnings]O despite [the trial]X . When a noun is attested more than once with a different grammatical role in the same sentence, we default to the role with the highest grammatical ranking: subjects are ranked higher than objects, which in turn are ranked higher than the rest. For example, the entityMicrosoft is mentioned twice in Sentence 1 with the grammatical rolesx (for Microsoft Corp.)a n ds (for the company ), but is represented only bys in the grid (see Tables 1 and 2). 3.2 Entity Grids as Feature Vectors Af u n d a m e n t a la s s u m p t i o nu n d e r l y i n go u ra p p r o a c hi st h a tt h ed i s t r i b u t i o no fe n t i t i e s in coherent texts exhibits certain regularities reﬂected in grid topology. Some of these regularities are formalized in Centering Theory as constraints on transitions of the local focus in adjacent sentences. Grids of coherent texts are likely to have some dense columns (i.e., columns with just a few gaps, such asMicrosoft in Table 1) and many sparse columns which will consist mostly of gaps (seemarkets and earnings in Table 1). One would further expect that entities corresponding to dense columns are more often subjects or objects. These characteristics will be less pronounced in low-coherence texts. Inspired by Centering Theory, our analysis revolves around patterns of local entity transitions. A local entity transitionis a sequence{S, O, X, –}n that represents entity occurrences and their syntactic roles inn adjacent sentences. Local transitions can be easily obtained from a grid as continuous subsequences of each column. Each transition will have a certain probability in a given grid. For instance, the probability of the transition [S –] in the grid from Table 1 is 0.08 (computed as a ratio of its frequency [i.e., six] divided by the total number of transitions of length two [i.e., 75]). Each text can thus be viewed as a distribution deﬁned over transition types. We can now go one step further and represent each text by a ﬁxed set of transition sequences using a standard feature vector notation. Each grid renderingj of a document d i corresponds to a feature vectorΦ(x ij ) = (p1(x ij ), p2(x ij ), ... , pm(x ij )), where m is the number of all predeﬁned entity transitions, andpt(x ij )t h ep r o b a b i l i t yo ft r a n s i t i o nt in gridx ij .T h i sf e a t u r ev e c t o rr e p r e s e n t a t i o ni su s e f u l l ya m e n a b l et om a c h i n el e a r n i n g algorithms (see our experiments in Sections 4–6). Furthermore, it allows the consid- eration of large numbers of transitions which could potentially uncover novel entity distribution patterns relevant for coherence assessment or other coherence-related tasks. Note that considerable latitude is available when specifying the transition types to be included in a feature vector. These can be all transitions of a given length (e.g., two or three) or the most frequent transitions within a document collection. An example of 7 Figure 24.9 A discourse with the entities marked and annotated with grammatical func- tions. Figure from Barzilay and Lapata (2008). resolution to cluster them into discourse entities (Chapter 23) as well as parsing the sentences to get grammatical roles. In the resulting grid, columns that are dense (like the column for Microsoft) in- dicate entities that are mentioned often in the texts; sparse columns (like the column for earnings) indicate entities that are mentioned rarely. In the entity grid model, coherence is measured by patterns oflocal entity tran- sition. For example, Department is a subject in sentence 1, and then not men- tioned in sentence 2; this is the transition [ S –]. The transitions are thus sequences {S,O X , –}n which can be extracted as continuous cells from each column. Each transition has a probability; the probability of [S –] in the grid from Fig. 24.8 is 0.08 (it occurs 6 times out of the 75 total transitions of length two). Fig. 24.10 shows the distribution over transitions of length 2 for the text of Fig. 24.9 (shown as the ﬁrst row d1), and 2 other documents. Computational Linguistics Volume 34, Number 1 af e a t u r es p a c ew i t ht r a n s i t i o n so fl e n g t ht w oi si l l u s t r a t e di nT a b l e3 .T h es e c o n dr o w (introduced byd1)i st h ef e a t u r ev e c t o rr e p r e s e n t a t i o no ft h eg r i di nT a b l e1 . 3.3 Grid Construction: Linguistic Dimensions One of the central research issues in developing entity-based models of coherence is determining what sources of linguistic knowledge are essential for accurate prediction, and how to encode them succinctly in a discourse representation. Previous approaches tend to agree on the features of entity distribution related to local coherence—the disagreement lies in the way these features are modeled. Our study of alternative encodings is not a mere duplication of previous ef- forts (Poesio et al. 2004) that focus on linguistic aspects of parameterization. Because we are interested in an automatically constructed model, we have to take into account com- putational and learning issues when considering alternative representations. Therefore, our exploration of the parameter space is guided by three considerations: the linguistic importance of a parameter, the accuracy of its automatic computation, and the size of the resulting feature space. From the linguistic side, we focus on properties of entity distri- bution that are tightly linked to local coherence, and at the same time allow for multiple interpretations during the encoding process. Computational considerations prevent us from considering discourse representations that cannot be computed reliably by exist- ing tools. For instance, we could not experiment with the granularity of an utterance— sentence versus clause—because available clause separators introduce substantial noise into a grid construction. Finally, we exclude representations that will explode the size of the feature space, thereby increasing the amount of data required for training the model. Entity Ex traction.The accurate computation of entity classes is key to computing mean- ingful entity grids. In previous implementations of entity-based models, classes of coref- erent nouns have been extracted manually (Miltsakaki and Kukich 2000; Karamanis et al. 2004; Poesio et al. 2004), but this is not an option for our model. An obvious solution for identifying entity classes is to employ an automatic coreference resolution tool that determines which noun phrases refer to the same entity in a document. Current approaches recast coreference resolution as a classiﬁcation task. A pair of NPs is classiﬁed as coreferring or not based on constraints that are learned from an annotated corpus. A separate clustering mechanism then coordinates the possibly contradictory pairwise classiﬁcations and constructs a partition on the set of NPs. In our experiments, we employ Ng and Cardie’s (2002) coreference resolution system. The system decides whether two NPs are coreferent by exploiting a wealth of lexical, grammatical, semantic, and positional features. It is trained on the MUC (6–7) data sets and yields state-of-the-art performance (70.4 F-measure on MUC-6 and 63.4 on MUC-7). Table 3 Example of a feature-vector document representation using all transitions of length two given syntactic categoriesS , O , X ,a n d–. SS SO SX S – OS OO OX O – XS XO XX X –– S – O – X –– d1 .01 .01 0 .08 .01 0 0 .09 0 0 0 .03 .05 .07 .03 .59 d2 .02 .01 .01 .02 0 .07 0 .02 .14 .14 .06 .04 .03 .07 0.1 .36 d3 .02 0 0 .03 .09 0 .09 .06 0 0 0 .05 .03 .07 .17 .39 8 Figure 24.10 A feature vector for representing documents using all transitions of length 2. Document d1 is the text in Fig. 24.9. Figure from Barzilay and Lapata (2008). The transitions and their probabilities can then be used as features for a machine learning model. This model can be a text classiﬁer trained to produce human-labeled coherence scores (for example from humans labeling each text as coherent or inco- herent). But such data is expensive to gather. Barzilay and Lapata (2005) introduced a simplifying innovation: coherence models can be trained by self-supervision: trained to distinguish the natural original order of sentences in a discourse from 544 CHAPTER 24 • D ISCOURSE COHERENCE a modiﬁed order (such as a randomized order). We turn to these evaluations in the next section. 24.3.3 Evaluating Neural and Entity-based coherence Entity-based coherence models, as well as the neural models we introduce in the next section, are generally evaluated in one of two ways. First, we can have humans rate the coherence of a document and train a classiﬁer to predict these human ratings, which can be categorial (high/low, or high/mid/low) or continuous. This is the best evaluation to use if we have some end task in mind, like essay grading, where human raters are the correct deﬁnition of the ﬁnal label. Alternatively, since it’s very expensive to get human labels, and we might not yet have an end-task in mind, we can use natural texts to do self-supervision. In self-supervision we pair up a natural discourse with a pseudo-document created by changing the ordering. Since naturally-ordered discourses are more coherent than random permutation (Lin et al., 2011), a successful coherence algorithm should pre- fer the original ordering. Self-supervision has been implemented in 3 ways. In the sentence order dis- crimination task (Barzilay and Lapata, 2005), we compare a document to a random permutation of its sentences. A model is considered correct for an (original, per- muted) test pair if it ranks the original document higher. Givenk documents, we can compute n permutations, resulting in kn pairs each with one original document and one permutation, to use in training and testing. In the sentence insertion task (Chen et al., 2007) we take a document, remove one of the n sentences s, and create n−1 copies of the document withs inserted into each position. The task is to decide which of the n documents is the one with the original ordering, distinguishing the original position for s from all other positions. Insertion is harder than discrimination since we are comparing documents that differ by only one sentence. Finally, in the sentence order reconstruction task (Lapata, 2003), we take a document, randomize the sentences, and train the model to put them back in the correct order. Again given k documents, we can compute n permutations, resulting in kn pairs each with one original document and one permutation, to use in training and testing. Reordering is of course a much harder task than simple classiﬁcation. 24.4 Representation learning models for local coherence The third kind of local coherence is topical or semantic ﬁeld coherence. Discourses cohere by talking about the same topics and subtopics, and drawing on the same semantic ﬁelds in doing so. The ﬁeld was pioneered by a series of unsupervised models in the 1990s of this kind of coherence that made use of lexical cohesion (Halliday and Hasan, 1976):lexical cohesion the sharing of identical or semantically related words in nearby sentences. Morris and Hirst (1991) computed lexical chains of words (like pine, bush trees, trunk) that occurred through a discourse and that were related in Roget’s Thesaurus (by being in the same category, or linked categories). They showed that the number and density of chain correlated with the topic structure. The TextTiling algorithm of HearstTextTiling (1997) computed the cosine between neighboring text spans (the normalized dot product of vectors of raw word counts), again showing that sentences or paragraph in 24.4 • R EPRESENTATION LEARNING MODELS FOR LOCAL COHERENCE 545 a subtopic have high cosine with each other, but not with sentences in a neighboring subtopic. A third early model, the LSA Coherence method of Foltz et al. (1998) was the ﬁrst to use embeddings, modeling the coherence between two sentences as the co- sine between their LSA sentence embedding vectors1, computing embeddings for a sentence s by summing the embeddings of its words w: sim(s,t) = cos(s,t) = cos( ∑ w∈s w, ∑ w∈t w) (24.31) and deﬁning the overall coherence of a text as the average similarity over all pairs of adjacent sentences si and si+1: coherence(T ) = 1 n −1 n−1∑ i=1 cos(si,si+1) (24.32) Modern neural representation-learning coherence models, beginning with Li et al. (2014), draw on the intuitions of these early unsupervised models for learning sen- tence representations and measuring how they change between neighboring sen- tences. But the new models also draw on the idea pioneered by Barzilay and Lapata (2005) of self-supervision. That is, unlike say coherence relation models, which train on hand-labeled representations for RST or PDTB, these models are trained to distinguish natural discourses from unnatural discourses formed by scrambling the order of sentences, thus using representation learning to discover the features that matter for at least the ordering aspect of coherence. Here we present one such model, the local coherence discriminator (LCD) (Xu et al., 2019). Like early models, LCD computes the coherence of a text as the av- erage of coherence scores between consecutive pairs of sentences. But unlike the early unsupervised models, LCD is a self-supervised model trained to discriminate consecutive sentence pairs (si,si+1) in the training documents (assumed to be coher- ent) from (constructed) incoherent pairs (si,s′). All consecutive pairs are positive examples, and the negative (incoherent) partner for a sentencesi is another sentence uniformly sampled from the same document as si. Fig. 24.11 describes the architecture of the model fθ , which takes a sentence pair and returns a score, higher scores for more coherent pairs. Given an input sentence pair s and t, the model computes sentence embeddings s and t (using any sentence embeddings algorithm), and then concatenates four features of the pair: (1) the concatenation of the two vectors (2) their difference s−t; (3) the absolute value of their difference \|s −t\|; (4) their element-wise product s ⊙t. These are passed through a one-layer feedforward network to output the coherence score. The model is trained to make this coherence score higher for real pairs than for negative pairs. More formally, the training objective for a corpusC of documents d, each of which consists of a list of sentences si, is: Lθ = ∑ d∈C ∑ si∈d E p(s′\|si) [L( fθ (si,si+1),fθ (si,s′))] (24.33) Ep(s′\|si) is the expectation with respect to the negative sampling distribution con- ditioned on si: given a sentence si the algorithms samples a negative sentence s′ 1 See Chapter 6 for more on LSA embeddings; they are computed by applying SVD to the term- document matrix (each cell weighted by log frequency and normalized by entropy), and then the ﬁrst 300 dimensions are used as the embedding. 546 CHAPTER 24 • D ISCOURSE COHERENCE 681 Loss function: The role of the loss function is to encouragef+ = f✓(si,s i+1) to be high while f = f✓(si,s 0) to be low. Common losses such as margin or log loss can all be used. Through exper- imental validation, we found that margin loss to be superior for this problem. Speciﬁcally,L takes on the form:L(f+,f ) = max(0,⌘ f+ + f) where ⌘ is the margin hyperparameter. Negative samples: Technically, we are free to choose any sentence s0 to form a negative pair with si. However, because of potential differ- ences in genre, topic and writing style, such neg- atives might cause the discriminative model to learn cues unrelated to coherence. Therefore, we only select sentences from the same document to construct negative pairs. Speciﬁcally, supposesi comes from document dk with length nk, then p(s0\|si) is a uniform distribution over thenk 1 sentences {sj }j 6= i from dk. For a document with n sentences, there are n1 positive pairs, and (n1)⇤(n2)/2 negative pairs. It turns out that the quadratic number of negatives provides a rich enough learning signal, while at the same time, is not too prohibitively large to be effectively cov- ered by a sampling procedure. In practice, we sample a new set of negatives each time we see a document, hence after many epochs, we can ef- fectively cover the space for even very long doc- uments. Section 5.7 discusses further details on sampling. 4.1 Model Architecture The speciﬁc neural architecture that we use forf✓ is illustrated in Figure1. We assume the use of some pre-trained sentence encoder, which is dis- cussed in the next section. Given an input sentence pair, the sentence en- coder maps the sentences to real-valued vectorsS and T . We then compute the concatenation of the following features: (1) concatenation of the two vectors (S, T); (2) element-wise differenceS T ; (3) element-wise productS ⇤T ; (4) absolute value of element-wise difference\|S T \|. The concate- nated feature representation is then fed to a one- layer MLP to output the coherence score. In practice, we make our overall coherence model bidirectional, by training a forward model with input(S, T) and a backward model with in- put (T,S ) with the same architecture but separate parameters. The coherence score is then the aver- age from the two models. Figure 1: Generic architecture for our proposed model. 4.2 Pre-trained Generative Model as the Sentence Encoder Our model can work with any pre-trained sen- tence encoder, ranging from the most simplistic average GloVe (Pennington et al., 2014) embed- dings to more sophisticated supervised or unsu- pervised pre-trained sentence encoders (Conneau et al., 2017). As mentioned in the introduction, since generative models can often be turned into sentence encoder, generative coherence model can be leveraged by our model to beneﬁt from the advantages of both generative and discriminative training, similar to (Kiros et al., 2015; Peters et al., 2018). After initialization, we freeze the genera- tive model parameters to avoid overﬁtting. In Section5, we will experimentally show that while we do beneﬁt from strong pre-trained en- coders, the fact that our local discriminative model improves over previous methods is independent of the choice of sentence encoder. 5 Experiments 5.1 Evaluation Tasks Following Nguyen and Joty(2017) and other pre- vious work, we evaluate our models on the dis- crimination and insertion tasks. Additionally, we evaluate on the paragraph reconstruction task in open-domain settings, in a similar manner toLi and Jurafsky(2017). In thediscrimination task, a document is com- pared to a random permutation of its sentences, and the model is considered correct if it scores the original document higher than the permuted one. Twenty permutations are used in the test set in ac- cordance with previous work. Figure 24.11 The architecture of the LCD model of document coherence, showing the computation of the score for a pair of sentences s and t. Figure from Xu et al. (2019). uniformly over the other sentences in the same document. L is a loss function that takes two scores, one for a positive pair and one for a negative pair, with the goal of encouraging f + = fθ (si,si+1) to be high and f −= fθ (si,s′)) to be low. Fig. 24.11 use the margin loss l( f +,f −) =max(0,η −f + + f −) where η is the margin hyper- parameter. Xu et al. (2019) also give a useful baseline algorithm that itself has quite high performance in measuring perplexity: train an RNN language model on the data, and compute the log likelihood of sentencesi in two ways, once given the preceding context (conditional log likelihood) and once with no context (marginal log likeli- hood). The difference between these values tells us how much the preceding context improved the predictability of si, a predictability measure of coherence. Training models to predict longer contexts than just consecutive pairs of sen- tences can result in even stronger discourse representations. For example a Trans- former language model trained with a contrastive sentence objective to predict text up to a distance of ±2 sentences improves performance on various discourse coher- ence tasks (Iter et al., 2020). Language-model style models are generally evaluated by the methods of Sec- tion 24.3.3, although they can also be evaluated on the RST and PDTB coherence relation tasks. 24.5 Global Coherence A discourse must also cohere globally rather than just at the level of pairs of sen- tences. Consider stories, for example. The narrative structure of stories is one of the oldest kinds of global coherence to be studied. In his inﬂuential Morphology of the Folktale, Propp (1968) models the discourse structure of Russian folktales via a kind of plot grammar. His model includes a set of character categories he called dramatis personae, like Hero, Villain, Donor, or Helper, and a set of events he called functions (like “Villain commits kidnapping”, “Donor tests Hero”, or “Hero is pursued”) that have to occur in particular order, along with other components. Propp shows that the plots of each of the fairy tales he studies can be represented as 24.5 • G LOBAL COHERENCE 547 a sequence of these functions, different tales choosing different subsets of functions, but always in the same order. Indeed Lakoff (1972) showed that Propp’s model amounted to a discourse grammar of stories, and in recent computational work Fin- layson (2016) demonstrates that some of these Proppian functions could be induced from corpora of folktale texts by detecting events that have similar actions across stories. Bamman et al. (2013) showed that generalizations over dramatis personae could be induced from movie plot summaries on Wikipedia. Their model induced latent personae from features like the actions the character takes (e.g., Villains stran- gle), the actions done to them (e.g., Villains are foiled and arrested) or the descriptive words used of them (Villains are evil). In this section we introduce two kinds of such global discourse structure that have been widely studied computationally. The ﬁrst is the structure of arguments: the way people attempt to convince each other in persuasive essays by offering claims and supporting premises. The second is somewhat related: the structure of scientiﬁc papers, and the way authors present their goals, results, and relationship to prior work in their papers. 24.5.1 Argumentation Structure The ﬁrst type of global discourse structure is the structure ofarguments. Analyzing people’s argumentation computationally is often calledargumentation mining.argumentation mining The study of arguments dates back to Aristotle, who in his Rhetorics described three components of a good argument: pathos (appealing to the emotions of thepathos listener), ethos (appealing to the speaker’s personal character), andlogos (the logicalethos logos structure of the argument). Most of the discourse structure studies of argumentation have focused on logos, particularly via building and training on annotated datasets of persuasive essays or other arguments (Reed et al. 2008, Stab and Gurevych 2014a, Peldszus and Stede 2016, Habernal and Gurevych 2017, Musi et al. 2018). Such corpora, for exam- ple, often include annotations of argumentative components like claims (the centralclaims component of the argument that is controversial and needs support) and premisespremises (the reasons given by the author to persuade the reader by supporting or attacking the claim or other premises), as well as the argumentative relations between themargumentative relations like SUPPORT and ATTACK. Consider the following example of a persuasive essay from Stab and Gurevych (2014b). The ﬁrst sentence (1) presents a claim (in bold). (2) and (3) present two premises supporting the claim. (4) gives a premise supporting premise (3). “(1) Museums and art galleries provide a better understanding about arts than Internet. (2) In most museums and art galleries, de- tailed descriptions in terms of the background, history and author are provided. (3) Seeing an artwork online is not the same as watching it with our own eyes, as (4) the picture online does not show the texture or three-dimensional structure of the art, which is important to study.” Thus this example has three argumentative relations:SUPPORT (2,1), SUPPORT (3,1) and SUPPORT (4,3). Fig. 24.12 shows the structure of a much more complex argu- ment. While argumentation mining is clearly related to rhetorical structure and other kinds of coherence relations, arguments tend to be much less local; often a persua- sive essay will have only a single main claim, with premises spread throughout the text, without the local coherence we see in coherence relations. 548 CHAPTER 24 • D ISCOURSE COHERENCE Stab and Gurevych Parsing Argumentation Structures cloning. This example illustrates that knowing argumentative relations is important for separating several arguments in a paragraph. The example also shows that argument components frequently exhibit preceding text units that are not relevant to the argument but helpful for recognizing the argument component type. For example, preceding dis- course connectors like “therefore”, “consequently”, or “thus” can signal a subsequent claim. Discourse markers like “because”, “since”, or “furthermore” could indicate a premise. Formally, thesepreceding tokensof an argument component starting at token ti are deﬁned as the tokens tim, ..., ti1 that are not covered by another argument component in the sentences = t1, t2, ..., tn where 1 i  n and i m 1. The third body paragraph illustrates a contra argument and argumentative attack relations: Admittedly, [cloning could be misused for military purposes]Claim5. For example, [ :it :::::could:::be:::::used ::to::::::::::manipulate:::::::human::::::genes ::in::::::order ::to::::::create::::::::obedient:::::::soldiers ::::with ::::::::::::extraordinary :::::::abilities]Premise9. However, because [::::moral::::and:::::::ethical ::::::values :::are ::::::::::::internationally::::::shared]Premise10,[ :it:::is ::::very::::::::unlikely ::::that :::::::cloning ::::will ::be::::::::misused:::for ::::::militant:::::::::objectives]Premise11. The paragraph begins withClaim5, which attacks the stance of the author. It is supported by Premise9 in the second sentence. The third sentence includes two premises, both of which defend the stance of the author.Premise11 is an attack ofClaim5, and Premise10 supports Premise11. The last paragraph (conclusion) restates the major claim and sum- marizes the main aspects of the essay: To sum up, although [ permitting cloning might bear some risks like misuse for military purposes]Claim6, I strongly believe that [this technology is beneﬁcial to humanity]MajorClaim2. It is likely that [this technology bears some important cures which will significantly improve life conditions]Claim7. The conclusion of the essay starts with an attacking claim followed by the restatement of the major claim. The last sentence includes another claim that summarizes the most im- portant points of the author’s argumentation. Figure 2 shows the entire argumentation structure of the example essay. Figure 2 Argumentation structure of the example essay. Arrows indicate argumentative relations. Arrowheads denote argumentative support relations and circleheads attack relations. Dashed lines indicate relations that are encoded in the stance attributes of claims. “P” denotes premises. 629 Figure 24.12 Argumentation structure of a persuasive essay. Arrows indicate argumentation relations, ei- ther of SUPPORT (with arrowheads) or ATTACK (with circleheads); P denotes premises. Figure from Stab and Gurevych (2017). Algorithms for detecting argumentation structure often include classiﬁers for distinguishing claims, premises, or non-argumentation, together with relation clas- siﬁers for deciding if two spans have the SUPPORT , ATTACK, or neither relation (Peldszus and Stede, 2013). While these are the main focus of much computational work, there is also preliminary efforts on annotating and detecting richer semantic relationships (Park and Cardie 2014, Hidey et al. 2017) such as detectingargumen- tation schemes, larger-scale structures for argument like argument from example,argumentation schemes or argument from cause to effect , or argument from consequences (Feng and Hirst, 2011). Another important line of research is studying how these argument structure (or other features) are associated with the success or persuasiveness of an argument (Habernal and Gurevych 2016, Tan et al. 2016, Hidey et al. 2017. Indeed, while it is Aristotle’s logos that is most related to discourse structure, Aristotle’s ethos and pathos techniques are particularly relevant in the detection of mechanisms of this sort of persuasion. For example scholars have investigated the linguistic realizationpersuasion of features studied by social scientists like reciprocity (people return favors), social proof (people follow others’ choices), authority (people are inﬂuenced by those with power), and scarcity (people value things that are scarce), all of which can be brought up in a persuasive argument (Cialdini, 1984). Rosenthal and McKeown (2017) showed that these features could be combined with argumentation structure to predict who inﬂuences whom on social media, Althoff et al. (2014) found that linguistic models of reciprocity and authority predicted success in online requests, while the semisupervised model of Yang et al. (2019) detected mentions of scarcity, commitment, and social identity to predict the success of peer-to-peer lending plat- forms. See Stede and Schneider (2018) for a comprehensive survey of argument mining. 24.5.2 The structure of scientiﬁc discourse Scientiﬁc papers have a very speciﬁc global structure: somewhere in the course of the paper the authors must indicate a scientiﬁc goal, develop a method for a solu- tion, provide evidence for the solution, and compare to prior work. One popular 24.6 • S UMMARY 549 annotation scheme for modeling these rhetorical goals is the argumentative zon- ing model of Teufel et al. (1999) and Teufel et al. (2009), which is informed by theargumentative zoning idea that each scientiﬁc paper tries to make a knowledge claim about a new piece of knowledge being added to the repository of the ﬁeld (Myers, 1992). Sentences in a scientiﬁc paper can be assigned one of 15 tags; Fig. 24.13 shows 7 (shortened) examples of labeled sentences. Category Description Example AIM Statement of speciﬁc research goal, or hypothesis of current paper “The aim of this process is to examine the role that training plays in the tagging process” OWN METHOD New Knowledge claim, own work: methods “In order for it to be useful for our purposes, the following extensions must be made:” OWN RESULTS Measurable/objective outcome of own work “All the curves have a generally upward trend but always lie far below backoff (51% error rate)” USE Other work is used in own work “We use the framework for the allocation and transfer of control of Whittaker....” GAP WEAK Lack of solution in ﬁeld, problem with other solutions “Here, we will produce experimental evidence suggesting that this simple model leads to serious overestimates” SUPPORT Other work supports current work or is supported by current work “Work similar to that described here has been car- ried out by Merialdo (1994), with broadly similar conclusions.” ANTISUPPORT Clash with other’s results or theory; su- periority of own work “This result challenges the claims of...” Figure 24.13 Examples for 7 of the 15 labels from the Argumentative Zoning labelset (Teufel et al., 2009). Teufel et al. (1999) and Teufel et al. (2009) develop labeled corpora of scientiﬁc articles from computational linguistics and chemistry, which can be used as supervi- sion for training standard sentence-classiﬁcation architecture to assign the 15 labels. 24.6 Summary In this chapter we introduced local and global models for discourse coherence. • Discourses are not arbitrary collections of sentences; they must be coherent. Among the factors that make a discourse coherent are coherence relations between the sentences, entity-based coherence, and topical coherence. • Various sets of coherence relations and rhetorical relations have been pro- posed. The relations in Rhetorical Structure Theory ( RST) hold between spans of text and are structured into a tree. Because of this, shift-reduce and other parsing algorithms are generally used to assign these structures. The Penn Discourse Treebank (PDTB) labels only relations between pairs of spans, and the labels are generally assigned by sequence models. • Entity-based coherence captures the intuition that discourses are about an entity, and continue mentioning the entity from sentence to sentence. Cen- tering Theory is a family of models describing how salience is modeled for discourse entities, and hence how coherence is achieved by virtue of keeping the same discourse entities salient over the discourse. The entity grid model gives a more bottom-up way to compute which entity realization transitions lead to coherence. 550 CHAPTER 24 • D ISCOURSE COHERENCE • Many different genres have different types of global coherence. Persuasive essays have claims and premises that are extracted in the ﬁeld of argument mining, scientiﬁc articles have structure related to aims, methods, results, and comparisons. Bibliographical and Historical Notes Coherence relations arose from the independent development of a number of schol- ars, including Hobbs (1979) idea that coherence relations play an inferential role for the hearer, and the investigations by Mann and Thompson (1987) of the discourse structure of large texts. Other approaches to coherence relations and their extrac- tion include Segmented Discourse Representation Theory (SDRT) (Asher and Las-SDRT carides 2003, Baldridge et al. 2007) and the Linguistic Discourse Model (Polanyi 1988, Scha and Polanyi 1988, Polanyi et al. 2004). Wolf and Gibson (2005) argue that coherence structure includes crossed bracketings, which make it impossible to represent as a tree, and propose a graph representation instead. A compendium of over 350 relations that have been proposed in the literature can be found in Hovy (1990). RST parsing was ﬁrst proposed by Marcu (1997), and early work was rule-based, focused on discourse markers (Marcu, 2000a). The creation of the RST Discourse TreeBank (Carlson et al. 2001, Carlson and Marcu 2001) enabled a wide variety of machine learning algorithms, beginning with the shift-reduce parser of Marcu (1999) that used decision trees to choose actions, and continuing with a wide variety of machine learned parsing methods (Soricut and Marcu 2003, Sagae 2009, Hernault et al. 2010, Feng and Hirst 2014, Surdeanu et al. 2015, Joty et al. 2015) and chunkers (Sporleder and Lapata, 2005). Subba and Di Eugenio (2009) integrated sophisticated semantic information into RST parsing. Ji and Eisenstein (2014) ﬁrst applied neural models to RST parsing neural models, leading to the modern set of neural RST models (Li et al. 2014, Li et al. 2016b, Braud et al. 2017, Yu et al. 2018, inter alia) as well as neural segmenters (Wang et al. 2018b). and neural PDTB parsing models (Ji and Eisenstein 2015, Qin et al. 2016, Qin et al. 2017). Barzilay and Lapata (2005) pioneered the idea of self-supervision for coher- ence: training a coherence model to distinguish true orderings of sentences from random permutations. Li et al. (2014) ﬁrst applied this paradigm to neural sentence- representation, and many neural self-supervised models followed (Li and Jurafsky 2017, Logeswaran et al. 2018, Lai and Tetreault 2018, Xu et al. 2019, Iter et al. 2020) Another aspect of global coherence is the global topic structure of a text, the way the topics shift over the course of the document. Barzilay and Lee (2004) introduced an HMM model for capturing topics for coherence, and later work expanded this intuition (Soricut and Marcu 2006, Elsner et al. 2007, Louis and Nenkova 2012, Li and Jurafsky 2017). The relationship between explicit and implicit discourse connectives has been a fruitful one for research. Marcu and Echihabi (2002) ﬁrst proposed to use sen- tences with explicit relations to help provide training data for implicit relations, by removing the explicit relations and trying to re-predict them as a way of improv- ing performance on implicit connectives; this idea was reﬁned by Sporleder and Lascarides (2005), (Pitler et al., 2009), and Rutherford and Xue (2015). This rela- BIBLIOGRAPHICAL AND HISTORICAL NOTES 551 tionship can also be used as a way to create discourse-aware representations. The DisSent algorithm (Nie et al., 2019) creates the task of predicting explicit discourse markers between two sentences. They show that representations learned to be good at this task also function as powerful sentence representations for other discourse tasks. The idea of entity-based coherence seems to have arisen in multiple ﬁelds in the mid-1970s, in functional linguistics (Chafe, 1976), in the psychology of discourse processing (Kintsch and Van Dijk, 1978), and in the roughly contemporaneous work of Grosz, Sidner, Joshi, and their colleagues. Grosz (1977a) addressed the focus of attention that conversational participants maintain as the discourse unfolds. She deﬁned two levels of focus; entities relevant to the entire discourse were said to be in global focus, whereas entities that are locally in focus (i.e., most central to a particular utterance) were said to be in immediate focus. Sidner 1979; 1983 de- scribed a method for tracking (immediate) discourse foci and their use in resolving pronouns and demonstrative noun phrases. She made a distinction between the cur- rent discourse focus and potential foci, which are the predecessors to the backward- and forward-looking centers of Centering theory, respectively. The name and further roots of the centering approach lie in papers by Joshi and Kuhn (1979) and Joshi and Weinstein (1981), who addressed the relationship between immediate focus and the inferences required to integrate the current utterance into the discourse model. Grosz et al. (1983) integrated this work with the prior work of Sidner and Grosz. This led to a manuscript on centering which, while widely circulated since 1986, remained unpublished until Grosz et al. (1995). A collection of centering papers appears in Walker et al. (1998b). See Karamanis et al. (2004) and Poesio et al. (2004) for a deeper exploration of centering and its parameterizations, and the History section of Chapter 23 for more on the use of centering on coreference. The grid model of entity-based coherence was ﬁrst proposed by Barzilay and Lapata (2005) drawing on earlier work by Lapata (2003) and Barzilay, and then extended by them Barzilay and Lapata (2008) and others with additional features (Elsner and Charniak 2008, 2011, Feng et al. 2014, Lin et al. 2011) a model that projects entities into a global graph for the discourse (Guinaudeau and Strube 2013, Mesgar and Strube 2016), and a convolutional model to capture longer-range entity dependencies (Nguyen and Joty, 2017). Theories of discourse coherence have also been used in algorithms for interpret- ing discourse-level linguistic phenomena, including verb phrase ellipsis and gap- ping (Asher 1993, Kehler 1993), and tense interpretation (Lascarides and Asher 1993, Kehler 1994, Kehler 2000). An extensive investigation into the relationship between coherence relations and discourse connectives can be found in Knott and Dale (1994). Useful surveys of discourse processing and structure include Stede (2011) and Webber et al. (2012). Andy Kehler wrote the Discourse chapter for the 2000 ﬁrst edition of this text- book, which we used as the starting point for the second-edition chapter, and there are some remnants of Andy’s lovely prose still in this third-edition coherence chap- ter. 552 CHAPTER 24 • D ISCOURSE COHERENCE Exercises 24.1 Finish the Centering Theory processing of the last two utterances of (24.30), and show how (24.29) would be processed. Does the algorithm indeed mark (24.29) as less coherent? 24.2 Select an editorial column from your favorite newspaper, and determine the discourse structure for a 10–20 sentence portion. What problems did you encounter? Were you helped by superﬁcial cues the speaker included (e.g., discourse connectives) in any places? Bibliography Abadi, M., A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. 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Subject Index ?, 9 +?, 9 .wav format, 336 10-fold cross-validation, 69 →(derives), 389 ˆ,58 * (RE Kleene ), 7 + (RE Kleene +), 7 . (RE any character), 7 $(RE end-of-line), 8 ((RE precedence symbol), 8 [(RE character disjunction), 6 \B (RE non word-boundary), 8 \b (RE word-boundary), 8 ](RE character disjunction), 6 ˆ(RE start-of-line), 8 [ˆ](single-char negation), 6 4-gram, 38 4-tuple, 392 5-gram, 38 A-D conversion, 335 AAC, 32 AAE, 15 AB test, 353 ablating, 248 absolute position, 198 absolute temporal expression, 452 abstract word, 485 accessible, 506 accessing a referent, 501 accomplishment expressions, 450 accuracy, 366 achievement expressions, 450 acknowledgment speech act, 312 activation, 133 activity expressions, 450 acute-eval, 325 ad hoc retrieval, 291 add gate, 172 add-k, 47 add-one smoothing, 46 adequacy, 280 adjacency pairs, 313 Adjectives, 364 adverb, 364 degree, 364 directional, 364 locative, 364 manner, 364 temporal, 364 Adverbs, 364 AED, 339 affective, 481 afﬁx, 24 agent, as thematic role, 462 agglutinative language, 267 AIFF ﬁle, 336 AISHELL-1, 334 aktionsart, 450 ALGOL, 409 algorithm byte-pair encoding, 22 CKY , 397 minimum edit distance, 28 naive Bayes classiﬁer, 57 pointwise mutual information, 114 semantic role labeling, 469 TextTiling, 544 Viterbi, 373 aligned, 249 alignment, 25, 342 in ASR, 346 minimum cost, 27 string, 25 via minimum edit distance, 27 Allen relations, 448 allocational harm, 126 ambiguity amount of part-of-speech in Brown corpus, 366 attachment, 396 coordination, 396 of referring expressions, 503 part-of-speech, 365 resolution of tag, 366 American Structuralism, 408 anaphor, 502 anaphora, 502 anaphoricity detector, 511 anchor texts, 520 anchors in regular expressions, 8, 29 anisotropy, 234 antecedent, 502 Apple AIFF, 336 approximate randomization, 71 arc eager, 423 arc standard, 417 argmax, 58 argumentation mining, 547 argumentation schemes, 548 argumentative relations, 547 argumentative zoning, 549 Aristotle, 362, 450 ARPA, 355 article (part-of-speech), 364 articulatory synthesis, 357 aspect, 450 ASR, 331 conﬁdence, 320 association, 103 ATIS corpus, 390 ATN, 478 ATRANS, 477 attachment ambiguity, 396 attention cross-attention, 272 encoder-decoder, 272 history in transformers, 202 attention head, 188 attention mechanism, 179 Attribution (as coherence relation), 534 augmentative communication, 32 authorship attribution, 56 autoregressive generation, 167, 207 Auxiliary, 365 B3, 524 Babbage, C., 332 backoff, 49 in smoothing, 48 backprop, 147 backpropagation through time, 161 backtrace in minimum edit distance, 29 backtranslation, 279 Backus-Naur form, 388 backward-looking center, 541 bag of words, 58, 59 in IR, 291 bakeoff, 355 speech recognition competition, 355 barged in, 326 base model, 249 basic emotions, 482 batch training, 94 Bayes’ rule,58 dropping denominator, 59, 372 Bayesian inference, 58 BDI, 329 beam search, 275, 424 beam width, 275, 424 Berkeley Restaurant Project, 36 Bernoulli naive Bayes, 75 BERT for affect, 497 best-worst scaling, 486 bias ampliﬁcation, 126 bias term, 79, 133 bidirectional RNN, 170 bigram, 34 binary branching, 394 binary naive Bayes, 63 binary tree, 394 BIO, 238, 368 BIO tagging, 238 for NER, 238, 368 BIOES, 238, 368 bitext, 270 bits for measuring entropy, 49 blank in CTC, 342 BM25, 291, 293 BNF (Backus-Naur form), 388 bootstrap, 73 bootstrap algorithm, 73 bootstrap test, 71 bootstrapping, 71 in IE, 441 bound pronoun, 504 BPE, 21 BPE, 22 bracketed notation, 391 bridging inference, 506 broadcast news speech recognition of, 355 Brown corpus, 13 original tagging of, 384 byte-pair encoding, 21 calibrated, 290 CALLHOME, 333 Candide, 287 Cantonese, 267 capture group, 12 cascade regular expression in ELIZA, 12 case sensitivity in regular expression search, 6 case folding, 23 case frame, 463, 478 CAT,263 cataphora, 504 CD (conceptual dependency), 477 Centering Theory, 532, 540 centroid, 117 cepstrum history, 355 CFG, see context-free grammar chain rule, 99, 148 chain-of-thought, 254 channels in stored waveforms, 336 chart parsing, 397 Chatbots, 309, 321 chatbots, 4 CHiME, 333 Chinese 585 586 Subject Index as verb-framed language, 267 words for brother, 266 Chomsky normal form, 394 Chomsky-adjunction, 395 chrF, 281 CIRCUS, 459 citation form, 102 Citizen Kane, 531 CKY algorithm, 387 claims, 547 class-based n-gram, 53 classiﬁer head, 235 clefts, 507 clitic, 19 origin of term, 362 closed book, 304 closed class, 363 cloze task, 226 cluster, 502 CNF, see Chomsky normal form Cocke-Kasami-Younger algorithm, see CKY code switching, 15 coherence, 531 entity-based, 540 relations, 533 cohesion lexical, 532, 544 ColBERT,300 cold languages, 268 collection in IR, 291 commissive speech act, 312 common crawl, 211 common ground, 312, 328 Common nouns, 363 complementizers, 364 componential analysis, 476 compression, 335 Computational Grammar Coder (CGC), 384 concatenation, 5, 29 conceptual dependency, 477 concrete word, 485 conditional generation, 204 conditional random ﬁeld, 376 conﬁdence, 285 ASR, 320 in relation extraction, 442 conﬁdence values, 442 conﬁguration, 417 confusion matrix, 66 Conjunctions, 364 connectionist, 157 connotation frame, 497 connotation frames, 479 connotations, 104, 482 constative speech act, 312 constituency, 388 constituent, 388 titles which are not, 387 Constraint Grammar, 433 content planning, 319 context embedding, 122 context-free grammar, 388, 392, 407 Chomsky normal form, 394 invention of, 409 non-terminal symbol, 389 productions, 388 rules, 388 terminal symbol, 389 weak and strong equivalence, 394 contextual embeddings, 186, 231 continued pretraining, 214 conversation, 309 conversation analysis, 313, 328 conversational implicature, 314 conversational speech, 333 convex, 90 coordination ambiguity, 396 copula, 365 CORAAL, 333 corefer, 501 coreference chain, 502 coreference resolution, 502 gender agreement, 508 Hobbs tree search algorithm, 528 number agreement, 507 person agreement, 508 recency preferences, 508 selectional restrictions, 509 syntactic (“binding”) constraints, 508 verb semantics, 509 corpora, 13 corpus, 13 ATIS, 390 Broadcast news, 355 Brown, 13, 384 ﬁsher, 355 LOB, 384 regular expression searching inside, 5 Switchboard, 13, 333, 335 TimeBank, 451 Wall Street Journal, 355 correction act detection, 319 cosine as a similarity metric, 110 cost function, 88 count nouns, 363 counters, 29 counts treating low as zero, 379 CRF, 376 compared to HMM, 376 inference, 380 Viterbi inference, 380 CRFs learning, 381 cross-attention, 272 cross-brackets, 406 cross-entropy, 51 cross-entropy loss, 88, 145 cross-validation, 69 10-fold, 69 crowdsourcing, 485 CTC, 341 datasheet, 16 dative alternation, 463 debiasing, 127 decision boundary, 80, 136 decoder-only model, 201 decoding, 207, 372 Viterbi, 372 deep neural networks, 132 deep learning, 132 deﬁnite reference, 504 degree adverb, 364 delexicalize, 320 demonstrations, 246 denoising, 226 dependency grammar, 411 dependency tree, 414 dependent, 412 derivation direct (in a formal language), 392 syntactic, 389, 389, 392, 392 Det, 388 determiner, 364, 388 Determiners, 364 development set, 38 development test set, 69 development test set (dev-test), 39 devset, see development test set (dev-test), 69 DFT, 338 dialogue, 309 dialogue act correction, 319 Dialogue acts, 318 dialogue policy, 319 dialogue systems, 309 design, 325 diathesis alternation, 463 diff program, 30 digit recognition, 332 digital divide, 263 digitization, 335 dilated convolutions, 352 dimension, 107 diphthong origin of term, 362 direct derivation (in a formal language), 392 directional adverb, 364 directive speech act, 312 disambiguation in parsing, 403 syntactic, 397 discount, 47, 49 discounting, 45 discourse, 531 segment, 534 discourse connectives, 535 discourse deixis, 503 discourse model, 501 discourse parsing, 536 discourse-new, 505 discourse-old, 505 discovery procedure, 408 discrete Fourier transform, 338 discriminative model, 78 disﬂuency, 13 disjunction, 29 pipe in regular expressions as, 8 square braces in regular expression as, 6 dispreferred response, 330 distant supervision, 443 distributional hypothesis, 101 distributional similarity, 408 divergences between languages in MT, 265 document in IR, 291 document frequency, 112 document vector, 117 domination in syntax, 389 dot product, 79, 110 dot-product attention, 180 Dragon Systems, 355 dropout, 151 duration temporal expression, 452 dynamic programming, 26 and parsing, 397 Viterbi as, 373 dynamic time warping, 355 edge-factored, 426 edit distance minimum algorithm, 26 EDU, 534 effect size, 70 efﬁciency costs, 317 Elaboration (as coherence relation), 533 ELIZA, 4 implementation, 12 sample conversation, 12 Elman Networks, 158 ELMo for affect, 497 EM for deleted interpolation, 48 embedding layer, 154 embeddings, 105 cosine for similarity, 110 skip-gram, learning, 120 sparse, 110 tf-idf, 112 word2vec, 117 emission probabilities, 370 EmoLex, 484 Subject Index 587 emotion, 482 Encoder-decoder, 175 encoder-decoder attention, 272 end-to-end training, 166 endpointing, 312 English lexical differences from French, 267 simpliﬁed grammar rules, 390 verb-framed, 267 entity dictionary, 379 entity grid, 542 Entity linking, 520 entity linking, 502 entity-based coherence, 540 entropy, 49 and perplexity, 49 cross-entropy, 51 per-word, 50 rate, 50 relative, 474 error backpropagation, 147 ESPnet, 356 ethos, 547 Euclidean distance in L2 regularization, 96 Eugene Onegin, 52 Euler’s formula,338 Europarl, 270 evalb, 406 evaluating parsers, 405 evaluation 10-fold cross-validation, 69 AB test, 353 comparing models, 41 cross-validation, 69 development test set, 39, 69 devset, 69 devset or development test set, 39 extrinsic, 38 ﬂuency in MT, 280 Matched-Pair Sentence Segment Word Error (MAPSSWE), 347 mean opinion score, 353 most frequent class baseline, 366 MT, 280 named entity recognition, 240, 381 of n-gram, 38 of n-grams via perplexity, 40 pseudoword, 476 relation extraction, 446 test set, 39 training on the test set, 39 training set, 39 TTS, 353 event coreference, 503 event extraction, 435, 446 events, 450 Evidence (as coherence relation), 533 evoking a referent, 501 execution accuracy, 256 expansion, 390, 391 expletive, 507 explicit conﬁrmation, 319 extraposition, 507 extrinsic evaluation, 38 F (for F-measure), 67 F-measure, 67 F-measure in NER, 240, 381 factoid question, 289 Faiss, 301 false negatives, 9 false positives, 9 Farsi, verb-framed, 267 fast Fourier transform, 338, 355 fasttext, 123 FASTUS, 457 feature cutoff, 379 feature interactions, 82 feature selection information gain, 76 feature template, 421 feature templates, 82 part-of-speech tagging, 378 feature vectors, 334 Federalist papers, 75 feedforward network, 138 fenceposts, 398 few-shot, 246 FFT, 338, 355 ﬁle format, .wav, 336 ﬁlled pause, 13 ﬁller, 13 ﬁnetuning, 213, 235 ﬁnetuning;supervsed, 249 ﬁrst-order co-occurrence, 124 ﬂuency, 280 in MT, 280 fold (in cross-validation), 69 forget gate, 172 formal language, 391 formant synthesis, 357 forward inference, 153 forward-looking centers, 541 Fosler, E., see Fosler-Lussier, E. foundation model, 222 fragment of word, 13 frame, 336 semantic, 467 frame elements, 467 FrameNet, 466 frames, 314 free word order, 411 Freebase, 437 freeze, 155, 214 French, 265 Frump, 459 fully-connected, 138 function word, 363, 383 fusion language, 267 Gaussian prior on weights, 96 gazetteer, 379 General Inquirer, 64, 484 generalize, 95 generalized semantic role, 464 generation of sentences to test a CFG grammar, 390 generative AI, 204 generative grammar, 391 generative model, 78 generative models, 59 generator, 389 generics, 507 German, 265 given-new,506 Godzilla, speaker as, 472 gold labels, 66 gradient, 90 Grammar Constraint, 433 Head-Driven Phrase Structure (HPSG), 406 Link, 433 grammar binary branching, 394 checking, 387 equivalence, 394 generative, 391 inversion transduction, 287 grammatical function, 412 grammatical relation, 412 grammatical sentences, 391 greedy decoding, 206 greedy RE patterns, 9 grep, 5, 5, 30 Gricean maxims, 314 grounding, 312 GUS, 314 hallucinate, 290 hallucination, 219 Hamilton, Alexander, 75 Hamming, 337 Hansard, 287 hanzi, 19 harmonic mean, 67 head, 188, 199, 406, 412 ﬁnding, 406 Head-Driven Phrase Structure Grammar (HPSG), 406 Heaps’ Law,14 Hearst patterns, 438 held-out, 48 Herdan’s Law,14 hidden, 370 hidden layer, 138 as representation of input, 139 hidden units, 138 Hindi, 265 Hindi, verb-framed, 267 HKUST, 334 HMM, 370 formal deﬁnition of, 370 history in speech recognition, 355 initial distribution, 370 observation likelihood, 370 observations, 370 simplifying assumptions for POS tagging, 372 states, 370 transition probabilities, 370 Hobbs algorithm, 528 Hobbs tree search algorithm for pronoun resolution, 528 homonymy, 232 hot languages, 268 Hungarian part-of-speech tagging, 382 hybrid, 356 hyperarticulation, 319 hypernym, 437 lexico-syntactic patterns for, 438 hyperparameter, 92 hyperparameters, 152 IBM Models, 287 IBM Thomas J. Watson Research Center, 53, 355 idf, 113 idf term weighting, 113, 292 immediately dominates, 389 implicature, 314 implicit argument, 479 implicit conﬁrmation, 320 in-context learning, 247 indeﬁnite reference, 504 induction heads, 247 inference-based learning, 429 infoboxes, 437 information structure, 505 status, 505 information extraction (IE), 435 bootstrapping, 441 information gain, 76 for feature selection, 76 Information retrieval, 108, 290 information retrieval, 290 initiative, 313 inner product, 110 instance, word, 14 588 Subject Index Institutional Review Board, 327 Instruction tuning, 249 intent determination, 316 intercept, 79 Interjections, 364 interpolated precision, 296 interpolation in smoothing, 48 interpretable, 98 interval algebra, 448 intrinsic evaluation, 38 inversion transduction grammar (ITG), 287 inverted index, 295 IO, 238, 368 IOB tagging for temporal expressions, 453 IR, 290 idf term weighting, 113, 292 term weighting, 291 vector space model, 107 IRB, 327 is-a, 437 ISO 8601, 454 isolating language, 267 iSRL, 479 ITG (inversion transduction grammar), 287 Japanese, 265, 267 Jay, John, 75 joint intention, 328 Kaldi, 356 KBP, 459 KenLM, 38, 53 key, 188 KL divergence, 474 Klatt formant synthesizer, 357 Kleene , 7 sneakiness of matching zero things, 7 Kleene +, 7 knowledge claim, 549 knowledge graphs, 435 Kullback-Leibler divergence, 474 KV cache, 217 L1 regularization, 96 L2 regularization, 96 labeled precision, 405 labeled recall, 405 language identiﬁcation, 354 universal, 264 language id, 56 language model, 32 language model:coined by, 53 language modeling head, 199 Laplace smoothing, 45 for PMI, 116 lasso regression, 96 latent semantic analysis, 130 layer norm, 192 LDC, 19 learning rate, 91 lemma, 15, 102 versus wordform, 15 Lemmatization, 23 lemmatization, 5 Levenshtein distance, 25 lexical category, 389 cohesion, 532, 544 gap, 267 semantics, 102 trigger, in IE, 452 lexico-syntactic pattern, 438 lexicon, 388 LibriSpeech, 333 light verbs, 447 likelihood, 59 linear chain CRF, 376, 377 linear classiﬁers, 60 linear interpolation for n-grams, 48 linearly separable, 136 Linguistic Data Consortium, 19 Linguistic Discourse model, 550 Link Grammar, 433 List (as coherence relation), 534 listen attend and spell, 339 LIWC, 64, 485 LM, 32 LOB corpus, 384 localization, 263 location-based attention, 351 locative, 364 locative adverb, 364 log why used for probabilities, 37 why used to compress speech, 336 log likelihood ratio, 493 log odds ratio, 493 log probabilities, 37, 37 logistic function, 79 logistic regression, 77 conditional maximum likelihood estimation, 88 Gaussian priors, 96 learning in, 87 regularization, 96 relation to neural networks, 140 logit, 80, 200 logit lens, 200 logos, 547 long short-term memory, 172 lookahead in regex, 13 LoRA, 218 loss, 88 low frame rate, 340 LPC (Linear Predictive Coding), 355 LSI, see latent semantic analysis LSTM, 385 LUNAR, 307 machine learning for NER, 382 textbooks, 75, 100 machine translation, 263 macroaveraging, 68 Madison, James, 75 MAE, 15 Mandarin, 265 Manhattan distance in L1 regularization, 96 manner adverb, 364 Markov, 34 assumption, 34 Markov assumption, 369 Markov chain, 52, 369 formal deﬁnition of, 370 initial distribution, 370 n-gram as, 369 states, 370 transition probabilities, 370 Markov model, 34 formal deﬁnition of, 370 history, 53 Marx, G., 387 Masked Language Modeling, 226 mass nouns, 363 maxent, 100 maxim, Gricean, 314 maximum entropy, 100 maximum spanning tree, 426 Mayan, 267 MBR, 277 McNemar’s test,348 mean element-wise, 167 mean average precision, 297 mean opinion score, 353 mean reciprocal rank, 305 mechanical indexing, 129 Mechanical Turk, 332 mel, 338 memory networks, 202 mention detection, 510 mention-pair, 513 mentions, 501 MERT, for training in MT, 287 MeSH (Medical Subject Headings), 57 Message Understanding Conference, 457 METEOR, 288 metonymy, 530 microaveraging, 68 Microsoft .wav format, 336 mini-batch, 94 Minimum Bayes risk, 277 minimum edit distance, 25, 25, 373 example of, 28 for speech recognition evaluation, 346 MINIMUM EDIT DISTANCE , 28 minimum edit distance algorithm, 26 Minimum Error Rate Training, 287 MLE for n-grams, 35 for n-grams, intuition, 36 MLM, 226 MLP, 138 MMLU, 258, 304 modal verb, 365 model alignment, 249 model card, 74 morpheme, 23 MOS (mean opinion score), 353 Moses, Michelangelo statue of, 309 Moses, MT toolkit, 287 MRR, 305 MS MARCO, 303 MT, 263 divergences, 265 post-editing, 263 mu-law, 336 MUC, 457, 459 MUC F-measure, 524 multi-head attention, 189 multi-hop, 303 multi-layer perceptrons, 138 multinomial logistic regression, 84 multinomial naive Bayes, 57 multinomial naive Bayes classiﬁer, 57 multiword expressions, 130 MWE, 130 n-best list, 341 n-gram, 32, 34 add-one smoothing, 45 as approximation, 34 as generators, 43 as Markov chain, 369 equation for, 35 example of, 36, 37 for Shakespeare, 43 history of, 53 interpolation, 48 KenLM, 38, 53 logprobs in, 37 normalizing, 36 parameter estimation, 35 sensitivity to corpus, 43 smoothing, 45 Subject Index 589 SRILM, 53 test set, 38 training set, 38 naive Bayes multinomial, 57 simplifying assumptions, 59 naive Bayes assumption, 59 naive Bayes classiﬁer use in text categorization, 57 named entity, 237, 362, 367 list of types, 238, 367 named entity recognition, 237, 367 natural language inference, 237 Natural Questions, 303 negative log likelihood loss, 88, 97, 146 NER, 237, 367 neural networks relation to logistic regression, 140 newline character, 10 Next Sentence Prediction, 228 NIST for MT evaluation, 288 noisy-or, 442 NomBank, 466 Nominal, 388 non-capturing group, 12 non-greedy, 9 non-standard words, 349 non-stationary process, 336 non-terminal symbols, 389, 390 normal form, 394, 394 normalization temporal, 453 word, 23 normalization of probabilities, 35 normalize, 83 normalizing, 140 noun abstract, 363 common, 363 count, 363 mass, 363 proper, 363 noun phrase, 388 constituents, 388 Nouns, 363 NP, 388, 390 nucleus, 533 null hypothesis, 70 Nyquist frequency, 335 observation likelihood role in Viterbi, 374 one-hot vector, 153, 197 open book, 304 open class, 363 open information extraction, 444 operation list, 25 operator precedence, 8, 9 optionality use of ? in regular expressions for, 6 output gate, 173 overﬁtting, 95 p-value, 71 Paired, 71 parallel corpus, 270 parallel distributed processing, 157 parallelogram model, 124 parameter-efﬁcient ﬁne tuning, 217 parse tree, 389, 391 PARSEV AL,405 parsing ambiguity, 395 CKY , 397 CYK, see CKY evaluation, 405 relation to grammars, 392 syntactic, 387 well-formed substring table, 409 part of speech as used in CFG, 389 part-of-speech adjective, 364 adverb, 364 closed class, 363 interjection, 364 noun, 363 open class, 363 particle, 364 subtle distinction between verb and noun, 364 verb, 364 part-of-speech tagger PARTS , 384 TAGGIT, 384 Part-of-speech tagging, 365 part-of-speech tagging ambiguity and, 365 amount of ambiguity in Brown corpus, 366 and morphological analysis, 382 feature templates, 378 history of, 384 Hungarian, 382 Turkish, 382 unknown words, 376 particle, 364 PARTS tagger, 384 parts of speech, 362 pathos, 547 pattern, regular expression, 5 PCM (Pulse Code Modulation), 336 PDP, 157 PDTB, 535 PEFT, 217 Penn Discourse TreeBank, 535 Penn Treebank, 393 tagset, 365, 365 Penn Treebank tokenization, 19 per-word entropy, 50 perceptron, 135 period disambiguation, 82 perplexity, 40, 52 as weighted average branching factor, 41 deﬁned via cross-entropy, 52 perplexity:coined by, 53 personal pronoun, 364 persuasion, 548 phrasal verb, 364 phrase-based translation, 287 phrase-structure grammar, 388 PII, 212 pipe, 8 planning and speech acts, 329 shared plans, 328 pleonastic, 507 Pointwise mutual information, 114 polysynthetic language, 267 pooling, 143, 166 max, 167 mean, 166 Porter stemmer, 24 POS, 362 position embeddings relative, 199 positional embeddings, 198 possessive pronoun, 364 post-editing, 263 post-training, 249 postings, 295 postposition, 265 Potts diagram, 492 PP, 390 PP-attachment ambiguity, 396 PPMI, 115 precedence, 8 precedence, operator, 8 Precision, 67 precision for MT evaluation, 288 in NER, 240, 381 precision-recall curve, 296 premises, 547 prepositional phrase constituency, 390 prepositions, 364 presequences, 313 pretraining, 145, 203 primitive decomposition, 476 principle of contrast, 103 prior probability, 59 pro-drop languages, 268 probabilistic context-free grammars, 409 productions, 388 projective, 414 prompt, 243 prompt engineering, 243 pronoun, 364 bound, 504 demonstrative, 505 non-binary, 508 personal, 364 possessive, 364 wh-, 364 PropBank, 465 proper noun, 363 PROTO -AGENT , 464 PROTO -PATIENT , 464 pseudoword, 476 PTRANS, 477 punctuation for numbers cross-linguistically, 19 for sentence segmentation, 24 tokenization, 19 treated as words, 13 treated as words in LM, 44 QA, 289 quantization, 335 query, 188, 291 in IR, 291 question factoid, 289 question answering factoid questions, 289 Radio Rex, 331 RAG, 290, 302 random sampling, 208 range, regular expression, 6 ranking, 281 rarefaction, 335 RDF, 437 RDF triple, 437 Read speech, 333 reading comprehension, 304 Reason (as coherence relation), 533 Recall, 67 recall for MT evaluation, 288 in NER, 240, 381 rectangular, 336 reference bound pronouns, 504 cataphora, 504 deﬁnite, 504 generics, 507 indeﬁnite, 504 reference point, 449 referent, 501 accessing of, 501 evoking of, 501 referential density, 268 590 Subject Index reﬂexive, 508 regex regular expression, 5 register in regex, 12 regression lasso, 96 ridge, 96 regular expression, 5, 29 substitutions, 11 regularization, 95 rejection conversation act, 320 relatedness, 103 relation extraction, 435 relative temporal expression, 452 relative entropy, 474 relative frequency, 36 relevance, 314 relexicalize, 321 ReLU, 134 reporting events, 447 representation learning, 101 representational harm, 127 representational harms, 73 rescore, 341 residual stream, 191 resolve, 366 Resource Management, 355 retrieval-augmented generation, 302 ReVerb, 445 rewrite, 389 Rhetorical Structure Theory, see RST Riau Indonesian, 364 ridge regression, 96 RLHF, 325 RNN-T, 345 role-ﬁller extraction, 457 Rosebud, sled named, 531 row vector, 108 RST, 533 TreeBank, 535, 550 rules context-free, 388 context-free, expansion, 389 context-free, sample, 390 Russian fusion language, 267 verb-framed, 267 S as start symbol in CFG, 390 salience, in discourse model, 506 Sampling, 42 sampling of analog waveform, 335 rate, 335 satellite, 267, 533 satellite-framed language, 267 saturated, 135 scaling laws, 216 SCISOR, 459 sclite, 347 sclite package, 30 script Schankian, 467 scripts, 456 SDRT (Segmented Discourse Representation Theory), 550 search engine, 290 search tree, 274 second-order co-occurrence, 124 seed pattern in IE, 441 seed tuples, 441 segmentation sentence, 24 word, 18 selectional association, 475 selectional preference strength, 474 selectional preferences pseudowords for evaluation, 476 selectional restriction, 472 representing with events, 473 violations in WSD, 474 self-supervision, 118, 163, 210 self-training, 155 semantic drift in IE, 442 semantic feature, 130 semantic ﬁeld, 103 semantic frame, 104 semantic relations in IE, 436 table, 437 semantic role, 462, 462, 464 Semantic role labeling, 468 semantics lexical, 102 sense word, 232 sentence error rate, 347 segmentation, 24 sentence realization, 320 sentence segmentation, 5 sentence separation, 176 SentencePiece, 270 sentiment, 104 origin of term, 500 sentiment analysis, 56 sentiment lexicons, 64 SentiWordNet, 490 sequence labeling, 362 SFT, 249 SGNS, 117 Shakespeare n-gram approximations to, 43 shallow discourse parsing, 539 shared plans, 328 side sequence, 313 sigmoid, 79, 133 signiﬁcance test MAPSSWE for ASR, 347 McNemar’s, 348 similarity, 103 cosine, 110 singleton, 502 singular they, 508 skip-gram, 117 slot error rate, 317 slot ﬁlling, 316, 459 slots, 314 smoothing, 45, 45 add-one, 45 interpolation, 48 Laplace, 45 linear interpolation, 48 softmax, 85, 140 SOV language, 265 spam detection, 56, 64 span, 403 Speaker diarization, 353 speaker identiﬁcation, 354 speaker recognition, 354 speaker veriﬁcation, 354 speech telephone bandwidth, 335 speech acts, 312 speech recognition architecture, 332, 339 history of, 354 speech synthesis, 332 split-half reliability, 487 SRILM, 53 SRL, 468 Stacked RNNs, 169 standardize, 82 start symbol, 389 states, 450 static embeddings, 118 stationary process, 336 stationary stochastic process, 51 statistical MT, 287 statistical signiﬁcance MAPSSWE for ASR, 347 McNemar’s test, 348 statistically signiﬁcant, 71 stative expressions, 450 stem, 23 Stemming, 5 stemming, 24 stop list, 294 stop words, 61 streaming, 345 stride, 336 structural ambiguity, 395 stupid backoff, 49 subdialogue, 313 subjectivity, 481, 500 substitutability, 408 substitution operator (regular expressions), 11 subword tokens, 18 subwords, 21 supervised ﬁnetuning, 249 supervised machine learning, 57 SVD, 130 SVO language, 265 Swedish, verb-framed, 267 Switchboard, 333 Switchboard Corpus, 13, 333, 335 synchronous grammar, 287 synonyms, 103 syntactic disambiguation, 397 syntax, 387 origin of term, 362 TAC KBP, 438 Tacotron2, 351 TACRED dataset, 437 TAGGIT, 384 tagset Penn Treebank, 365, 365 table of Penn Treebank tags, 365 Tamil, 267 tanh, 134 target embedding, 122 task error rate, 317 Tay,326 teacher forcing, 164, 178, 210, 274 technai, 362 telephone-bandwidth speech, 335 telic, 450 temperature sampling, 209 template, 245 template ﬁlling, 435, 456 template recognition, 456 template, in IE, 456 templates, 244 temporal adverb, 364 temporal anchor, 455 temporal expression absolute, 452 metaphor for, 449 relative, 452 temporal logic, 447 temporal normalization, 453 term in IR, 291 weight in IR, 291 term frequency, 112 term weight, 291 term-document matrix, 106 term-term matrix, 109 terminal symbol, 389 test set, 38 development, 39 how to choose, 39 text categorization, 56 bag-of-words assumption, 58 naive Bayes approach, 57 unknown words, 61 text normalization, 4, 16 text summarization, 205 text-to-speech, 332 Subject Index 591 TextTiling, 544 tf-idf, 113 The Pile, 211 thematic grid, 463 thematic role, 462 and diathesis alternation, 463 examples of, 462 problems, 464 theme, 462 theme, as thematic role, 462 TimeBank, 451 tokenization, 4 sentence, 24 word, 18 Top-k sampling, 208 top-p sampling, 209 topic models, 104 toxicity detection, 74 training oracle, 419 training set, 38 cross-validation, 69 how to choose, 39 transcription of speech, 331 reference, 346 transduction grammars, 287 transfer learning, 223 Transformations and Discourse Analysis Project (TDAP), 384 transition probability role in Viterbi, 374 transition-based, 416 translation divergences, 265 TREC, 308 treebank, 392 trigram, 38 TTS, 332 Turk, Mechanical, 332 Turkish agglutinative, 267 part-of-speech tagging, 382 turns, 311 TyDi QA, 304 typed dependency structure, 411 types word, 14 typology, 265 linguistic, 265 unembedding, 200 ungrammatical sentences, 391 unigram name of tokenization algorithm, 270 unit production, 397 unit vector, 111 Universal Dependencies, 413 universal, linguistic, 264 Unix, 5 unknown words in part-of-speech tagging, 376 in text categorization, 61 user-centered design, 325 utterance, 13 value, 188 value sensitive design, 326 vanishing gradient, 135 vanishing gradients, 172 Vauquois triangle, 286 vector, 107, 133 vector length, 110 Vector semantics, 105 vector semantics, 101 vector space, 107 vector space model, 107 verb copula, 365 modal, 365 phrasal, 364 verb alternations, 463 verb phrase, 390 verb-framed language, 267 Verbs, 364 Vietnamese, 267 Viterbi and beam search, 275 Viterbi algorithm, 26, 373 inference in CRF, 380 VITERBI ALGORITHM , 373 vocoder, 349 vocoding, 349 voice user interface, 325 VSO language, 265 wake word, 353 Wall Street Journal Wall Street Journal speech recognition of, 355 warping, 355 waveﬁle format, 336 WaveNet, 351 Wavenet, 351 weight tying, 165, 200 well-formed substring table, 409 WFST, 409 wh-pronoun, 364 wikiﬁcation, 520 wildcard, regular expression, 7 Winograd Schema, 525 Wizard-of-Oz system, 325 word boundary, regular expression notation, 8 closed class, 363 deﬁnition of, 13 error rate, 334, 346 fragment, 13 function, 363, 383 open class, 363 punctuation as, 13 tokens, 14 types, 14 word normalization, 23 word segmentation, 18, 20 word sense, 232 word sense disambiguation, 232, see WSD word shape, 378 word tokenization, 18 word-word matrix, 109 word2vec, 117 wordform, 15 and lemma, 102 versus lemma, 15 WordNet, 232 wordpiece, 269 WSD, 232 Yonkers Racetrack, 49 Yupik, 267 z-score, 82 zero anaphor, 505 zero-shot, 246 zero-width, 13 zeros, 45
foundations_bm25_review	Foundations and TrendsR⃝ in Information Retrieval Vol. 3, No. 4 (2009) 333–389 c⃝ 2009 S. Robertson and H. Zaragoza DOI: 10.1561/1500000019 The Probabilistic Relevance Framework: BM25 and Beyond By Stephen Robertson and Hugo Zaragoza Contents 1 Introduction 334 2 Development of the Basic Model 336 2.1 Information Needs and Queries 336 2.2 Binary Relevance 337 2.3 The Probability Ranking Principle 337 2.4 Some Notation 338 2.5 A Note on Probabilities and Rank Equivalence 345 3 Derived Models 347 3.1 The Binary Independence Model 347 3.2 Relevance Feedback and Query Expansion 349 3.3 Blind Feedback 351 3.4 The Eliteness Model and BM25 352 3.5 Uses of BM25 360 3.6 Multiple Streams and BM25F 361 3.7 Non-Textual Relevance Features 365 3.8 Positional Information 367 3.9 Open Source Implementations of BM25 and BM25F 369 4 Comparison with Other Models 371 4.1 Maron and Kuhns 371 4.2 The Uniﬁed Model 372 4.3 The Simple Language Model 374 4.4 The Relevance (Language) Model 375 4.5 Topic Models 375 4.6 Divergence from Randomness 376 5 Parameter Optimisation 377 5.1 Greedy Optimisation 378 5.2 Multidimensional Optimisation 380 5.3 Gradient Optimisation 382 6 Conclusions 384 References 385 Foundations and TrendsR⃝ in Information Retrieval Vol. 3, No. 4 (2009) 333–389 c⃝ 2009 S. Robertson and H. Zaragoza DOI: 10.1561/1500000019 The Probabilistic Relevance Framework: BM25 and Beyond Stephen Robertson1 and Hugo Zaragoza2 1 Microsoft Research,7JJ Thomson Avenue, Cambridge CB3 0FB, UK [email protected] 2 Yahoo! Research, Av. Diagonal 177, Barcelona 08028, Spain [email protected] Abstract The Probabilistic Relevance Framework (PRF) is a formal framework for document retrieval, grounded in work done in the 1970–1980s, which led to the development of one of the most successful text-retrieval algo- rithms, BM25. In recent years, research in the PRF has yielded new retrieval models capable of taking into account document meta-data (especially structure and link-graph information). Again, this has led to one of the most successful Web-search and corporate-search algo- rithms, BM25F. This work presents the PRF from a conceptual point of view, describing the probabilistic modelling assumptions behind the framework and the diﬀerent ranking algorithms that result from its application: the binary independence model, relevance feedback mod- els, BM25 and BM25F. It also discusses the relation between the PRF and other statistical models for IR, and covers some related topics, such as the use of non-textual features, and parameter optimisation for models with free parameters. 1 Introduction This monograph addresses theclassical probabilistic model of informa- tion retrieval. The model is characterised by including a speciﬁc notion of relevance, an explicit variable associated with a query–document pair, normally hidden in the sense ofnot observable. The model revolves around the notion of estimating a probability of relevance for each pair, and ranking documents in relation to a given query in descending order of probability of relevance. The best-known instantiation of the model is the BM25 term-weighting and document-scoring function. The model has been developed in stages over a period of about 30 years, with a precursor in 1960. A few of the main references are as follows: [30, 44, 46, 50, 52, 53, 58]; other surveys of a range of proba- bilistic approaches include [14, 17]. Some more detailed references are given below. There are a number of later developments of IR models which are also probabilistic but which diﬀer considerably from the models developed here — speciﬁcally and notably the language model (LM) approach [24, 26, 33] and thedivergence from randomness(DFR) mod- els [2]. For this reason we refer to the family of models developed here as the Probabilistic Relevance Framework(PRF), emphasising the 334 335 importance of the relevance variable in the development of the models. We do not cover the development of other probabilistic models in the present survey, but some points of comparison are made. This is not primarily an experimental survey; throughout, asser- tions will be made about techniques which are said to work well. In general such statements derive from experimental results, many exper- iments by many people over a long period, which will not in general be fully referenced. The emphasis is on the theoretical development of the methods, the logic and assumptions behind the models. The survey is organised as follows. In Section 2 we develop the most generic retrieval model, which subsumes a number of speciﬁc instanti- ations developed in Section 3. In Section 4 we discuss the similarities and diﬀerences with other retrieval frameworks. Finally in Section 5 we give an overview of optimisation techniques we have used to tune the diﬀerent parameters in the models and Section 6 concludes the survey. 2 Development of the Basic Model 2.1 Information Needs and Queries We start from a notion of information need. We take a query to be a representation (not necessarily very good or very complete) of an individual user’s information need or perhaps search intent. We take relevance to mean the relevance of a document to the information need, as judged by the user. We make no speciﬁc assumption about the conceptual nature of relevance; in particular, we donot assume rel- evance to be topical in nature. 1 We do, however, makes some assump- tions about the formal status of the relevance variable, given below, which might be taken to imply some assumptions about its conceptual nature. 1 The notion of topic, as used in TREC, is somewhat akin to information need, or at least to a more detailed and complete representation and speciﬁcation of an information need. The use of the term ‘topic’ for this purpose is a little unfortunate, as it seems to imply some kind of topical nature of relevance. However, we also note that the widespread use of the term following TREC also includes some examples of non-topical ‘topics’, for example the home-page topics used in the TREC Web track [59]. 336 2.2 Binary Relevance 337 2.2 Binary Relevance The assumptions about relevance are as follows: 1. Relevance is assumed to be a property of the document given information need only, assessable without reference to other documents; and 2. The relevance property is assumed to be binary. Either of these assumptions is at the least arguable. We might easily imagine situations in which one document’s relevance can only be per- ceived by the user in the context of another document, for example. Regarding the binary property, many recent experimental studies have preferred a graded notion of relevance. One might go further and sug- gest that diﬀerent documents may be relevant to a need in diﬀerent ways, not just to diﬀerent degrees. However, we emphasise that we consider the situation of a single information need (rather than the multiple needs or intents that might be represented by a single text query). If we take relevant to mean ‘the user would like to be pointed to this document’, the binary notion is at least moderately plausible. 2.3 The Probability Ranking Principle Given that an information retrieval system cannot know the values of the relevance property of each document, we assume that the infor- mation available to the system is at best probabilistic. That is, the known (to the system) properties of the document and the query may provide probabilistic or statistical evidence as to the relevance of that document to the underlying need. Potentially these properties may be rich and include a variety of diﬀerent kinds of evidence; the only infor- mation assumed to be absent is the actual relevance property itself. Given whatever information is available, the system may make some statistical statement about the possible value of the relevance property. Given a binary document-by-document relevance property, then this statistical information may be completely encapsulated in aprobability of relevance. The probability of relevance of a given document to a 338 Development of the Basic Model given query plays a central role in the present theory. We can in fact make a general statement about this: If retrieved documents are ordered by decreasing prob- ability of relevance on the data available, then the sys- tem’s eﬀectiveness is the best that can be obtained for the data. This is a statement of the Probability Ranking Principle (PRP), taken from [52], an abbreviated version of the one in [40]. The PRP can be justiﬁed under some further assumptions, for a range of speciﬁc measures of eﬀectiveness. It is not, however, completely general; one can also construct counter-examples. But these counter-examples depend on probabilities deﬁned over a population of users. The PRP is safe for the case considered: an individual user. It will be assumed for the remainder of this work. 2.4 Some Notation In this section, we introduce some of the notation that will be used throughout the survey. The notation assumes a single query q repre- senting a single information need. The ﬁrst symbol is used to indicate that two functions are equivalent as ranking functions. rank equivalence: ∝ q e.g. g() ∝q h() In developing the model, from the probability of relevance of a doc- ument to a term-weighting and document-scoring function, we make frequent use of transformations which thus preserve rank order. Such a transformation (in a document-scoring function, say) may be linear or non-linear, but must be strictly monotonic, so that if documents are ranked by the transformed function, they will be in the same rank order as if they had been ranked by the original function. The property of relevance is represented by a random variable Rel with two possible values: relevance Rel: rel , rel (relevant or not) 2.4 Some Notation 339 As discussed above, we assume that relevance is a binary property of a document (given an information need). We will use the short-hand notation P(rel\|d,q) to denote P(Rel = rel\|d,q). For documents and queries, we generally assume a bag or set of words model. We have a vocabulary of terms indexed into the set V, each of which may be present or absent (in the set of words model) or may be present with some frequency (bag of words model). In either case the objects (documents or queries) may be represented as vectors over the space deﬁned by the vocabulary. Thus a document is: document: d := (tf 1,..., tf \|V\|), where tf i normally represents the frequency of termti in the document. We will also need to distinguish between the random variable TFi and its observed value in a document, tf i. The random variable will usually refer to the term frequencies of a given term in the vocabulary. However, the formulation of the basic model is somewhat more general than this, and can accommodate any discrete variable as a feature (e.g., any discrete property or attribute of the document). Thus we can re-interpret tf as simply representing presence or absence of a term; this is the basis of Section 3.1 below. Continuous variables can also be accommodated by replacing probability mass functionsTF i (of discrete variables) by probability density functions (of continuous variables); although we do not present a formal development of the model with continuous variables, we will use this fact in Section 3.7 to introduce some non-textual continuous features into the model. A query is represented in two diﬀerent ways. In the ﬁrst, it is treated similarly as a vector: query: q := (qtf 1,..., qtf \|V\|) Here again the components qtf i may represent term frequencies in the query, or may represent a binary presence or absence feature. Throughout this survey we will need to sum or multiply variables of terms present in the query (i.e., with qtf i > 0). For this purpose we deﬁne the set of indices: query terms: q := {i\|qtf i > 0} (indices of terms in the query) 340 Development of the Basic Model 2.4.1 Preview of the Development The following provides an overview of the development of the basic model; the individual steps are discussed below. The main point to note is that we start with the very general PRP, and end up with an explicit document scoring function, composed of a simple sum of weights for the individual query terms present in the document. P(rel\|d,q) ∝ q P(rel\|d,q) P(rel\|d,q) (2.1) = P(d\|rel,q) P(d\|rel,q) P(rel\|q) P(rel\|q) (2.2) ∝q P(d\|rel,q) P(d\|rel,q) (2.3) ≈ ∏ i∈V P(TFi = tf i \|rel,q) P(TFi = tf i \|rel,q) (2.4) ≈ ∏ i∈q P(TFi = tf i \|rel) P(TFi = tf i \|rel) (2.5) ∝q ∑ q log P(TFi = tf i \|rel) P(TFi = tf i \|rel) (2.6) = ∑ q Ui(tf i) (2.7) = ∑ q,tf i>0 Ui(tf i)+ ∑ q,tf i=0 Ui(0) − ∑ q,tf i>0 Ui(0) + ∑ q,tf i>0 Ui(0) (2.8) = ∑ q,tf i>0 (Ui(tf i) − Ui(0)) + ∑ q Ui(0) (2.9) ∝q ∑ q,tf i>0 (Ui(tf i) − Ui(0)) (2.10) = ∑ q,tf i>0 wi (2.11) 2.4 Some Notation 341 2.4.2 Details We start with the probability of relevance of a given document and query. The ﬁrst three steps are simple algebraic transformations. In Step (2.1), we simply replace the probability by an odds-ratio. 2 In Step (2.2), we perform Bayesian inversions on both numerator and denominator. In Step (2.3), we drop the second component which is independent of the document, and therefore does not aﬀect the rank- ing of the documents. In Step (2.4) (term independence assumption), we expand each probability as a product over the terms of the vocabulary. This step depends on an assumption of statistical independence between terms — actually a pair of such assumptions, for the relevant and non-relevant cases, respectively. Note that we are not assuming unconditional term independence, but a weaker form, namely conditional independence. 3 This is clearly a much more arguable step: in general terms will not be statistically independent in this fashion. The obvious reason for taking this step is to lead us to a simple and tractable (even if approximate) scoring function. However, we may make some further arguments to support this step: 1. Models of this type in other domains, known as Na¨ıve Bayes models, are well known to be remarkably good, simple and robust, despite signiﬁcant deviations from independence. Experimental evidence in IR provides some support for this general statement. 2. The pair of assumptions is not in general equivalent to a blanket assumption of independence between terms over the whole collection. On the contrary, for a pair of query terms, both statistically correlated with relevance, the pair of assumptions predict a positive association between the two terms over the whole collection. In fact we often observe such a positive association. In eﬀect the model says that this 2 This transformation is order preserving; the odds-ratio of p is p 1−p , and this function is a monotonous increasing function of p ∈ [0, 1). 3 P (ab \|c)= P (a \|c)P (b \|c). 342 Development of the Basic Model positive association is induced by relevance to the query. 4 This is clearly an over-simpliﬁcation, but perhaps not such a bad one. 3. Cooper [11] has demonstrated that we can arrive at the same transformation on the basis of a weaker assumption, called linked dependence. 5 This is essentially that the degree of sta- tistical association between the terms is the same in the rel- evant as in the non-relevant subsets. Again, this theoretical result may help to explain the robustness of such a model. We may represent the independence assumptions by means of a graph- ical model, as in Figure 2.1. This diagram shows how the term frequen- cies (tf i)i∈V are assumed to be generated as stochastic observations dependant only on the state of the relevance variable Rel (a full expla- nation of graphical model diagrams is outside the scope of this paper, we refer the reader to [6]). In step (2.5) (query-terms assumption), we restrict ourselves to the query terms only: in eﬀect, we assume that for any non-query term, the ratio of conditional probabilities is constant and equal to one. 6 This might seem a drastic assumption (that no non-query terms have any association with relevance); however, this is not quite the explanation of the step. We have to consider the question of what information we Fig. 2.1 Graphical model indicating basic independence assumptions. 4 If P (a \|c) >P (a \|c) and similarly for b, then it follows that P (ab) >P (a)P (b) even under the conditional independence assumptions made. 5 P (a,b \|c) P (a,b \|c) = P (a \|c) P (a \|c) P (b \|c) P (b \|c) . 6 Note that if the ratio is constant, then it must be equal to one; otherwise we would have one of the probability distribution with probabilities always higher than another one, which is impossible since they both need to sum to 1. 2.4 Some Notation 343 have on which to base an estimate of the probability of relevance. It is a reasonable prior assumption, and turns out to be a good one, that in general query terms are (positively) correlated with relevance. However, we can make no such assumption about non-query terms (a term relating to a completely diﬀerent topic could well be negatively correlated with relevance). In the absence of any link to the query, the obvious vague prior assumption about a random vocabulary term must be that it is not correlated with relevance to this query. Later we will consider the issue of query expansion, adding new terms to the query, when we have evidence to link a non-query term with relevance to the query. Again, we can illustrate the assumptions by means of a graphical model, as in Figure 2.2. This diagram shows that in this model the relevance variable only aﬀects terms in the query. Starting in Step (2.6) we use the following short-hand notation: under the summation operator we will writeq (the starting set of values for i) followed by conditions that i should satisfy. For example: ∑ q should be read as ∑ i∈q, and ∑ q,tf i>0 should be read as ∑ {i \|i∈q,tf i>0}. In Step (2.6), we make a common, order-preserving transformation, namely we take a log. This allows us to express the product of proba- bilities as a sum of log probabilities — actually log-odds because of the ratio in the product. In Step (2.7), we rewrite the previous equation using a function U i(x): Ui(x): =l o gP(TFi = x\|rel) P(TFi = x\|rel) (2.12) Fig. 2.2 Graphical model for restriction to query terms. 344 Development of the Basic Model Note that this is not the log-odds function. Note also that in Step (2.7) each term frequencytf i will be applied to a diﬀerent function Ui(tf i). This is necessary since the weight of a term does not depend only on its frequency, but also in other factors such as the collection frequency, etc. The following two steps use an arithmetic trick (sometimes referred to as removing the zeros) which eliminates from the equation terms that are not present in the document. This is crucial for the eﬃcient implementation of PRF ranking functions using inverted indices. In Step (2.8), we do two things. In the ﬁrst line, the sum over query terms in Step (2.7) has been split into two groups: those terms that are present in the document (ﬁrst sum) and those that are absent (second sum). In the second line, we subtract and add the same quantity (the sum of zero frequency weights for terms in the document) leaving the result unchanged. The reason for doing this will become clear in the next step. In Step (2.9), we regroup the sums of Step (2.8). First we note that the ﬁrst and third sums in Step (2.8) are over the same terms, and can be grouped. This forms the ﬁrst sum in Step (2.9). Then we note that the second and fourth sums in Step (2.8) span all terms in the query. This forms the second sum in Step (2.9). In Step (2.10) we drop the last sum since its value is document- independent. We note that by doing so we are left with a single sum over terms present both in the document and in the query. We have removed terms in the query with zero frequency in the document. In Step (2.11), we again rewrite the equation using the short-hand notation: W i(x): = Ui(x) − Ui(0) (2.13) = log P(TFi = x\|rel)P(TFi =0 \|rel) P(TFi = x\|rel)P(TFi =0 \|rel) (2.14) wi := Wi(tf i) (2.15) Equation (2.14) is the formula for the basic weight of a query term in a document. BothUi and Wi are term-weighting functions which can be precomputed and stored at indexing time. The diﬀerence is that in 2.5 A Note on Probabilities and Rank Equivalence 345 order to use theUi in a document, we would have to sum over all query terms, whether or not they are present in the document. With Wi,w e need only to consider the weights wi of terms present in the document (in eﬀect the rewriting has deﬁned the weight of absent terms as zero). This ﬁts very well with the sparseness of the document-term matrix — we have no need to calculate scores of any document that contains no query terms. Indeed, this is a highly desirable property for a scoring function to be used in an inverted-ﬁle-based document retrieval system, since it means that we can easily calculate the score of all documents with non-zero scores by merging the inverted lists of the query terms. As indicated above, the model is not restricted to terms and term frequencies — any property, attribute or feature of the document, or of the document–query pair, which we reasonably expect to provide some evidence as to relevance, may be included. Below we will consider static features of documents — properties that are not dependent on the query — for inclusion. Any discrete property with a natural zero can be dealt with using the W i form of the weight — if we want to include a property without such a natural zero, we need to revert to the U i form. We note also that both forms are simple linear models — the combi- nation of evidence from the diﬀerent query terms is just by summation. This is not in itself an assumption — it arises naturally from the more basic assumptions of the model. In the sections which follow, we deﬁne various instantiations of this basic sum-of-weights scoring model. 2.5 A Note on Probabilities and Rank Equivalence One consequence of our reliance on the probability ranking principle is that we are enabled to make the very cavalier transformations dis- cussed above, on the basis that the only property we wish to preserve is the rank order of documents. This might be a reasonable assump- tion for traditional ad hoc retrieval, but does not work for all retrieval situations. In some, for example in adaptive ﬁltering [42], we ﬁnd it desirable or necessary to arrive at an explicit estimate of the proba- bility of relevance of each considered document. Unfortunately, while 346 Development of the Basic Model the above development allows us to serve the ranking purpose well, it is not easily reversible to give us such an explicit estimate. In particu- lar, some of the transformations involved dropping components which would not aﬀect the ranking, but would be required for a good proba- bility estimate. Often, as in the case of the component that we drop at Step (2.3), it would be very diﬃcult to estimate. Thus the above model has to be considerably modiﬁed if it is to be used in a situation which requires an explicit probability. This issue is not discussed further in the present survey. 3 Derived Models The models discussed in this section are all derived from the basic model presented in the previous section. We note again that the symbol TF in the basic weighting formula (2.14) can in general be any discrete property or attribute of the document. We start by interpreting it as a binary variable, indicating the presence or absence only of a query term in a document; in Section 3.4 we return to the more familiar term frequency. 3.1 The Binary Independence Model Suppose that TFi is a binary variable, having only the values zero and one. We can think of this, without loss of generality, as presence (one) or absence (zero) of a term: t i will refer to the event that the term is present in the document. The absence event is simply the complement of the presence event; probability of absence is one minus probability of presence. Now Equation (2.14) reduces to: w BIM i = logP(ti \|rel)(1 − P(ti \|rel)) (1 − P(ti \|rel))P(ti \|rel) (3.1) We now suppose that we do actually have some evidence on which to base estimates of these probabilities. Since they are conditional on 347 348 Derived Models the relevance property, we are assuming that we have some judgements of relevance. We ﬁrst imagine, unrealistically, that we have a random sample of the whole collection, which has been completely judged for relevance. We derive an estimator that will also be useful for more realistic cases. We use the following notation: N — Size of the judged sample n i — Number of documents in the judged sample containing ti R — Relevant set size (i.e., number of documents judged relevant) ri — Number of judged relevant docs containing ti Given this information, we would like to estimate in an appropri- ate fashion the four probabilities needed for the weights deﬁned in Equation (3.1). The standard maximum likelihood estimate of a prob- ability from trials is a simple ratio, e.g., P(t i \|rel) ≈ ri R . However, this is not a good estimator to plug into the weighting formula. A very obvious practical reason is that the combination of a ratio of proba- bilities and a log function may yield (positive or negative) inﬁnities as estimates. In fact there are also good theoretical reasons to modify the simple ratio estimates somewhat, as discussed in [44], to obtain a more robust estimator which introduces a small pseudo-count of frequency 0.5. The resulting formula is the well-known Robertson/Sprck Jones weight: w RSJ i = log(ri +0 .5)(N − R − ni + ri +0 .5) (ni − ri +0 .5)(R − ri +0 .5) (3.2) We next suppose that some documents, probably a small number, have been retrieved and judged – this is the usual relevance feedback scenario. In this case we might reasonably estimate the probability conditioned on (positive) relevance in the same way, from the known relevant documents. Estimation of the probabilities conditioned on non- relevance is more tricky. The obvious way, which is what Equation (3.2) ﬁrst suggests, would be to use the documents judged to be not relevant. However, we also know that (normally, for any reasonable query and any reasonable collection) the vast majority of documents are not rele- vant; those we have retrieved and judged are not only probably few in 3.2 Relevance Feedback and Query Expansion 349 number, they are also likely to be a very non-typical set. We can make use of this knowledge to get a better estimate (at the risk of intro- ducing a little bias) by assuming that any document not yet known to be relevant is non-relevant. This is known as thecomplement method. Under this assumption, the RSJ weighting formula deﬁned above still applies, with the following redeﬁnitions of N and n i. N — Size of the whole collection ni — Number of documents in the collection containing ti Experiments suggest that using this complement method gives better estimates than using judged documents only. Finally, we suppose that we have no relevance information at all (the more usual scenario). In this case, we can only assume that the relevance probability P(t i \|rel) is ﬁxed, but we can continue to use the complement method for the non-relevance probability — now we assume for this estimate that the entire collection is non-relevant. All this can be achieved by setting R = r i = 0 in the RSJ formula (3.2) — this is equivalent to settingP(ti \|rel) = 0.5 (other values can be used [15]). The resulting formula is a close approximation to classical idf (it can be made closer still by a slight modiﬁcation of the model [47]): wIDF i = logN − ni +0 .5 ni +0 .5 (3.3) 3.2 Relevance Feedback and Query Expansion The model thus far clearly contains a natural mechanism for rele- vance feedback — that is, for modifying the query based on relevance information. If we start with no relevance information, then we would weight the terms using the inverse document frequency (IDF) formula. Once the user makes some judgements of relevance, we should clearly reweight the terms according to the RSJ formula. But term reweighting is not in general an eﬀective method for improving search. Additionally, we have to consider expanding the query by adding new terms. At an early stage in the development of the basic model, rather than considering the entire vocabulary of terms in the estimation of 350 Derived Models probabilities, we restricted ourselves to query terms only. This was not because we assumed that non-query terms were incapable of giving us any useful information, but rather that in the absence of any evidence about either which terms might be useful, or how useful they might be, a reasonable neutral prior assumption was that all non-query terms had zero correlation with relevance. However, in the relevance feedback scenario discussed above, we do indeed have some evidence for the inclusion of non-query terms. Potentially we can treat any term that occurs in any of the relevant documents as possibly useful. However, it is clear that many such terms will be noise, so we will probably need a conservative rule for adding new terms to the query. For each candidate term (i.e., non-query term present in a document judged relevant) we can consider how useful it might be if added to the query. One measure of this is simply the RSJ weight as above. However, this will emphasise very rare terms (this is consistent with the IDF idea) — such terms may indeed be good evidence of relevance when they occur, but because they occur in so few documents, they will not have much overall eﬀect on retrieval. As an alternative, we look for a measure of the possible overall eﬀect of adding a term; we refer to this (following [53]) as an oﬀer weight. We could base an oﬀer weight formula on a number of diﬀerent models. One proposed in [41], to go with the binary independence model, is as follows. We look for terms that will maximally increase the diﬀerence in average score between relevant and non-relevant items. If we were to add term t i to the query with weight wi, then under the binary independence model (or indeed any additive term-weighting system based on term presence/absence only), it would increase the score of any document containing it by w i. The scores of other doc- uments would not be aﬀected, so the increase in average score could be calculated from the probability of the presence of t i. Thus the dif- ference in average score between relevant and non-relevant documents would be OWi =( P(ti \|rel) − P(ti \|rel))wi (3.4) 3.3 Blind Feedback 351 (note that this is not the same as wi itself). Further, an appropriate estimate of this quantity can be derived as follows: OWRSJ i ≈ P(ti \|rel)wi ≈ ri Rwi ∝q ri wRSJ i (3.5) The ﬁrst step approximates by ignoring the second probability (usually much smaller than the ﬁrst); the second replaces the probability by the obvious estimate; and the third multiplies by R, which is constant for a given query. The usual approach is to extract all terms from the relevant doc- uments, rank them in order of oﬀer weight by formula (3.5), and add only the top x terms from this ranked list ( x m a yb eo ft h e order of 10). This approach to query expansion was intended for the binary inde- pendence model and RSJ weighting, but has also been used with some success for BM25 (see Section 3.4 below). But it clearly has some limi- tations. As we add more and more terms to the query, we are likely to introduce synonyms or closely related words (indeed, this is why we do query expansion in the ﬁrst place). However, in [36, 37] authors argue that this query expansion may worsen the term independence assump- tion; they propose an extension of the PRF model which attempts to correct this by taking into account some of the semantic structure of the query. 3.3 Blind Feedback The same principles may be used in what is now known as pseudo- relevance or blind feedback. Here we assume that we have no actual relevance judgements, but we run an initial search on an initial version of the query (using only original query terms), take the top-ranked y documents (say 5 or 10), assume them to be relevant, and then follow the above relevance feedback procedures. 352 Derived Models We note, however, the following: 1. Blind feedback is generally known to be capable of improving search results on average, but tends to fail on some queries, particularly on queries that are diﬃcult to start with, where the top-ranked documents from the initial search may be poor. 2. The (true) relevance feedback procedure described above is in some sense correct in terms of the probabilistic model, on the assumption that the relevance judgements are good. In the blind feedback case, we have a set of documents whose relevance is (at best) likely rather than sure. It would be better to take account of the probability of relevance of each of these top-ranked documents, rather than simply assum- ing relevance. Such a method could be devised if we had a calibrated probability of relevance for each of these docu- ments. However, the fact that we do not normally have such a calibrated probability in the present model, as discussed in Section 2.5, makes it more diﬃcult to see how to accomplish this. 3.4 The Eliteness Model and BM25 We now re-introduce term frequencies into our model; this requires a model of how diﬀerent term frequencies might arise in a document (this model is originally due to Harter [19]). We suppose that for any document-term pair, there is a hidden property which we refer to as eliteness. This can be interpreted as a form of aboutness: if the term is elite in the document, in some sense the document isabout the concept denoted by the term. Now we assume that actual occurrences of the term in the document depend on eliteness, and that there may be an association between eliteness (to the term) and relevance (to the query). But we further assume that these relations are enough to explain the association between term frequency tf and relevance to the query — that is, given the two assumed dependencies, tf is independent of Rel. As before, we can illustrate the assumptions by means of a graphical model, as in Figure 3.1. 3.4 The Eliteness Model and BM25 353 Fig. 3.1 Graphical model of eliteness (E). We further assume that eliteness itself is binary. The following development, using Harter’s ideas in the context of the PRF, was originally proposed in part in [45] and in full (up to and including BM25) in [46]. 3.4.1 The 2-Poisson Model We introduce some new notation. The eliteness random variableE can take two values: Eliteness E: elite , elite (elite, not elite) We now decompose all the probabilities we want using these two disjoint events, following this pattern: P(α\|β)= P(α\|elite)P(elite\|β)+ P(α\|elite)P(elite\|β) The relationship between eliteness and relevance is denoted thus: pi1 := P(Ei = elite\|rel); pi0 := P(Ei = elite\|rel) The relationship between term frequencies and eliteness is denoted thus: Ei1(tf ): =P(TFi = tf \|Ei = elite); Ei0(tf ): =P(TFi = tf \|Ei = elite) Now, following the above pattern, we arrive at expressions for all the probabilities we are interested in relating the observedtf s to relevance 354 Derived Models like the following: P(TFi = tf \|rel) =pi1Ei1(tf )+( 1 − pi1)Ei0(tf ), etc. This gives us an equation for our term-weights: welite i = log(p1E1(tf )+( 1 − p1)E0(tf ))(p0E1( 0 )+( 1− p0)E0(0)) (p1E1( 0 )+( 1− p1)E0(0))(p0E1(tf )+( 1 − p0)E0(tf )) (3.6) (for readability, we dropped thei subscript from all the elements in the fraction). More speciﬁcally, we make distributional assumptions about these events. Again following Harter, we assume that the distributions of term frequencies across documents, conditioned on eliteness, are Poisson: E ie(tf ) ∼P (λie) (Poisson with mean λie), (3.7) where e ∈{ 0,1}. In general, we expect λi1 >λ i0. Thus in the entire collection of documents, which is a mixture of elite and non-elite doc- uments, tf is distributed as a mixture of two Poisson distributions — so that this model is known as the 2-Poisson model. The consequences of these distributional assumptions are analysed below. The nature of the 2-Poisson model deserves further discussion. In Harter’s original formulation, it was applied to abstracts rather than full-text documents, and indeed it can be said to assume that docu- ments are of ﬁxed length. We can interpret the model as follows. We assume that each document is generated by ﬁlling a certain number of word-positions (ﬁxed length) from a vocabulary of words. Further- more, we assume a simple multinomial distribution over words, so that for each position each word has a ﬁxed (small) probability of being chosen, independent of what other words have been chosen for other positions. Then it follows that the distribution of tf s for a given word is binomial, which approximates to a Poisson under these conditions [16, VI.5]. Now the eliteness model can be seen as a simple topical model which causes variation in the unigram distributions. The author is assumed ﬁrst to choose which topics to cover, i.e., which terms to treat as elite and which not. This deﬁnes speciﬁc probabilities for the unigram model, 3.4 The Eliteness Model and BM25 355 and the author then ﬁlls the word-positions according to this chosen model. This generative version of the 2-Poisson model (that is, a model for how documents are generated) ties it very closely with the language models and topical models discussed further in Sections 4.3 and 4.5. We note the following characteristics of this model: 1. The model of topicality is a very simple one — one word one topic. 2. There is no attempt to normalise the probabilities across the full unigram model for the document. 3. The model depends fairly crucially on the notion that all documents are of the same (ﬁxed) length. We do not in the present survey attempt to do anything about the ﬁrst two points — however, they do point forward to the more recent work on language models, discussed brieﬂy below. Concerning the third point, the issue of document-length normalisation is critical to the present model, and is discussed in Section 3.4.5. 3.4.2 Saturation If we plug the Poisson distributional assumptions into Equation (3.6), we can express the term weight as a function of the two meansλ e and the mixing proportion of elite and non-elite documents in the collection (as well as the observed tf ). This is a somewhat messy formula, and furthermore we do not in general know the values of these three param- eters, or have any easy way of estimating them. The next step in the development of the model was therefore to investigate the qualitative behaviour of the term-weighting function, under diﬀerent conditions, in the hope of arriving at a much simpler expression which would capture its dominant behaviour [46]. Clearly its exact behaviour depends on the parameters, but some generalisations are possible. We note in particular that: 1. w elite i (0) = 0 (this is by design); 2. welite i (tf ) increases monotonically with tf ; 356 Derived Models 3. ... but asymptotically approaches a maximum value as tf →∞ ; and 4. the asymptotic limit being lim tf →∞ welite i (tf ) = log p1(1 − p0) (1 − p1)p0 (3.8) = wBIM i . (3.9) This last formulation is the weight that the eliteness feature on its own would have. That is, if eliteness were observable, instead of being hidden, we could treat it like a simple binary attribute and weight it in exactly the same way as we weighted term presence in the binary independence model. This asymptotic property makes perfect sense. Given (as we have assumed) that the only association betweentf and relevance is via elite- ness, the best information we can hope to get from a term is that the document is indeed elite for that term. In reality our information on this score is probabilistic, and thus the term weight is correspondingly reduced. Although this behaviour of the weighting function has been established only for the 2-Poisson case, it seems likely thatany reason- able distributional assumptions would exhibit similar characteristics. We refer to this behaviour as saturation. That is, any one term’s contribution to the document score cannot exceed a saturation point (the asymptotic limit), however, frequently it occurs in the document. This turns out to be a very valuable property of the BM25 weighting function deﬁned below. 3.4.3 A Special Case There is one case in which the saturation limit does not apply. If we assume that the eliteness property for each query term coincides with relevance for the query/need, so thatp i1 = 1 andpi0 = 0, then the limit is inﬁnite, and the weight becomes linear in tf . Thus the commonly used term-weighting functions such as the traditional tf idf , linear in tf , seem to ﬁt with such a model. However, the non-linear, saturating function of tf developed below (also combined with anidf component) has frequently been shown to work better than traditional tf idf . 3.4 The Eliteness Model and BM25 357 3.4.4 BM25 Precursor We investigate the shape of the saturation function a little more closely. It is clear that the properties listed above severely limit the possible functions; nevertheless, there remain many possibilities, as illustrated for example in the left-hand graph in Figure 3.2. However, the 2-Poisson model generates much smoother functions, as shown in the right-hand graph. For most realistic combinations of the parameters the curve is convex, as the top two lines; for some combinations it has an initial concavity, as the bottom line. The next step in the development of BM25 is to approximate this shape. Lacking an appropriate generative corpus model from which to derive a convenient formula, the authors of BM25 decided to ﬁt a simple parametric curve to this shape. The following one-parameter function was chosen: tf k + tf for some k> 0 (3.10) This function satisﬁes the properties listed above, and ﬁts well the possible convex curves. We show values of this function for three diﬀer- ent values ofk in Figure 3.3; the middle line is fork = 1, the upper line for lower k and the lower line for higherk. Note that because we apply this to all terms, the absolute height does not matter; what matters is the relative increments for diﬀerent increments in tf . Thus for high k, increments in tf continue to contribute signiﬁcantly to the score, Fig. 3.2 Left: some possible saturation functions. Right: saturation functions generated by the 2-Poisson model. 358 Derived Models 123456789 1 0 0.0 0.2 0.4 0.6 0.8 1.0 x x / (x + k ) k = 0.2 k = 1 k = 3 k = 10 123456789 1 0 0.0 0.2 0.4 0.6 0.8 1.0 x x / (x + k ( (1 − b) + bdl/avdl) ) dl = avdl dl = avdl 0.1 dl = avdl * 10 Fig. 3.3 Saturation functions generated by Equation (3.10) with raw frequencies (left) and with frequencies normalised using Equation (3.13) (right). Stronger saturation is obtained with lower values of k (left) and with shorter documents (right). For the plot on the right we used k = 1 and b =0 .5. whereas for low k, the additional contribution of a newly observed occurrence tails oﬀ very rapidly. We obtain an early version of BM25 by combining the saturation function of Equation (3.10) with an approximation to the asymptotic maximum of Equation (3.9). The latter is obtained by using the old RSJ presence–absence term weight of Equation (3.2): w i(tf )= tf k + tf wRSJ i (3.11) (We will modify this below for the ﬁnal version of BM25.) The main thing missing so far from the analysis is the question of document length. 3.4.5 Document Length The ﬁxed-document-length assumption was made to allow a connection between a simple language model and BM25; we imagined an author ﬁlling a ﬁxed number of slots with a ﬁxed unigram model. Now we imagine instead that the author also chooses a document length. We suppose that there is something like a standard length for a document, but that an author may decide to make a document longer 3.4 The Eliteness Model and BM25 359 or shorter; we consider only the longer case. Why might an author so decide? We can postulate two extreme cases: Verbosity: Some authors are simply more verbose than others, using more words to say the same thing. Scope: Some authors have more to say: they may write a single document containing or covering more ground. An extreme version would have the author writing two or more documents and concatenating them. The verbosity hypothesis suggests that we should simply normalise any observed tf s by dividing by document length. The scope hypothesis, on the other hand, at least in its extreme version, suggests the opposite. In a real collection of documents we will observe variations in length, which might be due to either eﬀect, or to a combination. We suppose in general a combination: that each hypothesis represents some partial explanation for the observed variation. This in turn suggests that we should apply some kind of soft normalisation. We deﬁne document length in an obvious way: document length: dl := ∑ i∈V tf i and also an average document length for the collection: average doclength: avdl (average over collection) The length normalisation component will be deﬁned in relation to the average; this ensures that the deﬁnition of document length used is not critical. In practice, we could take (for example) the number of characters in the document, or the number of words before parsing, or even the number of unique terms, and still get very similar results. The soft length normalisation component is: B := ( (1 − b)+ b dl avdl ) , 0 ≤ b ≤ 1 (3.12) Thus setting b = 1 will perform full document-length normalisation, while b = 0 will switch normalisation oﬀ. Now we use B to normalise 360 Derived Models tf , before applying the saturation function, as follows: tf ′ = tf B (3.13) wBM25 i (tf )= tf ′ k1 + tf ′ wRSJ i (3.14) = tf k1 ( (1 − b)+ b dl avdl ) + tf wRSJ i (3.15) This is the classic BM25 term-weighting and document-scoring func- tion. As with all term-document weights deﬁned in this survey, the full document score is obtained by summing these term-weights over the (original or expanded) set of query terms. 3.5 Uses of BM25 In order to use BM25 as a ranking function for retrieval, we need to choose values for the internal parametersb and k 1, and also instantiate RSJ. Concerning the RSJ weight (Equation (3.2)), all the previous com- ments apply. In particular, it can be used with or without relevance information. In the absence of relevance information, it reduces as before to a form of idf . In this case, the BM25 weight looks very much like a traditional tf ∗idf weight — a product of two components, one based on tf and one onidf . However, there is one signiﬁcant diﬀerence. The tf component involves the saturation function discussed, and is therefore somewhat unlike most other tf functions seen in the litera- ture, where common choices are tf itself and (1 + log tf ). The latter has a somewhat similar shape curve, but does not have an asymptotic maximum — it goes to inﬁnity, even if somewhat slower thantf itself. Concerning the internal parameters, the model provides no guid- ance on how these should be set. This may be regarded as a limi- tation of the model. However, it provides an opportunity for optimi- sation, given some evaluated set of queries and relevance judgements in the traditional retrieval experiment style. A signiﬁcant number of such experiments have been done, and suggest that in general values s u c ha s0.5 <b< 0.8 and 1 .2 <k 1 < 2 are reasonably good in many 3.6 Multiple Streams and BM25F 361 circumstances. However, there is also evidence that optimal values do depend on other factors (such as the type of documents or queries). 3.5.1 Some Variations on BM25 Published versions of BM25 can vary somewhat (the original BM25 [46] was a little more complicated than that of Equation (3.15), for example). Here we indicate some diﬀerences that might be encountered in diﬀerent versions of the function in published sources. • The original had a component for within-query term fre- quency qtf , for longer queries where a term might occur mul- tiple times. In its full generality, this had a similar saturation function to that used for tf , but with its own k 3 constant. However, experiments suggested that the saturation eﬀect for qtf was unimportant, leading to a formula which was linear in qtf . In other words, one could simply treat multiple occur- rences of a term in the query as diﬀerent terms. • The original also had a further correction for document length, to the total document score. This correction was again found to be unimportant. • A common variant is to add a ( k1 + 1) component to the numerator of the saturation function. This is the same for all terms, and therefore does not aﬀect the ranking produced. The reason for including it was to make the ﬁnal formula more compatible with the RSJ weight used on its own. If it is included, then a single occurrence of a term would have the same weight in both schemes. • Some published versions are based on speciﬁc values assigned to b and k 1. A common combination would be b =0 .5 and k1 = 2. (However, many experiments suggest a somewhat lower value of k1 and a somewhat higher value of b.) 3.6 Multiple Streams and BM25F So far, all the arguments in this survey have been based on the idea that the document is a single body of text, unstructured and undiﬀerentiated 362 Derived Models in any way. However, it is commonplace in search systems to assume at least some minimal structure to documents. In this section we consider documents which are structured into a set of ﬁelds or streams. The assumption is that there is a single ﬂat stream structure, common to all documents. That is, there is a global set of labelled streams, and the text of each document is split between these streams. An obvious example is a title/abstract/body structure such as one might see in sci- entiﬁc papers. In the Web context, with extensive hyperlinks, it is usual to enhance the original texts with the anchor text of incoming links. This is clearly a very minimal kind of structure; one can imagine many document structures that do not ﬁt into this framework. Never- theless, such minimal structures have proved useful in search. The gen- eral idea, not at all conﬁned to the present framework but implemented in many diﬀerent ways in diﬀerent systems, is that some streams may be more predictive of relevance than others. In the above examples, a query match on the title might be expected to provide stronger evi- dence of possible relevance than an equivalent match on the body text. It is now well known in the Web context that matching on anchor text is a very strong signal. 3.6.1 Basic Idea Given a ranking algorithm or function that can be applied to a piece of undiﬀerentiated text, an obvious practical approach to such a stream structure would be to apply the function separately to each stream, and then combine these in some linear combination (with stream weights) for the ﬁnal document score. In terms of the eliteness model, this approach may be regarded as assuming a separate eliteness property for each term/stream pair. Thus for a given term, eliteness in title would be assumed to be a diﬀerent property from eliteness in body. Actu- ally, the assumption would be even stronger: we would have to apply the usual assumptions of independence (given relevance) between these distinct eliteness properties for the same term. This seems a little unreasonabl e—ab etter assumption might be that eliteness is a term/document property, shared across the streams of the document. We would then postulate that the relationship of 3.6 Multiple Streams and BM25F 363 eliteness to term frequency is stream-speciﬁc. In terms of the underlying language model discussed above, we might imagine that the author chooses a length for (say) the title and another for the body. Then, given eliteness (from the author’s choice of topics to cover), the unigram probabilities for the language model for ﬁlling the term-positions would also be stream-speciﬁc. In particular, there would be a much stronger bias to the elite terms when choosing words for the title than for the body (we expect a title to be much denser in topic-speciﬁc terms than an average body sentence). The consequence of this term-document eliteness property is that we should combine evidence across terms and streams in the opposite order to that suggested above: ﬁrst streams, then terms. That is, for each term, we should accumulate evidence for eliteness across all the streams. The saturation function should be applied at this stage, to the total evidence for each term. Then the ﬁnal document score should be derived by combination across the terms. 3.6.2 Notation We have a set of S streams, and we wish to assign relative weights v s to them. For a given document, each stream has its associated length (the total length of the document would normally be the sum of the stream lengths). Each term in the document may occur in any of the streams, with any frequency; the total across streams of these term– stream frequencies would be the usual term-document frequency. The entire document becomes a vector of vectors. streams s =1 ,...,S stream lengths sl s stream weights vs document ( tf1,..., tf\|V\|) vector of vectors tfi vector ( tf 1i,..., tf Si ) where tf si is the frequency of term i in stream s. 3.6.3 BM25F The simplest extension of BM25 to weighted streams is to calculate a weighted variant of the total term frequency. This also implies having 364 Derived Models a similarly weighted variant of the total document length: ˜tf i = S∑ s=1 vs tf si (3.16) ˜dl = S∑ s=1 vs sls (3.17) ˜avdl = average of ˜dl across documents wsimpleBM25F i = ˜tf i k1 ( (1 − b)+ b ˜dl ˜avdl ) + ˜tf i wRSJ i (3.18) However, we may also want to allow the parameters of BM25 to vary between streams — it may be that the diﬀerent streams have diﬀerent characteristics, e.g., in relation to verbosityv. scope (as deﬁned in Section 3.4.5). It is in fact possible to re-arrange formula (3.15) so as to include any of the following in the stream-speciﬁc part: k 1, b, wRSJ i or its non-feedback version wIDF i . We have in particular found it useful to allowb to be stream-speciﬁc; we present here the appropriate version of BM25F: ˜tf i = S∑ s=1 vs tf si Bs (3.19) Bs = ( (1 − bs)+ bs sls avsls ) , 0 ≤ bs ≤ 1 (3.20) wBM25F i = ˜tf i k1 + ˜tf i wRSJ i (3.21) This version was used in [62]; the simple version in [50]. As usual, in the absence of relevance feedback information,wRSJ i should be replaced by wIDF i . In [50, 62] we computed IDFs on the entire collection disregard- ing streams. This worked well in practice, but it can lead to degener- ate cases (e.g., when a stream is extremely verbose and contains most terms for most documents). The proper deﬁnition of IDF in this con- text requires further research (this is also discussed in [37] where the notion of an expected idf is introduced). 3.7 Non-Textual Relevance Features 365 Table 3.1. BM25F parameters reported in [62] for topic distillation (TD) and name page (NP) search tasks Parameter TD’03 NP’03 k1 27.5 4.9 btitle 0.95 0.6 bbody 0.7 0.5 banchor 0.6 0.6 vtitle 38.4 13.5 vbody 1.0 1.0 vanchor 35 11.5 As an illustration, we report in Table 3.1 the BM25Fweights reported in [62] for the 2003 TREC Web Search tasks. 3.6.4 Interpretation of the Simple Version It is worth mentioning a very transparent interpretation of the simple version — although it does not apply directly to the version with vari- able b, it may give some insight. If the stream weights v s are inte- gers, then we can see the simple BM25F formula as an ordinary BM25 function applied to a document in which some of the streams have been replicated. For example, if the streams and weights are{v title =5 , vabstract =2 , vbody =1}, then formula 3.18 is equivalent to 3.15 applied to a document in which the title has been replicated ﬁve times and the abstract twice. 3.7 Non-Textual Relevance Features In many collections there are other sources of relevance information besides the text. Things like the age of a ﬁle, its type or its link in-degree may provide useful information about the relevance of a document. The development of the PRF is quite general and does not make explicit reference to terms or text; it is therefore possible, at least in principle, to take non-textual features into account. We will make two assumptions here about non-textual features. First we make the usual assumption of feature independence with respect to relevance odds (as discussed in Section 2). Second, we will assume that all non-textual features provide relevance information 366 Derived Models which is independent of the query . With these two assumptions we can integrate non-textual features very easily into the PRF and BM25 scoring frameworks. It is possible in principle to relax these assumptions and derive more sophisticated models of relevance for non-textual features. Let us call c =( tf 1,..., tf \|V\|) the vector of term frequency counts of document d, and let us call f an extra vector of non-textual features f =( f1,...,f F ). We have that d =( c,f). Note that the initial PRF development (Equations (2.1–2.4) in Sec- tion 2) applies also to this extended version ofd. Equation (2.4) makes the assumption of feature independence, which carries over the non- textual features. Therefore the product in Equation 2.4 would include all non-textual features f. In Equation (2.5), where we drop all the non-query terms inc, none of the terms inf would be dropped — non-textual features are assumed to be correlated with relevance for all queries. After taking the log in Equation (2.6) we see that the addition of non-textual features simply adds a new term to the sum. Removing the zeros in Equations (2.7–2.9) does not aﬀect this term, so after (2.11) we may write: P(rel\|d,q) ∝ q ∑ q,tf i>0 Wi(tf i)+ F∑ i=1 λiVi(fi), (3.22) where Vi(fi): =l o gP(Fi = fi \|rel) P(Fi = fi \|rel) (3.23) We have artiﬁcially added a free parameter λi to account for re- scalings in the approximation of Wi and Vi. We note that features fi may well be multi-valued or continuous, and this implies the need for care in the choice of function V (fi) (just as we paid attention to ﬁnding a good function oftf ). This will depend on the nature of each non-textual feature fi and our prior knowledge about it. Here are some Vi functions that we have used with success in the past for diﬀerent features: • Logarithmic: log(λ′ i + fi) 3.8 Positional Information 367 • Rational: fi/(λ′ i + fi) • Sigmoid: [λ′ i + exp(−fi λ′′ i )]−1 This development can explain for example why simple scoring functions such as BM25F(q,d)+log(PageRank (d)) may work well in practice for Web Search retrieval. This can much improved by adding the scaling parameter λ, and it can be further improved (only slightly) by changing the log into a sigmoid and tuning the two extra parameters λ ′ and λ′′ [12]. In our work we developed more sophisticated ranking functions integrating several forms of non-textual information and using over a dozen parameters [12, 13]. The optimisation of these parameters is discussed in Section 5. 3.8 Positional Information For the most part the PRF ignores positional information: it cares only about the number of occurrences of a term, but not about their position. There are two important reasons that have held back the PRF from considering positional information: (i) it is extremely hard to develop a sound formal model of relevance which takes into account positional information without exploding the number of parameters, and (ii) position information has been shown to have surprisingly little eﬀect on retrieval accuracy on average. In this section we only give an overview of the existing approaches and discuss the main diﬃculties. We also propose some intuitions of why position may not be as crucial as it seems at ﬁrst sight. Why is it so hard to model relevance with respect to the position of terms in the query and document? Several problems appear. First, we need to deﬁne an appropriate universe of events. In the traditional PRF this universe is simplyN \|q\|, all possible term frequency vectors of terms in the query. The most natural way to consider positions would be to characterise all sequences of terms in the query separated by some num- ber of non-query terms. This leads to an excessively high-dimensional space, and one that is very hard to factor it in an appropriate way. In the traditional model the natural unit over which we build (factorise) the required quantities is the occurrence of a term a given number of 368 Derived Models times. What would be the natural unit over which to build (factorise) the probabilities required to model positions adequately? There have been a number of attempts to deal with these issues and introduce positional information into BM25 (see for example [3, 54, 51] and references therein). The three main approaches used are the following: 1. Indexing phrases as individual terms. This approach is ideal for multi-term tokens (such as White House ) for which a partial match would in fact be incorrect. Its implementation is extremely simple since one does not need to modify the ranking function in any way (each phrase would have its own tf and idf ). There are three problems however: (i) it does not take into account partial matches; (ii) it does not take into account terms which are close but not exactly adjacent; and (iii) one needs to deﬁne the valid phrases [48]. 2. Scoring spans of text instead of entire documents. This can be done explicitly, with a passage retrieval ranking func- tion [3], or implicitly by constructing a ranking function that integrates scores computed over many document spans [51]. 3. Deriving position features (such as the minimum and maxi- mum length of the document spans containing all the terms in the query) which can then be integrated into the scoring function as non-textual features (as those in Section 3.7) [54]. In our opinion none of these attempts would qualify as anatural exten- sion of the PRF, since it is not clear what the assumptions about rele- vance are. Metzler [32, 33] proposes a novel probabilistic retrieval model which makes clear assumptions about positions and relevance, and could perhaps be integrated within the PRF. The model estimates the probability of relevance of document and query jointly: P(Q,D \|Rel). This is done by a Markov Random Field (MRF) which can take into account term-positions in a natural way. The MRF can use any appro- priately deﬁned potentials: while the original work used LM-derived potentials, BM25-like potentials were used in [31]. However, even when using BM25-like potentials, we cannot call this model an extension of 3.9 Open Source Implementations of BM25 and BM25F 369 the PRF, since it models a diﬀerent distribution: P(D,Q \|Rel) instead of the posterior P(Rel\|D,Q). We end this section with a brief discussion of why position infor- mation may not be as important as it may seem at ﬁrst view. It is sobering to see how hard it has been in the past to eﬀectively use prox- imity in IR experiments. All the works referenced in this section claim statistically signiﬁcant improvements over non-positional baselines, but the improvements reported are small. We believe this is specially the case for small collections and high recall situations (typical of academic IR evaluations), since position information is a precision enhancement technique. But even in large collections with high-precision require- ments (such as realistic Web Search evaluations) the gains observed are small. Why is this? We do not know of theoretical or empirical studies about this, but we propose here two hypotheses. First, we believe that natural language, and queries in particular, are quite robust. We would argue that for many queries, a human could determine the relevance of a document to a query even after words in the document and query were scrambled. And for cases that this is not true, it is likely that the user would unconsciouslycorrect the query by adding terms to it that disambiguate it. This does not mean that all queries and documents are position-proof, but the fraction that require positions is small. Second, it should be noted that taking into account the structure of a document (e.g., in BM25F) implicitly rewards prox- imity within important short streams, such as the title. 3.9 Open Source Implementations of BM25 and BM25F We review here several implementations of BM25 and BM25F available as open source. This list is not exhaustive, there may be other search engines or extensions of them that implement BM25 and BM25F. As far as we know only the MG4J [9, 34] fully implements BM25F (version 2.2 or later). BM25 is implemented by Lemur, MG4J, Okapi, PF/Tija, Terrier, Wumpus, Xapian and Zettair [22, 27, 34, 35, 39, 56, 57, 60, 61, 63]. All these search engines are quite modular and could in principle be modiﬁed to extend BM25 in a number of ways, in particular to implement BM25F. Lucene [29] does not implement BM25 370 Derived Models but there exist a third party Lucene extensions that implement BM25 and BM25F [38]. We inspected the code of all these implementations and we believe they are correct. 1 (inspection was visual: we did not run tests ourselves; in two cases implementation was incorrect and we asked the authors to correct it, which they did). None of the systems above provide support for parameter optimisation, although it should not be diﬃcult to extend them for this. 1 There are some minor diﬀerences in BM25 implementations in these packages at the time of writing this survey. For example, PF/Tijah uses idf= logN +0.5 n+0.5 , Terrier does not allow modifying k programmatically and Xapian does not allow modifying any parameters pro- grammatically. 4 Comparison with Other Models The ﬁrst probabilistic model for retrieval was proposed by Maron and Kuhns in 1960 [30]. It was similarly based on a notion of probability of relevance; however, there was an interesting discrepancy between the Maron and Kuhns approach and that of Robertson and Sp¨arck Jones ﬁfteen years later. The discrepancy was the subject of research in the early 1980s on a uniﬁcation of the two. In order to explain both the discrepancy and the attempted uniﬁcation, we ﬁrst describe the Maron and Kuhns model. 4.1 Maron and Kuhns The situation envisaged by Maron and Kuhns was that of a librarian indexing a book (document). The idea was that indexing should antici- pate how people make requests in the ﬁrst place. Ideal indexing should match the requests of just those people who would want to be pointed to this monograph — those people who would ﬁnd it relevant to their needs. The system was assumed to be a system of subject headings, any of which might be assigned by the librarian to the book in ques- tion; a user request would take the form of a chosen subject heading to 371 372 Comparison with Other Models look under. Thus the librarian would be invited to estimate, in respect of each candidate subject heading, the probability that a user looking there would ﬁnd this particular book relevant. Thus far, the basis for the model looks very like the PRF deﬁned in Section 2. However, to better understand the nature of the probability of relevance as interpreted by Maron and Kuhns, we need to consider the event space in which it is deﬁned. This will give us the basis for a comparison in the following section. We have (at least in the mind of the librarian) a group of users (with information needs) who look under a particular subject heading. On the other hand, we have a single individual book in front of the librarian. Therefore the event space involved, in which the probability should be deﬁned, is a space of users. If we were to attempt to use feedback to estimate such a probability, we would ideally count users — that is, of all the users who express their queries by means of a particular subject heading, what proportion would like this monograph. This kind of approach has recently acquired new force, because the big Web Search engines typically see some queries (the so-called ‘head’ queries) repeated very many times by diﬀerent users. Click log data looks a little like relevance feedback on individual documents from a population of users with similar queries. However, the relation between this and the PRF discussed in this monograph needs further analysis. 4.2 The Uniﬁed Model We re-visit the original RSJ model, the foundation of the model pre- sented in this survey, in order to deﬁne it in similar terms. In this case, we start with a single individual user, who puts a request using cer- tain words. Now we ask the question, what is the probability that any arbitrary document matching one (or a combination) of these words is relevant to the user. Thus the event space here consists of documents, and if we want to use feedback to estimate the probability, we would count documents, as in Section 3.1. It immediately becomes clear that although both models refer to probability of relevance, they deﬁne their respective versions of this 4.2 The Uniﬁed Model 373 probability in diﬀerent event spaces. In fact, the two probabilities of relevance are actually quite distinct. The uniﬁcation attempt [43] was based on the following argument. We imagine a matrix of users-with-information-needs against docu- ments (individual users u, individual documents d). Ideally we would like to assign a meaningful probability of relevance to the speciﬁc com- bination of an individual user with an individual document:P(rel\|u,d). However, doing this directly looks diﬃcult — at least if we are looking for direct evidence (feedback). If we show d to u and get a judgement, we no longer need to assign a probability! Indeed, it only makes sense to do so when we have classes of similar events. Maron and Kuhns start by classifying users together, according to the subject headings they consult. Thus we are dealing with a class U of users, and ask about P(rel\|U,d). On the other hand, Robertson and Sprck Jones classify documents together in classes such as D, and ask about P(rel\|u,D). The uniﬁed model proceeded to deﬁne four diﬀerent probabilities of relevance. We might consider starting with P(rel\|U,D), which is a general model needing only feedback from similar users about similar documents (this is referred to as Model 0). If we have feedback about the particular document, we can improve this estimate by considering P(rel\|U,d) (Model 1). On the other hand, if we have feedback from the particular user, we should go for P(rel\|u,D) (Model 2). Finally, if we have both kinds of more specialist evidence simultaneously (Model 3), we might aim for an even better probability. However, its exact rela- tionship to P(rel\|u,d) is not quite obvious, because while it is based on feedback of the above three kinds, it would not actually have feedback on the exact individual pair. The uniﬁed model has been the subject of some more recent work [8], and we are now entering a domain in which many diﬀerent kinds of feedback may be possible, given the kind of logging of Web search- ing behaviour that is now the norm. Other authors (for example [14, 17], and later [25] in the context of the language model discussed below) have approached the problem by in eﬀect limiting themselves to Model 0, by considering onlyrepresentations of documents and queries, rather than individual instances. 374 Comparison with Other Models However, in the present survey we have restricted ourselves to the RSJ view of the probability of relevance. 4.3 The Simple Language Model The language model (LM) of IR is a more recent innovation [24, 26, 33], also with a strong probabilistic motivation. We ﬁrst describe the simple LM, and then some more sophisticated developments. In the LM, we regard any piece of text (document, query, etc.) as having been generated by a statistical process. Outside of IR, such models have been very successful in various areas, particularly in speech recognition, and the IR application has borrowed from that domain. In this particular view of text, it is regarded as being generated word- by-word in sequence, each word being chosen from a vocabulary. The simplest statistical model, so-calledunigram, has a ﬁxed probability dis- tribution over the vocabulary, applied to all word-positions (so actually the sequence is not important). n-gram models assume that the choice of the next word depends on the n − 1 previous words chosen. The simple LM approach to IR assumes that each document has its own model, and asks this question about each document: what is the probability that the query came from (was generated by) the same language model as the document (there is no separate query model in this approach). This probability is used to rank documents for a query. Thus there is no explicit notion of relevance; the implicit notion is that a document is relevant to a query if the query came from the same lan- guage model as the document. This approach also does not distinguish between individual users — a query is understood to be just a text, and each query–document pair is considered as an equivalent individual event. From an individual user point of view, the model implicitly assumes that there is just one relevant document (if this instance of a query was generated by the language model for document 1, it could not also have been generated by the diﬀerent language model for docu- ment 2). However, since the approach does not distinguish individuals, in eﬀect it represents a version of Model 0 (in the terms of the uniﬁed model, as discussed above). Diﬀerent instances of the same query can be generated from diﬀerent documents; in the (U,D) class, more than 4.4 The Relevance (Language) Model 375 one document can be relevant. But it follows that it makes no sense to consider individual feedback in the context of the simple LM. In more detail, the document language model is usually built by mixing (smoothing) probabilities derived from the document itself with those taken from a general background language model. One purpose of this smoothing is to avoid a zero probability being assigned to any term that does not occur in the document. In general, a rich fund of models and estimation methods has been mined within the general framework of the LM. We further explore only two of these developments in the following sections. 4.4 The Relevance (Language) Model Some of the limitations of the simple model are addressed in work on a relevance model for the LM framework [26, 33]. Here, by contrast, we assume that the query has (that is, is generated by) its own model, distinct from any particular document model. The initial source for this model is clearly the query itself; however, relevance feedback (from the individual user, for example) can provide additional evidence about it. With this approach, the document–query matching process becomes much less obvious. Note that both in the simple LM, and in the tradi- tional probabilistic relevance framework (PRF) described in this survey, the process of matching the query to the document is inherent in the model, entirely determined by the model itself. In this new context, no such matching process is deﬁned; it is necessary to choose one from outside the model. Given that both the document LM and the query (relevance) LM take the form of statistical distributions over a vocabulary, matching becomes a question of matching two statistical distributions. The most common way to do this is to use the Kullback–Leibler (KL) divergence measure. This has good empirical support. 4.5 Topic Models There is perhaps potential for some kind of bridge between the LM approach and the PRF in work on implicit topic models. Most such 376 Comparison with Other Models work has sought to discover the topics implicit in the statistical structure of the language of documents; examples are Latent Seman- tic Indexing (LSI) [18], Probabilistic LSI [21] and Latent Dirichlet Allocation [7]. The hidden eliteness variables postulated in the proba- bilistic relevance framework have some similarities (although much sim- pliﬁed by the assumption of one eliteness variable per term). A related approach is the parsimonious LM [20], which attempts to model in what ways a particular document or query is distinguished from the general background language. However, we appear to be some distance from a serious uniﬁcation of the LM and the PRF which is the main subject of this survey. 4.6 Divergence from Randomness The DFR models [2], like the language models, do not contain “relevance” as a primitive notion. Also like the language models, they concentrate on the statistical distribution of terms in documents. In general, they seek to identify the ways in which term distributions dif- fer from what one might expect on a random basis — evidence of such divergence is taken implicitly as evidence about relevance. It is possible to derive a variety of term-weighting and document ranking functions within this framework, including a formulation that is approximately the same as BM25. 5 Parameter Optimisation Like most IR models, the models in the PRF have free parameters that need to be set to appropriate values. The BM25 and BM25F models are known to be quite robust with respect to their parameters, mean- ing that small changes in the parameter values (or in the collection) do not produce large changes in accuracy or relevance. Nevertheless signiﬁcant gains in relevance can be obtained by properly optimising the parameters, specially when we deal with a new collection. Parameter optimisation comes with considerable costs: it will require the human evaluation of many query results, which is expensive, and the optimised parameters will be speciﬁc to the collection evalu- ated and may not work well for other collections. Furthermore, the optimisation procedure can be computationally costly, requiring more computing power that the search engine itself. For these reasons this approach is only appropriate for speciﬁc collections which merit the cost needed to optimise the ranking function. Examples of such collec- tions are the WWW, large corporate collections or high-value News or Help sites. Let us call θ the vector of all free parameters of the ranking model being tuned. In the case of BM25 this vector would have two 377 378 Parameter Optimisation components: θ =( k1,b). In the case of BM25F it would have more: θ =( k1,b1,...,b S ,v1,..., vS ). If we include non-textual features then we have even more parameters, the exact number depending on the trans- formation used for each non-textual feature. ‘Tuning’ the parameters of the model can be seen as an optimisa- tion problem, where we seek to maximise the relevance of the ranking model with respect to the model parameters θ, for a given set of rele- vance judgements. More speciﬁcally, if we ﬁx the document collection and a set of queries with their associated relevance judgements, then for any parameter setting θ we can compute the performance measure of our choice M(θ). What is left is to ﬁnd the parameter setting that maximises M(θ). This is exactly an n-dimensional optimisation prob- lem with M as the target function being optimised over the space of valid θ values. Optimising standard IR measures, 1 however, is not easy: they are very expensive to evaluate, they have local maxima andplateaus, they are not smooth and they don’t have gradients [49]. For these reasons, it is not easy to apply standard optimisation techniques. Even applying simple 0-th order optimisation techniques such as trusted region opti- misation is diﬃcult and expensive. In practice we use a number of ad hoc techniques to speed up exhaustive search. We will discuss these in the next subsection. Another alternative is to change the function being optimised. This approach is specially useful when one is optimising very many features, and is discussed in the last subsection. 5.1 Greedy Optimisation We discuss here a number of techniques (some heuristics, some imple- mentation tricks) we used in the past to speedup the exhaustive search and ﬁnd good parameter values. Caching: Because function evaluations are so expensive, we cache the evaluations. Indeed we may often re-visit a parameter value in our 1 Such as Average Precision, Precision@k, Mean Reciprocal Rank and Discounted Cumulative Gain. 5.1 Greedy Optimisation 379 search for the optimum. It would be wasteful to re-evaluate all the queries; instead, we store the resulting relevance in a hash table using the parameter values as the key: H[θ] ← M(θ). Grid: It is also useful to set a minimum resolution for every parame- ter, deﬁning a grid of allowed parameters values. For example, in most of our experiments we did not allow parameters to change beyond the second decimal. This has negligible aﬀect on the relevance performance and greatly increases the eﬀect of caching and the speed of conver- gence. Furthermore it makes it easier to report and share parameter values. 5.1.1 Robust Line Search We ﬁrst consider methods for optimising over a single parameter. Most parameters being optimised are positive but unbounded above. There- fore we do not know the region of interest of the parameter being opti- mised, nor do we know the required resolution (the size of the intervals to be tested). For this reason we developed the following search algo- rithm, which we call Robust Line Search (RLS). Call l and r the current left and right brackets of the 1 −D search space. Split the region inm equal parts of size (r − l)/m. Compute the performance on the m + 1 region boundaries, and call c the boundary with the highest performance. The idea is to move towards c by re-centering the search region around c and scaling it appropriately. If l<c<r we need to zoom in into c, by scaling down the search region. Since the function being optimised has local maxima we cannot zoom too much, or we may completely lose the global maximum; for this reason we tend to zoom very conservatively. On the other hand, if c equals r or l, it is most likely that the maximum is outside the current search region, and possibly far away. Therefore we increase the size of the search region. We iterate this until l and r are within the some minimum distance, typically below the minimum resolution set. An example optimisation is illustrated in Figure 5.1. 380 Parameter Optimisation Fig. 5.1 Greedy Optimisation example: robust line search. 5.2 Multidimensional Optimisation Any 1-D optimisation method can be generalised to n-D in several ways. We have used two methods to do this, both of which worked well in practice. Greedy Robust Line Search:Imagine that we want to run the RLS method on a 2-D problem, with parametersθ =( θ 1,θ2). Let’s leaveθ2 as a subordinate variable and run RLS onθ1. Every time that we need to compute the performance M(θ) at a given pointθ1 = z, we would need to ﬁx the subordinate θ2 at its optimal value. To ﬁnd out this value, we can run a local RLS to optimise θ2 locally while keeping θ1 = z. Generalising this, every line optimisation withi parameters ﬁres oﬀ an optimisation with i − 1 subordinate parameters and so on, recursively. Of course this is a greedy approximation to the exhaustive (exponen- tial) exploration of the parameter space, because we are running RLS. However, this approach remains exponentially expensive with respect to n because of its recursive nature, and therefore it is not practicable for large n (e.g., n> 3). Promising Directions: Another way we have used to carry out searches in n dimensions is the following. 2 Choose an initial point for 2 This method can be seen as a linear version oftrusted region optimisation methods [4, 5]; it has the advantage of requiring much fewer function evaluations, which are extremely expensive in our case. 5.2 Multidimensional Optimisation 381 each parameter (call the resulting the vector θ). Run n 1−D indepen- dent searches, to ﬁnd the optimum valueθ′ i of each parameter when all others are kept ﬁxed atθ. Each of these optimal values found deﬁnes a promising directionsin parameter space. Now consider the vector going from θ to θ′ =( θ′ 1,...,θ ′ n). We expect this vector to move through inter- esting regions of space if there is correlation between features. There- fore we do one more 1−D optimisation along this line. We do this by re-parameterising the system with a new variable that moves all param- eters linearly fromθ to θ ′. Finally we choose the best parameter setting found so far and we re-start the process. An example optimisation is illustrated in Figure 5.2, where we show the ﬁrst two iterations (noted 1 and 2), each consisting of three line searches (noted a, b and c). The total cost of an iteration is ( n +1 )1 −D optimisations. Therefore, it grows only linearly with n, but it may require very many iterations to complete. K1 Scaling: Note that in general k 1 will depend on the weights assigned to streams in BM25F, even if it is kept as a stream-independent parameter. This is because the stream weights in eﬀect rescale the tf values, and k 1 has to be adjusted accordingly. If we have a goodkBM25 1 value for the regular BM25 function (no streams), we can propose a good initial value ofk1 for BM25F by scaling it according to the change Fig. 5.2 Greedy optimisation example: promising directions. 382 Parameter Optimisation in average document length from the unweighted to the weighted form: kBM25F 1 ≈ kBM25 1 ∑ s vsavsls∑ s avsls (5.1) 5.2.1 Factoring the Search A very useful technique for speeding up the search in practice is to factor the search into batches, optimising together only the parame- ters that are known to be strongly dependent. When doing this it is important to choose the initial parameters and the order of the batches judiciously. A typical schedule for BM25F may be: 1. Compute the optimal k 1 and b (ignoring streams). This is equivalent to setting all vs = 1 and all bs = b. (Cost: 1×2D); 2. Optimise stream weights {w}s=1..S jointly. We use here thek1 re-scaling trick in Step (5.1). (Cost: 1×SD); and 3. Optimise k1 and {b}s=1..S independently (Cost: (S +1 )×1D). This schedule may be repeated until no further increase in performance is obtained. When dealing with non-textual features, the optimisation above is usually interleaved with the optimisation of the non-textual features, which can also be done independently or jointly by batches. 5.3 Gradient Optimisation A diﬀerent possibility is to choose a performance function that can be optimised directly by machine learning techniques. This approach was pursued with some success a few years ago [1, 10, 55], and has now become an active area of research (see for example the recent NIPS and SIGIR workshops on this topic [23, 28]). The main idea is to approximate rank-dependant relevance functions such as NDCG by a function with known gradients. Then we can apply the battery of gradient-based optimisation methods. The speed of these methods does not grow exponentially with the number of dimensions optimised, so one can optimise very many parameters simultaneously. Furthermore, if BM25F is being combined with a larger number of other (non-textual) 5.3 Gradient Optimisation 383 features, methods can be used to optimise jointly all parameters. For example, this is the case in [55] where the overall model is an artiﬁcial neural network which takes as input many features, one of them being BM25F. It is out of the scope of this survey to detail gradient-based optimisation methods. 6 Conclusions The classical probabilistic relevance framework has provided a series of well-founded scoring formula, as well as some signiﬁcant insights into diﬀerent aspects of search. One of the reasons of the success of the PRF, we believe, is the powerful combination of sound theoretical modelling and a pragmatic parameterisation that exploits our prior knowledge in IR. 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gray	Itanium — A System Implementor’s Tale Charles Gray† Matthew Chapman†‡ Peter Chubb†‡ David Mosberger-Tang§ Gernot Heiser†‡ † The University of New South Wales, Sydney, Australia ‡ National ICT Australia, Sydney, Australia § HP Labs, Palo Alto, CA [email protected] Abstract Itanium is a fairly new and rather unusual architecture. Its deﬁning feature is explicitly-parallel instruction-set computing (EPIC), which moves the onus for exploiting instruction-level parallelism (ILP) from the hardware to the code generator. Itanium theoretically supports high degrees of ILP, but in practice these are hard to achieve, as present compilers are often not up to the task. This is much more a problem for systems than for application code, as compiler writers’ efforts tend to be focused on SPEC benchmarks, which are not representative of op- erating systems code. As a result, good OS performance on Itanium is a serious challenge, but the potential re- wards are high. EPIC is not the only interesting and novel feature of Itanium. Others include an unusual MMU, a huge reg- ister set, and tricky virtualisation issues. We present a number of the challenges posed by the architecture, and show how they can be overcome by clever design and implementation. 1 Introduction Itanium [7] (also known as IA64) was introduced in 2000. It had been jointly developed by Intel and HP as Intel’s architecture for the next decades. At present, Ita- nium processors are used in high-end workstations and servers. Itanium’s strong ﬂoating-point performance is widely recognised, which makes it an increasingly popular plat- form for high-performance computing. Its small-scale integer performance is so far less impressive. This is partially a result of integer performance being very de- pendent on the ability of the hardware to exploit any instruction-level parallelism (ILP) available in the code. Most high-end architectures detect ILP in hardware, and re-order the instruction stream in order to maximise it. Itanium, by contrast, does no reordering, but instead relies on the code generator to identify ILP and repre- sent it in the instruction stream. This is called explicitly- parallel instruction-set computing (EPIC), and is based on the established (but to date not overly successful) very-long instruction word (VLIW) approach. EPIC is based on the realisation that the ILP that can be usefully exploited by reordering is limited, and aims at raising this limit. The performance of an EPIC machine is highly de- pendent on the quality of the compiler’s optimiser. Given the novelty of the architecture, it is not surpris- ing that contemporary compilers are not quite up to the challenge [22]. Furthermore, most work on compil- ers is focusing on application code (in fact, mostly on SPEC benchmarks), so compilers tend to perform even worse on systems code. Finally, of the various compilers around, by far the weakest, GCC, is presently the default for compiling the Linux kernel. This poses a number of challenges for system implementors who strive to obtain good OS performance on Itanium. Another challenge for the systems implementor is pre- sented by Itanium’s huge register ﬁle. This helps to keep the pipelines full when running CPU-bound ap- plications, but if all those registers must be saved and restored on a context switch, the costs will be signiﬁ- cant, Itanium’s high memory bandwidth notwithstand- ing. The architecture provides a register stack engine (RSE) which automatically ﬁlls/spills registers to mem- ory. This further complicates context switches, but has the potential for reducing register ﬁlling/spilling over- head [21]. The large register set, and the mechanisms for dealing with it, imply trade-offs that lead to different implementation strategies for a number of OS services, such as signal handling. Exceptions are expensive on processors with high ILP and deep pipelines, as they imply a break in the execu- tion ﬂow that requires ﬂushing the pipeline and wast- ing many issue slots. For most exceptions this is un- avoidable but irrelevant if the exceptions are relatively infrequent (like interrupts) or a result of program faults. System calls, however, which are treated as exceptions on most architectures, are not faults nor necessarily in- frequent, and must be fast. Itanium deals with this is- sue by providing a mechanism for increasing the priv- ilege level without an exception and the corresponding 2005 USENIX Annual Technical Conference USENIX Association 265 pipeline ﬂush, but it is subject to limitations which make it tricky to utilise. Itanium’s memory-management unit (MMU) also has some unusual properties which impact on OS design. Not only does it support a wide range of page sizes (which is nothing unusual), it also supports the choice of two different hardware page-table formats, a virtual linear array (called short VHPT format) and a hash ta- ble (called the long VHPT format). As the names imply, they have different size page table entries, and different performance and feature tradeoffs, including the support for superpages and the so-called protection keys. The hardware page-table walker can even be disabled, effec- tively producing a software-loaded TLB. Protection keys loosen the usual nexus between pro- tection and translation: access rights on pages are not only determined by access bits on page-table entries, but also by an orthogonal mechanism which allows group- ing sets of pages for access-control purposes. This mechanism also supports sharing of a single entry in the translation lookaside buffer (TLB) between processes sharing access to the page, even if their access rights dif- fer. The original architecture is disappointing in a rather surprising respect: it is not fully virtualisable. Virtual- machine monitors (VMMs) have gained signiﬁcant pop- ularity in recent years, and Itanium is almost, but not quite, virtualisable. This creates a signiﬁcant challenge for anyone who wants to develop an Itanium VMM. For- tunately, Intel recognised the deﬁciency and is address- ing it with an architecture-extension called Vanderpool Technology [10], which is to be implemented in future CPUs. This paper presents a discussion of the features of the Itanium architecture which present new and interesting challenges and design tradeoffs to the system implemen- tor. We will discuss the nature of those challenges, and how they can be dealt with in practice. First, however, we present an overview of the Itanium architecture in the next section. In Section 3 we discuss the most in- teresting features of the Itanium’s memory-management unit and the design tradeoffs it implies. In Section 4 we discuss issues with virtualisation of Itanium, while Sec- tion 5 presents a number of case studies of performance tradeoffs and micro-optimisation. Section 6 concludes the paper. 2 Itanium Architecture Overview 2.1 Explicitly-parallel instruction-set com- puting As stated in the Introduction, Itanium’s EPIC approach is based on VLIW principles, with several instructions contained in each instruction word. Scheduling of in- Issue Window 2 Bundles Inst 2 Inst 1 Inst 0 Templ. Instruction Buffer Bundle Backend Pipelines B I M F Figure 1: Instruction Issue structions, and speciﬁcation of ILP, becomes the duty of the compiler (or assembly coder). This means that details of the processor pipelines and instruction laten- cies must be exposed in the architecture, so the compiler can emit correct code without the processor needing to scoreboard instruction dependencies. The Itanium approach to EPIC aims at achieving this without overly limiting the design space of future pro- cessors, i.e., by describing ILP in a way that does not depend on the actual number of pipelines and func- tional units. The compiler is encouraged to maximise ILP in the code, in order to optimise performance for processors regardless of pipeline structure. The result is a greatly simpliﬁed instruction issue, with only a few pipeline stages dedicated to the processor front-end (two front-end and six back-end stages, ignoring ﬂoat- ing point, for Itanium 2). The shorter pipeline helps to reduce exception and mis-prediction penalties. Itanium presents a RISC-like load/store instruction set. Instructions are grouped into 128-bitbundles, which generally hold three instructions each. Several bundles form an instruction group delimited by stops. Present Itanium processors use a two-bundle issue window (re- sulting in an issue of six instructions per cycle). By def- inition, all instructions in a group are independent and can execute concurrently (subject to resource availabil- ity). Figure 1 shows the ﬁrst few stages of the Itanium pipeline. Bundles are placed into the instruction buffer speculatively and on demand. Each clock cycle, all in- structions in the issue window are dispersed into back- end pipelines (branch, memory, integer and ﬂoating- point) as directed by the template, unless a required pipeline is stalled or a stop is encountered in the instruc- tion stream. Each bundle has a 5-bit template ﬁeld which speci- ﬁes which instructions are to be dispersed into which pipeline types, allowing the instruction dispersal to be implemented by simple static logic. If there are not enough backend units of a particular type to disperse an instruction, split issue occurs; the preceding instructions 2005 USENIX Annual Technical Conference USENIX Association266 are issued but that instruction and subsequent instruc- tions must wait until the next cycle — Itanium issues strictly in order. This allows a compiler to optimise for a speciﬁc processor based on the knowledge of the number of pipelines, latencies etc., without leading to incorrect execution on earlier or later processors. One aspect of EPIC is to make even data and con- trol speculation explicit. Itanium supports this through speculative load instructions, which the compiler can move forward in the instruction stream without know- ing whether this is safe to do (the load could be through an invalid pointer or the memory location overwritten through an alias). Any exception resulting from a specu- lative load is deferred until the result is consumed. In or- der to support speculation, general registers are extended by an extra bit, theNaT (“not a thing”) bit, which is used to trap mis-speculated loads. 2.2 Register stack engine Itanium supports EPIC by a huge ﬁle of architected reg- isters, rather than relying on register renaming in the pipeline. There are 128 user-mode general registers (GRs), the ﬁrst 32 of which are global; 16 of these are banked (i.e., there is a separate copy for privileged mode). The remaining 96 registers are explicitly re- named by using register windows, similar to the SPARC [23]. Unlike the SPARC’s, Itanium’s register windows are of variable size. A function uses an allocinstruction to allocate local and output registers. On a function call via the br.callinstruction, the window is rotated up past the local registers leaving only the caller’s output registers exposed, which become the callee’s input reg- isters. The callee can then use allocto widen the win- dow for new local and output registers. On executing the br.ret instruction, the caller’s register window is restored. The second, and most important, difference to the SPARC is the Itanium’s register stack engine (RSE), which transparently spills or ﬁlls registers from memory when the register window overﬂows or underﬂows the available registers. This not only has the advantage of freeing the program from dealing with register-window exceptions. More importantly, it allows the processor designers to transparently add an arbitrary number of windowed registers, beyond the architected 96, in order to reduce memory trafﬁc from register ﬁlls/spills. It also supports lazy spilling and pre-ﬁlling by the hardware. Internally, the stack registers are partitioned into four categories — current, dirty, clean and invalid. Current registers are those in the active procedure context. Dirty registers are those in a parent context which have not yet been written to the backing store, while clean registers are parent registers with valid contents that have been written back (and can be discarded if necessary).Invalid registers contain undeﬁned data and are ready to be allo- cated or ﬁlled. The RSE operation is supported by a number of spe- cial instructions. The flushrs instruction is used to force the dirty section of registers to the backing store, as required on a context switch. Similarly, the loadrs instruction is used to reload registers on a context switch. The coverinstruction is used to allocate an empty reg- ister frame above the previously allocated frame, ensur- ing any previous frames are in the dirty or clean parti- tions. There is another form of register renaming: register rotation, which rotates registers within the current regis- ter window. This is used for so-called software pipelin- ing and supports optimisations of tight loops. As this is mostly relevant at application level it is not discussed further in this paper. 2.3 Fast system calls Traditionally, a system call is implemented by some form of invalid instruction exception that raises the priv- ilege level, saves some processor state and diverts to some handler code. This is essentially the same mecha- nism as an interrupt, except that it is synchronous (trig- gered by a speciﬁc instruction) and therefore often called a software interrupt. Such an exception is inherently expensive, as the pipeline must be ﬂushed, and speculation cannot be used to mitigate that cost. Itanium provides a mechanism for raising the privilege level without an exception, based on call gates. The MMU supports a special permission bit which allows designating a page as a gate page. If an epcinstruction in such a page is executed, the privilege level is raised without any other side effects. Code in the call page (or any code jumped to once in privileged mode) can access kernel data structures and thus imple- ment system calls. (Other architectures, such as IA-32, also provide gates. The Itanium version is more tricky to use, see Section 5.2). 2.4 Practical programming issues The explicit management of ILP makes Itanium per- formance critically dependent on optimal scheduling of instructions in the executable code, and thus puts a stronger emphasis on compiler optimisation (or hand- optimised assembler) than other architectures. In this section we discuss some of these issues. 2.4.1 Bundling and latencies The processor may issue less than a full (six instruction) issue window in a number of cases ( split issue). This can happen if the instructions cannot be issued concur- rently due to dependencies, in which case the compiler 2005 USENIX Annual Technical Conference USENIX Association 267 inserts stops which instruct the processor to split issue. Additionally, split issue will occur if the number of in- structions for a particular functional unit exceeds the (processor-dependent) number of corresponding back- end units available. Split issue may also occur in a num- ber of processor-speciﬁc cases. For example, the Ita- nium 2 processor splits issue directly after serialisation instructions (srlzand sync). Optimum scheduling also depends on accurate knowl- edge of instruction latency, deﬁned as the number of cy- cles of separation needed between a producing instruc- tion and a consuming instruction. Scheduling a consum- ing instruction within less than the producing instruc- tion’s latency does not lead to incorrect results, but stalls execution not only of this instruction, but also of all in the current and subsequent instruction-groups. ALU instructions as well as load instructions that hit in the L1 cache have single-cycle latencies. Thus the great majority of userspace code can be scheduled with- out much consideration of latencies — one simply needs to ensure that consumers are in instruction groups sub- sequent to producers. However, the situation is different for system instruc- tions, particularly those accessing control registers and application registers. On the Itanium 2 processor, many of these have latencies of 2–5 cycles, a few (processor- state register, RSE registers and kernel registers) have la- tencies of 12 cycles, some (timestamp counter, interrupt control and performance monitoring registers) have 36 cycle latencies. This makes scheduling of systems code difﬁcult, and the performance cost of getting it wrong very high. 2.4.2 Other pipeline stalls Normally latencies can be dealt with by overlapping execution of several bundles (Itanium supports out-of- order completion). However, some instructions can- not be overlapped, producing unconditional stalls. This naturally includes the various serialisation instructions (srlz, sync) but also instructions that force RSE ac- tivity (flushrs, loadrs). Exceptions and the rfi (return from exception) instruction also produce un- avoidable stalls, but these can be avoided for system calls by using epc. There also exist other constraints due to various re- source limitations. For example, while stores do not normally stall, they consume limited resources (store buffers and L2 request queue entries) and can therefore stall if too many of them are in progress. Similarly, the high-latency accesses to privileged registers are nor- mally queued to avoid stalls and allow overlapped ex- ecution. However, this queue is of limited size (8 en- tries on Itanium 2); only one result can be returned per cycle, and the results compete with loads for writeback resources. Moreover, accesses to the particularly slow registers (timestamp counter, interrupt control and per- formance monitoring registers) can only be issued every 6 cycles. A case study of minimising stalls resulting from la- tencies in system code is given in Section 5.3. 3 Memory-Management Unit 3.1 Address translation and protection As mentioned earlier, the memory-management unit (MMU) of the Itanium has a number of unusual features. The mechanics of address translation and access-right lookup are schematically shown in Figure 2. The top three bits of the 64-bit virtual address form the virtual region number, which is used to index into a set of eight region registers(RRs) which contain region IDs. The remaining 61 bits form the virtual page num- ber (VPN) and the page offset. Itanium 2 supports a wide range of page sizes, from 4kB to 4GB. The VPN is used together with the region ID to perform a fully- associative lookup of the translation lookaside buffer (TLB). The region ID serves as a generalisation of the address-space ID(ASID) tags found on many RISC pro- cessors. Like an ASID, the region ID supports the co-existence of mappings from different contexts without causing aliasing problems, but in addition allows for simple shar- ing of pages on a per-region basis: if two processes have the same region ID in one of their RRs, they share all mappings in that region. This provides a convenient way for sharing text segments, if one region is reserved for program code and a separate region ID is associated with each executable. Note that if region IDs are used for sharing, the processes not only share pages, but ac- tually share the TLB entries mapping those pages. This helps to reduce TLB pressure. A more unusual feature of the Itanium TLB is thepro- tection keytag on each entry (which is a generalisation of the protection-domain identiﬁers of the PA-RISC [24]). If protection keys are enabled, then the key ﬁeld of the matching TLB entry is used for an associative lookup of another data structure, a set of protection key registers (PKRs). The PKR contains a set of access rights which are combined with those found in the TLB to determine the legality of the attempted access. This can be used to implement write-only mappings (write-only mode is not supported by the rights ﬁeld in the TLB). Protection keys can be used to share individual (or sets of) pages with potentially different access rights. For ex- ample, if two processes share a page, one process with read-write access, the other read-only, then the page can be marked writable in the TLB, and given a protection key. In the one process’s context, the rights ﬁeld in the 2005 USENIX Annual Technical Conference USENIX Association268 Region ID Key Virtual Page # (VPN) Rights Physical Page # (PPN) Translation Lookaside Buffer (TLB) Region ID Region Registers Key Rights Protection Key Registers Virtual Address Physical Page # (PPN) Physical Address Offset Search Search Search Index Virtual Page # (VPN)Virtual Region # (VRN) Figure 2: Itanium address translation and memory protection. corresponding PKR would be set to read-write, while for the other process it would be set to read-only. The pro- cesses again share not only the page but also the actual TLB entries. The OS can even use the rights ﬁeld in the TLB to downgrade access rights for everybody, e.g. for implementing copy-on-write, or for temporarily dis- abling all access to the page. 3.2 Page tables The Itanium has hardware support for ﬁlling the TLB by walking a page table called the virtual hashed page ta- ble (VHPT). There are actually two hardware-supported page-table formats, called the short-format and long- format VHPT respectively. The hardware walker can also be completely turned off, requiring all TLB reloads to be done in software (from an arbitrary page table structure). Turning off the hardware walker is a bad idea. We measured the average TLB reﬁll cost in Linux to be around 45 cycles on an Itanium 2 with the hardware walker enabled, compared to around 160 cycles with the hardware walker disabled. A better way of supporting arbitrary page table formats is to use the VHPT as a hardware-walked software TLB [2] and reload from the page table proper on a miss. Figure 3 shows the format and access of the two types of page table. The short-format VHPT is, name notwith- standing, a linear virtual array page table [5, 12] that is indexed by the page number and maps a single re- gion, hence up to eight are required per process, and the size of each is determined by the page size. Each page table entry (PTE) is 8 bytes (one word) long. It con- tains a physical page number, access rights, caching at- tributes and software-maintained present, accessed and dirty bits, plus some more bits of information not rele- vant here. A region ID need not be speciﬁed in the short VHPT, as it is implicit in the access (each region uses a separate VHPT). The page size is also not speciﬁed in the PTE, instead it is taken from the preferred page size ﬁeld contained in the region register. This implies that when using the short VHPT, the hardware walker can be used for only one page size per region. Non-default page-sizes within a region would have to be handled by (slower) software ﬁlls. The PTE also contains no protection key, instead the architecture speciﬁes that the protection key is taken from the corresponding region register (and is therefore the same as the region ID, except that the two might be of different length). This makes it impossible to specify different protection keys in a region if the short-format VHPT is used. Hence, sharing TLB entries of selected (shared) pages within a region is not possible with this page table format. The long VHPT is a proper hashed page table, indexed by a hash of the page number. Its size can be an arbitrary power of two (within limits), and a single table can be used for all regions. Its entries are 32 bytes (4 words) long and contain all the information of the short VHPT entries, plus a page-size speciﬁcation, a protection key, a tag and a chain ﬁeld. Hence, the long VHPT supports a per-page speciﬁcation of page size and protection key. The tag ﬁeld is used to check for a match on a hashed ac- cess and must be generated by speciﬁc instructions. The chain ﬁeld is ignored by the hardware and can be used by the operating system to implement overﬂow chains. 2005 USENIX Annual Technical Conference USENIX Association 269 VPN 64 bits VPN Hash PPN PPN Tag PKEY psize Chain 4 x 64 bits Short Format Long Format Global VHPTPer−region VHPT Figure 3: Short and long VHPT formats. 3.3 VHPT tradeoffs The advantage of the short VHPT is that its entries are compact and highly localised. Since the Itanium’s L1 cache line size is 64 bytes, a cache line can hold 8 short entries, and as they form a linear array, the mappings for neighbouring pages have a high probability of lying in the same cache line. Hence, locality in the page working set translates into very high locality in the PTEs, and the number of data cache lines required for PTEs is small. In contrast, a long VHPT entry is four times as big, and only two ﬁt in a cache line. Furthermore, hashing destroys locality, and hence the probability of two PTEs sharing a cache line is small, unless the page table is small and the page working set large (a situation which will result in collisions and expensive software walks). Hence, the long VHPT format tends to be less cache- friendly than the short format. The long-format VHPT makes up for this by being more TLB friendly. For the short format, at least three TLB entries are generally required to map the page ta- ble working set of each process, one for code and data, one for shared libraries and one for the stack. Linux in fact, typically uses three regions for user code, and thus will require at least that many entries for mapping a sin- gle process’s page tables. In contrast, a process’s whole long-format VHPT can be mapped with a single large superpage mapping. Furthermore, a single long-format VHPT can be shared between all processes, reducing TLB entry consumption for page tables from ≥ 3 per process to one per CPU. This tradeoff is likely to favour the short-format VHPT in cases where TLB pressure is low, i.e., where the total page working set is smaller than the TLB ca- pacity. This is typically the case where processes have mostly small working sets and context switching rates are low to moderate. Many systems are likely to operate in that regime, which is the reason why present Linux only supports the short VHPT format. The most important aspect of the two page table for- mats is that the short format does not support many of the Itanium’s MMU features, in particular hardware- loaded mixed page sizes (superpages) within a region. Superpages have been shown to lead to signiﬁcant per- formance improvements [17] and given the overhead of handling TLB-misses in software, it is desirable to take advantage of the hardware walker. As Linux presently uses the short-format VHPT, doing so would require a switch of the VHPT format ﬁrst. This raises the question whether the potential performance gain might be offset by a performance loss resulting from the large page-table format. 3.4 Evaluation We did a comparison of page-table formats by imple- menting the long-format VHPT in the Linux 2.6.6 ker- nel. We ran the lmbench [15] suite as well as Suite IX of the aim benchmark [1], and the OSDL DBT-2 bench- mark [18]. Tests were run on a HP rx2600 server with dual 900MHz Itanium-2 CPUs. The processors have three levels of on-chip cache. The L1 is a split instruc- tion and data cache, each 16kB, 4-way associative with a line size of 64 bytes and a one-cycle hit latency. The L2 is a uniﬁed 256kB 8-way associative cache with 128B lines and a 5 cycle hit latency. The L3 is 1.5MB large, 6-way associative, with a 128B line size and 12 cycles hit latency. The memory latency with the HP zx1 chipset is around 100 cycles. The processors have separate fully-associative data and instruction TLBs, each structured as two-level caches with 32 L1 and 128 L2 entries. Using 16kB pages, the per-CPU long-format VHPT was sized at 16MB in our experiments, being four times the size needed to map the entire 2G physical memory. The results for the lmbench process and ﬁle- operation benchmarks are uninteresting. They show that the choice of page table has little impact on performance. This is not very surprising, as for these benchmarks there is no signiﬁcant space pressure on either the CPU caches or the TLB. Somewhat more interesting are the results of the lm- bench context-switching benchmarks, shown in Ta- ble 1. Here the long-format page table shows some no- ticeable performance advantage with a large number of processes but small working sets (and consequently high context-switching rates). This is most likely a result of the long-format VHPT reducing TLB pressure. The per- formance of the two systems becomes equal again when the working sets increase, probably a result of the bet- ter cache-friendliness of the short-format page table, and the reduced relative importance of TLB miss handling costs. The other lmbench runs as well as the aim bench- 2005 USENIX Annual Technical Conference USENIX Association270 Context switching with 0K 2proc 4proc 8proc 16proc 32proc 64proc 96proc U 0.98 1.00 0.95 0.88 0.98 1.44 1.34 M 0.94 0.96 0.95 0.96 1.23 1.30 1.27 Context switching with 4K 2proc 4proc 8proc 16proc 32proc 64proc 96proc U 0.97 0.99 0.97 0.95 1.17 1.20 1.09 M 0.95 0.61 0.78 0.87 1.11 1.13 1.09 Context switching with 8K 2proc 4proc 8proc 16proc 32proc 64proc 96proc U 0.99 0.98 0.96 0.97 1.31 1.17 1.08 M 0.95 0.91 0.96 1.00 1.29 1.15 1.06 Context switching with 16K 2proc 4proc 8proc 16prc 32prc 64prc 96prc U 0.99 0.98 0.96 0.97 1.31 1.17 1.08 M 0.95 0.91 0.96 1.00 1.29 1.15 1.06 Context switching with 32K 2proc 4proc 8proc 16prc 32prc 64prc 96prc U 0.98 0.99 1.04 1.30 1.04 1.03 1.00 M 0.94 0.96 1.00 1.01 0.87 1.00 1.00 Context switching with 64K 2proc 4proc 8proc 16prc 32prc 64prc 96prc U 1.00 0.98 0.94 0.94 1.00 1.00 1.00 M 0.97 0.98 1.06 1.22 0.94 0.99 0.98 Table 1: Lmbench context-switching results. Numbers indicate performance with a long-format VHPT relative to the short-format VHPT: a ﬁgure > 1.0 indicates bet- ter, < 1.0 worse performance than the short-format page table. Lines marked “U” are for a uniprocessor kernel, while “M” is the same for a multiprocessor kernel (on a two-CPU system). mark results were similarly unsurprising and are omit- ted for space reasons. Complete results can be found in a technical report [4]. The SPEC CPU2000 integer benchmarks, AIM7 and lmbench show no cases where the long-format VHPT resulted in signiﬁcantly worse performance than the short-format VHPT, provided the long-format VHPT is sized correctly (with the number of entries equal to four times the number of page frames). We also ran OSDL’s DBT-2 benchmark, which em- ulates a warehouse inventory system. This benchmark stresses the virtual memory system — it has a large res- ident set size, and has over 30 000 TLB misses per sec- ond. The results show no signiﬁcant performance differ- ence at an 85% conﬁdence level — for ﬁve samples, the long format VHPT gave 400(6) transactions per minute, and the short format page table gave 401(4) transactions per minute (standard deviation in the parentheses). We also investigated TLB entry sharing, but found no signiﬁcant beneﬁts with standard benchmarks [4]. Based on these experiments, we conclude that long- format VHPT can provide performance as good or better than short-format VHPT. Given that long-format VHPT also enables hardware-ﬁlled superpages and TLB-entry- sharing across address-spaces, we believe it may very well make sense to switch Linux to the long-format VHPT in the future. 4 Virtualisation Virtualisability of a processor architecture [20] generally depends on a clean separation between user and system state. Any instructions that inspect or modify the sys- tem state (sensitive instructions) must be privileged, so that the VMM can intervene and emulate their behaviour with respect to the simulated machine. Some exceptions to this may be permissible where the virtual machine monitor can ensure that the real state is synchronised with the simulated state. In one sense Itanium is simpler to virtualise than IA-32, since most of the instructions that inspect or modify system state are privileged by design. It seems likely that the original Itanium designers believed in this clear separation of user and system state which is nec- essary for virtualisation. Sadly, a small number of non- virtualisable features have crept into the architecture, as we discovered in our work on the vNUMA distributed virtual machine [3]. Some of these issues were also en- countered by the authors of vBlades [14], a recent virtual machine for the Itanium architecture. The cover instruction creates a new empty regis- ter stack frame, and thus is not privileged. However, when executed with interruption collection off (inter- ruption collection controls whether execution state is saved to the interruption registers on an exception), it has the side-effect of saving information about the previ- ous stack frame into the privileged interruption function state (IFS) register. Naturally, it would not be wise for a virtual machine monitor to actually turn off interruption collection at the behest of the guest operating system, and when the simulated interruption collection bit is off, there is no way for it to intercept the coverinstruction and perform the side-effect on the simulated copy of IFS. Hence, covermust be replaced with an instruction that faults to the VMM, either statically or at run time. The thash instruction, given an address, calculates the location of the corresponding hashtable entry in the VHPT. The ttaginstruction calculates the correspond- ing tag value. These instructions are, for some reason, unprivileged. However, they reveal processor memory management state, namely the pagetable base, size and format. When the guest OS uses these instructions, it obtains information about the real pagetables instead of its own pagetables. Therefore, as with cover, these in- structions must be replaced with faulting versions. 2005 USENIX Annual Technical Conference USENIX Association 271 Virtual memory semantics also need to be taken into account, since for a virtual machine to have reasonable performance, the majority of virtual memory accesses need to be handled by the hardware and should not trap to the VMM. For the Itanium architecture, most features can be mapped directly. However, a VMM will need to reserve some virtual address space (at least for excep- tion handlers). One simple way to do this is to report a smaller virtual address space than implemented on the real processor, thereby ensuring that the guest operating system will not use certain portions. On the other hand, the architecture deﬁnes a ﬁxed number of privilege lev- els (0 to 3). Since the most privileged level must be re- served for the VMM, this means that the four privilege levels in the guest must be mapped onto three real priv- ilege levels (a common technique known as ring com- pression). This means there may be some loss of protec- tion, though most operating systems do not use all four privilege levels. The Itanium architecture provides separate control over instruction translation, data translation and register- stack translation. For example, it is possible to have register-stack translation on (virtual) and data translation off (physical). There is no way to efﬁciently replicate this in virtual mode, since register-stack references and data references access the same virtual address space. Finally, if a fault is taken while the register-stack en- gine is ﬁlling the current frame, the RSE is halted and the exception handler is executed with an incomplete frame. As soon as the exception handler returns, the RSE re- sumes trying to load the frame. This poses difﬁculties if the exception handler needs to return to the guest kernel (at user-level) to handle the fault. Future Itanium processors will have enhanced virtual- isation support known as Vanderpool Technology. This provides a new processor operating mode in which sen- sitive instructions are properly isolated. Additionally, this mode is designed so as to allow the guest operat- ing system to run at its normal privilege level (0) with- out compromising protection, negating the need for ring compression. Vanderpool Technology also provides fa- cilities for some of the virtualisation to be handled in hardware or ﬁrmware ( virtualisation acceleration). In concert these features should provide for simpler and more efﬁcient virtualisation. Nevertheless, there remain some architectural features which are difﬁcult to virtu- alise efﬁciently and require special treatment, in partic- ular the translation modes and the RSE issue described above. 5 Case studies In this section we present three implementation studies which we believe are representative of the approaches that need to be taken to develop well-performing sys- tems software on Itanium. The ﬁrst example, implemen- tation of signals in Linux, illustrates that Itanium fea- tures (in this case, the large register ﬁle) lead to differ- ent tradeoffs from these on other architectures. The sec- ond example investigates the use of the fast system-call mechanism in Linux. The third, micro-optimisation of a fast system-call path, illustrates the challenges of EPIC (and the cost of insufﬁcient documentation). 5.1 Efﬁcient signal delivery In this section we explore a technique to accelerate sig- nal delivery in Linux. This is an exercise in intelli- gent state-management, necessitated by the large regis- ter ﬁle of the Itanium processor, and relies heavily on exploiting the software conventions established for the Itanium architecture [8]. The techniques described here not only improved signal-delivery performance on Ita- nium Linux, but also simpliﬁed the kernel. In this section we use standard Itanium terminology. We usescratch registerto refer to a caller-saved register, i.e., a register whose contents is not preserved across a function-call. Analogously, we use preserved register to refer to a callee-saved register, i.e., a register whose contents is preserved across a function-call. 5.1.1 Linux signal delivery The canonical way for delivering a signal in Linux con- sists of the following steps: • On any entry into the kernel (e.g., due to system call, device interrupt, or page-fault), Linux saves the scratch registers at the top of the kernel-stack in a structure called pt regs. • Right before returning to user level, the kernel checks whether the current process has a signal pending. If so, the kernel: 1. saves the contents of the preserved regis- ters on the kernel-stack in a structure called switch stack (on some architectures, the switch stackstructure is an implicit part of pt regs but for the discussion here, it’s easier to treat it as separate); 2. calls the routine to deliver the signal, which may ignore the signal, terminate the process, create a core dump, or arrange for a signal handler to be invoked. The important point here is that the combination of the pt regs and switch stack structures con- tain the full user-level state (machine context). The pt regs structure obviously contains user-level state, since it is created right on entry to the kernel. For the switch stackstructure, this is also true but less ob- vious: it is true because at the time theswitch stack structure is created, the kernel stack is empty apart from 2005 USENIX Annual Technical Conference USENIX Association272 preserved registers (user state) user stack kernel stack 1 4 sigcontext switch_stack prepare for signal delivery return to user 2 invoke signal handler 3 return from signal pt_regs Figure 4: Steps taken during signal delivery the pt regsstructure. Since there are no intermediate call frames, the preserved registers must by deﬁnition contain the original user-level state. Signal-delivery requires access to the full user-level state for two reasons: 1. if the signal results in a core dump, the user-level state needs to be written to the core ﬁle; 2. if the signal results in the invocation of a signal han- dler, the user-level state needs to be stored in the sigcontextstructure. 5.1.2 Performance Considerations The problem with the canonical way of delivering a sig- nal is that it entails a fair amount of redundant moving of state between registers and memory. For example, as illustrated in Figure 4, the preserved registers: 1. get saved on the kernel stack in preparation for sig- nal delivery 2. get copied to the user-level stack in preparation for invoking a signal handler 3. get copied back to the kernel stack on return from a signal-handler 4. need to be restored from the kernel-stack upon re- turning execution to user level. On architectures with small numbers of architected reg- isters, redundant copying of registers is not a big issue, particularly since their contents is likely to be hot in the cache anyway. However, with Itanium’s large register ﬁle, the cost of copying registers can be high. When faced with this challenge, we decided that rather than trying to micro-optimise the moving of the state, a better approach would be to avoid the redundant moves in the ﬁrst place. This was helped by the follow- ing observations: • For a core dump, the preserved registers can be re- constructed after the fact with the help of a kernel- stack unwinder. Speciﬁcally, when the kernel needs to create a core dump, it can take a snapshot of the current registers and then walk the kernel stack. In each stack frame, it can update the snapshot with the contents of the registers saved in that stack frame. When reaching the top of the kernel-stack, the snapshot contains the desired user-level state. • There is noinherent reason for saving the preserved registers in the sigcontext structure. While it is customary to do so, there is nothing in the Single-UNIX Speciﬁcation [19] or the POSIX stan- dard that would require this. The reason it is not necessary to include the preserved registers in the sigcontext structure is that the signal handler (and its callees) automatically save preserved regis- ters before using them and restore them before re- turning. Thus, there is no need to create a copy of these registers in the sigcontext structure. In- stead, we can just leave them alone. In combination, these two observations make it possible to completely eliminate the switch stack structure from the signal subsystem. We made this change for Itanium Linux in Decem- ber 2000. At that time, there was some concern about the existence of applications which rely on having the full machine-state available in sigcontext and for this reason, we left the door open for a user-level compatibility-layer which would make it appear as if the kernel had saved the full state [16]. Fortunately, in the four years since making the change, we have not heard of a need to activate the compatibility layer. To quantify the performance effect of saving only the minimal state, we forward-ported the original signal- handling code to a recent kernel (v2.6.9-rc3) and found it to be 23–34% slower. This relative slowdown var- ied with kernel-conﬁguration (uni- vs. multi-processor) and chip generation (Itanium vs. Itanium 2). The abso- lute slowdown was about 1,400 cycles for Itanium and 700 cycles for Itanium 2. We should point out that, had it not been for backwards-compatibility, sigcontext could have been shrunk considerably and fewer cache- lines would have to be touched during signal delivery. In other words, in a design free from compatibility con- cerns, the savings could be even bigger. Table 2 shows that saving the minimal state yields signal-delivery performance that is comparable to other architectures: even a 1GHz Itanium 2 can deliver signals about as fast as a 2.66GHz Pentium 4. Apart from substantially speeding up signal delivery, this technique (which is not Itanium-speciﬁc) simpliﬁed the kernel considerably: it eliminated the need to main- tain the switch stack in the signal subsystem and 2005 USENIX Annual Technical Conference USENIX Association 273 SMP UP Chip cycles µs cycles µs Itanium 2 1.00 GHz 3,087 3.1 2,533 2.5 Pentium 4 2.66 GHz 8,320 3.2 6,500 2.4 Table 2: Signalling times with Linux kernel v2.6.9-rc3. (SMP = multiprocessor kernel, UP = uniprocessor kernel) Dynamic Static System Call break epc break epc getpid() 294 18 287 12 getppid() 299 77 290 54 gettimeofday() 442 174 432 153 Table 3: Comparison of system call costs (in cycles) us- ing the standard ( break) and fast ( epc) mechanism, both with dynamically and statically linked binaries removed all implicit dependencies on the existence of this structure. 5.2 Fast system-call implementation 5.2.1 Fast system calls in Linux As discussed in Section 2.3, Itanium provides gate pages and the epc instruction for getting into kernel mode without a costly exception. Here we discuss the prac- ticalities of using this mechanism in Linux. After executing the epc instruction, the program is executing in privileged mode, but still uses the user’s stack and register-backing store. These cannot be trusted by the kernel, and therefore such a system call is very limited, until it loads a sane stack and RSE backing-store pointer. This is presently not supported in Linux, and thus the fast system-call mechanism is restricted by the following conditions: • the code cannot perform a normal function call (which would create a new register stack frame and could lead to a spill to the RSE backing store); • the code must not cause an exception, because nor- mal exception handling spills registers. This means that all user arguments must be carefully checked, including checking for a possible NaT consumption exception (which could normally be handled trans- parently). As a result, fast system calls are presently restricted to handcrafted assembly language code, and function- ality that is essentially limited to passing data between the kernel and the user. System calls ﬁtting those re- quirements are inherently short, and thus normally dom- inated by the exception overhead, so good candidates for implementing in an exception-less way. So far we implemented the trivial system calls getpid() and getppid(), and the somewhat less trivial gettimeofday(), and rt sigprocmask(). The beneﬁt is signiﬁcant, as shown in Table 3: we see close to a factor of three improvement for the most complicated system call. The performance of rt sigprocmask() is not shown. Currently glibc does not implement rt sigprocmask(), so it is not possible to make a meaningful comparison. 5.3 Fast message-passing implementation Linux, owing to its size and complexity, is not the best vehicle for experimenting with fast system calls. The L4 microkernel [11] is a much simpler platform for such work, and also one where system-call performance is much more critical. Message-passing inter-process communication (IPC) is the operation used to invoke any service in an L4-based system, and the IPC oper- ation is therefore highly critical to the performance of such systems. While there is a generic (architecture- independent) implementation of this primitive, for the common (and fastest) case it is usually replaced in each port by a carefully-optimised architecture-speciﬁc ver- sion. This so-called IPC fast path is usually written in assembler and tends to be of the order of 100 in- structions. Here we describe our experience with micro- optimising L4’s IPC operation. 5.3.1 Logical control ﬂow The logical operation of the IPC fast path is as follows, assuming that a sender invokes the ipc() system call and the receiver is already blocked waiting to receive: 1. enter kernel mode (using epc); 2. inspect the thread control blocks (TCBs) of source and destination threads; 3. check that fast path conditions hold, otherwise call the generic “slow path” (written in C++); 4. copy message (if the whole message does not ﬁt in registers); 5. switch the register stack and several other registers to the receiver’s state (most registers are either used to transfer the message or clobbered during the op- eration); 6. switch the address space (by switching the page ta- ble pointer); 7. update some state in the TCBs and the pointer to the current thread; 8. return (in the receiver’s context). The original implementation of this operation (a com- bination of C++ code, compiled by GCC 3.2, and some assembly code to implement context switching) exe- cuted in 508 cycles with hot caches on an Itanium-2 ma- chine. An initial assembler fast path to transfer up to 8 words, only loosely optimised, brought this down to 170 2005 USENIX Annual Technical Conference USENIX Association274 56 BACK_END_BUBBLE.ALL 30 BE_EXE_BUBBLE.ALL 16 BE_EXE_BUBBLE.GRALL 14 BE_EXE_BUBBLE.ARCR 15 BE_L1D_FPU_BUBBLE.ALL 10 BE_L1D_FPU_BUBBLE.L1D_DCURECIR 5 BE_L1D_FPU_BUBBLE.L1D_STBUFRECIR 11 BE_RSE_BUBBLE.ALL 4 BE_RSE_BUBBLE.AR_DEP 6 BE_RSE_BUBBLE.LOADRS 1 BE_RSE_BUBBLE.OVERFLOW Figure 5: Breakdown of bubbles provided by the PMU. cycles. While this is a factor of three faster, it is still on the high side; on RISC architectures the operation tends to take 70–150 cycles [13].1 5.3.2 Manual optimisation An inspection of the code showed that it consisted of only 83 instruction groups, hence 87 cycles were lost to bubbles. Rescheduling instructions to eliminate bubbles would potentially double performance! An attempt at manual scheduling resulted not only in an elimination of bubbles, but also a reduction of the number of instruction groups (mostly achieved by rear- ranging the instructions to make better use of the avail- able instruction templates). The result was 39 instruc- tion groups executing in 95 cycles. This means that there were still 56 bubbles, accounting for just under 60% of execution time. The reason could only be that some instructions had latencies that were much higher than expected. Unfortu- nately, Intel documentation contains very little informa- tion on instruction latencies, and did not help us further. Using the perfmon utility [6] to access Itanium’sper- formance monitoring unit(PMU) we obtained the break- down of the bubbles summarised in Figure 5. The data in the ﬁgure is to be read as follows: 56 bubbles were recorded by the counter back end bubble.all. This consists of 30 bubbles forbe exe bubble.all, 15 bubbles for be l1d fpu bubble.all and 11 bubbles for be rse bubble.all. Each of these is broken down further as per the ﬁgure. Unfortunately, the Itanium 2 Processor Reference Manual [9] is not very helpful here, it typically gives a one-line summary for each PMU counter, which is insuf- ﬁcient to understand what is happening. What was clear, however, was the register stack engine was a signiﬁcant cause of latency. 5.3.3 Fighting the documentation gap Register-stack-engine stalls In order to obtain the in- formation required to optimise the code further, we saw no alternative to systematically measuring the latencies between any two instructions which involve the RSE. The results of those measurements are summarised in Table 4. Some of those ﬁgures are surprising, with some seemingly innocent instructions having latencies in ex- cess of 10 cycles. Thus attention to this table is impor- tant when scheduling RSE instructions. Using Table 4 we were able to reschedule in- structions such that almost all RSE-related bub- bles were eliminated, that is, all of the ones recorded by counters be exe bubble.arcr and be rse bubble.ar dep, plus most of be rse bubble.loadrs. In total, 23 of the 25 RSE-related bubbles were eliminated, resulting in a total execution time of 72 cycles. The remaining 2 bubbles (from loadrsand flushrsinstructions) are unavoidable (see Section 2.4.2). System-instruction latencies Of the re- maining 31 bubbles, 16 are due to counter be exe bubble.grall. These relate to gen- eral register scoreboard stalls, which in this case result from accesses to long-latency registers such as the kernel register that is used to hold the current thread ID. Hence we measured latencies of system instructions and registers. For this we used a modiﬁed Linux kernel, where we made use of gate pages to execute privileged instructions from a user-level program. The modiﬁed Linux kernel allows user-space code to create gate pages using mprotect(). Executing privileged instructions from user-level code greatly simpliﬁed taking the required measurements. Our results are summarised in Table 5. Fortunately, register latencies are now provided in the latest version of the Itanium 2 Processor Reference Manual [9], so they are not included in this table. Unlike the RSE-induced latencies, our coverage of system-instruction latencies is presently still incomplete, but sufﬁcient for the case at hand. Using this information we eliminated the 16 remaining execution-unit-related bubbles, by scheduling useful work instead of allowing the processor to stall. Data-load stalls This leaves 15 bubbles due to data load pipeline stalls, counted as be l1d fpu bubble.l1d dcurecir and be l1d fpu bubble.l1d stbufrecir. The Itanium 2 Processor Reference Manual explains the former as “back-end was stalled by L1D due to DCU recirculating” and the latter as “back-end was stalled by L1D due to store buffer cancel needing recirculate”, which is hardly enlightening. We determined that the store buffer recirculation was most likely due to address conﬂicts between loads and stores (a load following a store to the same cache line within 3 cycles), due to the way we had scheduled loads and stores in parallel. Even 2005 USENIX Annual Technical Conference USENIX Association 275 From To cyc PMU counter mov ar.rsc= RSE AR 13 BE RSE BUBBLE.AR DEP mov ar.bspstore= RSE AR 6 BE RSE BUBBLE.AR DEP mov =ar.bspstore mov ar.rnat= 8 BE EXE BUBBLE.ARCR mov =ar.bsp mov ar.rnat= 8 BE EXE BUBBLE.ARCR mov =ar.rnat/ar.unat mov ar.rnat/ar.unat= 6 BE EXE BUBBLE.ARCR mov ar.rnat/ar.unat= mov =ar.rnat/ar.unat 6 BE EXE BUBBLE.ARCR mov =ar.unat FP OP 6 BE EXE BUBBLE.ARCR mov ar.bspstore= flushrs 12 BE RSE BUBBLE.OVERFLOW mov ar.rsc= loadrs 12 BE RSE BUBBLE.LOADRS mov ar.bspstore= loadrs 12 BE RSE BUBBLE.LOADRS mov =ar.bspstore loadrs 2 BE RSE BUBBLE.LOADRS loadrs loadrs 8 BE RSE BUBBLE.LOADRS Table 4: Experimentally-determined latencies for all combinations of two instructions involving the RSE. RSE AR means any access to one of the registers ar.rsc, ar.bspstore, ar.bsp,o r ar.rnat. From To cyc PMU counter epc ANY 1 - bsw ANY 6 BE RSE BUBBLE.BANK SWITCH rfi ANY 13 BE FLUSH BUBBLE.BRU (1), BE FLUSH BUBBLE.XPN (8), BACK END BUBBLE.FE (3) srlz.d ANY 1 - srlz.i ANY 12 BE FLUSH BUBBLE.XPN (8), BACK END BUBBLE.FE (3) sum/rum/mov psr.um= ANY 5 BE EXE BUBBLE.ARCR sum/rum/mov psr.um= srlz 10 BE EXE BUBBLE.ARCR ssm/rsm/mov psr.l= srlz 5 BE EXE BUBBLE.ARCR mov =psr.um/psr srlz 2 BE EXE BUBBLE.ARCR mov pkr/rr= srlz/sync/fwb/ 14 BE EXE BUBBLE.ARCR mf/invala M0 probe/tpa/tak/thash/ttag USE 5 BE EXE BUBBLE.GRALL Table 5: Experimentally-determined latencies for system instructions (incomplete).ANYmeans any instruction, while USEmeans any instruction consuming the result. after eliminating this, there were still DCU recirculation stalls remaining. While investigating this we noticed a few other undoc- umented features of the Itanium pipeline. It seems that most application register(AR) and control register(CR) accesses are issued to a limited-size buffer (of apparently 8 entries), with a “DCS stall” occurring when that buffer is full. No explanation of the acronym “DCS” is given in the Itanium manuals. It also seems that a DCU recir- culation stall occurs if a DCS data return coincides with two L1 data-cache returns, which points to a limitation in the number of writeback ports. We also found that a DCU recirculation stall occurs if there is a load or store exactly 5 cycles after a move to a region register (RR) or protection-key register (PKR). These facts allowed us to identify the remaining stalls, but there may be other cases as well. We also found a number of undocumented special split-issue cases. Split issue occurs after srlz, sync and mov=ar.unat and before mfinstructions. It also occurs between amov=ar.bsp and any B-unit instruc- tion, as well as between an M-unit and an fwbinstruc- tion. There may be other cases. We also found a case where the documentation on mapping of instruction templates to functional units is clearly incorrect. The manual says “ M AMLI − MSMAI gets mapped to ports M2 M0 I0 – M3 M1 I1. If MS is a getf instruction, a split issue will occur.” However, our experiments show that the mapping is re- ally M1 M0 I0 – M2 M3 I1, and no split issue occurs in this case. It seems that in general the load subtype is allocated ﬁrst. 2005 USENIX Annual Technical Conference USENIX Association276 Version cycles inst. grps bubbles C++ generic 508 231 277 Initial asm 170 83 87 Optimised 95 39 56 Final 36 33 3 Optimal 34 32 2 Archit. limit 9 9 0 Table 6: Comparison of IPC path optimisation, starting with the generic C++ implementation. Optimised refers to the version achieved using publicly available docu- mentation, ﬁnal denotes what was achieved after system- atically measuring latencies. Optimal is what could be achieved on the present hardware with perfect instruc- tion scheduling, while the architectural limit assumes unlimited resources and only single-cycle latencies. 5.3.4 Final optimisation Armed with this knowledge we were able to eliminate all but one of the 15 data-load stalls, resulting in only 3 bubbles and a ﬁnal execution time of 36 cycles, or 24ns on a 1.5GHz Itanium 2. This is extremely fast, in fact unrivalled on any other architecture. In terms of cycle times this is about a factor of two faster than the fastest RISC architecture (Alpha 21264) to which the kernel has been ported so far, and in terms of absolute time it is well beyond anything we have seen so far. This is a clear indication of the excellent performance potential of the Itanium architecture. The achieved time of 36 cycles (including 3 bubbles) is actually still slightly short of the optimal solution on the present Itanium. The optimal solution can be found by examining the critical path of operations, which turns out to be 34 cycles (including 2 unavoidable bubbles for flushrsand loadrs). Signiﬁcant manual reschedul- ing the code would (yet again) be necessary to achieve this 2 cycle improvement. The bottlenecks preventing optimisation past 34 cy- cles are the kernel register read to obtain the current thread ID, which has a 12 cycle latency, and the latency of 12 cycles between mov ar.bspstore=(chang- ing the RSE backing store pointer) and the following loadrs instruction. Also, since many of the instruc- tions are system instructions which can only execute on a particular unit (M2), the availability of that unit be- comes limiting. Additionally, it seems to be impossible to avoid a branch misprediction on return to user mode, as the predicted return address comes from the return stack buffer, but the nature of IPC is that it returns to a different thread. Eliminating those latencies would get us close to the architectural limit of Itanium, which is characterised as having unlimited resources (functional units) and only single-cycle latencies. This limit is a mind-boggling 9 cycles! The achieved and theoretical execution times are summarised in Table 6. The almost threefold speedup from 95 to 36 cycles made a signiﬁcant difference for the performance of driver benchmarks within our component system. It would not have been possible without the powerful per- formance monitoring support on the Itanium processor, particularly the ability to break down stall events. The PMU allowed us to discover and explain all of the stalls involved. This experience also helped us to appreciate the chal- lenges facing compiler writers on Itanium. Without in- formation such as that of Tables 4 and 5 it is impossi- ble to generate truly efﬁcient code. A compiler could use this information to drive its code optimisation, elim- inating the need for labour-intensive hand-scheduled as- sembler code. Present compilers seem to be far away from being able to achieve this. While we have not anal- ysed system-call code from other operating systems to the same degree, we would expect them to suffer from the same problems, and beneﬁt from the same solutions. However, system-call performance is particularly criti- cal in a microkernel, owing to the high frequency of ker- nel invocations. 6 Conclusion As has been shown, the Itanium is a very interesting plat- form for systems programming. It presents a number of unusual features, such as its approach to address trans- lation and memory protection, which are creating a new design space for systems builders. The architecture provides plenty of challenges too, in- cluding managing its large register set efﬁciently, and overcoming hurdles to virtualisation. However, the most signiﬁcant challenge of the architecture to systems im- plementors is the more mundane one of optimising the code. The EPIC approach has proven a formidable chal- lenge to compiler writers, and almost ﬁve years after the architecture was ﬁrst introduced, the quality of code produced by the available compilers is often very poor for systems code. Given this time scale, the situation is not likely to improve signiﬁcantly for quite a number of years. In the meantime, systems implementors who want to tap into the great performance potential of the architec- ture have to resort to hand-tuned assembler code, written with a thorough understanding of the architecture and its complex instruction scheduling rules. Performance improvements by factors of 2–3 are not unusual in this situation, and we have experienced cases where perfor- mance could be improved by an order of magnitude over GCC-generated code. Such manual micro-optimisation is made harder by the unavailability of sufﬁciently detailed documentation. 2005 USENIX Annual Technical Conference USENIX Association 277 This, at least, seems be something the manufacturer should be able to resolve quickly. Acknowledgements This work was supported by a Linkage Grant from the Australian Research Council (ARC) and a grant from HP Company via the Gelato.org project, as well as hard- ware grants from HP and Intel. National ICT Australia is funded by the Australia Government’s Department of Communications, Information Technology, and the Arts and the ARC throughBacking Australia’s Abilityand the ICT Research Centre of Excellence programs. We would also like to thank UNSW Gelato staff Ian Wienand and Darren Williams for their help with bench- marking. Notes 1. The results in [13] were obtained kernels that were not fully functional and are thus somewhat optimistic. Also the processors used had shorter pipelines than modern high-end CPUs and hence lower hardware-dictated con- text switching costs. The ﬁgure of 70–150 cycles reﬂects (yet) unpublished measurements performed in our lab on optimised kernels for ARM, MIPS, Alpha and Power 4. References [1] Aim benchmarks. http://sourceforge.net/ projects/aimbench. [2] Kavita Bala, M. Frans Kaashoek, and William E. Weihl. Software prefetching and caching for translation looka- side buffers. In Proc. 1st OSDI , pages 243–253, Mon- terey, CA, USA, 1994. USENIX/ACM/IEEE. [3] Matthew Chapman and Gernot Heiser. Implementing transparent shared memory on clusters using virtual ma- chines. In Proc. 2005 USENIX Techn. Conf., Anaheim, CA, USA, Apr 2005. [4] Matthew Chapman, Ian Wienand, and Gernot Heiser. Ita- nium page tables and TLB. Technical Report UNSW- CSE-TR-0307, School Comp. Sci. & Engin., University NSW, Sydney 2052, Australia, May 2003. [5] Douglas W. Clark and Joel S. Emer. 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[23] SPARC International Inc., Menlo Park, CA, USA.The SPARC Architecture Manual, Version 8, 1991. http: //www.sparc.org/standards.html. [24] John Wilkes and Bart Sears. A comparison of protection lookaside buffers and the PA-RISC protection architec- ture. Technical Report HPL-92-55, HP Labs, Palo Alto, CA, USA, Mar 1992. 2005 USENIX Annual Technical Conference USENIX Association278
lec05	Lecture 5: Introduction to (Robertson/Sp¨arck Jones) Probabilistic Retrieval Scribes: Ellis Weng, Andrew Owens February 11, 2010 1 Introduction In this lecture, we will introduce our second paradigm for document retrieval: probabilistic retrieval. We will focus on Roberston and Sp¨arck Jones’ 1976 version, presented in the paperRelevance Weighting of Search Terms1. This was an inﬂuential paper that was published when the Vector Space Model was ﬁrst being developed — it is important to keep in mind the differences and similarities between these two models and the motivations for each. Recall that the Vector Space Model was originally a representation model. The retrieval scheme of the Vector Space Model was empirically-driven and chosen in a fairly atheoretical manner. In contrast, proba- bilistic retrieval is more principled and theoretically-driven. On the other hand, many statistical estimations and empirical substitutions will drive the derivation of this paradigm. 2 Notation 2.1 Problem First, let us assume a binary label set, L, where the documents are either relevant, r, or not relevant, r. (The binary distinction is not so important at this time but simpliﬁes presentation.) L = {r,r}(relevant/irrelevant) (1) The main goal of probabilistic retrieval is to rank documents by the probability that they are relevant. Intuitively, this can be represented in the following way: Pr(r\|d, q), (2) where d is the document and q is the query. It is important to note that the score of a document ought to be a probability between 0 and 1 (inclusive), not a simple binary relevance score of 0 or 1, for this to be a meaningful scoring function. 2.2 Uncertainty Question: Why is this not 0 or 1? Shouldn’t a document just be either relevant or irrelevant? Answer: There is uncertainty associated with probabilistic retrieval. Uncertainty can arise from any of the following: 1. Uncertainty with respect to a particular user. A user’s judgment for a document relevancy might change from time to time depending on context. The resulting sample space for this uncertainty is 1 Q ×D ×L ×F, (3) where Q is the set of all possible queries (or information need), D is the set of all possible documents, L is the label set, and F is the set of all other factors that might change your threshold. 2. Uncertainty with respect to different users. For a single document-query pair, there is variation in determining if the document is relevant or not among different users. Thus, we have to take different users into account. The underlying sample space for this uncertainty is Q ×D ×L ×U, (4) where U is the set of all possible users. 3. Uncertainty with respect to “lossy” input representation. Depending on the representation, a document can either be relevant or not. It is impossible to take into account all the features of a document. Thus, several document-query pairs yield the same rep- resentation; however, these different documents do not necessarily have to be both relevant or not relevant. The resulting sample space for this uncertainty is still Q ×D ×L, (5) but the “observables” induced from this are different. 4. Uncertainty with respect to to system uncertainty. The retrieval system itself may be a source of uncertainty. For example, the system might trade off accuracy for speed by using an approximation algorithm or run on a network that is prone to errors. Question: Why is the type of uncertainty important? Answer: The type of uncertainty presumably affects the estima- tion and derivation of the probabilistic model. The ultimate goal of document retrieval is to ﬁnd documents that are relevant for a different users. One can imagine document retrieval systems that were tailored to a speciﬁc user’s needs and moods (thus eliminating type-a uncertainty) or document retrieval systems that captured all the important features of the documents and queries (thus eliminating type-c uncertainty); however, document retrieval systems are typically used by more than one user. Solving the problem with respect to b-type uncertainty thus seems to be the goal. Unfortunately, this problem has the worst sample space and seemingly requires extensive user annotation by many users. Because of these problems, our derivation will be based on representational uncertainty (type-c uncertainty). (Note: Throughout the derivation, it is worthwhile to ask how this model can apply to these other types of uncertainty.) 2.3 Representation Assume we have a feature-vector representation (which is not implied by any principles we have men- tioned so far). A document can be represented as a set of values of m feature functions. A feature function fj : D →R describes the important features of the document. These functions are based on attribute fre- quency, so 0 must indicate the absence of a particular feature. These feature functions are analogous to the term counts in the VSM, but unlike term counts, these feature functions can, in principle, take more into account than simply the number of times words appear. For example: f14(d) =the number of times “juice” appears in the document (6) 2 f16(d) = { 1 if d contains “Cornell” and “best” 0 otherwise (7) ⃗fd =   f1(d) ... f2(d) ... fm(d)   (8) Similarly, query can be represented as a document: ⃗ q=   f1(q) ... f2(q) ... fm(q)   (9) Note that we will sometimes use the notation fd[j] for fj(d) for readability. Let ⃗F = (F[1], F[2], ..., F[m])T be a feature vector. We can write ⃗F = ⃗fd to mean that a feature vector matches the description for document d. Under this notation, we can rewrite our probabilistic scoring function for a given query and a given document d as Pr(r\|⃗F = ⃗fd, ⃗ q) (10) 2.4 Comparison of Scoring Functions Probabilistic Retrieval Vector Space Model Pr(r\|⃗F = ⃗fd, ⃗ q) ⃗d ·⃗ q Goal-driven. The goal of a probabilistic retrieval model is clearly to retrieve the documents with the highest probability of relevance to the given query. More Arbitrary. The inner product makes intu- itive sense, but we have no further justiﬁcation for this besides empirical evidence. Lack of Data. There seemingly needs to be data on what documents are relevant (to which queries) in order to compute these probabilities. There can potentially be no data in regards to rel- evance. Computable. No additional information is needed in order to compute these values, assum- ing a typical representation scheme. 3 Computation Assume the query ⃗ qis ﬁxed. We want to compute Pr(r\|⃗F = ⃗fd, ⃗ q) for each document. There are many challenges to this computation. One problem is that relevance is unlabeled in our dataset, so we do not know which documents are relevant for a given query. Another problem is the sparse data for ⃗F = ⃗fd. There may be very few documents d and d′ such that ⃗fd = ⃗fd′ . To deal with the ﬁrst problem, we will use the “strategy of wishful thinking,” a surprisingly powerful principle. We will condition on r, as though we had the full labeled dataset! First, we present some probability background material. If we have random variables a and b, we can apply the Bayes Flip 3 Pr(a\|b) =Pr(a, b) Pr(b) = Pr(b\|a) Pr(a) Pr(b) (11) Terminology: Pr(a, b) is the joint probability of a and b, and Pr(b) is the marginal probabilityof b. Applying the Bayes ﬂip, we get Pr( ⃗F = ⃗fd\|r, ⃗ q) Pr(r\|⃗ q) Pr( ⃗Fd = ⃗fd\|⃗ q) (12) Thankfully, we can simplify this equation.Pr(r\|⃗ q) is the probability that a randomly-chosen document is relevant to the user, given his or her query. Since this value is independent of ⃗fd, the equation is equivalent under ranking to Pr( ⃗F = ⃗fd\|r, ⃗ q) Pr( ⃗F = ⃗fd\|⃗ q) (13) Note that ⃗F = ⃗fd is independent of the the query for systems where the document’s representation is not affected by the user’s query. So, we can rewrite the above as Pr( ⃗F = ⃗fd\|r, ⃗ q) Pr( ⃗F = ⃗fd) (14) Now consider the denominator Pr( ⃗F = ⃗fd). This is the probability that a randomly-chosen document is represented as fd in our retrieval system, given the query. We could estimate this probability by counting the fraction of our documents in our database that are represented as fd. Unfortunately, our data is sparse, so almost always Pr( ⃗F = ⃗fd) = 1/n (where n is the number of documents) because it is unlikely two documents have the same representation. To deal with this issue, we assume (as in Cooper 2) that there exists a constant k >0 such that Pr( ⃗F = ⃗f) = k ∏ j Pr( ⃗F[j] = ⃗fd[j]). If k = 1, then we are assuming that each Pr( ⃗F[j] = ⃗fd[j]) is an independent event. We run into data sparsity issues in the numerator as well, so we make a similar assumption. We assume there is a k′ > 0 such that Pr( ⃗F = ⃗fd\|r, ⃗ q) =k′ ∏ j Pr( ⃗F[j] = ⃗fd[j]\|r, ⃗ q). If k′ = 1then this is a Naive Bayes assumption. (Aside: we can avoid making one of the assumptions if we start with a slightly more complex scoring function.) Under these assumptions, we can rewrite the above as ∏ j Pr( ⃗F[j] = ⃗fd[j]\|r, ⃗ q) Pr( ⃗F[j] = ⃗fd[j]) k′ k (15) Note that we can remove the constant k′ k and get an equation that is ranking-equivalent to this one. At this point, we still have no information on the relevance of the documents. Now we try to simplify the equation further by looking at the query, since the terms in the query are the only information we have regarding relevance. We distinguish between terms that appear in the query and those that don’t. ∏ j:q[j]̸=0 Pr( ⃗F[j] = ⃗fd[j]\|r, ⃗ q) Pr( ⃗F[j] = ⃗fd[j]) ∏ j:q[j]=0 Pr( ⃗F[j] = ⃗fd[j]\|r, ⃗ q) Pr( ⃗F[j] = ⃗fd[j]) (16) Let us compare the numerator and denominator of the second product. The numerator represents the probability that a relevant document has a certain feature value for a feature that is not in the query, while the denominator represents the probability thatany document has a certain value for a feature that is not in the query. One can argue that these two are similar because words that are left out of a query are usually not 4 intentionally left out by a user. For example, when a user searches for the word ”computer”, the user is not intentionally leaving out the words “PC”, “Mac”, “Desktop”, etc. It is impossible to list all the terms that are of interest to a user, so the words that are left out do not necessarily indicate that the user thinks they are irrelevant. So, we don’t know whether these terms occur more in relevant or in general documents. Also in large corpora, factoring in the features that are not in a query is not practical, because the vocabulary or set of features of all the documents is massive. Thus, one can make an assumption that the words that are not present in the query are distributed across relevant documents the same as they are distributed across the whole corpus, since we lack any other information to the contrary. This yields: ∏ j:q[j]̸=0 Pr( ⃗F[j] = ⃗fd[j]\|r, ⃗ q) Pr( ⃗F[j] = ⃗fd[j]) (17) Although this equation does not seem better than the original probablistic model, there are many terms that were eliminated using assumptions. In the following lectures, we will see the beneﬁts of having the equation in this form. 4 Questions Recall that the derivation of the probabilistic model required a number of assumptions and independence arguments. This exercise will help you understand the meaning of the assumptions and how each of these assumptions affects the relevance scores of the documents. Suppose we have a set of 10 documents and 5 queries. Each document can have multiple representations as a vector of feature functions. Assume that the feature functions are binary word counts (the ﬁrst feature function is whether the word ”car” is in the document). Therefore, each document can be represented as a vector of 30 feature functions (there are 30 words in this corpus, so there will be one feature function for each word). The queries can also be represented as documents (also a vector of 30 feature functions). The following table shows the documents (rows), queries (columns), and whether a document is relevant with respect to the query (1 or 0). In practice, we would not have these labels, so it would not be possible to compute these probabilities. In future lectures we will discuss ways to deal with this problem. car toyota park green car low milage toyota brand car this red car is fast and new toyota is a good brand of car 1 1 0 0 1 this automobile is red and the brand is toyota it is not very fast but it is cheap 1 1 0 0 1 the green car is fast and the mileage is low 1 0 0 1 0 the car with low mileage is under a tree 1 0 0 0 0 the green tree is in ithe park 0 0 1 0 0 the automobile is green and this car is cheap 1 0 0 0 0 park the car under the tree 1 0 1 0 0 the toyota has low mileage and is very cheap 1 1 0 0 0 the car is not a good brand 1 0 0 0 0 there is a automobile that is red and in the park 1 0 1 0 0 1. Using the provided table, estimate the relevance of the ﬁrst document in the ﬁrst row with the query “toyota brand car”, using the ﬁnal equation presented in this lecture: ∏ j:q[j]̸=0 Pr( ⃗F[j] = ⃗fd[j]\|r, ⃗ q) Pr( ⃗F[j] = ⃗fd[j]) 5 2. Estimate the relevance of the ﬁrst document using the equation without the assumption that the only signiﬁcant features are those in the query (i.e. use equation (16) instead of equation (17)). ∏ j:q[j]̸=0 Pr( ⃗F[j] = ⃗fd[j]\|r, ⃗ q) Pr( ⃗F[j] = ⃗fd[j]) ∏ j:q[j]=0 Pr( ⃗F[j] = ⃗fd[j]\|r, ⃗ q) Pr( ⃗F[j] = ⃗fd[j]) Feel free to use a computer program. 3. Compare and contrast the following two tables. An entry is a relevance score for a given (document, query) pair. In the ﬁrst table, the only features considered were those in the query (as in the ﬁrst exercise), while the second table uses all of the features (as in the second exercise). car toyota park green car low milage toyota brand car this red car is fast and new toyota is a good brand of car 1.11 3.33 0 0 Answer 1 this automobile is red and the brand is toyota it is not very fast but it is cheap 0.83 3.33 0 0 13.89 the green car is fast and the mileage is low 1.11 0 0 61.73 0 the car with low mileage is under a tree 1.11 0 0 0 0 the green tree is in ithe park 0.83 0 3.33 0 0 the automobile is green and this car is cheap 1.11 0 0 0 0 park the car under the tree 1.11 0 3.33 0 0 the toyota has low mileage and is very cheap 0.83 3.33 0 0 0 the car is not a good brand 1.11 0 0 0 0 there is a automobile that is red and in the park 0.83 0 3.33 0 0 car toyota park green car low milage toyota brand car this red car is fast and new toyota is a good brand of car 3.74 468.17 0 0 Answer 2 this automobile is red and the brand is toyota it is not very fast but it is cheap 3.18 40782.92 0 0 353160.66 the green car is fast and the mileage is low 1.08 0 0 136672.91 0 the car with low mileage is under a tree 0.85 0 0 0 0 the green tree is in ithe park 0.07 0 2031.53 0 0 the automobile is green and this car is cheap 0.92 0 0 0 0 park the car under the tree 0.35 0 652.99 0 0 the toyota has low mileage and is very cheap 1.45 26.01 0 0 0 the car is not a good brand 1.60 0 0 0 0 there is a automobile that is red and in the park 0.42 0 2571.15 0 0 4. Why do the tables contain values greater than 1? 5. One problem with the approach we have described is that every time a user performs a query, we must check whether each document is relevant. This approach does not scale well. It would be nice if we could process a small subset of the documents instead. We assumed that the features in the query are the only features that inﬂuence the probability calculation. Let’s make an additional assumption in this same spirit: for any relevant document d, there is a feature j such that fj(d) ̸= 0and fj(q) ̸= 0. Is this assumption reasonable? Explain how you could build a more efﬁcient information retrieval system that only looked at a subset of the documents for a given query. 6 5 Answers 1. This equation is only concerned with words that are in the query, so we only have to take the product of the probabilities of three words. For the word “toyota”, the probability of seeing the word “toyota” in the relevant documents is 1. The probability of seeing the word “toyota” in this corpus is 3/10. The word “toyota” contributes 10/3 to the product (note that this is greater than 1). For the word “brand”, the probability of seeing the word “brand” in the relevant documents is 1. The probability of seeing the word “brand” in this corpus is 3/10. The word “brand” contributes 10/3 to the product. For the word “car”, the probability of seeing the word “car” in the relevant documents is 1/2. The probability of seeing the word “car” in this corpus is 6/10. The word “car” contributes 10/12 to the product. Multiplying these terms yields 9.26. 2. We used a computer program to compute the result: 64866.24. The source code for this program is included at the end of this document. 3. The scores of the two tables differ greatly, but for the most part, the document ranking has not changed much for the two tables. The document ranking changed the most with the query “car”. For example, in the ﬁrst table, the second document is tied with being the least relevant document; however, in the second table, it is the second most relevant document. This particular example shows that terms not in the query can have different distribution in relevant documents than in documents overall. Also, notice that the word “car” occurs in most of the queries. Because the query “car” is so generic, almost all the documents were relevant to some extent. In the ﬁrst table, it is impossible to distinguish between documents with just one common word. In the second table, there is more variation among the documents’ relevance scores for the “car” column. This variation is due to the fact that we are using features that were not present in the query. This implies that sometimes the words in the query itself are not enough information to decide if a document is relevant or not. 4. We are no longer computing the probability that the document is relevant. Instead, we are computing a quantity that is equivalent to this probability under rank. As a result of some of the simpliﬁcations that we made (e.g. dropping the k′/k term) it is possible to get a relevance score that is greater than 1. 5. This assumption seems reasonable for many real world situations (in fact, it is built into the VSM). When users search for documents, they usually want documents that contain one of the terms they searched for, for example. You could implement a more efﬁcient retrieval system by precomputing an index that tells you which documents have each feature. Then you could use this index to obtain all of the documents that have at least one feature in common with the query and estimate the relevance of the documents in this subset. 6 References 1. Stephen Robertson and Karen Sp ¨arck Jones. Relevance weighting of search terms. Journal of the American Society for Information Science 27(3): 129-46 (1976). The probabilistic argument is pre- sented in the appendix. 2. William S. Cooper. Some inconsistencies and misidentiﬁed modeling assumptions in probabilistic information retrieval. ACM Transactions on Information Systems (TOIS), pp. 100-111, 1995. 7 Code Listing 1: Python code for problem 1b from collections import defaultdict docs = ["this red car is fast and new toyota is a good brand of car", "this automobile is red and the brand is toyota it is not very fast but it is cheap", "this green car is fast and the mileage is low", "the car with low mileage is under a tree", "the green tree is in the park", "the automobile is green and this car is cheap", "park the car under the tree", "the toyota has low mileage and is very cheap", "this car is not a good brand", "there is a automobile that is red and in the park"] # split each doc into words docs = [d.split() for d in docs] queries = ["car", "toyota", "park", "green car low mileage", "toyota brand car"] queries = [q.split() for q in queries] # (i, j) is in the list if query qi is relevant to doc j query_doc_relevant = [(1,1), (2,1), (5,1), (1,2), (2,2), (5,2), (1,3), (4,3), (1,4), (3,5), (1,6), (1,7), (3,7), (1,8), (2,8), (1,9), (1,10), (3,10)] query_doc_relevant = [(x-1, y-1) for x, y in query_doc_relevant] def word_probs(docs): """ word_probs(docs)[w] is the fraction of docs that contain w """ probs = defaultdict(float) for d in docs: for w in set(d): probs[w] += 1.0 / len(docs) return probs def make_table(only_terms_in_query): """ Returns d such that d[i][j] is the probability document i is relevant to query j using the RSJ model. If only_terms_in_query is True, then equation 16 is used, otherwise equation 17 is used.""" # estimate Pr(F[j]) word_priors = word_probs(docs) doc_query_prob = defaultdict(lambda: defaultdict(float)) for qi, query in enumerate(queries): rel_docs = [docs[dj] for qj, dj in query_doc_relevant if qj == qi] word_given_rel = word_probs(rel_docs) # estimate probability doc di is relevant to query qi # this is the product of for di, doc in enumerate(docs): doc_query_prob[qi][di] = 1.0 # only use the words in the query if only_terms_in_query is True for w in (query if only_terms_in_query else word_priors.keys()): if w in set(doc): # Pr(F[j] = 0 \| r, q)/Pr(F[j] = 0) doc_query_prob[qi][di] = word_given_rel[w] / word_priors[w] else: # Pr(F[j] = 1 \| r, q)/Pr(F[j] = 1) doc_query_prob[qi][di] = \ (1.0 - word_given_rel[w])/(1.0 - word_priors[w]) 8 return doc_query_prob def print_table(doc_query_probs): for di, doc in enumerate(docs): row_probs = [doc_query_probs[qi][di] for qi in xrange(len(queries))] print ' '.join(("%.2f" % p).ljust(9) for p in row_probs) print 'the entry at row i and col j is the probability doc i is relevant to query j' print 'just terms in query' print_table(make_table(True)) print 'all terms' print_table(make_table(False)) 9
main_notes	CS229 Lecture Notes Andrew Ng and Tengyu Ma June 11, 2023 Contents I Supervised learning 5 1 Linear regression 8 1.1 LMS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 The normal equations . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.1 Matrix derivatives . . . . . . . . . . . . . . . . . . . . . 13 1.2.2 Least squares revisited . . . . . . . . . . . . . . . . . . 14 1.3 Probabilistic interpretation . . . . . . . . . . . . . . . . . . . . 15 1.4 Locally weighted linear regression (optional reading) . . . . . . 17 2 Classiﬁcation and logistic regression 20 2.1 Logistic regression . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Digression: the perceptron learning algorithm . . . . . . . . . 23 2.3 Multi-class classiﬁcation . . . . . . . . . . . . . . . . . . . . . 24 2.4 Another algorithm for maximizing ℓ(θ) . . . . . . . . . . . . . 27 3 Generalized linear models 29 3.1 The exponential family . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Constructing GLMs . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.1 Ordinary least squares . . . . . . . . . . . . . . . . . . 32 3.2.2 Logistic regression . . . . . . . . . . . . . . . . . . . . 33 4 Generative learning algorithms 34 4.1 Gaussian discriminant analysis . . . . . . . . . . . . . . . . . . 35 4.1.1 The multivariate normal distribution . . . . . . . . . . 35 4.1.2 The Gaussian discriminant analysis model . . . . . . . 38 4.1.3 Discussion: GDA and logistic regression . . . . . . . . 40 4.2 Naive bayes (Option Reading) . . . . . . . . . . . . . . . . . . 41 4.2.1 Laplace smoothing . . . . . . . . . . . . . . . . . . . . 44 4.2.2 Event models for text classiﬁcation . . . . . . . . . . . 46 1 CS229 Spring 20223 2 5 Kernel methods 48 5.1 Feature maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2 LMS (least mean squares) with features . . . . . . . . . . . . . 49 5.3 LMS with the kernel trick . . . . . . . . . . . . . . . . . . . . 49 5.4 Properties of kernels . . . . . . . . . . . . . . . . . . . . . . . 53 6 Support vector machines 59 6.1 Margins: intuition . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.2 Notation (option reading) . . . . . . . . . . . . . . . . . . . . 61 6.3 Functional and geometric margins (option reading) . . . . . . 61 6.4 The optimal margin classiﬁer (option reading) . . . . . . . . . 63 6.5 Lagrange duality (optional reading) . . . . . . . . . . . . . . . 65 6.6 Optimal margin classiﬁers: the dual form (option reading) . . 68 6.7 Regularization and the non-separable case (optional reading) . 72 6.8 The SMO algorithm (optional reading) . . . . . . . . . . . . . 73 6.8.1 Coordinate ascent . . . . . . . . . . . . . . . . . . . . . 74 6.8.2 SMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 II Deep learning 79 7 Deep learning 80 7.1 Supervised learning with non-linear models . . . . . . . . . . . 80 7.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.3 Modules in Modern Neural Networks . . . . . . . . . . . . . . 92 7.4 Backpropagation . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.4.1 Preliminaries on partial derivatives . . . . . . . . . . . 99 7.4.2 General strategy of backpropagation . . . . . . . . . . 102 7.4.3 Backward functions for basic modules . . . . . . . . . . 105 7.4.4 Back-propagation for MLPs . . . . . . . . . . . . . . . 107 7.5 Vectorization over training examples . . . . . . . . . . . . . . 109 III Generalization and regularization 112 8 Generalization 113 8.1 Bias-variance tradeoﬀ . . . . . . . . . . . . . . . . . . . . . . . 115 8.1.1 A mathematical decomposition (for regression) . . . . . 120 8.2 The double descent phenomenon . . . . . . . . . . . . . . . . . 121 8.3 Sample complexity bounds (optional readings) . . . . . . . . . 126 CS229 Spring 20223 3 8.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 126 8.3.2 The case of ﬁnite H. . . . . . . . . . . . . . . . . . . . 128 8.3.3 The case of inﬁnite H . . . . . . . . . . . . . . . . . . 131 9 Regularization and model selection 135 9.1 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.2 Implicit regularization eﬀect (optional reading) . . . . . . . . . 137 9.3 Model selection via cross validation . . . . . . . . . . . . . . . 139 9.4 Bayesian statistics and regularization . . . . . . . . . . . . . . 142 IV Unsupervised learning 144 10 Clustering and the k-means algorithm 145 11 EM algorithms 148 11.1 EM for mixture of Gaussians . . . . . . . . . . . . . . . . . . . 148 11.2 Jensen’s inequality . . . . . . . . . . . . . . . . . . . . . . . . 151 11.3 General EM algorithms . . . . . . . . . . . . . . . . . . . . . . 152 11.3.1 Other interpretation of ELBO . . . . . . . . . . . . . . 158 11.4 Mixture of Gaussians revisited . . . . . . . . . . . . . . . . . . 158 11.5 Variational inference and variational auto-encoder (optional reading) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 12 Principal components analysis 165 13 Independent components analysis 171 13.1 ICA ambiguities . . . . . . . . . . . . . . . . . . . . . . . . . . 172 13.2 Densities and linear transformations . . . . . . . . . . . . . . . 173 13.3 ICA algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 14 Self-supervised learning and foundation models 177 14.1 Pretraining and adaptation . . . . . . . . . . . . . . . . . . . . 177 14.2 Pretraining methods in computer vision . . . . . . . . . . . . . 179 14.3 Pretrained large language models . . . . . . . . . . . . . . . . 181 14.3.1 Open up the blackbox of Transformers . . . . . . . . . 183 14.3.2 Zero-shot learning and in-context learning . . . . . . . 186 CS229 Spring 20223 4 V Reinforcement Learning and Control 188 15 Reinforcement learning 189 15.1 Markov decision processes . . . . . . . . . . . . . . . . . . . . 190 15.2 Value iteration and policy iteration . . . . . . . . . . . . . . . 192 15.3 Learning a model for an MDP . . . . . . . . . . . . . . . . . . 194 15.4 Continuous state MDPs . . . . . . . . . . . . . . . . . . . . . 196 15.4.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . 196 15.4.2 Value function approximation . . . . . . . . . . . . . . 199 15.5 Connections between Policy and Value Iteration (Optional) . . 203 16 LQR, DDP and LQG 206 16.1 Finite-horizon MDPs . . . . . . . . . . . . . . . . . . . . . . . 206 16.2 Linear Quadratic Regulation (LQR) . . . . . . . . . . . . . . . 210 16.3 From non-linear dynamics to LQR . . . . . . . . . . . . . . . 213 16.3.1 Linearization of dynamics . . . . . . . . . . . . . . . . 214 16.3.2 Diﬀerential Dynamic Programming (DDP) . . . . . . . 214 16.4 Linear Quadratic Gaussian (LQG) . . . . . . . . . . . . . . . . 216 17 Policy Gradient (REINFORCE) 220 Part I Supervised learning 5 6 Let’s start by talking about a few examples of supervised learning prob- lems. Suppose we have a dataset giving the living areas and prices of 47 houses from Portland, Oregon: Living area (feet2) Price (1000$s) 2104 400 1600 330 2400 369 1416 232 3000 540 ... ... We can plot this data: 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 100 200 300 400 500 600 700 800 900 1000 housing prices square feet price (in $1000) Given data like this, how can we learn to predict the prices of other houses in Portland, as a function of the size of their living areas? To establish notation for future use, we’ll use x(i) to denote the “input” variables (living area in this example), also called input features, and y(i) to denote the “output” or target variable that we are trying to predict (price). A pair ( x(i),y(i)) is called a training example , and the dataset that we’ll be using to learn—a list of n training examples {(x(i),y(i)); i = 1,...,n }—is called a training set. Note that the superscript “( i)” in the notation is simply an index into the training set, and has nothing to do with exponentiation. We will also use Xdenote the space of input values, and Y the space of output values. In this example, X= Y= R. To describe the supervised learning problem slightly more formally, our goal is, given a training set, to learn a function h: X↦→Y so that h(x) is a “good” predictor for the corresponding value of y. For historical reasons, this 7 function h is called a hypothesis. Seen pictorially, the process is therefore like this: Training set house.) (living area of Learning algorithm h predicted y x (predicted price) of house) When the target variable that we’re trying to predict is continuous, such as in our housing example, we call the learning problem a regression prob- lem. When y can take on only a small number of discrete values (such as if, given the living area, we wanted to predict if a dwelling is a house or an apartment, say), we call it a classiﬁcation problem. Chapter 1 Linear regression To make our housing example more interesting, let’s consider a slightly richer dataset in which we also know the number of bedrooms in each house: Living area (feet2) #bedrooms Price (1000$s) 2104 3 400 1600 3 330 2400 3 369 1416 2 232 3000 4 540 ... ... ... Here, the x’s are two-dimensional vectors in R2. For instance, x(i) 1 is the living area of the i-th house in the training set, and x(i) 2 is its number of bedrooms. (In general, when designing a learning problem, it will be up to you to decide what features to choose, so if you are out in Portland gathering housing data, you might also decide to include other features such as whether each house has a ﬁreplace, the number of bathrooms, and so on. We’ll say more about feature selection later, but for now let’s take the features as given.) To perform supervised learning, we must decide how we’re going to rep- resent functions/hypotheses h in a computer. As an initial choice, let’s say we decide to approximate y as a linear function of x: hθ(x) = θ0 + θ1x1 + θ2x2 Here, the θi’s are the parameters (also called weights) parameterizing the space of linear functions mapping from X to Y. When there is no risk of 8 9 confusion, we will drop the θ subscript in hθ(x), and write it more simply as h(x). To simplify our notation, we also introduce the convention of letting x0 = 1 (this is the intercept term), so that h(x) = d∑ i=0 θixi = θTx, where on the right-hand side above we are viewing θ and x both as vectors, and here d is the number of input variables (not counting x0). Now, given a training set, how do we pick, or learn, the parameters θ? One reasonable method seems to be to make h(x) close to y, at least for the training examples we have. To formalize this, we will deﬁne a function that measures, for each value of the θ’s, how close the h(x(i))’s are to the corresponding y(i)’s. We deﬁne the cost function: J(θ) = 1 2 n∑ i=1 (hθ(x(i)) −y(i))2. If you’ve seen linear regression before, you may recognize this as the familiar least-squares cost function that gives rise to the ordinary least squares regression model. Whether or not you have seen it previously, let’s keep going, and we’ll eventually show this to be a special case of a much broader family of algorithms. 1.1 LMS algorithm We want to choose θ so as to minimize J(θ). To do so, let’s use a search algorithm that starts with some “initial guess” for θ, and that repeatedly changes θ to make J(θ) smaller, until hopefully we converge to a value of θ that minimizes J(θ). Speciﬁcally, let’s consider the gradient descent algorithm, which starts with some initial θ, and repeatedly performs the update: θj := θj −α ∂ ∂θj J(θ). (This update is simultaneously performed for all values of j = 0 ,...,d .) Here, α is called the learning rate. This is a very natural algorithm that repeatedly takes a step in the direction of steepest decrease of J. In order to implement this algorithm, we have to work out what is the partial derivative term on the right hand side. Let’s ﬁrst work it out for the 10 case of if we have only one training example ( x,y), so that we can neglect the sum in the deﬁnition of J. We have: ∂ ∂θj J(θ) = ∂ ∂θj 1 2 (hθ(x) −y)2 = 2 ·1 2 (hθ(x) −y) · ∂ ∂θj (hθ(x) −y) = ( hθ(x) −y) · ∂ ∂θj ( d∑ i=0 θixi −y ) = ( hθ(x) −y) xj For a single training example, this gives the update rule: 1 θj := θj + α ( y(i) −hθ(x(i)) ) x(i) j . The rule is called theLMS update rule (LMS stands for “least mean squares”), and is also known as the Widrow-Hoﬀ learning rule. This rule has several properties that seem natural and intuitive. For instance, the magnitude of the update is proportional to the error term (y(i) −hθ(x(i))); thus, for in- stance, if we are encountering a training example on which our prediction nearly matches the actual value of y(i), then we ﬁnd that there is little need to change the parameters; in contrast, a larger change to the parameters will be made if our prediction hθ(x(i)) has a large error (i.e., if it is very far from y(i)). We’d derived the LMS rule for when there was only a single training example. There are two ways to modify this method for a training set of more than one example. The ﬁrst is replace it with the following algorithm: Repeat until convergence { θj := θj + α n∑ i=1 ( y(i) −hθ(x(i)) ) x(i) j ,(for every j) (1.1) } 1We use the notation “ a := b” to denote an operation (in a computer program) in which we set the value of a variable a to be equal to the value of b. In other words, this operation overwrites awith the value of b. In contrast, we will write “ a= b” when we are asserting a statement of fact, that the value of a is equal to the value of b. 11 By grouping the updates of the coordinates into an update of the vector θ, we can rewrite update (1.1) in a slightly more succinct way: θ:= θ+ α n∑ i=1 ( y(i) −hθ(x(i)) ) x(i) The reader can easily verify that the quantity in the summation in the update rule above is just ∂J(θ)/∂θj (for the original deﬁnition of J). So, this is simply gradient descent on the original cost function J. This method looks at every example in the entire training set on every step, and is called batch gradient descent. Note that, while gradient descent can be susceptible to local minima in general, the optimization problem we have posed here for linear regression has only one global, and no other local, optima; thus gradient descent always converges (assuming the learning rate α is not too large) to the global minimum. Indeed, J is a convex quadratic function. Here is an example of gradient descent as it is run to minimize a quadratic function. 5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50 The ellipses shown above are the contours of a quadratic function. Also shown is the trajectory taken by gradient descent, which was initialized at (48,30). The x’s in the ﬁgure (joined by straight lines) mark the successive values of θ that gradient descent went through. When we run batch gradient descent to ﬁt θ on our previous dataset, to learn to predict housing price as a function of living area, we obtain θ0 = 71.27, θ1 = 0.1345. If we plot hθ(x) as a function of x (area), along with the training data, we obtain the following ﬁgure: 12 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 100 200 300 400 500 600 700 800 900 1000 housing prices square feet price (in $1000) If the number of bedrooms were included as one of the input features as well, we get θ0 = 89.60,θ1 = 0.1392, θ2 = −8.738. The above results were obtained with batch gradient descent. There is an alternative to batch gradient descent that also works very well. Consider the following algorithm: Loop { for i= 1 to n, { θj := θj + α ( y(i) −hθ(x(i)) ) x(i) j , (for every j) (1.2) } } By grouping the updates of the coordinates into an update of the vector θ, we can rewrite update (1.2) in a slightly more succinct way: θ:= θ+ α ( y(i) −hθ(x(i)) ) x(i) In this algorithm, we repeatedly run through the training set, and each time we encounter a training example, we update the parameters according to the gradient of the error with respect to that single training example only. This algorithm is called stochastic gradient descent (also incremental gradient descent). Whereas batch gradient descent has to scan through the entire training set before taking a single step—a costly operation if n is large—stochastic gradient descent can start making progress right away, and 13 continues to make progress with each example it looks at. Often, stochastic gradient descent gets θ “close” to the minimum much faster than batch gra- dient descent. (Note however that it may never “converge” to the minimum, and the parameters θ will keep oscillating around the minimum of J(θ); but in practice most of the values near the minimum will be reasonably good approximations to the true minimum.2) For these reasons, particularly when the training set is large, stochastic gradient descent is often preferred over batch gradient descent. 1.2 The normal equations Gradient descent gives one way of minimizing J. Let’s discuss a second way of doing so, this time performing the minimization explicitly and without resorting to an iterative algorithm. In this method, we will minimize J by explicitly taking its derivatives with respect to the θj’s, and setting them to zero. To enable us to do this without having to write reams of algebra and pages full of matrices of derivatives, let’s introduce some notation for doing calculus with matrices. 1.2.1 Matrix derivatives For a function f : Rn×d ↦→R mapping from n-by-d matrices to the real numbers, we deﬁne the derivative of f with respect to A to be: ∇Af(A) =   ∂f ∂A11 ··· ∂f ∂A1d ... ... ... ∂f ∂An1 ··· ∂f ∂And   Thus, the gradient ∇Af(A) is itself an n-by-dmatrix, whose (i,j)-element is ∂f/∂Aij. For example, suppose A = [ A11 A12 A21 A22 ] is a 2-by-2 matrix, and the function f : R2×2 ↦→R is given by f(A) = 3 2A11 + 5A2 12 + A21A22. 2By slowly letting the learning rate α decrease to zero as the algorithm runs, it is also possible to ensure that the parameters will converge to the global minimum rather than merely oscillate around the minimum. 14 Here, Aij denotes the (i,j) entry of the matrix A. We then have ∇Af(A) = [ 3 2 10A12 A22 A21 ] . 1.2.2 Least squares revisited Armed with the tools of matrix derivatives, let us now proceed to ﬁnd in closed-form the value of θ that minimizes J(θ). We begin by re-writing J in matrix-vectorial notation. Given a training set, deﬁne thedesign matrix X to be the n-by-dmatrix (actually n-by-d + 1, if we include the intercept term) that contains the training examples’ input values in its rows: X =   — (x(1))T — — (x(2))T — ... — (x(n))T —  . Also, let ⃗ ybe the n-dimensional vector containing all the target values from the training set: ⃗ y=   y(1) y(2) ... y(n)  . Now, since hθ(x(i)) = (x(i))Tθ, we can easily verify that Xθ −⃗ y=   (x(1))Tθ ... (x(n))Tθ  −   y(1) ... y(n)   =   hθ(x(1)) −y(1) ... hθ(x(n)) −y(n)  . Thus, using the fact that for a vector z, we have that zTz = ∑ iz2 i: 1 2(Xθ −⃗ y)T(Xθ −⃗ y) = 1 2 n∑ i=1 (hθ(x(i)) −y(i))2 = J(θ) 15 Finally, to minimize J, let’s ﬁnd its derivatives with respect to θ. Hence, ∇θJ(θ) = ∇θ 1 2(Xθ −⃗ y)T(Xθ −⃗ y) = 1 2∇θ ( (Xθ)TXθ −(Xθ)T⃗ y−⃗ yT(Xθ) + ⃗ yT⃗ y ) = 1 2∇θ ( θT(XTX)θ−⃗ yT(Xθ) −⃗ yT(Xθ) ) = 1 2∇θ ( θT(XTX)θ−2(XT⃗ y)Tθ ) = 1 2 ( 2XTXθ −2XT⃗ y ) = XTXθ −XT⃗ y In the third step, we used the fact that aTb = bTa, and in the ﬁfth step used the facts ∇xbTx= b and ∇xxTAx= 2Ax for symmetric matrix A (for more details, see Section 4.3 of “Linear Algebra Review and Reference”). To minimize J, we set its derivatives to zero, and obtain thenormal equations: XTXθ = XT⃗ y Thus, the value of θ that minimizes J(θ) is given in closed form by the equation θ= (XTX)−1XT⃗ y.3 1.3 Probabilistic interpretation When faced with a regression problem, why might linear regression, and speciﬁcally why might the least-squares cost function J, be a reasonable choice? In this section, we will give a set of probabilistic assumptions, under which least-squares regression is derived as a very natural algorithm. Let us assume that the target variables and the inputs are related via the equation y(i) = θTx(i) + ϵ(i), 3Note that in the above step, we are implicitly assuming that XTX is an invertible matrix. This can be checked before calculating the inverse. If either the number of linearly independent examples is fewer than the number of features, or if the features are not linearly independent, then XTX will not be invertible. Even in such cases, it is possible to “ﬁx” the situation with additional techniques, which we skip here for the sake of simplicty. 16 where ϵ(i) is an error term that captures either unmodeled eﬀects (such as if there are some features very pertinent to predicting housing price, but that we’d left out of the regression), or random noise. Let us further assume that the ϵ(i) are distributed IID (independently and identically distributed) according to a Gaussian distribution (also called a Normal distribution) with mean zero and some variance σ2. We can write this assumption as “ ϵ(i) ∼ N(0,σ2).” I.e., the density of ϵ(i) is given by p(ϵ(i)) = 1√ 2πσ exp ( −(ϵ(i))2 2σ2 ) . This implies that p(y(i)\|x(i); θ) = 1√ 2πσ exp ( −(y(i) −θTx(i))2 2σ2 ) . The notation “ p(y(i)\|x(i); θ)” indicates that this is the distribution of y(i) given x(i) and parameterized by θ. Note that we should not condition on θ (“p(y(i)\|x(i),θ)”), since θ is not a random variable. We can also write the distribution of y(i) as y(i) \|x(i); θ∼N(θTx(i),σ2). Given X (the design matrix, which contains all the x(i)’s) and θ, what is the distribution of the y(i)’s? The probability of the data is given by p(⃗ y\|X; θ). This quantity is typically viewed a function of ⃗ y(and perhaps X), for a ﬁxed value of θ. When we wish to explicitly view this as a function of θ, we will instead call it the likelihood function: L(θ) = L(θ; X,⃗ y) = p(⃗ y\|X; θ). Note that by the independence assumption on the ϵ(i)’s (and hence also the y(i)’s given the x(i)’s), this can also be written L(θ) = n∏ i=1 p(y(i) \|x(i); θ) = n∏ i=1 1√ 2πσ exp ( −(y(i) −θTx(i))2 2σ2 ) . Now, given this probabilistic model relating the y(i)’s and the x(i)’s, what is a reasonable way of choosing our best guess of the parameters θ? The principal of maximum likelihood says that we should choose θ so as to make the data as high probability as possible. I.e., we should choose θ to maximize L(θ). 17 Instead of maximizing L(θ), we can also maximize any strictly increasing function of L(θ). In particular, the derivations will be a bit simpler if we instead maximize the log likelihood ℓ(θ): ℓ(θ) = log L(θ) = log n∏ i=1 1√ 2πσ exp ( −(y(i) −θTx(i))2 2σ2 ) = n∑ i=1 log 1√ 2πσ exp ( −(y(i) −θTx(i))2 2σ2 ) = nlog 1√ 2πσ − 1 σ2 ·1 2 n∑ i=1 (y(i) −θTx(i))2. Hence, maximizing ℓ(θ) gives the same answer as minimizing 1 2 n∑ i=1 (y(i) −θTx(i))2, which we recognize to be J(θ), our original least-squares cost function. To summarize: Under the previous probabilistic assumptions on the data, least-squares regression corresponds to ﬁnding the maximum likelihood esti- mate of θ. This is thus one set of assumptions under which least-squares re- gression can be justiﬁed as a very natural method that’s just doing maximum likelihood estimation. (Note however that the probabilistic assumptions are by no means necessary for least-squares to be a perfectly good and rational procedure, and there may—and indeed there are—other natural assumptions that can also be used to justify it.) Note also that, in our previous discussion, our ﬁnal choice of θ did not depend on what was σ2, and indeed we’d have arrived at the same result even if σ2 were unknown. We will use this fact again later, when we talk about the exponential family and generalized linear models. 1.4 Locally weighted linear regression (optional reading) Consider the problem of predicting y from x∈R. The leftmost ﬁgure below shows the result of ﬁtting a y= θ0 + θ1x to a dataset. We see that the data doesn’t really lie on straight line, and so the ﬁt is not very good. 18 0 1 2 3 4 5 6 7 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x y 0 1 2 3 4 5 6 7 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x y 0 1 2 3 4 5 6 7 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x y Instead, if we had added an extra feature x2, and ﬁt y= θ0 + θ1x+ θ2x2, then we obtain a slightly better ﬁt to the data. (See middle ﬁgure) Naively, it might seem that the more features we add, the better. However, there is also a danger in adding too many features: The rightmost ﬁgure is the result of ﬁtting a 5-th order polynomial y= ∑5 j=0 θjxj. We see that even though the ﬁtted curve passes through the data perfectly, we would not expect this to be a very good predictor of, say, housing prices ( y) for diﬀerent living areas (x). Without formally deﬁning what these terms mean, we’ll say the ﬁgure on the left shows an instance of underﬁtting—in which the data clearly shows structure not captured by the model—and the ﬁgure on the right is an example of overﬁtting. (Later in this class, when we talk about learning theory we’ll formalize some of these notions, and also deﬁne more carefully just what it means for a hypothesis to be good or bad.) As discussed previously, and as shown in the example above, the choice of features is important to ensuring good performance of a learning algorithm. (When we talk about model selection, we’ll also see algorithms for automat- ically choosing a good set of features.) In this section, let us brieﬂy talk about the locally weighted linear regression (LWR) algorithm which, assum- ing there is suﬃcient training data, makes the choice of features less critical. This treatment will be brief, since you’ll get a chance to explore some of the properties of the LWR algorithm yourself in the homework. In the original linear regression algorithm, to make a prediction at a query point x (i.e., to evaluate h(x)), we would: 1. Fit θ to minimize ∑ i(y(i) −θTx(i))2. 2. Output θTx. In contrast, the locally weighted linear regression algorithm does the fol- lowing: 1. Fit θ to minimize ∑ iw(i)(y(i) −θTx(i))2. 2. Output θTx. 19 Here, the w(i)’s are non-negative valued weights. Intuitively, if w(i) is large for a particular value of i, then in picking θ, we’ll try hard to make ( y(i) − θTx(i))2 small. If w(i) is small, then the ( y(i) −θTx(i))2 error term will be pretty much ignored in the ﬁt. A fairly standard choice for the weights is 4 w(i) = exp ( −(x(i) −x)2 2τ2 ) Note that the weights depend on the particular point xat which we’re trying to evaluate x. Moreover, if \|x(i) −x\|is small, then w(i) is close to 1; and if \|x(i) −x\|is large, then w(i) is small. Hence, θ is chosen giving a much higher “weight” to the (errors on) training examples close to the query point x. (Note also that while the formula for the weights takes a form that is cosmetically similar to the density of a Gaussian distribution, the w(i)’s do not directly have anything to do with Gaussians, and in particular the w(i) are not random variables, normally distributed or otherwise.) The parameter τ controls how quickly the weight of a training example falls oﬀ with distance of its x(i) from the query point x; τ is called the bandwidth parameter, and is also something that you’ll get to experiment with in your homework. Locally weighted linear regression is the ﬁrst example we’re seeing of a non-parametric algorithm. The (unweighted) linear regression algorithm that we saw earlier is known as a parametric learning algorithm, because it has a ﬁxed, ﬁnite number of parameters (the θi’s), which are ﬁt to the data. Once we’ve ﬁt the θi’s and stored them away, we no longer need to keep the training data around to make future predictions. In contrast, to make predictions using locally weighted linear regression, we need to keep the entire training set around. The term “non-parametric” (roughly) refers to the fact that the amount of stuﬀ we need to keep in order to represent the hypothesis h grows linearly with the size of the training set. 4If xis vector-valued, this is generalized to be w(i) = exp(−(x(i) −x)T(x(i) −x)/(2τ2)), or w(i) = exp(−(x(i) −x)TΣ−1(x(i) −x)/(2τ2)), for an appropriate choice of τ or Σ. Chapter 2 Classiﬁcation and logistic regression Let’s now talk about the classiﬁcation problem. This is just like the regression problem, except that the values y we now want to predict take on only a small number of discrete values. For now, we will focus on the binary classiﬁcation problem in which y can take on only two values, 0 and 1. (Most of what we say here will also generalize to the multiple-class case.) For instance, if we are trying to build a spam classiﬁer for email, then x(i) may be some features of a piece of email, and y may be 1 if it is a piece of spam mail, and 0 otherwise. 0 is also called the negative class, and 1 the positive class, and they are sometimes also denoted by the symbols “-” and “+.” Given x(i), the corresponding y(i) is also called the label for the training example. 2.1 Logistic regression We could approach the classiﬁcation problem ignoring the fact that y is discrete-valued, and use our old linear regression algorithm to try to predict y given x. However, it is easy to construct examples where this method performs very poorly. Intuitively, it also doesn’t make sense for hθ(x) to take values larger than 1 or smaller than 0 when we know that y∈{0,1}. To ﬁx this, let’s change the form for our hypotheseshθ(x). We will choose hθ(x) = g(θTx) = 1 1 + e−θTx, where g(z) = 1 1 + e−z 20 21 is called the logistic function or the sigmoid function . Here is a plot showing g(z): −5 −4 −3 −2 −1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z g(z) Notice that g(z) tends towards 1 as z →∞, and g(z) tends towards 0 as z →−∞. Moreover, g(z), and hence also h(x), is always bounded between 0 and 1. As before, we are keeping the convention of letting x0 = 1, so that θTx= θ0 + ∑d j=1 θjxj. For now, let’s take the choice ofgas given. Other functions that smoothly increase from 0 to 1 can also be used, but for a couple of reasons that we’ll see later (when we talk about GLMs, and when we talk about generative learning algorithms), the choice of the logistic function is a fairly natural one. Before moving on, here’s a useful property of the derivative of the sigmoid function, which we write as g′: g′(z) = d dz 1 1 + e−z = 1 (1 + e−z)2 ( e−z) = 1 (1 + e−z) · ( 1 − 1 (1 + e−z) ) = g(z)(1 −g(z)). So, given the logistic regression model, how do we ﬁt θ for it? Following how we saw least squares regression could be derived as the maximum like- lihood estimator under a set of assumptions, let’s endow our classiﬁcation model with a set of probabilistic assumptions, and then ﬁt the parameters via maximum likelihood. 22 Let us assume that P(y= 1 \|x; θ) = hθ(x) P(y= 0 \|x; θ) = 1 −hθ(x) Note that this can be written more compactly as p(y\|x; θ) = (hθ(x))y (1 −hθ(x))1−y Assuming that the n training examples were generated independently, we can then write down the likelihood of the parameters as L(θ) = p(⃗ y\|X; θ) = n∏ i=1 p(y(i) \|x(i); θ) = n∏ i=1 ( hθ(x(i)) )y(i) ( 1 −hθ(x(i)) )1−y(i) As before, it will be easier to maximize the log likelihood: ℓ(θ) = log L(θ) = n∑ i=1 y(i) log h(x(i)) + (1−y(i)) log(1−h(x(i))) (2.1) How do we maximize the likelihood? Similar to our derivation in the case of linear regression, we can use gradient ascent. Written in vectorial notation, our updates will therefore be given by θ := θ+ α∇θℓ(θ). (Note the positive rather than negative sign in the update formula, since we’re maximizing, rather than minimizing, a function now.) Let’s start by working with just one training example ( x,y), and take derivatives to derive the stochastic gradient ascent rule: ∂ ∂θj ℓ(θ) = ( y 1 g(θTx) −(1 −y) 1 1 −g(θTx) ) ∂ ∂θj g(θTx) = ( y 1 g(θTx) −(1 −y) 1 1 −g(θTx) ) g(θTx)(1 −g(θTx)) ∂ ∂θj θTx = ( y(1 −g(θTx)) −(1 −y)g(θTx) ) xj = (y−hθ(x)) xj (2.2) 23 Above, we used the fact that g′(z) = g(z)(1 −g(z)). This therefore gives us the stochastic gradient ascent rule θj := θj + α ( y(i) −hθ(x(i)) ) x(i) j If we compare this to the LMS update rule, we see that it looks identical; but this is not the same algorithm, because hθ(x(i)) is now deﬁned as a non-linear function of θTx(i). Nonetheless, it’s a little surprising that we end up with the same update rule for a rather diﬀerent algorithm and learning problem. Is this coincidence, or is there a deeper reason behind this? We’ll answer this when we get to GLM models. Remark 2.1.1: An alternative notational viewpoint of the same loss func- tion is also useful, especially for Section 7.1 where we study nonlinear models. Let ℓlogistic : R ×{0,1}→ R≥0 be the logistic loss deﬁned as ℓlogistic(t,y) ≜ ylog(1 + exp(−t)) + (1−y) log(1 + exp(t)) . (2.3) One can verify by plugging in hθ(x) = 1 /(1 + e−θ⊤x) that the negative log- likelihood (the negation of ℓ(θ) in equation (2.1)) can be re-written as −ℓ(θ) = ℓlogistic(θ⊤x,y). (2.4) Oftentimes θ⊤x or t is called the logit. Basic calculus gives us that ∂ℓlogistic(t,y) ∂t = y −exp(−t) 1 + exp(−t) + (1 −y) 1 1 + exp(−t) (2.5) = 1/(1 + exp(−t)) −y. (2.6) Then, using the chain rule, we have that ∂ ∂θj ℓ(θ) = −∂ℓlogistic(t,y) ∂t · ∂t ∂θj (2.7) = (y−1/(1 + exp(−t))) ·xj = (y−hθ(x))xj , (2.8) which is consistent with the derivation in equation (2.2). We will see this viewpoint can be extended nonlinear models in Section 7.1. 2.2 Digression: the perceptron learning algo- rithm We now digress to talk brieﬂy about an algorithm that’s of some historical interest, and that we will also return to later when we talk about learning 24 theory. Consider modifying the logistic regression method to “force” it to output values that are either 0 or 1 or exactly. To do so, it seems natural to change the deﬁnition of g to be the threshold function: g(z) = { 1 if z ≥0 0 if z <0 If we then let hθ(x) = g(θTx) as before but using this modiﬁed deﬁnition of g, and if we use the update rule θj := θj + α ( y(i) −hθ(x(i)) ) x(i) j . then we have the perceptron learning algorithn. In the 1960s, this “perceptron” was argued to be a rough model for how individual neurons in the brain work. Given how simple the algorithm is, it will also provide a starting point for our analysis when we talk about learning theory later in this class. Note however that even though the perceptron may be cosmetically similar to the other algorithms we talked about, it is actually a very diﬀerent type of algorithm than logistic regression and least squares linear regression; in particular, it is diﬃcult to endow the perceptron’s predic- tions with meaningful probabilistic interpretations, or derive the perceptron as a maximum likelihood estimation algorithm. 2.3 Multi-class classiﬁcation Consider a classiﬁcation problem in which the response variableycan take on any one of kvalues, so y∈{1,2,...,k }. For example, rather than classifying emails into the two classes spam or not-spam—which would have been a binary classiﬁcation problem—we might want to classify them into three classes, such as spam, personal mails, and work-related mails. The label / response variable is still discrete, but can now take on more than two values. We will thus model it as distributed according to a multinomial distribution. In this case, p(y\|x; θ) is a distribution over k possible discrete outcomes and is thus a multinomial distribution. Recall that a multinomial distribu- tion involves k numbers φ1,...,φ k specifying the probability of each of the outcomes. Note that these numbers must satisfy ∑k i=1 φi = 1. We will de- sign a parameterized model that outputs φ1,...,φ k satisfying this constraint given the input x. We introduce k groups of parameters θ1,...,θ k, each of them being a vector in Rd. Intuitively, we would like to use θ⊤ 1 x,...,θ ⊤ k x to represent 25 φ1,...,φ k, the probabilities P(y = 1 \|x; θ),...,P (y = k \|x; θ). However, there are two issues with such a direct approach. First, θ⊤ j x is not neces- sarily within [0 ,1]. Second, the summation of θ⊤ j x’s is not necessarily 1. Thus, instead, we will use the softmax function to turn ( θ⊤ 1 x,··· ,θ⊤ k x) into a probability vector with nonnegative entries that sum up to 1. Deﬁne the softmax function softmax : Rk →Rk as softmax(t1,...,t k) =   exp(t1)∑k j=1 exp(tj) ... exp(tk)∑k j=1 exp(tj)  . (2.9) The inputs to the softmax function, the vector t here, are often called log- its. Note that by deﬁnition, the output of the softmax function is always a probability vector whose entries are nonnegative and sum up to 1. Let ( t1,...,t k) = ( θ⊤ 1 x,··· ,θ⊤ k x). We apply the softmax function to (t1,...,t k), and use the output as the probabilitiesP(y= 1 \|x; θ),...,P (y= k\|x; θ). We obtain the following probabilistic model:   P(y= 1 \|x; θ) ... P(y= k\|x; θ)  = softmax(t1,··· ,tk) =   exp(θ⊤ 1 x)∑k j=1 exp(θ⊤ j x) ... exp(θ⊤ k x)∑k j=1 exp(θ⊤ j x)  . (2.10) For notational convenience, we will let φi = exp(ti)∑k j=1 exp(tj) . More succinctly, the equation above can be written as: P(y= i\|x; θ) = φi = exp(ti)∑k j=1 exp(tj) = exp(θ⊤ i x)∑k j=1 exp(θ⊤ j x) . (2.11) Next, we compute the negative log-likelihood of a single example ( x,y). −log p(y\|x,θ) = −log ( exp(ty)∑k j=1 exp(tj) ) = −log ( exp(θ⊤ y x) ∑k j=1 exp(θ⊤ j x) ) (2.12) Thus, the loss function, the negative log-likelihood of the training data, is given as ℓ(θ) = n∑ i=1 −log ( exp(θ⊤ y(i) x(i)) ∑k j=1 exp(θ⊤ j x(i)) ) . (2.13) 26 It’s convenient to deﬁne the cross-entropy loss ℓce : Rk ×{1,...,k }→ R≥0, which modularizes in the complex equation above: 1 ℓce((t1,...,t k),y) = −log ( exp(ty)∑k j=1 exp(tj) ) . (2.14) With this notation, we can simply rewrite equation (2.13) as ℓ(θ) = n∑ i=1 ℓce((θ⊤ 1 x(i),...,θ ⊤ k x(i)),y(i)) . (2.15) Moreover, conveniently, the cross-entropy loss also has a simple gradient. Let t= (t1,...,t k), and recall φi = exp(ti)∑k j=1 exp(tj) . By basic calculus, we can derive ∂ℓce(t,y) ∂ti = φi −1{y= i}, (2.16) where 1 {·}is the indicator function, that is, 1 {y = i}= 1 if y = i, and 1{y = i}= 0 if y ̸= i. Alternatively, in vectorized notations, we have the following form which will be useful for Chapter 7: ∂ℓce(t,y) ∂t = φ−ey , (2.17) where es ∈Rk is the s-th natural basis vector (where the s-th entry is 1 and all other entries are zeros.) Using Chain rule, we have that ∂ℓce((θ⊤ 1 x,...,θ ⊤ k x),y) ∂θi = ∂ℓ(t,y) ∂ti ·∂ti ∂θi = (φi −1{y= i}) ·x. (2.18) Therefore, the gradient of the loss with respect to the part of parameter θi is ∂ℓ(θ) ∂θi = n∑ j=1 (φ(j) i −1{y(j) = i}) ·x(j) , (2.19) where φ(j) i = exp(θ⊤ i x(j))∑k s=1 exp(θ⊤s x(j)) is the probability that the model predicts item i for example x(j). With the gradients above, one can implement (stochastic) gradient descent to minimize the loss function ℓ(θ). 1There are some ambiguity in the naming here. Some people call the cross-entropy loss the function that maps the probability vector (the φ in our language) and label y to the ﬁnal real number, and call our version of cross-entropy loss softmax-cross-entropy loss. We choose our current naming convention because it’s consistent with the naming of most modern deep learning library such as PyTorch and Jax. 27 1 1.5 2 2.5 3 3.5 4 4.5 5 −10 0 10 20 30 40 50 60 x f(x) 1 1.5 2 2.5 3 3.5 4 4.5 5 −10 0 10 20 30 40 50 60 x f(x) 1 1.5 2 2.5 3 3.5 4 4.5 5 −10 0 10 20 30 40 50 60 x f(x) 2.4 Another algorithm for maximizing ℓ(θ) Returning to logistic regression with g(z) being the sigmoid function, let’s now talk about a diﬀerent algorithm for maximizing ℓ(θ). To get us started, let’s consider Newton’s method for ﬁnding a zero of a function. Speciﬁcally, suppose we have some function f : R ↦→R, and we wish to ﬁnd a value of θ so that f(θ) = 0. Here, θ ∈R is a real number. Newton’s method performs the following update: θ:= θ−f(θ) f′(θ). This method has a natural interpretation in which we can think of it as approximating the function f via a linear function that is tangent to f at the current guess θ, solving for where that linear function equals to zero, and letting the next guess for θ be where that linear function is zero. Here’s a picture of the Newton’s method in action: In the leftmost ﬁgure, we see the function f plotted along with the line y= 0. We’re trying to ﬁnd θso that f(θ) = 0; the value ofθthat achieves this is about 1.3. Suppose we initialized the algorithm with θ = 4.5. Newton’s method then ﬁts a straight line tangent to f at θ = 4.5, and solves for the where that line evaluates to 0. (Middle ﬁgure.) This give us the next guess for θ, which is about 2.8. The rightmost ﬁgure shows the result of running one more iteration, which the updates θ to about 1.8. After a few more iterations, we rapidly approach θ= 1.3. Newton’s method gives a way of getting to f(θ) = 0. What if we want to use it to maximize some function ℓ? The maxima of ℓ correspond to points where its ﬁrst derivative ℓ′(θ) is zero. So, by letting f(θ) = ℓ′(θ), we can use the same algorithm to maximize ℓ, and we obtain update rule: θ:= θ−ℓ′(θ) ℓ′′(θ). (Something to think about: How would this change if we wanted to use Newton’s method to minimize rather than maximize a function?) 28 Lastly, in our logistic regression setting, θ is vector-valued, so we need to generalize Newton’s method to this setting. The generalization of Newton’s method to this multidimensional setting (also called the Newton-Raphson method) is given by θ:= θ−H−1∇θℓ(θ). Here, ∇θℓ(θ) is, as usual, the vector of partial derivatives of ℓ(θ) with respect to the θi’s; and His an d-by-dmatrix (actually,d+1−by−d+1, assuming that we include the intercept term) called the Hessian, whose entries are given by Hij = ∂2ℓ(θ) ∂θi∂θj . Newton’s method typically enjoys faster convergence than (batch) gra- dient descent, and requires many fewer iterations to get very close to the minimum. One iteration of Newton’s can, however, be more expensive than one iteration of gradient descent, since it requires ﬁnding and inverting an d-by-d Hessian; but so long as d is not too large, it is usually much faster overall. When Newton’s method is applied to maximize the logistic regres- sion log likelihood function ℓ(θ), the resulting method is also called Fisher scoring. Chapter 3 Generalized linear models So far, we’ve seen a regression example, and a classiﬁcation example. In the regression example, we had y\|x; θ ∼N(µ,σ2), and in the classiﬁcation one, y\|x; θ∼Bernoulli(φ), for some appropriate deﬁnitions ofµand φas functions of x and θ. In this section, we will show that both of these methods are special cases of a broader family of models, called Generalized Linear Models (GLMs).1 We will also show how other models in the GLM family can be derived and applied to other classiﬁcation and regression problems. 3.1 The exponential family To work our way up to GLMs, we will begin by deﬁning exponential family distributions. We say that a class of distributions is in the exponential family if it can be written in the form p(y; η) = b(y) exp(ηTT(y) −a(η)) (3.1) Here, ηis called the natural parameter (also called the canonical param- eter) of the distribution; T(y) is the suﬃcient statistic (for the distribu- tions we consider, it will often be the case that T(y) = y); and a(η) is the log partition function. The quantity e−a(η) essentially plays the role of a nor- malization constant, that makes sure the distribution p(y; η) sums/integrates over y to 1. A ﬁxed choice of T, aand bdeﬁnes a family (or set) of distributions that is parameterized by η; as we vary η, we then get diﬀerent distributions within this family. 1The presentation of the material in this section takes inspiration from Michael I. Jordan, Learning in graphical models(unpublished book draft), and also McCullagh and Nelder, Generalized Linear Models (2nd ed.). 29 30 We now show that the Bernoulli and the Gaussian distributions are ex- amples of exponential family distributions. The Bernoulli distribution with mean φ, written Bernoulli(φ), speciﬁes a distribution over y∈{0,1}, so that p(y = 1; φ) = φ; p(y = 0; φ) = 1 −φ. As we vary φ, we obtain Bernoulli distributions with diﬀerent means. We now show that this class of Bernoulli distributions, ones obtained by varying φ, is in the exponential family; i.e., that there is a choice of T, a and b so that Equation (3.1) becomes exactly the class of Bernoulli distributions. We write the Bernoulli distribution as: p(y; φ) = φy(1 −φ)1−y = exp( ylog φ+ (1 −y) log(1−φ)) = exp (( log ( φ 1 −φ )) y+ log(1 −φ) ) . Thus, the natural parameter is given by η= log(φ/(1 −φ)). Interestingly, if we invert this deﬁnition for η by solving for φ in terms of η, we obtain φ= 1/(1 + e−η). This is the familiar sigmoid function! This will come up again when we derive logistic regression as a GLM. To complete the formulation of the Bernoulli distribution as an exponential family distribution, we also have T(y) = y a(η) = −log(1 −φ) = log(1 + eη) b(y) = 1 This shows that the Bernoulli distribution can be written in the form of Equation (3.1), using an appropriate choice of T, a and b. Let’s now move on to consider the Gaussian distribution. Recall that, when deriving linear regression, the value of σ2 had no eﬀect on our ﬁnal choice of θand hθ(x). Thus, we can choose an arbitrary value for σ2 without changing anything. To simplify the derivation below, let’s set σ2 = 1.2 We 2If we leave σ2 as a variable, the Gaussian distribution can also be shown to be in the exponential family, where η∈R2 is now a 2-dimension vector that depends on both µand σ. For the purposes of GLMs, however, theσ2 parameter can also be treated by considering a more general deﬁnition of the exponential family: p(y; η,τ) = b(a,τ) exp((ηTT(y) − a(η))/c(τ)). Here, τ is called thedispersion parameter, and for the Gaussian,c(τ) = σ2; but given our simpliﬁcation above, we won’t need the more general deﬁnition for the examples we will consider here. 31 then have: p(y; µ) = 1√ 2πexp ( −1 2(y−µ)2 ) = 1√ 2πexp ( −1 2y2 ) ·exp ( µy−1 2µ2 ) Thus, we see that the Gaussian is in the exponential family, with η = µ T(y) = y a(η) = µ2/2 = η2/2 b(y) = (1 / √ 2π) exp(−y2/2). There’re many other distributions that are members of the exponen- tial family: The multinomial (which we’ll see later), the Poisson (for mod- elling count-data; also see the problem set); the gamma and the exponen- tial (for modelling continuous, non-negative random variables, such as time- intervals); the beta and the Dirichlet (for distributions over probabilities); and many more. In the next section, we will describe a general “recipe” for constructing models in which y (given x and θ) comes from any of these distributions. 3.2 Constructing GLMs Suppose you would like to build a model to estimate the number y of cus- tomers arriving in your store (or number of page-views on your website) in any given hour, based on certain features xsuch as store promotions, recent advertising, weather, day-of-week, etc. We know that the Poisson distribu- tion usually gives a good model for numbers of visitors. Knowing this, how can we come up with a model for our problem? Fortunately, the Poisson is an exponential family distribution, so we can apply a Generalized Linear Model (GLM). In this section, we will we will describe a method for constructing GLM models for problems such as these. More generally, consider a classiﬁcation or regression problem where we would like to predict the value of some random variable y as a function of x. To derive a GLM for this problem, we will make the following three assumptions about the conditional distribution of y given x and about our model: 32 1. y\|x; θ∼ExponentialFamily(η). I.e., given x and θ, the distribution of y follows some exponential family distribution, with parameter η. 2. Given x, our goal is to predict the expected value of T(y) given x. In most of our examples, we will have T(y) = y, so this means we would like the prediction h(x) output by our learned hypothesis h to satisfy h(x) = E[ y\|x]. (Note that this assumption is satisﬁed in the choices for hθ(x) for both logistic regression and linear regression. For instance, in logistic regression, we had hθ(x) = p(y= 1\|x; θ) = 0 ·p(y= 0\|x; θ) + 1·p(y= 1\|x; θ) = E[y\|x; θ].) 3. The natural parameter ηand the inputs xare related linearly: η= θTx. (Or, if η is vector-valued, then ηi = θT i x.) The third of these assumptions might seem the least well justiﬁed of the above, and it might be better thought of as a “design choice” in our recipe for designing GLMs, rather than as an assumption per se. These three assumptions/design choices will allow us to derive a very elegant class of learning algorithms, namely GLMs, that have many desirable properties such as ease of learning. Furthermore, the resulting models are often very eﬀective for modelling diﬀerent types of distributions over y; for example, we will shortly show that both logistic regression and ordinary least squares can both be derived as GLMs. 3.2.1 Ordinary least squares To show that ordinary least squares is a special case of the GLM family of models, consider the setting where the target variable y (also called the response variable in GLM terminology) is continuous, and we model the conditional distribution of y given x as a Gaussian N(µ,σ2). (Here, µ may depend x.) So, we let the ExponentialFamily(η) distribution above be the Gaussian distribution. As we saw previously, in the formulation of the Gaussian as an exponential family distribution, we had µ= η. So, we have hθ(x) = E[y\|x; θ] = µ = η = θTx. The ﬁrst equality follows from Assumption 2, above; the second equality follows from the fact thaty\|x; θ∼N(µ,σ2), and so its expected value is given 33 by µ; the third equality follows from Assumption 1 (and our earlier derivation showing that µ = η in the formulation of the Gaussian as an exponential family distribution); and the last equality follows from Assumption 3. 3.2.2 Logistic regression We now consider logistic regression. Here we are interested in binary classiﬁ- cation, so y∈{0,1}. Given that yis binary-valued, it therefore seems natural to choose the Bernoulli family of distributions to model the conditional dis- tribution of y given x. In our formulation of the Bernoulli distribution as an exponential family distribution, we had φ = 1/(1 + e−η). Furthermore, note that if y\|x; θ∼Bernoulli(φ), then E[y\|x; θ] = φ. So, following a similar derivation as the one for ordinary least squares, we get: hθ(x) = E[y\|x; θ] = φ = 1 /(1 + e−η) = 1 /(1 + e−θTx) So, this gives us hypothesis functions of the form hθ(x) = 1/(1 + e−θTx). If you are previously wondering how we came up with the form of the logistic function 1/(1 + e−z), this gives one answer: Once we assume that y condi- tioned on xis Bernoulli, it arises as a consequence of the deﬁnition of GLMs and exponential family distributions. To introduce a little more terminology, the function g giving the distri- bution’s mean as a function of the natural parameter ( g(η) = E[ T(y); η]) is called the canonical response function . Its inverse, g−1, is called the canonical link function . Thus, the canonical response function for the Gaussian family is just the identify function; and the canonical response function for the Bernoulli is the logistic function. 3 3Many texts use gto denote the link function, and g−1 to denote the response function; but the notation we’re using here, inherited from the early machine learning literature, will be more consistent with the notation used in the rest of the class. Chapter 4 Generative learning algorithms So far, we’ve mainly been talking about learning algorithms that model p(y\|x; θ), the conditional distribution of y given x. For instance, logistic regression modeled p(y\|x; θ) as hθ(x) = g(θTx) where g is the sigmoid func- tion. In these notes, we’ll talk about a diﬀerent type of learning algorithm. Consider a classiﬁcation problem in which we want to learn to distinguish between elephants ( y = 1) and dogs ( y = 0), based on some features of an animal. Given a training set, an algorithm like logistic regression or the perceptron algorithm (basically) tries to ﬁnd a straight line—that is, a decision boundary—that separates the elephants and dogs. Then, to classify a new animal as either an elephant or a dog, it checks on which side of the decision boundary it falls, and makes its prediction accordingly. Here’s a diﬀerent approach. First, looking at elephants, we can build a model of what elephants look like. Then, looking at dogs, we can build a separate model of what dogs look like. Finally, to classify a new animal, we can match the new animal against the elephant model, and match it against the dog model, to see whether the new animal looks more like the elephants or more like the dogs we had seen in the training set. Algorithms that try to learn p(y\|x) directly (such as logistic regression), or algorithms that try to learn mappings directly from the space of inputs X to the labels {0,1}, (such as the perceptron algorithm) are called discrim- inative learning algorithms. Here, we’ll talk about algorithms that instead try to model p(x\|y) (and p(y)). These algorithms are called generative learning algorithms. For instance, if y indicates whether an example is a dog (0) or an elephant (1), then p(x\|y = 0) models the distribution of dogs’ features, and p(x\|y= 1) models the distribution of elephants’ features. After modeling p(y) (called the class priors) and p(x\|y), our algorithm 34 35 can then use Bayes rule to derive the posterior distribution on y given x: p(y\|x) = p(x\|y)p(y) p(x) . Here, the denominator is given by p(x) = p(x\|y = 1) p(y = 1) + p(x\|y = 0)p(y = 0) (you should be able to verify that this is true from the standard properties of probabilities), and thus can also be expressed in terms of the quantities p(x\|y) and p(y) that we’ve learned. Actually, if were calculating p(y\|x) in order to make a prediction, then we don’t actually need to calculate the denominator, since arg max y p(y\|x) = arg max y p(x\|y)p(y) p(x) = arg max y p(x\|y)p(y). 4.1 Gaussian discriminant analysis The ﬁrst generative learning algorithm that we’ll look at is Gaussian discrim- inant analysis (GDA). In this model, we’ll assume that p(x\|y) is distributed according to a multivariate normal distribution. Let’s talk brieﬂy about the properties of multivariate normal distributions before moving on to the GDA model itself. 4.1.1 The multivariate normal distribution The multivariate normal distribution in d-dimensions, also called the multi- variate Gaussian distribution, is parameterized by a mean vector µ ∈Rd and a covariance matrix Σ ∈Rd×d, where Σ ≥0 is symmetric and positive semi-deﬁnite. Also written “ N(µ,Σ)”, its density is given by: p(x; µ,Σ) = 1 (2π)d/2\|Σ\|1/2 exp ( −1 2(x−µ)TΣ−1(x−µ) ) . In the equation above, “ \|Σ\|” denotes the determinant of the matrix Σ. For a random variable X distributed N(µ,Σ), the mean is (unsurpris- ingly) given by µ: E[X] = ∫ x xp(x; µ,Σ)dx= µ The covarianceof a vector-valued random variableZis deﬁned as Cov(Z) = E[(Z −E[Z])(Z −E[Z])T]. This generalizes the notion of the variance of a 36 real-valued random variable. The covariance can also be deﬁned as Cov(Z) = E[ZZT] −(E[Z])(E[Z])T. (You should be able to prove to yourself that these two deﬁnitions are equivalent.) If X ∼N(µ,Σ), then Cov(X) = Σ. Here are some examples of what the density of a Gaussian distribution looks like: −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.05 0.1 0.15 0.2 0.25 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.05 0.1 0.15 0.2 0.25 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.05 0.1 0.15 0.2 0.25 The left-most ﬁgure shows a Gaussian with mean zero (that is, the 2x1 zero-vector) and covariance matrix Σ = I (the 2x2 identity matrix). A Gaus- sian with zero mean and identity covariance is also called thestandard nor- mal distribution. The middle ﬁgure shows the density of a Gaussian with zero mean and Σ = 0.6I; and in the rightmost ﬁgure shows one with , Σ = 2I. We see that as Σ becomes larger, the Gaussian becomes more “spread-out,” and as it becomes smaller, the distribution becomes more “compressed.” Let’s look at some more examples. −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.05 0.1 0.15 0.2 0.25 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.05 0.1 0.15 0.2 0.25 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.05 0.1 0.15 0.2 0.25 The ﬁgures above show Gaussians with mean 0, and with covariance matrices respectively Σ = [ 1 0 0 1 ] ; Σ = [ 1 0.5 0.5 1 ] ; Σ = [ 1 0.8 0.8 1 ] . The leftmost ﬁgure shows the familiar standard normal distribution, and we see that as we increase the oﬀ-diagonal entry in Σ, the density becomes more 37 “compressed” towards the 45◦line (given by x1 = x2). We can see this more clearly when we look at the contours of the same three densities: −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 Here’s one last set of examples generated by varying Σ: −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 The plots above used, respectively, Σ = [ 1 -0.5 -0.5 1 ] ; Σ = [ 1 -0.8 -0.8 1 ] ; Σ = [ 3 0.8 0.8 1 ] . From the leftmost and middle ﬁgures, we see that by decreasing the oﬀ- diagonal elements of the covariance matrix, the density now becomes “com- pressed” again, but in the opposite direction. Lastly, as we vary the pa- rameters, more generally the contours will form ellipses (the rightmost ﬁgure showing an example). As our last set of examples, ﬁxing Σ = I, by varying µ, we can also move the mean of the density around. −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.05 0.1 0.15 0.2 0.25 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.05 0.1 0.15 0.2 0.25 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.05 0.1 0.15 0.2 0.25 38 The ﬁgures above were generated using Σ = I, and respectively µ= [ 1 0 ] ; µ= [ -0.5 0 ] ; µ= [ -1 -1.5 ] . 4.1.2 The Gaussian discriminant analysis model When we have a classiﬁcation problem in which the input features x are continuous-valued random variables, we can then use the Gaussian Discrim- inant Analysis (GDA) model, which models p(x\|y) using a multivariate nor- mal distribution. The model is: y ∼ Bernoulli(φ) x\|y= 0 ∼ N(µ0,Σ) x\|y= 1 ∼ N(µ1,Σ) Writing out the distributions, this is: p(y) = φy(1 −φ)1−y p(x\|y= 0) = 1 (2π)d/2\|Σ\|1/2 exp ( −1 2(x−µ0)TΣ−1(x−µ0) ) p(x\|y= 1) = 1 (2π)d/2\|Σ\|1/2 exp ( −1 2(x−µ1)TΣ−1(x−µ1) ) Here, the parameters of our model are φ, Σ, µ0 and µ1. (Note that while there’re two diﬀerent mean vectors µ0 and µ1, this model is usually applied using only one covariance matrix Σ.) The log-likelihood of the data is given by ℓ(φ,µ0,µ1,Σ) = log n∏ i=1 p(x(i),y(i); φ,µ0,µ1,Σ) = log n∏ i=1 p(x(i)\|y(i); µ0,µ1,Σ)p(y(i); φ). 39 By maximizing ℓwith respect to the parameters, we ﬁnd the maximum like- lihood estimate of the parameters (see problem set 1) to be: φ = 1 n n∑ i=1 1{y(i) = 1} µ0 = ∑n i=1 1{y(i) = 0}x(i) ∑n i=1 1{y(i) = 0} µ1 = ∑n i=1 1{y(i) = 1}x(i) ∑n i=1 1{y(i) = 1} Σ = 1 n n∑ i=1 (x(i) −µy(i) )(x(i) −µy(i) )T. Pictorially, what the algorithm is doing can be seen in as follows: −2 −1 0 1 2 3 4 5 6 7 −7 −6 −5 −4 −3 −2 −1 0 1 Shown in the ﬁgure are the training set, as well as the contours of the two Gaussian distributions that have been ﬁt to the data in each of the two classes. Note that the two Gaussians have contours that are the same shape and orientation, since they share a covariance matrix Σ, but they have diﬀerent means µ0 and µ1. Also shown in the ﬁgure is the straight line giving the decision boundary at which p(y = 1 \|x) = 0 .5. On one side of the boundary, we’ll predict y= 1 to be the most likely outcome, and on the other side, we’ll predict y= 0. 40 4.1.3 Discussion: GDA and logistic regression The GDA model has an interesting relationship to logistic regression. If we view the quantity p(y= 1\|x; φ,µ0,µ1,Σ) as a function of x, we’ll ﬁnd that it can be expressed in the form p(y= 1\|x; φ,Σ,µ0,µ1) = 1 1 + exp(−θTx), where θis some appropriate function of φ,Σ,µ0,µ1.1 This is exactly the form that logistic regression—a discriminative algorithm—used to model p(y = 1\|x). When would we prefer one model over another? GDA and logistic regres- sion will, in general, give diﬀerent decision boundaries when trained on the same dataset. Which is better? We just argued that if p(x\|y) is multivariate gaussian (with shared Σ), then p(y\|x) necessarily follows a logistic function. The converse, however, is not true; i.e., p(y\|x) being a logistic function does not imply p(x\|y) is multivariate gaussian. This shows that GDA makes stronger modeling as- sumptions about the data than does logistic regression. It turns out that when these modeling assumptions are correct, then GDA will ﬁnd better ﬁts to the data, and is a better model. Speciﬁcally, when p(x\|y) is indeed gaus- sian (with shared Σ), then GDA is asymptotically eﬃcient. Informally, this means that in the limit of very large training sets (large n), there is no algorithm that is strictly better than GDA (in terms of, say, how accurately they estimate p(y\|x)). In particular, it can be shown that in this setting, GDA will be a better algorithm than logistic regression; and more generally, even for small training set sizes, we would generally expect GDA to better. In contrast, by making signiﬁcantly weaker assumptions, logistic regres- sion is also more robust and less sensitive to incorrect modeling assumptions. There are many diﬀerent sets of assumptions that would lead top(y\|x) taking the form of a logistic function. For example, if x\|y = 0 ∼Poisson(λ0), and x\|y = 1 ∼Poisson(λ1), then p(y\|x) will be logistic. Logistic regression will also work well on Poisson data like this. But if we were to use GDA on such data—and ﬁt Gaussian distributions to such non-Gaussian data—then the results will be less predictable, and GDA may (or may not) do well. To summarize: GDA makes stronger modeling assumptions, and is more data eﬃcient (i.e., requires less training data to learn “well”) when the mod- eling assumptions are correct or at least approximately correct. Logistic 1This uses the convention of redeﬁning the x(i)’s on the right-hand-side to be ( d+ 1)- dimensional vectors by adding the extra coordinate x(i) 0 = 1; see problem set 1. 41 regression makes weaker assumptions, and is signiﬁcantly more robust to deviations from modeling assumptions. Speciﬁcally, when the data is in- deed non-Gaussian, then in the limit of large datasets, logistic regression will almost always do better than GDA. For this reason, in practice logistic re- gression is used more often than GDA. (Some related considerations about discriminative vs. generative models also apply for the Naive Bayes algo- rithm that we discuss next, but the Naive Bayes algorithm is still considered a very good, and is certainly also a very popular, classiﬁcation algorithm.) 4.2 Naive bayes (Option Reading) In GDA, the feature vectors x were continuous, real-valued vectors. Let’s now talk about a diﬀerent learning algorithm in which the xj’s are discrete- valued. For our motivating example, consider building an email spam ﬁlter using machine learning. Here, we wish to classify messages according to whether they are unsolicited commercial (spam) email, or non-spam email. After learning to do this, we can then have our mail reader automatically ﬁlter out the spam messages and perhaps place them in a separate mail folder. Classifying emails is one example of a broader set of problems called text classiﬁcation. Let’s say we have a training set (a set of emails labeled as spam or non- spam). We’ll begin our construction of our spam ﬁlter by specifying the features xj used to represent an email. We will represent an email via a feature vector whose length is equal to the number of words in the dictionary. Speciﬁcally, if an email contains the j-th word of the dictionary, then we will set xj = 1; otherwise, we let xj = 0. For instance, the vector x=   1 0 0 ... 1 ... 0   a aardvark aardwolf ... buy ... zygmurgy is used to represent an email that contains the words “a” and “buy,” but not 42 “aardvark,” “aardwolf” or “zygmurgy.”2 The set of words encoded into the feature vector is called the vocabulary, so the dimension of x is equal to the size of the vocabulary. Having chosen our feature vector, we now want to build a generative model. So, we have to model p(x\|y). But if we have, say, a vocabulary of 50000 words, then x∈{0,1}50000 (xis a 50000-dimensional vector of 0’s and 1’s), and if we were to modelxexplicitly with a multinomial distribution over the 250000 possible outcomes, then we’d end up with a (250000 −1)-dimensional parameter vector. This is clearly too many parameters. To modelp(x\|y), we will therefore make a very strong assumption. We will assume that the xi’s are conditionally independent given y. This assumption is called theNaive Bayes (NB) assumption, and the resulting algorithm is called the Naive Bayes classiﬁer. For instance, if y= 1 means spam email; “buy” is word 2087 and “price” is word 39831; then we are assuming that if I tell you y = 1 (that a particular piece of email is spam), then knowledge of x2087 (knowledge of whether “buy” appears in the message) will have no eﬀect on your beliefs about the value of x39831 (whether “price” appears). More formally, this can be written p(x2087\|y) = p(x2087\|y,x39831). (Note that this is not the same as saying that x2087 and x39831 are independent, which would have been written “ p(x2087) = p(x2087\|x39831)”; rather, we are only assuming that x2087 and x39831 are conditionally independent given y.) We now have: p(x1,...,x 50000\|y) = p(x1\|y)p(x2\|y,x1)p(x3\|y,x1,x2) ···p(x50000\|y,x1,...,x 49999) = p(x1\|y)p(x2\|y)p(x3\|y) ···p(x50000\|y) = d∏ j=1 p(xj\|y) The ﬁrst equality simply follows from the usual properties of probabilities, and the second equality used the NB assumption. We note that even though 2Actually, rather than looking through an English dictionary for the list of all English words, in practice it is more common to look through our training set and encode in our feature vector only the words that occur at least once there. Apart from reducing the number of words modeled and hence reducing our computational and space requirements, this also has the advantage of allowing us to model/include as a feature many words that may appear in your email (such as “cs229”) but that you won’t ﬁnd in a dictionary. Sometimes (as in the homework), we also exclude the very high frequency words (which will be words like “the,” “of,” “and”; these high frequency, “content free” words are called stop words) since they occur in so many documents and do little to indicate whether an email is spam or non-spam. 43 the Naive Bayes assumption is an extremely strong assumptions, the resulting algorithm works well on many problems. Our model is parameterized by φj\|y=1 = p(xj = 1\|y= 1), φj\|y=0 = p(xj = 1\|y = 0), and φy = p(y = 1). As usual, given a training set {(x(i),y(i)); i = 1,...,n }, we can write down the joint likelihood of the data: L(φy,φj\|y=0,φj\|y=1) = n∏ i=1 p(x(i),y(i)). Maximizing this with respect to φy,φj\|y=0 and φj\|y=1 gives the maximum likelihood estimates: φj\|y=1 = ∑n i=1 1{x(i) j = 1 ∧y(i) = 1}∑n i=1 1{y(i) = 1} φj\|y=0 = ∑n i=1 1{x(i) j = 1 ∧y(i) = 0}∑n i=1 1{y(i) = 0} φy = ∑n i=1 1{y(i) = 1} n In the equations above, the “ ∧” symbol means “and.” The parameters have a very natural interpretation. For instance, φj\|y=1 is just the fraction of the spam (y= 1) emails in which word j does appear. Having ﬁt all these parameters, to make a prediction on a new example with features x, we then simply calculate p(y= 1\|x) = p(x\|y= 1)p(y= 1) p(x) = (∏d j=1 p(xj\|y= 1) ) p(y= 1) (∏d j=1 p(xj\|y= 1) ) p(y= 1) + (∏d j=1 p(xj\|y= 0) ) p(y= 0) , and pick whichever class has the higher posterior probability. Lastly, we note that while we have developed the Naive Bayes algorithm mainly for the case of problems where the features xj are binary-valued, the generalization to where xj can take values in {1,2,...,k j}is straightforward. Here, we would simply modelp(xj\|y) as multinomial rather than as Bernoulli. Indeed, even if some original input attribute (say, the living area of a house, as in our earlier example) were continuous valued, it is quite common to discretize it—that is, turn it into a small set of discrete values—and apply Naive Bayes. For instance, if we use some feature xj to represent living area, we might discretize the continuous values as follows: 44 Living area (sq. feet) < 400 400-800 800-1200 1200-1600 >1600 xi 1 2 3 4 5 Thus, for a house with living area 890 square feet, we would set the value of the corresponding feature xj to 3. We can then apply the Naive Bayes algorithm, and model p(xj\|y) with a multinomial distribution, as described previously. When the original, continuous-valued attributes are not well- modeled by a multivariate normal distribution, discretizing the features and using Naive Bayes (instead of GDA) will often result in a better classiﬁer. 4.2.1 Laplace smoothing The Naive Bayes algorithm as we have described it will work fairly well for many problems, but there is a simple change that makes it work much better, especially for text classiﬁcation. Let’s brieﬂy discuss a problem with the algorithm in its current form, and then talk about how we can ﬁx it. Consider spam/email classiﬁcation, and let’s suppose that, we are in the year of 20xx, after completing CS229 and having done excellent work on the project, you decide around May 20xx to submit work you did to the NeurIPS conference for publication. 3 Because you end up discussing the conference in your emails, you also start getting messages with the word “neurips” in it. But this is your ﬁrst NeurIPS paper, and until this time, you had not previously seen any emails containing the word “neurips”; in particular “neurips” did not ever appear in your training set of spam/non-spam emails. Assuming that “neurips” was the 35000th word in the dictionary, your Naive Bayes spam ﬁlter therefore had picked its maximum likelihood estimates of the parameters φ35000\|y to be φ35000\|y=1 = ∑n i=1 1{x(i) 35000 = 1 ∧y(i) = 1}∑n i=1 1{y(i) = 1} = 0 φ35000\|y=0 = ∑n i=1 1{x(i) 35000 = 1 ∧y(i) = 0}∑n i=1 1{y(i) = 0} = 0 I.e., because it has never seen “neurips” before in either spam or non-spam training examples, it thinks the probability of seeing it in either type of email is zero. Hence, when trying to decide if one of these messages containing 3NeurIPS is one of the top machine learning conferences. The deadline for submitting a paper is typically in May-June. 45 “neurips” is spam, it calculates the class posterior probabilities, and obtains p(y= 1\|x) = ∏d j=1 p(xj\|y= 1)p(y= 1) ∏d j=1 p(xj\|y= 1)p(y= 1) + ∏d j=1 p(xj\|y= 0)p(y= 0) = 0 0. This is because each of the terms “∏d j=1 p(xj\|y)” includes a termp(x35000\|y) = 0 that is multiplied into it. Hence, our algorithm obtains 0 /0, and doesn’t know how to make a prediction. Stating the problem more broadly, it is statistically a bad idea to esti- mate the probability of some event to be zero just because you haven’t seen it before in your ﬁnite training set. Take the problem of estimating the mean of a multinomial random variable z taking values in {1,...,k }. We can pa- rameterize our multinomial with φj = p(z = j). Given a set of nindependent observations {z(1),...,z (n)}, the maximum likelihood estimates are given by φj = ∑n i=1 1{z(i) = j} n . As we saw previously, if we were to use these maximum likelihood estimates, then some of the φj’s might end up as zero, which was a problem. To avoid this, we can use Laplace smoothing , which replaces the above estimate with φj = 1 + ∑n i=1 1{z(i) = j} k+ n . Here, we’ve added 1 to the numerator, and k to the denominator. Note that∑k j=1 φj = 1 still holds (check this yourself!), which is a desirable property since the φj’s are estimates for probabilities that we know must sum to 1. Also, φj ̸= 0 for all values of j, solving our problem of probabilities being estimated as zero. Under certain (arguably quite strong) conditions, it can be shown that the Laplace smoothing actually gives the optimal estimator of the φj’s. Returning to our Naive Bayes classiﬁer, with Laplace smoothing, we therefore obtain the following estimates of the parameters: φj\|y=1 = 1 + ∑n i=1 1{x(i) j = 1 ∧y(i) = 1} 2 + ∑n i=1 1{y(i) = 1} φj\|y=0 = 1 + ∑n i=1 1{x(i) j = 1 ∧y(i) = 0} 2 + ∑n i=1 1{y(i) = 0} 46 (In practice, it usually doesn’t matter much whether we apply Laplace smooth- ing to φy or not, since we will typically have a fair fraction each of spam and non-spam messages, so φy will be a reasonable estimate of p(y= 1) and will be quite far from 0 anyway.) 4.2.2 Event models for text classiﬁcation To close oﬀ our discussion of generative learning algorithms, let’s talk about one more model that is speciﬁcally for text classiﬁcation. While Naive Bayes as we’ve presented it will work well for many classiﬁcation problems, for text classiﬁcation, there is a related model that does even better. In the speciﬁc context of text classiﬁcation, Naive Bayes as presented uses the what’s called the Bernoulli event model (or sometimes multi-variate Bernoulli event model). In this model, we assumed that the way an email is generated is that ﬁrst it is randomly determined (according to the class priors p(y)) whether a spammer or non-spammer will send you your next message. Then, the person sending the email runs through the dictionary, deciding whether to include each word j in that email independently and according to the probabilities p(xj = 1\|y) = φj\|y. Thus, the probability of a message was given by p(y) ∏d j=1 p(xj\|y). Here’s a diﬀerent model, called the Multinomial event model . To describe this model, we will use a diﬀerent notation and set of features for representing emails. We let xj denote the identity of the j-th word in the email. Thus, xj is now an integer taking values in {1,..., \|V\|}, where \|V\| is the size of our vocabulary (dictionary). An email of d words is now rep- resented by a vector ( x1,x2,...,x d) of length d; note that d can vary for diﬀerent documents. For instance, if an email starts with “A NeurIPS . . . ,” then x1 = 1 (“a” is the ﬁrst word in the dictionary), and x2 = 35000 (if “neurips” is the 35000th word in the dictionary). In the multinomial event model, we assume that the way an email is generated is via a random process in which spam/non-spam is ﬁrst deter- mined (according to p(y)) as before. Then, the sender of the email writes the email by ﬁrst generating x1 from some multinomial distribution over words (p(x1\|y)). Next, the second word x2 is chosen independently of x1 but from the same multinomial distribution, and similarly for x3, x4, and so on, until all dwords of the email have been generated. Thus, the overall probability of a message is given by p(y) ∏d j=1 p(xj\|y). Note that this formula looks like the one we had earlier for the probability of a message under the Bernoulli event model, but that the terms in the formula now mean very diﬀerent things. In particular xj\|y is now a multinomial, rather than a Bernoulli distribution. 47 The parameters for our new model are φy = p(y) as before, φk\|y=1 = p(xj = k\|y= 1) (for any j) and φk\|y=0 = p(xj = k\|y= 0). Note that we have assumed that p(xj\|y) is the same for all values of j (i.e., that the distribution according to which a word is generated does not depend on its position j within the email). If we are given a training set {(x(i),y(i)); i = 1 ,...,n } where x(i) = (x(i) 1 ,x(i) 2 ,...,x (i) di ) (here, di is the number of words in thei-training example), the likelihood of the data is given by L(φy,φk\|y=0,φk\|y=1) = n∏ i=1 p(x(i),y(i)) = n∏ i=1 (di∏ j=1 p(x(i) j \|y; φk\|y=0,φk\|y=1) ) p(y(i); φy). Maximizing this yields the maximum likelihood estimates of the parameters: φk\|y=1 = ∑n i=1 ∑di j=1 1{x(i) j = k∧y(i) = 1}∑n i=1 1{y(i) = 1}di φk\|y=0 = ∑n i=1 ∑di j=1 1{x(i) j = k∧y(i) = 0}∑n i=1 1{y(i) = 0}di φy = ∑n i=1 1{y(i) = 1} n . If we were to apply Laplace smoothing (which is needed in practice for good performance) when estimating φk\|y=0 and φk\|y=1, we add 1 to the numerators and \|V\|to the denominators, and obtain: φk\|y=1 = 1 + ∑n i=1 ∑di j=1 1{x(i) j = k∧y(i) = 1} \|V\|+ ∑n i=1 1{y(i) = 1}di φk\|y=0 = 1 + ∑n i=1 ∑di j=1 1{x(i) j = k∧y(i) = 0} \|V\|+ ∑n i=1 1{y(i) = 0}di . While not necessarily the very best classiﬁcation algorithm, the Naive Bayes classiﬁer often works surprisingly well. It is often also a very good “ﬁrst thing to try,” given its simplicity and ease of implementation. Chapter 5 Kernel methods 5.1 Feature maps Recall that in our discussion about linear regression, we considered the prob- lem of predicting the price of a house (denoted by y) from the living area of the house (denoted by x), and we ﬁt a linear function of x to the training data. What if the price y can be more accurately represented as a non-linear function of x? In this case, we need a more expressive family of models than linear models. We start by considering ﬁtting cubic functions y= θ3x3 + θ2x2 + θ1x+ θ0. It turns out that we can view the cubic function as a linear function over the a diﬀerent set of feature variables (deﬁned below). Concretely, let the function φ: R →R4 be deﬁned as φ(x) =   1 x x2 x3  ∈R4. (5.1) Let θ ∈R4 be the vector containing θ0,θ1,θ2,θ3 as entries. Then we can rewrite the cubic function in x as: θ3x3 + θ2x2 + θ1x+ θ0 = θTφ(x) Thus, a cubic function of the variable x can be viewed as a linear function over the variables φ(x). To distinguish between these two sets of variables, in the context of kernel methods, we will call the “original” input value the input attributes of a problem (in this case, x, the living area). When the 48 49 original input is mapped to some new set of quantitiesφ(x), we will call those new quantities the features variables. (Unfortunately, diﬀerent authors use diﬀerent terms to describe these two things in diﬀerent contexts.) We will call φ a feature map, which maps the attributes to the features. 5.2 LMS (least mean squares) with features We will derive the gradient descent algorithm for ﬁtting the model θTφ(x). First recall that for ordinary least square problem where we were to ﬁt θTx, the batch gradient descent update is (see the ﬁrst lecture note for its deriva- tion): θ:= θ+ α n∑ i=1 ( y(i) −hθ(x(i)) ) x(i) := θ+ α n∑ i=1 ( y(i) −θTx(i)) x(i). (5.2) Let φ : Rd →Rp be a feature map that maps attribute x (in Rd) to the features φ(x) in Rp. (In the motivating example in the previous subsection, we have d= 1 and p= 4.) Now our goal is to ﬁt the function θTφ(x), with θ being a vector in Rp instead of Rd. We can replace all the occurrences of x(i) in the algorithm above by φ(x(i)) to obtain the new update: θ:= θ+ α n∑ i=1 ( y(i) −θTφ(x(i)) ) φ(x(i)) (5.3) Similarly, the corresponding stochastic gradient descent update rule is θ:= θ+ α ( y(i) −θTφ(x(i)) ) φ(x(i)) (5.4) 5.3 LMS with the kernel trick The gradient descent update, or stochastic gradient update above becomes computationally expensive when the features φ(x) is high-dimensional. For example, consider the direct extension of the feature map in equation (5.1) to high-dimensional input x: suppose x∈Rd, and let φ(x) be the vector that 50 contains all the monomials of x with degree ≤3 φ(x) =   1 x1 x2 ... x2 1 x1x2 x1x3 ... x2x1 ... x3 1 x2 1x2 ...   . (5.5) The dimension of the features φ(x) is on the order of d3.1 This is a pro- hibitively long vector for computational purpose — when d = 1000, each update requires at least computing and storing a 1000 3 = 109 dimensional vector, which is 10 6 times slower than the update rule for for ordinary least squares updates (5.2). It may appear at ﬁrst that such d3 runtime per update and memory usage are inevitable, because the vector θitself is of dimension p≈d3, and we may need to update every entry of θ and store it. However, we will introduce the kernel trick with which we will not need to storeθexplicitly, and the runtime can be signiﬁcantly improved. For simplicity, we assume the initialize the value θ = 0, and we focus on the iterative update (5.3). The main observation is that at any time, θ can be represented as a linear combination of the vectors φ(x(1)),...,φ (x(n)). Indeed, we can show this inductively as follows. At initialization, θ = 0 =∑n i=1 0 ·φ(x(i)). Assume at some point, θ can be represented as θ= n∑ i=1 βiφ(x(i)) (5.6) 1Here, for simplicity, we include all the monomials with repetitions (so that, e.g.,x1x2x3 and x2x3x1 both appear in φ(x)). Therefore, there are totally 1 + d+ d2 + d3 entries in φ(x). 51 for some β1,...,β n ∈R. Then we claim that in the next round, θ is still a linear combination of φ(x(1)),...,φ (x(n)) because θ:= θ+ α n∑ i=1 ( y(i) −θTφ(x(i)) ) φ(x(i)) = n∑ i=1 βiφ(x(i)) + α n∑ i=1 ( y(i) −θTφ(x(i)) ) φ(x(i)) = n∑ i=1 (βi + α ( y(i) −θTφ(x(i)) ) )   new βi φ(x(i)) (5.7) You may realize that our general strategy is to implicitly represent the p- dimensional vector θ by a set of coeﬃcients β1,...,β n. Towards doing this, we derive the update rule of the coeﬃcients β1,...,β n. Using the equation above, we see that the new βi depends on the old one via βi := βi + α ( y(i) −θTφ(x(i)) ) (5.8) Here we still have the old θ on the RHS of the equation. Replacing θ by θ= ∑n j=1 βjφ(x(j)) gives ∀i∈{1,...,n },βi := βi + α ( y(i) − n∑ j=1 βjφ(x(j)) T φ(x(i)) ) We often rewrite φ(x(j)) T φ(x(i)) as ⟨φ(x(j)),φ(x(i))⟩to emphasize that it’s the inner product of the two feature vectors. Viewing βi’s as the new representa- tion of θ, we have successfully translated the batch gradient descent algorithm into an algorithm that updates the value of β iteratively. It may appear that at every iteration, we still need to compute the values of ⟨φ(x(j)),φ(x(i))⟩for all pairs of i,j, each of which may take roughly O(p) operation. However, two important properties come to rescue: 1. We can pre-compute the pairwise inner products ⟨φ(x(j)),φ(x(i))⟩for all pairs of i,j before the loop starts. 2. For the feature map φ deﬁned in (5.5) (or many other interesting fea- ture maps), computing ⟨φ(x(j)),φ(x(i))⟩can be eﬃcient and does not 52 necessarily require computing φ(x(i)) explicitly. This is because: ⟨φ(x),φ(z)⟩= 1 + d∑ i=1 xizi + ∑ i,j∈{1,...,d} xixjzizj + ∑ i,j,k∈{1,...,d} xixjxkzizjzk = 1 + d∑ i=1 xizi + ( d∑ i=1 xizi )2 + ( d∑ i=1 xizi )3 = 1 + ⟨x,z⟩+ ⟨x,z⟩2 + ⟨x,z⟩3 (5.9) Therefore, to compute ⟨φ(x),φ(z)⟩, we can ﬁrst compute ⟨x,z⟩with O(d) time and then take another constant number of operations to com- pute 1 + ⟨x,z⟩+ ⟨x,z⟩2 + ⟨x,z⟩3. As you will see, the inner products between the features ⟨φ(x),φ(z)⟩are essential here. We deﬁne the Kernel corresponding to the feature map φas a function that maps X×X→ R satisfying: 2 K(x,z) ≜ ⟨φ(x),φ(z)⟩ (5.10) To wrap up the discussion, we write the down the ﬁnal algorithm as follows: 1. Compute all the values K(x(i),x(j)) ≜ ⟨φ(x(i)),φ(x(j))⟩ using equa- tion (5.9) for all i,j ∈{1,...,n }. Set β := 0. 2. Loop: ∀i∈{1,...,n },βi := βi + α ( y(i) − n∑ j=1 βjK(x(i),x(j)) ) (5.11) Or in vector notation, letting K be the n×n matrix with Kij = K(x(i),x(j)), we have β := β+ α(⃗ y−Kβ) With the algorithm above, we can update the representation β of the vector θeﬃciently with O(n) time per update. Finally, we need to show that 2Recall that Xis the space of the input x. In our running example, X= Rd 53 the knowledge of the representation β suﬃces to compute the prediction θTφ(x). Indeed, we have θTφ(x) = n∑ i=1 βiφ(x(i)) T φ(x) = n∑ i=1 βiK(x(i),x) (5.12) You may realize that fundamentally all we need to know about the feature map φ(·) is encapsulated in the corresponding kernel function K(·,·). We will expand on this in the next section. 5.4 Properties of kernels In the last subsection, we started with an explicitly deﬁned feature map φ, which induces the kernel function K(x,z) ≜ ⟨φ(x),φ(z)⟩. Then we saw that the kernel function is so intrinsic so that as long as the kernel function is deﬁned, the whole training algorithm can be written entirely in the language of the kernel without referring to the feature map φ, so can the prediction of a test example x (equation (5.12).) Therefore, it would be tempted to deﬁne other kernel function K(·,·) and run the algorithm (5.11). Note that the algorithm (5.11) does not need to explicitly access the feature map φ, and therefore we only need to ensure the existence of the feature map φ, but do not necessarily need to be able to explicitly write φ down. What kinds of functions K(·,·) can correspond to some feature mapφ? In other words, can we tell if there is some feature mapping φso that K(x,z) = φ(x)Tφ(z) for all x, z? If we can answer this question by giving a precise characterization of valid kernel functions, then we can completely change the interface of selecting feature maps φ to the interface of selecting kernel function K. Concretely, we can pick a function K, verify that it satisﬁes the characterization (so that there exists a feature map φ that K corresponds to), and then we can run update rule (5.11). The beneﬁt here is that we don’t have to be able to compute φ or write it down analytically, and we only need to know its existence. We will answer this question at the end of this subsection after we go through several concrete examples of kernels. Suppose x,z ∈Rd, and let’s ﬁrst consider the function K(·,·) deﬁned as: K(x,z) = (xTz)2. 54 We can also write this as K(x,z) = ( d∑ i=1 xizi )( d∑ j=1 xjzj ) = d∑ i=1 d∑ j=1 xixjzizj = d∑ i,j=1 (xixj)(zizj) Thus, we see that K(x,z) = ⟨φ(x),φ(z)⟩is the kernel function that corre- sponds to the the feature mapping φgiven (shown here for the case of d= 3) by φ(x) =   x1x1 x1x2 x1x3 x2x1 x2x2 x2x3 x3x1 x3x2 x3x3   . Revisiting the computational eﬃciency perspective of kernel, note that whereas calculating the high-dimensional φ(x) requires O(d2) time, ﬁnding K(x,z) takes only O(d) time—linear in the dimension of the input attributes. For another related example, also consider K(·,·) deﬁned by K(x,z) = ( xTz+ c)2 = d∑ i,j=1 (xixj)(zizj) + d∑ i=1 ( √ 2cxi)( √ 2czi) + c2. (Check this yourself.) This function K is a kernel function that corresponds 55 to the feature mapping (again shown for d= 3) φ(x) =   x1x1 x1x2 x1x3 x2x1 x2x2 x2x3 x3x1 x3x2 x3x3√ 2cx1√ 2cx2√ 2cx3 c   , and the parameter c controls the relative weighting between the xi (ﬁrst order) and the xixj (second order) terms. More broadly, the kernel K(x,z) = ( xTz+ c)k corresponds to a feature mapping to an (d+k k ) feature space, corresponding of all monomials of the form xi1 xi2 ...x ik that are up to order k. However, despite working in this O(dk)-dimensional space, computing K(x,z) still takes only O(d) time, and hence we never need to explicitly represent feature vectors in this very high dimensional feature space. Kernels as similarity metrics. Now, let’s talk about a slightly diﬀerent view of kernels. Intuitively, (and there are things wrong with this intuition, but nevermind), if φ(x) and φ(z) are close together, then we might expect K(x,z) = φ(x)Tφ(z) to be large. Conversely, if φ(x) and φ(z) are far apart— say nearly orthogonal to each other—thenK(x,z) = φ(x)Tφ(z) will be small. So, we can think of K(x,z) as some measurement of how similar are φ(x) and φ(z), or of how similar are x and z. Given this intuition, suppose that for some learning problem that you’re working on, you’ve come up with some functionK(x,z) that you think might be a reasonable measure of how similar x and z are. For instance, perhaps you chose K(x,z) = exp ( −\|\|x−z\|\|2 2σ2 ) . This is a reasonable measure of x and z’s similarity, and is close to 1 when x and z are close, and near 0 when x and z are far apart. Does there exist 56 a feature map φ such that the kernel K deﬁned above satisﬁes K(x,z) = φ(x)Tφ(z)? In this particular example, the answer is yes. This kernel is called the Gaussian kernel, and corresponds to an inﬁnite dimensional feature mapping φ. We will give a precise characterization about what properties a function K needs to satisfy so that it can be a valid kernel function that corresponds to some feature map φ. Necessary conditions for valid kernels. Suppose for now that K is indeed a valid kernel corresponding to some feature mapping φ, and we will ﬁrst see what properties it satisﬁes. Now, consider some ﬁnite set of npoints (not necessarily the training set) {x(1),...,x (n)}, and let a square, n-by-n matrix K be deﬁned so that its ( i,j)-entry is given by Kij = K(x(i),x(j)). This matrix is called the kernel matrix. Note that we’ve overloaded the notation and used K to denote both the kernel function K(x,z) and the kernel matrix K, due to their obvious close relationship. Now, if K is a valid kernel, then Kij = K(x(i),x(j)) = φ(x(i))Tφ(x(j)) = φ(x(j))Tφ(x(i)) = K(x(j),x(i)) = Kji, and hence Kmust be symmetric. More- over, letting φk(x) denote the k-th coordinate of the vector φ(x), we ﬁnd that for any vector z, we have zTKz = ∑ i ∑ j ziKijzj = ∑ i ∑ j ziφ(x(i))Tφ(x(j))zj = ∑ i ∑ j zi ∑ k φk(x(i))φk(x(j))zj = ∑ k ∑ i ∑ j ziφk(x(i))φk(x(j))zj = ∑ k (∑ i ziφk(x(i)) )2 ≥ 0. The second-to-last step uses the fact that ∑ i,j aiaj = ( ∑ iai)2 for ai = ziφk(x(i)). Since z was arbitrary, this shows that K is positive semi-deﬁnite (K ≥0). Hence, we’ve shown that if K is a valid kernel (i.e., if it corresponds to some feature mapping φ), then the corresponding kernel matrix K ∈Rn×n is symmetric positive semideﬁnite. 57 Suﬃcient conditions for valid kernels. More generally, the condition above turns out to be not only a necessary, but also a suﬃcient, condition for K to be a valid kernel (also called a Mercer kernel). The following result is due to Mercer. 3 Theorem (Mercer). Let K : Rd ×Rd ↦→ R be given. Then for K to be a valid (Mercer) kernel, it is necessary and suﬃcient that for any {x(1),...,x (n)}, (n< ∞), the corresponding kernel matrix is symmetric pos- itive semi-deﬁnite. Given a function K, apart from trying to ﬁnd a feature mapping φ that corresponds to it, this theorem therefore gives another way of testing if it is a valid kernel. You’ll also have a chance to play with these ideas more in problem set 2. In class, we also brieﬂy talked about a couple of other examples of ker- nels. For instance, consider the digit recognition problem, in which given an image (16x16 pixels) of a handwritten digit (0-9), we have to ﬁgure out which digit it was. Using either a simple polynomial kernel K(x,z) = (xTz)k or the Gaussian kernel, SVMs were able to obtain extremely good perfor- mance on this problem. This was particularly surprising since the input attributes x were just 256-dimensional vectors of the image pixel intensity values, and the system had no prior knowledge about vision, or even about which pixels are adjacent to which other ones. Another example that we brieﬂy talked about in lecture was that if the objects x that we are trying to classify are strings (say, x is a list of amino acids, which strung together form a protein), then it seems hard to construct a reasonable, “small” set of features for most learning algorithms, especially if diﬀerent strings have dif- ferent lengths. However, consider letting φ(x) be a feature vector that counts the number of occurrences of each length- k substring in x. If we’re consid- ering strings of English letters, then there are 26 k such strings. Hence, φ(x) is a 26 k dimensional vector; even for moderate values of k, this is probably too big for us to eﬃciently work with. (e.g., 26 4 ≈460000.) However, using (dynamic programming-ish) string matching algorithms, it is possible to ef- ﬁciently compute K(x,z) = φ(x)Tφ(z), so that we can now implicitly work in this 26k-dimensional feature space, but without ever explicitly computing feature vectors in this space. 3Many texts present Mercer’s theorem in a slightly more complicated form involving L2 functions, but when the input attributes take values in Rd, the version given here is equivalent. 58 Application of kernel methods: We’ve seen the application of kernels to linear regression. In the next part, we will introduce the support vector machines to which kernels can be directly applied. dwell too much longer on it here. In fact, the idea of kernels has signiﬁcantly broader applicability than linear regression and SVMs. Speciﬁcally, if you have any learning algorithm that you can write in terms of only inner products ⟨x,z⟩between input attribute vectors, then by replacing this with K(x,z) where K is a kernel, you can “magically” allow your algorithm to work eﬃciently in the high dimensional feature space corresponding to K. For instance, this kernel trick can be applied with the perceptron to derive a kernel perceptron algorithm. Many of the algorithms that we’ll see later in this class will also be amenable to this method, which has come to be known as the “kernel trick.” Chapter 6 Support vector machines This set of notes presents the Support Vector Machine (SVM) learning al- gorithm. SVMs are among the best (and many believe are indeed the best) “oﬀ-the-shelf” supervised learning algorithms. To tell the SVM story, we’ll need to ﬁrst talk about margins and the idea of separating data with a large “gap.” Next, we’ll talk about the optimal margin classiﬁer, which will lead us into a digression on Lagrange duality. We’ll also see kernels, which give a way to apply SVMs eﬃciently in very high dimensional (such as inﬁnite- dimensional) feature spaces, and ﬁnally, we’ll close oﬀ the story with the SMO algorithm, which gives an eﬃcient implementation of SVMs. 6.1 Margins: intuition We’ll start our story on SVMs by talking about margins. This section will give the intuitions about margins and about the “conﬁdence” of our predic- tions; these ideas will be made formal in Section 6.3. Consider logistic regression, where the probability p(y = 1\|x; θ) is mod- eled by hθ(x) = g(θTx). We then predict “1” on an input x if and only if hθ(x) ≥0.5, or equivalently, if and only if θTx ≥0. Consider a positive training example (y= 1). The larger θTxis, the larger also is hθ(x) = p(y= 1\|x; θ), and thus also the higher our degree of “conﬁdence” that the label is 1. Thus, informally we can think of our prediction as being very conﬁdent that y = 1 if θTx ≫0. Similarly, we think of logistic regression as conﬁdently predicting y= 0, if θTx≪0. Given a training set, again informally it seems that we’d have found a good ﬁt to the training data if we can ﬁnd θ so that θTx(i) ≫0 whenever y(i) = 1, and θTx(i) ≪0 whenever y(i) = 0, since this would reﬂect a very conﬁdent (and correct) set of classiﬁcations for all the 59 60 training examples. This seems to be a nice goal to aim for, and we’ll soon formalize this idea using the notion of functional margins. For a diﬀerent type of intuition, consider the following ﬁgure, in which x’s represent positive training examples, o’s denote negative training examples, a decision boundary (this is the line given by the equation θTx = 0, and is also called the separating hyperplane) is also shown, and three points have also been labeled A, B and C. /0 /1 /0 /1 /0 /1 B A C Notice that the point A is very far from the decision boundary. If we are asked to make a prediction for the value of y at A, it seems we should be quite conﬁdent that y = 1 there. Conversely, the point C is very close to the decision boundary, and while it’s on the side of the decision boundary on which we would predict y= 1, it seems likely that just a small change to the decision boundary could easily have caused out prediction to be y = 0. Hence, we’re much more conﬁdent about our prediction at A than at C. The point B lies in-between these two cases, and more broadly, we see that if a point is far from the separating hyperplane, then we may be signiﬁcantly more conﬁdent in our predictions. Again, informally we think it would be nice if, given a training set, we manage to ﬁnd a decision boundary that allows us to make all correct and conﬁdent (meaning far from the decision boundary) predictions on the training examples. We’ll formalize this later using the notion of geometric margins. 61 6.2 Notation (option reading) To make our discussion of SVMs easier, we’ll ﬁrst need to introduce a new notation for talking about classiﬁcation. We will be considering a linear classiﬁer for a binary classiﬁcation problem with labels y and features x. From now, we’ll use y∈{−1,1}(instead of {0,1}) to denote the class labels. Also, rather than parameterizing our linear classiﬁer with the vector θ, we will use parameters w,b, and write our classiﬁer as hw,b(x) = g(wTx+ b). Here, g(z) = 1 if z ≥0, and g(z) = −1 otherwise. This “ w,b” notation allows us to explicitly treat the intercept term b separately from the other parameters. (We also drop the convention we had previously of lettingx0 = 1 be an extra coordinate in the input feature vector.) Thus, btakes the role of what was previously θ0, and w takes the role of [ θ1 ...θ d]T. Note also that, from our deﬁnition of g above, our classiﬁer will directly predict either 1 or −1 (cf. the perceptron algorithm), without ﬁrst going through the intermediate step of estimating p(y= 1) (which is what logistic regression does). 6.3 Functional and geometric margins (op- tion reading) Let’s formalize the notions of the functional and geometric margins. Given a training example (x(i),y(i)), we deﬁne the functional margin of (w,b) with respect to the training example as ˆγ(i) = y(i)(wTx(i) + b). Note that if y(i) = 1, then for the functional margin to be large (i.e., for our prediction to be conﬁdent and correct), we need wTx(i) + bto be a large positive number. Conversely, if y(i) = −1, then for the functional margin to be large, we need wTx(i) + b to be a large negative number. Moreover, if y(i)(wTx(i) + b) >0, then our prediction on this example is correct. (Check this yourself.) Hence, a large functional margin represents a conﬁdent and a correct prediction. For a linear classiﬁer with the choice of g given above (taking values in {−1,1}), there’s one property of the functional margin that makes it not a very good measure of conﬁdence, however. Given our choice ofg, we note that 62 if we replace wwith 2wand bwith 2b, then since g(wTx+b) = g(2wTx+2b), this would not change hw,b(x) at all. I.e., g, and hence also hw,b(x), depends only on the sign, but not on the magnitude, of wTx+ b. However, replacing (w,b) with (2 w,2b) also results in multiplying our functional margin by a factor of 2. Thus, it seems that by exploiting our freedom to scale w and b, we can make the functional margin arbitrarily large without really changing anything meaningful. Intuitively, it might therefore make sense to impose some sort of normalization condition such as that \|\|w\|\|2 = 1; i.e., we might replace ( w,b) with ( w/\|\|w\|\|2,b/\|\|w\|\|2), and instead consider the functional margin of (w/\|\|w\|\|2,b/\|\|w\|\|2). We’ll come back to this later. Given a training set S = {(x(i),y(i)); i = 1 ,...,n }, we also deﬁne the function margin of ( w,b) with respect to S as the smallest of the functional margins of the individual training examples. Denoted by ˆγ, this can therefore be written: ˆγ = min i=1,...,n ˆγ(i). Next, let’s talk about geometric margins. Consider the picture below: wA γ B (i) The decision boundary corresponding to ( w,b) is shown, along with the vector w. Note that w is orthogonal (at 90 ◦) to the separating hyperplane. (You should convince yourself that this must be the case.) Consider the point at A, which represents the input x(i) of some training example with label y(i) = 1. Its distance to the decision boundary, γ(i), is given by the line segment AB. How can we ﬁnd the value of γ(i)? Well, w/\|\|w\|\|is a unit-length vector pointing in the same direction as w. Since A represents x(i), we therefore 63 ﬁnd that the point B is given by x(i) −γ(i) ·w/\|\|w\|\|. But this point lies on the decision boundary, and all points xon the decision boundary satisfy the equation wTx+ b= 0. Hence, wT ( x(i) −γ(i) w \|\|w\|\| ) + b= 0. Solving for γ(i) yields γ(i) = wTx(i) + b \|\|w\|\| = ( w \|\|w\|\| )T x(i) + b \|\|w\|\|. This was worked out for the case of a positive training example at A in the ﬁgure, where being on the “positive” side of the decision boundary is good. More generally, we deﬁne the geometric margin of ( w,b) with respect to a training example (x(i),y(i)) to be γ(i) = y(i) (( w \|\|w\|\| )T x(i) + b \|\|w\|\| ) . Note that if \|\|w\|\|= 1, then the functional margin equals the geometric margin—this thus gives us a way of relating these two diﬀerent notions of margin. Also, the geometric margin is invariant to rescaling of the parame- ters; i.e., if we replace w with 2w and b with 2b, then the geometric margin does not change. This will in fact come in handy later. Speciﬁcally, because of this invariance to the scaling of the parameters, when trying to ﬁt wand b to training data, we can impose an arbitrary scaling constraint on wwithout changing anything important; for instance, we can demand that \|\|w\|\|= 1, or \|w1\|= 5, or \|w1 + b\|+ \|w2\|= 2, and any of these can be satisﬁed simply by rescaling w and b. Finally, given a training set S = {(x(i),y(i)); i= 1,...,n }, we also deﬁne the geometric margin of ( w,b) with respect to S to be the smallest of the geometric margins on the individual training examples: γ = min i=1,...,n γ(i). 6.4 The optimal margin classiﬁer (option read- ing) Given a training set, it seems from our previous discussion that a natural desideratum is to try to ﬁnd a decision boundary that maximizes the (ge- ometric) margin, since this would reﬂect a very conﬁdent set of predictions 64 on the training set and a good “ﬁt” to the training data. Speciﬁcally, this will result in a classiﬁer that separates the positive and the negative training examples with a “gap” (geometric margin). For now, we will assume that we are given a training set that is linearly separable; i.e., that it is possible to separate the positive and negative ex- amples using some separating hyperplane. How will we ﬁnd the one that achieves the maximum geometric margin? We can pose the following opti- mization problem: maxγ,w,b γ s.t. y(i)(wTx(i) + b) ≥γ, i = 1,...,n \|\|w\|\|= 1. I.e., we want to maximize γ, subject to each training example having func- tional margin at least γ. The \|\|w\|\|= 1 constraint moreover ensures that the functional margin equals to the geometric margin, so we are also guaranteed that all the geometric margins are at least γ. Thus, solving this problem will result in (w,b) with the largest possible geometric margin with respect to the training set. If we could solve the optimization problem above, we’d be done. But the “\|\|w\|\|= 1” constraint is a nasty (non-convex) one, and this problem certainly isn’t in any format that we can plug into standard optimization software to solve. So, let’s try transforming the problem into a nicer one. Consider: maxˆγ,w,b ˆγ \|\|w\|\| s.t. y(i)(wTx(i) + b) ≥ˆγ, i = 1,...,n Here, we’re going to maximize ˆγ/\|\|w\|\|, subject to the functional margins all being at least ˆγ. Since the geometric and functional margins are related by γ = ˆγ/\|\|w\|, this will give us the answer we want. Moreover, we’ve gotten rid of the constraint \|\|w\|\|= 1 that we didn’t like. The downside is that we now have a nasty (again, non-convex) objective ˆγ \|\|w\|\| function; and, we still don’t have any oﬀ-the-shelf software that can solve this form of an optimization problem. Let’s keep going. Recall our earlier discussion that we can add an arbi- trary scaling constraint on w and b without changing anything. This is the key idea we’ll use now. We will introduce the scaling constraint that the functional margin of w,b with respect to the training set must be 1: ˆγ = 1. 65 Since multiplying w and bby some constant results in the functional margin being multiplied by that same constant, this is indeed a scaling constraint, and can be satisﬁed by rescaling w,b. Plugging this into our problem above, and noting that maximizing ˆγ/\|\|w\|\|= 1/\|\|w\|\|is the same thing as minimizing \|\|w\|\|2, we now have the following optimization problem: minw,b 1 2\|\|w\|\|2 s.t. y(i)(wTx(i) + b) ≥1, i = 1,...,n We’ve now transformed the problem into a form that can be eﬃciently solved. The above is an optimization problem with a convex quadratic ob- jective and only linear constraints. Its solution gives us the optimal mar- gin classiﬁer. This optimization problem can be solved using commercial quadratic programming (QP) code. 1 While we could call the problem solved here, what we will instead do is make a digression to talk about Lagrange duality. This will lead us to our optimization problem’s dual form, which will play a key role in allowing us to use kernels to get optimal margin classiﬁers to work eﬃciently in very high dimensional spaces. The dual form will also allow us to derive an eﬃcient algorithm for solving the above optimization problem that will typically do much better than generic QP software. 6.5 Lagrange duality (optional reading) Let’s temporarily put aside SVMs and maximum margin classiﬁers, and talk about solving constrained optimization problems. Consider a problem of the following form: minw f(w) s.t. hi(w) = 0, i = 1,...,l. Some of you may recall how the method of Lagrange multipliers can be used to solve it. (Don’t worry if you haven’t seen it before.) In this method, we deﬁne the Lagrangian to be L(w,β) = f(w) + l∑ i=1 βihi(w) 1You may be familiar with linear programming, which solves optimization problems that have linear objectives and linear constraints. QP software is also widely available, which allows convex quadratic objectives and linear constraints. 66 Here, the βi’s are called the Lagrange multipliers. We would then ﬁnd and set L’s partial derivatives to zero: ∂L ∂wi = 0; ∂L ∂βi = 0, and solve for w and β. In this section, we will generalize this to constrained optimization prob- lems in which we may have inequality as well as equality constraints. Due to time constraints, we won’t really be able to do the theory of Lagrange duality justice in this class, 2 but we will give the main ideas and results, which we will then apply to our optimal margin classiﬁer’s optimization problem. Consider the following, which we’ll call theprimal optimization problem: minw f(w) s.t. gi(w) ≤0, i = 1,...,k hi(w) = 0, i = 1,...,l. To solve it, we start by deﬁning the generalized Lagrangian L(w,α,β ) = f(w) + k∑ i=1 αigi(w) + l∑ i=1 βihi(w). Here, the αi’s and βi’s are the Lagrange multipliers. Consider the quantity θP(w) = max α,β: αi≥0 L(w,α,β ). Here, the “ P” subscript stands for “primal.” Let some w be given. If w violates any of the primal constraints (i.e., if either gi(w) > 0 or hi(w) ̸= 0 for some i), then you should be able to verify that θP(w) = max α,β: αi≥0 f(w) + k∑ i=1 αigi(w) + l∑ i=1 βihi(w) (6.1) = ∞. (6.2) Conversely, if the constraints are indeed satisﬁed for a particular value of w, then θP(w) = f(w). Hence, θP(w) = { f(w) if w satisﬁes primal constraints ∞ otherwise. 2Readers interested in learning more about this topic are encouraged to read, e.g., R. T. Rockarfeller (1970), Convex Analysis, Princeton University Press. 67 Thus, θP takes the same value as the objective in our problem for all val- ues of w that satisﬁes the primal constraints, and is positive inﬁnity if the constraints are violated. Hence, if we consider the minimization problem min w θP(w) = min w max α,β: αi≥0 L(w,α,β ), we see that it is the same problem (i.e., and has the same solutions as) our original, primal problem. For later use, we also deﬁne the optimal value of the objective to be p∗ = minwθP(w); we call this the value of the primal problem. Now, let’s look at a slightly diﬀerent problem. We deﬁne θD(α,β) = min w L(w,α,β ). Here, the “ D” subscript stands for “dual.” Note also that whereas in the deﬁnition of θP we were optimizing (maximizing) with respect to α,β, here we are minimizing with respect to w. We can now pose the dual optimization problem: max α,β: αi≥0 θD(α,β) = max α,β: αi≥0 min w L(w,α,β ). This is exactly the same as our primal problem shown above, except that the order of the “max” and the “min” are now exchanged. We also deﬁne the optimal value of the dual problem’s objective to be d∗= maxα,β: αi≥0 θD(w). How are the primal and the dual problems related? It can easily be shown that d∗= max α,β: αi≥0 min w L(w,α,β ) ≤min w max α,β: αi≥0 L(w,α,β ) = p∗. (You should convince yourself of this; this follows from the “max min” of a function always being less than or equal to the “min max.”) However, under certain conditions, we will have d∗= p∗, so that we can solve the dual problem in lieu of the primal problem. Let’s see what these conditions are. Suppose f and the gi’s are convex, 3 and the hi’s are aﬃne. 4 Suppose further that the constraints gi are (strictly) feasible; this means that there exists some w so that gi(w) <0 for all i. 3When f has a Hessian, then it is convex if and only if the Hessian is positive semi- deﬁnite. For instance, f(w) = wTw is convex; similarly, all linear (and aﬃne) functions are also convex. (A function f can also be convex without being diﬀerentiable, but we won’t need those more general deﬁnitions of convexity here.) 4I.e., there exists ai, bi, so that hi(w) = aT i w+ bi. “Aﬃne” means the same thing as linear, except that we also allow the extra intercept term bi. 68 Under our above assumptions, there must existw∗,α∗,β∗so that w∗is the solution to the primal problem, α∗,β∗ are the solution to the dual problem, and moreover p∗ = d∗ = L(w∗,α∗,β∗). Moreover, w∗,α∗ and β∗ satisfy the Karush-Kuhn-Tucker (KKT) conditions, which are as follows: ∂ ∂wi L(w∗,α∗,β∗) = 0 , i = 1,...,d (6.3) ∂ ∂βi L(w∗,α∗,β∗) = 0 , i = 1,...,l (6.4) α∗ igi(w∗) = 0 , i = 1,...,k (6.5) gi(w∗) ≤ 0, i = 1,...,k (6.6) α∗ ≥ 0, i = 1,...,k (6.7) Moreover, if somew∗,α∗,β∗ satisfy the KKT conditions, then it is also a solution to the primal and dual problems. We draw attention to Equation (6.5), which is called the KKT dual complementarity condition. Speciﬁcally, it implies that if α∗ i > 0, then gi(w∗) = 0. (I.e., the “ gi(w) ≤0” constraint is active, meaning it holds with equality rather than with inequality.) Later on, this will be key for showing that the SVM has only a small number of “support vectors”; the KKT dual complementarity condition will also give us our convergence test when we talk about the SMO algorithm. 6.6 Optimal margin classiﬁers: the dual form (option reading) Note: The equivalence of optimization problem (6.8) and the optimization problem (6.12), and the relationship between the primary and dual variables in equation (6.10) are the most important take home messages of this section. Previously, we posed the following (primal) optimization problem for ﬁnd- ing the optimal margin classiﬁer: minw,b 1 2\|\|w\|\|2 (6.8) s.t. y(i)(wTx(i) + b) ≥1, i = 1,...,n We can write the constraints as gi(w) = −y(i)(wTx(i) + b) + 1 ≤0. 69 We have one such constraint for each training example. Note that from the KKT dual complementarity condition, we will have αi >0 only for the train- ing examples that have functional margin exactly equal to one (i.e., the ones corresponding to constraints that hold with equality, gi(w) = 0). Consider the ﬁgure below, in which a maximum margin separating hyperplane is shown by the solid line. The points with the smallest margins are exactly the ones closest to the decision boundary; here, these are the three points (one negative and two pos- itive examples) that lie on the dashed lines parallel to the decision boundary. Thus, only three of the αi’s—namely, the ones corresponding to these three training examples—will be non-zero at the optimal solution to our optimiza- tion problem. These three points are called the support vectors in this problem. The fact that the number of support vectors can be much smaller than the size the training set will be useful later. Let’s move on. Looking ahead, as we develop the dual form of the prob- lem, one key idea to watch out for is that we’ll try to write our algorithm in terms of only the inner product ⟨x(i),x(j)⟩(think of this as ( x(i))Tx(j)) between points in the input feature space. The fact that we can express our algorithm in terms of these inner products will be key when we apply the kernel trick. When we construct the Lagrangian for our optimization problem we have: L(w,b,α ) = 1 2\|\|w\|\|2 − n∑ i=1 αi [ y(i)(wTx(i) + b) −1 ] . (6.9) Note that there’re only “ αi” but no “ βi” Lagrange multipliers, since the problem has only inequality constraints. 70 Let’s ﬁnd the dual form of the problem. To do so, we need to ﬁrst minimize L(w,b,α ) with respect to w and b (for ﬁxed α), to get θD, which we’ll do by setting the derivatives of Lwith respect to w and b to zero. We have: ∇wL(w,b,α ) = w− n∑ i=1 αiy(i)x(i) = 0 This implies that w= n∑ i=1 αiy(i)x(i). (6.10) As for the derivative with respect to b, we obtain ∂ ∂bL(w,b,α ) = n∑ i=1 αiy(i) = 0. (6.11) If we take the deﬁnition of w in Equation (6.10) and plug that back into the Lagrangian (Equation 6.9), and simplify, we get L(w,b,α ) = n∑ i=1 αi −1 2 n∑ i,j=1 y(i)y(j)αiαj(x(i))Tx(j) −b n∑ i=1 αiy(i). But from Equation (6.11), the last term must be zero, so we obtain L(w,b,α ) = n∑ i=1 αi −1 2 n∑ i,j=1 y(i)y(j)αiαj(x(i))Tx(j). Recall that we got to the equation above by minimizing Lwith respect to w and b. Putting this together with the constraints αi ≥0 (that we always had) and the constraint (6.11), we obtain the following dual optimization problem: maxα W(α) = n∑ i=1 αi −1 2 n∑ i,j=1 y(i)y(j)αiαj⟨x(i),x(j)⟩. (6.12) s.t. αi ≥0, i = 1,...,n n∑ i=1 αiy(i) = 0, You should also be able to verify that the conditions required for p∗= d∗ and the KKT conditions (Equations 6.3–6.7) to hold are indeed satisﬁed in 71 our optimization problem. Hence, we can solve the dual in lieu of solving the primal problem. Speciﬁcally, in the dual problem above, we have a maximization problem in which the parameters are the αi’s. We’ll talk later about the speciﬁc algorithm that we’re going to use to solve the dual problem, but if we are indeed able to solve it (i.e., ﬁnd the α’s that maximize W(α) subject to the constraints), then we can use Equation (6.10) to go back and ﬁnd the optimal w’s as a function of theα’s. Having found w∗, by considering the primal problem, it is also straightforward to ﬁnd the optimal value for the intercept term b as b∗= −maxi:y(i)=−1 w∗Tx(i) + mini:y(i)=1 w∗Tx(i) 2 . (6.13) (Check for yourself that this is correct.) Before moving on, let’s also take a more careful look at Equation (6.10), which gives the optimal value of w in terms of (the optimal value of) α. Suppose we’ve ﬁt our model’s parameters to a training set, and now wish to make a prediction at a new point input x. We would then calculate wTx+ b, and predict y = 1 if and only if this quantity is bigger than zero. But using (6.10), this quantity can also be written: wTx+ b = ( n∑ i=1 αiy(i)x(i) )T x+ b (6.14) = n∑ i=1 αiy(i)⟨x(i),x⟩+ b. (6.15) Hence, if we’ve found the αi’s, in order to make a prediction, we have to calculate a quantity that depends only on the inner product between x and the points in the training set. Moreover, we saw earlier that the αi’s will all be zero except for the support vectors. Thus, many of the terms in the sum above will be zero, and we really need to ﬁnd only the inner products between x and the support vectors (of which there is often only a small number) in order calculate (6.15) and make our prediction. By examining the dual form of the optimization problem, we gained sig- niﬁcant insight into the structure of the problem, and were also able to write the entire algorithm in terms of only inner products between input feature vectors. In the next section, we will exploit this property to apply the ker- nels to our classiﬁcation problem. The resulting algorithm, support vector machines, will be able to eﬃciently learn in very high dimensional spaces. 72 6.7 Regularization and the non-separable case (optional reading) The derivation of the SVM as presented so far assumed that the data is linearly separable. While mapping data to a high dimensional feature space via φ does generally increase the likelihood that the data is separable, we can’t guarantee that it always will be so. Also, in some cases it is not clear that ﬁnding a separating hyperplane is exactly what we’d want to do, since that might be susceptible to outliers. For instance, the left ﬁgure below shows an optimal margin classiﬁer, and when a single outlier is added in the upper-left region (right ﬁgure), it causes the decision boundary to make a dramatic swing, and the resulting classiﬁer has a much smaller margin. To make the algorithm work for non-linearly separable datasets as well as be less sensitive to outliers, we reformulate our optimization (using ℓ1 regularization) as follows: minγ,w,b 1 2\|\|w\|\|2 + C n∑ i=1 ξi s.t. y(i)(wTx(i) + b) ≥1 −ξi, i = 1,...,n ξi ≥0, i = 1,...,n. Thus, examples are now permitted to have (functional) margin less than 1, and if an example has functional margin 1 −ξi (with ξ >0), we would pay a cost of the objective function being increased by Cξi. The parameter C controls the relative weighting between the twin goals of making the \|\|w\|\|2 small (which we saw earlier makes the margin large) and of ensuring that most examples have functional margin at least 1. 73 As before, we can form the Lagrangian: L(w,b,ξ,α,r ) = 1 2wTw+ C n∑ i=1 ξi− n∑ i=1 αi [ y(i)(xTw+ b) −1 + ξi ] − n∑ i=1 riξi. Here, the αi’s and ri’s are our Lagrange multipliers (constrained to be ≥0). We won’t go through the derivation of the dual again in detail, but after setting the derivatives with respect to wand bto zero as before, substituting them back in, and simplifying, we obtain the following dual form of the problem: maxα W(α) = n∑ i=1 αi −1 2 n∑ i,j=1 y(i)y(j)αiαj⟨x(i),x(j)⟩ s.t. 0 ≤αi ≤C, i = 1,...,n n∑ i=1 αiy(i) = 0, As before, we also have that w can be expressed in terms of the αi’s as given in Equation (6.10), so that after solving the dual problem, we can con- tinue to use Equation (6.15) to make our predictions. Note that, somewhat surprisingly, in adding ℓ1 regularization, the only change to the dual prob- lem is that what was originally a constraint that 0 ≤αi has now become 0 ≤αi ≤C. The calculation for b∗also has to be modiﬁed (Equation 6.13 is no longer valid); see the comments in the next section/Platt’s paper. Also, the KKT dual-complementarity conditions (which in the next sec- tion will be useful for testing for the convergence of the SMO algorithm) are: αi = 0 ⇒ y(i)(wTx(i) + b) ≥1 (6.16) αi = C ⇒ y(i)(wTx(i) + b) ≤1 (6.17) 0 <αi <C ⇒ y(i)(wTx(i) + b) = 1. (6.18) Now, all that remains is to give an algorithm for actually solving the dual problem, which we will do in the next section. 6.8 The SMO algorithm (optional reading) The SMO (sequential minimal optimization) algorithm, due to John Platt, gives an eﬃcient way of solving the dual problem arising from the derivation 74 of the SVM. Partly to motivate the SMO algorithm, and partly because it’s interesting in its own right, let’s ﬁrst take another digression to talk about the coordinate ascent algorithm. 6.8.1 Coordinate ascent Consider trying to solve the unconstrained optimization problem max α W(α1,α2,...,α n). Here, we think of W as just some function of the parametersαi’s, and for now ignore any relationship between this problem and SVMs. We’ve already seen two optimization algorithms, gradient ascent and Newton’s method. The new algorithm we’re going to consider here is called coordinate ascent: Loop until convergence: { For i= 1,...,n , { αi := arg maxˆαi W(α1,...,α i−1,ˆαi,αi+1,...,α n). } } Thus, in the innermost loop of this algorithm, we will hold all the variables except for some αi ﬁxed, and reoptimize W with respect to just the parameter αi. In the version of this method presented here, the inner-loop reoptimizes the variables in orderα1,α2,...,α n,α1,α2,... . (A more sophisticated version might choose other orderings; for instance, we may choose the next variable to update according to which one we expect to allow us to make the largest increase in W(α).) When the function W happens to be of such a form that the “arg max” in the inner loop can be performed eﬃciently, then coordinate ascent can be a fairly eﬃcient algorithm. Here’s a picture of coordinate ascent in action: 75 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 The ellipses in the ﬁgure are the contours of a quadratic function that we want to optimize. Coordinate ascent was initialized at (2 ,−2), and also plotted in the ﬁgure is the path that it took on its way to the global maximum. Notice that on each step, coordinate ascent takes a step that’s parallel to one of the axes, since only one variable is being optimized at a time. 6.8.2 SMO We close oﬀ the discussion of SVMs by sketching the derivation of the SMO algorithm. Here’s the (dual) optimization problem that we want to solve: maxα W(α) = n∑ i=1 αi −1 2 n∑ i,j=1 y(i)y(j)αiαj⟨x(i),x(j)⟩. (6.19) s.t. 0 ≤αi ≤C, i = 1,...,n (6.20) n∑ i=1 αiy(i) = 0. (6.21) Let’s say we have set of αi’s that satisfy the constraints (6.20-6.21). Now, suppose we want to hold α2,...,α n ﬁxed, and take a coordinate ascent step and reoptimize the objective with respect to α1. Can we make any progress? The answer is no, because the constraint (6.21) ensures that α1y(1) = − n∑ i=2 αiy(i). 76 Or, by multiplying both sides by y(1), we equivalently have α1 = −y(1) n∑ i=2 αiy(i). (This step used the fact that y(1) ∈{−1,1}, and hence ( y(1))2 = 1.) Hence, α1 is exactly determined by the other αi’s, and if we were to hold α2,...,α n ﬁxed, then we can’t make any change to α1 without violating the con- straint (6.21) in the optimization problem. Thus, if we want to update some subject of the αi’s, we must update at least two of them simultaneously in order to keep satisfying the constraints. This motivates the SMO algorithm, which simply does the following: Repeat till convergence { 1. Select some pair αi and αj to update next (using a heuristic that tries to pick the two that will allow us to make the biggest progress towards the global maximum). 2. Reoptimize W(α) with respect to αi and αj, while holding all the other αk’s (k̸= i,j) ﬁxed. } To test for convergence of this algorithm, we can check whether the KKT conditions (Equations 6.16-6.18) are satisﬁed to within some tol. Here, tol is the convergence tolerance parameter, and is typically set to around 0.01 to 0.001. (See the paper and pseudocode for details.) The key reason that SMO is an eﬃcient algorithm is that the update to αi, αj can be computed very eﬃciently. Let’s now brieﬂy sketch the main ideas for deriving the eﬃcient update. Let’s say we currently have some setting of the αi’s that satisfy the con- straints (6.20-6.21), and suppose we’ve decided to hold α3,...,α n ﬁxed, and want to reoptimize W(α1,α2,...,α n) with respect to α1 and α2 (subject to the constraints). From (6.21), we require that α1y(1) + α2y(2) = − n∑ i=3 αiy(i). Since the right hand side is ﬁxed (as we’ve ﬁxed α3,...α n), we can just let it be denoted by some constant ζ: α1y(1) + α2y(2) = ζ. (6.22) We can thus picture the constraints on α1 and α2 as follows: 77 α 2 α 1 α 1 α 2 C C (1) + (2)y y =ζH L From the constraints (6.20), we know that α1 and α2 must lie within the box [0,C]×[0,C] shown. Also plotted is the line α1y(1) +α2y(2) = ζ, on which we know α1 and α2 must lie. Note also that, from these constraints, we know L ≤α2 ≤H; otherwise, ( α1,α2) can’t simultaneously satisfy both the box and the straight line constraint. In this example, L= 0. But depending on what the line α1y(1) + α2y(2) = ζ looks like, this won’t always necessarily be the case; but more generally, there will be some lower-bound L and some upper-bound H on the permissible values for α2 that will ensure that α1, α2 lie within the box [0 ,C] ×[0,C]. Using Equation (6.22), we can also write α1 as a function of α2: α1 = (ζ−α2y(2))y(1). (Check this derivation yourself; we again used the fact that y(1) ∈{−1,1}so that (y(1))2 = 1.) Hence, the objective W(α) can be written W(α1,α2,...,α n) = W((ζ−α2y(2))y(1),α2,...,α n). Treating α3,...,α n as constants, you should be able to verify that this is just some quadratic function in α2. I.e., this can also be expressed in the form aα2 2 + bα2 + c for some appropriate a, b, and c. If we ignore the “box” constraints (6.20) (or, equivalently, that L ≤α2 ≤H), then we can easily maximize this quadratic function by setting its derivative to zero and solving. We’ll let αnew,unclipped 2 denote the resulting value of α2. You should also be able to convince yourself that if we had instead wanted to maximize W with respect to α2 but subject to the box constraint, then we can ﬁnd the resulting value optimal simply by taking αnew,unclipped 2 and “clipping” it to lie in the 78 [L,H] interval, to get αnew 2 =    H if αnew,unclipped 2 >H αnew,unclipped 2 if L≤αnew,unclipped 2 ≤H L if αnew,unclipped 2 <L Finally, having found the αnew 2 , we can use Equation (6.22) to go back and ﬁnd the optimal value of αnew 1 . There’re a couple more details that are quite easy but that we’ll leave you to read about yourself in Platt’s paper: One is the choice of the heuristics used to select the next αi, αj to update; the other is how to update b as the SMO algorithm is run. Part II Deep learning 79 Chapter 7 Deep learning We now begin our study of deep learning. In this set of notes, we give an overview of neural networks, discuss vectorization and discuss training neural networks with backpropagation. 7.1 Supervised learning with non-linear mod- els In the supervised learning setting (predicting y from the input x), suppose our model/hypothesis is hθ(x). In the past lectures, we have considered the cases when hθ(x) = θ⊤x(in linear regression) or hθ(x) = θ⊤φ(x) (where φ(x) is the feature map). A commonality of these two models is that they are linear in the parameters θ. Next we will consider learning general family of models that are non-linear in both the parameters θand the inputs x. The most common non-linear models are neural networks, which we will deﬁne staring from the next section. For this section, it suﬃces to think hθ(x) as an abstract non-linear model. 1 Suppose {(x(i),y(i))}n i=1 are the training examples. We will deﬁne the nonlinear model and the loss/cost function for learning it. Regression problems. For simplicity, we start with the case where the output is a real number, that is, y(i) ∈R, and thus the model hθ also outputs a real number hθ(x) ∈R. We deﬁne the least square cost function for the 1If a concrete example is helpful, perhaps think about the model hθ(x) = θ2 1x2 1 +θ2 2x2 2 + ··· + θ2 dx2 d in this subsection, even though it’s not a neural network. 80 81 i-th example (x(i),y(i)) as J(i)(θ) = 1 2(hθ(x(i)) −y(i))2 , (7.1) and deﬁne the mean-square cost function for the dataset as J(θ) = 1 n n∑ i=1 J(i)(θ) , (7.2) which is same as in linear regression except that we introduce a constant 1/n in front of the cost function to be consistent with the convention. Note that multiplying the cost function with a scalar will not change the local minima or global minima of the cost function. Also note that the underlying parameterization for hθ(x) is diﬀerent from the case of linear regression, even though the form of the cost function is the same mean-squared loss. Throughout the notes, we use the words “loss” and “cost” interchangeably. Binary classiﬁcation. Next we deﬁne the model and loss function for binary classiﬁcation. Suppose the inputs x ∈Rd. Let ¯hθ : Rd →R be a parameterized model (the analog of θ⊤x in logistic linear regression). We call the output ¯hθ(x) ∈R the logit. Analogous to Section 2.1, we use the logistic function g(·) to turn the logit ¯hθ(x) to a probability hθ(x) ∈[0,1]: hθ(x) = g(¯hθ(x)) = 1/(1 + exp(−¯hθ(x)) . (7.3) We model the conditional distribution of y given x and θ by P(y= 1 \|x; θ) = hθ(x) P(y= 0 \|x; θ) = 1 −hθ(x) Following the same derivation in Section 2.1 and using the derivation in Remark 2.1.1, the negative likelihood loss function is equal to: J(i)(θ) = −log p(y(i) \|x(i); θ) = ℓlogistic(¯hθ(x(i)),y(i)) (7.4) As done in equation (7.2), the total loss function is also deﬁned as the average of the loss function over individual training examples, J(θ) = 1 n ∑n i=1 J(i)(θ). 82 Multi-class classiﬁcation. Following Section 2.3, we consider a classiﬁca- tion problem where the response variable y can take on any one of k values, i.e. y ∈{1,2,...,k }. Let ¯hθ : Rd →Rk be a parameterized model. We call the outputs ¯hθ(x) ∈Rk the logits. Each logit corresponds to the predic- tion for one of the k classes. Analogous to Section 2.3, we use the softmax function to turn the logits ¯hθ(x) into a probability vector with non-negative entries that sum up to 1: P(y= j \|x; θ) = exp(¯hθ(x)j) ∑k s=1 exp(¯hθ(x)s) , (7.5) where ¯hθ(x)s denotes the s-th coordinate of ¯hθ(x). Similarly to Section 2.3, the loss function for a single training example (x(i),y(i)) is its negative log-likelihood: J(i)(θ) = −log p(y(i) \|x(i); θ) = −log ( exp(¯hθ(x(i))y(i) ) ∑k s=1 exp(¯hθ(x(i))s) ) . (7.6) Using the notations of Section 2.3, we can simply write in an abstract way: J(i)(θ) = ℓce(¯hθ(x(i)),y(i)). (7.7) The loss function is also deﬁned as the average of the loss function of indi- vidual training examples, J(θ) = 1 n ∑n i=1 J(i)(θ). We also note that the approach above can also be generated to any con- ditional probabilistic model where we have an exponential distribution for y, Exponential-family( y; η), where η = ¯hθ(x) is a parameterized nonlinear function of x. However, the most widely used situations are the three cases discussed above. Optimizers (SGD). Commonly, people use gradient descent (GD), stochas- tic gradient (SGD), or their variants to optimize the loss functionJ(θ). GD’s update rule can be written as 2 θ:= θ−α∇θJ(θ) (7.8) where α >0 is often referred to as the learning rate or step size. Next, we introduce a version of the SGD (Algorithm 1), which is lightly diﬀerent from that in the ﬁrst lecture notes. 2Recall that, as deﬁned in the previous lecture notes, we use the notation “ a:= b” to denote an operation (in a computer program) in which we set the value of a variable ato be equal to the value of b. In other words, this operation overwrites a with the value of b. In contrast, we will write “ a= b” when we are asserting a statement of fact, that the value of a is equal to the value of b. 83 Algorithm 1 Stochastic Gradient Descent 1: Hyperparameter: learning rate α, number of total iteration niter. 2: Initialize θ randomly. 3: for i= 1 to niter do 4: Sample j uniformly from {1,...,n }, and update θ by θ:= θ−α∇θJ(j)(θ) (7.9) Oftentimes computing the gradient of B examples simultaneously for the parameter θ can be faster than computing B gradients separately due to hardware parallelization. Therefore, a mini-batch version of SGD is most commonly used in deep learning, as shown in Algorithm 2. There are also other variants of the SGD or mini-batch SGD with slightly diﬀerent sampling schemes. Algorithm 2 Mini-batch Stochastic Gradient Descent 1: Hyperparameters: learning rate α, batch size B, # iterations niter. 2: Initialize θ randomly 3: for i= 1 to niter do 4: Sample B examples j1,...,j B (without replacement) uniformly from {1,...,n }, and update θ by θ:= θ−α B B∑ k=1 ∇θJ(jk)(θ) (7.10) With these generic algorithms, a typical deep learning model is learned with the following steps. 1. Deﬁne a neural network parametrization hθ(x), which we will introduce in Section 7.2, and 2. write the backpropagation algorithm to compute the gradient of the loss function J(j)(θ) eﬃciently, which will be covered in Section 7.4, and 3. run SGD or mini-batch SGD (or other gradient-based optimizers) with the loss function J(θ). 84 7.2 Neural networks Neural networks refer to a broad type of non-linear models/parametrizations ¯hθ(x) that involve combinations of matrix multiplications and other entry- wise non-linear operations. To have a uniﬁed treatment for regression prob- lem and classiﬁcation problem, here we consider ¯hθ(x) as the output of the neural network. For regression problem, the ﬁnal prediction hθ(x) = ¯hθ(x), and for classiﬁcation problem,¯hθ(x) is the logits and the predicted probability will be hθ(x) = 1/(1+exp( −¯hθ(x)) (see equation 7.3) for binary classiﬁcation or hθ(x) = softmax(¯hθ(x)) for multi-class classiﬁcation (see equation 7.5). We will start small and slowly build up a neural network, step by step. A Neural Network with a Single Neuron. Recall the housing price prediction problem from before: given the size of the house, we want to predict the price. We will use it as a running example in this subsection. Previously, we ﬁt a straight line to the graph of size vs. housing price. Now, instead of ﬁtting a straight line, we wish to prevent negative housing prices by setting the absolute minimum price as zero. This produces a “kink” in the graph as shown in Figure 7.1. How do we represent such a function with a single kink as ¯hθ(x) with unknown parameter? (After doing so, we can invoke the machinery in Section 7.1.) We deﬁne a parameterized function ¯hθ(x) with input x, parameterized by θ, which outputs the price of the house y. Formally, ¯hθ : x →y. Perhaps one of the simplest parametrization would be ¯hθ(x) = max(wx+ b,0), where θ= (w,b) ∈R2 (7.11) Here ¯hθ(x) returns a single value: (wx+b) or zero, whichever is greater. In the context of neural networks, the function max{t,0}is called a ReLU (pro- nounced “ray-lu”), or rectiﬁed linear unit, and often denoted by ReLU( t) ≜ max{t,0}. Generally, a one-dimensional non-linear function that mapsR to R such as ReLU is often referred to as anactivation function. The model ¯hθ(x) is said to have a single neuron partly because it has a single non-linear activation function. (We will discuss more about why a non-linear activation is called neuron.) When the input x ∈Rd has multiple dimensions, a neural network with a single neuron can be written as ¯hθ(x) = ReLU(w⊤x+ b), where w∈Rd, b∈R, and θ= (w,b) (7.12) 85 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 100 200 300 400 500 600 700 800 900 1000 housing prices square feet price (in $1000) Figure 7.1: Housing prices with a “kink” in the graph. The term bis often referred to as the “bias”, and the vector w is referred to as the weight vector. Such a neural network has 1 layer. (We will deﬁne what multiple layers mean in the sequel.) Stacking Neurons. A more complex neural network may take the single neuron described above and “stack” them together such that one neuron passes its output as input into the next neuron, resulting in a more complex function. Let us now deepen the housing prediction example. In addition to the size of the house, suppose that you know the number of bedrooms, the zip code and the wealth of the neighborhood. Building neural networks is analogous to Lego bricks: you take individual bricks and stack them together to build complex structures. The same applies to neural networks: we take individual neurons and stack them together to create complex neural networks. Given these features (size, number of bedrooms, zip code, and wealth), we might then decide that the price of the house depends on the maximum family size it can accommodate. Suppose the family size is a function of the size of the house and number of bedrooms (see Figure 7.2). The zip code may provide additional information such as how walkable the neighborhood is (i.e., can you walk to the grocery store or do you need to drive everywhere). Combining the zip code with the wealth of the neighborhood may predict the quality of the local elementary school. Given these three derived features (family size, walkable, school quality), we may conclude that the price of the 86 home ultimately depends on these three features. Family Size School Quality Walkable Size # Bedrooms Zip Code Wealth Price y Figure 7.2: Diagram of a small neural network for predicting housing prices. Formally, the input to a neural network is a set of input features x1,x2,x3,x4. We denote the intermediate variables for “family size”, “walk- able”, and “school quality” by a1,a2,a3 (these ai’s are often referred to as “hidden units” or “hidden neurons”). We represent each of the ai’s as a neu- ral network with a single neuron with a subset of x1,...,x 4 as inputs. Then as in Figure 7.1, we will have the parameterization: a1 = ReLU(θ1x1 + θ2x2 + θ3) a2 = ReLU(θ4x3 + θ5) a3 = ReLU(θ6x3 + θ7x4 + θ8) where (θ1,··· ,θ8) are parameters. Now we represent the ﬁnal output ¯hθ(x) as another linear function with a1,a2,a3 as inputs, and we get 3 ¯hθ(x) = θ9a1 + θ10a2 + θ11a3 + θ12 (7.13) where θ contains all the parameters ( θ1,··· ,θ12). Now we represent the output as a quite complex function of x with pa- rameters θ. Then you can use this parametrization ¯hθ with the machinery of Section 7.1 to learn the parameters θ. Inspiration from Biological Neural Networks. As the name suggests, artiﬁcial neural networks were inspired by biological neural networks. The hidden units a1,...,a m correspond to the neurons in a biological neural net- work, and the parameters θi’s correspond to the synapses. However, it’s unclear how similar the modern deep artiﬁcial neural networks are to the bi- ological ones. For example, perhaps not many neuroscientists think biological 3Typically, for multi-layer neural network, at the end, near the output, we don’t apply ReLU, especially when the output is not necessarily a positive number. 87 neural networks could have 1000 layers, while some modern artiﬁcial neural networks do (we will elaborate more on the notion of layers.) Moreover, it’s an open question whether human brains update their neural networks in a way similar to the way that computer scientists learn artiﬁcial neural net- works (using backpropagation, which we will introduce in the next section.). Two-layer Fully-Connected Neural Networks. We constructed the neural network in equation (7.13) using a signiﬁcant amount of prior knowl- edge/belief about how the “family size”, “walkable”, and “school quality” are determined by the inputs. We implicitly assumed that we know the family size is an important quantity to look at and that it can be determined by only the “size” and “# bedrooms”. Such a prior knowledge might not be available for other applications. It would be more ﬂexible and general to have a generic parameterization. A simple way would be to write the intermediate variable a1 as a function of all x1,...,x 4: a1 = ReLU(w⊤ 1 x+ b1), where w1 ∈R4 and b1 ∈R (7.14) a2 = ReLU(w⊤ 2 x+ b2), where w2 ∈R4 and b2 ∈R a3 = ReLU(w⊤ 3 x+ b3), where w3 ∈R4 and b3 ∈R We still deﬁne ¯hθ(x) using equation (7.13) with a1,a2,a3 being deﬁned as above. Thus we have a so-called fully-connected neural network because all the intermediate variables ai’s depend on all the inputs xi’s. For full generality, a two-layer fully-connected neural network with m hidden units and d dimensional input x∈Rd is deﬁned as ∀j ∈[1,...,m ], z j = w[1] j ⊤ x+ b[1] j where w[1] j ∈Rd,b[1] j ∈R (7.15) aj = ReLU(zj), a= [a1,...,a m]⊤∈Rm ¯hθ(x) = w[2]⊤ a+ b[2] where w[2] ∈Rm,b[2] ∈R, (7.16) Note that by default the vectors in Rd are viewed as column vectors, and in particular ais a column vector with components a1,a2,...,a m. The indices [1] and [2] are used to distinguish two sets of parameters: the w[1] j ’s (each of which is a vector in Rd) and w[2] (which is a vector in Rm). We will have more of these later. Vectorization. Before we introduce neural networks with more layers and more complex structures, we will simplify the expressions for neural networks 88 with more matrix and vector notations. Another important motivation of vectorization is the speed perspective in the implementation. In order to implement a neural network eﬃciently, one must be careful when using for loops. The most natural way to implement equation (7.15) in code is perhaps to use a for loop. In practice, the dimensionalities of the inputs and hidden units are high. As a result, code will run very slowly if you use for loops. Leveraging the parallelism in GPUs is/was crucial for the progress of deep learning. This gave rise to vectorization. Instead of using for loops, vectorization takes advantage of matrix algebra and highly optimized numerical linear algebra packages (e.g., BLAS) to make neural network computations run quickly. Before the deep learning era, a for loop may have been suﬃcient on smaller datasets, but modern deep networks and state-of-the-art datasets will be infeasible to run with for loops. We vectorize the two-layer fully-connected neural network as below. We deﬁne a weight matrix W[1] in Rm×d as the concatenation of all the vectors w[1] j ’s in the following way: W[1] =   — w[1] 1 ⊤ — — w[1] 2 ⊤ — ... — w[1] m ⊤ —   ∈Rm×d (7.17) Now by the deﬁnition of matrix vector multiplication, we can write z = [z1,...,z m]⊤∈Rm as   z1 ... ... zm      z ∈Rm×1 =   — w[1] 1 ⊤ — — w[1] 2 ⊤ — ... — w[1] m ⊤ —      W[1] ∈Rm×d   x1 x2 ... xd      x∈Rd×1 +   b[1] 1 b[1] 2 ... b[1] m      b[1] ∈Rm×1 (7.18) Or succinctly, z = W[1]x+ b[1] (7.19) We remark again that a vector in Rd in this notes, following the conventions previously established, is automatically viewed as a column vector, and can 89 also be viewed as a d×1 dimensional matrix. (Note that this is diﬀerent from numpy where a vector is viewed as a row vector in broadcasting.) Computing the activations a ∈Rm from z ∈Rm involves an element- wise non-linear application of the ReLU function, which can be computed in parallel eﬃciently. Overloading ReLU for element-wise application of ReLU (meaning, for a vector t ∈Rd, ReLU( t) is a vector such that ReLU( t)i = ReLU(ti)), we have a= ReLU(z) (7.20) Deﬁne W[2] = [ w[2]⊤ ] ∈ R1×m similarly. Then, the model in equa- tion (7.16) can be summarized as a= ReLU(W[1]x+ b[1]) ¯hθ(x) = W[2]a+ b[2] (7.21) Here θ consists of W[1],W[2] (often referred to as the weight matrices) and b[1],b[2] (referred to as the biases). The collection of W[1],b[1] is referred to as the ﬁrst layer, andW[2],b[2] the second layer. The activation ais referred to as the hidden layer. A two-layer neural network is also called one-hidden-layer neural network. Multi-layer fully-connected neural networks. With this succinct no- tations, we can stack more layers to get a deeper fully-connected neu- ral network. Let r be the number of layers (weight matrices). Let W[1],...,W [r],b[1],...,b [r] be the weight matrices and biases of all the layers. Then a multi-layer neural network can be written as a[1] = ReLU(W[1]x+ b[1]) a[2] = ReLU(W[2]a[1] + b[2]) ··· a[r−1] = ReLU(W[r−1]a[r−2] + b[r−1]) ¯hθ(x) = W[r]a[r−1] + b[r] (7.22) We note that the weight matrices and biases need to have compatible dimensions for the equations above to make sense. If a[k] has dimension mk, then the weight matrix W[k] should be of dimension mk×mk−1, and the bias b[k] ∈Rmk. Moreover, W[1] ∈Rm1×d and W[r] ∈R1×mr−1 . 90 The total number of neurons in the network is m1 + ··· + mr, and the total number of parameters in this network is (d+ 1)m1 + (m1 + 1)m2 + ···+ (mr−1 + 1)mr. Sometimes for notational consistency we also write a[0] = x, and a[r] = hθ(x). Then we have simple recursion that a[k] = ReLU(W[k]a[k−1] + b[k]),∀k= 1,...,r −1 (7.23) Note that this would have be true for k = r if there were an additional ReLU in equation (7.22), but often people like to make the last layer linear (aka without a ReLU) so that negative outputs are possible and it’s easier to interpret the last layer as a linear model. (More on the interpretability at the “connection to kernel method” paragraph of this section.) Other activation functions. The activation function ReLU can be re- placed by many other non-linear function σ(·) that maps R to R such as σ(z) = 1 1 + e−z (sigmoid) (7.24) σ(z) = ez −e−z ez + e−z (tanh) (7.25) σ(z) = max{z,γz},γ ∈(0,1) (leaky ReLU) (7.26) σ(z) = z 2 [ 1 + erf( z√ 2) ] (GELU) (7.27) σ(z) = 1 β log(1 + exp(βz)),β >0 (Softplus) (7.28) The activation functions are plotted in Figure 7.3. Sigmoid and tanh are less and less used these days partly because their are bounded from both sides and the gradient of them vanishes as z goes to both positive and negative inﬁnity (whereas all the other activation functions still have gradients as the input goes to positive inﬁnity.) Softplus is not used very often either in practice and can be viewed as a smoothing of the ReLU so that it has a proper second order derivative. GELU and leaky ReLU are both variants of ReLU but they have some non-zero gradient even when the input is negative. GELU (or its slight variant) is used in NLP models such as BERT and GPT (which we will discuss in Chapter 14.) Why do we not use the identity function for σ(z)? That is, why not use σ(z) = z? Assume for sake of argument that b[1] and b[2] are zeros. 91 Figure 7.3: Activation functions in deep learning. Suppose σ(z) = z, then for two-layer neural network, we have that ¯hθ(x) = W[2]a[1] (7.29) = W[2]σ(z[1]) by deﬁnition (7.30) = W[2]z[1] since σ(z) = z (7.31) = W[2]W[1]x from Equation (7.18) (7.32) = ˜Wx where ˜W = W[2]W[1] (7.33) Notice how W[2]W[1] collapsed into ˜W. This is because applying a linear function to another linear function will result in a linear function over the original input (i.e., you can construct a ˜W such that ˜Wx = W[2]W[1]x). This loses much of the representational power of the neural network as often times the output we are trying to predict has a non-linear relationship with the inputs. Without non-linear activation functions, the neural network will simply perform linear regression. Connection to the Kernel Method. In the previous lectures, we covered the concept of feature maps. Recall that the main motivation for feature maps is to represent functions that are non-linear in the input x by θ⊤φ(x), where θ are the parameters and φ(x), the feature map, is a handcrafted function non-linear in the raw input x. The performance of the learning algorithms can signiﬁcantly depends on the choice of the feature map φ(x). Oftentimes people use domain knowledge to design the feature mapφ(x) that 92 suits the particular applications. The process of choosing the feature maps is often referred to as feature engineering. We can view deep learning as a way to automatically learn the right feature map (sometimes also referred to as “the representation”) as follows. Suppose we denote by β the collection of the parameters in a fully-connected neural networks (equation (7.22)) except those in the last layer. Then we can abstract right a[r−1] as a function of the input x and the parameters in β: a[r−1] = φβ(x). Now we can write the model as ¯hθ(x) = W[r]φβ(x) + b[r] (7.34) When β is ﬁxed, then φβ(·) can viewed as a feature map, and therefore ¯hθ(x) is just a linear model over the features φβ(x). However, we will train the neural networks, both the parameters in β and the parameters W[r],b[r] are optimized, and therefore we are not learning a linear model in the feature space, but also learning a good feature map φβ(·) itself so that it’s possi- ble to predict accurately with a linear model on top of the feature map. Therefore, deep learning tends to depend less on the domain knowledge of the particular applications and requires often less feature engineering. The penultimate layer a[r] is often (informally) referred to as the learned features or representations in the context of deep learning. In the example of house price prediction, a fully-connected neural network does not need us to specify the intermediate quantity such “family size”, and may automatically discover some useful features in the last penultimate layer (the activation a[r−1]), and use them to linearly predict the housing price. Often the feature map / representation obtained from one datasets (that is, the function φβ(·) can be also useful for other datasets, which indicates they contain essential information about the data. However, oftentimes, the neural network will discover complex features which are very useful for predicting the output but may be diﬃcult for a human to understand or interpret. This is why some people refer to neural networks as a black box , as it can be diﬃcult to understand the features it has discovered. 7.3 Modules in Modern Neural Networks The multi-layer neural network introduced in equation (7.22) of Section 7.2 is often called multi-layer perceptron (MLP) these days. Modern neural net- works used in practice are often much more complex and consist of multiple building blocks or multiple layers of building blocks. In this section, we will 93 introduce some of the other building blocks and discuss possible ways to combine them. First, each matrix multiplication can be viewed as a building block. Con- sider a matrix multiplication operation with parameters ( W,b) where W is the weight matrix and b is the bias vector, operating on an input z, MMW,b(z) = Wz + b. (7.35) Note that we implicitly assume all the dimensions are chosen to be compat- ible. We will also drop the subscripts under MM when they are clear in the context or just for convenience when they are not essential to the discussion. Then, the MLP can be written as as a composition of multiple matrix multiplication modules and nonlinear activation modules (which can also be viewed as a building block): MLP(x) = MMW[r],b[r] (σ(MMW[r−1],b[r−1] (σ(···MMW[1],b[1] (x)))). (7.36) Alternatively, when we drop the subscripts that indicate the parameters for convenience, we can write MLP(x) = MM(σ(MMσ(···MM(x)))). (7.37) Note that in this lecture notes, by default, all the modules have diﬀerent sets of parameters, and the dimensions of the parameters are chosen such that the composition is meaningful. Larger modules can be deﬁned via smaller modules as well, e.g., one activation layer σ and a matrix multiplication layer MM are often combined and called a “layer” in many papers. People often draw the architecture with the basic modules in a ﬁgure by indicating the dependency between these modules. E.g., see an illustration of an MLP in Figure 7.4, Left. Residual connections. One of the very inﬂuential neural network archi- tecture for vision application is ResNet, which uses the residual connections that are essentially used in almost all large-scale deep learning architectures these days. Using our notation above, a very much simpliﬁed residual block can be deﬁned as Res(z) = z+ σ(MM(σ(MM(z)))). (7.38) A much simpliﬁed ResNet is a composition of many residual blocks followed by a matrix multiplication, ResNet-S(x) = MM(Res(Res(···Res(x)))). (7.39) 94 𝑥 Layer 𝑟−1 Layer 𝑖...Layer 1 MLP(𝑥)... Layer𝑖 MM!["],#["] 𝜎 MM![$],#[$] 𝑥 Res Res...Res ResNet-S(𝑥)...Res MM𝜎MM𝜎 Figure 7.4: Illustrative Figures for Architecture. Left: An MLP with r layers. Right: A residual network. We also draw the dependency of these modules in Figure 7.4, Right. We note that the ResNet-S is still not the same as the ResNet architec- ture introduced in the seminal paper [He et al., 2016] because ResNet uses convolution layers instead of vanilla matrix multiplication, and adds batch normalization between convolutions and activations. We will introduce con- volutional layers and some variants of batch normalization below. ResNet-S and layer normalization are part of the Transformer architecture that are widely used in modern large language models. Layer normalization. Layer normalization, denoted by LN in this text, is a module that maps a vector z ∈Rm to a more normalized vector LN(z) ∈ Rm. It is oftentimes used after the nonlinear activations. We ﬁrst deﬁne a sub-module of the layer normalization, denoted by LN-S. LN-S(z) =   z1−ˆµ ˆσz2−ˆµ ˆσ ... zm−ˆµ ˆσ  , (7.40) where ˆµ= ∑m i=1 zi m is the empirical mean of the vector zand ˆσ= √∑m i=1(zi−ˆµ2) m is the empirical standard deviation of the entries of z.4 Intuitively, LN-S(z) is a vector that is normalized to having empirical mean zero and empirical standard deviation 1. 4Note that we divide by m instead of m−1 in the empirical standard deviation here because we are interested in making the output of LN-S( z) have sum of squares equal to 1 (as opposed to estimating the standard deviation in statistics.) 95 Oftentimes zero mean and standard deviation 1 is not the most desired normalization scheme, and thus layernorm introduces to parameters learnable scalars βand γas the desired mean and standard deviation, and use an aﬃne transformation to turn the output of LN-S(z) into a vector with mean β and standard deviation γ. LN(z) = β+ γ·LN-S(z) =   β+ γ (z1−ˆµ ˆσ ) β+ γ (z2−ˆµ ˆσ ) ... β+ γ (zm−ˆµ ˆσ )  . (7.41) Here the ﬁrst occurrence of β should be technically interpreted as a vector with all the entries being β. in We also note that ˆµand ˆσ are also functions of zand shouldn’t be treated as constants when computing the derivatives of layernorm. Moreover, β and γ are learnable parameters and thus layernorm is a parameterized module (as opposed to the activation layer which doesn’t have any parameters.) Scaling-invariant property. One important property of layer normalization is that it will make the model invariant to scaling of the parameters in the following sense. Suppose we consider composing LN with MM W,b and get a subnetwork LN(MM W,b(z)). Then, we have that the output of this sub- network does not change when the parameter in MM W,b is scaled: LN(MMαW,αb(z)) = LN(MMW,b(z)),∀α> 0. (7.42) To see this, we ﬁrst know that LN-S( ·) is scale-invariant LN-S(αz) =   αz1−αˆµ αˆσαz2−αˆµ αˆσ ... αzm−αˆµ αˆσ  =   z1−ˆµ ˆσz2−ˆµ ˆσ ... zm−ˆµ ˆσ  = LN-S(z). (7.43) Then we have LN(MMαW,αb(z)) = β+ γLN-S(MMαW,αb(z)) (7.44) = β+ γLN-S(αMMW,b(z)) (7.45) = β+ γLN-S(MMW,b(z)) (7.46) = LN(MMW,b(z)). (7.47) Due to this property, most of the modern DL architectures for large-scale computer vision and language applications have the following scale-invariant 96 property w.r.t all the weights that are not at the last layer. Suppose the network f has last layer’ weights Wlast, and all the rest of the weights are denote by W. Then, we have fWlast,αW(x) = fWlast,W(x) for all α >0. Here, the last layers weights are special because there are typically no layernorm or batchnorm after the last layer’s weights. Other normalization layers. There are several other normalization layers that aim to normalize the intermediate layers of the neural networks to a more ﬁxed and controllable scaling, such as batch-normalization [ ?], and group normalization [ ?]. Batch normalization and group normalization are more often used in computer vision applications whereas layer norm is used more often in language applications. Convolutional Layers. Convolutional Neural Networks are neural net- works that consist of convolution layers (and many other modules), and are particularly useful for computer vision applications. For the simplicity of exposition, we focus on 1-D convolution in this text and only brieﬂy mention 2-D convolution informally at the end of this subsection. (2-D convolution is more suitable for images which have two dimensions. 1-D convolution is also used in natural language processing.) We start by introducing a simpliﬁed version of the 1-D convolution layer, denoted by Conv1D-S(·) which is a type of matrix multiplication layer with a special structure. The parameters of Conv1D-S are a ﬁlter vector w ∈Rk where k is called the ﬁlter size (oftentimes k ≪m), and a bias scalar b. Oftentimes the ﬁlter is also called a kernel (but it does not have much to do with the kernel in kernel method.) For simplicity, we assume k = 2ℓ+ 1 is an odd number. We ﬁrst pad zeros to the input vector z in the sense that we let z1−ℓ = z1−ℓ+1 = ..= z0 = 0 and zm+1 = zm+2 = ..= zm+ℓ = 0, and treat z as an (m+ 2ℓ)-dimension vector. Conv1D-S outputs a vector of dimension Rm where each output dimension is a linear combination of subsets of zj’s with coeﬃcients from w, Conv1D-S(z)i = w1zi−ℓ + w2zi−ℓ+1 + ··· + w2ℓ+1zi+ℓ = 2ℓ+1∑ j=1 wjzi−ℓ+(j−1). (7.48) Therefore, one can view Conv1D-S as a matrix multiplication with shared 97 parameters: Conv1D-S(z) = Qz, where Q=   wℓ+1 ··· w2ℓ+1 0 0 ··· ··· ··· ··· ··· ··· 0 wℓ ··· w2ℓ w2ℓ+1 0 ··· ··· ··· ··· ··· ··· 0 ... w1 ··· wℓ+1 ··· ··· ··· w2ℓ+1 0 ··· ··· ··· 0 0 w1 ··· ··· ··· ··· w2ℓ w2ℓ+1 0 ··· ··· 0 ... ... 0 ··· ··· ··· ··· ··· 0 w1 ··· ··· w2ℓ+1 ... 0 ··· ··· ··· ··· ··· ··· ··· 0 w1 ··· wℓ+1   . (7.49) Note that Qi,j = Qi−1,j−1 for all i,j ∈{2,...,m }, and thus convoluation is a matrix multiplication with parameter sharing. We also note that computing the convolution only takes O(km) times but computing a generic matrix multiplication takes O(m2) time. Convolution has k parameters but generic matrix multiplication will havem2 parameters. Thus convolution is supposed to be much more eﬃcient than a generic matrix multiplication (as long as the additional structure imposed does not hurt the ﬂexibility of the model to ﬁt the data). We also note that in practice there are many variants of the convolutional layers that we deﬁne here, e.g., there are other ways to pad zeros or sometimes the dimension of the output of the convolutional layers could be diﬀerent from the input. We omit some of this subtleties here for simplicity. The convolutional layers used in practice have also many “channels” and the simpliﬁed version above corresponds to the 1-channel version. Formally, Conv1D takes in C vectors z1,...,z C ∈Rm as inputs, where C is referred to as the number of channels. In other words, the more general version, denoted by Conv1D, takes in a matrix as input, which is the concatenation of z1,...,z C and has dimension m×C. It can output C′vectors of dimension m, denoted by Conv1D(z)1,..., Conv1D(z)C′, where C′ is referred to as the output channel, or equivalently a matrix of dimension m×C′. Each of the output is a sum of the simpliﬁed convolutions applied on various channels. ∀i∈[C′],Conv1D(z)i = C∑ j=1 Conv1D-Si,j(zj). (7.50) Note that each Conv1D-S i,j are modules with diﬀerent parameters, and thus the total number of parameters is k (the number of parameters in a Conv1D-S) ×CC′ (the number of Conv1D-S i.j’s) = kCC′. In contrast, a generic linear mapping from Rm×C and Rm×C′ has m2CC′ parameters. The 98 parameters can also be represented as a three-dimensional tensor of dimen- sion k×C×C′. 2-D convolution (brief). A 2-D convolution with one channel, denoted by Conv2D-S, is analogous to the Conv1D-S, but takes a 2-dimensional input z ∈Rm×m and applies a ﬁlter of size k ×k, and outputs Conv2D-S( z) ∈ Rm×m. The full 2-D convolutional layer, denoted by Conv2D, takes in a sequence of matrices z1,...,z C ∈ Rm×m, or equivalently a 3-D ten- sor z = ( z1,...,z C) ∈ Rm×m×C and outputs a sequence of matrices, Conv2D(z)1,..., Conv2D(z)C′ ∈Rm×m, which can also be viewed as a 3D tensor in Rm×m×C′ . Each channel of the output is sum of the outcomes of applying Conv2D-S layers on all the input channels. ∀i∈[C′],Conv2D(z)i = C∑ j=1 Conv2D-Si,j(zj). (7.51) Because there are CC′ number of Conv2D-S modules and each of the Conv2D-S module has k2 parameters, the total number of parameters is CC′k2. The parameters can also be viewed as a 4D tensor of dimension C×C′×k×k. 7.4 Backpropagation In this section, we introduce backpropgation or auto-diﬀerentiation, which computes the gradient of the loss ∇J(θ) eﬃciently. We will start with an informal theorem that states that as long as a real-valued function f can be eﬃciently computed/evaluated by a diﬀerentiable network or circuit, then its gradient can be eﬃciently computed in a similar time. We will then show how to do this concretely for neural networks. Because the formality of the general theorem is not the main focus here, we will introduce the terms with informal deﬁnitions. By a diﬀerentiable circuit or a diﬀerentiable network, we mean a composition of a sequence of diﬀerentiable arithmetic operations (additions, subtraction, multiplication, divisions, etc) and elementary diﬀerentiable functions (ReLU, exp, log, sin, cos, etc.). Let the size of the circuit be the total number of such operations and elementary functions. We assume that each of the operations and func- tions, and their derivatives or partial derivatives ecan be computed in O(1) time. Theorem 7.4.1: [backpropagation or auto-diﬀerentiation, informally stated] Suppose a diﬀerentiable circuit of size N computes a real-valued function 99 f : Rℓ →R. Then, the gradient ∇f can be computed in time O(N), by a circuit of size O(N).5 We note that the loss function J(j)(θ) for j-th example can be indeed computed by a sequence of operations and functions involving additions, subtraction, multiplications, and non-linear activations. Thus the theorem suggests that we should be able to compute the ∇J(j)(θ) in a similar time to that for computing J(j)(θ) itself. This does not only apply to the fully- connected neural network introduced in the Section 7.2, but also many other types of neural networks that uses more advance modules. We remark that auto-diﬀerentiation or backpropagation is already imple- mented in all the deep learning packages such as tensorﬂow and pytorch, and thus in practice, in most of cases a researcher does not need to write their backpropagation algorithms. However, understanding it is very helpful for gaining insights into the working of deep learning. Organization of the rest of the section. In Section 7.4.1, we will start review- ing the basic Chain rule with a new perspective that is particularly useful for understanding backpropgation. Section 7.4.2 will introduce the general strategy for backpropagation. Section 7.4.2 will discuss how to compute the so-called backward function for basic modules used in neural networks, and Section 7.4.4 will put everything together to get a concrete backprop algo- rithm for MLPs. 7.4.1 Preliminaries on partial derivatives Suppose a scalar variable J depend on some variables z (which could be a scalar, matrix, or high-order tensor), we write ∂J ∂z as the partial derivatives of J w.r.t to the variable z. We stress that the convention here is that ∂J ∂z has exactly the same dimension as z itself. For example, if z ∈Rm×n, then ∂J ∂z ∈Rm×n, and the ( i,j)-entry of ∂J ∂z is equal to ∂J ∂zij . Remark 7.4.2 :When both J and zare not scalars, the partial derivatives of J w.r.t z becomes either a matrix or tensor and the notation becomes some- what tricky. Besides the mathematical or notational challenges in dealing 5We note if the output of the function f does not depend on some of the input co- ordinates, then we set by default the gradient w.r.t that coordinate to zero. Setting to zero does not count towards the total runtime here in our accounting scheme. This is why when N ≤ℓ, we can compute the gradient in O(N) time, which might be potentially even less than ℓ. 100 with these partial derivatives of multi-variate functions, they are also expen- sive to compute and store, and thus rarely explicitly constructed empirically. The experience of authors of this note is that it’s generally more productive to think only about derivatives of scalar function w.r.t to vector, matrices, or tensors. For example, in this note, we will not deal with derivatives of multi-variate functions. Chain rule. We review the chain rule in calculus but with a perspective and notions that are more relevant for auto-diﬀerentiation. Consider a scalar variable J which is obtained by the composition of f and g on some variable z, z ∈Rm u= g(z) ∈Rn J = f(u) ∈R. (7.52) The same derivations below can be easily extend to the cases when z and u are matrices or tensors; but we insist that the ﬁnal variableJ is a scalar. (See also Remark 7.4.2.) Let u = (u1,...,u n) and let g(z) = ( g1(z),··· ,gn(z)). Then, the standard chain rule gives us that ∀i∈{1,...,m }, ∂J ∂zi = n∑ j=1 ∂J ∂uj ·∂gj ∂zi . (7.53) Alternatively, when z and u are both vectors, in a vectorized notation: ∂J ∂z =   ∂g1 ∂z1 ··· ∂gn ∂z1 ... ... ... ∂g1 ∂zm ··· ∂gn ∂zm  ·∂J ∂u . (7.54) In other words, the backward function is always a linear map from ∂J ∂u to ∂J ∂z, though note that the mapping itself can depend on z in complex ways. The matrix on the RHS of (7.54) is actually the transpose of the Jacobian matrix of the functiong. However, we do not discuss in-depth about Jacobian matrices to avoid complications. Part of the reason is that when zis a matrix (or tensor), to write an analog of equation (7.54), one has to either ﬂatten z into a vector or introduce additional notations on tensor-matrix product. In this sense, equation (7.53) is more convenient and eﬀective to use in all cases. For example, whenz ∈Rr×s is a matrix, we can easily rewrite equation (7.53) 101 to ∀i,k, ∂J ∂zik = n∑ j=1 ∂J ∂uj ·∂gj ∂zik . (7.55) which will indeed be used in some of the derivations in Section 7.4.3. Key interpretation of the chain rule. We can view the formula above (equa- tion (7.53) or (7.54)) as a way to compute ∂J ∂z from ∂J ∂u. Consider the following abstract problem. Suppose Jdepends on zvia uas deﬁned in equation (7.52). However, suppose the function f is not given or the function f is complex, but we are given the value of ∂J ∂u. Then, the formula in equation (7.54) gives us a way to compute ∂J ∂z from ∂J ∂u. ∂J ∂u chain rule, formula (7.54) = = = = = = = = = = = = = = = = = = = =⇒ only requires info about g(·) and z ∂J ∂z . (7.56) Moreover, this formula only involves knowledge aboutg(more precisely ∂gj ∂zi ). We will repeatedly use this fact in situations where g is a building blocks of a complex network f. Empirically, it’s often useful to modularized the mapping in (7.53) or (7.54) into a black-box, and mathematically it’s also convenient to deﬁne a notation for it. 6 We use B[g,z] to deﬁne the function that maps ∂J ∂u to ∂J ∂z, and write ∂J ∂z = B[g,z] (∂J ∂u ) . (7.57) We call B[g,z] the backward function for the module g. Note that when z is ﬁxed, B[g,z] is merely a linear map from Rn to Rm. Using equation (7.53), we have (B[g,z](v))i = m∑ j=1 ∂gj ∂zi ·vj . (7.58) Or in vectorized notation, using (7.54), we have B[g,z](v) =   ∂g1 ∂z1 ··· ∂gn ∂z1 ... ... ... ∂g1 ∂zm ··· ∂gn ∂zm  ·v. (7.59) 6e.g., the function is the .backward() method of the module in pytorch. 102 and therefore B[g,z] can be viewed as a matrix. However, in reality, zwill be changing and thus the backward mapping has to be recomputed for diﬀerent z’s while g is often ﬁxed. Thus, empirically, the backward function B[g,z](v) is often viewed as a function which takes in z (=the input to g) and v (=a vector that is supposed to be the gradient of some variable J w.r.t to the output of g) as the inputs, and outputs a vector that is supposed to be the gradient of J w.r.t to z. 7.4.2 General strategy of backpropagation We discuss the general strategy of auto-diﬀerentiation in this section to build a high-level understanding. Then, we will instantiate the approach to con- crete neural networks. We take the viewpoint that neural networks are com- plex compositions of small building blocks such as MM, σ, Conv2D, LN, etc., deﬁned in Section 7.3. Note that the losses (e.g., mean-squared loss, or the cross-entropy loss) can also be abstractly viewed as additional modules. Thus, we can abstractly write the loss function J (on a single example (x,y)) as a composition of many modules: 7 J = Mk(Mk−1(···M1(x))) . (7.60) For example, for a binary classiﬁcation problem with a MLP ¯hθ(x) (de- ﬁned in equation (7.36) and (7.37)), the loss function has ber written in the form of equation (7.60) with M1 = MMW[1],b[1] , M2 = σ, M3 = MMW[2],b[2] , ... , and Mk−1 = MMW[r],b[r] and Mk = ℓlogistic. We can see from this example that some modules involve parameters, and other modules might only involve a ﬁxed set of operations. For generality, we assume that eachj Mi involves a set of parameters θ[i], though θ[i] could possibly be an empty set when Mi is a ﬁxed operation such as the nonlinear activations. We will discuss more on the granularity of the modularization, but so far we assume all the modules Mi’s are simple enough. We introduce the intermediate variables for the computation in (7.60). 7Technically, we should write J = Mk(Mk−1(···M1(x)),y). However, y is treated as a constant for the purpose of computing the derivatives w.r.t to the parameters, and thus we can view it as part of Mk for the sake of simplicity of notations. 103 Let u[0] = x u[1] = M1(u[0]) u[2] = M2(u[1]) ... J = u[k] = Mk(u[k−1]) . (F) Backpropgation consists of two passes, the forward pass and backward pass. In the forward pass, the algorithm simply computes u[1],...,u [k] from i = 1,...,k , sequentially using the deﬁnition in (F), and save all the in- termediate variables u[i]’s in the memory. In the backward pass , we ﬁrst compute the derivatives w.r.t to the intermediate variables, that is, ∂J ∂u[k] ,..., ∂J ∂u[1] , sequentially in this backward order, and then compute the derivatives of the parameters ∂J ∂θ[i] from ∂J ∂u[i] and u[i−1]. These two type of computations can be also interleaved with each other because ∂J ∂θ[i] only depends on ∂J ∂u[i] and u[i−1] but not any ∂J ∂u[k] with k <i. We ﬁrst see why ∂J ∂u[i−1] can be computed eﬃciently from ∂J ∂u[i] and u[i−1] by invoking the discussion in Section 7.4.1 on the chain rule. We in- stantiate the discussion by setting u = u[i] and z = u[i−1], and f(u) = Mk(Mk−1(···Mi+1(u[i]))), and g(·) = Mi(·). Note that f is very complex but we don’t need any concrete information about f. Then, the conclusive equation (7.56) corresponds to ∂J ∂u[i] chain rule = = = = = = = = = = = = = = = = = = = = = = = = = =⇒ only requires info aboutMi(·) and u[i−1] ∂J ∂u[i−1] . (7.61) More precisely, we can write, following equation (7.57) ∂J ∂u[i−1] = B[Mi,u[i−1]] ( ∂J ∂u[i] ) . (B1) Instantiating the chain rule with z = θ[i] and u= u[i], we also have ∂J ∂θ[i] = B[Mi,θ[i]] ( ∂J ∂u[i] ) . (B2) See Figure 7.5 for an illustration of the algorithm. 104 𝑥 ... 𝑀! 𝐽...𝑀" 𝑢[!] 𝑢["%!] 𝜕𝐽𝜕𝐽ℬ[𝑀",𝑢["%!]]𝜕𝐽𝜕𝑢!"# 𝑢[&] 𝑀& 𝑢[&%!] ℬ[𝑀&,𝑢[&%!]]𝜕𝐽𝜕𝑢$"# 𝜕𝐽𝜕𝑢$ ... 𝜕𝐽𝜕𝑢# ... Forward pass Backward pass ℬ[𝑀!,𝜃!]𝜕𝐽𝜕𝜃# ℬ[𝑀&,𝜃&]𝜕𝐽𝜕𝜃$ ℬ[𝑀",𝜃"]𝜕𝐽𝜕𝜃! Figure 7.5: Back-propagation. Remark 7.4.3: [Computational eﬃciency and granularity of the modules] The main underlying purpose of treating a complex network as compositions of small modules is that small modules tend to have eﬃciently implementable backward function. In fact, the backward functions of all the atomic modules such as addition, multiplication and ReLU can be computed as eﬃciently as the the evaluation of these modules (up to multiplicative constant factor). Using this fact, we can prove Theorem 7.4.1 by viewing neural networks as compositions of many atomic operations, and invoking the backpropagation discussed above. However, in practice, it’s oftentimes more convenient to modularize the networks using modules on the level of matrix multiplication, layernorm, etc. As we will see, naive implementation of these operations’ backward functions also have the same runtime as the evaluation of these functions. 105 7.4.3 Backward functions for basic modules Using the general strategy in Section 7.4.2, it suﬃces to compute the back- ward function for all modules Mi’s used in the networks. We compute the backward function for the basic module MM, activationsσ, and loss functions in this section. Backward function for MM. Suppose MMW,b(z) = Wz + bis a matrix multi- plication module where z ∈Rm and W ∈Rn×m. Then, using equation (7.59), we have for v∈Rn B[MM,z](v) =   ∂(Wz+b)1 ∂z1 ··· ∂(Wz+b)n ∂z1 ... ... ... ∂(Wz+b)1 ∂zm ··· ∂(Wz+b)n ∂zm  v. (7.62) Using the fact that ∀i ∈[m],j ∈[n],∂(Wz+b)j ∂zi = ∂bj+∑m k=1 Wjkzk ∂zi = Wji, we have B[MM,z](v) = W⊤v∈Rm. (7.63) In the derivation above, we have treated MM as a function of z. If we treat MM as a function of W and b, then we can also compute the backward function for the parameter variables W and b. It’s less convenient to use equation (7.59) because the variable W is a matrix and the matrix in (7.59) will be a 4-th order tensor that is challenging for us to mathematically write down. We use (7.58) instead: (B[MM,W](v))ij = m∑ k=1 ∂(Wz + b)k ∂Wij ·vk = m∑ k=1 ∂∑m s=1 Wkszs ∂Wij ·vk = vizj . (7.64) In vectorized notation, we have B[MM,W](v) = vz⊤∈Rn××m. (7.65) Using equation (7.59) for the variable b, we have, B[MM,b](v) =   ∂(Wz+b)1 ∂b1 ··· ∂(Wz+b)n ∂b1 ... ... ... ∂(Wz+b)1 ∂bn ··· ∂(Wz+b)n ∂bn  v= v. (7.66) 106 Here we used that ∂(Wz+b)j ∂bi = 0 if i̸= j and ∂(Wz+b)j ∂bi = 1 if i= j. The computational eﬃciency for computing the backward function is O(mn), the same as evaluating the result of matrix multiplication up to constant factor. Backward function for the activations. Suppose M(z) = σ(z) where σ is an element-wise activation function and z ∈Rm. Then, using equation (7.59), we have B[σ,z](v) =   ∂σ(z1) ∂z1 ··· ∂σ(zm) ∂z1 ... ... ... ∂σ(z1) ∂zm ··· ∂σ(zm) ∂zm  v (7.67) = diag(σ′(z1),··· ,σ′(zm))v (7.68) = σ′(z) ⊙v∈Rm. (7.69) Here, we used the fact that ∂σ(zj) ∂zi = 0 when j ̸= i, diag(λ1,...,λ m) denotes the diagonal matrix with λ1,...,λ m on the diagonal, and ⊙denotes the element-wise product of two vectors with the same dimension, and σ′(·) is the element-wise application of the derivative of the activation function σ. Regarding computation eﬃciency, we note that at the ﬁrst sight, equa- tion (7.67) appears to indicate the backward function takes O(m2) time, but equation (7.69) shows that it’s implementable in O(m) time (which is the same as the time for evaluating of the function.) We are not supposed to be surprised by that the possibility of simplifying equation (7.67) to (7.69)—if we use smaller modules, that is, treating the vector-to-vector nonlinear ac- tivation as m scalar-to-scalar non-linear activation, then it’s more obvious that the backward pass should have similar time to the forward pass. Backward function for loss functions. When a module M takes in a vector z and outputs a scalar, by equation (7.59), the backward function takes in a scalar v and outputs a vector with entries ( B[M,z](v))i = ∂M ∂zi v. Therefore, in vectorized notation, B[M,z](v) = ∂M ∂z ·v. Recall that squared loss ℓMSE(z,y) = 1 2 (z −y)2. Thus, B[ℓMSE,z](v) = ∂1 2 (z−y)2 ∂z ·v= (z−y) ·v. For logistics loss, by equation (2.6), we have B[ℓlogistic,t](v) = ∂ℓlogistic(t,y) ∂t ·v= (1/(1 + exp(−t)) −y) ·v. (7.70) 107 For cross-entropy loss, by equation (2.17), we have B[ℓce,t](v) = ∂ℓce(t,y) ∂t ·v= (φ−ey) ·v, (7.71) where φ= softmax(t). 7.4.4 Back-propagation for MLPs Given the backward functions for every module needed in evaluating the loss of an MLP, we follow the strategy in Section 7.4.2 to compute the gradient of the loss w.r.t to the hidden activations and the parameters. We consider the an r-layer MLP with a logistic loss. The loss function can be computed via a sequence of operations (that is, the forward pass), z[1] = MMW[1],b[1] (x), a[1] = σ(z[1]) z[2] = MMW[2],b[2] (a[1]) a[2] = σ(z[2]) ... z[r] = MMW[r],b[r] (a[r−1]) J = ℓlogistic(z[r],y) . (7.72) We apply the backward function sequentially in a backward order. First, we have that ∂J ∂z[r] = B[ℓlogistic,z[r]] (∂J ∂J ) = B[ℓlogistic,z[r]](1) . (7.73) Then, we iteratively compute ∂J ∂a[i] and ∂J ∂z[i] ’s by repeatedly invoking the chain rule (equation (7.58)), ∂J ∂a[r−1] = B[MM,a[r−1]] ( ∂J ∂z[r] ) ∂J ∂z[r−1] = B[σ,z[r−1]] ( ∂J ∂a[r−1] ) ... ∂J ∂z[1] = B[σ,z[1]] ( ∂J ∂a[1] ) . (7.74) 108 Numerically, we compute these quantities by repeatedly invoking equa- tions (7.69) and (7.63) with diﬀerent choices of variables. We note that the intermediate values of a[i] and z[i] are used in the back- propagation (equation (7.74)), and therefore these values need to be stored in the memory after the forward pass. Next, we compute the gradient of the parameters by invoking equa- tions (7.65) and (7.66), ∂J ∂W[r] = B[MM,W[r]] ( ∂J ∂z[r] ) ∂J ∂b[r] = B[MM,b[r]] ( ∂J ∂z[r] ) ... ∂J ∂W[1] = B[MM,W[1]] ( ∂J ∂z[1] ) ∂J ∂b[1] = B[MM,b[1]] ( ∂J ∂z[1] ) . (7.75) We also note that the block of computations in equations (7.75) can be interleaved with the block of computation in equations (7.74) because the ∂J ∂W[i] and ∂J ∂b[i] can be computed as soon as ∂J ∂z[i] is computed. Putting all of these together, and explicitly invoking the equa- tions (7.72), (7.74) and (7.75), we have the following algorithm (Algorithm 3). 109 Algorithm 3 Back-propagation for multi-layer neural networks. 1: Forward pass. Compute and store the values of a[k]’s, z[k]’s, and J using the equations (7.72). 2: Backward pass. Compute the gradient of loss J with respect to z[r]: ∂J ∂z[r] = B[ℓlogistic,z[r]](1) = ( 1/(1 + exp(−z[r])) −y ) . (7.76) 3: for k= r−1 to 0 do 4: Compute the gradient with respect to parameters W[k+1] and b[k+1]. ∂J ∂W[k+1] = B[MM,W[k+1]] ( ∂J ∂z[k+1] ) = ∂J ∂z[k+1] a[k]⊤ . (7.77) ∂J ∂b[k+1] = B[MM,b[k+1]] ( ∂J ∂z[k+1] ) = ∂J ∂z[k+1] . (7.78) 5: When k≥1, compute the gradient with respect to z[k] and a[k]. ∂J ∂a[k] = B[σ,a[k]] ( ∂J ∂z[k+1] ) = W[k+1]⊤ ∂J ∂z[k+1] . (7.79) ∂J ∂z[k] = B[σ,z[k]] ( ∂J ∂a[k] ) = σ′(z[k]) ⊙ ∂J ∂a[k] . (7.80) 7.5 Vectorization over training examples As we discussed in Section 7.1, in the implementation of neural networks, we will leverage the parallelism across the multiple examples. This means that we will need to write the forward pass (the evaluation of the outputs) of the neural network and the backward pass (backpropagation) for multiple 110 training examples in matrix notation. The basic idea. The basic idea is simple. Suppose you have a training set with three examples x(1),x(2),x(3). The ﬁrst-layer activations for each example are as follows: z[1](1) = W[1]x(1) + b[1] z[1](2) = W[1]x(2) + b[1] z[1](3) = W[1]x(3) + b[1] Note the diﬀerence between square brackets [·], which refer to the layer num- ber, and parenthesis ( ·), which refer to the training example number. In- tuitively, one would implement this using a for loop. It turns out, we can vectorize these operations as well. First, deﬁne: X =   \| \| \| x(1) x(2) x(3) \| \| \|  ∈Rd×3 (7.81) Note that we are stacking training examples in columns and not rows. We can then combine this into a single uniﬁed formulation: Z[1] =   \| \| \| z[1](1) z[1](2) z[1](3) \| \| \|  = W[1]X+ b[1] (7.82) You may notice that we are attempting to add b[1] ∈ R4×1 to W[1]X ∈ R4×3. Strictly following the rules of linear algebra, this is not allowed. In practice however, this addition is performed using broadcasting. We create an intermediate ˜b[1] ∈R4×3: ˜b[1] =   \| \| \| b[1] b[1] b[1] \| \| \|   (7.83) We can then perform the computation: Z[1] = W[1]X+ ˜b[1]. Often times, it is not necessary to explicitly construct ˜b[1]. By inspecting the dimensions in (7.82), you can assume b[1] ∈R4×1 is correctly broadcast to W[1]X ∈R4×3. The matricization approach as above can easily generalize to multiple layers, with one subtlety though, as discussed below. 111 Complications/Subtlety in the Implementation. All the deep learn- ing packages or implementations put the data points in the rows of a data matrix. (If the data point itself is a matrix or tensor, then the data are con- centrated along the zero-th dimension.) However, most of the deep learning papers use a similar notation to these notes where the data points are treated as column vectors.8 There is a simple conversion to deal with the mismatch: in the implementation, all the columns become row vectors, row vectors be- come column vectors, all the matrices are transposed, and the orders of the matrix multiplications are ﬂipped. In the example above, using the row ma- jor convention, the data matrix is X ∈R3×d, the ﬁrst layer weight matrix has dimensionality d×m (instead of m×d as in the two layer neural net section), and the bias vector b[1] ∈R1×m. The computation for the hidden activation becomes Z[1] = XW[1] + b[1] ∈R3×m (7.84) 8The instructor suspects that this is mostly because in mathematics we naturally mul- tiply a matrix to a vector on the left hand side. Part III Generalization and regularization 112 Chapter 8 Generalization This chapter discusses tools to analyze and understand the generaliza- tion of machine learning models, i.e, their performances on unseen test examples. Recall that for supervised learning problems, given a train- ing dataset {(x(i),y(i))}n i=1, we typically learn a model hθ by minimizing a loss/cost function J(θ), which encourages hθ to ﬁt the data. E.g., when the loss function is the least square loss (aka mean squared error), we have J(θ) = 1 n ∑n i=1(y(i) −hθ(x(i)))2. This loss function for training purposes is oftentimes referred to as the training loss/error/cost. However, minimizing the training loss is not our ultimate goal—it is merely our approach towards the goal of learning a predictive model. The most important evaluation metric of a model is the loss on unseen test exam- ples, which is oftentimes referred to as the test error. Formally, we sample a test example ( x,y) from the so-called test distribution D, and measure the model’s error on it, by, e.g., the mean squared error, ( hθ(x) −y)2. The ex- pected loss/error over the randomness of the test example is called the test loss/error,1 L(θ) = E(x,y)∼D[(y−hθ(x))2] (8.1) Note that the measurement of the error involves computing the expectation, and in practice, it can be approximated by the average error on many sampled test examples, which are referred to as the test dataset. Note that the key diﬀerence here between training and test datasets is that the test examples 1In theoretical and statistical literature, we oftentimes call the uniform distribution over the training set {(x(i),y(i))}n i=1, denoted by ˆD, an empirical distribution, and call Dthe population distribution. Partly because of this, the training loss is also referred to as the empirical loss/risk/error, and the test loss is also referred to as the population loss/risk/error. 113 114 are unseen, in the sense that the training procedure has not used the test examples. In classical statistical learning settings, the training examples are also drawn from the same distribution as the test distribution D, but still the test examples are unseen by the learning procedure whereas the training examples are seen.2 Because of this key diﬀerence between training and test datasets, even if they are both drawn from the same distribution D, the test error is not necessarily always close to the training error. 3 As a result, successfully min- imizing the training error may not always lead to a small test error. We typically say the model overﬁts the data if the model predicts accurately on the training dataset but doesn’t generalize well to other test examples, that is, if the training error is small but the test error is large. We say the model underﬁts the data if the training error is relatively large 4 (and in this case, typically the test error is also relatively large.) This chapter studies how the test error is inﬂuenced by the learning pro- cedure, especially the choice of model parameterizations. We will decompose the test error into “bias” and “variance” terms and study how each of them is aﬀected by the choice of model parameterizations and their tradeoﬀs. Using the bias-variance tradeoﬀ, we will discuss when overﬁtting and underﬁtting will occur and be avoided. We will also discuss the double descent phe- nomenon in Section 8.2 and some classical theoretical results in Section 8.3. 2These days, researchers have increasingly been more interested in the setting with “domain shift”, that is, the training distribution and test distribution are diﬀerent. 3the diﬀerence between test error and training error is often referred to as the gener- alization gap. The term generalization error in some literature means the test error, and in some other literature means the generalization gap. 4e.g., larger than the intrinsic noise level of the data in regression problems. 115 8.1 Bias-variance tradeoﬀ 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y training dataset training data ground truth h * 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y test dataset test data ground truth h * Figure 8.1: A running example of training and test dataset for this section. As an illustrating example, we consider the following training dataset and test dataset, which are also shown in Figure 8.1. The training inputsx(i)’s are randomly chosen and the outputs y(i) are generated by y(i) = h⋆(x(i)) + ξ(i) where the function h⋆(·) is a quadratic function and is shown in Figure 8.1 as the solid line, and ξ(i) is the a observation noise assumed to be generated from ∼ N(0,σ2). A test example ( x,y) also has the same input-output relationship y= h⋆(x) +ξ where ξ ∼N(0,σ2). It’s impossible to predict the noise ξ, and therefore essentially our goal is to recover the function h⋆(·). We will consider the test error of learning various types of models. When talking about linear regression, we discussed the problem of whether to ﬁt a “simple” model such as the linear “ y = θ0 + θ1x,” or a more “complex” model such as the polynomial “ y= θ0 + θ1x+ ···θ5x5.” We start with ﬁtting a linear model, as shown in Figure 8.2. The best ﬁtted linear model cannot predict y from x accurately even on the training dataset, let alone on the test dataset. This is because the true relationship between y and x is not linear—any linear model is far away from the true function h⋆(·). As a result, the training error is large and this is a typical situation of underﬁtting. 116 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y training data best fit linear model 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y test data best fit linear model Figure 8.2: The best ﬁt linear model has large training and test errors. The issue cannot be mitigated with more training examples—even with a very large amount of, or even inﬁnite training examples, the best ﬁtted linear model is still inaccurate and fails to capture the structure of the data (Figure 8.3). Even if the noise is not present in the training data, the issue still occurs (Figure 8.4). Therefore, the fundamental bottleneck here is the linear model family’s inability to capture the structure in the data—linear models cannot represent the true quadratic function h⋆—, but not the lack of the data. Informally, we deﬁne the bias of a model to be the test error even if we were to ﬁt it to a very (say, inﬁnitely) large training dataset. Thus, in this case, the linear model suﬀers from large bias, and underﬁts (i.e., fails to capture structure exhibited by) the data. 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y fitting linear models on a large dataset training data ground truth h * best fit linear model Figure 8.3: The best ﬁt linear model on a much larger dataset still has a large training error. 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y fitting linear models on a noiseless dataset training data ground truth h * best fit linear model Figure 8.4: The best ﬁt linear model on a noiseless dataset also has a large training/test error. Next, we ﬁt a 5th-degree polynomial to the data. Figure 8.5 shows that it fails to learn a good model either. However, the failure pattern is diﬀerent from the linear model case. Speciﬁcally, even though the learnt 5th-degree 117 polynomial did a very good job predicting y(i)’s from x(i)’s for training ex- amples, it does not work well on test examples (Figure 8.5). In other words, the model learnt from the training set does not generalize well to other test examples—the test error is high. Contrary to the behavior of linear models, the bias of the 5-th degree polynomials is small—if we were to ﬁt a 5-th de- gree polynomial to an extremely large dataset, the resulting model would be close to a quadratic function and be accurate (Figure 8.6). This is because the family of 5-th degree polynomials contains all the quadratic functions (setting θ5 = θ4 = θ3 = 0 results in a quadratic function), and, therefore, 5-th degree polynomials are in principle capable of capturing the structure of the data. 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y training data best fit 5-th degree model 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y test data ground truth h * best fit 5-th degree model Figure 8.5: Best ﬁt 5-th degree polynomial has zero training error, but still has a large test error and does not recover the the ground truth. This is a classic situation of overﬁtting. 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y training data best fit 5-th degree model ground truth h * fitting 5-th degree model on large dataset Figure 8.6: The best ﬁt 5-th degree polynomial on a huge dataset nearly recovers the ground-truth—suggesting that the culprit in Figure 8.5 is the variance (or lack of data) but not bias. The failure of ﬁtting 5-th degree polynomials can be captured by another 118 component of the test error, called variance of a model ﬁtting procedure. Speciﬁcally, when ﬁtting a 5-th degree polynomial as in Figure 8.7, there is a large risk that we’re ﬁtting patterns in the data that happened to be present in our small, ﬁnite training set, but that do not reﬂect the wider pattern of the relationship between x and y. These “spurious” patterns in the training set are (mostly) due to the observation noise ξ(i), and ﬁtting these spurious patters results in a model with large test error. In this case, we say the model has a large variance. 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y training data best fit 5-th degree model 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y training data best fit 5-th degree model 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y training data best fit 5-th degree model fitting 5-th degree model on different datasets Figure 8.7: The best ﬁt 5-th degree models on three diﬀerent datasets gen- erated from the same distribution behave quite diﬀerently, suggesting the existence of a large variance. The variance can be intuitively (and mathematically, as shown in Sec- tion 8.1.1) characterized by the amount of variations across models learnt on multiple diﬀerent training datasets (drawn from the same underlying dis- tribution). The “spurious patterns” are speciﬁc to the randomness of the noise (and inputs) in a particular dataset, and thus are diﬀerent across mul- tiple training datasets. Therefore, overﬁtting to the “spurious patterns” of multiple datasets should result in very diﬀerent models. Indeed, as shown in Figure 8.7, the models learned on the three diﬀerent training datasets are quite diﬀerent, overﬁtting to the “spurious patterns” of each datasets. Often, there is a tradeoﬀ between bias and variance. If our model is too “simple” and has very few parameters, then it may have large bias (but small variance), and it typically may suﬀer from underﬁttng. If it is too “complex” and has very many parameters, then it may suﬀer from large variance (but have smaller bias), and thus overﬁtting. See Figure 8.8 for a typical tradeoﬀ between bias and variance. 119 Model Complexity Error Bias2 Variance Test Error (= Bias2 +Variance)Optimal Tradeoff Figure 8.8: An illustration of the typical bias-variance tradeoﬀ. As we will see formally in Section 8.1.1, the test error can be decomposed as a summation of bias and variance. This means that the test error will have a convex curve as the model complexity increases, and in practice we should tune the model complexity to achieve the best tradeoﬀ. For instance, in the example above, ﬁtting a quadratic function does better than either of the extremes of a ﬁrst or a 5-th degree polynomial, as shown in Figure 8.9. 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y training data best fit quadratic model 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5y test data best fit quadratic model ground truth h * Figure 8.9: Best ﬁt quadratic model has small training and test error because quadratic model achieves a better tradeoﬀ. Interestingly, the bias-variance tradeoﬀ curves or the test error curves do not universally follow the shape in Figure 8.8, at least not universally when the model complexity is simply measured by the number of parameters. (We will discuss the so-called double descent phenomenon in Section 8.2.) Nevertheless, the principle of bias-variance tradeoﬀ is perhaps still the ﬁrst resort when analyzing and predicting the behavior of test errors. 120 8.1.1 A mathematical decomposition (for regression) To formally state the bias-variance tradeoﬀ for regression problems, we con- sider the following setup (which is an extension of the beginning paragraph of Section 8.1). • Draw a training dataset S = {x(i),y(i)}n i=1 such that y(i) = h⋆(x(i)) +ξ(i) where ξ(i) ∈N(0,σ2). • Train a model on the dataset S, denoted by ˆhS. • Take a test example (x,y) such that y= h⋆(x) + ξ where ξ ∼N(0,σ2), and measure the expected test error (averaged over the random draw of the training set S and the randomness of ξ)56 MSE(x) = ES,ξ[(y−hS(x))2] (8.2) We will decompose the MSE into a bias and variance term. We start by stating a following simple mathematical tool that will be used twice below. Claim 8.1.1: Suppose A and B are two independent real random variables and E[A] = 0. Then, E[(A+ B)2] = E[A2] + E[B2]. As a corollary, because a random variable A is independent with a con- stant c, when E[A] = 0, we have E[(A+ c)2] = E[A2] + c2. The proof of the claim follows from expanding the square: E[(A+ B)2] = E[A2] +E[B2] + 2E[AB] = E[A2] +E[B2]. Here we used the independence to show that E[AB] = E[A]E[B] = 0. Using Claim 8.1.1 with A= ξ and B = h⋆(x) −ˆhS(x), we have MSE(x) = E[(y−hS(x))2] = E[(ξ+ (h⋆(x) −hS(x)))2] (8.3) = E[ξ2] + E[(h⋆(x) −hS(x))2] (by Claim 8.1.1) = σ2 + E[(h⋆(x) −hS(x))2] (8.4) Then, let’s deﬁne havg(x) = ES[hS(x)] as the “average model”—the model obtained by drawing an inﬁnite number of datasets, training on them, and averaging their predictions on x. Note that havg is a hypothetical model for analytical purposes that can not be obtained in reality (because we don’t 5For simplicity, the test input xis considered to be ﬁxed here, but the same conceptual message holds when we average over the choice of x’s. 6The subscript under the expectation symbol is to emphasize the variables that are considered as random by the expectation operation. 121 have inﬁnite number of datasets). It turns out that for many cases, havg is (approximately) equal to the the model obtained by training on a single dataset with inﬁnite samples. Thus, we can also intuitively interprethavg this way, which is consistent with our intuitive deﬁnition of bias in the previous subsection. We can further decompose MSE(x) by lettingc= h⋆(x)−havg(x) (which is a constant that does not depend on the choice ofS!) and A= havg(x)−hS(x) in the corollary part of Claim 8.1.1: MSE(x) = σ2 + E[(h⋆(x) −hS(x))2] (8.5) = σ2 + (h⋆(x) −havg(x))2 + E[(havg −hS(x))2] (8.6) = σ2  unavoidable + (h⋆(x) −havg(x))2    ≜ bias2 + var(hS(x))   ≜ variance (8.7) We call the second term the bias (square) and the third term the variance. As discussed before, the bias captures the part of the error that are introduced due to the lack of expressivity of the model. Recall that havg can be thought of as the best possible model learned even with inﬁnite data. Thus, the bias is not due to the lack of data, but is rather caused by that the family of models fundamentally cannot approximate the h⋆. For example, in the illustrating example in Figure 8.2, because any linear model cannot approximate the true quadratic function h⋆, neither can havg, and thus the bias term has to be large. The variance term captures how the random nature of the ﬁnite dataset introduces errors in the learned model. It measures the sensitivity of the learned model to the randomness in the dataset. It often decreases as the size of the dataset increases. There is nothing we can do about the ﬁrst term σ2 as we can not predict the noise ξ by deﬁnition. Finally, we note that the bias-variance decomposition for classiﬁcation is much less clear than for regression problems. There have been several proposals, but there is as yet no agreement on what is the “right” and/or the most useful formalism. 8.2 The double descent phenomenon Model-wise double descent. Recent works have demonstrated that the test error can present a “double descent” phenomenon in a range of machine 122 learning models including linear models and deep neural networks. 7 The conventional wisdom, as discussed in Section 8.1, is that as we increase the model complexity, the test error ﬁrst decreases and then increases, as illus- trated in Figure 8.8. However, in many cases, we empirically observe that the test error can have a second descent—it ﬁrst decreases, then increases to a peak around when the model size is large enough to ﬁt all the training data very well, and then decreases again in the so-called overparameterized regime, where the number of parameters is larger than the number of data points. See Figure 8.10 for an illustration of the typical curves of test errors against model complexity (measured by the number of parameters). To some extent, the overparameterized regime with the second descent is considered as new to the machine learning community—partly because lightly-regularized, overparameterized models are only extensively used in the deep learning era. A practical implication of the phenomenon is that one should not hold back from scaling into and experimenting with over-parametrized models because the test error may well decrease again to a level even smaller than the previ- ous lowest point. Actually, in many cases, larger overparameterized models always lead to a better test performance (meaning there won’t be a second ascent after the second descent). # parameters test error typically when # parameters is sufficient to fit the data classical regime: bias-variance tradeoff modern regime: over-parameterization Figure 8.10: A typical model-wise double descent phenomenon. As the num- ber of parameters increases, the test error ﬁrst decreases when the number of parameters is smaller than the training data. Then in the overparameterized regime, the test error decreases again. 7The discovery of the phenomenon perhaps dates back to Opper [1995, 2001], and has been recently popularized by Belkin et al. [2020], Hastie et al. [2019], etc. 123 Sample-wise double descent. A priori, we would expect that more training examples always lead to smaller test errors—more samples give strictly more information for the algorithm to learn from. However, recent work [Nakkiran, 2019] observes that the test error is not monotonically de- creasing as we increase the sample size. Instead, as shown in Figure 8.11, the test error decreases, and then increases and peaks around when the number of examples (denoted by n) is similar to the number of parameters (denoted by d), and then decreases again. We refer to this as the sample-wise dou- ble descent phenomenon. To some extent, sample-wise double descent and model-wise double descent are essentially describing similar phenomena—the test error is peaked when n≈d. Explanation and mitigation strategy. The sample-wise double descent, or, in particular, the peak of test error at n ≈d, suggests that the existing training algorithms evaluated in these experiments are far from optimal when n ≈d. We will be better oﬀ by tossing away some examples and run the algorithms with a smaller sample size to steer clear of the peak. In other words, in principle, there are other algorithms that can achieve smaller test error when n≈d, but the algorithms evaluated in these experiments fail to do so. The sub-optimality of the learning procedure appears to be the culprit of the peak in both sample-wise and model-wise double descent. Indeed, with an optimally-tuned regularization (which will be discussed more in Section 9), the test error in the n ≈d regime can be dramatically improved, and the model-wise and sample-wise double descent are both mit- igated. See Figure 8.11. The intuition above only explains the peak in the model-wise and sample- wise double descent, but does not explain the second descent in the model- wise double descent—why overparameterized models are able to generalize so well. The theoretical understanding of overparameterized models is an ac- tive research area with many recent advances. A typical explanation is that the commonly-used optimizers such as gradient descent provide an implicit regularization eﬀect (which will be discussed in more detail in Section 9.2). In other words, even in the overparameterized regime and with an unregular- ized loss function, the model is still implicitly regularized, and thus exhibits a better test performance than an arbitrary solution that ﬁts the data. For example, for linear models, when n≪d, the gradient descent optimizer with zero initialization ﬁnds the minimum norm solution that ﬁts the data (in- stead of an arbitrary solution that ﬁts the data), and the minimum norm reg- ularizer turns out to be a suﬃciently good for the overparameterized regime (but it’s not a good regularizer when n ≈d, resulting in the peak of test 124 error). 0 200 400 600 800 1000 Num Samples 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 T est Error T est Error vs. # Samples T est Error Figure 8.11: Left: The sample-wise double descent phenomenon for linear models. Right: The sample-wise double descent with diﬀerent regularization strength for linear models. Using the optimal regularization parameter λ (optimally tuned for each n, shown in green solid curve) mitigates double descent. Setup: The data distribution of ( x,y) is x ∼N (0,Id) and y ∼ x⊤β+ N(0,σ2) where d= 500,σ = 0.5 and ∥β∥2 = 1.8 Finally, we also remark that the double descent phenomenon has been mostly observed when the model complexity is measured by the number of parameters. It is unclear if and when the number of parameters is the best complexity measure of a model. For example, in many situations, the norm of the models is used as a complexity measure. As shown in Figure 8.12 right, for a particular linear case, if we plot the test error against the norm of the learnt model, the double descent phenomenon no longer occurs. This is partly because the norm of the learned model is also peaked around n≈d (See Figure 8.12 (middle) or Belkin et al. [2019], Mei and Montanari [2022], and discussions in Section 10.8 of James et al. [2021]). For deep neural networks, the correct complexity measure is even more elusive. The study of double descent phenomenon is an active research topic. 8The ﬁgure is reproduced from Figure 1 of Nakkiran et al. [2020]. Similar phenomenon are also observed in Hastie et al. [2022], Mei and Montanari [2022] 125 0 250 500 750 1000 # parameters 0.0 0.2 0.4 0.6 0.8 1.0 test error test error vs. # params 0 200 400 600 800 1000 # parameters 0 10 20 30 40 norm norm vs. # params 0 10 20 30 40 norm 0.0 0.2 0.4 0.6 0.8 1.0 test error d=n # parameters test error vs. norm 0 200 400 600 800 1000 Figure 8.12: Left: The double descent phenomenon, where the number of pa- rameters is used as the model complexity. Middle: The norm of the learned model is peaked around n ≈d. Right: The test error against the norm of the learnt model. The color bar indicate the number of parameters and the arrows indicates the direction of increasing model size. Their relationship are closer to the convention wisdom than to a double descent. Setup: We consider a linear regression with a ﬁxed dataset of size n = 500. The input x is a random ReLU feature on Fashion-MNIST, and output y ∈R10 is the one-hot label. This is the same setting as in Section 5.2 of Nakkiran et al. [2020]. 126 8.3 Sample complexity bounds (optional readings) 8.3.1 Preliminaries In this set of notes, we begin our foray into learning theory. Apart from being interesting and enlightening in its own right, this discussion will also help us hone our intuitions and derive rules of thumb about how to best apply learning algorithms in diﬀerent settings. We will also seek to answer a few questions: First, can we make formal the bias/variance tradeoﬀ that was just discussed? This will also eventually lead us to talk about model selection methods, which can, for instance, automatically decide what order polynomial to ﬁt to a training set. Second, in machine learning it’s really generalization error that we care about, but most learning algorithms ﬁt their models to the training set. Why should doing well on the training set tell us anything about generalization error? Speciﬁcally, can we relate error on the training set to generalization error? Third and ﬁnally, are there conditions under which we can actually prove that learning algorithms will work well? We start with two simple but very useful lemmas. Lemma. (The union bound). Let A1,A2,...,A k be k diﬀerent events (that may not be independent). Then P(A1 ∪···∪ Ak) ≤P(A1) + ... + P(Ak). In probability theory, the union bound is usually stated as an axiom (and thus we won’t try to prove it), but it also makes intuitive sense: The probability of any one of k events happening is at most the sum of the probabilities of the k diﬀerent events. Lemma. (Hoeﬀding inequality) Let Z1,...,Z n be n independent and iden- tically distributed (iid) random variables drawn from a Bernoulli( φ) distri- bution. I.e., P(Zi = 1) = φ, and P(Zi = 0) = 1 −φ. Let ˆφ= (1/n) ∑n i=1 Zi be the mean of these random variables, and let any γ >0 be ﬁxed. Then P(\|φ−ˆφ\|>γ ) ≤2 exp(−2γ2n) This lemma (which in learning theory is also called theChernoﬀ bound) says that if we take ˆφ—the average of n Bernoulli(φ) random variables—to be our estimate of φ, then the probability of our being far from the true value is small, so long as nis large. Another way of saying this is that if you have a biased coin whose chance of landing on heads is φ, then if you toss it n 127 times and calculate the fraction of times that it came up heads, that will be a good estimate of φ with high probability (if n is large). Using just these two lemmas, we will be able to prove some of the deepest and most important results in learning theory. To simplify our exposition, let’s restrict our attention to binary classiﬁca- tion in which the labels are y∈{0,1}. Everything we’ll say here generalizes to other problems, including regression and multi-class classiﬁcation. We assume we are given a training setS = {(x(i),y(i)); i= 1,...,n }of size n, where the training examples (x(i),y(i)) are drawn iid from some probability distribution D. For a hypothesis h, we deﬁne the training error (also called the empirical risk or empirical error in learning theory) to be ˆε(h) = 1 n n∑ i=1 1{h(x(i)) ̸= y(i)}. This is just the fraction of training examples that h misclassiﬁes. When we want to make explicit the dependence of ˆε(h) on the training set S, we may also write this a ˆεS(h). We also deﬁne the generalization error to be ε(h) = P(x,y)∼D(h(x) ̸= y). I.e. this is the probability that, if we now draw a new example ( x,y) from the distribution D, h will misclassify it. Note that we have assumed that the training data was drawn from the same distribution Dwith which we’re going to evaluate our hypotheses (in the deﬁnition of generalization error). This is sometimes also referred to as one of the PAC assumptions.9 Consider the setting of linear classiﬁcation, and let hθ(x) = 1{θTx≥0}. What’s a reasonable way of ﬁtting the parameters θ? One approach is to try to minimize the training error, and pick ˆθ= arg min θ ˆε(hθ). We call this processempirical risk minimization(ERM), and the resulting hypothesis output by the learning algorithm is ˆh = hˆθ. We think of ERM as the most “basic” learning algorithm, and it will be this algorithm that we 9PAC stands for “probably approximately correct,” which is a framework and set of assumptions under which numerous results on learning theory were proved. Of these, the assumption of training and testing on the same distribution, and the assumption of the independently drawn training examples, were the most important. 128 focus on in these notes. (Algorithms such as logistic regression can also be viewed as approximations to empirical risk minimization.) In our study of learning theory, it will be useful to abstract away from the speciﬁc parameterization of hypotheses and from issues such as whether we’re using a linear classiﬁer. We deﬁne the hypothesis class Hused by a learning algorithm to be the set of all classiﬁers considered by it. For linear classiﬁcation, H= {hθ : hθ(x) = 1 {θTx ≥0},θ ∈Rd+1}is thus the set of all classiﬁers over X(the domain of the inputs) where the decision boundary is linear. More broadly, if we were studying, say, neural networks, then we could let Hbe the set of all classiﬁers representable by some neural network architecture. Empirical risk minimization can now be thought of as a minimization over the class of functions H, in which the learning algorithm picks the hypothesis: ˆh= arg min h∈H ˆε(h) 8.3.2 The case of ﬁnite H Let’s start by considering a learning problem in which we have a ﬁnite hy- pothesis class H= {h1,...,h k}consisting of khypotheses. Thus, His just a set of kfunctions mapping from Xto {0,1}, and empirical risk minimization selects ˆhto be whichever of these k functions has the smallest training error. We would like to give guarantees on the generalization error of ˆh. Our strategy for doing so will be in two parts: First, we will show that ˆ ε(h) is a reliable estimate of ε(h) for all h. Second, we will show that this implies an upper-bound on the generalization error of ˆh. Take any one, ﬁxed, hi ∈H. Consider a Bernoulli random variable Z whose distribution is deﬁned as follows. We’re going to sample ( x,y) ∼D. Then, we set Z = 1 {hi(x) ̸= y}. I.e., we’re going to draw one example, and let Z indicate whether hi misclassiﬁes it. Similarly, we also deﬁne Zj = 1{hi(x(j)) ̸= y(j)}. Since our training set was drawn iid from D, Z and the Zj’s have the same distribution. We see that the misclassiﬁcation probability on a randomly drawn example—that is, ε(h)—is exactly the expected value of Z (and Zj). More- over, the training error can be written ˆε(hi) = 1 n n∑ j=1 Zj. Thus, ˆε(hi) is exactly the mean of the nrandom variables Zj that are drawn iid from a Bernoulli distribution with mean ε(hi). Hence, we can apply the 129 Hoeﬀding inequality, and obtain P(\|ε(hi) −ˆε(hi)\|>γ ) ≤2 exp(−2γ2n). This shows that, for our particular hi, training error will be close to generalization error with high probability, assuming nis large. But we don’t just want to guarantee thatε(hi) will be close to ˆε(hi) (with high probability) for just only one particular hi. We want to prove that this will be true simultaneously for all h∈H. To do so, let Ai denote the event that \|ε(hi) − ˆε(hi)\|> γ. We’ve already shown that, for any particular Ai, it holds true that P(Ai) ≤2 exp(−2γ2n). Thus, using the union bound, we have that P(∃h∈H.\|ε(hi) −ˆε(hi)\|>γ ) = P(A1 ∪···∪ Ak) ≤ k∑ i=1 P(Ai) ≤ k∑ i=1 2 exp(−2γ2n) = 2 kexp(−2γ2n) If we subtract both sides from 1, we ﬁnd that P(¬∃h∈H.\|ε(hi) −ˆε(hi)\|>γ ) = P(∀h∈H.\|ε(hi) −ˆε(hi)\|≤ γ) ≥ 1 −2kexp(−2γ2n) (The “ ¬” symbol means “not.”) So, with probability at least 1 − 2kexp(−2γ2n), we have that ε(h) will be within γ of ˆε(h) for all h ∈H. This is called a uniform convergence result, because this is a bound that holds simultaneously for all (as opposed to just one) h∈H. In the discussion above, what we did was, for particular values of n and γ, give a bound on the probability that for some h ∈H, \|ε(h) −ˆε(h)\|> γ. There are three quantities of interest here: n, γ, and the probability of error; we can bound either one in terms of the other two. For instance, we can ask the following question: Given γ and some δ >0, how large must n be before we can guarantee that with probability at least 1 −δ, training error will be within γ of generalization error? By setting δ = 2kexp(−2γ2n) and solving for n, [you should convince yourself this is the right thing to do!], we ﬁnd that if n≥ 1 2γ2 log 2k δ , 130 then with probability at least 1 −δ, we have that \|ε(h) −ˆε(h)\|≤ γ for all h∈H. (Equivalently, this shows that the probability that \|ε(h) −ˆε(h)\|>γ for some h ∈ His at most δ.) This bound tells us how many training examples we need in order make a guarantee. The training set size n that a certain method or algorithm requires in order to achieve a certain level of performance is also called the algorithm’s sample complexity. The key property of the bound above is that the number of training examples needed to make this guarantee is only logarithmic in k, the number of hypotheses in H. This will be important later. Similarly, we can also hold n and δ ﬁxed and solve for γ in the previous equation, and show [again, convince yourself that this is right!] that with probability 1 −δ, we have that for all h∈H, \|ˆε(h) −ε(h)\|≤ √ 1 2nlog 2k δ . Now, let’s assume that uniform convergence holds, i.e., that\|ε(h)−ˆε(h)\|≤ γ for all h∈H. What can we prove about the generalization of our learning algorithm that picked ˆh= arg minh∈Hˆε(h)? Deﬁne h∗= arg minh∈Hε(h) to be the best possible hypothesis inH. Note that h∗ is the best that we could possibly do given that we are using H, so it makes sense to compare our performance to that of h∗. We have: ε(ˆh) ≤ ˆε(ˆh) + γ ≤ ˆε(h∗) + γ ≤ ε(h∗) + 2γ The ﬁrst line used the fact that\|ε(ˆh)−ˆε(ˆh)\|≤ γ(by our uniform convergence assumption). The second used the fact that ˆh was chosen to minimize ˆε(h), and hence ˆε(ˆh) ≤ˆε(h) for all h, and in particular ˆε(ˆh) ≤ˆε(h∗). The third line used the uniform convergence assumption again, to show that ˆ ε(h∗) ≤ ε(h∗) + γ. So, what we’ve shown is the following: If uniform convergence occurs, then the generalization error of ˆh is at most 2 γ worse than the best possible hypothesis in H! Let’s put all this together into a theorem. Theorem. Let \|H\|= k, and let any n,δ be ﬁxed. Then with probability at least 1 −δ, we have that ε(ˆh) ≤ ( min h∈H ε(h) ) + 2 √ 1 2nlog 2k δ . 131 This is proved by letting γ equal the √·term, using our previous argu- ment that uniform convergence occurs with probability at least 1 −δ, and then noting that uniform convergence implies ε(h) is at most 2γ higher than ε(h∗) = minh∈Hε(h) (as we showed previously). This also quantiﬁes what we were saying previously saying about the bias/variance tradeoﬀ in model selection. Speciﬁcally, suppose we have some hypothesis class H, and are considering switching to some much larger hy- pothesis class H′ ⊇H. If we switch to H′, then the ﬁrst term min hε(h) can only decrease (since we’d then be taking a min over a larger set of func- tions). Hence, by learning using a larger hypothesis class, our “bias” can only decrease. However, if k increases, then the second 2 √·term would also increase. This increase corresponds to our “variance” increasing when we use a larger hypothesis class. By holding γ and δ ﬁxed and solving for nlike we did before, we can also obtain the following sample complexity bound: Corollary. Let \|H\| = k, and let any δ,γ be ﬁxed. Then for ε(ˆh) ≤ minh∈Hε(h) + 2γ to hold with probability at least 1 −δ, it suﬃces that n ≥ 1 2γ2 log 2k δ = O (1 γ2 log k δ ) , 8.3.3 The case of inﬁnite H We have proved some useful theorems for the case of ﬁnite hypothesis classes. But many hypothesis classes, including any parameterized by real numbers (as in linear classiﬁcation) actually contain an inﬁnite number of functions. Can we prove similar results for this setting? Let’s start by going through something that is not the “right” argument. Better and more general arguments exist , but this will be useful for honing our intuitions about the domain. Suppose we have an Hthat is parameterized by dreal numbers. Since we are using a computer to represent real numbers, and IEEE double-precision ﬂoating point (double’s in C) uses 64 bits to represent a ﬂoating point num- ber, this means that our learning algorithm, assuming we’re using double- precision ﬂoating point, is parameterized by 64 d bits. Thus, our hypothesis class really consists of at most k= 264d diﬀerent hypotheses. From the Corol- lary at the end of the previous section, we therefore ﬁnd that, to guarantee 132 ε(ˆh) ≤ε(h∗)+2 γ, with to hold with probability at least 1 −δ, it suﬃces that n ≥O ( 1 γ2 log 264d δ ) = O ( d γ2 log 1 δ ) = Oγ,δ(d). (The γ,δ subscripts indicate that the last big- O is hiding constants that may depend on γ and δ.) Thus, the number of training examples needed is at most linear in the parameters of the model. The fact that we relied on 64-bit ﬂoating point makes this argument not entirely satisfying, but the conclusion is nonetheless roughly correct: If what we try to do is minimize training error, then in order to learn “well” using a hypothesis class that has dparameters, generally we’re going to need on the order of a linear number of training examples in d. (At this point, it’s worth noting that these results were proved for an al- gorithm that uses empirical risk minimization. Thus, while the linear depen- dence of sample complexity on ddoes generally hold for most discriminative learning algorithms that try to minimize training error or some approxima- tion to training error, these conclusions do not always apply as readily to discriminative learning algorithms. Giving good theoretical guarantees on many non-ERM learning algorithms is still an area of active research.) The other part of our previous argument that’s slightly unsatisfying is that it relies on the parameterization of H. Intuitively, this doesn’t seem like it should matter: We had written the class of linear classiﬁers as hθ(x) = 1{θ0 + θ1x1 + ···θdxd ≥0}, with n+ 1 parameters θ0,...,θ d. But it could also be written hu,v(x) = 1 {(u2 0 −v2 0) + (u2 1 −v2 1)x1 + ···(u2 d −v2 d)xd ≥0} with 2d+ 2 parameters ui,vi. Yet, both of these are just deﬁning the same H: The set of linear classiﬁers in d dimensions. To derive a more satisfying argument, let’s deﬁne a few more things. Given a set S = {x(i),...,x (D)}(no relation to the training set) of points x(i) ∈X , we say that Hshatters S if Hcan realize any labeling on S. I.e., if for any set of labels {y(1),...,y (D)}, there exists some h∈H so that h(x(i)) = y(i) for all i= 1,... D. Given a hypothesis class H, we then deﬁne its Vapnik-Chervonenkis dimension, written VC(H), to be the size of the largest set that is shattered by H. (If Hcan shatter arbitrarily large sets, then VC( H) = ∞.) For instance, consider the following set of three points: 133 /0 /1 /0 /1 /0 /1 x x1 2 Can the set Hof linear classiﬁers in two dimensions (h(x) = 1{θ0 +θ1x1 + θ2x2 ≥0}) can shatter the set above? The answer is yes. Speciﬁcally, we see that, for any of the eight possible labelings of these points, we can ﬁnd a linear classiﬁer that obtains “zero training error” on them: x x1 2 x x1 2 x x1 2 x x1 2 x x1 2 x x1 2 x x1 2 x x1 2 Moreover, it is possible to show that there is no set of 4 points that this hypothesis class can shatter. Thus, the largest set that Hcan shatter is of size 3, and hence VC( H) = 3. Note that the VC dimension of Hhere is 3 even though there may be sets of size 3 that it cannot shatter. For instance, if we had a set of three points lying in a straight line (left ﬁgure), then there is no way to ﬁnd a linear separator for the labeling of the three points shown below (right ﬁgure): 134 x x1 2 /0 /1 /0 /1 /0 /1 x x1 2 In order words, under the deﬁnition of the VC dimension, in order to prove that VC(H) is at least D, we need to show only that there’s at least one set of size D that Hcan shatter. The following theorem, due to Vapnik, can then be shown. (This is, many would argue, the most important theorem in all of learning theory.) Theorem. Let Hbe given, and let D = VC(H). Then with probability at least 1 −δ, we have that for all h∈H, \|ε(h) −ˆε(h)\|≤ O (√ D n log n D + 1 nlog 1 δ ) . Thus, with probability at least 1 −δ, we also have that: ε(ˆh) ≤ε(h∗) + O (√ D n log n D + 1 nlog 1 δ ) . In other words, if a hypothesis class has ﬁnite VC dimension, then uniform convergence occurs as n becomes large. As before, this allows us to give a bound on ε(h) in terms of ε(h∗). We also have the following corollary: Corollary. For \|ε(h) −ˆε(h)\|≤ γ to hold for all h ∈H (and hence ε(ˆh) ≤ ε(h∗) + 2γ) with probability at least 1 −δ, it suﬃces that n= Oγ,δ(D). In other words, the number of training examples needed to learn “well” using His linear in the VC dimension of H. It turns out that, for “most” hypothesis classes, the VC dimension (assuming a “reasonable” parameter- ization) is also roughly linear in the number of parameters. Putting these together, we conclude that for a given hypothesis class H(and for an algo- rithm that tries to minimize training error), the number of training examples needed to achieve generalization error close to that of the optimal classiﬁer is usually roughly linear in the number of parameters of H. Chapter 9 Regularization and model selection 9.1 Regularization Recall that as discussed in Section 8.1, overftting is typically a result of using too complex models, and we need to choose a proper model complexity to achieve the optimal bias-variance tradeoﬀ. When the model complexity is measured by the number of parameters, we can vary the size of the model (e.g., the width of a neural net). However, the correct, informative complex- ity measure of the models can be a function of the parameters (e.g., ℓ2 norm of the parameters), which may not necessarily depend on the number of pa- rameters. In such cases, we will use regularization, an important technique in machine learning, control the model complexity and prevent overﬁtting. Regularization typically involves adding an additional term, called a reg- ularizer and denoted by R(θ) here, to the training loss/cost function: Jλ(θ) = J(θ) + λR(θ) (9.1) Here Jλ is often called the regularized loss, and λ ≥0 is called the regular- ization parameter. The regularizer R(θ) is a nonnegative function (in almost all cases). In classical methods, R(θ) is purely a function of the parameter θ, but some modern approach allows R(θ) to depend on the training dataset. 1 The regularizer R(θ) is typically chosen to be some measure of the com- plexity of the model θ. Thus, when using the regularized loss, we aim to ﬁnd a model that both ﬁt the data (a small loss J(θ)) and have a small 1Here our notations generally omit the dependency on the training dataset for simplicity—we writeJ(θ) even though it obviously needs to depend on the training dataset. 135 136 model complexity (a small R(θ)). The balance between the two objectives is controlled by the regularization parameter λ. When λ = 0, the regularized loss is equivalent to the original loss. When λ is a suﬃciently small positive number, minimizing the regularized loss is eﬀectively minimizing the original loss with the regularizer as the tie-breaker. When the regularizer is extremely large, then the original loss is not eﬀective (and likely the model will have a large bias.) The most commonly used regularization is perhaps ℓ2 regularization, where R(θ) = 1 2 ∥θ∥2 2. It encourages the optimizer to ﬁnd a model with small ℓ2 norm. In deep learning, it’s oftentimes referred to as weight de- cay, because gradient descent with learning rate η on the regularized loss Rλ(θ) is equivalent to shrinking/decaying θ by a scalar factor of 1 −ηλ and then applying the standard gradient θ←θ−η∇Jλ(θ) = θ−ηλθ−η∇J(θ) = (1 −λη)θ   decaying weights −η∇J(θ) (9.2) Besides encouraging simpler models, regularization can also impose in- ductive biases or structures on the model parameters. For example, suppose we had a prior belief that the number of non-zeros in the ground-truth model parameters is small,2—which is oftentimes called sparsity of the model—, we can impose a regularization on the number of non-zeros in θ, denoted by ∥θ∥0, to leverage such a prior belief. Imposing additional structure of the parameters narrows our search space and makes the complexity of the model family smaller,—e.g., the family of sparse models can be thought of as having lower complexity than the family of all models—, and thus tends to lead to a better generalization. On the other hand, imposing additional structure may risk increasing the bias. For example, if we regularize the sparsity strongly but no sparse models can predict the label accurately, we will suﬀer from large bias (analogously to the situation when we use linear models to learn data than can only be represented by quadratic functions in Section 8.1.) The sparsity of the parameters is not a continuous function of the param- eters, and thus we cannot optimize it with (stochastic) gradient descent. A common relaxation is to use R(θ) = ∥θ∥1 as a continuous surrogate.3 2For linear models, this means the model just uses a few coordinates of the inputs to make an accurate prediction. 3There has been a rich line of theoretical work that explains why ∥θ∥1 is a good sur- rogate for encouraging sparsity, but it’s beyond the scope of this course. An intuition is: assuming the parameter is on the unit sphere, the parameter with smallest ℓ1 norm also 137 The R(θ) = ∥θ∥1 (also called LASSO) and R(θ) = 1 2 ∥θ∥2 2 are perhaps among the most commonly used regularizers for linear models. Other norm and powers of norms are sometimes also used. The ℓ2 norm regularization is much more commonly used with kernel methods because ℓ1 regularization is typically not compatible with the kernel trick (the optimal solution cannot be written as functions of inner products of features.) In deep learning, the most commonly used regularizer is ℓ2 regularization or weight decay. Other common ones include dropout, data augmentation, regularizing the spectral norm of the weight matrices, and regularizing the Lipschitzness of the model, etc. Regularization in deep learning is an ac- tive research area, and it’s known that there is another implicit source of regularization, as discussed in the next section. 9.2 Implicit regularization eﬀect (optional reading) The implicit regularization eﬀect of optimizers, or implicit bias or algorithmic regularization, is a new concept/phenomenon observed in the deep learning era. It largely refers to that the optimizers can implicitly impose structures on parameters beyond what has been imposed by the regularized loss. In most classical settings, the loss or regularized loss has a unique global minimum, and thus any reasonable optimizer should converge to that global minimum and cannot impose any additional preferences. However, in deep learning, oftentimes the loss or regularized loss has more than one (approx- imate) global minima, and diﬀerence optimizers may converge to diﬀerent global minima. Though these global minima have the same or similar train- ing losses, they may be of diﬀerent nature and have dramatically diﬀerent generalization performance. See Figures 9.1 and 9.2 and its caption for an illustration and some experiment results. For example, it’s possible that one global minimum gives a much more Lipschitz or sparse model than others and thus has a better test error. It turns out that many commonly-used op- timizers (or their components) prefer or bias towards ﬁnding global minima of certain properties, leading to a better test performance. happen to be the sparsest parameter with only 1 non-zero coordinate. Thus, sparsity and ℓ1 norm gives the same extremal points to some extent. 138 θ loss Figure 9.1: An Illustration that diﬀerent global minima of the training loss can have diﬀerent test performance. Figure 9.2: Left: Performance of neural networks trained by two diﬀerent learning rates schedules on the CIFAR-10 dataset. Although both exper- iments used exactly the same regularized losses and the optimizers ﬁt the training data perfectly, the models’ generalization performance diﬀer much. Right: On a diﬀerent synthetic dataset, optimizers with diﬀerent initializa- tions have the same training error but diﬀerent generalization performance. 4 In summary, the takehome message here is that the choice of optimizer does not only aﬀect minimizing the training loss, but also imposes implicit regularization and aﬀects the generalization of the model. Even if your cur- rent optimizer already converges to a small training error perfectly, you may still need to tune your optimizer for a better generalization, . 4The setting is the same as in Woodworth et al. [2020], HaoChen et al. [2020] 139 One may wonder which components of the optimizers bias towards what type of global minima and what type of global minima may generalize bet- ter. These are open questions that researchers are actively investigating. Empirical and theoretical research have oﬀered some clues and heuristics. In many (but deﬁnitely far from all) situations, among those setting where optimization can succeed in minimizing the training loss, the use of larger initial learning rate, smaller initialization, smaller batch size, and momen- tum appears to help with biasing towards more generalizable solutions. A conjecture (that can be proven in certain simpliﬁed case) is that stochas- ticity in the optimization process help the optimizer to ﬁnd ﬂatter global minima (global minima where the curvature of the loss is small), and ﬂat global minima tend to give more Lipschitz models and better generalization. Characterizing the implicit regularization eﬀect formally is still a challenging open research question. 9.3 Model selection via cross validation Suppose we are trying select among several diﬀerent models for a learning problem. For instance, we might be using a polynomial regression model hθ(x) = g(θ0 + θ1x+ θ2x2 + ··· + θkxk), and wish to decide if k should be 0, 1, . . . , or 10. How can we automatically select a model that represents a good tradeoﬀ between the twin evils of bias and variance 5? Alternatively, suppose we want to automatically choose the bandwidth parameter τ for locally weighted regression, or the parameter C for our ℓ1-regularized SVM. How can we do that? For the sake of concreteness, in these notes we assume we have some ﬁnite set of models M= {M1,...,M d}that we’re trying to select among. For instance, in our ﬁrst example above, the model Mi would be an i-th degree polynomial regression model. (The generalization to inﬁnite Mis not hard.6) Alternatively, if we are trying to decide between using an SVM, a neural network or logistic regression, then Mmay contain these models. 5Given that we said in the previous set of notes that bias and variance are two very diﬀerent beasts, some readers may be wondering if we should be calling them “twin” evils here. Perhaps it’d be better to think of them as non-identical twins. The phrase “the fraternal twin evils of bias and variance” doesn’t have the same ring to it, though. 6If we are trying to choose from an inﬁnite set of models, say corresponding to the possible values of the bandwidth τ ∈R+, we may discretize τ and consider only a ﬁnite number of possible values for it. More generally, most of the algorithms described here can all be viewed as performing optimization search in the space of models, and we can perform this search over inﬁnite model classes as well. 140 Cross validation. Lets suppose we are, as usual, given a training set S. Given what we know about empirical risk minimization, here’s what might initially seem like a algorithm, resulting from using empirical risk minimiza- tion for model selection: 1. Train each model Mi on S, to get some hypothesis hi. 2. Pick the hypotheses with the smallest training error. This algorithm does not work. Consider choosing the degree of a poly- nomial. The higher the degree of the polynomial, the better it will ﬁt the training set S, and thus the lower the training error. Hence, this method will always select a high-variance, high-degree polynomial model, which we saw previously is often poor choice. Here’s an algorithm that works better. In hold-out cross validation (also called simple cross validation), we do the following: 1. Randomly split S into Strain (say, 70% of the data) and Scv (the remain- ing 30%). Here, Scv is called the hold-out cross validation set. 2. Train each model Mi on Strain only, to get some hypothesis hi. 3. Select and output the hypothesis hi that had the smallest error ˆεScv (hi) on the hold out cross validation set. (Here ˆ εScv (h) denotes the average error of h on the set of examples in Scv.) The error on the hold out validation set is also referred to as the validation error. By testing/validating on a set of examples Scv that the models were not trained on, we obtain a better estimate of each hypothesis hi’s true general- ization/test error. Thus, this approach is essentially picking the model with the smallest estimated generalization/test error. The size of the validation set depends on the total number of available examples. Usually, somewhere between 1/4−1/3 of the data is used in the hold out cross validation set, and 30% is a typical choice. However, when the total dataset is huge, validation set can be a smaller fraction of the total examples as long as the absolute number of validation examples is decent. For example, for the ImageNet dataset that has about 1M training images, the validation set is sometimes set to be 50K images, which is only about 5% of the total examples. Optionally, step 3 in the algorithm may also be replaced with selecting the model Mi according to arg mini ˆεScv (hi), and then retraining Mi on the entire training set S. (This is often a good idea, with one exception being learning algorithms that are be very sensitive to perturbations of the initial 141 conditions and/or data. For these methods, Mi doing well on Strain does not necessarily mean it will also do well on Scv, and it might be better to forgo this retraining step.) The disadvantage of using hold out cross validation is that it “wastes” about 30% of the data. Even if we were to take the optional step of retraining the model on the entire training set, it’s still as if we’re trying to ﬁnd a good model for a learning problem in which we had 0.7ntraining examples, rather than n training examples, since we’re testing models that were trained on only 0.7n examples each time. While this is ﬁne if data is abundant and/or cheap, in learning problems in which data is scarce (consider a problem with n= 20, say), we’d like to do something better. Here is a method, called k-fold cross validation , that holds out less data each time: 1. Randomly split S into kdisjoint subsets of m/ktraining examples each. Lets call these subsets S1,...,S k. 2. For each model Mi, we evaluate it as follows: For j = 1,...,k Train the model Mi on S1 ∪···∪ Sj−1 ∪Sj+1 ∪··· Sk (i.e., train on all the data except Sj) to get some hypothesis hij. Test the hypothesis hij on Sj, to get ˆεSj(hij). The estimated generalization error of model Mi is then calculated as the average of the ˆεSj(hij)’s (averaged over j). 3. Pick the model Mi with the lowest estimated generalization error, and retrain that model on the entire training setS. The resulting hypothesis is then output as our ﬁnal answer. A typical choice for the number of folds to use here would be k = 10. While the fraction of data held out each time is now 1 /k—much smaller than before—this procedure may also be more computationally expensive than hold-out cross validation, since we now need train to each model k times. While k = 10 is a commonly used choice, in problems in which data is really scarce, sometimes we will use the extreme choice of k = m in order to leave out as little data as possible each time. In this setting, we would repeatedly train on all but one of the training examples inS, and test on that held-out example. The resulting m= k errors are then averaged together to obtain our estimate of the generalization error of a model. This method has 142 its own name; since we’re holding out one training example at a time, this method is called leave-one-out cross validation. Finally, even though we have described the diﬀerent versions of cross vali- dation as methods for selecting a model, they can also be used more simply to evaluate a single model or algorithm. For example, if you have implemented some learning algorithm and want to estimate how well it performs for your application (or if you have invented a novel learning algorithm and want to report in a technical paper how well it performs on various test sets), cross validation would give a reasonable way of doing so. 9.4 Bayesian statistics and regularization In this section, we will talk about one more tool in our arsenal for our battle against overﬁtting. At the beginning of the quarter, we talked about parameter ﬁtting using maximum likelihood estimation (MLE), and chose our parameters according to θMLE = arg max θ n∏ i=1 p(y(i)\|x(i); θ). Throughout our subsequent discussions, we viewed θ as an unknown param- eter of the world. This view of the θ as being constant-valued but unknown is taken in frequentist statistics. In the frequentist this view of the world, θ is not random—it just happens to be unknown—and it’s our job to come up with statistical procedures (such as maximum likelihood) to try to estimate this parameter. An alternative way to approach our parameter estimation problems is to take the Bayesian view of the world, and think of θ as being a random variable whose value is unknown. In this approach, we would specify a prior distribution p(θ) on θ that expresses our “prior beliefs” about the parameters. Given a training set S = {(x(i),y(i))}n i=1, when we are asked to make a prediction on a new value of x, we can then compute the posterior distribution on the parameters p(θ\|S) = p(S\|θ)p(θ) p(S) = (∏n i=1 p(y(i)\|x(i),θ) ) p(θ)∫ θ(∏n i=1 p(y(i)\|x(i),θ)p(θ)) dθ (9.3) In the equation above, p(y(i)\|x(i),θ) comes from whatever model you’re using 143 for your learning problem. For example, if you are using Bayesian logistic re- gression, then you might choosep(y(i)\|x(i),θ) = hθ(x(i))y(i) (1−hθ(x(i)))(1−y(i)), where hθ(x(i)) = 1/(1 + exp(−θTx(i))).7 When we are given a new test example xand asked to make it prediction on it, we can compute our posterior distribution on the class label using the posterior distribution on θ: p(y\|x,S) = ∫ θ p(y\|x,θ)p(θ\|S)dθ (9.4) In the equation above, p(θ\|S) comes from Equation (9.3). Thus, for example, if the goal is to the predict the expected value of y given x, then we would output8 E[y\|x,S] = ∫ y yp(y\|x,S)dy The procedure that we’ve outlined here can be thought of as doing “fully Bayesian” prediction, where our prediction is computed by taking an average with respect to the posterior p(θ\|S) over θ. Unfortunately, in general it is computationally very diﬃcult to compute this posterior distribution. This is because it requires taking integrals over the (usually high-dimensional) θ as in Equation (9.3), and this typically cannot be done in closed-form. Thus, in practice we will instead approximate the posterior distribution for θ. One common approximation is to replace our posterior distribution for θ (as in Equation 9.4) with a single point estimate. The MAP (maximum a posteriori) estimate for θ is given by θMAP = arg max θ n∏ i=1 p(y(i)\|x(i),θ)p(θ). (9.5) Note that this is the same formulas as for the MLE (maximum likelihood) estimate for θ, except for the prior p(θ) term at the end. In practical applications, a common choice for the prior p(θ) is to assume that θ∼N(0,τ2I). Using this choice of prior, the ﬁtted parametersθMAP will have smaller norm than that selected by maximum likelihood. In practice, this causes the Bayesian MAP estimate to be less susceptible to overﬁtting than the ML estimate of the parameters. For example, Bayesian logistic regression turns out to be an eﬀective algorithm for text classiﬁcation, even though in text classiﬁcation we usually have d≫n. 7Since we are now viewing θ as a random variable, it is okay to condition on it value, and write “p(y\|x,θ)” instead of “ p(y\|x; θ).” 8The integral below would be replaced by a summation if y is discrete-valued. Part IV Unsupervised learning 144 Chapter 10 Clustering and the k-means algorithm In the clustering problem, we are given a training set {x(1),...,x (n)}, and want to group the data into a few cohesive “clusters.” Here, x(i) ∈ Rd as usual; but no labels y(i) are given. So, this is an unsupervised learning problem. The k-means clustering algorithm is as follows: 1. Initialize cluster centroids µ1,µ2,...,µ k ∈Rd randomly. 2. Repeat until convergence: { For every i, set c(i) := arg min j \|\|x(i) −µj\|\|2. For each j, set µj := ∑n i=1 1{c(i) = j}x(i) ∑n i=1 1{c(i) = j} . } In the algorithm above, k (a parameter of the algorithm) is the number of clusters we want to ﬁnd; and the cluster centroids µj represent our current guesses for the positions of the centers of the clusters. To initialize the cluster centroids (in step 1 of the algorithm above), we could choose k training examples randomly, and set the cluster centroids to be equal to the values of these k examples. (Other initialization methods are also possible.) The inner-loop of the algorithm repeatedly carries out two steps: (i) “Assigning” each training example x(i) to the closest cluster centroid µj, and 145 146 Figure 10.1: K-means algorithm. Training examples are shown as dots, and cluster centroids are shown as crosses. (a) Original dataset. (b) Random ini- tial cluster centroids (in this instance, not chosen to be equal to two training examples). (c-f) Illustration of running two iterations of k-means. In each iteration, we assign each training example to the closest cluster centroid (shown by “painting” the training examples the same color as the cluster centroid to which is assigned); then we move each cluster centroid to the mean of the points assigned to it. (Best viewed in color.) Images courtesy Michael Jordan. (ii) Moving each cluster centroid µj to the mean of the points assigned to it. Figure 10.1 shows an illustration of running k-means. Is the k-means algorithm guaranteed to converge? Yes it is, in a certain sense. In particular, let us deﬁne the distortion function to be: J(c,µ) = n∑ i=1 \|\|x(i) −µc(i) \|\|2 Thus, J measures the sum of squared distances between each training exam- ple x(i) and the cluster centroid µc(i) to which it has been assigned. It can be shown that k-means is exactly coordinate descent on J. Speciﬁcally, the inner-loop of k-means repeatedly minimizes J with respect to cwhile holding µﬁxed, and then minimizes J with respect to µwhile holding cﬁxed. Thus, 147 J must monotonically decrease, and the value of J must converge. (Usu- ally, this implies that c and µ will converge too. In theory, it is possible for k-means to oscillate between a few diﬀerent clusterings—i.e., a few diﬀerent values for cand/or µ—that have exactly the same value of J, but this almost never happens in practice.) The distortion function J is a non-convex function, and so coordinate descent on J is not guaranteed to converge to the global minimum. In other words, k-means can be susceptible to local optima. Very often k-means will work ﬁne and come up with very good clusterings despite this. But if you are worried about getting stuck in bad local minima, one common thing to do is run k-means many times (using diﬀerent random initial values for the cluster centroids µj). Then, out of all the diﬀerent clusterings found, pick the one that gives the lowest distortion J(c,µ). Chapter 11 EM algorithms In this set of notes, we discuss the EM (Expectation-Maximization) algorithm for density estimation. 11.1 EM for mixture of Gaussians Suppose that we are given a training set {x(1),...,x (n)}as usual. Since we are in the unsupervised learning setting, these points do not come with any labels. We wish to model the data by specifying a joint distributionp(x(i),z(i)) = p(x(i)\|z(i))p(z(i)). Here, z(i) ∼Multinomial(φ) (where φj ≥0, ∑k j=1 φj = 1, and the parameter φj gives p(z(i) = j)), and x(i)\|z(i) = j ∼N (µj,Σj). We let k denote the number of values that the z(i)’s can take on. Thus, our model posits that each x(i) was generated by randomly choosing z(i) from {1,...,k }, and then x(i) was drawn from one of k Gaussians depending on z(i). This is called the mixture of Gaussians model. Also, note that the z(i)’s are latent random variables, meaning that they’re hidden/unobserved. This is what will make our estimation problem diﬃcult. The parameters of our model are thus φ, µand Σ. To estimate them, we can write down the likelihood of our data: ℓ(φ,µ, Σ) = n∑ i=1 log p(x(i); φ,µ, Σ) = n∑ i=1 log k∑ z(i)=1 p(x(i)\|z(i); µ,Σ)p(z(i); φ). However, if we set to zero the derivatives of this formula with respect to 148 149 the parameters and try to solve, we’ll ﬁnd that it is not possible to ﬁnd the maximum likelihood estimates of the parameters in closed form. (Try this yourself at home.) The random variables z(i) indicate which of the k Gaussians each x(i) had come from. Note that if we knew what the z(i)’s were, the maximum likelihood problem would have been easy. Speciﬁcally, we could then write down the likelihood as ℓ(φ,µ, Σ) = n∑ i=1 log p(x(i)\|z(i); µ,Σ) + logp(z(i); φ). Maximizing this with respect to φ, µ and Σ gives the parameters: φj = 1 n n∑ i=1 1{z(i) = j}, µj = ∑n i=1 1{z(i) = j}x(i) ∑n i=1 1{z(i) = j} , Σj = ∑n i=1 1{z(i) = j}(x(i) −µj)(x(i) −µj)T ∑n i=1 1{z(i) = j} . Indeed, we see that if the z(i)’s were known, then maximum likelihood estimation becomes nearly identical to what we had when estimating the parameters of the Gaussian discriminant analysis model, except that here the z(i)’s playing the role of the class labels. 1 However, in our density estimation problem, the z(i)’s are not known. What can we do? The EM algorithm is an iterative algorithm that has two main steps. Applied to our problem, in the E-step, it tries to “guess” the values of the z(i)’s. In the M-step, it updates the parameters of our model based on our guesses. Since in the M-step we are pretending that the guesses in the ﬁrst part were correct, the maximization becomes easy. Here’s the algorithm: Repeat until convergence: { (E-step) For each i,j, set w(i) j := p(z(i) = j\|x(i); φ,µ, Σ) 1There are other minor diﬀerences in the formulas here from what we’d obtained in PS1 with Gaussian discriminant analysis, ﬁrst because we’ve generalized the z(i)’s to be multinomial rather than Bernoulli, and second because here we are using a diﬀerent Σ j for each Gaussian. 150 (M-step) Update the parameters: φj := 1 n n∑ i=1 w(i) j , µj := ∑n i=1 w(i) j x(i) ∑n i=1 w(i) j , Σj := ∑n i=1 w(i) j (x(i) −µj)(x(i) −µj)T ∑n i=1 w(i) j } In the E-step, we calculate the posterior probability of our parameters the z(i)’s, given the x(i) and using the current setting of our parameters. I.e., using Bayes rule, we obtain: p(z(i) = j\|x(i); φ,µ, Σ) = p(x(i)\|z(i) = j; µ,Σ)p(z(i) = j; φ)∑k l=1 p(x(i)\|z(i) = l; µ,Σ)p(z(i) = l; φ) Here, p(x(i)\|z(i) = j; µ,Σ) is given by evaluating the density of a Gaussian with mean µj and covariance Σj at x(i); p(z(i) = j; φ) is given by φj, and so on. The values w(i) j calculated in the E-step represent our “soft” guesses2 for the values of z(i). Also, you should contrast the updates in the M-step with the formulas we had when the z(i)’s were known exactly. They are identical, except that in- stead of the indicator functions “1{z(i) = j}” indicating from which Gaussian each datapoint had come, we now instead have the w(i) j ’s. The EM-algorithm is also reminiscent of the K-means clustering algo- rithm, except that instead of the “hard” cluster assignments c(i), we instead have the “soft” assignments w(i) j . Similar to K-means, it is also susceptible to local optima, so reinitializing at several diﬀerent initial parameters may be a good idea. It’s clear that the EM algorithm has a very natural interpretation of repeatedly trying to guess the unknown z(i)’s; but how did it come about, and can we make any guarantees about it, such as regarding its convergence? In the next set of notes, we will describe a more general view of EM, one 2The term “soft” refers to our guesses being probabilities and taking values in [0 ,1]; in contrast, a “hard” guess is one that represents a single best guess (such as taking values in {0,1}or {1,...,k }). 151 that will allow us to easily apply it to other estimation problems in which there are also latent variables, and which will allow us to give a convergence guarantee. 11.2 Jensen’s inequality We begin our discussion with a very useful result calledJensen’s inequality Let f be a function whose domain is the set of real numbers. Recall that f is a convex function if f′′(x) ≥0 (for all x ∈R). In the case of f taking vector-valued inputs, this is generalized to the condition that its hessian H is positive semi-deﬁnite ( H ≥0). If f′′(x) > 0 for all x, then we say f is strictly convex (in the vector-valued case, the corresponding statement is that H must be positive deﬁnite, written H >0). Jensen’s inequality can then be stated as follows: Theorem. Let f be a convex function, and let X be a random variable. Then: E[f(X)] ≥f(EX). Moreover, if f is strictly convex, then E[ f(X)] = f(EX) holds true if and only if X = E[X] with probability 1 (i.e., if X is a constant). Recall our convention of occasionally dropping the parentheses when writ- ing expectations, so in the theorem above, f(EX) = f(E[X]). For an interpretation of the theorem, consider the ﬁgure below. a E[X] b f(a) f(b) f(EX) E[f(X)] f Here, f is a convex function shown by the solid line. Also, X is a random variable that has a 0.5 chance of taking the value a, and a 0.5 chance of 152 taking the value b (indicated on the x-axis). Thus, the expected value of X is given by the midpoint between a and b. We also see the values f(a), f(b) and f(E[X]) indicated on the y-axis. Moreover, the value E[f(X)] is now the midpoint on the y-axis between f(a) and f(b). From our example, we see that because f is convex, it must be the case that E[f(X)] ≥f(EX). Incidentally, quite a lot of people have trouble remembering which way the inequality goes, and remembering a picture like this is a good way to quickly ﬁgure out the answer. Remark. Recall that f is [strictly] concave if and only if −f is [strictly] convex (i.e., f′′(x) ≤0 or H ≤0). Jensen’s inequality also holds for concave functions f, but with the direction of all the inequalities reversed (E[f(X)] ≤ f(EX), etc.). 11.3 General EM algorithms Suppose we have an estimation problem in which we have a training set {x(1),...,x (n)}consisting of nindependent examples. We have a latent vari- able model p(x,z; θ) with z being the latent variable (which for simplicity is assumed to take ﬁnite number of values). The density for xcan be obtained by marginalized over the latent variable z: p(x; θ) = ∑ z p(x,z; θ) (11.1) We wish to ﬁt the parameters θ by maximizing the log-likelihood of the data, deﬁned by ℓ(θ) = n∑ i=1 log p(x(i); θ) (11.2) We can rewrite the objective in terms of the joint density p(x,z; θ) by ℓ(θ) = n∑ i=1 log p(x(i); θ) (11.3) = n∑ i=1 log ∑ z(i) p(x(i),z(i); θ). (11.4) But, explicitly ﬁnding the maximum likelihood estimates of the parameters θ may be hard since it will result in diﬃcult non-convex optimization prob- 153 lems.3 Here, the z(i)’s are the latent random variables; and it is often the case that if the z(i)’s were observed, then maximum likelihood estimation would be easy. In such a setting, the EM algorithm gives an eﬃcient method for max- imum likelihood estimation. Maximizing ℓ(θ) explicitly might be diﬃcult, and our strategy will be to instead repeatedly construct a lower-bound on ℓ (E-step), and then optimize that lower-bound (M-step). 4 It turns out that the summation ∑n i=1 is not essential here, and towards a simpler exposition of the EM algorithm, we will ﬁrst consider optimizing the the likelihood logp(x) for a single example x. After we derive the algorithm for optimizing log p(x), we will convert it to an algorithm that works for n examples by adding back the sum to each of the relevant equations. Thus, now we aim to optimize log p(x; θ) which can be rewritten as log p(x; θ) = log ∑ z p(x,z; θ) (11.5) Let Q be a distribution over the possible values of z. That is, ∑ z Q(z) = 1, Q(z) ≥0). Consider the following:5 log p(x; θ) = log ∑ z p(x,z; θ) = log ∑ z Q(z)p(x,z; θ) Q(z) (11.6) ≥ ∑ z Q(z) log p(x,z; θ) Q(z) (11.7) The last step of this derivation used Jensen’s inequality. Speciﬁcally, f(x) = log x is a concave function, since f′′(x) = −1/x2 <0 over its domain 3It’s mostly an empirical observation that the optimization problem is diﬃcult to op- timize. 4Empirically, the E-step and M-step can often be computed more eﬃciently than op- timizing the function ℓ(·) directly. However, it doesn’t necessarily mean that alternating the two steps can always converge to the global optimum of ℓ(·). Even for mixture of Gaussians, the EM algorithm can either converge to a global optimum or get stuck, de- pending on the properties of the training data. Empirically, for real-world data, often EM can converge to a solution with relatively high likelihood (if not the optimum), and the theory behind it is still largely not understood. 5If z were continuous, then Q would be a density, and the summations over z in our discussion are replaced with integrals over z. 154 x∈R+. Also, the term ∑ z Q(z) [p(x,z; θ) Q(z) ] in the summation is just an expectation of the quantity [p(x,z; θ)/Q(z)] with respect to z drawn according to the distribution given by Q.6 By Jensen’s inequality, we have f ( Ez∼Q [p(x,z; θ) Q(z) ]) ≥Ez∼Q [ f (p(x,z; θ) Q(z) )] , where the “z ∼Q” subscripts above indicate that the expectations are with respect to z drawn from Q. This allowed us to go from Equation (11.6) to Equation (11.7). Now, for any distribution Q, the formula (11.7) gives a lower-bound on log p(x; θ). There are many possible choices for the Q’s. Which should we choose? Well, if we have some current guess θ of the parameters, it seems natural to try to make the lower-bound tight at that value of θ. I.e., we will make the inequality above hold with equality at our particular value of θ. To make the bound tight for a particular value of θ, we need for the step involving Jensen’s inequality in our derivation above to hold with equality. For this to be true, we know it is suﬃcient that the expectation be taken over a “constant”-valued random variable. I.e., we require that p(x,z; θ) Q(z) = c for some constant c that does not depend on z. This is easily accomplished by choosing Q(z) ∝p(x,z; θ). Actually, since we know ∑ z Q(z) = 1 (because it is a distribution), this further tells us that Q(z) = p(x,z; θ)∑ z p(x,z; θ) = p(x,z; θ) p(x; θ) = p(z\|x; θ) (11.8) 6We note that the notion p(x,z;θ) Q(z) only makes sense if Q(z) ̸= 0 whenever p(x,z; θ) ̸= 0. Here we implicitly assume that we only consider those Q with such a property. 155 Thus, we simply set the Q’s to be the posterior distribution of the z’s given x and the setting of the parameters θ. Indeed, we can directly verify that when Q(z) = p(z\|x; θ), then equa- tion (11.7) is an equality because ∑ z Q(z) log p(x,z; θ) Q(z) = ∑ z p(z\|x; θ) log p(x,z; θ) p(z\|x; θ) = ∑ z p(z\|x; θ) log p(z\|x; θ)p(x; θ) p(z\|x; θ) = ∑ z p(z\|x; θ) logp(x; θ) = log p(x; θ) ∑ z p(z\|x; θ) = log p(x; θ) (because ∑ z p(z\|x; θ) = 1) For convenience, we call the expression in Equation (11.7) the evidence lower bound (ELBO) and we denote it by ELBO(x; Q,θ) = ∑ z Q(z) log p(x,z; θ) Q(z) (11.9) With this equation, we can re-write equation (11.7) as ∀Q,θ,x, log p(x; θ) ≥ELBO(x; Q,θ) (11.10) Intuitively, the EM algorithm alternatively updates Q and θ by a) set- ting Q(z) = p(z\|x; θ) following Equation (11.8) so that ELBO( x; Q,θ) = log p(x; θ) for xand the current θ, and b) maximizing ELBO( x; Q,θ) w.r.t θ while ﬁxing the choice of Q. Recall that all the discussion above was under the assumption that we aim to optimize the log-likelihood log p(x; θ) for a single example x. It turns out that with multiple training examples, the basic idea is the same and we only needs to take a sum over examples at relevant places. Next, we will build the evidence lower bound for multiple training examples and make the EM algorithm formal. Recall we have a training set{x(1),...,x (n)}. Note that the optimal choice of Qis p(z\|x; θ), and it depends on the particular example x. Therefore here we will introduce n distributions Q1,...,Q n, one for each example x(i). For each example x(i), we can build the evidence lower bound log p(x(i); θ) ≥ELBO(x(i); Qi,θ) = ∑ z(i) Qi(z(i)) log p(x(i),z(i); θ) Qi(z(i)) 156 Taking sum over all the examples, we obtain a lower bound for the log- likelihood ℓ(θ) ≥ ∑ i ELBO(x(i); Qi,θ) (11.11) = ∑ i ∑ z(i) Qi(z(i)) log p(x(i),z(i); θ) Qi(z(i)) For any set of distributions Q1,...,Q n, the formula (11.11) gives a lower- bound on ℓ(θ), and analogous to the argument around equation (11.8), the Qi that attains equality satisﬁes Qi(z(i)) = p(z(i)\|x(i); θ) Thus, we simply set the Qi’s to be the posterior distribution of the z(i)’s given x(i) with the current setting of the parameters θ. Now, for this choice of the Qi’s, Equation (11.11) gives a lower-bound on the loglikelihood ℓ that we’re trying to maximize. This is the E-step. In the M-step of the algorithm, we then maximize our formula in Equation (11.11) with respect to the parameters to obtain a new setting of the θ’s. Repeatedly carrying out these two steps gives us the EM algorithm, which is as follows: Repeat until convergence { (E-step) For each i, set Qi(z(i)) := p(z(i)\|x(i); θ). (M-step) Set θ:= arg max θ n∑ i=1 ELBO(x(i); Qi,θ) = arg max θ ∑ i ∑ z(i) Qi(z(i)) log p(x(i),z(i); θ) Qi(z(i)) . (11.12) } How do we know if this algorithm will converge? Well, suppose θ(t) and θ(t+1) are the parameters from two successive iterations of EM. We will now prove that ℓ(θ(t)) ≤ℓ(θ(t+1)), which shows EM always monotonically im- proves the log-likelihood. The key to showing this result lies in our choice of 157 the Qi’s. Speciﬁcally, on the iteration of EM in which the parameters had started out as θ(t), we would have chosen Q(t) i (z(i)) := p(z(i)\|x(i); θ(t)). We saw earlier that this choice ensures that Jensen’s inequality, as applied to get Equation (11.11), holds with equality, and hence ℓ(θ(t)) = n∑ i=1 ELBO(x(i); Q(t) i ,θ(t)) (11.13) The parameters θ(t+1) are then obtained by maximizing the right hand side of the equation above. Thus, ℓ(θ(t+1)) ≥ n∑ i=1 ELBO(x(i); Q(t) i ,θ(t+1)) (because ineqaulity (11.11) holds for all Q and θ) ≥ n∑ i=1 ELBO(x(i); Q(t) i ,θ(t)) (see reason below) = ℓ(θ(t)) (by equation (11.13)) where the last inequality follows from that θ(t+1) is chosen explicitly to be arg max θ n∑ i=1 ELBO(x(i); Q(t) i ,θ) Hence, EM causes the likelihood to converge monotonically. In our de- scription of the EM algorithm, we said we’d run it until convergence. Given the result that we just showed, one reasonable convergence test would be to check if the increase in ℓ(θ) between successive iterations is smaller than some tolerance parameter, and to declare convergence if EM is improving ℓ(θ) too slowly. Remark. If we deﬁne (by overloading ELBO( ·)) ELBO(Q,θ) = n∑ i=1 ELBO(x(i); Qi,θ) = ∑ i ∑ z(i) Qi(z(i)) log p(x(i),z(i); θ) Qi(z(i)) (11.14) then we know ℓ(θ) ≥ELBO(Q,θ) from our previous derivation. The EM can also be viewed an alternating maximization algorithm on ELBO( Q,θ), in which the E-step maximizes it with respect to Q(check this yourself), and the M-step maximizes it with respect to θ. 158 11.3.1 Other interpretation of ELBO Let ELBO( x; Q,θ) = ∑ z Q(z) log p(x,z;θ) Q(z) be deﬁned as in equation (11.9). There are several other forms of ELBO. First, we can rewrite ELBO(x; Q,θ) = Ez∼Q[log p(x,z; θ)] −Ez∼Q[log Q(z)] = Ez∼Q[log p(x\|z; θ)] −DKL(Q∥pz) (11.15) where we use pz to denote the marginal distribution of z (under the distri- bution p(x,z; θ)), and DKL() denotes the KL divergence DKL(Q∥pz) = ∑ z Q(z) log Q(z) p(z) (11.16) In many cases, the marginal distribution of z does not depend on the param- eter θ. In this case, we can see that maximizing ELBO over θ is equivalent to maximizing the ﬁrst term in (11.15). This corresponds to maximizing the conditional likelihood of xconditioned on z, which is often a simpler question than the original question. Another form of ELBO( ·) is (please verify yourself) ELBO(x; Q,θ) = log p(x) −DKL(Q∥pz\|x) (11.17) where pz\|x is the conditional distribution of z given x under the parameter θ. This forms shows that the maximizer of ELBO( Q,θ) over Q is obtained when Q= pz\|x, which was shown in equation (11.8) before. 11.4 Mixture of Gaussians revisited Armed with our general deﬁnition of the EM algorithm, let’s go back to our old example of ﬁtting the parameters φ, µ and Σ in a mixture of Gaussians. For the sake of brevity, we carry out the derivations for the M-step updates only for φand µj, and leave the updates for Σ j as an exercise for the reader. The E-step is easy. Following our algorithm derivation above, we simply calculate w(i) j = Qi(z(i) = j) = P(z(i) = j\|x(i); φ,µ, Σ). Here, “Qi(z(i) = j)” denotes the probability of z(i) taking the value j under the distribution Qi. 159 Next, in the M-step, we need to maximize, with respect to our parameters φ,µ, Σ, the quantity n∑ i=1 ∑ z(i) Qi(z(i)) log p(x(i),z(i); φ,µ, Σ) Qi(z(i)) = n∑ i=1 k∑ j=1 Qi(z(i) = j) log p(x(i)\|z(i) = j; µ,Σ)p(z(i) = j; φ) Qi(z(i) = j) = n∑ i=1 k∑ j=1 w(i) j log 1 (2π)d/2\|Σj\|1/2 exp ( −1 2 (x(i) −µj)TΣ−1 j (x(i) −µj) ) ·φj w(i) j Let’s maximize this with respect to µl. If we take the derivative with respect to µl, we ﬁnd ∇µl n∑ i=1 k∑ j=1 w(i) j log 1 (2π)d/2\|Σj\|1/2 exp ( −1 2 (x(i) −µj)TΣ−1 j (x(i) −µj) ) ·φj w(i) j = −∇µl n∑ i=1 k∑ j=1 w(i) j 1 2(x(i) −µj)TΣ−1 j (x(i) −µj) = 1 2 n∑ i=1 w(i) l ∇µl2µT l Σ−1 l x(i) −µT l Σ−1 l µl = n∑ i=1 w(i) l ( Σ−1 l x(i) −Σ−1 l µl ) Setting this to zero and solving for µl therefore yields the update rule µl := ∑n i=1 w(i) l x(i) ∑n i=1 w(i) l , which was what we had in the previous set of notes. Let’s do one more example, and derive the M-step update for the param- eters φj. Grouping together only the terms that depend on φj, we ﬁnd that we need to maximize n∑ i=1 k∑ j=1 w(i) j log φj. However, there is an additional constraint that the φj’s sum to 1, since they represent the probabilities φj = p(z(i) = j; φ). To deal with the constraint 160 that ∑k j=1 φj = 1, we construct the Lagrangian L(φ) = n∑ i=1 k∑ j=1 w(i) j log φj + β( k∑ j=1 φj −1), where β is the Lagrange multiplier.7 Taking derivatives, we ﬁnd ∂ ∂φj L(φ) = n∑ i=1 w(i) j φj + β Setting this to zero and solving, we get φj = ∑n i=1 w(i) j −β I.e., φj ∝∑n i=1 w(i) j . Using the constraint that ∑ j φj = 1, we easily ﬁnd that −β = ∑n i=1 ∑k j=1 w(i) j = ∑n i=1 1 = n. (This used the fact that w(i) j = Qi(z(i) = j), and since probabilities sum to 1, ∑ j w(i) j = 1.) We therefore have our M-step updates for the parameters φj: φj := 1 n n∑ i=1 w(i) j . The derivation for the M-step updates to Σ j are also entirely straightfor- ward. 11.5 Variational inference and variational auto-encoder (optional reading) Loosely speaking, variational auto-encoder Kingma and Welling [2013] gen- erally refers to a family of algorithms that extend the EM algorithms to more complex models parameterized by neural networks. It extends the technique of variational inference with the additional “re-parametrization trick” which will be introduced below. Variational auto-encoder may not give the best performance for many datasets, but it contains several central ideas about how to extend EM algorithms to high-dimensional continuous latent variables 7We don’t need to worry about the constraint that φj ≥0, because as we’ll shortly see, the solution we’ll ﬁnd from this derivation will automatically satisfy that anyway. 161 with non-linear models. Understanding it will likely give you the language and backgrounds to understand various recent papers related to it. As a running example, we will consider the following parameterization of p(x,z; θ) by a neural network. Let θ be the collection of the weights of a neural network g(z; θ) that maps z ∈Rk to Rd. Let z ∼N(0,Ik×k) (11.18) x\|z ∼N(g(z; θ),σ2Id×d) (11.19) Here Ik×k denotes identity matrix of dimension k by k, and σ is a scalar that we assume to be known for simplicity. For the Gaussian mixture models in Section 11.4, the optimal choice of Q(z) = p(z\|x; θ) for each ﬁxed θ, that is the posterior distribution of z, can be analytically computed. In many more complex models such as the model (11.19), it’s intractable to compute the exact the posterior distribution p(z\|x; θ). Recall that from equation (11.10), ELBO is always a lower bound for any choice of Q, and therefore, we can also aim for ﬁnding an approximation of the true posterior distribution. Often, one has to use some particular form to approximate the true posterior distribution. Let Qbe a family of Q’s that we are considering, and we will aim to ﬁnd a Qwithin the family of Qthat is closest to the true posterior distribution. To formalize, recall the deﬁnition of the ELBO lower bound as a function of Q and θ deﬁned in equation (11.14) ELBO(Q,θ) = n∑ i=1 ELBO(x(i); Qi,θ) = ∑ i ∑ z(i) Qi(z(i)) log p(x(i),z(i); θ) Qi(z(i)) Recall that EM can be viewed as alternating maximization of ELBO(Q,θ). Here instead, we optimize the EBLO over Q∈Q max Q∈Q max θ ELBO(Q,θ) (11.20) Now the next question is what form of Q(or what structural assumptions to make aboutQ) allows us to eﬃciently maximize the objective above. When the latent variable z are high-dimensional discrete variables, one popular as- sumption is the mean ﬁeld assumption, which assumes that Qi(z) gives a distribution with independent coordinates, or in other words, Qi can be de- composed into Qi(z) = Q1 i(z1) ···Qk i(zk). There are tremendous applications of mean ﬁeld assumptions to learning generative models with discrete latent variables, and we refer to Blei et al. [2017] for a survey of these models and 162 their impact to a wide range of applications including computational biology, computational neuroscience, social sciences. We will not get into the details about the discrete latent variable cases, and our main focus is to deal with continuous latent variables, which requires not only mean ﬁeld assumptions, but additional techniques. When z ∈Rk is a continuous latent variable, there are several decisions to make towards successfully optimizing (11.20). First we need to give a succinct representation of the distribution Qi because it is over an inﬁnite number of points. A natural choice is to assume Qi is a Gaussian distribution with some mean and variance. We would also like to have more succinct representation of the means of Qi of all the examples. Note that Qi(z(i)) is supposed to approximate p(z(i)\|x(i); θ). It would make sense let all the means of the Qi’s be some function of x(i). Concretely, let q(·; φ),v(·; φ) be two functions that map from dimension dto k, which are parameterized by φand ψ, we assume that Qi = N(q(x(i); φ),diag(v(x(i); ψ))2) (11.21) Here diag(w) means the k×k matrix with the entries of w ∈Rk on the diagonal. In other words, the distribution Qi is assumed to be a Gaussian distribution with independent coordinates, and the mean and standard de- viations are governed by q and v. Often in variational auto-encoder, q and v are chosen to be neural networks. 8 In recent deep learning literature, often q,v are called encoder (in the sense of encoding the data into latent code), whereas g(z; θ) if often referred to as the decoder. We remark thatQi of such form in many cases are very far from a good ap- proximation of the true posterior distribution. However, some approximation is necessary for feasible optimization. In fact, the form of Qi needs to satisfy other requirements (which happened to be satisﬁed by the form (11.21)) Before optimizing the ELBO, let’s ﬁrst verify whether we can eﬃciently evaluate the value of the ELBO for ﬁxed Q of the form (11.21) and θ. We rewrite the ELBO as a function of φ,ψ,θ by ELBO(φ,ψ,θ ) = n∑ i=1 Ez(i)∼Qi [ log p(x(i),z(i); θ) Qi(z(i)) ] , (11.22) where Qi = N(q(x(i); φ),diag(v(x(i); ψ))2) Note that to evaluate Qi(z(i)) inside the expectation, we should be able to compute the density of Qi. To estimate the expectation E z(i)∼Qi, we 8q and v can also share parameters. We sweep this level of details under the rug in this note. 163 should be able to sample from distribution Qi so that we can build an empirical estimator with samples. It happens that for Gaussian distribution Qi = N(q(x(i); φ),diag(v(x(i); ψ))2), we are able to be both eﬃciently. Now let’s optimize the ELBO. It turns out that we can run gradient ascent over φ,ψ,θ instead of alternating maximization. There is no strong need to compute the maximum over each variable at a much greater cost. (For Gaus- sian mixture model in Section 11.4, computing the maximum is analytically feasible and relatively cheap, and therefore we did alternating maximization.) Mathematically, let η be the learning rate, the gradient ascent step is θ:= θ+ η∇θELBO(φ,ψ,θ ) φ:= φ+ η∇φELBO(φ,ψ,θ ) ψ:= ψ+ η∇ψELBO(φ,ψ,θ ) Computing the gradient over θ is simple because ∇θELBO(φ,ψ,θ ) = ∇θ n∑ i=1 Ez(i)∼Qi [ log p(x(i),z(i); θ) Qi(z(i)) ] = ∇θ n∑ i=1 Ez(i)∼Qi [ log p(x(i),z(i); θ) ] = n∑ i=1 Ez(i)∼Qi [ ∇θlog p(x(i),z(i); θ) ] , (11.23) But computing the gradient over φ and ψ is tricky because the sam- pling distribution Qi depends on φ and ψ. (Abstractly speaking, the is- sue we face can be simpliﬁed as the problem of computing the gradi- ent E z∼Qφ[f(φ)] with respect to variable φ. We know that in general, ∇Ez∼Qφ[f(φ)] ̸= Ez∼Qφ[∇f(φ)] because the dependency of Qφ on φhas to be taken into account as well. ) The idea that comes to rescue is the so-called re-parameterization trick: we rewrite z(i) ∼Qi = N(q(x(i); φ),diag(v(x(i); ψ))2) in an equivalent way: z(i) = q(x(i); φ) + v(x(i); ψ) ⊙ξ(i) where ξ(i) ∼N(0,Ik×k) (11.24) Here x⊙y denotes the entry-wise product of two vectors of the same dimension. Here we used the fact that x ∼N(µ,σ2) is equivalent to that x= µ+ξσ with ξ ∼N(0,1). We mostly just used this fact in every dimension simultaneously for the random variable z(i) ∼Qi. 164 With this re-parameterization, we have that Ez(i)∼Qi [ log p(x(i),z(i); θ) Qi(z(i)) ] (11.25) = Eξ(i)∼N(0,1) [ log p(x(i),q(x(i); φ) + v(x(i); ψ) ⊙ξ(i); θ) Qi(q(x(i); φ) + v(x(i); ψ) ⊙ξ(i)) ] It follows that ∇φEz(i)∼Qi [ log p(x(i),z(i); θ) Qi(z(i)) ] = ∇φEξ(i)∼N(0,1) [ log p(x(i),q(x(i); φ) + v(x(i); ψ) ⊙ξ(i); θ) Qi(q(x(i); φ) + v(x(i); ψ) ⊙ξ(i)) ] = Eξ(i)∼N(0,1) [ ∇φlog p(x(i),q(x(i); φ) + v(x(i); ψ) ⊙ξ(i); θ) Qi(q(x(i); φ) + v(x(i); ψ) ⊙ξ(i)) ] We can now sample multiple copies of ξ(i)’s to estimate the the expecta- tion in the RHS of the equation above. 9 We can estimate the gradient with respect to ψ similarly, and with these, we can implement the gradient ascent algorithm to optimize the ELBO over φ,ψ,θ. There are not many high-dimensional distributions with analytically com- putable density function are known to be re-parameterizable. We refer to Kingma and Welling [2013] for a few other choices that can replace Gaussian distribution. 9Empirically people sometimes just use one sample to estimate it for maximum com- putational eﬃciency. Chapter 12 Principal components analysis In this set of notes, we will develop a method, Principal Components Analysis (PCA), that tries to identify the subspace in which the data approximately lies. PCA is computationally eﬃcient: it will require only an eigenvector calculation (easily done with the eig function in Matlab). Suppose we are given a dataset {x(i); i= 1,...,n }of attributes of n dif- ferent types of automobiles, such as their maximum speed, turn radius, and so on. Let x(i) ∈Rd for each i (d ≪n). But unknown to us, two diﬀerent attributes—some xi and xj—respectively give a car’s maximum speed mea- sured in miles per hour, and the maximum speed measured in kilometers per hour. These two attributes are therefore almost linearly dependent, up to only small diﬀerences introduced by rounding oﬀ to the nearest mph or kph. Thus, the data really lies approximately on an n−1 dimensional subspace. How can we automatically detect, and perhaps remove, this redundancy? For a less contrived example, consider a dataset resulting from a survey of pilots for radio-controlled helicopters, where x(i) 1 is a measure of the piloting skill of pilot i, and x(i) 2 captures how much he/she enjoys ﬂying. Because RC helicopters are very diﬃcult to ﬂy, only the most committed students, ones that truly enjoy ﬂying, become good pilots. So, the two attributes x1 and x2 are strongly correlated. Indeed, we might posit that that the data actually likes along some diagonal axis (the u1 direction) capturing the intrinsic piloting “karma” of a person, with only a small amount of noise lying oﬀ this axis. (See ﬁgure.) How can we automatically compute this u1 direction? 165 166 x1 x2 (enjoyment) (skill) 1 u u 2 We will shortly develop the PCA algorithm. But prior to running PCA per se, typically we ﬁrst preprocess the data by normalizing each feature to have mean 0 and variance 1. We do this by subtracting the mean and dividing by the empirical standard deviation: x(i) j ←x(i) j −µj σj where µj = 1 n ∑n i=1 x(i) j and σ2 j = 1 n ∑n i=1(x(i) j −µj)2 are the mean variance of feature j, respectively. Subtracting µj zeros out the mean and may be omitted for data known to have zero mean (for instance, time series corresponding to speech or other acoustic signals). Dividing by the standard deviation σj rescales each coor- dinate to have unit variance, which ensures that diﬀerent attributes are all treated on the same “scale.” For instance, if x1 was cars’ maximum speed in mph (taking values in the high tens or low hundreds) and x2 were the num- ber of seats (taking values around 2-4), then this renormalization rescales the diﬀerent attributes to make them more comparable. This rescaling may be omitted if we had a priori knowledge that the diﬀerent attributes are all on the same scale. One example of this is if each data point represented a grayscale image, and each x(i) j took a value in {0,1,..., 255}corresponding to the intensity value of pixel j in image i. Now, having normalized our data, how do we compute the “major axis of variation” u—that is, the direction on which the data approximately lies? One way is to pose this problem as ﬁnding the unit vector u so that when 167 the data is projected onto the direction corresponding to u, the variance of the projected data is maximized. Intuitively, the data starts oﬀ with some amount of variance/information in it. We would like to choose a direction u so that if we were to approximate the data as lying in the direction/subspace corresponding to u, as much as possible of this variance is still retained. Consider the following dataset, on which we have already carried out the normalization steps: Now, suppose we pick u to correspond the the direction shown in the ﬁgure below. The circles denote the projections of the original data onto this line. 168 /0 /0 /1 /1 /0/0 /0/0 /1/1 /1/1 /0 /1 /0/0 /0/0 /1/1 /1/1 /0 /1/0/0 /0/0 /1/1 /1/1 /0/0 /0/0 /0/0 /0/0 /0/0 /0/0 /0/0 /1/1 /1/1 /1/1 /1/1 /1/1 /1/1 /1/1 /0/0 /0/0 /0/0 /0/0 /1/1 /1/1 /1/1 /1/1 /0/0/0 /0/0/0 /0/0/0 /0/0/0 /0/0/0 /0/0/0 /0/0/0 /1/1/1 /1/1/1 /1/1/1 /1/1/1 /1/1/1 /1/1/1 /1/1/1 /0/0 /0/0 /0/0 /0/0 /1/1 /1/1 /1/1 /1/1 We see that the projected data still has a fairly large variance, and the points tend to be far from zero. In contrast, suppose had instead picked the following direction: /0/0 /1/1 /0/0 /0/0 /1/1 /1/1 /0 /1 /0/0 /0/0 /1/1 /1/1 /0 /0 /1 /1 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0 /0/0/0/0/0/0 /0/0/0/0/0/0 /0/0/0/0/0/0 /0/0/0/0/0/0 /0/0/0/0/0/0 /0/0/0/0/0/0 /0/0/0/0/0/0 /0/0/0/0/0/0 /0/0/0/0/0/0 /0/0/0/0/0/0 /1/1/1/1/1/1 /1/1/1/1/1/1 /1/1/1/1/1/1 /1/1/1/1/1/1 /1/1/1/1/1/1 /1/1/1/1/1/1 /1/1/1/1/1/1 /1/1/1/1/1/1 /1/1/1/1/1/1 /1/1/1/1/1/1 /1/1/1/1/1/1 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /1/1/1/1/1/1/1/1 /0/0/0/0/0 /0/0/0/0/0 /0/0/0/0/0 /0/0/0/0/0 /0/0/0/0/0 /0/0/0/0/0 /0/0/0/0/0 /0/0/0/0/0 /1/1/1/1/1 /1/1/1/1/1 /1/1/1/1/1 /1/1/1/1/1 /1/1/1/1/1 /1/1/1/1/1 /1/1/1/1/1 /1/1/1/1/1 Here, the projections have a signiﬁcantly smaller variance, and are much closer to the origin. We would like to automatically select the direction u corresponding to the ﬁrst of the two ﬁgures shown above. To formalize this, note that given a 169 unit vector uand a point x, the length of the projection of xonto uis given by xTu. I.e., if x(i) is a point in our dataset (one of the crosses in the plot), then its projection onto u(the corresponding circle in the ﬁgure) is distance xTufrom the origin. Hence, to maximize the variance of the projections, we would like to choose a unit-length u so as to maximize: 1 n n∑ i=1 (x(i)T u)2 = 1 n n∑ i=1 uTx(i)x(i)T u = uT ( 1 n n∑ i=1 x(i)x(i)T ) u. We easily recognize that the maximizing this subject to ∥u∥2 = 1 gives the principal eigenvector of Σ = 1 n ∑n i=1 x(i)x(i)T , which is just the empirical covariance matrix of the data (assuming it has zero mean). 1 To summarize, we have found that if we wish to ﬁnd a 1-dimensional subspace with with to approximate the data, we should choose u to be the principal eigenvector of Σ. More generally, if we wish to project our data into a k-dimensional subspace (k <d), we should choose u1,...,u k to be the top k eigenvectors of Σ. The ui’s now form a new, orthogonal basis for the data.2 Then, to represent x(i) in this basis, we need only compute the corre- sponding vector y(i) =   uT 1 x(i) uT 2 x(i) ... uT kx(i)  ∈Rk. Thus, whereas x(i) ∈Rd, the vector y(i) now gives a lower, k-dimensional, approximation/representation for x(i). PCA is therefore also referred to as a dimensionality reduction algorithm. The vectors u1,...,u k are called the ﬁrst k principal components of the data. Remark. Although we have shown it formally only for the case of k = 1, using well-known properties of eigenvectors it is straightforward to show that 1If you haven’t seen this before, try using the method of Lagrange multipliers to max- imize uTΣusubject to that uTu= 1. You should be able to show that Σ u= λu, for some λ, which implies u is an eigenvector of Σ, with eigenvalue λ. 2Because Σ is symmetric, the ui’s will (or always can be chosen to be) orthogonal to each other. 170 of all possible orthogonal bases u1,...,u k, the one that we have chosen max- imizes ∑ i∥y(i)∥2 2. Thus, our choice of a basis preserves as much variability as possible in the original data. PCA can also be derived by picking the basis that minimizes the ap- proximation error arising from projecting the data onto the k-dimensional subspace spanned by them. (See more in homework.) PCA has many applications; we will close our discussion with a few exam- ples. First, compression—representing x(i)’s with lower dimension y(i)’s—is an obvious application. If we reduce high dimensional data to k= 2 or 3 di- mensions, then we can also plot the y(i)’s to visualize the data. For instance, if we were to reduce our automobiles data to 2 dimensions, then we can plot it (one point in our plot would correspond to one car type, say) to see what cars are similar to each other and what groups of cars may cluster together. Another standard application is to preprocess a dataset to reduce its dimension before running a supervised learning learning algorithm with the x(i)’s as inputs. Apart from computational beneﬁts, reducing the data’s dimension can also reduce the complexity of the hypothesis class considered and help avoid overﬁtting (e.g., linear classiﬁers over lower dimensional input spaces will have smaller VC dimension). Lastly, as in our RC pilot example, we can also view PCA as a noise reduction algorithm. In our example it, estimates the intrinsic “piloting karma” from the noisy measures of piloting skill and enjoyment. In class, we also saw the application of this idea to face images, resulting in eigenfaces method. Here, each point x(i) ∈R100×100 was a 10000 dimensional vector, with each coordinate corresponding to a pixel intensity value in a 100x100 image of a face. Using PCA, we represent each image x(i) with a much lower- dimensional y(i). In doing so, we hope that the principal components we found retain the interesting, systematic variations between faces that capture what a person really looks like, but not the “noise” in the images introduced by minor lighting variations, slightly diﬀerent imaging conditions, and so on. We then measure distances between faces iand j by working in the reduced dimension, and computing ∥y(i) −y(j)∥2. This resulted in a surprisingly good face-matching and retrieval algorithm. Chapter 13 Independent components analysis Our next topic is Independent Components Analysis (ICA). Similar to PCA, this will ﬁnd a new basis in which to represent our data. However, the goal is very diﬀerent. As a motivating example, consider the “cocktail party problem.” Here, d speakers are speaking simultaneously at a party, and any microphone placed in the room records only an overlapping combination of thedspeakers’ voices. But lets say we haveddiﬀerent microphones placed in the room, and because each microphone is a diﬀerent distance from each of the speakers, it records a diﬀerent combination of the speakers’ voices. Using these microphone record- ings, can we separate out the original d speakers’ speech signals? To formalize this problem, we imagine that there is some data s ∈Rd that is generated via d independent sources. What we observe is x= As, where Ais an unknown square matrix called the mixing matrix. Repeated observations gives us a dataset {x(i); i= 1,...,n }, and our goal is to recover the sources s(i) that had generated our data ( x(i) = As(i)). In our cocktail party problem, s(i) is an d-dimensional vector, and s(i) j is the sound that speaker jwas uttering at timei. Also, x(i) in an d-dimensional vector, and x(i) j is the acoustic reading recorded by microphone j at time i. Let W = A−1 be the unmixing matrix. Our goal is to ﬁnd W, so that given our microphone recordings x(i), we can recover the sources by computing s(i) = Wx(i). For notational convenience, we also let wT i denote 171 172 the i-th row of W, so that W =   — wT 1 — ... — wT d —  . Thus, wi ∈Rd, and the j-th source can be recovered as s(i) j = wT j x(i). 13.1 ICA ambiguities To what degree can W = A−1 be recovered? If we have no prior knowledge about the sources and the mixing matrix, it is easy to see that there are some inherent ambiguities in Athat are impossible to recover, given only thex(i)’s. Speciﬁcally, let P be any d-by-d permutation matrix. This means that each row and each column of P has exactly one “1.” Here are some examples of permutation matrices: P =   0 1 0 1 0 0 0 0 1  ; P = [ 0 1 1 0 ] ; P = [ 1 0 0 1 ] . If z is a vector, then Pz is another vector that contains a permuted version of z’s coordinates. Given only the x(i)’s, there will be no way to distinguish between W and PW. Speciﬁcally, the permutation of the original sources is ambiguous, which should be no surprise. Fortunately, this does not matter for most applications. Further, there is no way to recover the correct scaling of the wi’s. For in- stance, if Awere replaced with 2A, and every s(i) were replaced with (0.5)s(i), then our observed x(i) = 2A·(0.5)s(i) would still be the same. More broadly, if a single column of A were scaled by a factor of α, and the corresponding source were scaled by a factor of 1/α, then there is again no way to determine that this had happened given only the x(i)’s. Thus, we cannot recover the “correct” scaling of the sources. However, for the applications that we are concerned with—including the cocktail party problem—this ambiguity also does not matter. Speciﬁcally, scaling a speaker’s speech signal s(i) j by some positive factor αaﬀects only the volume of that speaker’s speech. Also, sign changes do not matter, and s(i) j and −s(i) j sound identical when played on a speaker. Thus, if the wi found by an algorithm is scaled by any non-zero real number, the corresponding recovered source si = wT i x will be scaled by the 173 same factor; but this usually does not matter. (These comments also apply to ICA for the brain/MEG data that we talked about in class.) Are these the only sources of ambiguity in ICA? It turns out that they are, so long as the sources si are non-Gaussian. To see what the diﬃculty is with Gaussian data, consider an example in which n = 2, and s ∼N(0,I). Here, I is the 2x2 identity matrix. Note that the contours of the density of the standard normal distribution N(0,I) are circles centered on the origin, and the density is rotationally symmetric. Now, suppose we observe some x = As, where A is our mixing matrix. Then, the distribution of x will be Gaussian, x∼N(0,AAT), since Es∼N(0,I)[x] = E[As] = AE[s] = 0 Cov[x] = Es∼N(0,I)[xxT] = E[AssTAT] = AE[ssT]AT = A·Cov[s] ·AT = AAT Now, let R be an arbitrary orthogonal (less formally, a rotation/reﬂection) matrix, so that RRT = RTR = I, and let A′ = AR. Then if the data had been mixed according to A′ instead of A, we would have instead observed x′ = A′s. The distribution of x′ is also Gaussian, x′ ∼N (0,AAT), since Es∼N(0,I)[x′(x′)T] = E[ A′ssT(A′)T] = E[ ARssT(AR)T] = ARRTAT = AAT. Hence, whether the mixing matrix is A or A′, we would observe data from a N(0,AAT) distribution. Thus, there is no way to tell if the sources were mixed using A and A′. There is an arbitrary rotational component in the mixing matrix that cannot be determined from the data, and we cannot recover the original sources. Our argument above was based on the fact that the multivariate standard normal distribution is rotationally symmetric. Despite the bleak picture that this paints for ICA on Gaussian data, it turns out that, so long as the data is not Gaussian, it is possible, given enough data, to recover the dindependent sources. 13.2 Densities and linear transformations Before moving on to derive the ICA algorithm proper, we ﬁrst digress brieﬂy to talk about the eﬀect of linear transformations on densities. Suppose a random variable s is drawn according to some density ps(s). For simplicity, assume for now that s ∈R is a real number. Now, let the random variable xbe deﬁned according to x= As(here, x∈R,A ∈R). Let px be the density of x. What is px? Let W = A−1. To calculate the “probability” of a particular value of x, it is tempting to compute s= Wx, then then evaluate ps at that point, and 174 conclude that “ px(x) = ps(Wx).” However, this is incorrect . For example, let s ∼Uniform[0,1], so ps(s) = 1{0 ≤s ≤1}. Now, let A = 2, so x = 2s. Clearly, x is distributed uniformly in the interval [0 ,2]. Thus, its density is given by px(x) = (0 .5)1{0 ≤x ≤2}. This does not equal ps(Wx), where W = 0.5 = A−1. Instead, the correct formula is px(x) = ps(Wx)\|W\|. More generally, if s is a vector-valued distribution with density ps, and x= As for a square, invertible matrix A, then the density of x is given by px(x) = ps(Wx) ·\|W\|, where W = A−1. Remark. If you’re seen the result that Amaps [0,1]d to a set of volume \|A\|, then here’s another way to remember the formula forpx given above, that also generalizes our previous 1-dimensional example. Speciﬁcally, let A∈Rd×d be given, and let W = A−1 as usual. Also let C1 = [0,1]d be the d-dimensional hypercube, and deﬁne C2 = {As : s ∈C1}⊆ Rd to be the image of C1 under the mapping given by A. Then it is a standard result in linear algebra (and, indeed, one of the ways of deﬁning determinants) that the volume of C2 is given by \|A\|. Now, suppose s is uniformly distributed in [0 ,1]d, so its density is ps(s) = 1 {s ∈C1}. Then clearly x will be uniformly distributed in C2. Its density is therefore found to be px(x) = 1{x∈C2}/vol(C2) (since it must integrate over C2 to 1). But using the fact that the determinant of the inverse of a matrix is just the inverse of the determinant, we have 1/vol(C2) = 1/\|A\|= \|A−1\|= \|W\|. Thus, px(x) = 1{x∈C2}\|W\|= 1{Wx ∈ C1}\|W\|= ps(Wx)\|W\|. 13.3 ICA algorithm We are now ready to derive an ICA algorithm. We describe an algorithm by Bell and Sejnowski, and we give an interpretation of their algorithm as a method for maximum likelihood estimation. (This is diﬀerent from their orig- inal interpretation involving a complicated idea called the infomax principal which is no longer necessary given the modern understanding of ICA.) We suppose that the distribution of each source sj is given by a density ps, and that the joint distribution of the sources s is given by p(s) = d∏ j=1 ps(sj). 175 Note that by modeling the joint distribution as a product of marginals, we capture the assumption that the sources are independent. Using our formulas from the previous section, this implies the following density on x = As = W−1s: p(x) = d∏ j=1 ps(wT j x) ·\|W\|. All that remains is to specify a density for the individual sources ps. Recall that, given a real-valued random variable z, its cumulative distri- bution function (cdf) F is deﬁned by F(z0) = P(z ≤z0) = ∫z0 −∞pz(z)dz and the density is the derivative of the cdf: pz(z) = F′(z). Thus, to specify a density for the si’s, all we need to do is to specify some cdf for it. A cdf has to be a monotonic function that increases from zero to one. Following our previous discussion, we cannot choose the Gaussian cdf, as ICA doesn’t work on Gaussian data. What we’ll choose instead as a reasonable “default” cdf that slowly increases from 0 to 1, is the sigmoid function g(s) = 1/(1 + e−s). Hence, ps(s) = g′(s).1 The square matrix W is the parameter in our model. Given a training set {x(i); i= 1,...,n }, the log likelihood is given by ℓ(W) = n∑ i=1 ( d∑ j=1 log g′(wT j x(i)) + log\|W\| ) . We would like to maximize this in terms W. By taking derivatives and using the fact (from the ﬁrst set of notes) that ∇W\|W\|= \|W\|(W−1)T, we easily derive a stochastic gradient ascent learning rule. For a training example x(i), the update rule is: W := W + α     1 −2g(wT 1 x(i)) 1 −2g(wT 2 x(i)) ... 1 −2g(wT dx(i))  x(i)T + (WT)−1  , 1If you have prior knowledge that the sources’ densities take a certain form, then it is a good idea to substitute that in here. But in the absence of such knowledge, the sigmoid function can be thought of as a reasonable default that seems to work well for many problems. Also, the presentation here assumes that either the data x(i) has been preprocessed to have zero mean, or that it can naturally be expected to have zero mean (such as acoustic signals). This is necessary because our assumption that ps(s) = g′(s) implies E[ s] = 0 (the derivative of the logistic function is a symmetric function, and hence gives a density corresponding to a random variable with zero mean), which implies E[x] = E[As] = 0. 176 where α is the learning rate. After the algorithm converges, we then compute s(i) = Wx(i) to recover the original sources. Remark. When writing down the likelihood of the data, we implicitly as- sumed that the x(i)’s were independent of each other (for diﬀerent values of i; note this issue is diﬀerent from whether the diﬀerent coordinates of x(i) are independent), so that the likelihood of the training set was given by ∏ ip(x(i); W). This assumption is clearly incorrect for speech data and other time series where the x(i)’s are dependent, but it can be shown that having correlated training examples will not hurt the performance of the al- gorithm if we have suﬃcient data. However, for problems where successive training examples are correlated, when implementing stochastic gradient as- cent, it sometimes helps accelerate convergence if we visit training examples in a randomly permuted order. (I.e., run stochastic gradient ascent on a randomly shuﬄed copy of the training set.) Chapter 14 Self-supervised learning and foundation models Despite its huge success, supervised learning with neural networks typically relies on the availability of a labeled dataset of decent size, which is some- times costly to collect. Recently, AI and machine learning are undergoing a paradigm shift with the rise of models (e.g., BERT [Devlin et al., 2019] and GPT-3 [Brown et al., 2020]) that are pre-trained on broad data at scale and are adaptable to a wide range of downstream tasks. These models, called foundation models by Bommasani et al. [2021], oftentimes leverage massive unlabeled data so that much fewer labeled data in the downstream tasks are needed. Moreover, though foundation models are based on standard deep learning and transfer learning, their scale results in new emergent capabil- ities. These models are typically (pre-)trained by self-supervised learning methods where the supervisions/labels come from parts of the inputs. This chapter will introduce the paradigm of foundation models and basic related concepts. 14.1 Pretraining and adaptation The foundation models paradigm consists of two phases: pretraining (or sim- ply training) and adaptation. We ﬁrst pretrain a large model on a massive unlabeled dataset (e.g., billions of unlabeled images). 1 Then, we adapt the pretrained model to a downstream task (e.g., detecting cancer from scan im- ages). These downstream tasks are often prediction tasks with limited or 1Sometimes, pretraining can involve large-scale labeled datasets as well (e.g., the Ima- geNet dataset). 177 178 even no labeled data. The intuition is that the pretrained models learn good representations that capture intrinsic semantic structure/ information about the data, and the adaptation phase customizes the model to a particular downstream task by, e.g., retrieving the information speciﬁc to it. For ex- ample, a model pretrained on massive unlabeled image data may learn good general visual representations/features, and we adapt the representations to solve biomedical imagining tasks. We formalize the two phases below. Pretraining. Suppose we have an unlabeled pretraining dataset {x(1),x(2) ··· ,x(n)}that consists of nexamples in Rd. Let φθ be a model that is parameterized by θand maps the input xto some m-dimensional represen- tation φθ(x). (People also call φθ(x) ∈Rm the embedding or features of the example x.) We pretrain the model θ with a pretraining loss, which is often an average of loss functions on all the examples:Lpre(θ) = 1 n ∑n i=1 ℓpre(θ,x(i)). Here ℓpre is a so-called self-supervised loss on a single datapoint x(i), because as shown later, e.g., in Section 14.3, the “supervision” comes from the data point x(i) itself. It is also possible that the pretraining loss is not a sum of losses on individual examples. We will discuss two pretraining losses in Section 14.2 and Section 14.3. We use some optimizers (mostly likely SGD or ADAM [Kingma and Ba, 2014]) to minimize Lpre(θ). We denote the obtained pretrained model by ˆθ. Adaptation. For a downstream task, we usually have a labeled dataset {(x(1) task,y(1) task),··· ,(x(ntask) task ,y(ntask) task )}with ntask examples. The setting when ntask = 0 is called zero-shot learning—the downstream task doesn’t have any labeled examples. When ntask is relatively small (say, between 1 and 50), the setting is called few-shot learning. It’s also pretty common to have a larger ntask on the order of ranging from hundreds to tens of thousands. An adaptation algorithm generally takes in a downstream dataset and the pretrained model ˆθ, and outputs a variant of ˆθ that solves the downstream task. We will discuss below two popular and general adaptation methods, linear probe and ﬁnetuning. In addition, two other methods speciﬁc to lan- guage problems are introduced in 14.3.2. The linear probe approach uses a linear head on top of the representation to predict the downstream labels. Mathematically, the adapted model out- puts w⊤φˆθ(x), where w ∈Rm is a parameter to be learned, and ˆθ is exactly the pretrained model (ﬁxed). We can use SGD (or other optimizers) to train 179 w on the downstream task loss to predict the task label min w∈Rm 1 ntask ntask∑ i=1 ℓtask(y(i) task,w⊤φˆθ(x(i) task)) (14.1) E.g., if the downstream task is a regression problem, we will have ℓtask(ytask,w⊤φˆθ(xtask)) = (ytask −w⊤φˆθ(xtask))2. The ﬁnetuning algorithm uses a similar structure for the downstream prediction model, but also further ﬁnetunes the pretrained model (instead of keeping it ﬁxed). Concretely, the prediction model is w⊤φθ(x) with pa- rameters w and θ. We optimize both w and θ to ﬁt the downstream data, but initialize θ with the pretrained model ˆθ. The linear head w is usually initialized randomly. minimize w,θ 1 ntask ntask∑ i=1 ℓtask(y(i) task,w⊤φθ(x(i) task)) (14.2) with initialization w←random vector (14.3) θ←ˆθ (14.4) Various other adaptation methods exists and are sometimes specialized to the particular pretraining methods. We will discuss one of them in Sec- tion 14.3.2. 14.2 Pretraining methods in computer vision This section introduces two concrete pretraining methods for computer vi- sion: supervised pretraining and contrastive learning. Supervised pretraining. Here, the pretraining dataset is a large-scale labeled dataset (e.g., ImageNet), and the pretrained models are simply a neural network trained with vanilla supervised learning (with the last layer being removed). Concretely, suppose we write the learned neural network as Uφˆθ(x), where U is the last (fully-connected) layer parameters, ˆθcorresponds to the parameters of all the other layers, and φˆθ(x) are the penultimate activations layer (which serves as the representation). We simply discard U and use φˆθ(x) as the pretrained model. Contrastive learning. Contrastive learning is a self-supervised pretraining method that uses only unlabeled data. The main intuition is that a good representation function φθ(·) should map semantically similar images to sim- ilar representations, and that random pair of images should generally have 180 distinct representations. E.g., we may want to map images of two huskies to similar representations, but a husky and an elephant should have diﬀerent representations. One deﬁnition of similarity is that images from the same class are similar. Using this deﬁnition will result in the so-called supervised contrastive algorithms that work well when labeled pretraining datasets are available. Without labeled data, we can use data augmentation to generate a pair of “similar” augmented images given an original image x. Data augmenta- tion typically means that we apply random cropping, ﬂipping, and/or color transformation on the original image x to generate a variant. We can take two random augmentations, denoted by ˆx and ˜x, of the same original image x, and call them a positive pair. We observe that positive pairs of images are often semantically related because they are augmentations of the same image. We will design a loss function for θ such that the representations of a positive pair, φθ(ˆx),φθ(˜x), as close to each other as possible. On the other hand, we can also take another random image z from the pretraining dataset and generate an augmentation ˆz from z. Note that (ˆx,ˆz) are from diﬀerent images; therefore, with a good chance, they are not seman- tically related. We call (ˆx,ˆz) a negative or random pair. 2 We will design a loss to push the representation of random pairs, φθ(ˆx),φθ(ˆz), far away from each other. There are many recent algorithms based on the contrastive learning prin- ciple, and here we introduce SIMCLR [Chen et al., 2020] as an concrete example. The loss function is deﬁned on a batch of examples ( x1,··· ,x(B)) with batch size B. The algorithm computes two random augmentations for each example x(i) in the batch, denoted by ˆ x(i) and ˜x(i). As a result, we have the augmented batch of 2 B examples: ˆx1,··· ,ˆx(B), ˜x1,··· ,˜x(B). The SIMCLR loss is deﬁned as 3 Lpre(θ) = − B∑ i=1 log exp ( φθ(ˆx(i))⊤φθ(˜x(i)) ) exp (φθ(ˆx(i))⊤φθ(˜x(i))) + ∑ j̸=iexp (φθ(ˆx(i))⊤φθ(˜x(j))). The intuition is as follows. The loss is increasing in φθ(ˆx(i))⊤φθ(˜x(j)), and thus minimizing the loss encourages φθ(ˆx(i))⊤φθ(˜x(j)) to be small, making φθ(ˆx(i)) far away from φθ(˜x(j)). On the other hand, the loss is decreasing in 2Random pair may be a more accurate term because it’s still possible (though not likely) that x and z are semantically related, so are ˆx and ˆz. But in the literature, the term negative pair seems to be also common. 3This is a variant and simpliﬁcation of the original loss that does not change the essence (but may change the eﬃciency slightly). 181 φθ(ˆx(i))⊤φθ(˜x(i)), and thus minimizing the loss encourages φθ(ˆx(i))⊤φθ(˜x(i)) to be large, resulting in φθ(ˆx(i)) and φθ(˜x(i)) to be close. 4 14.3 Pretrained large language models Natural language processing is another area where pretraining models are particularly successful. In language problems, an example typically corre- sponds to a document or generally a sequence/trunk of words, 5 denoted by x = ( x1,··· ,xT) where T is the length of the document/sequence, xi ∈{1,··· ,V }are words in the document, and V is the vocabulary size. 6 A language model is a probabilistic model representing the probability of a document, denoted by p(x1,··· ,xT). This probability distribution is very complex because its support size is VT — exponential in the length of the document. Instead of modeling the distribution of a document itself, we can apply the chain rule of conditional probability to decompose it as follows: p(x1,··· ,xT) = p(x1)p(x2\|x1) ···p(xT\|x1,··· ,xT−1). (14.5) Now the support size of each of the conditional probability p(xt\|x1,··· ,xt−1) is V. We will model the conditional probability p(xt\|x1,··· ,xt−1) as a function of x1,...,x t−1 parameterized by some parameter θ. A parameterized model takes in numerical inputs and therefore we ﬁrst introduce embeddings or representations fo the words. Let ei ∈Rd be the embedding of the word i ∈{1,2,··· ,V }. We call [ e1,··· ,eV] ∈Rd×V the embedding matrix. The most commonly used model is Transformer [Vaswani et al., 2017]. In this subsection, we will introduce the input-output interface of a Transformer, but treat the intermediate computation in the Transformer as a blackbox. We refer the students to the transformer paper or more advanced courses for more details. As shown in Figure 14.1, given a document ( x1,··· ,xT), we ﬁrst translate the sequence of discrete variables into a sequence of corresponding 4To see this, you can verify that the function−log p p+q is decreasing in p, and increasing in q when p,q >0. 5In the practical implementations, typically all the data are concatenated into a single sequence in some order, and each example typically corresponds a sub-sequence of consec- utive words which may corresponds to a subset of a document or may span across multiple documents. 6Technically, words may be decomposed into tokens which could be words or sub-words (combinations of letters), but this note omits this technicality. In fact most commons words are a single token themselves. 182 word embeddings ( ex1 ,··· ,exT). We also introduce a ﬁxed special token x0 = ⊥in the vocabulary with corresponding embedding ex0 to mark the beginning of a document. Then, the word embeddings are passed into a Transformer model, which takes in a sequence of vectors ( ex0 ,ex1 ,··· ,exT) and outputs a sequence of vectors ( u1,u2,··· ,uT+1), where ut ∈RV will be interpreted as the logits for the probability distribution of the next word. Here we use the autoregressive version of the Transformers which by design ensures ut only depends on x1,··· ,xt−1 (note that this property does not hold in masked language models [Devlin et al., 2019] where the losses are also diﬀerent.) We view the whole mapping from x’s to u’s a blackbox in this subsection and call it a Transformer, denoted it by fθ, where θ include both the parameters in the Transformer and the input embeddings. We write ut = fθ(x0,x1,...,x t−1) where fθ denotes the mapping from the input to the outputs. 𝑥!𝑥" 𝑥#𝑒$!𝑒$" 𝑒$#… Transformer 𝑓%(𝑥) 𝑥&𝑒$$ 𝑢"𝑢' 𝑢#(!𝑢! … Figure 14.1: The inputs and outputs of a Transformer model. The conditional probabilityp(xt\|x1,··· ,xt−1) is the softtmax of the logits:   p(xt = 1\|x1 ··· ,xt−1) p(xt = 2\|x1 ··· ,xt−1) ... p(xt = V\|x1 ··· ,xt−1)  = softmax(ut) ∈RV (14.6) = softmax(fθ(x0,...,x t−1)) (14.7) We train the Transformer parameter θ by minimizing the negative log- likelihood of seeing the data under the probabilistic model deﬁned by θ, 183 which is the cross-entropy loss on the logitis. loss(θ) = 1 T T∑ t=1 −log(pθ(xt\|x1,...,x t−1)) (14.8) = 1 T T∑ t=1 ℓce(fθ(x0,x1,··· ,xt−1),xt) = 1 T T∑ t=1 −log(softmax(fθ(x0,x1,··· ,xt−1))xt) . Autoregressive text decoding / generation. Given a autoregressive Transformer, we can simply sample text from it sequentially. Given a pre- ﬁx x1,...x t, we generate text completion xt+1,...x T sequentially using the conditional distribution. xt+1 ∼softmax(fθ(x0,x1,··· ,xt)) (14.9) xt+2 ∼softmax(fθ(x0,x1,··· ,xt+1)) (14.10) ... (14.11) xT ∼softmax(fθ(x0,x1,··· ,xT−1)) . (14.12) Note that each generated token is used as the input to the model when gen- erating the following tokens. In practice, people often introduce a parameter τ > 0 named temperature to further adjust the entropy/sharpness of the generated distribution, xt+1 ∼softmax(fθ(x0,x1,··· ,xt)/τ) (14.13) xt+2 ∼softmax(fθ(x0,x1,··· ,xt+1)/τ) (14.14) ... (14.15) xT ∼softmax(fθ(x0,x1,··· ,xT−1)/τ) . (14.16) When τ = 1, the text is sampled from the original conditional probability deﬁned by the model. With a decreasing τ, the generated text gradually becomes more “deterministic”. τ →0 reduces to greedy decoding, where we generate the most probable next token from the conditional probability. 14.3.1 Zero-shot learning and in-context learning For language models, there are many ways to adapt a pretrained model to downstream tasks. In this notes, we discuss three of them: ﬁnetuning, zero- shot learning, and in-context learning. 184 Finetuning is not very common for the autoregressive language models that we introduced in Section 14.3 but much more common for other variants such as masked language models which has similar input-output interfaces but are pretrained diﬀerently [Devlin et al., 2019]. The ﬁnetuning method is the same as introduced generally in Section 14.1—the only question is how we deﬁne the prediction task with an additional linear head. One option is to treat cT+1 = φθ(x1,··· ,xT) as the representation and use w⊤cT+1 = w⊤φθ(x1,··· ,xT) to predict task label. As described in Section 14.1, we initialize θ to the pretrained model ˆθ and then optimize both w and θ. Zero-shot adaptation or zero-shot learning is the setting where there is no input-output pairs from the downstream tasks. For language problems tasks, typically the task is formatted as a question or a cloze test form via natural language. For example, we can format an example as a question: xtask = (xtask,1,··· ,xtask,T) = “Is the speed of light a universal constant?” Then, we compute the most likely next word predicted by the lan- guage model given this question, that is, computing argmax xT+1 p(xT+1 \| xtask,1,··· ,xtask,T). In this case, if the most likely next word xT+1 is “No”, then we solve the task. (The speed of light is only a constant in vacuum). We note that there are many ways to decode the answer from the language models, e.g., instead of computing the argmax, we may use the language model to generate a few words word. It is an active research question to ﬁnd the best way to utilize the language models. In-context learning is mostly used for few-shot settings where we have a few labeled examples (x(1) task,y(1) task),··· ,(x(ntask) task ,y(ntask) task ). Given a test example xtest, we construct a document ( x1,··· ,xT), which is more commonly called a “prompt” in this context, by concatenating the labeled examples and the text example in some format. For example, we may construct the prompt as follows x1,··· ,xT = “Q: 2 ∼3 = ? x(1) task A: 5 y(1) task Q: 6 ∼7 = ? x(2) task A: 13 y(2) task ··· Q: 15 ∼2 = ?” xtest 185 Then, we let the pretrained model generate the most likely xT+1,xT+2,··· . In this case, if the model can “learn” that the symbol ∼means addition from the few examples, we will obtain the following which suggests the answer is 17. xT+1,xT+2,··· = “A: 17”. The area of foundation models is very new and quickly growing. The notes here only attempt to introduce these models on a conceptual level with a signiﬁcant amount of simpliﬁcation. We refer the readers to other materials, e.g., Bommasani et al. [2021], for more details. Part V Reinforcement Learning and Control 186 Chapter 15 Reinforcement learning We now begin our study of reinforcement learning and adaptive control. In supervised learning, we saw algorithms that tried to make their outputs mimic the labels y given in the training set. In that setting, the labels gave an unambiguous “right answer” for each of the inputs x. In contrast, for many sequential decision making and control problems, it is very diﬃcult to provide this type of explicit supervision to a learning algorithm. For example, if we have just built a four-legged robot and are trying to program it to walk, then initially we have no idea what the “correct” actions to take are to make it walk, and so do not know how to provide explicit supervision for a learning algorithm to try to mimic. In the reinforcement learning framework, we will instead provide our al- gorithms only a reward function, which indicates to the learning agent when it is doing well, and when it is doing poorly. In the four-legged walking ex- ample, the reward function might give the robot positive rewards for moving forwards, and negative rewards for either moving backwards or falling over. It will then be the learning algorithm’s job to ﬁgure out how to choose actions over time so as to obtain large rewards. Reinforcement learning has been successful in applications as diverse as autonomous helicopter ﬂight, robot legged locomotion, cell-phone network routing, marketing strategy selection, factory control, and eﬃcient web-page indexing. Our study of reinforcement learning will begin with a deﬁnition of the Markov decision processes (MDP), which provides the formalism in which RL problems are usually posed. 187 188 15.1 Markov decision processes A Markov decision process is a tuple ( S,A, {Psa},γ,R ), where: • S is a set of states. (For example, in autonomous helicopter ﬂight, S might be the set of all possible positions and orientations of the heli- copter.) • Ais a set of actions. (For example, the set of all possible directions in which you can push the helicopter’s control sticks.) • Psa are the state transition probabilities. For each state s ∈S and action a∈A, Psa is a distribution over the state space. We’ll say more about this later, but brieﬂy, Psa gives the distribution over what states we will transition to if we take action a in state s. • γ ∈[0,1) is called the discount factor. • R: S×A↦→R is the reward function. (Rewards are sometimes also written as a function of a state S only, in which case we would have R: S ↦→R). The dynamics of an MDP proceeds as follows: We start in some state s0, and get to choose some action a0 ∈Ato take in the MDP. As a result of our choice, the state of the MDP randomly transitions to some successor state s1, drawn according to s1 ∼Ps0a0 . Then, we get to pick another action a1. As a result of this action, the state transitions again, now to some s2 ∼Ps1a1 . We then pick a2, and so on. . . . Pictorially, we can represent this process as follows: s0 a0 −→s1 a1 −→s2 a2 −→s3 a3 −→... Upon visiting the sequence of states s0,s1,... with actions a0,a1,... , our total payoﬀ is given by R(s0,a0) + γR(s1,a1) + γ2R(s2,a2) + ··· . Or, when we are writing rewards as a function of the states only, this becomes R(s0) + γR(s1) + γ2R(s2) + ··· . For most of our development, we will use the simpler state-rewards R(s), though the generalization to state-action rewards R(s,a) oﬀers no special diﬃculties. 189 Our goal in reinforcement learning is to choose actions over time so as to maximize the expected value of the total payoﬀ: E [ R(s0) + γR(s1) + γ2R(s2) + ··· ] Note that the reward at timestep tis discounted by a factor of γt. Thus, to make this expectation large, we would like to accrue positive rewards as soon as possible (and postpone negative rewards as long as possible). In economic applications where R(·) is the amount of money made, γ also has a natural interpretation in terms of the interest rate (where a dollar today is worth more than a dollar tomorrow). A policy is any function π : S ↦→A mapping from the states to the actions. We say that we are executing some policy π if, whenever we are in state s, we take action a = π(s). We also deﬁne the value function for a policy π according to Vπ(s) = E [ R(s0) + γR(s1) + γ2R(s2) + ··· ⏐⏐s0 = s,π]. Vπ(s) is simply the expected sum of discounted rewards upon starting in state s, and taking actions according to π.1 Given a ﬁxed policy π, its value function Vπ satisﬁes the Bellman equa- tions: Vπ(s) = R(s) + γ ∑ s′∈S Psπ(s)(s′)Vπ(s′). This says that the expected sum of discounted rewards Vπ(s) for starting in s consists of two terms: First, the immediate reward R(s) that we get right away simply for starting in state s, and second, the expected sum of future discounted rewards. Examining the second term in more detail, we see that the summation term above can be rewritten E s′∼Psπ(s) [Vπ(s′)]. This is the expected sum of discounted rewards for starting in state s′, where s′ is distributed according Psπ(s), which is the distribution over where we will end up after taking the ﬁrst action π(s) in the MDP from state s. Thus, the second term above gives the expected sum of discounted rewards obtained after the ﬁrst step in the MDP. Bellman’s equations can be used to eﬃciently solve for Vπ. Speciﬁcally, in a ﬁnite-state MDP ( \|S\|< ∞), we can write down one such equation for Vπ(s) for every state s. This gives us a set of \|S\|linear equations in \|S\| variables (the unknown Vπ(s)’s, one for each state), which can be eﬃciently solved for the Vπ(s)’s. 1This notation in which we condition on π isn’t technically correct because π isn’t a random variable, but this is quite standard in the literature. 190 We also deﬁne the optimal value function according to V∗(s) = max π Vπ(s). (15.1) In other words, this is the best possible expected sum of discounted rewards that can be attained using any policy. There is also a version of Bellman’s equations for the optimal value function: V∗(s) = R(s) + max a∈A γ ∑ s′∈S Psa(s′)V∗(s′). (15.2) The ﬁrst term above is the immediate reward as before. The second term is the maximum over all actions a of the expected future sum of discounted rewards we’ll get upon after action a. You should make sure you understand this equation and see why it makes sense. We also deﬁne a policy π∗: S ↦→A as follows: π∗(s) = arg max a∈A ∑ s′∈S Psa(s′)V∗(s′). (15.3) Note that π∗(s) gives the action a that attains the maximum in the “max” in Equation (15.2). It is a fact that for every state s and every policy π, we have V∗(s) = Vπ∗ (s) ≥Vπ(s). The ﬁrst equality says that the Vπ∗ , the value function for π∗, is equal to the optimal value function V∗ for every state s. Further, the inequality above says that π∗’s value is at least a large as the value of any other other policy. In other words, π∗ as deﬁned in Equation (15.3) is the optimal policy. Note that π∗ has the interesting property that it is the optimal policy for all states s. Speciﬁcally, it is not the case that if we were starting in some state s then there’d be some optimal policy for that state, and if we were starting in some other state s′then there’d be some other policy that’s optimal policy for s′. The same policy π∗ attains the maximum in Equa- tion (15.1) for all states s. This means that we can use the same policy π∗ no matter what the initial state of our MDP is. 15.2 Value iteration and policy iteration We now describe two eﬃcient algorithms for solving ﬁnite-state MDPs. For now, we will consider only MDPs with ﬁnite state and action spaces ( \|S\|< 191 ∞, \|A\|< ∞). In this section, we will also assume that we know the state transition probabilities {Psa}and the reward function R. The ﬁrst algorithm, value iteration, is as follows: Algorithm 4 Value Iteration 1: For each state s, initialize V(s) := 0. 2: for until convergence do 3: For every state, update V(s) := R(s) + max a∈A γ ∑ s′ Psa(s′)V(s′). (15.4) This algorithm can be thought of as repeatedly trying to update the estimated value function using Bellman Equations (15.2). There are two possible ways of performing the updates in the inner loop of the algorithm. In the ﬁrst, we can ﬁrst compute the new values for V(s) for every state s, and then overwrite all the old values with the new values. This is called a synchronous update. In this case, the algorithm can be viewed as implementing a “Bellman backup operator” that takes a current estimate of the value function, and maps it to a new estimate. (See homework problem for details.) Alternatively, we can also perform asynchronous updates. Here, we would loop over the states (in some order), updating the values one at a time. Under either synchronous or asynchronous updates, it can be shown that value iteration will cause V to converge to V∗. Having found V∗, we can then use Equation (15.3) to ﬁnd the optimal policy. Apart from value iteration, there is a second standard algorithm for ﬁnd- ing an optimal policy for an MDP. The policy iteration algorithm proceeds as follows: Thus, the inner-loop repeatedly computes the value function for the cur- rent policy, and then updates the policy using the current value function. (The policy π found in step (b) is also called the policy that is greedy with respect to V.) Note that step (a) can be done via solving Bellman’s equa- tions as described earlier, which in the case of a ﬁxed policy, is just a set of \|S\|linear equations in \|S\|variables. After at most a ﬁnite number of iterations of this algorithm, V will con- verge to V∗, and π will converge to π∗.2 2Note that value iteration cannot reach the exact V∗ in a ﬁnite number of iterations, 192 Algorithm 5 Policy Iteration 1: Initialize π randomly. 2: for until convergence do 3: Let V := Vπ. ⊿ typically by linear system solver 4: For each state s, let π(s) := arg max a∈A ∑ s′ Psa(s′)V(s′). Both value iteration and policy iteration are standard algorithms for solv- ing MDPs, and there isn’t currently universal agreement over which algo- rithm is better. For small MDPs, policy iteration is often very fats and converges with very few iterations. However, for MDPs with large state spaces, solving for Vπ explicitly would involve solving a large system of lin- ear equations, and could be diﬃcult (and note that one has to solve the linear system multiple times in policy iteration). In these problems, value iteration may be preferred. For this reason, in practice value iteration seems to be used more often than policy iteration. For some more discussions on the comparison and connection of value iteration and policy iteration, please see Section 15.5. 15.3 Learning a model for an MDP So far, we have discussed MDPs and algorithms for MDPs assuming that the state transition probabilities and rewards are known. In many realistic prob- lems, we are not given state transition probabilities and rewards explicitly, but must instead estimate them from data. (Usually, S,A and γ are known.) For example, suppose that, for the inverted pendulum problem (see prob- whereas policy iteration with an exact linear system solver, can. This is because when the actions space and policy space are discrete and ﬁnite, and once the policy reaches the optimal policy in policy iteration, then it will not change at all. On the other hand, even though value iteration will converge to the V∗, but there is always some non-zero error in the learned value function. 193 lem set 4), we had a number of trials in the MDP, that proceeded as follows: s(1) 0 a(1) 0 −→s(1) 1 a(1) 1 −→s(1) 2 a(1) 2 −→s(1) 3 a(1) 3 −→... s(2) 0 a(2) 0 −→s(2) 1 a(2) 1 −→s(2) 2 a(2) 2 −→s(2) 3 a(2) 3 −→... ... Here, s(j) i is the state we were at time i of trial j, and a(j) i is the cor- responding action that was taken from that state. In practice, each of the trials above might be run until the MDP terminates (such as if the pole falls over in the inverted pendulum problem), or it might be run for some large but ﬁnite number of timesteps. Given this “experience” in the MDP consisting of a number of trials, we can then easily derive the maximum likelihood estimates for the state transition probabilities: Psa(s′) = #times took we action a in state s and got to s′ #times we took action a in state s (15.5) Or, if the ratio above is “0/0”—corresponding to the case of never having taken action ain state sbefore—the we might simply estimate Psa(s′) to be 1/\|S\|. (I.e., estimate Psa to be the uniform distribution over all states.) Note that, if we gain more experience (observe more trials) in the MDP, there is an eﬃcient way to update our estimated state transition probabilities using the new experience. Speciﬁcally, if we keep around the counts for both the numerator and denominator terms of (15.5), then as we observe more trials, we can simply keep accumulating those counts. Computing the ratio of these counts then given our estimate of Psa. Using a similar procedure, if Ris unknown, we can also pick our estimate of the expected immediate reward R(s) in state s to be the average reward observed in state s. Having learned a model for the MDP, we can then use either value it- eration or policy iteration to solve the MDP using the estimated transition probabilities and rewards. For example, putting together model learning and value iteration, here is one possible algorithm for learning in an MDP with unknown state transition probabilities: 1. Initialize π randomly. 2. Repeat { (a) Execute π in the MDP for some number of trials. 194 (b) Using the accumulated experience in the MDP, update our esti- mates for Psa (and R, if applicable). (c) Apply value iteration with the estimated state transition probabil- ities and rewards to get a new estimated value function V. (d) Update π to be the greedy policy with respect to V. } We note that, for this particular algorithm, there is one simple optimiza- tion that can make it run much more quickly. Speciﬁcally, in the inner loop of the algorithm where we apply value iteration, if instead of initializing value iteration with V = 0, we initialize it with the solution found during the pre- vious iteration of our algorithm, then that will provide value iteration with a much better initial starting point and make it converge more quickly. 15.4 Continuous state MDPs So far, we’ve focused our attention on MDPs with a ﬁnite number of states. We now discuss algorithms for MDPs that may have an inﬁnite number of states. For example, for a car, we might represent the state as (x,y,θ, ˙x, ˙y, ˙θ), comprising its position (x,y); orientation θ; velocity in the xand ydirections ˙x and ˙y; and angular velocity ˙θ. Hence, S = R6 is an inﬁnite set of states, because there is an inﬁnite number of possible positions and orientations for the car. 3 Similarly, the inverted pendulum you saw in PS4 has states (x,θ, ˙x, ˙θ), where θ is the angle of the pole. And, a helicopter ﬂying in 3d space has states of the form (x,y,z,φ,θ,ψ, ˙x, ˙y, ˙z, ˙φ, ˙θ, ˙ψ), where here the roll φ, pitch θ, and yaw ψ angles specify the 3d orientation of the helicopter. In this section, we will consider settings where the state space is S = Rd, and describe ways for solving such MDPs. 15.4.1 Discretization Perhaps the simplest way to solve a continuous-state MDP is to discretize the state space, and then to use an algorithm like value iteration or policy iteration, as described previously. For example, if we have 2d states ( s1,s2), we can use a grid to discretize the state space: 3Technically,θis an orientation and so the range of θis better written θ∈[−π,π) than θ∈R; but for our purposes, this distinction is not important. 195 [t] Here, each grid cell represents a separate discrete state ¯ s. We can then approximate the continuous-state MDP via a discrete-state one ( ¯S,A, {P¯sa},γ,R ), where ¯S is the set of discrete states, {P¯sa}are our state transition probabilities over the discrete states, and so on. We can then use value iteration or policy iteration to solve for the V∗(¯s) and π∗(¯s) in the discrete state MDP ( ¯S,A, {P¯sa},γ,R ). When our actual system is in some continuous-valued state s∈S and we need to pick an action to execute, we compute the corresponding discretized state ¯s, and execute action π∗(¯s). This discretization approach can work well for many problems. However, there are two downsides. First, it uses a fairly naive representation for V∗ (and π∗). Speciﬁcally, it assumes that the value function is takes a constant value over each of the discretization intervals (i.e., that the value function is piecewise constant in each of the gridcells). To better understand the limitations of such a representation, consider a supervised learning problem of ﬁtting a function to this dataset: [t] 1 2 3 4 5 6 7 8 1.5 2 2.5 3 3.5 4 4.5 5 5.5 x y 196 Clearly, linear regression would do ﬁne on this problem. However, if we instead discretize the x-axis, and then use a representation that is piecewise constant in each of the discretization intervals, then our ﬁt to the data would look like this: [t] 1 2 3 4 5 6 7 8 1.5 2 2.5 3 3.5 4 4.5 5 5.5 x y This piecewise constant representation just isn’t a good representation for many smooth functions. It results in little smoothing over the inputs, and no generalization over the diﬀerent grid cells. Using this sort of representation, we would also need a very ﬁne discretization (very small grid cells) to get a good approximation. A second downside of this representation is called the curse of dimen- sionality. Suppose S = Rd, and we discretize each of theddimensions of the state into k values. Then the total number of discrete states we have is kd. This grows exponentially quickly in the dimension of the state space d, and thus does not scale well to large problems. For example, with a 10d state, if we discretize each state variable into 100 values, we would have 10010 = 1020 discrete states, which is far too many to represent even on a modern desktop computer. As a rule of thumb, discretization usually works extremely well for 1d and 2d problems (and has the advantage of being simple and quick to im- plement). Perhaps with a little bit of cleverness and some care in choosing the discretization method, it often works well for problems with up to 4d states. If you’re extremely clever, and somewhat lucky, you may even get it to work for some 6d problems. But it very rarely works for problems any higher dimensional than that. 197 15.4.2 Value function approximation We now describe an alternative method for ﬁnding policies in continuous- state MDPs, in which we approximate V∗ directly, without resorting to dis- cretization. This approach, called value function approximation, has been successfully applied to many RL problems. Using a model or simulator To develop a value function approximation algorithm, we will assume that we have a model, or simulator, for the MDP. Informally, a simulator is a black-box that takes as input any (continuous-valued) state st and action at, and outputs a next-state st+1 sampled according to the state transition probabilities Pstat: [t] There are several ways that one can get such a model. One is to use physics simulation. For example, the simulator for the inverted pendulum in PS4 was obtained by using the laws of physics to calculate what position and orientation the cart/pole will be in at time t+ 1, given the current state at time t and the action a taken, assuming that we know all the parameters of the system such as the length of the pole, the mass of the pole, and so on. Alternatively, one can also use an oﬀ-the-shelf physics simulation software package which takes as input a complete physical description of a mechanical system, the current state st and action at, and computes the state st+1 of the system a small fraction of a second into the future. 4 An alternative way to get a model is to learn one from data collected in the MDP. For example, suppose we execute ntrials in which we repeatedly take actions in an MDP, each trial for T timesteps. This can be done picking actions at random, executing some speciﬁc policy, or via some other way of 4Open Dynamics Engine (http://www.ode.com) is one example of a free/open-source physics simulator that can be used to simulate systems like the inverted pendulum, and that has been a reasonably popular choice among RL researchers. 198 choosing actions. We would then observe nstate sequences like the following: s(1) 0 a(1) 0 −→s(1) 1 a(1) 1 −→s(1) 2 a(1) 2 −→··· a(1) T−1 −→s(1) T s(2) 0 a(2) 0 −→s(2) 1 a(2) 1 −→s(2) 2 a(2) 2 −→··· a(2) T−1 −→s(2) T ··· s(n) 0 a(n) 0 −→s(n) 1 a(n) 1 −→s(n) 2 a(n) 2 −→··· a(n) T−1 −→s(n) T We can then apply a learning algorithm to predict st+1 as a function of st and at. For example, one may choose to learn a linear model of the form st+1 = Ast + Bat, (15.6) using an algorithm similar to linear regression. Here, the parameters of the model are the matrices A and B, and we can estimate them using the data collected from our n trials, by picking arg min A,B n∑ i=1 T−1∑ t=0 s(i) t+1 − ( As(i) t + Ba(i) t ) 2 2 . We could also potentially use other loss functions for learning the model. For example, it has been found in recent work Luo et al. [2018] that using ∥·∥2 norm (without the square) may be helpful in certain cases. Having learned A and B, one option is to build a deterministic model, in which given an input st and at, the output st+1 is exactly determined. Speciﬁcally, we always compute st+1 according to Equation (15.6). Alter- natively, we may also build a stochastic model, in which st+1 is a random function of the inputs, by modeling it as st+1 = Ast + Bat + ϵt, where here ϵt is a noise term, usually modeled as ϵt ∼N(0,Σ). (The covari- ance matrix Σ can also be estimated from data in a straightforward way.) Here, we’ve written the next-state st+1 as a linear function of the current state and action; but of course, non-linear functions are also possible. Specif- ically, one can learn a model st+1 = Aφs(st) + Bφa(at), where φs and φa are some non-linear feature mappings of the states and actions. Alternatively, one can also use non-linear learning algorithms, such as locally weighted lin- ear regression, to learn to estimate st+1 as a function of st and at. These approaches can also be used to build either deterministic or stochastic sim- ulators of an MDP. 199 Fitted value iteration We now describe the ﬁtted value iteration algorithm for approximating the value function of a continuous state MDP. In the sequel, we will assume that the problem has a continuous state space S = Rd, but that the action space A is small and discrete. 5 Recall that in value iteration, we would like to perform the update V(s) := R(s) + γmax a ∫ s′ Psa(s′)V(s′)ds′ (15.7) = R(s) + γmax a Es′∼Psa[V(s′)] (15.8) (In Section 15.2, we had written the value iteration update with a summation V(s) := R(s) + γmaxa ∑ s′Psa(s′)V(s′) rather than an integral over states; the new notation reﬂects that we are now working in continuous states rather than discrete states.) The main idea of ﬁtted value iteration is that we are going to approxi- mately carry out this step, over a ﬁnite sample of states s(1),...,s (n). Specif- ically, we will use a supervised learning algorithm—linear regression in our description below—to approximate the value function as a linear or non-linear function of the states: V(s) = θTφ(s). Here, φ is some appropriate feature mapping of the states. For each state s in our ﬁnite sample of n states, ﬁtted value iteration will ﬁrst compute a quantity y(i), which will be our approximation to R(s) + γmaxaEs′∼Psa[V(s′)] (the right hand side of Equation 15.8). Then, it will apply a supervised learning algorithm to try to get V(s) close to R(s) + γmaxaEs′∼Psa[V(s′)] (or, in other words, to try to get V(s) close to y(i)). In detail, the algorithm is as follows: 1. Randomly sample n states s(1),s(2),...s (n) ∈S. 2. Initialize θ:= 0. 3. Repeat { For i= 1,...,n { 5In practice, most MDPs have much smaller action spaces than state spaces. E.g., a car has a 6d state space, and a 2d action space (steering and velocity controls); the inverted pendulum has a 4d state space, and a 1d action space; a helicopter has a 12d state space, and a 4d action space. So, discretizing this set of actions is usually less of a problem than discretizing the state space would have been. 200 For each action a∈A { Sample s′ 1,...,s ′ k ∼Ps(i)a (using a model of the MDP). Set q(a) = 1 k ∑k j=1 R(s(i)) + γV(s′ j) // Hence, q(a) is an estimate of R(s(i)) + γEs′∼Ps(i)a [V(s′)]. } Set y(i) = maxaq(a). // Hence, y(i) is an estimate of R(s(i)) + γmaxaEs′∼Ps(i)a [V(s′)]. } // In the original value iteration algorithm (over discrete states) // we updated the value function according to V(s(i)) := y(i). // In this algorithm, we want V(s(i)) ≈y(i), which we’ll achieve // using supervised learning (linear regression). Set θ:= arg minθ 1 2 ∑n i=1 ( θTφ(s(i)) −y(i))2 } Above, we had written out ﬁtted value iteration using linear regression as the algorithm to try to make V(s(i)) close to y(i). That step of the algo- rithm is completely analogous to a standard supervised learning (regression) problem in which we have a training set (x(1),y(1)),(x(2),y(2)),..., (x(n),y(n)), and want to learn a function mapping from xto y; the only diﬀerence is that here splays the role of x. Even though our description above used linear re- gression, clearly other regression algorithms (such as locally weighted linear regression) can also be used. Unlike value iteration over a discrete set of states, ﬁtted value iteration cannot be proved to always to converge. However, in practice, it often does converge (or approximately converge), and works well for many problems. Note also that if we are using a deterministic simulator/model of the MDP, then ﬁtted value iteration can be simpliﬁed by settingk= 1 in the algorithm. This is because the expectation in Equation (15.8) becomes an expectation over a deterministic distribution, and so a single example is suﬃcient to exactly compute that expectation. Otherwise, in the algorithm above, we had to draw k samples, and average to try to approximate that expectation (see the deﬁnition of q(a), in the algorithm pseudo-code). 201 Finally, ﬁtted value iteration outputs V, which is an approximation to V∗. This implicitly deﬁnes our policy. Speciﬁcally, when our system is in some state s, and we need to choose an action, we would like to choose the action arg max a Es′∼Psa[V(s′)] (15.9) The process for computing/approximating this is similar to the inner-loop of ﬁtted value iteration, where for each action, we sample s′ 1,...,s ′ k ∼Psa to approximate the expectation. (And again, if the simulator is deterministic, we can set k= 1.) In practice, there are often other ways to approximate this step as well. For example, one very common case is if the simulator is of the form st+1 = f(st,at) + ϵt, where f is some deterministic function of the states (such as f(st,at) = Ast + Bat), and ϵ is zero-mean Gaussian noise. In this case, we can pick the action given by arg max a V(f(s,a)). In other words, here we are just setting ϵt = 0 (i.e., ignoring the noise in the simulator), and setting k = 1. Equivalent, this can be derived from Equation (15.9) using the approximation Es′[V(s′)] ≈ V(Es′[s′]) (15.10) = V(f(s,a)), (15.11) where here the expectation is over the random s′∼Psa. So long as the noise terms ϵt are small, this will usually be a reasonable approximation. However, for problems that don’t lend themselves to such approximations, having to sample k\|A\|states using the model, in order to approximate the expectation above, can be computationally expensive. 15.5 Connections between Policy and Value Iteration (Optional) In the policy iteration, line 3 of Algorithm 5, we typically use linear system solver to compute Vπ. Alternatively, one can also the iterative Bellman updates, similarly to the value iteration, to evaluate Vπ, as in the Procedure VE(·) in Line 1 of Algorithm 6 below. Here if we take option 1 in Line 2 of the Procedure VE, then the diﬀerence between the Procedure VE from the 202 Algorithm 6 Variant of Policy Iteration 1: procedure VE(π, k) ⊿ To evaluate Vπ 2: Option 1: initialize V(s) := 0; Option 2: Initialize from the current V in the main algorithm. 3: for i= 0 to k−1 do 4: For every state s, update V(s) := R(s) + γ ∑ s′ Psπ(s)(s′)V(s′). (15.12) return V 5: Require: hyperparameter k. 6: Initialize π randomly. 7: for until convergence do 8: Let V = VE(π,k). 9: For each state s, let π(s) := arg max a∈A ∑ s′ Psa(s′)V(s′). (15.13) 203 value iteration (Algorithm 4) is that on line 4, the procedure is using the action from π instead of the greedy action. Using the Procedure VE, we can build Algorithm 6, which is a variant of policy iteration that serves an intermediate algorithm that connects pol- icy iteration and value iteration. Here we are going to use option 2 in VE to maximize the re-use of knowledge learned before. One can verify indeed that if we take k = 1 and use option 2 in Line 2 in Algorithm 6, then Algo- rithm 6 is semantically equivalent to value iteration (Algorithm 4). In other words, both Algorithm 6 and value iteration interleave the updates in (15.13) and (15.12). Algorithm 6 alternate between k steps of update (15.12) and one step of (15.13), whereas value iteration alternates between 1 steps of up- date (15.12) and one step of (15.13). Therefore generally Algorithm 6 should not be faster than value iteration, because assuming that update (15.12) and (15.13) are equally useful and time-consuming, then the optimal balance of the update frequencies could be just k= 1 or k≈1. On the other hand, if k steps of update (15.12) can be done much faster than k times a single step of (15.12), then taking additional steps of equa- tion (15.12) in group might be useful. This is what policy iteration is lever- aging — the linear system solver can give us the result of Procedure VE with k = ∞much faster than using the Procedure VE for a large k. On the ﬂip side, when such a speeding-up eﬀect no longer exists, e.g.,, when the state space is large and linear system solver is also not fast, then value iteration is more preferable. Chapter 16 LQR, DDP and LQG 16.1 Finite-horizon MDPs In Chapter 15, we deﬁned Markov Decision Processes (MDPs) and covered Value Iteration / Policy Iteration in a simpliﬁed setting. More speciﬁcally we introduced the optimal Bellman equation that deﬁnes the optimal value function Vπ∗ of the optimal policy π∗. Vπ∗ (s) = R(s) + max a∈A γ ∑ s′∈S Psa(s′)Vπ∗ (s′) Recall that from the optimal value function, we were able to recover the optimal policy π∗ with π∗(s) = argmaxa∈A ∑ s′∈S Psa(s′)V∗(s′) In this chapter, we’ll place ourselves in a more general setting: 1. We want to write equations that make sense for both the discrete and the continuous case. We’ll therefore write Es′∼Psa [ Vπ∗ (s′) ] instead of∑ s′∈S Psa(s′)Vπ∗ (s′) meaning that we take the expectation of the value function at the next state. In the ﬁnite case, we can rewrite the expectation as a sum over 204 205 states. In the continuous case, we can rewrite the expectation as an integral. The notation s′∼Psa means that the state s′is sampled from the distribution Psa. 2. We’ll assume that the rewards depend on both states and actions. In other words, R: S×A→ R. This implies that the previous mechanism for computing the optimal action is changed into π∗(s) = argmaxa∈AR(s,a) + γEs′∼Psa [ Vπ∗ (s′) ] 3. Instead of considering an inﬁnite horizon MDP, we’ll assume that we have a ﬁnite horizon MDP that will be deﬁned as a tuple (S,A,Psa,T,R ) with T >0 the time horizon (for instance T = 100). In this setting, our deﬁnition of payoﬀ is going to be (slightly) diﬀerent: R(s0,a0) + R(s1,a1) + ··· + R(sT,aT) instead of (inﬁnite horizon case) R(s0,a0) + γR(s1,a1) + γ2R(s2,a2) + ... ∞∑ t=0 R(st,at)γt What happened to the discount factor γ? Remember that the intro- duction of γ was (partly) justiﬁed by the necessity of making sure that the inﬁnite sum would be ﬁnite and well-deﬁned. If the rewards are bounded by a constant ¯R, the payoﬀ is indeed bounded by \| ∞∑ t=0 R(st)γt\|≤ ¯R ∞∑ t=0 γt and we recognize a geometric sum! Here, as the payoﬀ is a ﬁnite sum, the discount factor γ is not necessary anymore. 206 In this new setting, things behave quite diﬀerently. First, the optimal policy π∗might be non-stationary, meaning thatit changes over time. In other words, now we have π(t) : S→A where the superscript (t) denotes the policy at time step t. The dynam- ics of the ﬁnite horizon MDP following policy π(t) proceeds as follows: we start in some state s0, take some action a0 := π(0)(s0) according to our policy at time step 0. The MDP transitions to a successor s1, drawn according to Ps0a0 . Then, we get to pick another action a1 := π(1)(s1) following our new policy at time step 1 and so on... Why does the optimal policy happen to be non-stationary in the ﬁnite- horizon setting? Intuitively, as we have a ﬁnite numbers of actions to take, we might want to adopt diﬀerent strategies depending on where we are in the environment and how much time we have left. Imagine a grid with 2 goals with rewards +1 and +10. At the beginning, we might want to take actions to aim for the +10 goal. But if after some steps, dynamics somehow pushed us closer to the +1 goal and we don’t have enough steps left to be able to reach the +10 goal, then a better strategy would be to aim for the +1 goal... 4. This observation allows us to use time dependent dynamics st+1 ∼P(t) st,at meaning that the transition’s distribution P(t) st,at changes over time. The same thing can be said about R(t). Note that this setting is a better model for real life. In a car, the gas tank empties, traﬃc changes, etc. Combining the previous remarks, we’ll use the following general formulation for our ﬁnite horizon MDP ( S,A,P(t) sa ,T,R (t)) Remark: notice that the above formulation would be equivalent to adding the time into the state. 207 The value function at time t for a policy π is then deﬁned in the same way as before, as an expectation over trajectories generated following policy π starting in state s. Vt(s) = E [ R(t)(st,at) + ··· + R(T)(sT,aT)\|st = s,π ] Now, the question is In this ﬁnite-horizon setting, how do we ﬁnd the optimal value function V∗ t (s) = max π Vπ t (s) It turns out that Bellman’s equation for Value Iteration is made for Dy- namic Programming. This may come as no surprise as Bellman is one of the fathers of dynamic programming and the Bellman equation is strongly related to the ﬁeld. To understand how we can simplify the problem by adopting an iteration-based approach, we make the following observations: 1. Notice that at the end of the game (for time step T), the optimal value is obvious ∀s∈S : V∗ T(s) := max a∈A R(T)(s,a) (16.1) 2. For another time step 0 ≤ t < T, if we suppose that we know the optimal value function for the next time step V∗ t+1, then we have ∀t<T,s ∈S : V∗ t (s) := max a∈A [ R(t)(s,a) + Es′∼P(t) sa [ V∗ t+1(s′) ]] (16.2) With these observations in mind, we can come up with a clever algorithm to solve for the optimal value function: 1. compute V∗ T using equation (16.1). 2. for t= T −1,..., 0: compute V∗ t using V∗ t+1 using equation (16.2) 208 Side note We can interpret standard value iteration as a special case of this general case, but without keeping track of time. It turns out that in the standard setting, if we run value iteration for T steps, we get a γT approximation of the optimal value iteration (geometric convergence). See problem set 4 for a proof of the following result: Theorem Let B denote the Bellman update and \|\|f(x)\|\|∞:= supx\|f(x)\|. If Vt denotes the value function at the t-th step, then \|\|Vt+1 −V∗\|\|∞= \|\|B(Vt) −V∗\|\|∞ ≤γ\|\|Vt −V∗\|\|∞ ≤γt\|\|V1 −V∗\|\|∞ In other words, the Bellman operator B is a γ-contracting operator. 16.2 Linear Quadratic Regulation (LQR) In this section, we’ll cover a special case of the ﬁnite-horizon setting described in Section 16.1, for which the exact solution is (easily) tractable. This model is widely used in robotics, and a common technique in many problems is to reduce the formulation to this framework. First, let’s describe the model’s assumptions. We place ourselves in the continuous setting, with S= Rd, A= Rd and we’ll assume linear transitions (with noise) st+1 = Atst + Btat + wt where At ∈Rd×d,Bt ∈Rd×d are matrices and wt ∼N (0,Σt) is some gaussian noise (with zero mean). As we’ll show in the following paragraphs, it turns out that the noise, as long as it has zero mean, does not impact the optimal policy! We’ll also assume quadratic rewards R(t)(st,at) = −s⊤ t Utst −a⊤ t Wtat 209 where Ut ∈Rd×n,Wt ∈Rd×d are positive deﬁnite matrices (meaning that the reward is always negative). Remark Note that the quadratic formulation of the reward is equivalent to saying that we want our state to be close to the origin (where the reward is higher). For example, if Ut = Id (the identity matrix) and Wt = Id, then Rt = −\|\|st\|\|2 −\|\|at\|\|2, meaning that we want to take smooth actions (small norm of at) to go back to the origin (small norm of st). This could model a car trying to stay in the middle of lane without making impulsive moves... Now that we have deﬁned the assumptions of our LQR model, let’s cover the 2 steps of the LQR algorithm step 1 suppose that we don’t know the matrices A,B, Σ. To esti- mate them, we can follow the ideas outlined in the Value Ap- proximation section of the RL notes. First, collect transitions from an arbitrary policy. Then, use linear regression to ﬁnd argminA,B ∑n i=1 ∑T−1 t=0 s(i) t+1 − ( As(i) t + Ba(i) t ) 2 . Finally, use a tech- nique seen in Gaussian Discriminant Analysis to learn Σ. step 2 assuming that the parameters of our model are known (given or esti- mated with step 1), we can derive the optimal policy using dynamic programming. In other words, given { st+1 = Atst + Btat + wt At,Bt,Ut,Wt,Σt known R(t)(st,at) = −s⊤ t Utst −a⊤ t Wtat we want to compute V∗ t . If we go back to section 16.1, we can apply dynamic programming, which yields 1. Initialization step For the last time step T, V∗ T(sT) = max aT∈A RT(sT,aT) = max aT∈A −s⊤ TUTsT −a⊤ TWtaT = −s⊤ TUtsT (maximized for aT = 0) 210 2. Recurrence step Let t<T . Suppose we know V∗ t+1. Fact 1: It can be shown that if V∗ t+1 is a quadratic function in st, then V∗ t is also a quadratic function. In other words, there exists some matrix Φ and some scalar Ψ such that if V∗ t+1(st+1) = s⊤ t+1Φt+1st+1 + Ψt+1 then V∗ t (st) = s⊤ t Φtst + Ψt For time step t= T, we had Φt = −UT and ΨT = 0. Fact 2: We can show that the optimal policy is just a linear function of the state. Knowing V∗ t+1 is equivalent to knowing Φ t+1 and Ψt+1, so we just need to explain how we compute Φt and Ψt from Φt+1 and Ψt+1 and the other parameters of the problem. V∗ t (st) = s⊤ t Φtst + Ψt = max at [ R(t)(st,at) + Est+1∼P(t) st,at [V∗ t+1(st+1)] ] = max at [ −s⊤ t Utst −a⊤ t Vtat + Est+1∼N(Atst+Btat,Σt)[s⊤ t+1Φt+1st+1 + Ψt+1] ] where the second line is just the deﬁnition of the optimal value function and the third line is obtained by plugging in the dynamics of our model along with the quadratic assumption. Notice that the last expression is a quadratic function in at and can thus be (easily) optimized 1. We get the optimal action a∗ t a∗ t = [ (B⊤ t Φt+1Bt −Vt)−1BtΦt+1At ] ·st = Lt ·st where Lt := [ (B⊤ t Φt+1Bt −Wt)−1BtΦt+1At ] 1Use the identity E [ w⊤ t Φt+1wt ] = Tr(ΣtΦt+1) with wt ∼N (0,Σt) 211 which is an impressive result: our optimal policy is linear in st. Given a∗ t we can solve for Φ t and Ψ t. We ﬁnally get the Discrete Ricatti equations Φt = A⊤ t ( Φt+1 −Φt+1Bt ( B⊤ t Φt+1Bt −Wt )−1 BtΦt+1 ) At −Ut Ψt = −tr (ΣtΦt+1) + Ψt+1 Fact 3: we notice that Φt depends on neither Ψ nor the noise Σ t! As Lt is a function of At,Bt and Φt+1, it implies that the optimal policy also does not depend on the noise ! (But Ψ t does depend on Σ t, which implies that V∗ t depends on Σt.) Then, to summarize, the LQR algorithm works as follows 1. (if necessary) estimate parameters At,Bt,Σt 2. initialize Φ T := −UT and ΨT := 0. 3. iterate from t = T −1 ... 0 to update Φ t and Ψt using Φt+1 and Ψt+1 using the discrete Ricatti equations. If there exists a policy that drives the state towards zero, then convergence is guaranteed! Using Fact 3, we can be even more clever and make our algorithm run (slightly) faster! As the optimal policy does not depend on Ψ t, and the update of Φt only depends on Φ t, it is suﬃcient to update only Φt! 16.3 From non-linear dynamics to LQR It turns out that a lot of problems can be reduced to LQR, even if dynamics are non-linear. While LQR is a nice formulation because we are able to come up with a nice exact solution, it is far from being general. Let’s take for instance the case of the inverted pendulum. The transitions between states look like   xt+1 ˙xt+1 θt+1 ˙θt+1  = F     xt ˙xt θt ˙θt  ,at   where the function F depends on the cos of the angle etc. Now, the question we may ask is Can we linearize this system? 212 16.3.1 Linearization of dynamics Let’s suppose that at time t, the system spends most of its time in some state ¯st and the actions we perform are around ¯at. For the inverted pendulum, if we reached some kind of optimal, this is true: our actions are small and we don’t deviate much from the vertical. We are going to use Taylor expansion to linearize the dynamics. In the simple case where the state is one-dimensional and the transition function F does not depend on the action, we would write something like st+1 = F(st) ≈F( ¯st) + F′( ¯st) ·(st −¯st) In the more general setting, the formula looks the same, with gradients instead of simple derivatives st+1 ≈F( ¯st, ¯at) + ∇sF( ¯st, ¯at) ·(st −¯st) + ∇aF( ¯st, ¯at) ·(at −¯at) (16.3) and now, st+1 is linear in st and at, because we can rewrite equation (16.3) as st+1 ≈Ast + Bst + κ where κis some constant and A,B are matrices. Now, this writing looks awfully similar to the assumptions made for LQR. We just have to get rid of the constant term κ! It turns out that the constant term can be absorbed into st by artiﬁcially increasing the dimension by one. This is the same trick that we used at the beginning of the class for linear regression... 16.3.2 Diﬀerential Dynamic Programming (DDP) The previous method works well for cases where the goal is to stay around some state s∗ (think about the inverted pendulum, or a car having to stay in the middle of a lane). However, in some cases, the goal can be more complicated. We’ll cover a method that applies when our system has to follow some trajectory (think about a rocket). This method is going to discretize the trajectory into discrete time steps, and create intermediary goals around which we will be able to use the previous technique! This method is called Diﬀerential Dynamic Programming. The main steps are 213 step 1 come up with a nominal trajectory using a naive controller, that approx- imate the trajectory we want to follow. In other words, our controller is able to approximate the gold trajectory with s∗ 0,a∗ 0 →s∗ 1,a∗ 1 →... step 2 linearize the dynamics around each trajectory point s∗ t, in other words st+1 ≈F(s∗ t,a∗ t) + ∇sF(s∗ t,a∗ t)(st −s∗ t) + ∇aF(s∗ t,a∗ t)(at −a∗ t) where st,at would be our current state and action. Now that we have a linear approximation around each of these points, we can use the previous section and rewrite st+1 = At ·st + Bt ·at (notice that in that case, we use the non-stationary dynamics setting that we mentioned at the beginning of these lecture notes) Note We can apply a similar derivation for the reward R(t), with a second-order Taylor expansion. R(st,at) ≈R(s∗ t,a∗ t) + ∇sR(s∗ t,a∗ t)(st −s∗ t) + ∇aR(s∗ t,a∗ t)(at −a∗ t) + 1 2(st −s∗ t)⊤Hss(st −s∗ t) + (st −s∗ t)⊤Hsa(at −a∗ t) + 1 2(at −a∗ t)⊤Haa(at −a∗ t) where Hxy refers to the entry of the Hessian of Rwith respect to xand y evaluated in (s∗ t,a∗ t) (omitted for readability). This expression can be re-written as Rt(st,at) = −s⊤ t Utst −a⊤ t Wtat for some matrices Ut,Wt, with the same trick of adding an extra dimen- sion of ones. To convince yourself, notice that ( 1 x ) · (a b b c ) · (1 x ) = a+ 2bx+ cx2 214 step 3 Now, you can convince yourself that our problem is strictly re-written in the LQR framework. Let’s just use LQR to ﬁnd the optimal policy πt. As a result, our new controller will (hopefully) be better! Note: Some problems might arise if the LQR trajectory deviates too much from the linearized approximation of the trajectory, but that can be ﬁxed with reward-shaping... step 4 Now that we get a new controller (our new policy πt), we use it to produce a new trajectory s∗ 0,π0(s∗ 0) →s∗ 1,π1(s∗ 1) →... →s∗ T note that when we generate this new trajectory, we use the real F and not its linear approximation to compute transitions, meaning that s∗ t+1 = F(s∗ t,a∗ t) then, go back to step 2 and repeat until some stopping criterion. 16.4 Linear Quadratic Gaussian (LQG) Often, in the real word, we don’t get to observe the full statest. For example, an autonomous car could receive an image from a camera, which is merely an observation, and not the full state of the world. So far, we assumed that the state was available. As this might not hold true for most of the real-world problems, we need a new tool to model this situation: Partially Observable MDPs. A POMDP is an MDP with an extra observation layer. In other words, we introduce a new variable ot, that follows some conditional distribution given the current state st ot\|st ∼O(o\|s) Formally, a ﬁnite-horizon POMDP is given by a tuple (S,O,A,Psa,T,R ) Within this framework, the general strategy is to maintain a belief state (distribution over states) based on the observation o1,...,o t. Then, a policy in a POMDP maps this belief states to actions. 215 In this section, we’ll present a extension of LQR to this new setting. Assume that we observe yt ∈Rn with m<n such that { yt = C·st + vt st+1 = A·st + B·at + wt where C ∈Rn×d is a compression matrix and vt is the sensor noise (also gaussian, like wt). Note that the reward function R(t) is left unchanged, as a function of the state (not the observation) and action. Also, as distributions are gaussian, the belief state is also going to be gaussian. In this new frame- work, let’s give an overview of the strategy we are going to adopt to ﬁnd the optimal policy: step 1 ﬁrst, compute the distribution on the possible states (the belief state), based on the observations we have. In other words, we want to compute the mean st\|t and the covariance Σt\|t of st\|y1,...,y t ∼N ( st\|t,Σt\|t ) to perform the computation eﬃciently over time, we’ll use the Kalman Filter algorithm (used on-board Apollo Lunar Module!). step 2 now that we have the distribution, we’ll use the mean st\|t as the best approximation for st step 3 then set the action at := Ltst\|t where Lt comes from the regular LQR algorithm. Intuitively, to understand why this works, notice that st\|t is a noisy ap- proximation of st (equivalent to adding more noise to LQR) but we proved that LQR is independent of the noise! Step 1 needs to be explicated. We’ll cover a simple case where there is no action dependence in our dynamics (but the general case follows the same idea). Suppose that { st+1 = A·st + wt, w t ∼N(0,Σs) yt = C·st + vt, v t ∼N(0,Σy) As noises are Gaussians, we can easily prove that the joint distribution is also Gaussian 216   s1 ... st y1 ... yt   ∼N (µ,Σ) for some µ,Σ then, using the marginal formulas of gaussians (see Factor Analysis notes), we would get st\|y1,...,y t ∼N ( st\|t,Σt\|t ) However, computing the marginal distribution parameters using these formulas would be computationally expensive! It would require manipulating matrices of shape t×t. Recall that inverting a matrix can be done in O(t3), and it would then have to be repeated over the time steps, yielding a cost in O(t4)! The Kalman ﬁlter algorithm provides a much better way of computing the mean and variance, by updating them over time in constant time in t! The kalman ﬁlter is based on two basics steps. Assume that we know the distribution of st\|y1,...,y t: predict step compute st+1\|y1,...,y t update step compute st+1\|y1,...,y t+1 and iterate over time steps! The combination of the predict and update steps updates our belief states. In other words, the process looks like (st\|y1,...,y t) predict − −−− →(st+1\|y1,...,y t) update − −−− →(st+1\|y1,...,y t+1) predict − −−− →... predict step Suppose that we know the distribution of st\|y1,...,y t ∼N ( st\|t,Σt\|t ) then, the distribution over the next state is also a gaussian distribution st+1\|y1,...,y t ∼N ( st+1\|t,Σt+1\|t ) where 217 { st+1\|t = A·st\|t Σt+1\|t = A·Σt\|t ·A⊤+ Σs update step given st+1\|t and Σt+1\|t such that st+1\|y1,...,y t ∼N ( st+1\|t,Σt+1\|t ) we can prove that st+1\|y1,...,y t+1 ∼N ( st+1\|t+1,Σt+1\|t+1 ) where { st+1\|t+1 = st+1\|t + Kt(yt+1 −Cst+1\|t) Σt+1\|t+1 = Σt+1\|t −Kt ·C·Σt+1\|t with Kt := Σt+1\|tC⊤(CΣt+1\|tC⊤+ Σy)−1 The matrix Kt is called the Kalman gain. Now, if we have a closer look at the formulas, we notice that we don’t need the observations prior to time step t! The update steps only depends on the previous distribution. Putting it all together, the algorithm ﬁrst runs a forward pass to compute the Kt, Σ t\|t and st\|t (sometimes referred to as ˆs in the literature). Then, it runs a backward pass (the LQR updates) to compute the quantities Ψt,Ψt and Lt. Finally, we recover the optimal policy with a∗ t = Ltst\|t. Chapter 17 Policy Gradient (REINFORCE) We will present a model-free algorithm called REINFORCE that does not require the notion of value functions andQfunctions. It turns out to be more convenient to introduce REINFORCE in the ﬁnite horizon case, which will be assumed throughout this note: we use τ = (s0,a0,...,s T−1,aT−1,sT) to denote a trajectory, where T <∞is the length of the trajectory. Moreover, REINFORCE only applies to learning arandomized policy. We use πθ(a\|s) to denote the probability of the policy πθ outputting the action aat state s. The other notations will be the same as in previous lecture notes. The advantage of applying REINFORCE is that we only need to assume that we can sample from the transition probabilities {Psa}and can query the reward function R(s,a) at state sand action a,1 but we do not need to know the analytical form of the transition probabilities or the reward function. We do not explicitly learn the transition probabilities or the reward function either. Let s0 be sampled from some distribution µ. We consider optimizing the expected total payoﬀ of the policy πθ over the parameter θ deﬁned as. η(θ) ≜ E [T−1∑ t=0 γtR(st,at) ] (17.1) Recall that st ∼ Pst−1at−1 and at ∼ πθ(·\|st). Also note that η(θ) = Es0∼P [Vπθ(s0)] if we ignore the diﬀerence between ﬁnite and inﬁnite hori- zon. 1In this notes we will work with the general setting where the reward depends on both the state and the action. 218 219 We aim to use gradient ascent to maximize η(θ). The main challenge we face here is to compute (or estimate) the gradient of η(θ) without the knowledge of the form of the reward function and the transition probabilities. Let Pθ(τ) denote the distribution of τ (generated by the policy πθ), and let f(τ) = ∑T−1 t=0 γtR(st,at). We can rewrite η(θ) as η(θ) = Eτ∼Pθ [f(τ)] (17.2) We face a similar situations in the variational auto-encoder (VAE) setting covered in the previous lectures, where the we need to take the gradient w.r.t to a variable that shows up under the expectation — the distribution Pθ depends on θ. Recall that in VAE, we used the re-parametrization techniques to address this problem. However it does not apply here because we do know not how to compute the gradient of the function f. (We only have an eﬃcient way to evaluate the function f by taking a weighted sum of the observed rewards, but we do not necessarily know the reward function itself to compute the gradient.) The REINFORCE algorithm uses an another approach to estimate the gradient of η(θ). We start with the following derivation: ∇θEτ∼Pθ [f(τ)] = ∇θ ∫ Pθ(τ)f(τ)dτ = ∫ ∇θ(Pθ(τ)f(τ))dτ (swap integration with gradient) = ∫ (∇θPθ(τ))f(τ)dτ (becaue f does not depend on θ) = ∫ Pθ(τ)(∇θlog Pθ(τ))f(τ)dτ (because ∇log Pθ(τ) = ∇Pθ(τ) Pθ(τ) ) = Eτ∼Pθ [(∇θlog Pθ(τ))f(τ)] (17.3) Now we have a sample-based estimator for ∇θEτ∼Pθ [f(τ)]. Let τ(1),...,τ (n) be n empirical samples from Pθ (which are obtained by running the policy πθ for n times, with T steps for each run). We can estimate the gradient of η(θ) by ∇θEτ∼Pθ [f(τ)] = Eτ∼Pθ [(∇θlog Pθ(τ))f(τ)] (17.4) ≈1 n n∑ i=1 (∇θlog Pθ(τ(i)))f(τ(i)) (17.5) 220 The next question is how to compute log Pθ(τ). We derive an analyt- ical formula for log Pθ(τ) and compute its gradient w.r.t θ (using auto- diﬀerentiation). Using the deﬁnition of τ, we have Pθ(τ) = µ(s0)πθ(a0\|s0)Ps0a0 (s1)πθ(a1\|s1)Ps1a1 (s2) ···PsT−1aT−1 (sT) (17.6) Here recall that µ to used to denote the density of the distribution of s0. It follows that log Pθ(τ) = log µ(s0) + logπθ(a0\|s0) + logPs0a0 (s1) + logπθ(a1\|s1) + logPs1a1 (s2) + ··· + logPsT−1aT−1 (sT) (17.7) Taking gradient w.r.t to θ, we obtain ∇θlog Pθ(τ) = ∇θlog πθ(a0\|s0) + ∇θlog πθ(a1\|s1) + ··· + ∇θlog πθ(aT−1\|sT−1) Note that many of the terms disappear because they don’t depend on θ and thus have zero gradients. (This is somewhat important — we don’t know how to evaluate those terms such as log Ps0a0 (s1) because we don’t have access to the transition probabilities, but luckily those terms have zero gradients!) Plugging the equation above into equation (17.4), we conclude that ∇θη(θ) = ∇θEτ∼Pθ [f(τ)] = Eτ∼Pθ [(T−1∑ t=0 ∇θlog πθ(at\|st) ) ·f(τ) ] = Eτ∼Pθ [(T−1∑ t=0 ∇θlog πθ(at\|st) ) · (T−1∑ t=0 γtR(st,at) )] (17.8) We estimate the RHS of the equation above by empirical sample trajectories, and the estimate is unbiased. The vanilla REINFORCE algorithm iteratively updates the parameter by gradient ascent using the estimated gradients. Interpretation of the policy gradient formula (17.8). The quantity ∇θPθ(τ) = ∑T−1 t=0 ∇θlog πθ(at\|st) is intuitively the direction of the change of θ that will make the trajectory τ more likely to occur (or increase the probability of choosing action a0,...,a t−1), and f(τ) is the total payoﬀ of this trajectory. Thus, by taking a gradient step, intuitively we are trying to improve the likelihood of all the trajectories, but with a diﬀerent emphasis or weight for each τ (or for each set of actions a0,a1,...,a t−1). If τ is very rewarding (that is, f(τ) is large), we try very hard to move in the direction 221 that can increase the probability of the trajectory τ (or the direction that increases the probability of choosing a0,...,a t−1), and if τ has low payoﬀ, we try less hard with a smaller weight. An interesting fact that follows from formula (17.3) is that Eτ∼Pθ [T−1∑ t=0 ∇θlog πθ(at\|st) ] = 0 (17.9) To see this, we take f(τ) = 1 (that is, the reward is always a constant), then the LHS of (17.8) is zero because the payoﬀ is always a ﬁxed constant∑T t=0 γt. Thus the RHS of (17.8) is also zero, which implies (17.9). In fact, one can verify that E at∼πθ(·\|st)∇θlog πθ(at\|st) = 0 for any ﬁxed t and st.2 This fact has two consequences. First, we can simplify formula (17.8) to ∇θη(θ) = T−1∑ t=0 Eτ∼Pθ [ ∇θlog πθ(at\|st) · (T−1∑ j=0 γjR(sj,aj) )] = T−1∑ t=0 Eτ∼Pθ [ ∇θlog πθ(at\|st) · (T−1∑ j≥t γjR(sj,aj) )] (17.10) where the second equality follows from Eτ∼Pθ [ ∇θlog πθ(at\|st) · (∑ 0≤j<t γjR(sj,aj) )] = E [ E [∇θlog πθ(at\|st)\|s0,a0,...,s t−1,at−1,st] · (∑ 0≤j<t γjR(sj,aj) )] = 0 (because E [ ∇θlog πθ(at\|st)\|s0,a0,...,s t−1,at−1,st] = 0) Note that here we used the law of total expectation. The outer expecta- tion in the second line above is over the randomness of s0,a0,...,a t−1,st, whereas the inner expectation is over the randomness of at (conditioned on s0,a0,...,a t−1,st.) We see that we’ve made the estimator slightly simpler. The second consequence of Eat∼πθ(·\|st)∇θlog πθ(at\|st) = 0 is the following: for any value B(st) that only depends on st, it holds that Eτ∼Pθ [∇θlog πθ(at\|st) ·B(st)] = E [E [∇θlog πθ(at\|st)\|s0,a0,...,s t−1,at−1,st] B(st)] = 0 (because E [ ∇θlog πθ(at\|st)\|s0,a0,...,s t−1,at−1,st] = 0) 2In general, it’s true that E x∼pθ [∇log pθ(x)] = 0. 222 Again here we used the law of total expectation. The outer expecta- tion in the second line above is over the randomness of s0,a0,...,a t−1,st, whereas the inner expectation is over the randomness of at (conditioned on s0,a0,...,a t−1,st.) It follows from equation (17.10) and the equation above that ∇θη(θ) = T−1∑ t=0 Eτ∼Pθ [ ∇θlog πθ(at\|st) · (T−1∑ j≥t γjR(sj,aj) −γtB(st) )] = T−1∑ t=0 Eτ∼Pθ [ ∇θlog πθ(at\|st) ·γt (T−1∑ j≥t γj−tR(sj,aj) −B(st) )] (17.11) Therefore, we will get a diﬀerent estimator for estimating the ∇η(θ) with a diﬀerence choice of B(·). The beneﬁt of introducing a proper B(·) — which is often referred to as a baseline — is that it helps reduce the variance of the estimator.3 It turns out that a near optimal estimator would be the expected future payoﬀ E [∑T−1 j≥t γj−tR(sj,aj)\|st ] , which is pretty much the same as the value function Vπθ(st) (if we ignore the diﬀerence between ﬁnite and inﬁnite horizon.) Here one could estimate the value function Vπθ(·) in a crude way, because its precise value doesn’t inﬂuence the mean of the estimator but only the variance. This leads to a policy gradient algorithm with baselines stated in Algorithm 7.4 3As a heuristic but illustrating example, suppose for a ﬁxed t, the future reward∑T−1 j≥t γj−tR(sj,aj) randomly takes two values 1000 + 1 and 1000 −2 with equal proba- bility, and the corresponding values for ∇θlog πθ(at\|st) are vector z and −z. (Note that because E [∇θlog πθ(at\|st)] = 0, if ∇θlog πθ(at\|st) can only take two values uniformly, then the two values have to two vectors in an opposite direction.) In this case, without subtracting the baseline, the estimators take two values (1000 + 1) z and −(1000 −2)z, whereas after subtracting a baseline of 1000, the estimator has two values z and 2z. The latter estimator has much lower variance compared to the original estimator. 4We note that the estimator of the gradient in the algorithm does not exactly match the equation 17.11. If we multiply γt in the summand of equation (17.13), then they will exactly match. Removing such discount factors empirically works well because it gives a large update. 223 Algorithm 7 Vanilla policy gradient with baseline for i= 1,··· do Collect a set of trajectories by executing the current policy. 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nist.fips.197	Date updated: May 9, 2023 Withdrawn NIST Technical Series Publication Warning Notice The attached publication has been withdrawn (archived), and is provided solely for historical purposes. It may have been superseded by another publication (indicated below). Withdrawn Publication Series/Number Federal Information Processing Standards (FIPS) Publication 197 Title Advanced Encryption Standard (AES) Publication Date(s) November 26, 2001 Withdrawal Date May 9, 2023 Withdrawal Note FIPS 197 is updated by NIST FIPS 197-upd1 Superseding Publication(s) (if applicable) The attached publication has been superseded by the following publication(s): Series/Number NIST FIPS 197-upd1 Title Advanced Encryption Standard (AES) Author(s) National Institute of Standards and Technology Publication Date(s) November 26, 2001; Updated May 9, 2023 URL/DOI https://doi.org/10.6028/NIST.FIPS.197-upd1 Additional Information (if applicable) Contact Computer Security Division (Information Technology Laboratory) Latest revision of the attached publication Related Information This update makes no technical changes to the algorithm specified in the original (2001) release of this standard. This update includes extensive editorial improvements to the original version. Withdrawal Announcement Link https://csrc.nist.gov/News/2023/nist-updates-fips-197-advanced-encryption- standard Federal Information Processing Standards Publication 197 November 26, 2001 Announcing the ADVANCED ENCRYPTION STANDARD (AES) Federal Information Processing Standards Publications (FIPS PUBS) are issued by the National Institute of Standards and Technology (NIST) after approval by the Secretary of Commerce pursuant to Section 5131 of the Information Technology Management Reform Act of 1996 (Public Law 104-106) and the Computer Security Act of 1987 (Public Law 100-235). 1. Name of Standard. Advanced Encryption Standard (AES) (FIPS PUB 197). 2. Category of Standard. Computer Security Standard, Cryptography. 3. Explanation. The Advanced Encryption Standard (AES) specifies a FIPS-approved cryptographic algorithm that can be used to protect electronic data. The AES algorithm is a symmetric block cipher that can encrypt (encipher) and decrypt (decipher) information. Encryption converts data to an unintelligible form called ciphertext; decrypting the ciphertext converts the data back into its original form, called plaintext. The AES algorithm is capable of using cryptographic keys of 128, 192, and 256 bits to encrypt and decrypt data in blocks of 128 bits. 4. Approving Authority. Secretary of Commerce. 5. Maintenance Agency. Department of Commerce, National Institute of Standards and Technology, Information Technology Laboratory (ITL). 6. Applicability. This standard may be used by Federal departments and agencies when an agency determines that sensitive (unclassified) information (as defined in P. L. 100-235) requires cryptographic protection. Other FIPS-approved cryptographic algorithms may be used in addition to, or in lieu of, this standard. Federal agencies or departments that use cryptographic devices for protecting classified information can use those devices for protecting sensitive (unclassified) information in lieu of this standard. In addition, this standard may be adopted and used by non-Federal Government organizations. Such use is encouraged when it provides the desired security for commercial and private organizations. 7. Specifications. Federal Information Processing Standard (FIPS) 197, Advanced Encryption Standard (AES) (affixed). 8. Implementations. The algorithm specified in this standard may be implemented in software, firmware, hardware, or any combination thereof. The specific implementation may depend on several factors such as the application, the environment, the technology used, etc. The algorithm shall be used in conjunction with a FIPS approved or NIST recommended mode of operation. Object Identifiers (OIDs) and any associated parameters for AES used in these modes are available at the Computer Security Objects Register (CSOR), located at http://csrc.nist.gov/csor/ [2]. Implementations of the algorithm that are tested by an accredited laboratory and validated will be considered as complying with this standard. Since cryptographic security depends on many factors besides the correct implementation of an encryption algorithm, Federal Government employees, and others, should also refer to NIST Special Publication 800-21, Guideline for Implementing Cryptography in the Federal Government, for additional information and guidance (NIST SP 800-21 is available at http://csrc.nist.gov/publications/). 9. Implementation Schedule. This standard becomes effective on May 26, 2002. 10. Patents. Implementations of the algorithm specified in this standard may be covered by U.S. and foreign patents. 11. Export Control. Certain cryptographic devices and technical data regarding them are subject to Federal export controls. Exports of cryptographic modules implementing this standard and technical data regarding them must comply with these Federal regulations and be licensed by the Bureau of Export Administration of the U.S. Department of Commerce. Applicable Federal government export controls are specified in Title 15, Code of Federal Regulations (CFR) Part 740.17; Title 15, CFR Part 742; and Title 15, CFR Part 774, Category 5, Part 2. 12. Qualifications. NIST will continue to follow developments in the analysis of the AES algorithm. As with its other cryptographic algorithm standards, NIST will formally reevaluate this standard every five years. Both this standard and possible threats reducing the security provided through the use of this standard will undergo review by NIST as appropriate, taking into account newly available analysis and technology. In addition, the awareness of any breakthrough in technology or any mathematical weakness of the algorithm will cause NIST to reevaluate this standard and provide necessary revisions. 13. Waiver Procedure. Under certain exceptional circumstances, the heads of Federal agencies, or their delegates, may approve waivers to Federal Information Processing Standards (FIPS). The heads of such agencies may redelegate such authority only to a senior official designated pursuant to Section 3506(b) of Title 44, U.S. Code. Waivers shall be granted only when compliance with this standard would a. adversely affect the accomplishment of the mission of an operator of Federal computer system or b. cause a major adverse financial impact on the operator that is not offset by government- wide savings. ii Agency heads may act upon a written waiver request containing the information detailed above. Agency heads may also act without a written waiver request when they determine that conditions for meeting the standard cannot be met. Agency heads may approve waivers only by a written decision that explains the basis on which the agency head made the required finding(s). A copy of each such decision, with procurement sensitive or classified portions clearly identified, shall be sent to: National Institute of Standards and Technology; ATTN: FIPS Waiver Decision, Information Technology Laboratory, 100 Bureau Drive, Stop 8900, Gaithersburg, MD 20899 8900. In addition, notice of each waiver granted and each delegation of authority to approve waivers shall be sent promptly to the Committee on Government Operations of the House of Representatives and the Committee on Government Affairs of the Senate and shall be published promptly in the Federal Register. When the determination on a waiver applies to the procurement of equipment and/or services, a notice of the waiver determination must be published in the Commerce Business Daily as a part of the notice of solicitation for offers of an acquisition or, if the waiver determination is made after that notice is published, by amendment to such notice. A copy of the waiver, any supporting documents, the document approving the waiver and any supporting and accompanying documents, with such deletions as the agency is authorized and decides to make under Section 552(b) of Title 5, U.S. Code, shall be part of the procurement documentation and retained by the agency. 14. Where to obtain copies. This publication is available electronically by accessing http://csrc.nist.gov/publications/. A list of other available computer security publications, including ordering information, can be obtained from NIST Publications List 91, which is available at the same web site. Alternatively, copies of NIST computer security publications are available from: National Technical Information Service (NTIS), 5285 Port Royal Road, Springfield, VA 22161. iii iv Federal Information Processing Standards Publication 197 November 26, 2001 Specification for the ADVANCED ENCRYPTION STANDARD (AES) Table of Contents 1. INTRODUCTION ............................................................................................................................................. 5 2. DEFINITIONS .................................................................................................................................................. 5 2.1 G LOSSARY OF T ERMS AND A CRONYMS ........................................................................................................... 5 2.2 A LGORITHM P ARAMETERS , S YMBOLS , AND F UNCTIONS ................................................................................. 6 3. NOTATION AND CONVENTIONS ............................................................................................................... 7 3.1 I NPUTS AND O UTPUTS ..................................................................................................................................... 7 3.2 B YTES ............................................................................................................................................................. 8 3.3 A RRAYS OF B YTES .......................................................................................................................................... 8 3.4 T HE S TATE ...................................................................................................................................................... 9 3.5 T HE S TATE AS AN A RRAY OF C OLUMNS ........................................................................................................ 10 4. MATHEMATICAL PRELIMINARIES ....................................................................................................... 10 4.1 A DDITION ...................................................................................................................................................... 10 4.2 M ULTIPLICATION .......................................................................................................................................... 10 4.2.1 Multiplication by x .............................................................................................................................. 11 4.3 P OLYNOMIALS WITH C OEFFICIENTS IN GF(2 8 ) .............................................................................................. 12 5. ALGORITHM SPECIFICATION................................................................................................................. 13 5.1 C IPHER .......................................................................................................................................................... 14 5.1.1 SubBytes()Transformation ............................................................................................................ 15 5.1.2 ShiftRows() Transformation ........................................................................................................ 17 5.1.3 MixColumns() Transformation ...................................................................................................... 17 5.1.4 AddRoundKey() Transformation .................................................................................................. 18 5.2 K EY E XPANSION ........................................................................................................................................... 19 5.3 I NVERSE C IPHER ............................................................................................................................................ 20 5.3.1 InvShiftRows() Transformation ................................................................................................. 21 5.3.2 InvSubBytes() Transformation ................................................................................................... 22 5.3.3 InvMixColumns() Transformation ............................................................................................... 23 5.3.4 Inverse of the AddRoundKey() Transformation ............................................................................. 23 5.3.5 Equivalent Inverse Cipher .................................................................................................................. 23 6. IMPLEMENTATION ISSUES ...................................................................................................................... 25 6.1 K EY L ENGTH R EQUIREMENTS ....................................................................................................................... 25 6.2 K EYING R ESTRICTIONS ................................................................................................................................. 26 6.3 P ARAMETERIZATION OF K EY L ENGTH , B LOCK S IZE , AND R OUND N UMBER ................................................. 26 6.4 I MPLEMENTATION S UGGESTIONS R EGARDING V ARIOUS P LATFORMS ........................................................... 26 APPENDIX A - KEY EXPANSION EXAMPLES ................................................................................................ 27 A.1 E XPANSION OF A 128- BIT C IPHER K EY .......................................................................................................... 27 A.2 E XPANSION OF A 192- BIT C IPHER K EY .......................................................................................................... 28 A.3 E XPANSION OF A 256- BIT C IPHER K EY .......................................................................................................... 30 APPENDIX B – CIPHER EXAMPLE .................................................................................................................... 33 APPENDIX C – EXAMPLE VECTORS ................................................................................................................ 35 C.1 AES-128 (N K =4, N R =10) .............................................................................................................................. 35 C.2 AES-192 (N K =6, N R =12) .............................................................................................................................. 38 C.3 AES-256 (N K =8, N R =14) .............................................................................................................................. 42 APPENDIX D - REFERENCES .............................................................................................................................. 47 2 Table of Figures Figure 1. Hexadecimal representation of bit patterns. ................................................................. 8 Figure 2. Indices for Bytes and Bits. ........................................................................................... 9 Figure 3. State array input and output. ........................................................................................ 9 Figure 4. Key-Block-Round Combinations............................................................................... 14 Figure 5. Pseudo Code for the Cipher. ...................................................................................... 15 Figure 6. SubBytes() applies the S-box to each byte of the State. ...................................... 16 Figure 7. S-box: substitution values for the byte xy (in hexadecimal format). ....................... 16 Figure 8. ShiftRows() cyclically shifts the last three rows in the State .............................. 17 Figure 9. MixColumns() operates on the State column-by-column. .................................... 18 Figure 10. AddRoundKey() XORs each column of the State with a word from the key schedule. ...................................................................................................................... 19 Figure 11. Pseudo Code for Key Expansion. ............................................................................... 20 Figure 12. Pseudo Code for the Inverse Cipher. .......................................................................... 21 Figure 13. InvShiftRows()cyclically shifts the last three rows in the State. ....................... 22 Figure 14. Inverse S-box: substitution values for the byte xy (in hexadecimal format). ............ 22 Figure 15. Pseudo Code for the Equivalent Inverse Cipher. ........................................................ 25 3 4 1. Introduction This standard specifies the Rijndael algorithm ([3] and [4]), a symmetric block cipher that can process data blocks of 128 bits, using cipher keys with lengths of 128, 192, and 256 bits. Rijndael was designed to handle additional block sizes and key lengths, however they are not adopted in this standard. Throughout the remainder of this standard, the algorithm specified herein will be referred to as “the AES algorithm.” The algorithm may be used with the three different key lengths indicated above, and therefore these different “flavors” may be referred to as “AES-128”, “AES-192”, and “AES-256”. This specification includes the following sections: 2. Definitions of terms, acronyms, and algorithm parameters, symbols, and functions; 3. Notation and conventions used in the algorithm specification, including the ordering and numbering of bits, bytes, and words; 4. Mathematical properties that are useful in understanding the algorithm; 5. Algorithm specification, covering the key expansion, encryption, and decryption routines; 6. Implementation issues, such as key length support, keying restrictions, and additional block/key/round sizes. The standard concludes with several appendices that include step-by-step examples for Key Expansion and the Cipher, example vectors for the Cipher and Inverse Cipher, and a list of references. 2. Definitions 2.1 Glossary of Terms and Acronyms The following definitions are used throughout this standard: AES Advanced Encryption Standard Affine A transformation consisting of multiplication by a matrix followed by Transformation the addition of a vector. Array An enumerated collection of identical entities (e.g., an array of bytes). Bit A binary digit having a value of 0 or 1. Block Sequence of binary bits that comprise the input, output, State, and Round Key. The length of a sequence is the number of bits it contains. Blocks are also interpreted as arrays of bytes. Byte A group of eight bits that is treated either as a single entity or as an array of 8 individual bits. 5 Cipher Series of transformations that converts plaintext to ciphertext using the Cipher Key. Cipher Key Secret, cryptographic key that is used by the Key Expansion routine to generate a set of Round Keys; can be pictured as a rectangular array of bytes, having four rows and Nk columns. Ciphertext Data output from the Cipher or input to the Inverse Cipher. Inverse Cipher Series of transformations that converts ciphertext to plaintext using the Cipher Key. Key Expansion Routine used to generate a series of Round Keys from the Cipher Key. Plaintext Data input to the Cipher or output from the Inverse Cipher. Rijndael Cryptographic algorithm specified in this Advanced Encryption Standard (AES). Round Key Round keys are values derived from the Cipher Key using the Key Expansion routine; they are applied to the State in the Cipher and Inverse Cipher. State Intermediate Cipher result that can be pictured as a rectangular array of bytes, having four rows and Nb columns. S-box Non-linear substitution table used in several byte substitution transformations and in the Key Expansion routine to perform a one- for-one substitution of a byte value. Word A group of 32 bits that is treated either as a single entity or as an array of 4 bytes. 2.2 Algorithm Parameters, Symbols, and Functions The following algorithm parameters, symbols, and functions are used throughout this standard: AddRoundKey() Transformation in the Cipher and Inverse Cipher in which a Round Key is added to the State using an XOR operation. The length of a Round Key equals the size of the State (i.e., for Nb = 4, the Round Key length equals 128 bits/16 bytes). InvMixColumns()Transformation in the Inverse Cipher that is the inverse of MixColumns(). InvShiftRows() Transformation in the Inverse Cipher that is the inverse of ShiftRows(). InvSubBytes() Transformation in the Inverse Cipher that is the inverse of SubBytes(). K Cipher Key. 6 MixColumns() Transformation in the Cipher that takes all of the columns of the State and mixes their data (independently of one another) to produce new columns. Nb Number of columns (32-bit words) comprising the State. For this standard, Nb = 4. (Also see Sec. 6.3.) Nk Number of 32-bit words comprising the Cipher Key. For this standard, Nk = 4, 6, or 8. (Also see Sec. 6.3.) Nr Number of rounds, which is a function of Nk and Nb (which is fixed). For this standard, Nr = 10, 12, or 14. (Also see Sec. 6.3.) Rcon[] The round constant word array. RotWord() Function used in the Key Expansion routine that takes a four-byte word and performs a cyclic permutation. ShiftRows() Transformation in the Cipher that processes the State by cyclically shifting the last three rows of the State by different offsets. SubBytes() Transformation in the Cipher that processes the State using a non linear byte substitution table (S-box) that operates on each of the State bytes independently. SubWord() Function used in the Key Expansion routine that takes a four-byte input word and applies an S-box to each of the four bytes to produce an output word. XOR Exclusive-OR operation. ¯ Exclusive-OR operation. ˜ Multiplication of two polynomials (each with degree < 4) modulo x 4 + 1. • Finite field multiplication. 3. Notation and Conventions 3.1 Inputs and Outputs The input and output for the AES algorithm each consist of sequences of 128 bits (digits with values of 0 or 1). These sequences will sometimes be referred to as blocks and the number of bits they contain will be referred to as their length. The Cipher Key for the AES algorithm is a sequence of 128, 192 or 256 bits. Other input, output and Cipher Key lengths are not permitted by this standard. The bits within such sequences will be numbered starting at zero and ending at one less than the sequence length (block length or key length). The number i attached to a bit is known as its index and will be in one of the ranges 0 £ i < 128, 0 £ i < 192 or 0 £ i < 256 depending on the block length and key length (specified above). 7 3.2 Bytes The basic unit for processing in the AES algorithm is a byte, a sequence of eight bits treated as a single entity. The input, output and Cipher Key bit sequences described in Sec. 3.1 are processed as arrays of bytes that are formed by dividing these sequences into groups of eight contiguous bits to form arrays of bytes (see Sec. 3.3). For an input, output or Cipher Key denoted by a, the bytes in the resulting array will be referenced using one of the two forms, a n or a[ n], where n will be in one of the following ranges: Key length = 128 bits, 0 £ n < 16; Block length = 128 bits, 0 £ n < 16; Key length = 192 bits, 0 £ n < 24; Key length = 256 bits, 0 £ n < 32. All byte values in the AES algorithm will be presented as the concatenation of its individual bit values (0 or 1) between braces in the order {b 7 , b 6 , b 5 , b 4 , b 3 , b 2 , b 1 , b 0 }. These bytes are interpreted as finite field elements using a polynomial representation: 7 7 6 5 4 3 2 ib x + b x + b x + b x + b x + b x + b x + b = �b i x . (3.1)7 6 5 4 3 2 1 0 i =0 For example, {01100011} identifies the specific finite field element x 6 + x 5 + x +1 . It is also convenient to denote byte values using hexadecimal notation with each of two groups of four bits being denoted by a single character as in Fig. 1. Bit Pattern Character 0000 0 0001 1 0010 2 0011 3 Bit Pattern Character 0100 4 0101 5 0110 6 0111 7 Bit Pattern Character 1000 8 1001 9 1010 a 1011 b Bit Pattern Character 1100 c 1101 d 1110 e 1111 f Figure 1. Hexadecimal representation of bit patterns. Hence the element {01100011} can be represented as {63}, where the character denoting the four-bit group containing the higher numbered bits is again to the left. Some finite field operations involve one additional bit (b 8 ) to the left of an 8-bit byte. Where this extra bit is present, it will appear as ‘{01}’ immediately preceding the 8-bit byte; for example , a 9-bit sequence will be presented as {01}{1b}. 3.3 Arrays of Bytes Arrays of bytes will be represented in the following form: a a a ... a0 1 2 15 The bytes and the bit ordering within bytes are derived from the 128-bit input sequence input 0 input 1 input 2 … input 126 input 127 as follows: 8 a 0 = {input 0 , input 1 , …, input 7 }; a 1 = {input 8 , input 9 , …, input 15 }; M a 15 = {input 120 , input 121 , …, input 127 }. The pattern can be extended to longer sequences (i.e., for 192- and 256-bit keys), so that, in general, a n = {input 8 n , input 8 n+1 , …, input 8 n +7 }. (3.2) Taking Sections 3.2 and 3.3 together, Fig. 2 shows how bits within each byte are numbered. Input bit sequence 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 … Byte number 0 1 2 … Bit numbers in byte 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 … Figure 2. Indices for Bytes and Bits. 3.4 The State Internally, the AES algorithm’s operations are performed on a two-dimensional array of bytes called the State. The State consists of four rows of bytes, each containing Nb bytes, where Nb is the block length divided by 32. In the State array denoted by the symbol s , each individual byte has two indices, with its row number r in the range 0 £ r < 4 and its column number c in the range 0 £ c < Nb. This allows an individual byte of the State to be referred to as either s r ,c or s [ r ,c ]. For this standard, Nb=4, i.e., 0 £ c < 4 (also see Sec. 6.3). At the start of the Cipher and Inverse Cipher described in Sec. 5, the input – the array of bytes in 0 , in 1 , … in 15 – is copied into the State array as illustrated in Fig. 3. The Cipher or Inverse Cipher operations are then conducted on this State array, after which its final value is copied to the output – the array of bytes out 0 , out 1 , … out 15 . input bytes State array output bytes in 0 in 4 in 8 in 12 in 1 in 5 in 9 in 1 3 in 2 in 6 in 10 in 14 in 3 in 7 in 11 in 15 s 0,0 s 0,1 s 0,2 s 0,3 s 1,0 s 1,1 s 1,2 s 1,3 s 2,0 s 2,1 s 2,2 s 2,3 s 3,0 s 3,1 s 3,2 s 3,3 out 0 out 4 out 8 out 12 out 1 out 5 out 9 out 13 out 2 out 6 out 10 out 14 out 3 out 7 out 11 out 15 � � Figure 3. State array input and output. Hence, at the beginning of the Cipher or Inverse Cipher, the input array, in, is copied to the State array according to the scheme: s [ r , c ] = in[ r + 4c ] for 0 £ r < 4 and 0 £ c < Nb, (3.3) 9 and at the end of the Cipher and Inverse Cipher, the State is copied to the output array out as follows: out[ r + 4c ] = s [ r , c ] for 0 £ r < 4 and 0 £ c < Nb. (3.4) 3.5 The State as an Array of Columns The four bytes in each column of the State array form 32-bit words, where the row number r provides an index for the four bytes within each word. The state can hence be interpreted as a one-dimensional array of 32 bit words (columns), w 0 ...w 3 , where the column number c provides an index into this array. Hence, for the example in Fig. 3, the State can be considered as an array of four words, as follows: w 0 = s 0,0 s 1,0 s 2,0 s 3,0 w 2 = s 0,2 s 1,2 s 2,2 s 3,2 w 1 = s 0,1 s 1,1 s 2,1 s 3,1 w 3 = s 0,3 s 1,3 s 2,3 s 3,3 . (3.5) 4. Mathematical Preliminaries All bytes in the AES algorithm are interpreted as finite field elements using the notation introduced in Sec. 3.2. Finite field elements can be added and multiplied, but these operations are different from those used for numbers. The following subsections introduce the basic mathematical concepts needed for Sec. 5. 4.1 Addition The addition of two elements in a finite field is achieved by “adding” the coefficients for the corresponding powers in the polynomials for the two elements. The addition is performed with the XOR operation (denoted by ¯ ) - i.e., modulo 2 - so that 1 ¯1 = 0 , 1 ¯ 0 = 1 , and 0 ¯ 0 = 0 . Consequently, subtraction of polynomials is identical to addition of polynomials. Alternatively, addition of finite field elements can be described as the modulo 2 addition of corresponding bits in the byte. For two bytes {a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 } and {b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 }, the sum is {c 7 c 6 c 5 c 4 c 3 c 2 c 1 c 0 }, where each c i = a i ¯ b i (i.e., c 7 = a 7 ¯ b 7 , c 6 = a 6 ¯ b 6 , ...c 0 = a 0 ¯ b 0 ) . For example, the following expressions are equivalent to one another: 6 4 2 7 7 6 4 2( x + x + x + x +1) + ( x + x +1) = x + x + x + x (polynomial notation); {01010111} ¯ {10000011} = {11010100} (binary notation); {57} ¯ {83} = {d4} (hexadecimal notation). 4.2 Multiplication In the polynomial representation, multiplication in GF(2 8 ) (denoted by •) corresponds with the multiplication of polynomials modulo an irreducible polynomial of degree 8. A polynomial is irreducible if its only divisors are one and itself. For the AES algorithm, this irreducible polynomial is 8 4 3m ( x ) = x + x + x + x +1 , (4.1) 10 or {01}{1b} in hexadecimal notation. For example, {57} • {83} = {c1}, because 6 4 2 7 13 11 9 8 7( x + x + x + x +1) ( x + x +1) = x + x + x + x + x + 7 5 3 2x + x + x + x + x + 6 4 2x + x + x + x +1 13 11 9 8 6 5 4 3= x + x + x + x + x + x + x + x +1 and 13 11 9 8 6 5 4 3 8 4 3x + x + x + x + x + x + x + x +1 modulo ( x + x + x + x +1 ) = x 7 + x 6 +1 . The modular reduction by m ( x ) ensures that the result will be a binary polynomial of degree less than 8, and thus can be represented by a byte. Unlike addition, there is no simple operation at the byte level that corresponds to this multiplication. The multiplication defined above is associative, and the element {01} is the multiplicative identity. For any non-zero binary polynomial b( x ) of degree less than 8, the multiplicative inverse of b( x ), denoted b -1 ( x ), can be found as follows: the extended Euclidean algorithm [7] is used to compute polynomials a( x ) and c ( x ) such that b ( x ) a ( x ) + m ( x ) c ( x ) = 1 . (4.2) Hence, a ( x ) • b ( x ) mod m ( x ) = 1 , which means -b 1 ( x ) = a ( x ) mod m ( x ) . (4.3) Moreover, for any a ( x ), b( x ) and c ( x ) in the field, it holds that a ( x ) • ( b ( x ) + c ( x )) = a ( x ) • b ( x ) + a ( x ) • c ( x ) . It follows that the set of 256 possible byte values, with XOR used as addition and the multiplication defined as above, has the structure of the finite field GF(2 8 ). 4.2.1 Multiplication by x Multiplying the binary polynomial defined in equation (3.1) with the polynomial x results in 8 7 6 5 4 3 2b 7 x + b 6 x + b 5 x + b 4 x + b 3 x + b 2 x + b 1 x + b 0 x . (4.4) The result x • b ( x ) is obtained by reducing the above result modulo m ( x ), as defined in equation (4.1). If b 7 = 0, the result is already in reduced form. If b 7 = 1, the reduction is accomplished by subtracting (i.e., XORing) the polynomial m ( x ). It follows that multiplication by x (i.e., {00000010} or {02}) can be implemented at the byte level as a left shift and a subsequent conditional bitwise XOR with {1b}. This operation on bytes is denoted by xtime(). Multiplication by higher powers of x can be implemented by repeated application of xtime(). By adding intermediate results, multiplication by any constant can be implemented. For example, {57} • {13} = {fe} because 11 {57} • {02} = xtime({57}) = {ae} {57} • {04} = xtime({ae}) = {47} {57} • {08} = xtime({47}) = {8e} {57} • {10} = xtime({8e}) = {07}, thus, {57} • {13} = { 57} • ({01} ¯ {02} ¯ {10}) = { 57} ¯ {ae} ¯ {07} = { fe}. 4.3 Polynomials with Coefficients in GF(2 8 ) Four-term polynomials can be defined - with coefficients that are finite field elements - as: a ( x ) = a 3 x 3 + a 2 x 2 + a 1 x + a 0 (4.5) which will be denoted as a word in the form [ a 0 , a 1 , a 2 , a 3 ]. Note that the polynomials in this section behave somewhat differently than the polynomials used in the definition of finite field elements, even though both types of polynomials use the same indeterminate, x . The coefficients in this section are themselves finite field elements, i.e., bytes, instead of bits; also, the multiplication of four-term polynomials uses a different reduction polynomial, defined below. The distinction should always be clear from the context. To illustrate the addition and multiplication operations, let b ( x ) = b x 3 + b x 2 + b x + b (4.6)3 2 1 0 define a second four-term polynomial. Addition is performed by adding the finite field coefficients of like powers of x . This addition corresponds to an XOR operation between the corresponding bytes in each of the words – in other words, the XOR of the complete word values. Thus, using the equations of (4.5) and (4.6), a ( x ) + b ( x ) = ( a ¯ b ) x 3 + ( a ¯ b ) x 2 + ( a ¯ b ) x + ( a ¯ b ) (4.7)3 3 2 2 1 1 0 0 Multiplication is achieved in two steps. In the first step, the polynomial product c ( x ) = a( x ) • b( x ) is algebraically expanded, and like powers are collected to give 6 5 4 3 2c ( x ) = c x + c x + c x + c x + c x + c x + c (4.8)6 5 4 3 2 1 0 where c = a • b c = a • b ¯ a • b ¯ a • b0 0 0 4 3 1 2 2 1 3 c 1 = a 1 • b 0 ¯ a 0 • b 1 c 5 = a 3 • b 2 ¯ a 2 • b 3 c 2 = a 2 • b 0 ¯ a 1 • b 1 ¯ a 0 • b 2 c 6 = a 3 • b 3 (4.9) 12 c = a • b ¯ a • b ¯ a • b ¯ a • b .3 3 0 2 1 1 2 0 3 The result, c ( x ), does not represent a four-byte word. Therefore, the second step of the multiplication is to reduce c ( x ) modulo a polynomial of degree 4; the result can be reduced to a polynomial of degree less than 4. For the AES algorithm, this is accomplished with the polynomial x 4 + 1, so that i 4 i mod 4x mod( x +1) = x . (4.10) The modular product of a( x ) and b( x ), denoted by a( x ) ˜ b( x ), is given by the four-term polynomial d( x ), defined as follows: d ( x ) = d 3 x 3 + d 2 x 2 + d 1 x + d 0 (4.11) with d = ( a • b ) ¯ ( a • b ) ¯ ( a • b ) ¯ ( a • b )0 0 0 3 1 2 2 1 3 d = ( a • b ) ¯ ( a • b ) ¯ ( a • b ) ¯ ( a • b ) (4.12)1 1 0 0 1 3 2 2 3 d = ( a • b ) ¯ ( a • b ) ¯ ( a • b ) ¯ ( a • b )2 2 0 1 1 0 2 3 3 = ( a • b ) ¯ ( a • b ) ¯ ( a • b ) ¯ ( a • b )d 3 3 0 2 1 1 2 0 3 When a( x ) is a fixed polynomial, the operation defined in equation (4.11) can be written in matrix form as: Ød 0 ø Ø a 0 a 3 a 2 a 1 ø Øb 0 ø Œ œ Œ œ Œ œd a a a a b1 1 0 3 2 1Œ œ = Œ œ Œ œ (4.13)Œd œ Œ a a a a œ Œb œ2 2 1 0 3 2 Œ œ Œ œ Œ œ aºd 3 ß º a 3 2 a 1 a 0 ß ºb 3 ß Because x 4 +1 is not an irreducible polynomial over GF(2 8 ), multiplication by a fixed four-term polynomial is not necessarily invertible. However, the AES algorithm specifies a fixed four-term polynomial that does have an inverse (see Sec. 5.1.3 and Sec. 5.3.3): a( x ) = {03} x 3 + {01}x 2 + {01}x + {02} (4.14) a -1 ( x ) = {0b}x 3 + {0d} x 2 + {09}x + {0e}. (4.15) Another polynomial used in the AES algorithm (see the RotWord() function in Sec. 5.2) has a 0 = a 1 = a 2 = {00} and a 3 = {01}, which is the polynomial x 3 . Inspection of equation (4.13) above will show that its effect is to form the output word by rotating bytes in the input word. This means that [b 0 , b 1 , b 2 , b 3 ] is transformed into [b 1 , b 2 , b 3 , b 0 ]. 5. Algorithm Specification For the AES algorithm, the length of the input block, the output block and the State is 128 bits. This is represented by Nb = 4, which reflects the number of 32-bit words (number of columns) in the State. 13 For the AES algorithm, the length of the Cipher Key, K , is 128, 192, or 256 bits. The key length is represented by Nk = 4, 6, or 8, which reflects the number of 32-bit words (number of columns) in the Cipher Key. For the AES algorithm, the number of rounds to be performed during the execution of the algorithm is dependent on the key size. The number of rounds is represented by Nr, where Nr = 10 when Nk = 4, Nr = 12 when Nk = 6, and Nr = 14 when Nk = 8. The only Key-Block-Round combinations that conform to this standard are given in Fig. 4. For implementation issues relating to the key length, block size and number of rounds, see Sec. 6.3. Key Length (Nk words) Block Size (Nb words) Number of Rounds (Nr) AES-128 4 4 10 AES-192 6 4 12 AES-256 8 4 14 Figure 4. Key-Block-Round Combinations. For both its Cipher and Inverse Cipher, the AES algorithm uses a round function that is composed of four different byte-oriented transformations: 1) byte substitution using a substitution table (S-box), 2) shifting rows of the State array by different offsets, 3) mixing the data within each column of the State array, and 4) adding a Round Key to the State. These transformations (and their inverses) are described in Sec. 5.1.1-5.1.4 and 5.3.1-5.3.4. The Cipher and Inverse Cipher are described in Sec. 5.1 and Sec. 5.3, respectively, while the Key Schedule is described in Sec. 5.2. 5.1 Cipher At the start of the Cipher, the input is copied to the State array using the conventions described in Sec. 3.4. After an initial Round Key addition, the State array is transformed by implementing a round function 10, 12, or 14 times (depending on the key length), with the final round differing slightly from the first Nr -1 rounds. The final State is then copied to the output as described in Sec. 3.4. The round function is parameterized using a key schedule that consists of a one-dimensional array of four-byte words derived using the Key Expansion routine described in Sec. 5.2. The Cipher is described in the pseudo code in Fig. 5. The individual transformations - SubBytes(), ShiftRows(), MixColumns(), and AddRoundKey() – process the State and are described in the following subsections. In Fig. 5, the array w[] contains the key schedule, which is described in Sec. 5.2. As shown in Fig. 5, all Nr rounds are identical with the exception of the final round, which does not include the MixColumns() transformation. 14 Appendix B presents an example of the Cipher, showing values for the State array at the beginning of each round and after the application of each of the four transformations described in the following sections. Cipher(byte in[4Nb], byte out[4Nb], word w[Nb(Nr+1)]) begin byte state[4,Nb] state = in AddRoundKey(state, w[0, Nb-1]) // See Sec. 5.1.4 for round = 1 step 1 to Nr–1 SubBytes(state) // See Sec. 5.1.1 ShiftRows(state) // See Sec. 5.1.2 MixColumns(state) // See Sec. 5.1.3 AddRoundKey(state, w[roundNb, (round+1)Nb-1]) end for SubBytes(state) ShiftRows(state) AddRoundKey(state, w[NrNb, (Nr+1)Nb-1]) out = state end Figure 5. Pseudo Code for the Cipher. 1 5.1.1 SubBytes()Transformation The SubBytes() transformation is a non-linear byte substitution that operates independently on each byte of the State using a substitution table (S-box). This S-box (Fig. 7), which is invertible, is constructed by composing two transformations: 1. Take the multiplicative inverse in the finite field GF(2 8 ), described in Sec. 4.2; the element {00} is mapped to itself. 2. Apply the following affine transformation (over GF(2) ): b ' = b ¯ b ¯ b ¯ b ¯ b ¯ c (5.1)i i ( i +4) mod 8 ( i +5) mod 8 ( i +6) mod 8 ( i +7) mod 8 i for 0 £ i < 8 , where b i is the i th bit of the byte, and c i is the i th bit of a byte c with the value {63} or {01100011}. Here and elsewhere, a prime on a variable (e.g., b ¢ ) indicates that the variable is to be updated with the value on the right. In matrix form, the affine transformation element of the S-box can be expressed as: 1 The various transformations (e.g., SubBytes(), ShiftRows(), etc.) act upon the State array that is addressed by the ‘state’ pointer. AddRoundKey() uses an additional pointer to address the Round Key. 15 1 , 1s 1 , 1s 'Ø ø Ø ø Ø 1 0 0 0 1 1 1 1 ø Ø 1 b 0b 0 ø ' 1 1 0 0 0 1 1 1 1 b 1b 1 ' Figure 6 illustrates the effect of the SubBytes() transformation on the State. Œ Œ Œ Œ Œ Œ Œ Œ Œ Œ Œº Œ Œ Œ Œ Œ Œ Œ Œ Œ Œ Œº Œ Œ Œ Œ Œ Œ Œ Œ Œ Œ Œº Œ Œ Œ Œ Œ Œ Œ Œ Œ Œ Œº œ œ œ œ œ œ œ œ œ œ œß œ œ œ œ œ œ œ œ œ œ ßœ œ œ œ œ œ œ œ œ œ œ ßœ œ œ œ œ œ œ œ œ œ œ ßœ 1 1 1 0 0 0 1 1 b 2 0 b 2 ' 1 1 1 1 0 0 0 1 b 3 0 b 3 . (5.2) + = ' 1 1 1 1 1 0 0 0 b 4 0 b 4 ' 0 1 1 1 1 1 0 0 b 5 1 b 5 ' 0 0 1 1 1 1 1 0 b 6 1 b 6 ' 0 0 0 1 1 1 1 1 b 7 0 b 7 s 0,0 s 0,1 s 0,2 s 0,3 S-Box ' s 0,0 ' s 0,1 ' s 0,2 ' s 0,3 ' ' ' ' s 1,0 rs s 1,2 c s 1,3 s 1,0 ' s s 1,2 s 1,3 , r ,c ' ' ' ' s 2,0 s 2,1 s 2,2 s 2,3 s 2,0 s 2,1 s 2,2 s 2,3 ' ' ' ' s 3,0 s 3,1 s 3,2 s 3,3 s 3,0 s 3,1 s 3,2 s 3,3 Figure 6. SubBytes() applies the S-box to each byte of the State. The S-box used in the SubBytes() transformation is presented in hexadecimal form in Fig. 7. For example, if s 1,1 = {53}, then the substitution value would be determined by the intersection of the row with index ‘5’ and the column with index ‘3’ in Fig. 7. This would result in s 1¢ ,1 having a value of {ed}. y 0 1 2 3 4 5 6 7 8 9 a b c d e f x 0 63 7c 77 7b f2 6b 6f c5 30 01 67 2b fe d7 ab 76 1 ca 82 c9 7d fa 59 47 f0 ad d4 a2 af 9c a4 72 c0 2 b7 fd 93 26 36 3f f7 cc 34 a5 e5 f1 71 d8 31 15 3 04 c7 23 c3 18 96 05 9a 07 12 80 e2 eb 27 b2 75 4 09 83 2c 1a 1b 6e 5a a0 52 3b d6 b3 29 e3 2f 84 5 53 d1 00 ed 20 fc b1 5b 6a cb be 39 4a 4c 58 cf 6 d0 ef aa fb 43 4d 33 85 45 f9 02 7f 50 3c 9f a8 7 51 a3 40 8f 92 9d 38 f5 bc b6 da 21 10 ff f3 d2 8 cd 0c 13 ec 5f 97 44 17 c4 a7 7e 3d 64 5d 19 73 9 60 81 4f dc 22 2a 90 88 46 ee b8 14 de 5e 0b db a e0 32 3a 0a 49 06 24 5c c2 d3 ac 62 91 95 e4 79 b e7 c8 37 6d 8d d5 4e a9 6c 56 f4 ea 65 7a ae 08 c ba 78 25 2e 1c a6 b4 c6 e8 dd 74 1f 4b bd 8b 8a d 70 3e b5 66 48 03 f6 0e 61 35 57 b9 86 c1 1d 9e e e1 f8 98 11 69 d9 8e 94 9b 1e 87 e9 ce 55 28 df f 8c a1 89 0d bf e6 42 68 41 99 2d 0f b0 54 bb 16 Figure 7. S-box: substitution values for the byte xy (in hexadecimal format). 16 5.1.2 ShiftRows() Transformation In the ShiftRows() transformation, the bytes in the last three rows of the State are cyclically shifted over different numbers of bytes (offsets). The first row, r = 0, is not shifted. Specifically, the ShiftRows() transformation proceeds as follows: s ' = s for 0 < r < 4 and 0 £ c < Nb, (5.3)r , c r ,( c + shift ( r , Nb )) mod Nb where the shift value shift( r ,Nb) depends on the row number, r , as follows (recall that Nb = 4): shift (1 ,4) = 1 ; shift (2,4) = 2 ; shift (3,4) = 3 . (5.4) This has the effect of moving bytes to “lower” positions in the row (i.e., lower values of c in a given row), while the “lowest” bytes wrap around into the “top” of the row (i.e., higher values of c in a given row). Figure 8 illustrates the ShiftRows() transformation. ShiftRows () 0 ,rs 1 ,rs 2 ,rs 3 ,rs ' 0 ,rs ' 2 ,rs ' 3 ,rs ' 1 ,rs S S ’ s 0,0 s 0,1 s 0,2 s 0,3 s 0,0 s 0,1 s 0,2 s 0,3 s 1,0 s 1,1 s 1,2 s 1,3 s 1,1 s 1,2 s 1,3 s 1,0 s 2,0 s 2,1 s 2,2 s 2,3 s 2,2 s 2,3 s 2,0 s 2,1 s 3,0 s 3,1 s 3,2 s 3,3 s 3,3 s 3,0 s 3,1 s 3,2 Figure 8. ShiftRows() cyclically shifts the last three rows in the State. 5.1.3 MixColumns() Transformation The MixColumns() transformation operates on the State column-by-column, treating each column as a four-term polynomial as described in Sec. 4.3. The columns are considered as polynomials over GF(2 8 ) and multiplied modulo x 4 + 1 with a fixed polynomial a( x ), given by a( x ) = {03} x 3 + {01}x 2 + {01}x + {02} . (5.5) As described in Sec. 4.3, this can be written as a matrix multiplication. Let s ¢( x ) = a ( x ) ˜ s ( x ) : 17 1 , 0s 1 , 0s 1 , 1s 1 , 1s 1 , 2s 1 , 2s 1 , 3s 1 , 3s 'Ø Ø 02 03 01 Ø ø s 0, ø s 0, cc ' 01 02 03 s 1, s 1, c for 0 £ c < Nb. (5.6) c ' 01 01 01 01 02 03 ø œ œ œ œ Œ Œ Œ Œ Œº Œ Œ Œ Œ Œ Œ Œ Œ º = œ œ œ œ ßœ œ œ œ œ ss 2,2, cc ' 03 01 01 02 ºß s 3, ß s 3, cc As a result of this multiplication, the four bytes in a column are replaced by the following: 0, 2, 3, ¢ ¢ ¢ ¢ ({02} • s ) ¯ ({03} • s ) ¯ 0, c 1, c ¯ = s s s2, 3,c c c ¯ ({02} • s ) ¯ ({03} • s ) ¯ 1, c 2, c = s 1, s s0, 3,c c c ¯ ¯ ({02} • s ) ¯ ({03} • s )2, c 3, c = s s s 1,0,c c c ({03} • s 0, c ) ¯ ¯ ¯ ({02} • s 3, c ).= s s 1, s 2,c c c Figure 9 illustrates the MixColumns() transformation. 0 , 0s 2 , 0s 3 , 0s ' 0 , 0s ' ' 2 , 0s ' 3 , 0s 0 , 1s 2 , 1s 3 , 1s ' 0 , 1s ' ' 2 , 1s ' 3 , 1s 0 , 2s 2 , 2s 3 , 2s ' 0 , 2s ' ' 2 , 2s ' 3 , 2s 0 , 3s 2 , 3s 3 , 3s ' 0 , 3s ' ' 2 , 3s ' 3 , 3s MixColumns () cs , 0 cs , 1 cs , 2 cs , 3 ' , 0cs ' , 1cs ' , 2cs ' , 3cs Figure 9. MixColumns() operates on the State column-by-column. 5.1.4 AddRoundKey() Transformation In the AddRoundKey() transformation, a Round Key is added to the State by a simple bitwise XOR operation. Each Round Key consists of Nb words from the key schedule (described in Sec. 5.2). Those Nb words are each added into the columns of the State, such that ¯ [ w round Nb +c ] for 0 £ c < Nb, (5.7) [ ' , ' , ' , ' ] [ , 1, , , ]= s s s s s s s s0, 1, 2, 3, 0, 2, 3,c c c c c c c c where [w i] are the key schedule words described in Sec. 5.2, and round is a value in the range 0 £ round £ Nr. In the Cipher, the initial Round Key addition occurs when round = 0, prior to the first application of the round function (see Fig. 5). The application of the AddRoundKey() transformation to the Nr rounds of the Cipher occurs when 1 £ round £ Nr. The action of this transformation is illustrated in Fig. 10, where l = round * Nb. The byte address within words of the key schedule was described in Sec. 3.1. 18 1 , 0s 0s 1 , 0s s 1 , 1s 1s 1 , 1s s 1 , 2s s 1 , 2s s 1 , 3s 3s 1+lw lw 1 , 3s s 0 , 0s 2 , 3 , 0s ' 0 , 0s ' ' 2 , 0 ' 3 , 0s 0 , 1s 2 , 3 , 1 s ' 0 , 1s ' ' 2 , 1 ' 3 , 1s ' 0 , 2s0 , 2s 2 , 2 3 , 2s ' ' 2 , 2 ' 3 , 2s 0 , 3s 2 , 3 , 3s lw 2+ 3+lw ' 0 , 3s ' ' 2 , 3 ' 3 , 3s ¯ cs , 0 cs , 1 cs , 2 cs , 3 ' , 0cs ' , 1cs ' , 2cs ' , 3cs w l+c Nbroundl = Figure 10. AddRoundKey() XORs each column of the State with a word from the key schedule. 5.2 Key Expansion The AES algorithm takes the Cipher Key, K , and performs a Key Expansion routine to generate a key schedule. The Key Expansion generates a total of Nb ( Nr + 1) words: the algorithm requires an initial set of Nb words, and each of the Nr rounds requires Nb words of key data. The resulting key schedule consists of a linear array of 4-byte words, denoted [ w i ], with i in the range 0 £ i < Nb( Nr + 1). The expansion of the input key into the key schedule proceeds according to the pseudo code in Fig. 11. SubWord() is a function that takes a four-byte input word and applies the S-box (Sec. 5.1.1, Fig. 7) to each of the four bytes to produce an output word. The function RotWord() takes a word [a 0 ,a 1 ,a 2 ,a 3 ] as input, performs a cyclic permutation, and returns the word [a 1 ,a 2 ,a 3 ,a 0 ]. The round constant word array, Rcon[i], contains the values given by [x i-1 ,{00},{00},{00}], with x i -1 being powers of x ( x is denoted as {02}) in the field GF(2 8 ), as discussed in Sec. 4.2 (note that i starts at 1, not 0). From Fig. 11, it can be seen that the first Nk words of the expanded key are filled with the Cipher Key. Every following word, w [ [i ] ], is equal to the XOR of the previous word, w [ [i-1 ] ], and the word Nk positions earlier, w [ [i-Nk ] ]. For words in positions that are a multiple of Nk, a transformation is applied to w [ [i-1 ] ] prior to the XOR, followed by an XOR with a round constant, Rcon[i]. This transformation consists of a cyclic shift of the bytes in a word ( RotWord()), followed by the application of a table lookup to all four bytes of the word ( SubWord()). It is important to note that the Key Expansion routine for 256-bit Cipher Keys (Nk = 8) is slightly different than for 128- and 192-bit Cipher Keys. If Nk = 8 and i-4 is a multiple of Nk, then SubWord() is applied to w [ [i-1 ] ] prior to the XOR. 19 KeyExpansion(byte key[4Nk], word w[Nb(Nr+1)], Nk) begin word temp i = 0 while (i < Nk) w[i] = word(key[4i], key[4i+1], key[4i+2], key[4i+3]) i = i+1 end while i = Nk while (i < Nb (Nr+1)] temp = w[i-1] if (i mod Nk = 0) temp = SubWord(RotWord(temp)) xor Rcon[i/Nk] else if (Nk > 6 and i mod Nk = 4) temp = SubWord(temp) end if w[i] = w[i-Nk] xor temp i = i + 1 end while end Note that Nk=4, 6, and 8 do not all have to be implemented; they are all included in the conditional statement above for conciseness. Specific implementation requirements for the Cipher Key are presented in Sec. 6.1. Figure 11. Pseudo Code for Key Expansion. 2 Appendix A presents examples of the Key Expansion. 5.3 Inverse Cipher The Cipher transformations in Sec. 5.1 can be inverted and then implemented in reverse order to produce a straightforward Inverse Cipher for the AES algorithm. The individual transformations used in the Inverse Cipher - InvShiftRows(), InvSubBytes(),InvMixColumns(), and AddRoundKey() – process the State and are described in the following subsections. The Inverse Cipher is described in the pseudo code in Fig. 12. In Fig. 12, the array w[] contains the key schedule, which was described previously in Sec. 5.2. 2 The functions SubWord() and RotWord() return a result that is a transformation of the function input, whereas the transformations in the Cipher and Inverse Cipher (e.g., ShiftRows(), SubBytes(), etc.) transform the State array that is addressed by the ‘state’ pointer. 20 InvCipher(byte in[4Nb], byte out[4Nb], word w[Nb(Nr+1)]) begin byte state[4,Nb] state = in AddRoundKey(state, w[NrNb, (Nr+1)Nb-1]) // See Sec. 5.1.4 for round = Nr-1 step -1 downto 1 InvShiftRows(state) // See Sec. 5.3.1 InvSubBytes(state) // See Sec. 5.3.2 AddRoundKey(state, w[roundNb, (round+1)Nb-1]) InvMixColumns(state) // See Sec. 5.3.3 end for InvShiftRows(state) InvSubBytes(state) AddRoundKey(state, w[0, Nb-1]) out = state end Figure 12. Pseudo Code for the Inverse Cipher. 3 5.3.1 InvShiftRows() Transformation InvShiftRows() is the inverse of the ShiftRows() transformation. The bytes in the last three rows of the State are cyclically shifted over different numbers of bytes (offsets). The first row, r = 0, is not shifted. The bottom three rows are cyclically shifted by Nb - shift ( r , Nb ) bytes, where the shift value shift(r,Nb) depends on the row number, and is given in equation (5.4) (see Sec. 5.1.2). Specifically, the InvShiftRows() transformation proceeds as follows: s ' = s for 0 < r < 4 and 0 £ c < Nb (5.8)r ,( c + shift ( r , Nb )) mod Nb r , c Figure 13 illustrates the InvShiftRows() transformation. 3 The various transformations (e.g., InvSubBytes(), InvShiftRows(), etc.) act upon the State array that is addressed by the ‘state’ pointer. AddRoundKey() uses an additional pointer to address the Round Key. 21 InvShiftRows () 0 ,rs 1 ,rs 2 ,rs 3 ,rs ' 0 ,rs ' 2 ,rs ' 3 ,rs ' 1 ,rs S S ’ s 0,0 s 0,1 s 0,2 s 0,3 s 1,0 s 1,1 s 1,2 s 1,3 s 2,0 s 2,1 s 2,2 s 2,3 s 3,0 s 3,1 s 3,2 s 3,3 s 0,0 s 0,1 s 0,2 s 0,3 s 1,3 s 1,0 s 1,1 s 1,2 s 2,2 s 2,3 s 2,0 s 2,1 s 3,1 s 3,2 s 3,3 s 3,0 Figure 13. InvShiftRows()cyclically shifts the last three rows in the State. 5.3.2 InvSubBytes() Transformation InvSubBytes() is the inverse of the byte substitution transformation, in which the inverse S- box is applied to each byte of the State. This is obtained by applying the inverse of the affine transformation (5.1) followed by taking the multiplicative inverse in GF(2 8 ). The inverse S-box used in the InvSubBytes() transformation is presented in Fig. 14: y 0 1 2 3 4 5 6 7 8 9 a b c d e f x 0 52 09 6a d5 30 36 a5 38 bf 40 a3 9e 81 f3 d7 fb 1 7c e3 39 82 9b 2f ff 87 34 8e 43 44 c4 de e9 cb 2 54 7b 94 32 a6 c2 23 3d ee 4c 95 0b 42 fa c3 4e 3 08 2e a1 66 28 d9 24 b2 76 5b a2 49 6d 8b d1 25 4 72 f8 f6 64 86 68 98 16 d4 a4 5c cc 5d 65 b6 92 5 6c 70 48 50 fd ed b9 da 5e 15 46 57 a7 8d 9d 84 6 90 d8 ab 00 8c bc d3 0a f7 e4 58 05 b8 b3 45 06 7 d0 2c 1e 8f ca 3f 0f 02 c1 af bd 03 01 13 8a 6b 8 3a 91 11 41 4f 67 dc ea 97 f2 cf ce f0 b4 e6 73 9 96 ac 74 22 e7 ad 35 85 e2 f9 37 e8 1c 75 df 6e a 47 f1 1a 71 1d 29 c5 89 6f b7 62 0e aa 18 be 1b b fc 56 3e 4b c6 d2 79 20 9a db c0 fe 78 cd 5a f4 c 1f dd a8 33 88 07 c7 31 b1 12 10 59 27 80 ec 5f d 60 51 7f a9 19 b5 4a 0d 2d e5 7a 9f 93 c9 9c ef e a0 e0 3b 4d ae 2a f5 b0 c8 eb bb 3c 83 53 99 61 f 17 2b 04 7e ba 77 d6 26 e1 69 14 63 55 21 0c 7d Figure 14. Inverse S-box: substitution values for the byte xy (in hexadecimal format). 22 5.3.3 InvMixColumns() Transformation InvMixColumns() is the inverse of the MixColumns() transformation. InvMixColumns() operates on the State column-by-column, treating each column as a four- term polynomial as described in Sec. 4.3. The columns are considered as polynomials over GF(2 8 ) and multiplied modulo x 4 + 1 with a fixed polynomial a -1 ( x ), given by a -1 ( x ) = {0b}x 3 + {0d} x 2 + {09}x + {0e}. (5.9) As described in Sec. 4.3, this can be written as a matrix multiplication. Let s ( x )¢ = a -1 ( x ) ˜ s ( x ) : 'Ø Ø 0e 0 b 0 d 09Ø ø s ø s ø 0, s 1, 0, cc Œ Œ Œ Œ Œº Œ Œ Œ Œ Œ Œ Œ Œ º = œ œ œ œ ßœ œ œ œ œ œ œ œ œ ' 09 0e 0 b 0 d 0 d 09 0e 0 b s 1, c for 0 £ c < Nb. (5.10) c ' ss 2,2, cc ' 0 b 0 d 09 0e ºß s 3, ß s 3, cc As a result of this multiplication, the four bytes in a column are replaced by the following: 0,¢ ({0e} • s 0, c ) ¯ ({0b} • ¯ •({ }0d ) ¯ ({09} • ) ) = s s 1, s s2, 3,c c c c ({09} • s ) ¯ ({0e} • s ) ¯ ({0b} • s ) ¯ ({0d} • s )0, c 1, c 2, c 3, cs 1, 2, 3, ¢ ¢ ¢ = c ({0d} • s ) ¯ ({09} • s ) ¯ ({0e} • s ) ¯ ({0b} • s )0, c 1, c 2, c 3, c = s c ({0b} • s 0, c ¯ •) ({ }0d s 1, c ) ¯ ({09} • s ) ¯ ({0e} • s )2, c 3, c = s c 5.3.4 Inverse of the AddRoundKey() Transformation AddRoundKey(), which was described in Sec. 5.1.4, is its own inverse, since it only involves an application of the XOR operation. 5.3.5 Equivalent Inverse Cipher In the straightforward Inverse Cipher presented in Sec. 5.3 and Fig. 12, the sequence of the transformations differs from that of the Cipher, while the form of the key schedules for encryption and decryption remains the same. However, several properties of the AES algorithm allow for an Equivalent Inverse Cipher that has the same sequence of transformations as the Cipher (with the transformations replaced by their inverses). This is accomplished with a change in the key schedule. The two properties that allow for this Equivalent Inverse Cipher are as follows: 1. The SubBytes() and ShiftRows() transformations commute; that is, a SubBytes() transformation immediately followed by a ShiftRows() transformation is equivalent to a ShiftRows() transformation immediately followed buy a SubBytes() transformation. The same is true for their inverses, InvSubBytes() and InvShiftRows. 23 2. The column mixing operations - MixColumns() and InvMixColumns() - are linear with respect to the column input, which means InvMixColumns(state XOR Round Key) = InvMixColumns(state) XOR InvMixColumns(Round Key). These properties allow the order of InvSubBytes() and InvShiftRows() transformations to be reversed. The order of the AddRoundKey() and InvMixColumns() transformations can also be reversed, provided that the columns (words) of the decryption key schedule are modified using the InvMixColumns() transformation. The equivalent inverse cipher is defined by reversing the order of the InvSubBytes() and InvShiftRows() transformations shown in Fig. 12, and by reversing the order of the AddRoundKey() and InvMixColumns() transformations used in the “round loop” after first modifying the decryption key schedule for round = 1 to Nr-1 using the InvMixColumns() transformation. The first and last Nb words of the decryption key schedule shall not be modified in this manner. Given these changes, the resulting Equivalent Inverse Cipher offers a more efficient structure than the Inverse Cipher described in Sec. 5.3 and Fig. 12. Pseudo code for the Equivalent Inverse Cipher appears in Fig. 15. (The word array dw[] contains the modified decryption key schedule. The modification to the Key Expansion routine is also provided in Fig. 15.) 24 EqInvCipher(byte in[4Nb], byte out[4Nb], word dw[Nb(Nr+1)]) begin byte state[4,Nb] state = in AddRoundKey(state, dw[NrNb, (Nr+1)Nb-1]) for round = Nr-1 step -1 downto 1 InvSubBytes(state) InvShiftRows(state) InvMixColumns(state) AddRoundKey(state, dw[roundNb, (round+1)Nb-1]) end for InvSubBytes(state) InvShiftRows(state) AddRoundKey(state, dw[0, Nb-1]) out = state end For the Equivalent Inverse Cipher, the following pseudo code is added at the end of the Key Expansion routine (Sec. 5.2): for i = 0 step 1 to (Nr+1)Nb-1 dw[i] = w[i] end for for round = 1 step 1 to Nr-1 InvMixColumns(dw[roundNb, (round+1)*Nb-1]) // note change of type end for Note that, since InvMixColumns operates on a two-dimensional array of bytes while the Round Keys are held in an array of words, the call to InvMixColumns in this code sequence involves a change of type (i.e. the input to InvMixColumns() is normally the State array, which is considered to be a two-dimensional array of bytes, whereas the input here is a Round Key computed as a one-dimensional array of words). Figure 15. Pseudo Code for the Equivalent Inverse Cipher. 6. Implementation Issues 6.1 Key Length Requirements An implementation of the AES algorithm shall support at least one of the three key lengths specified in Sec. 5: 128, 192, or 256 bits (i.e., Nk = 4, 6, or 8, respectively). Implementations 25 may optionally support two or three key lengths, which may promote the interoperability of algorithm implementations. 6.2 Keying Restrictions No weak or semi-weak keys have been identified for the AES algorithm, and there is no restriction on key selection. 6.3 Parameterization of Key Length, Block Size, and Round Number This standard explicitly defines the allowed values for the key length (Nk), block size (Nb), and number of rounds (Nr) – see Fig. 4. However, future reaffirmations of this standard could include changes or additions to the allowed values for those parameters. Therefore, implementers may choose to design their AES implementations with future flexibility in mind. 6.4 Implementation Suggestions Regarding Various Platforms Implementation variations are possible that may, in many cases, offer performance or other advantages. Given the same input key and data (plaintext or ciphertext), any implementation that produces the same output (ciphertext or plaintext) as the algorithm specified in this standard is an acceptable implementation of the AES. Reference [3] and other papers located at Ref. [1] include suggestions on how to efficiently implement the AES algorithm on a variety of platforms. 26 Appendix A - Key Expansion Examples This appendix shows the development of the key schedule for various key sizes. Note that multi- byte values are presented using the notation described in Sec. 3. The intermediate values produced during the development of the key schedule (see Sec. 5.2) are given in the following table (all values are in hexadecimal format, with the exception of the index column (i)). A.1 Expansion of a 128-bit Cipher Key This section contains the key expansion of the following cipher key: Cipher Key = 2b 7e 15 16 28 ae d2 a6 ab f7 15 88 09 cf 4f 3c for Nk = 4, which results in w 0 = 2b7e1516 w 1 = 28aed2a6 w 2 = abf71588 w 3 = 09cf4f3c i (dec) temp After RotWord() After SubWord() Rcon[i/Nk] After XOR with Rcon w[i–Nk] w[i]= temp XOR w[i-Nk] 4 09cf4f3c cf4f3c09 8a84eb01 01000000 8b84eb01 2b7e1516 a0fafe17 5 a0fafe17 28aed2a6 88542cb1 6 88542cb1 abf71588 23a33939 7 23a33939 09cf4f3c 2a6c7605 8 2a6c7605 6c76052a 50386be5 02000000 52386be5 a0fafe17 f2c295f2 9 f2c295f2 88542cb1 7a96b943 10 7a96b943 23a33939 5935807a 11 5935807a 2a6c7605 7359f67f 12 7359f67f 59f67f73 cb42d28f 04000000 cf42d28f f2c295f2 3d80477d 13 3d80477d 7a96b943 4716fe3e 14 4716fe3e 5935807a 1e237e44 15 1e237e44 7359f67f 6d7a883b 16 6d7a883b 7a883b6d dac4e23c 08000000 d2c4e23c 3d80477d ef44a541 17 ef44a541 4716fe3e a8525b7f 18 a8525b7f 1e237e44 b671253b 19 b671253b 6d7a883b db0bad00 20 db0bad00 0bad00db 2b9563b9 10000000 3b9563b9 ef44a541 d4d1c6f8 21 d4d1c6f8 a8525b7f 7c839d87 22 7c839d87 b671253b caf2b8bc 23 caf2b8bc db0bad00 11f915bc 27 24 11f915bc f915bc11 99596582 20000000 b9596582 d4d1c6f8 6d88a37a 25 6d88a37a 7c839d87 110b3efd 26 110b3efd caf2b8bc dbf98641 27 dbf98641 11f915bc ca0093fd 28 ca0093fd 0093fdca 63dc5474 40000000 23dc5474 6d88a37a 4e54f70e 29 4e54f70e 110b3efd 5f5fc9f3 30 5f5fc9f3 dbf98641 84a64fb2 31 84a64fb2 ca0093fd 4ea6dc4f 32 4ea6dc4f a6dc4f4e 2486842f 80000000 a486842f 4e54f70e ead27321 33 ead27321 5f5fc9f3 b58dbad2 34 b58dbad2 84a64fb2 312bf560 35 312bf560 4ea6dc4f 7f8d292f 36 7f8d292f 8d292f7f 5da515d2 1b000000 46a515d2 ead27321 ac7766f3 37 ac7766f3 b58dbad2 19fadc21 38 19fadc21 312bf560 28d12941 39 28d12941 7f8d292f 575c006e 40 575c006e 5c006e57 4a639f5b 36000000 7c639f5b ac7766f3 d014f9a8 41 d014f9a8 19fadc21 c9ee2589 42 c9ee2589 28d12941 e13f0cc8 43 e13f0cc8 575c006e b6630ca6 A.2 Expansion of a 192-bit Cipher Key This section contains the key expansion of the following cipher key: Cipher Key = 8e 73 b0 f7 da 0e 64 52 c8 10 f3 2b 80 90 79 e5 62 f8 ea d2 52 2c 6b 7b for Nk = 6, which results in w 0 = 8e73b0f7 w 1 = da0e6452 w 2 = c810f32b w 3 = 809079e5 w 4 = 62f8ead2 w 5 = 522c6b7b i (dec) temp After RotWord() After SubWord() Rcon[i/Nk] After XOR with Rcon w[i–Nk] w[i]= temp XOR w[i-Nk] 6 522c6b7b 2c6b7b52 717f2100 01000000 707f2100 8e73b0f7 fe0c91f7 7 fe0c91f7 da0e6452 2402f5a5 8 2402f5a5 c810f32b ec12068e 28 9 ec12068e 809079e5 6c827f6b 10 6c827f6b 62f8ead2 0e7a95b9 11 0e7a95b9 522c6b7b 5c56fec2 12 5c56fec2 56fec25c b1bb254a 02000000 b3bb254a fe0c91f7 4db7b4bd 13 4db7b4bd 2402f5a5 69b54118 14 69b54118 ec12068e 85a74796 15 85a74796 6c827f6b e92538fd 16 e92538fd 0e7a95b9 e75fad44 17 e75fad44 5c56fec2 bb095386 18 bb095386 095386bb 01ed44ea 04000000 05ed44ea 4db7b4bd 485af057 19 485af057 69b54118 21efb14f 20 21efb14f 85a74796 a448f6d9 21 a448f6d9 e92538fd 4d6dce24 22 4d6dce24 e75fad44 aa326360 23 aa326360 bb095386 113b30e6 24 113b30e6 3b30e611 e2048e82 08000000 ea048e82 485af057 a25e7ed5 25 a25e7ed5 21efb14f 83b1cf9a 26 83b1cf9a a448f6d9 27f93943 27 27f93943 4d6dce24 6a94f767 28 6a94f767 aa326360 c0a69407 29 c0a69407 113b30e6 d19da4e1 30 d19da4e1 9da4e1d1 5e49f83e 10000000 4e49f83e a25e7ed5 ec1786eb 31 ec1786eb 83b1cf9a 6fa64971 32 6fa64971 27f93943 485f7032 33 485f7032 6a94f767 22cb8755 34 22cb8755 c0a69407 e26d1352 35 e26d1352 d19da4e1 33f0b7b3 36 33f0b7b3 f0b7b333 8ca96dc3 20000000 aca96dc3 ec1786eb 40beeb28 37 40beeb28 6fa64971 2f18a259 38 2f18a259 485f7032 6747d26b 39 6747d26b 22cb8755 458c553e 40 458c553e e26d1352 a7e1466c 41 a7e1466c 33f0b7b3 9411f1df 42 9411f1df 11f1df94 82a19e22 40000000 c2a19e22 40beeb28 821f750a 43 821f750a 2f18a259 ad07d753 29 44 ad07d753 6747d26b ca400538 45 ca400538 458c553e 8fcc5006 46 8fcc5006 a7e1466c 282d166a 47 282d166a 9411f1df bc3ce7b5 48 bc3ce7b5 3ce7b5bc eb94d565 80000000 6b94d565 821f750a e98ba06f 49 e98ba06f ad07d753 448c773c 50 448c773c ca400538 8ecc7204 51 8ecc7204 8fcc5006 01002202 A.3 Expansion of a 256-bit Cipher Key This section contains the key expansion of the following cipher key: Cipher Key = 60 3d eb 10 15 ca 71 be 2b 73 ae f0 85 7d 77 81 1f 35 2c 07 3b 61 08 d7 2d 98 10 a3 09 14 df f4 for Nk = 8, which results in w 0 = 603deb10 w 1 = 15ca71be w 2 = 2b73aef0 w 3 = 857d7781 w 4 = 1f352c07 w 5 = 3b6108d7 w 6 = 2d9810a3 w 7 = 0914dff4 i (dec) temp After RotWord() After SubWord() Rcon[i/Nk] After XOR with Rcon w[i–Nk] w[i]= temp XOR w[i-Nk] 8 0914dff4 14dff409 fa9ebf01 01000000 fb9ebf01 603deb10 9ba35411 9 9ba35411 15ca71be 8e6925af 10 8e6925af 2b73aef0 a51a8b5f 11 a51a8b5f 857d7781 2067fcde 12 2067fcde b785b01d 1f352c07 a8b09c1a 13 a8b09c1a 3b6108d7 93d194cd 14 93d194cd 2d9810a3 be49846e 15 be49846e 0914dff4 b75d5b9a 16 b75d5b9a 5d5b9ab7 4c39b8a9 02000000 4e39b8a9 9ba35411 d59aecb8 17 d59aecb8 8e6925af 5bf3c917 18 5bf3c917 a51a8b5f fee94248 19 fee94248 2067fcde de8ebe96 20 de8ebe96 1d19ae90 a8b09c1a b5a9328a 21 b5a9328a 93d194cd 2678a647 22 2678a647 be49846e 98312229 30 23 98312229 b75d5b9a 2f6c79b3 24 2f6c79b3 6c79b32f 50b66d15 04000000 54b66d15 d59aecb8 812c81ad 25 812c81ad 5bf3c917 dadf48ba 26 dadf48ba fee94248 24360af2 27 24360af2 de8ebe96 fab8b464 28 fab8b464 2d6c8d43 b5a9328a 98c5bfc9 29 98c5bfc9 2678a647 bebd198e 30 bebd198e 98312229 268c3ba7 31 268c3ba7 2f6c79b3 09e04214 32 09e04214 e0421409 e12cfa01 08000000 e92cfa01 812c81ad 68007bac 33 68007bac dadf48ba b2df3316 34 b2df3316 24360af2 96e939e4 35 96e939e4 fab8b464 6c518d80 36 6c518d80 50d15dcd 98c5bfc9 c814e204 37 c814e204 bebd198e 76a9fb8a 38 76a9fb8a 268c3ba7 5025c02d 39 5025c02d 09e04214 59c58239 40 59c58239 c5823959 a61312cb 10000000 b61312cb 68007bac de136967 41 de136967 b2df3316 6ccc5a71 42 6ccc5a71 96e939e4 fa256395 43 fa256395 6c518d80 9674ee15 44 9674ee15 90922859 c814e204 5886ca5d 45 5886ca5d 76a9fb8a 2e2f31d7 46 2e2f31d7 5025c02d 7e0af1fa 47 7e0af1fa 59c58239 27cf73c3 48 27cf73c3 cf73c327 8a8f2ecc 20000000 aa8f2ecc de136967 749c47ab 49 749c47ab 6ccc5a71 18501dda 50 18501dda fa256395 e2757e4f 51 e2757e4f 9674ee15 7401905a 52 7401905a 927c60be 5886ca5d cafaaae3 53 cafaaae3 2e2f31d7 e4d59b34 54 e4d59b34 7e0af1fa 9adf6ace 55 9adf6ace 27cf73c3 bd10190d 56 bd10190d 10190dbd cad4d77a 40000000 8ad4d77a 749c47ab fe4890d1 57 fe4890d1 18501dda e6188d0b 31 58 e6188d0b e2757e4f 046df344 59 046df344 7401905a 706c631e 32 Appendix B – Cipher Example The following diagram shows the values in the State array as the Cipher progresses for a block length and a Cipher Key length of 16 bytes each (i.e., Nb = 4 and Nk = 4). Input = 32 43 f6 a8 88 5a 30 8d 31 31 98 a2 e0 37 07 34 Cipher Key = 2b 7e 15 16 28 ae d2 a6 ab f7 15 88 09 cf 4f 3c The Round Key values are taken from the Key Expansion example in Appendix A. Round Start of After After After Round Key Number Round SubBytes ShiftRows MixColumns Value 32 88 31 e0 43 5a 31 37 f6 30 98 07 a8 8d a2 34 2b 28 ab 09 7e ae f7 cf 15 d2 15 4f 16 a6 88 3c input = ¯ ¯ 19 a0 9a e9 3d f4 c6 f8 e3 e2 8d 48 be 2b 2a 08 d4 e0 b8 1e 27 bf b4 41 11 98 5d 52 ae f1 e5 30 d4 e0 b8 1e bf b4 41 27 5d 52 11 98 30 ae f1 e5 04 e0 48 28 66 cb f8 06 81 19 d3 26 e5 9a 7a 4c a0 88 23 2a fa 54 a3 6c fe 2c 39 76 17 b1 39 05 1 = ¯ ¯ 2 a4 68 6b 02 9c 9f 5b 6a 7f 35 ea 50 f2 2b 43 49 = 49 45 7f 77 de db 39 02 d2 96 87 53 89 f1 1a 3b ¯ ¯ 49 45 7f 77 db 39 02 de 87 53 d2 96 3b 89 f1 1a f2 7a 59 73 c2 96 35 59 95 b9 80 f6 f2 43 7a 7f 58 1b db 1b 4d 4b e7 6b ca 5a ca b0 f1 ac a8 e5 3 aa 61 82 68 8f dd d2 32 5f e3 4a 46 03 ef d2 9a = ac ef 13 45 73 c1 b5 23 cf 11 d6 5a 7b df b5 b8 ¯ ¯ ac ef 13 45 c1 b5 23 73 d6 5a cf 11 b8 7b df b5 3d 47 1e 6d 80 16 23 7a 47 fe 7e 88 7d 3e 44 3b 75 20 53 bb ec 0b c0 25 09 63 cf d0 93 33 7c dc 4 48 67 4d d6 6c 1d e3 5f 4e 9d b1 58 ee 0d 38 e7 = 52 85 e3 f6 50 a4 11 cf 2f 5e c8 6a 28 d7 07 94 ¯ ¯ 52 85 e3 f6 a4 11 cf 50 c8 6a 2f 5e 94 28 d7 07 ef a8 b6 db 44 52 71 0b a5 5b 25 ad 41 7f 3b 00 0f 60 6f 5e d6 31 c0 b3 da 38 10 13 a9 bf 6b 01 5 = ¯ ¯ e0 c8 d9 85 92 63 b1 b8 7f 63 35 be e8 c0 50 01 e1 e8 35 97 4f fb c8 6c d2 fb 96 ae 9b ba 53 7c e1 e8 35 97 fb c8 6c 4f 96 ae d2 fb 7c 9b ba 53 25 bd b6 4c d1 11 3a 4c a9 d1 33 c0 ad 68 8e b0 d4 7c ca 11 d1 83 f2 f9 c6 9d b8 15 f8 87 bc bc 33 6 10 = ¯ ¯ f1 c1 7c 5d 00 92 c8 b5 6f 4c 8b d5 55 ef 32 0c a1 78 10 4c 63 4f e8 d5 a8 29 3d 03 fc df 23 fe a1 78 10 4c 4f e8 d5 63 3d 03 a8 29 fe fc df 23 4b 2c 33 37 86 4a 9d d2 8d 89 f4 18 6d 80 e8 d8 6d 11 db ca 88 0b f9 00 a3 3e 86 93 7a fd 41 fd 26 3d e8 fd 0e 41 64 d2 2e b7 72 8b 17 7d a9 25 f7 27 9b 54 ab 83 43 b5 31 a9 40 3d f0 ff d3 3f f7 27 9b 54 83 43 b5 ab 40 3d 31 a9 3f f0 ff d3 14 46 27 34 15 16 46 2a b5 15 56 d8 bf ec d7 43 4e 5f 84 4e 54 5f a6 a6 f7 c9 4f dc 0e f3 b2 4f 7 = ¯ ¯ 8 5a 19 a3 7a 41 49 e0 8c 42 dc 19 04 b1 1f 65 0c = be d4 0a da 83 3b e1 64 2c 86 d4 f2 c8 c0 4d fe ¯ ¯ be d4 0a da 3b e1 64 83 d4 f2 2c 86 fe c8 c0 4d ea b5 31 7f d2 8d 2b 8d 73 ba f5 29 21 d2 60 2f 00 b1 54 fa 51 c8 76 1b 2f 89 6d 99 d1 ff cd ea 9 = ¯ ¯ ea 04 65 85 83 45 5d 96 5c 33 98 b0 f0 2d ad c5 87 f2 4d 97 ec 6e 4c 90 4a c3 46 e7 8c d8 95 a6 87 f2 4d 97 6e 4c 90 ec 46 e7 4a c3 a6 8c d8 95 47 40 a3 4c 37 d4 70 9f 94 e4 3a 42 ed a5 a6 bc ac 19 28 57 77 fa d1 5c 66 dc 29 00 f3 21 41 6e eb 59 8b 1b 40 2e a1 c3 f2 38 13 42 1e 84 e7 d2 e9 cb 3d af 09 31 32 2e 89 07 7d 2c 72 5f 94 b5 e9 cb 3d af 31 32 2e 09 7d 2c 89 07 b5 72 5f 94 d0 c9 e1 b6 14 ee 3f 63 f9 25 0c 0c a8 89 c8 a6 = ¯ ¯ 39 02 dc 19 25 dc 11 6a 84 09 85 0b 1d fb 97 32 output 34 Appendix C – Example Vectors This appendix contains example vectors, including intermediate values – for all three AES key lengths (Nk = 4, 6, and 8), for the Cipher, Inverse Cipher, and Equivalent Inverse Cipher that are described in Sec. 5.1, 5.3, and 5.3.5, respectively. Additional examples may be found at [1] and [5]. All vectors are in hexadecimal notation, with each pair of characters giving a byte value in which the left character of each pair provides the bit pattern for the 4 bit group containing the higher numbered bits using the notation explained in Sec. 3.2, while the right character provides the bit pattern for the lower-numbered bits. The array index for all bytes (groups of two hexadecimal digits) within these test vectors starts at zero and increases from left to right. Legend for CIPHER (ENCRYPT) (round number r = 0 to 10, 12 or 14): input: cipher input start: state at start of round[r] s_box: state after SubBytes() s_row: state after ShiftRows() m_col: state after MixColumns() k_sch: key schedule value for round[r] output: cipher output Legend for INVERSE CIPHER (DECRYPT) (round number r = 0 to 10, 12 or 14): iinput: inverse cipher input istart: state at start of round[r] is_box: state after InvSubBytes() is_row: state after InvShiftRows() ik_sch: key schedule value for round[r] ik_add: state after AddRoundKey() ioutput: inverse cipher output Legend for EQUIVALENT INVERSE CIPHER (DECRYPT) (round number r = 0 to 10, 12 or 14): iinput: inverse cipher input istart: state at start of round[r] is_box: state after InvSubBytes() is_row: state after InvShiftRows() im_col: state after InvMixColumns() ik_sch: key schedule value for round[r] ioutput: inverse cipher output C.1 AES-128 (Nk=4, Nr=10) PLAINTEXT: 00112233445566778899aabbccddeeff KEY: 000102030405060708090a0b0c0d0e0f CIPHER (ENCRYPT): 35 round[ 0].input 00112233445566778899aabbccddeeff round[ 0].k_sch 000102030405060708090a0b0c0d0e0f round[ 1].start 00102030405060708090a0b0c0d0e0f0 round[ 1].s_box 63cab7040953d051cd60e0e7ba70e18c round[ 1].s_row 6353e08c0960e104cd70b751bacad0e7 round[ 1].m_col 5f72641557f5bc92f7be3b291db9f91a round[ 1].k_sch d6aa74fdd2af72fadaa678f1d6ab76fe round[ 2].start 89d810e8855ace682d1843d8cb128fe4 round[ 2].s_box a761ca9b97be8b45d8ad1a611fc97369 round[ 2].s_row a7be1a6997ad739bd8c9ca451f618b61 round[ 2].m_col ff87968431d86a51645151fa773ad009 round[ 2].k_sch b692cf0b643dbdf1be9bc5006830b3fe round[ 3].start 4915598f55e5d7a0daca94fa1f0a63f7 round[ 3].s_box 3b59cb73fcd90ee05774222dc067fb68 round[ 3].s_row 3bd92268fc74fb735767cbe0c0590e2d round[ 3].m_col 4c9c1e66f771f0762c3f868e534df256 round[ 3].k_sch b6ff744ed2c2c9bf6c590cbf0469bf41 round[ 4].start fa636a2825b339c940668a3157244d17 round[ 4].s_box 2dfb02343f6d12dd09337ec75b36e3f0 round[ 4].s_row 2d6d7ef03f33e334093602dd5bfb12c7 round[ 4].m_col 6385b79ffc538df997be478e7547d691 round[ 4].k_sch 47f7f7bc95353e03f96c32bcfd058dfd round[ 5].start 247240236966b3fa6ed2753288425b6c round[ 5].s_box 36400926f9336d2d9fb59d23c42c3950 round[ 5].s_row 36339d50f9b539269f2c092dc4406d23 round[ 5].m_col f4bcd45432e554d075f1d6c51dd03b3c round[ 5].k_sch 3caaa3e8a99f9deb50f3af57adf622aa round[ 6].start c81677bc9b7ac93b25027992b0261996 round[ 6].s_box e847f56514dadde23f77b64fe7f7d490 round[ 6].s_row e8dab6901477d4653ff7f5e2e747dd4f round[ 6].m_col 9816ee7400f87f556b2c049c8e5ad036 round[ 6].k_sch 5e390f7df7a69296a7553dc10aa31f6b round[ 7].start c62fe109f75eedc3cc79395d84f9cf5d round[ 7].s_box b415f8016858552e4bb6124c5f998a4c round[ 7].s_row b458124c68b68a014b99f82e5f15554c round[ 7].m_col c57e1c159a9bd286f05f4be098c63439 round[ 7].k_sch 14f9701ae35fe28c440adf4d4ea9c026 round[ 8].start d1876c0f79c4300ab45594add66ff41f round[ 8].s_box 3e175076b61c04678dfc2295f6a8bfc0 round[ 8].s_row 3e1c22c0b6fcbf768da85067f6170495 round[ 8].m_col baa03de7a1f9b56ed5512cba5f414d23 round[ 8].k_sch 47438735a41c65b9e016baf4aebf7ad2 round[ 9].start fde3bad205e5d0d73547964ef1fe37f1 round[ 9].s_box 5411f4b56bd9700e96a0902fa1bb9aa1 round[ 9].s_row 54d990a16ba09ab596bbf40ea111702f round[ 9].m_col e9f74eec023020f61bf2ccf2353c21c7 round[ 9].k_sch 549932d1f08557681093ed9cbe2c974e round[10].start bd6e7c3df2b5779e0b61216e8b10b689 round[10].s_box 7a9f102789d5f50b2beffd9f3dca4ea7 round[10].s_row 7ad5fda789ef4e272bca100b3d9ff59f round[10].k_sch 13111d7fe3944a17f307a78b4d2b30c5 round[10].output 69c4e0d86a7b0430d8cdb78070b4c55a INVERSE CIPHER (DECRYPT): round[ 0].iinput 69c4e0d86a7b0430d8cdb78070b4c55a round[ 0].ik_sch 13111d7fe3944a17f307a78b4d2b30c5 round[ 1].istart 7ad5fda789ef4e272bca100b3d9ff59f 36 round[ 1].is_row 7a9f102789d5f50b2beffd9f3dca4ea7 round[ 1].is_box bd6e7c3df2b5779e0b61216e8b10b689 round[ 1].ik_sch 549932d1f08557681093ed9cbe2c974e round[ 1].ik_add e9f74eec023020f61bf2ccf2353c21c7 round[ 2].istart 54d990a16ba09ab596bbf40ea111702f round[ 2].is_row 5411f4b56bd9700e96a0902fa1bb9aa1 round[ 2].is_box fde3bad205e5d0d73547964ef1fe37f1 round[ 2].ik_sch 47438735a41c65b9e016baf4aebf7ad2 round[ 2].ik_add baa03de7a1f9b56ed5512cba5f414d23 round[ 3].istart 3e1c22c0b6fcbf768da85067f6170495 round[ 3].is_row 3e175076b61c04678dfc2295f6a8bfc0 round[ 3].is_box d1876c0f79c4300ab45594add66ff41f round[ 3].ik_sch 14f9701ae35fe28c440adf4d4ea9c026 round[ 3].ik_add c57e1c159a9bd286f05f4be098c63439 round[ 4].istart b458124c68b68a014b99f82e5f15554c round[ 4].is_row b415f8016858552e4bb6124c5f998a4c round[ 4].is_box c62fe109f75eedc3cc79395d84f9cf5d round[ 4].ik_sch 5e390f7df7a69296a7553dc10aa31f6b round[ 4].ik_add 9816ee7400f87f556b2c049c8e5ad036 round[ 5].istart e8dab6901477d4653ff7f5e2e747dd4f round[ 5].is_row e847f56514dadde23f77b64fe7f7d490 round[ 5].is_box c81677bc9b7ac93b25027992b0261996 round[ 5].ik_sch 3caaa3e8a99f9deb50f3af57adf622aa round[ 5].ik_add f4bcd45432e554d075f1d6c51dd03b3c round[ 6].istart 36339d50f9b539269f2c092dc4406d23 round[ 6].is_row 36400926f9336d2d9fb59d23c42c3950 round[ 6].is_box 247240236966b3fa6ed2753288425b6c round[ 6].ik_sch 47f7f7bc95353e03f96c32bcfd058dfd round[ 6].ik_add 6385b79ffc538df997be478e7547d691 round[ 7].istart 2d6d7ef03f33e334093602dd5bfb12c7 round[ 7].is_row 2dfb02343f6d12dd09337ec75b36e3f0 round[ 7].is_box fa636a2825b339c940668a3157244d17 round[ 7].ik_sch b6ff744ed2c2c9bf6c590cbf0469bf41 round[ 7].ik_add 4c9c1e66f771f0762c3f868e534df256 round[ 8].istart 3bd92268fc74fb735767cbe0c0590e2d round[ 8].is_row 3b59cb73fcd90ee05774222dc067fb68 round[ 8].is_box 4915598f55e5d7a0daca94fa1f0a63f7 round[ 8].ik_sch b692cf0b643dbdf1be9bc5006830b3fe round[ 8].ik_add ff87968431d86a51645151fa773ad009 round[ 9].istart a7be1a6997ad739bd8c9ca451f618b61 round[ 9].is_row a761ca9b97be8b45d8ad1a611fc97369 round[ 9].is_box 89d810e8855ace682d1843d8cb128fe4 round[ 9].ik_sch d6aa74fdd2af72fadaa678f1d6ab76fe round[ 9].ik_add 5f72641557f5bc92f7be3b291db9f91a round[10].istart 6353e08c0960e104cd70b751bacad0e7 round[10].is_row 63cab7040953d051cd60e0e7ba70e18c round[10].is_box 00102030405060708090a0b0c0d0e0f0 round[10].ik_sch 000102030405060708090a0b0c0d0e0f round[10].ioutput 00112233445566778899aabbccddeeff EQUIVALENT INVERSE CIPHER (DECRYPT): round[ 0].iinput 69c4e0d86a7b0430d8cdb78070b4c55a round[ 0].ik_sch 13111d7fe3944a17f307a78b4d2b30c5 round[ 1].istart 7ad5fda789ef4e272bca100b3d9ff59f round[ 1].is_box bdb52189f261b63d0b107c9e8b6e776e round[ 1].is_row bd6e7c3df2b5779e0b61216e8b10b689 round[ 1].im_col 4773b91ff72f354361cb018ea1e6cf2c 37 round[ 1].ik_sch 13aa29be9c8faff6f770f58000f7bf03 round[ 2].istart 54d990a16ba09ab596bbf40ea111702f round[ 2].is_box fde596f1054737d235febad7f1e3d04e round[ 2].is_row fde3bad205e5d0d73547964ef1fe37f1 round[ 2].im_col 2d7e86a339d9393ee6570a1101904e16 round[ 2].ik_sch 1362a4638f2586486bff5a76f7874a83 round[ 3].istart 3e1c22c0b6fcbf768da85067f6170495 round[ 3].is_box d1c4941f7955f40fb46f6c0ad68730ad round[ 3].is_row d1876c0f79c4300ab45594add66ff41f round[ 3].im_col 39daee38f4f1a82aaf432410c36d45b9 round[ 3].ik_sch 8d82fc749c47222be4dadc3e9c7810f5 round[ 4].istart b458124c68b68a014b99f82e5f15554c round[ 4].is_box c65e395df779cf09ccf9e1c3842fed5d round[ 4].is_row c62fe109f75eedc3cc79395d84f9cf5d round[ 4].im_col 9a39bf1d05b20a3a476a0bf79fe51184 round[ 4].ik_sch 72e3098d11c5de5f789dfe1578a2cccb round[ 5].istart e8dab6901477d4653ff7f5e2e747dd4f round[ 5].is_box c87a79969b0219bc2526773bb016c992 round[ 5].is_row c81677bc9b7ac93b25027992b0261996 round[ 5].im_col 18f78d779a93eef4f6742967c47f5ffd round[ 5].ik_sch 2ec410276326d7d26958204a003f32de round[ 6].istart 36339d50f9b539269f2c092dc4406d23 round[ 6].is_box 2466756c69d25b236e4240fa8872b332 round[ 6].is_row 247240236966b3fa6ed2753288425b6c round[ 6].im_col 85cf8bf472d124c10348f545329c0053 round[ 6].ik_sch a8a2f5044de2c7f50a7ef79869671294 round[ 7].istart 2d6d7ef03f33e334093602dd5bfb12c7 round[ 7].is_box fab38a1725664d2840246ac957633931 round[ 7].is_row fa636a2825b339c940668a3157244d17 round[ 7].im_col fc1fc1f91934c98210fbfb8da340eb21 round[ 7].ik_sch c7c6e391e54032f1479c306d6319e50c round[ 8].istart 3bd92268fc74fb735767cbe0c0590e2d round[ 8].is_box 49e594f755ca638fda0a59a01f15d7fa round[ 8].is_row 4915598f55e5d7a0daca94fa1f0a63f7 round[ 8].im_col 076518f0b52ba2fb7a15c8d93be45e00 round[ 8].ik_sch a0db02992286d160a2dc029c2485d561 round[ 9].istart a7be1a6997ad739bd8c9ca451f618b61 round[ 9].is_box 895a43e485188fe82d121068cbd8ced8 round[ 9].is_row 89d810e8855ace682d1843d8cb128fe4 round[ 9].im_col ef053f7c8b3d32fd4d2a64ad3c93071a round[ 9].ik_sch 8c56dff0825dd3f9805ad3fc8659d7fd round[10].istart 6353e08c0960e104cd70b751bacad0e7 round[10].is_box 0050a0f04090e03080d02070c01060b0 round[10].is_row 00102030405060708090a0b0c0d0e0f0 round[10].ik_sch 000102030405060708090a0b0c0d0e0f round[10].ioutput 00112233445566778899aabbccddeeff C.2 AES-192 (Nk=6, Nr=12) PLAINTEXT: 00112233445566778899aabbccddeeff KEY: 000102030405060708090a0b0c0d0e0f1011121314151617 CIPHER (ENCRYPT): round[ 0].input 00112233445566778899aabbccddeeff round[ 0].k_sch 000102030405060708090a0b0c0d0e0f round[ 1].start 00102030405060708090a0b0c0d0e0f0 38 round[ 1].s_box 63cab7040953d051cd60e0e7ba70e18c round[ 1].s_row 6353e08c0960e104cd70b751bacad0e7 round[ 1].m_col 5f72641557f5bc92f7be3b291db9f91a round[ 1].k_sch 10111213141516175846f2f95c43f4fe round[ 2].start 4f63760643e0aa85aff8c9d041fa0de4 round[ 2].s_box 84fb386f1ae1ac977941dd70832dd769 round[ 2].s_row 84e1dd691a41d76f792d389783fbac70 round[ 2].m_col 9f487f794f955f662afc86abd7f1ab29 round[ 2].k_sch 544afef55847f0fa4856e2e95c43f4fe round[ 3].start cb02818c17d2af9c62aa64428bb25fd7 round[ 3].s_box 1f770c64f0b579deaaac432c3d37cf0e round[ 3].s_row 1fb5430ef0accf64aa370cde3d77792c round[ 3].m_col b7a53ecbbf9d75a0c40efc79b674cc11 round[ 3].k_sch 40f949b31cbabd4d48f043b810b7b342 round[ 4].start f75c7778a327c8ed8cfebfc1a6c37f53 round[ 4].s_box 684af5bc0acce85564bb0878242ed2ed round[ 4].s_row 68cc08ed0abbd2bc642ef555244ae878 round[ 4].m_col 7a1e98bdacb6d1141a6944dd06eb2d3e round[ 4].k_sch 58e151ab04a2a5557effb5416245080c round[ 5].start 22ffc916a81474416496f19c64ae2532 round[ 5].s_box 9316dd47c2fa92834390a1de43e43f23 round[ 5].s_row 93faa123c2903f4743e4dd83431692de round[ 5].m_col aaa755b34cffe57cef6f98e1f01c13e6 round[ 5].k_sch 2ab54bb43a02f8f662e3a95d66410c08 round[ 6].start 80121e0776fd1d8a8d8c31bc965d1fee round[ 6].s_box cdc972c53854a47e5d64c765904cc028 round[ 6].s_row cd54c7283864c0c55d4c727e90c9a465 round[ 6].m_col 921f748fd96e937d622d7725ba8ba50c round[ 6].k_sch f501857297448d7ebdf1c6ca87f33e3c round[ 7].start 671ef1fd4e2a1e03dfdcb1ef3d789b30 round[ 7].s_box 8572a1542fe5727b9e86c8df27bc1404 round[ 7].s_row 85e5c8042f8614549ebca17b277272df round[ 7].m_col e913e7b18f507d4b227ef652758acbcc round[ 7].k_sch e510976183519b6934157c9ea351f1e0 round[ 8].start 0c0370d00c01e622166b8accd6db3a2c round[ 8].s_box fe7b5170fe7c8e93477f7e4bf6b98071 round[ 8].s_row fe7c7e71fe7f807047b95193f67b8e4b round[ 8].m_col 6cf5edf996eb0a069c4ef21cbfc25762 round[ 8].k_sch 1ea0372a995309167c439e77ff12051e round[ 9].start 7255dad30fb80310e00d6c6b40d0527c round[ 9].s_box 40fc5766766c7bcae1d7507f09700010 round[ 9].s_row 406c501076d70066e17057ca09fc7b7f round[ 9].m_col 7478bcdce8a50b81d4327a9009188262 round[ 9].k_sch dd7e0e887e2fff68608fc842f9dcc154 round[10].start a906b254968af4e9b4bdb2d2f0c44336 round[10].s_box d36f3720907ebf1e8d7a37b58c1c1a05 round[10].s_row d37e3705907a1a208d1c371e8c6fbfb5 round[10].m_col 0d73cc2d8f6abe8b0cf2dd9bb83d422e round[10].k_sch 859f5f237a8d5a3dc0c02952beefd63a round[11].start 88ec930ef5e7e4b6cc32f4c906d29414 round[11].s_box c4cedcabe694694e4b23bfdd6fb522fa round[11].s_row c494bffae62322ab4bb5dc4e6fce69dd round[11].m_col 71d720933b6d677dc00b8f28238e0fb7 round[11].k_sch de601e7827bcdf2ca223800fd8aeda32 round[12].start afb73eeb1cd1b85162280f27fb20d585 round[12].s_box 79a9b2e99c3e6cd1aa3476cc0fb70397 round[12].s_row 793e76979c3403e9aab7b2d10fa96ccc 39 round[12].k_sch a4970a331a78dc09c418c271e3a41d5d round[12].output dda97ca4864cdfe06eaf70a0ec0d7191 INVERSE CIPHER (DECRYPT): round[ 0].iinput dda97ca4864cdfe06eaf70a0ec0d7191 round[ 0].ik_sch a4970a331a78dc09c418c271e3a41d5d round[ 1].istart 793e76979c3403e9aab7b2d10fa96ccc round[ 1].is_row 79a9b2e99c3e6cd1aa3476cc0fb70397 round[ 1].is_box afb73eeb1cd1b85162280f27fb20d585 round[ 1].ik_sch de601e7827bcdf2ca223800fd8aeda32 round[ 1].ik_add 71d720933b6d677dc00b8f28238e0fb7 round[ 2].istart c494bffae62322ab4bb5dc4e6fce69dd round[ 2].is_row c4cedcabe694694e4b23bfdd6fb522fa round[ 2].is_box 88ec930ef5e7e4b6cc32f4c906d29414 round[ 2].ik_sch 859f5f237a8d5a3dc0c02952beefd63a round[ 2].ik_add 0d73cc2d8f6abe8b0cf2dd9bb83d422e round[ 3].istart d37e3705907a1a208d1c371e8c6fbfb5 round[ 3].is_row d36f3720907ebf1e8d7a37b58c1c1a05 round[ 3].is_box a906b254968af4e9b4bdb2d2f0c44336 round[ 3].ik_sch dd7e0e887e2fff68608fc842f9dcc154 round[ 3].ik_add 7478bcdce8a50b81d4327a9009188262 round[ 4].istart 406c501076d70066e17057ca09fc7b7f round[ 4].is_row 40fc5766766c7bcae1d7507f09700010 round[ 4].is_box 7255dad30fb80310e00d6c6b40d0527c round[ 4].ik_sch 1ea0372a995309167c439e77ff12051e round[ 4].ik_add 6cf5edf996eb0a069c4ef21cbfc25762 round[ 5].istart fe7c7e71fe7f807047b95193f67b8e4b round[ 5].is_row fe7b5170fe7c8e93477f7e4bf6b98071 round[ 5].is_box 0c0370d00c01e622166b8accd6db3a2c round[ 5].ik_sch e510976183519b6934157c9ea351f1e0 round[ 5].ik_add e913e7b18f507d4b227ef652758acbcc round[ 6].istart 85e5c8042f8614549ebca17b277272df round[ 6].is_row 8572a1542fe5727b9e86c8df27bc1404 round[ 6].is_box 671ef1fd4e2a1e03dfdcb1ef3d789b30 round[ 6].ik_sch f501857297448d7ebdf1c6ca87f33e3c round[ 6].ik_add 921f748fd96e937d622d7725ba8ba50c round[ 7].istart cd54c7283864c0c55d4c727e90c9a465 round[ 7].is_row cdc972c53854a47e5d64c765904cc028 round[ 7].is_box 80121e0776fd1d8a8d8c31bc965d1fee round[ 7].ik_sch 2ab54bb43a02f8f662e3a95d66410c08 round[ 7].ik_add aaa755b34cffe57cef6f98e1f01c13e6 round[ 8].istart 93faa123c2903f4743e4dd83431692de round[ 8].is_row 9316dd47c2fa92834390a1de43e43f23 round[ 8].is_box 22ffc916a81474416496f19c64ae2532 round[ 8].ik_sch 58e151ab04a2a5557effb5416245080c round[ 8].ik_add 7a1e98bdacb6d1141a6944dd06eb2d3e round[ 9].istart 68cc08ed0abbd2bc642ef555244ae878 round[ 9].is_row 684af5bc0acce85564bb0878242ed2ed round[ 9].is_box f75c7778a327c8ed8cfebfc1a6c37f53 round[ 9].ik_sch 40f949b31cbabd4d48f043b810b7b342 round[ 9].ik_add b7a53ecbbf9d75a0c40efc79b674cc11 round[10].istart 1fb5430ef0accf64aa370cde3d77792c round[10].is_row 1f770c64f0b579deaaac432c3d37cf0e round[10].is_box cb02818c17d2af9c62aa64428bb25fd7 round[10].ik_sch 544afef55847f0fa4856e2e95c43f4fe round[10].ik_add 9f487f794f955f662afc86abd7f1ab29 round[11].istart 84e1dd691a41d76f792d389783fbac70 40 round[11].is_row 84fb386f1ae1ac977941dd70832dd769 round[11].is_box 4f63760643e0aa85aff8c9d041fa0de4 round[11].ik_sch 10111213141516175846f2f95c43f4fe round[11].ik_add 5f72641557f5bc92f7be3b291db9f91a round[12].istart 6353e08c0960e104cd70b751bacad0e7 round[12].is_row 63cab7040953d051cd60e0e7ba70e18c round[12].is_box 00102030405060708090a0b0c0d0e0f0 round[12].ik_sch 000102030405060708090a0b0c0d0e0f round[12].ioutput 00112233445566778899aabbccddeeff EQUIVALENT INVERSE CIPHER (DECRYPT): round[ 0].iinput dda97ca4864cdfe06eaf70a0ec0d7191 round[ 0].ik_sch a4970a331a78dc09c418c271e3a41d5d round[ 1].istart 793e76979c3403e9aab7b2d10fa96ccc round[ 1].is_box afd10f851c28d5eb62203e51fbb7b827 round[ 1].is_row afb73eeb1cd1b85162280f27fb20d585 round[ 1].im_col 122a02f7242ac8e20605afce51cc7264 round[ 1].ik_sch d6bebd0dc209ea494db073803e021bb9 round[ 2].istart c494bffae62322ab4bb5dc4e6fce69dd round[ 2].is_box 88e7f414f532940eccd293b606ece4c9 round[ 2].is_row 88ec930ef5e7e4b6cc32f4c906d29414 round[ 2].im_col 5cc7aecce3c872194ae5ef8309a933c7 round[ 2].ik_sch 8fb999c973b26839c7f9d89d85c68c72 round[ 3].istart d37e3705907a1a208d1c371e8c6fbfb5 round[ 3].is_box a98ab23696bd4354b4c4b2e9f006f4d2 round[ 3].is_row a906b254968af4e9b4bdb2d2f0c44336 round[ 3].im_col b7113ed134e85489b20866b51d4b2c3b round[ 3].ik_sch f77d6ec1423f54ef5378317f14b75744 round[ 4].istart 406c501076d70066e17057ca09fc7b7f round[ 4].is_box 72b86c7c0f0d52d3e0d0da104055036b round[ 4].is_row 7255dad30fb80310e00d6c6b40d0527c round[ 4].im_col ef3b1be1b9b0e64bdcb79f1e0a707fbb round[ 4].ik_sch 1147659047cf663b9b0ece8dfc0bf1f0 round[ 5].istart fe7c7e71fe7f807047b95193f67b8e4b round[ 5].is_box 0c018a2c0c6b3ad016db7022d603e6cc round[ 5].is_row 0c0370d00c01e622166b8accd6db3a2c round[ 5].im_col 592460b248832b2952e0b831923048f1 round[ 5].ik_sch dcc1a8b667053f7dcc5c194ab5423a2e round[ 6].istart 85e5c8042f8614549ebca17b277272df round[ 6].is_box 672ab1304edc9bfddf78f1033d1e1eef round[ 6].is_row 671ef1fd4e2a1e03dfdcb1ef3d789b30 round[ 6].im_col 0b8a7783417ae3a1f9492dc0c641a7ce round[ 6].ik_sch c6deb0ab791e2364a4055fbe568803ab round[ 7].istart cd54c7283864c0c55d4c727e90c9a465 round[ 7].is_box 80fd31ee768c1f078d5d1e8a96121dbc round[ 7].is_row 80121e0776fd1d8a8d8c31bc965d1fee round[ 7].im_col 4ee1ddf9301d6352c9ad769ef8d20515 round[ 7].ik_sch dd1b7cdaf28d5c158a49ab1dbbc497cb round[ 8].istart 93faa123c2903f4743e4dd83431692de round[ 8].is_box 2214f132a896251664aec94164ff749c round[ 8].is_row 22ffc916a81474416496f19c64ae2532 round[ 8].im_col 1008ffe53b36ee6af27b42549b8a7bb7 round[ 8].ik_sch 78c4f708318d3cd69655b701bfc093cf round[ 9].istart 68cc08ed0abbd2bc642ef555244ae878 round[ 9].is_box f727bf53a3fe7f788cc377eda65cc8c1 round[ 9].is_row f75c7778a327c8ed8cfebfc1a6c37f53 round[ 9].im_col 7f69ac1ed939ebaac8ece3cb12e159e3 41 round[ 9].ik_sch 60dcef10299524ce62dbef152f9620cf round[10].istart 1fb5430ef0accf64aa370cde3d77792c round[10].is_box cbd264d717aa5f8c62b2819c8b02af42 round[10].is_row cb02818c17d2af9c62aa64428bb25fd7 round[10].im_col cfaf16b2570c18b52e7fef50cab267ae round[10].ik_sch 4b4ecbdb4d4dcfda5752d7c74949cbde round[11].istart 84e1dd691a41d76f792d389783fbac70 round[11].is_box 4fe0c9e443f80d06affa76854163aad0 round[11].is_row 4f63760643e0aa85aff8c9d041fa0de4 round[11].im_col 794cf891177bfd1d8a327086f3831b39 round[11].ik_sch 1a1f181d1e1b1c194742c7d74949cbde round[12].istart 6353e08c0960e104cd70b751bacad0e7 round[12].is_box 0050a0f04090e03080d02070c01060b0 round[12].is_row 00102030405060708090a0b0c0d0e0f0 round[12].ik_sch 000102030405060708090a0b0c0d0e0f round[12].ioutput 00112233445566778899aabbccddeeff C.3 AES-256 (Nk=8, Nr=14) PLAINTEXT: 00112233445566778899aabbccddeeff KEY: 000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f CIPHER (ENCRYPT): round[ 0].input 00112233445566778899aabbccddeeff round[ 0].k_sch 000102030405060708090a0b0c0d0e0f round[ 1].start 00102030405060708090a0b0c0d0e0f0 round[ 1].s_box 63cab7040953d051cd60e0e7ba70e18c round[ 1].s_row 6353e08c0960e104cd70b751bacad0e7 round[ 1].m_col 5f72641557f5bc92f7be3b291db9f91a round[ 1].k_sch 101112131415161718191a1b1c1d1e1f round[ 2].start 4f63760643e0aa85efa7213201a4e705 round[ 2].s_box 84fb386f1ae1ac97df5cfd237c49946b round[ 2].s_row 84e1fd6b1a5c946fdf4938977cfbac23 round[ 2].m_col bd2a395d2b6ac438d192443e615da195 round[ 2].k_sch a573c29fa176c498a97fce93a572c09c round[ 3].start 1859fbc28a1c00a078ed8aadc42f6109 round[ 3].s_box adcb0f257e9c63e0bc557e951c15ef01 round[ 3].s_row ad9c7e017e55ef25bc150fe01ccb6395 round[ 3].m_col 810dce0cc9db8172b3678c1e88a1b5bd round[ 3].k_sch 1651a8cd0244beda1a5da4c10640bade round[ 4].start 975c66c1cb9f3fa8a93a28df8ee10f63 round[ 4].s_box 884a33781fdb75c2d380349e19f876fb round[ 4].s_row 88db34fb1f807678d3f833c2194a759e round[ 4].m_col b2822d81abe6fb275faf103a078c0033 round[ 4].k_sch ae87dff00ff11b68a68ed5fb03fc1567 round[ 5].start 1c05f271a417e04ff921c5c104701554 round[ 5].s_box 9c6b89a349f0e18499fda678f2515920 round[ 5].s_row 9cf0a62049fd59a399518984f26be178 round[ 5].m_col aeb65ba974e0f822d73f567bdb64c877 round[ 5].k_sch 6de1f1486fa54f9275f8eb5373b8518d round[ 6].start c357aae11b45b7b0a2c7bd28a8dc99fa round[ 6].s_box 2e5bacf8af6ea9e73ac67a34c286ee2d round[ 6].s_row 2e6e7a2dafc6eef83a86ace7c25ba934 round[ 6].m_col b951c33c02e9bd29ae25cdb1efa08cc7 round[ 6].k_sch c656827fc9a799176f294cec6cd5598b round[ 7].start 7f074143cb4e243ec10c815d8375d54c round[ 7].s_box d2c5831a1f2f36b278fe0c4cec9d0329 42 round[ 7].s_row d22f0c291ffe031a789d83b2ecc5364c round[ 7].m_col ebb19e1c3ee7c9e87d7535e9ed6b9144 round[ 7].k_sch 3de23a75524775e727bf9eb45407cf39 round[ 8].start d653a4696ca0bc0f5acaab5db96c5e7d round[ 8].s_box f6ed49f950e06576be74624c565058ff round[ 8].s_row f6e062ff507458f9be50497656ed654c round[ 8].m_col 5174c8669da98435a8b3e62ca974a5ea round[ 8].k_sch 0bdc905fc27b0948ad5245a4c1871c2f round[ 9].start 5aa858395fd28d7d05e1a38868f3b9c5 round[ 9].s_box bec26a12cfb55dff6bf80ac4450d56a6 round[ 9].s_row beb50aa6cff856126b0d6aff45c25dc4 round[ 9].m_col 0f77ee31d2ccadc05430a83f4ef96ac3 round[ 9].k_sch 45f5a66017b2d387300d4d33640a820a round[10].start 4a824851c57e7e47643de50c2af3e8c9 round[10].s_box d61352d1a6f3f3a04327d9fee50d9bdd round[10].s_row d6f3d9dda6279bd1430d52a0e513f3fe round[10].m_col bd86f0ea748fc4f4630f11c1e9331233 round[10].k_sch 7ccff71cbeb4fe5413e6bbf0d261a7df round[11].start c14907f6ca3b3aa070e9aa313b52b5ec round[11].s_box 783bc54274e280e0511eacc7e200d5ce round[11].s_row 78e2acce741ed5425100c5e0e23b80c7 round[11].m_col af8690415d6e1dd387e5fbedd5c89013 round[11].k_sch f01afafee7a82979d7a5644ab3afe640 round[12].start 5f9c6abfbac634aa50409fa766677653 round[12].s_box cfde0208f4b418ac5309db5c338538ed round[12].s_row cfb4dbedf4093808538502ac33de185c round[12].m_col 7427fae4d8a695269ce83d315be0392b round[12].k_sch 2541fe719bf500258813bbd55a721c0a round[13].start 516604954353950314fb86e401922521 round[13].s_box d133f22a1aed2a7bfa0f44697c4f3ffd round[13].s_row d1ed44fd1a0f3f2afa4ff27b7c332a69 round[13].m_col 2c21a820306f154ab712c75eee0da04f round[13].k_sch 4e5a6699a9f24fe07e572baacdf8cdea round[14].start 627bceb9999d5aaac945ecf423f56da5 round[14].s_box aa218b56ee5ebeacdd6ecebf26e63c06 round[14].s_row aa5ece06ee6e3c56dde68bac2621bebf round[14].k_sch 24fc79ccbf0979e9371ac23c6d68de36 round[14].output 8ea2b7ca516745bfeafc49904b496089 INVERSE CIPHER (DECRYPT): round[ 0].iinput 8ea2b7ca516745bfeafc49904b496089 round[ 0].ik_sch 24fc79ccbf0979e9371ac23c6d68de36 round[ 1].istart aa5ece06ee6e3c56dde68bac2621bebf round[ 1].is_row aa218b56ee5ebeacdd6ecebf26e63c06 round[ 1].is_box 627bceb9999d5aaac945ecf423f56da5 round[ 1].ik_sch 4e5a6699a9f24fe07e572baacdf8cdea round[ 1].ik_add 2c21a820306f154ab712c75eee0da04f round[ 2].istart d1ed44fd1a0f3f2afa4ff27b7c332a69 round[ 2].is_row d133f22a1aed2a7bfa0f44697c4f3ffd round[ 2].is_box 516604954353950314fb86e401922521 round[ 2].ik_sch 2541fe719bf500258813bbd55a721c0a round[ 2].ik_add 7427fae4d8a695269ce83d315be0392b round[ 3].istart cfb4dbedf4093808538502ac33de185c round[ 3].is_row cfde0208f4b418ac5309db5c338538ed round[ 3].is_box 5f9c6abfbac634aa50409fa766677653 round[ 3].ik_sch f01afafee7a82979d7a5644ab3afe640 round[ 3].ik_add af8690415d6e1dd387e5fbedd5c89013 43 round[ 4].istart 78e2acce741ed5425100c5e0e23b80c7 round[ 4].is_row 783bc54274e280e0511eacc7e200d5ce round[ 4].is_box c14907f6ca3b3aa070e9aa313b52b5ec round[ 4].ik_sch 7ccff71cbeb4fe5413e6bbf0d261a7df round[ 4].ik_add bd86f0ea748fc4f4630f11c1e9331233 round[ 5].istart d6f3d9dda6279bd1430d52a0e513f3fe round[ 5].is_row d61352d1a6f3f3a04327d9fee50d9bdd round[ 5].is_box 4a824851c57e7e47643de50c2af3e8c9 round[ 5].ik_sch 45f5a66017b2d387300d4d33640a820a round[ 5].ik_add 0f77ee31d2ccadc05430a83f4ef96ac3 round[ 6].istart beb50aa6cff856126b0d6aff45c25dc4 round[ 6].is_row bec26a12cfb55dff6bf80ac4450d56a6 round[ 6].is_box 5aa858395fd28d7d05e1a38868f3b9c5 round[ 6].ik_sch 0bdc905fc27b0948ad5245a4c1871c2f round[ 6].ik_add 5174c8669da98435a8b3e62ca974a5ea round[ 7].istart f6e062ff507458f9be50497656ed654c round[ 7].is_row f6ed49f950e06576be74624c565058ff round[ 7].is_box d653a4696ca0bc0f5acaab5db96c5e7d round[ 7].ik_sch 3de23a75524775e727bf9eb45407cf39 round[ 7].ik_add ebb19e1c3ee7c9e87d7535e9ed6b9144 round[ 8].istart d22f0c291ffe031a789d83b2ecc5364c round[ 8].is_row d2c5831a1f2f36b278fe0c4cec9d0329 round[ 8].is_box 7f074143cb4e243ec10c815d8375d54c round[ 8].ik_sch c656827fc9a799176f294cec6cd5598b round[ 8].ik_add b951c33c02e9bd29ae25cdb1efa08cc7 round[ 9].istart 2e6e7a2dafc6eef83a86ace7c25ba934 round[ 9].is_row 2e5bacf8af6ea9e73ac67a34c286ee2d round[ 9].is_box c357aae11b45b7b0a2c7bd28a8dc99fa round[ 9].ik_sch 6de1f1486fa54f9275f8eb5373b8518d round[ 9].ik_add aeb65ba974e0f822d73f567bdb64c877 round[10].istart 9cf0a62049fd59a399518984f26be178 round[10].is_row 9c6b89a349f0e18499fda678f2515920 round[10].is_box 1c05f271a417e04ff921c5c104701554 round[10].ik_sch ae87dff00ff11b68a68ed5fb03fc1567 round[10].ik_add b2822d81abe6fb275faf103a078c0033 round[11].istart 88db34fb1f807678d3f833c2194a759e round[11].is_row 884a33781fdb75c2d380349e19f876fb round[11].is_box 975c66c1cb9f3fa8a93a28df8ee10f63 round[11].ik_sch 1651a8cd0244beda1a5da4c10640bade round[11].ik_add 810dce0cc9db8172b3678c1e88a1b5bd round[12].istart ad9c7e017e55ef25bc150fe01ccb6395 round[12].is_row adcb0f257e9c63e0bc557e951c15ef01 round[12].is_box 1859fbc28a1c00a078ed8aadc42f6109 round[12].ik_sch a573c29fa176c498a97fce93a572c09c round[12].ik_add bd2a395d2b6ac438d192443e615da195 round[13].istart 84e1fd6b1a5c946fdf4938977cfbac23 round[13].is_row 84fb386f1ae1ac97df5cfd237c49946b round[13].is_box 4f63760643e0aa85efa7213201a4e705 round[13].ik_sch 101112131415161718191a1b1c1d1e1f round[13].ik_add 5f72641557f5bc92f7be3b291db9f91a round[14].istart 6353e08c0960e104cd70b751bacad0e7 round[14].is_row 63cab7040953d051cd60e0e7ba70e18c round[14].is_box 00102030405060708090a0b0c0d0e0f0 round[14].ik_sch 000102030405060708090a0b0c0d0e0f round[14].ioutput 00112233445566778899aabbccddeeff EQUIVALENT INVERSE CIPHER (DECRYPT): 44 round[ 0].iinput 8ea2b7ca516745bfeafc49904b496089 round[ 0].ik_sch 24fc79ccbf0979e9371ac23c6d68de36 round[ 1].istart aa5ece06ee6e3c56dde68bac2621bebf round[ 1].is_box 629deca599456db9c9f5ceaa237b5af4 round[ 1].is_row 627bceb9999d5aaac945ecf423f56da5 round[ 1].im_col e51c9502a5c1950506a61024596b2b07 round[ 1].ik_sch 34f1d1ffbfceaa2ffce9e25f2558016e round[ 2].istart d1ed44fd1a0f3f2afa4ff27b7c332a69 round[ 2].is_box 5153862143fb259514920403016695e4 round[ 2].is_row 516604954353950314fb86e401922521 round[ 2].im_col 91a29306cc450d0226f4b5eaef5efed8 round[ 2].ik_sch 5e1648eb384c350a7571b746dc80e684 round[ 3].istart cfb4dbedf4093808538502ac33de185c round[ 3].is_box 5fc69f53ba4076bf50676aaa669c34a7 round[ 3].is_row 5f9c6abfbac634aa50409fa766677653 round[ 3].im_col b041a94eff21ae9212278d903b8a63f6 round[ 3].ik_sch c8a305808b3f7bd043274870d9b1e331 round[ 4].istart 78e2acce741ed5425100c5e0e23b80c7 round[ 4].is_box c13baaeccae9b5f6705207a03b493a31 round[ 4].is_row c14907f6ca3b3aa070e9aa313b52b5ec round[ 4].im_col 638357cec07de6300e30d0ec4ce2a23c round[ 4].ik_sch b5708e13665a7de14d3d824ca9f151c2 round[ 5].istart d6f3d9dda6279bd1430d52a0e513f3fe round[ 5].is_box 4a7ee5c9c53de85164f348472a827e0c round[ 5].is_row 4a824851c57e7e47643de50c2af3e8c9 round[ 5].im_col ca6f71058c642842a315595fdf54f685 round[ 5].ik_sch 74da7ba3439c7e50c81833a09a96ab41 round[ 6].istart beb50aa6cff856126b0d6aff45c25dc4 round[ 6].is_box 5ad2a3c55fe1b93905f3587d68a88d88 round[ 6].is_row 5aa858395fd28d7d05e1a38868f3b9c5 round[ 6].im_col ca46f5ea835eab0b9537b6dbb221b6c2 round[ 6].ik_sch 3ca69715d32af3f22b67ffade4ccd38e round[ 7].istart f6e062ff507458f9be50497656ed654c round[ 7].is_box d6a0ab7d6cca5e695a6ca40fb953bc5d round[ 7].is_row d653a4696ca0bc0f5acaab5db96c5e7d round[ 7].im_col 2a70c8da28b806e9f319ce42be4baead round[ 7].ik_sch f85fc4f3374605f38b844df0528e98e1 round[ 8].istart d22f0c291ffe031a789d83b2ecc5364c round[ 8].is_box 7f4e814ccb0cd543c175413e8307245d round[ 8].is_row 7f074143cb4e243ec10c815d8375d54c round[ 8].im_col f0073ab7404a8a1fc2cba0b80df08517 round[ 8].ik_sch de69409aef8c64e7f84d0c5fcfab2c23 round[ 9].istart 2e6e7a2dafc6eef83a86ace7c25ba934 round[ 9].is_box c345bdfa1bc799e1a2dcaab0a857b728 round[ 9].is_row c357aae11b45b7b0a2c7bd28a8dc99fa round[ 9].im_col 3225fe3686e498a32593c1872b613469 round[ 9].ik_sch aed55816cf19c100bcc24803d90ad511 round[10].istart 9cf0a62049fd59a399518984f26be178 round[10].is_box 1c17c554a4211571f970f24f0405e0c1 round[10].is_row 1c05f271a417e04ff921c5c104701554 round[10].im_col 9d1d5c462e655205c4395b7a2eac55e2 round[10].ik_sch 15c668bd31e5247d17c168b837e6207c round[11].istart 88db34fb1f807678d3f833c2194a759e round[11].is_box 979f2863cb3a0fc1a9e166a88e5c3fdf round[11].is_row 975c66c1cb9f3fa8a93a28df8ee10f63 round[11].im_col d24bfb0e1f997633cfce86e37903fe87 round[11].ik_sch 7fd7850f61cc991673db890365c89d12 45 round[12].istart ad9c7e017e55ef25bc150fe01ccb6395 round[12].is_box 181c8a098aed61c2782ffba0c45900ad round[12].is_row 1859fbc28a1c00a078ed8aadc42f6109 round[12].im_col aec9bda23e7fd8aff96d74525cdce4e7 round[12].ik_sch 2a2840c924234cc026244cc5202748c4 round[13].istart 84e1fd6b1a5c946fdf4938977cfbac23 round[13].is_box 4fe0210543a7e706efa476850163aa32 round[13].is_row 4f63760643e0aa85efa7213201a4e705 round[13].im_col 794cf891177bfd1ddf67a744acd9c4f6 round[13].ik_sch 1a1f181d1e1b1c191217101516131411 round[14].istart 6353e08c0960e104cd70b751bacad0e7 round[14].is_box 0050a0f04090e03080d02070c01060b0 round[14].is_row 00102030405060708090a0b0c0d0e0f0 round[14].ik_sch 000102030405060708090a0b0c0d0e0f round[14].ioutput 00112233445566778899aabbccddeeff 46 Appendix D - References [1] AES page available via http://www.nist.gov/CryptoToolkit. 4 [2] Computer Security Objects Register (CSOR): http://csrc.nist.gov/csor/. [3] J. Daemen and V. Rijmen, AES Proposal: Rijndael , AES Algorithm Submission, September 3, 1999, available at [1]. [4] J. Daemen and V. Rijmen, The block cipher Rijndael, Smart Card research and Applications, LNCS 1820, Springer-Verlag, pp. 288-296. [5] B. Gladman’s AES related home page http://fp.gladman.plus.com/cryptography_technology/. [6] A. Lee, NIST Special Publication 800-21, Guideline for Implementing Cryptography in the Federal Government, National Institute of Standards and Technology, November 1999. [7] A. Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography, CRC Press, New York, 1997, p. 81-83. [8] J. Nechvatal, et. al., Report on the Development of the Advanced Encryption Standard (AES), National Institute of Standards and Technology, October 2, 2000, available at [1]. 4 A complete set of documentation from the AES development effort – including announcements, public comments, analysis papers, conference proceedings, etc. – is available from this site. 47
nistspecialpublication800-38a	NIST Special Publication 800-38A Recommendation for Block 2001 Edition Cipher Modes of Operation M ethods and Techniques Morris Dworkin C O M P U T E R S E C U R I T Y ii C O M P U T E R S E C U R I T Y Computer Security Division Information Technology Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899-8930 December 2001 U.S. Department of Commerce Donald L. Evans, Secretary Technology Administration Phillip J. Bond, Under Secretary of Commerce for Technology National Institute of Standards and Technology Arden L. Bement, Jr., Director iii Reports on Information Security Technology The Information Technology Laboratory (ITL) at the National Institute of Standards and Technology (NIST) promotes the U.S. economy and public welfare by providing technical leadership for the Nation’s measurement and standards infrastructure. ITL develops tests, test methods, reference data, proof of concept implementations, and technical analyses to advance the development and productive use of information technology. ITL’s responsibilities include the development of technical, physical, administrative, and management standards and guidelines for the cost-effective security and privacy of sensitive unclassified information in Federal computer systems. This Special Publication 800-series reports on ITL’s research, guidance, and outreach efforts in computer security, and its collaborative activities with industry, government, and academic organizations. Certain commercial entities, equipment, or materials may be identified in this document in order to describe an experimental procedure or concept adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the entities, materials, or equipment are necessarily the best available for the purpose. National Institute of Standards and Technology Special Publication 800-38A 2001 ED Natl. Inst. Stand. Technol. Spec. Publ. 800-38A 2001 ED, 66 pages (December 2001) CODEN: NSPUE2 U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 2001 For sale by the Superintendent of Documents, U.S. Government Printing Office Internet: bookstore.gpo.gov — Phone: (202) 512-1800 — Fax: (202) 512-2250 Mail: Stop SSOP, Washington, DC 20402-0001 iv Abstract This recommendation defines five confidentiality modes of operation for use with an underlying symmetric key block cipher algorithm: Electronic Codebook (ECB), Cipher Block Chaining (CBC), Cipher Feedback (CFB), Output Feedback (OFB), and Counter (CTR). Used with an underlying block cipher algorithm that is approved in a Federal Information Processing Standard (FIPS), these modes can provide cryptographic protection for sensitive, but unclassified, computer data. KEY WORDS: Computer security; cryptography; data security; block cipher; encryption; Federal Information Processing Standard; mode of operation. v Table of Contents 1 PURPOSE .......................................................................................................................................................... 1 2 AUTHORITY .................................................................................................................................................... 1 3 INTRODUCTION ................................................................................................................... .......................... 1 4 DEFINITIONS, ABBREVIATIONS, AND SYMBOLS................................................................................. 3 4.1 DEFINITIONS AND ABBREVIATIONS ............................................................................................................ 3 4.2 SYMBOLS.................................................................................................................................................... 5 4.2.1 Variables ............................................................................................................................................... 5 4.2.2 Operations and Functions....................................................................................................... .............. 5 5 PRELIMINARIES.................................................................................................................. ........................... 7 5.1 UNDERLYING BLOCK CIPHER ALGORITHM................................................................................................. 7 5.2 REPRESENTATION OF THE PLAINTEXT AND THE CIPHERTEXT ..................................................................... 7 5.3 INITIALIZATION VECTORS........................................................................................................................... 8 5.4 EXAMPLES OF OPERATIONS AND FUNCTIONS ............................................................................................. 8 6 BLOCK CIPHER MODES OF OPERATION ............................................................................................... 9 6.1 THE ELECTRONIC CODEBOOK MODE.......................................................................................................... 9 6.2 THE CIPHER BLOCK CHAINING MODE ...................................................................................................... 10 6.3 THE CIPHER FEEDBACK MODE ................................................................................................................. 11 6.4 THE OUTPUT FEEDBACK MODE................................................................................................................ 13 6.5 THE COUNTER MODE ............................................................................................................................... 15 APPENDIX A: PADDING ........................................................................................................... ........................... 17 APPENDIX B: GENERATION OF COUNTER BLOCKS ................................................................................. 18 B.1 THE STANDARD INCREMENTING FUNCTION ............................................................................................. 18 B.2 CHOOSING INITIAL COUNTER BLOCKS ..................................................................................................... 19 APPENDIX C: GENERATION OF INITIALIZATION VECTORS ................................................................. 20 APPENDIX D: ERROR PROPERTIES .................................................................................................. .............. 21 APPENDIX E: MODES OF TRIPLE DES............................................................................................... ............. 23 APPENDIX F: EXAMPLE VECTORS FOR MODES OF OPERATION OF THE AES ................................ 24 F.1 ECB EXAMPLE VECTORS ......................................................................................................................... 24 F.1.1 ECB-AES128.Encrypt ............................................................................................................. ............ 24 F.1.2 ECB-AES128.Decrypt ............................................................................................................. ............ 24 F.1.3 ECB-AES192.Encrypt ............................................................................................................. ............ 25 F.1.4 ECB-AES192.Decrypt ............................................................................................................. ............ 25 F.1.5 ECB-AES256.Encrypt ............................................................................................................. ............ 26 F.1.6 ECB-AES256.Decrypt ............................................................................................................. ............ 26 F.2 CBC EXAMPLE VECTORS ......................................................................................................................... 27 F.2.1 CBC-AES128.Encrypt ............................................................................................................. ............ 27 F.2.2 CBC-AES128.Decrypt......................................................................................................................... 27 F.2.3 CBC-AES192.Encrypt ............................................................................................................. ............ 28 F.2.4 CBC-AES192.Decrypt......................................................................................................................... 28 vi F.2.5 CBC-AES256.Encrypt ............................................................................................................. ............ 28 F.2.6 CBC-AES256.Decrypt......................................................................................................................... 29 F.3 CFB EXAMPLE VECTORS ......................................................................................................................... 29 F.3.1 CFB1-AES128.Encrypt ............................................................................................................ ........... 29 F.3.2 CFB1-AES128.Decrypt ............................................................................................................ ........... 31 F.3.3 CFB1-AES192.Encrypt ............................................................................................................ ........... 33 F.3.4 CFB1-AES192.Decrypt ............................................................................................................ ........... 34 F.3.5 CFB1-AES256.Encrypt ............................................................................................................ ........... 36 F.3.6 CFB1-AES256.Decrypt ............................................................................................................ ........... 37 F.3.7 CFB8-AES128.Encrypt ............................................................................................................ ........... 39 F.3.8 CFB8-AES128.Decrypt ............................................................................................................ ........... 41 F.3.9 CFB8-AES192.Encrypt ............................................................................................................ ........... 42 F.3.10 CFB8-AES192.Decrypt ............................................................................................................ ...... 44 F.3.11 CFB8-AES256.Encrypt ............................................................................................................ ...... 46 F.3.12 CFB8-AES256.Decrypt ............................................................................................................ ...... 48 F.3.13 CFB128-AES128.Encrypt .......................................................................................................... .... 50 F.3.14 CFB128-AES128.Decrypt .......................................................................................................... .... 50 F.3.15 CFB128-AES192.Encrypt .......................................................................................................... .... 50 F.3.16 CFB128-AES192.Decrypt .......................................................................................................... .... 51 F.3.17 CFB128-AES256.Encrypt .......................................................................................................... .... 51 F.3.18 CFB128-AES256.Decrypt .......................................................................................................... .... 52 F.4 OFB EXAMPLE VECTORS ......................................................................................................................... 52 F.4.1 OFB-AES128.Encrypt ............................................................................................................. ............ 52 F.4.2 OFB-AES128.Decrypt......................................................................................................................... 53 F.4.3 OFB-AES192.Encrypt ............................................................................................................. ............ 53 F.4.4 OFB-AES192.Decrypt......................................................................................................................... 54 F.4.5 OFB-AES256.Encrypt ............................................................................................................. ............ 54 F.4.6 OFB-AES256.Decrypt......................................................................................................................... 55 F.5 CTR EXAMPLE VECTORS ......................................................................................................................... 55 F.5.1 CTR-AES128.Encrypt ............................................................................................................. ............ 55 F.5.2 CTR-AES128.Decrypt ............................................................................................................. ............ 56 F.5.3 CTR-AES192.Encrypt ............................................................................................................. ............ 56 F.5.4 CTR-AES192.Decrypt ............................................................................................................. ............ 57 F.5.5 CTR-AES256.Encrypt ............................................................................................................. ............ 57 F.5.6 CTR-AES256.Decrypt ............................................................................................................. ............ 57 APPENDIX G: REFERENCES ........................................................................................................ ...................... 59 Table of Figures Figure 1: The ECB Mode ................................................................................................................9 Figure 2: The CBC Mode..............................................................................................................10 Figure 3: The CFB Mode ..............................................................................................................12 Figure 4: The OFB Mode ..............................................................................................................14 Figure 5: The CTR Mode ..............................................................................................................16 vii 1 Purpose This publication provides recommendations regarding modes of operation to be used with symmetric key block cipher algorithms. 2 Authority This document has been developed by the National Institute of Standards and Technology (NIST) in furtherance of its statutory responsibilities under the Computer Security Act of 1987 (Public Law 100-235) and the Information Technology Management Reform Act of 1996, specifically 15 U.S.C. 278 g-3(a)(5). This is not a guideline within the meaning of 15 U.S.C. 278 g-3 (a)(5). This recommendation is neither a standard nor a guideline, and as such, is neither mandatory nor binding on Federal agencies. Federal agencies and non-government organizations may use this recommendation on a voluntary basis. It is not subject to copyright. Nothing in this recommendation should be taken to contradict standards and guidelines that have been made mandatory and binding upon Federal agencies by the Secretary of Commerce under his statutory authority. Nor should this recommendation be interpreted as altering or superseding the existing authorities of the Secretary of Commerce, the Director of the Office of Management and Budget, or any other Federal official. Conformance testing for implementations of the modes of operation that are specified in this recommendation will be conducted within the framework of the Cryptographic Module Validation Program (CMVP), a joint effort of the NIST and the Communications Security Establishment of the Government of Canada. An implementation of a mode of operation must adhere to the requirements in this recommendation in order to be validated under the CMVP. 3 Introduction This recommendation specifies five confidentiality modes of operation for symmetric key block cipher algorithms, such as the algorithm specified in FIPS Pub. 197, the Advanced Encryption Standard (AES) [2]. The modes may be used in conjunction with any symmetric key block cipher algorithm that is approved by a Federal Information Processing Standard (FIPS). The five modes—the Electronic Codebook (ECB), Cipher Block Chaining (CBC), Cipher Feedback (CFB), Output Feedback (OFB), and Counter (CTR) modes—can provide data confidentiality. Two FIPS publications already approve confidentiality modes of operation for two particular block cipher algorithms. FIPS Pub. 81 [4] specifies the ECB, CBC, CFB, and OFB modes of the Data Encryption Standard (DES). FIPS Pub. 46-3 [3] approves the seven modes that are specified in ANSI X9.52 [1]. Four of these modes are equivalent to the ECB, CBC, CFB, and OFB modes with the Triple DES algorithm (TDEA) as the underlying block cipher; the other 1 three modes in ANSI X9.52 are variants of the CBC, CFB, and OFB modes of Triple DES that use interleaving or pipelining. Thus, there are three new elements in this recommendation: 1) the extension of the four confidentiality modes in FIPS Pub 81 for use with any FIPS-approved block cipher; 2) the revision of the requirements for these modes; and 3) the specification of an additional confidentiality mode, the CTR mode, for use with any FIPS-approved block cipher. 2 4 Definitions, Abbreviations, and Symbols 4.1 Definitions and Abbreviations Bit A binary digit: 0 or 1. Bit Error The substitution of a ‘0’ bit for a ‘1’ bit, or vice versa. Bit String An ordered sequence of 0’s and 1’s. Block Cipher A family of functions and their inverse functions that is parameterized by cryptographic keys; the functions map bit strings of a fixed length to bit strings of the same length. Block Size The number of bits in an input (or output) block of the block cipher. CBC Cipher Block Chaining. CFB Cipher Feedback. Ciphertext Encrypted data. Confidentiality Mode A mode that is used to encipher plaintext and decipher ciphertext. The confidentiality modes in this recommendation are the ECB, CBC, CFB, OFB, and CTR modes. CTR Counter. Cryptographic Key A parameter used in the block cipher algorithm that determines the forward cipher operation and the inverse cipher operation. Data Block (Block) A sequence of bits whose length is the block size of the block cipher. Data Segment (Segment) In the CFB mode, a sequence of bits whose length is a parameter that does not exceed the block size. Decryption (Deciphering) The process of a confidentiality mode that transforms encrypted data into the original usable data. ECB Electronic Codebook. Encryption (Enciphering) The process of a confidentiality mode that transforms usable data into an unreadable form. 3 Exclusive-OR The bitwise addition, modulo 2, of two bit strings of equal length. FIPS Federal Information Processing Standard. Forward Cipher Function (Forward Cipher Operation) One of the two functions of the block cipher algorithm that is selected by the cryptographic key. Initialization Vector (IV) A data block that some modes of operation require as an additional initial input. Input Block A data block that is an input to either the forward cipher function or the inverse cipher function of the block cipher algorithm. Inverse Cipher Function (Inverse Cipher Operation) The function that reverses the transformation of the forward cipher function when the same cryptographic key is used. Least Significant Bit(s) The right-most bit(s) of a bit string. Mode of Operation (Mode) An algorithm for the cryptographic transformation of data that features a symmetric key block cipher algorithm. Most Significant Bit(s) The left-most bit(s) of a bit string. Nonce A value that is used only once. Octet A group of eight binary digits. OFB Output Feedback. Output Block A data block that is an output of either the forward cipher function or the inverse cipher function of the block cipher algorithm. Plaintext Usable data that is formatted as input to a mode. 4 4.2 Symbols 4.2.1 Variables b The block size, in bits. j The index to a sequence of data blocks or data segments ordered from left to right. n The number of data blocks or data segments in the plaintext. s The number of bits in a data segment. u The number of bits in the last plaintext or ciphertext block. C j The j th ciphertext block. C # The j th ciphertext segment. C * j The last block of the ciphertext, which may be a partial block. Ij The j th input block. IV The initialization vector. K The secret key. Oj The j th output block. n Pj The j th plaintext block. P # The j th plaintext segment. P * j The last block of the plaintext, which may be a partial block. Tj The j th counter block. n 4.2.2 Operations and Functions X \| Y The concatenation of two bit strings X and Y. X ⊕ Y The bitwise exclusive-OR of two bit strings X and Y of the same length. CIPHK(X) The forward cipher function of the block cipher algorithm under the key K applied to the data block X. 5 CIPH -1 (X) The inverse cipher function of the block cipher algorithm under the key K appliedK to the data block X. LSB (X) The bit string consisting of the m least significant bits of the bit string X.m MSB (X) The bit string consisting of the m most significant bits of the bit string X.m [x] The binary representation of the non-negative integer x, in m bits, where x<2 m .m 6 5 Preliminaries 5.1 Underlying Block Cipher Algorithm This recommendation assumes that a FIPS-approved symmetric key block cipher algorithm has been chosen as the underlying algorithm, and that a secret, random key, denoted K, has been established among all of the parties to the communication. The cryptographic key regulates the functioning of the block cipher algorithm and, thus, by extension, regulates the functioning of the mode. The specifications of the block cipher and algorithms and the modes are public, so the security of the mode depends, at a minimum, on the secrecy of the key. A confidentiality mode of operation of the block cipher algorithm consists of two processes that are inverses of each other: encryption and decryption. Encryption is the transformation of a usable message, called the plaintext, into an unreadable form, called the ciphertext; decryption is the transformation that recovers the plaintext from the ciphertext. For any given key, the underlying block cipher algorithm of the mode also consists of two functions that are inverses of each other. These two functions are often called encryption and decryption, but in this recommendation, those terms are reserved for the processes of the confidentiality modes. Instead, as part of the choice of the block cipher algorithm, one of the two functions is designated as the forward cipher function, denoted CIPH K; the other function is then called the inverse cipher function, denoted CIPH –1 . The inputs and outputs of both functions areK called input blocks and output blocks. The input and output blocks of the block cipher algorithm have the same bit length, called the block size, denoted b. 5.2 Representation of the Plaintext and the Ciphertext For all of the modes in this recommendation, the plaintext must be represented as a sequence of bit strings; the requirements on the lengths of the bit strings vary according to the mode: For the ECB and CBC modes, the total number of bits in the plaintext must be a multiple of the block size, b; in other words, for some positive integer n, the total number of bits in the plaintext must be nb. The plaintext consists of a sequence of n bit strings, each with bit length b. The bit strings in the sequence are called data blocks, and the plaintext is denoted P 1, P2,…, P .n For the CFB mode, the total number of bits in the plaintext must be a multiple of a parameter, denoted s, that does not exceed the block size; in other words, for some positive integer n, the total number of bits in the message must be ns. The plaintext consists of a sequence of n bit strings, each with bit length s. The bit strings in the sequence are called data segments, and the plaintext is denoted P # 1, P # 2,…, P # .n For the OFB and CTR modes, the plaintext need not be a multiple of the block size. Let n and u denote the unique pair of positive integers such that the total number of bits in the message is (n-1)b+u, where 1 ≤ u≤ b. The plaintext consists of a sequence of n bit strings, in which the bit length of the last bit string is u, and the bit length of the other bit strings is b. The sequence is denoted P 1, P2,…, Pn-1, P * , and the bit strings are called data blocks, although the last bit string,n 7 P * , may not be a complete block.n For each mode, the encryption process transforms every plaintext data block or segment into a corresponding ciphertext data block or segment with the same bit length, so that the ciphertext is a sequence of data blocks or segments. The ciphertext is denoted as follows: for the ECB and CBC modes, C 1, C2,…, C ; for the CFB mode, C # 1, C # 2,…, C # ; and, for the OFB and CTR modes, C1, C2,…, Cn-1, C * , where n C * may be a partial block. n n n The formatting of the plaintext, including in some cases the appending of padding bits to form complete data blocks or data segments, is outside the scope of this recommendation. Padding is discussed in Appendix A. 5.3 Initialization Vectors The input to the encryption processes of the CBC, CFB, and OFB modes includes, in addition to the plaintext, a data block called the initialization vector (IV), denoted IV. The IV is used in an initial step in the encryption of a message and in the corresponding decryption of the message. The IV need not be secret; however, for the CBC and CFB modes, the IV for any particular execution of the encryption process must be unpredictable, and, for the OFB mode, unique IVs must be used for each execution of the encryption process. The generation of IVs is discussed in Appendix C. 5.4 Examples of Operations and Functions The concatenation operation on bit strings is denoted \| ; for example, 001 \| 10111 = 00110111. Given bit strings of equal length, the exclusive-OR operation, denoted ⊕ , specifies the addition, modulo 2, of the bits in each bit position, i.e., without carries. Thus, 10011 ⊕ 10101= 00110, for example. The functions LSB and MSB return the s least significant bits and the s most significant bits of s s their arguments. For example, LSB3(111011010) = 010, and MSB4(111011010) = 1110. Given a positive integer m and a non-negative (decimal) integer x that is less than 2 m , the binary representation of x in m bits is denoted [x] . For example, [45]8 = 00101101.m 8 6 Block Cipher Modes of Operation The mathematical specifications of the five modes are given in Sections 6.1-6.5, along with descriptions, illustrations, and comments on the potential for parallel processing. 6.1 The Electronic Codebook Mode The Electronic Codebook (ECB) mode is a confidentiality mode that features, for a given key, the assignment of a fixed ciphertext block to each plaintext block, analogous to the assignment of code words in a codebook. The Electronic Codebook (ECB) mode is defined as follows: ECB Encryption: C j = CIPHK(Pj) for j = 1 … n. ECB Decryption: Pj = CIPH -1 (Cj) for j = 1 … n. In ECB encryption, the forward cipher function is applied directly and independently to each block of the plaintext. The resulting sequence of output blocks is the ciphertext. In ECB decryption, the inverse cipher function is applied directly and independently to each block of the ciphertext. The resulting sequence of output blocks is the plaintext. K ECB Encryption ECB Decryption PLAINTEXT CIPH K INPUT BLOCK OUTPUT BLOCK CIPHERTEXT CIPH -1 K INPUT BLOCK OUTPUT BLOCK Figure 1: The ECB Mode In ECB encryption and ECB decryption, multiple forward cipher functions and inverse cipher functions can be computed in parallel. CIPHERTEXT PLAINTEXT In the ECB mode, under a given key, any given plaintext block always gets encrypted to the 9 same ciphertext block. If this property is undesirable in a particular application, the ECB mode should not be used. The ECB mode is illustrated in Figure 1. 6.2 The Cipher Block Chaining Mode The Cipher Block Chaining (CBC) mode is a confidentiality mode whose encryption process features the combining (“ chaining”) of the plaintext blocks with the previous ciphertext blocks. The CBC mode requires an IV to combine with the first plaintext block. The IV need not be secret, but it must be unpredictable; the generation of such IVs is discussed in Appendix C. Also, the integrity of the IV should be protected, as discussed in Appendix D. The CBC mode is defined as follows: CBC Encryption: C 1 = CIPHK(P1 ⊕ IV); Cj = CIPHK(Pj ⊕ Cj-1) for j = 2 … n. CBC Decryption: P1 = CIPH -1 (C1) ⊕ IV;K Pj = CIPH -1 (Cj) ⊕ Cj-1 for j = 2 … n.K CIPH K OUTPUT BLOCK 1 ⊕ OUTPUT BLOCK 2 ⊕ OUTPUT BLOCK n ⊕ PLAINTEXT 1 PLAINTEXT 2 INITIALIZATION VECTOR INPUT BLOCK 1 INPUT BLOCK 2 INPUT BLOCK n PLAINTEXT n ENCRYPTDECRYPT ⊕ ⊕ ⊕ INPUT BLOCK 1 INPUT BLOCK 2 INPUT BLOCK n CIPH -1 K PLAINTEXT 1 PLAINTEXT 2 PLAINTEXT n OUTPUT BLOCK 1 OUTPUT BLOCK 2 OUTPUT BLOCK n INITIALIZATION VECTOR CIPHERTEXT 1 CIPHERTEXT 2 CIPHERTEXT 1 CIPHERTEXT 2 CIPHERTEXT n CIPHERTEXT n CIPH K CIPH K CIPH -1 K CIPH -1 K Figure 2: The CBC Mode In CBC encryption, the first input block is formed by exclusive-ORing the first block of the plaintext with the IV. The forward cipher function is applied to the first input block, and the 10 resulting output block is the first block of the ciphertext. This output block is also exclusive- ORed with the second plaintext data block to produce the second input block, and the forward cipher function is applied to produce the second output block. This output block, which is the second ciphertext block, is exclusive-ORed with the next plaintext block to form the next input block. Each successive plaintext block is exclusive-ORed with the previous output/ciphertext block to produce the new input block. The forward cipher function is applied to each input block to produce the ciphertext block. In CBC decryption, the inverse cipher function is applied to the first ciphertext block, and the resulting output block is exclusive-ORed with the initialization vector to recover the first plaintext block. The inverse cipher function is also applied to the second ciphertext block, and the resulting output block is exclusive-ORed with the first ciphertext block to recover the second plaintext block. In general, to recover any plaintext block (except the first), the inverse cipher function is applied to the corresponding ciphertext block, and the resulting block is exclusive- ORed with the previous ciphertext block. In CBC encryption, the input block to each forward cipher operation (except the first) depends on the result of the previous forward cipher operation, so the forward cipher operations cannot be performed in parallel. In CBC decryption, however, the input blocks for the inverse cipher function, i.e., the ciphertext blocks, are immediately available, so that multiple inverse cipher operations can be performed in parallel. The CBC mode is illustrated in Figure 2. 6.3 The Cipher Feedback Mode The Cipher Feedback (CFB) mode is a confidentiality mode that features the feedback of successive ciphertext segments into the input blocks of the forward cipher to generate output blocks that are exclusive-ORed with the plaintext to produce the ciphertext, and vice versa. The CFB mode requires an IV as the initial input block. The IV need not be secret, but it must be unpredictable; the generation of such IVs is discussed in Appendix C. The CFB mode also requires an integer parameter, denoted s, such that 1 ≤ s ≤ b. In the specification of the CFB mode below, each plaintext segment ( P # ) and ciphertext segment ( C # )j j consists of s bits. The value of s is sometimes incorporated into the name of the mode, e.g., the 1-bit CFB mode, the 8-bit CFB mode, the 64-bit CFB mode, or the 128-bit CFB mode. The CFB mode is defined as follows: CFB Encryption: I 1 = IV; Ij = LSBb-s(Ij –1) \| C # j -1 for j = 2 … n; Oj = CIPHK(Ij) for j = 1, 2 … n; C # j = P # j ⊕ MSB (Oj) for j = 1, 2 … n.s CFB Decryption: I1 = IV; Ij = LSBb-s(Ij -1 )\| C # j -1 for j = 2 … n; 11 Oj = CIPHK(Ij) for j = 1, 2 … n; P # = C # j ⊕ MSB (Oj) for j = 1, 2 … n.j s In CFB encryption, the first input block is the IV, and the forward cipher operation is applied to the IV to produce the first output block. The first ciphertext segment is produced by exclusive- ORing the first plaintext segment with the s most significant bits of the first output block. (The remaining b-s bits of the first output block are discarded.) The b-s least significant bits of the IV are then concatenated with the s bits of the first ciphertext segment to form the second input block. An alternative description of the formation of the second input block is that the bits of the first input block circularly shift s positions to the left, and then the ciphertext segment replaces the s least significant bits of the result. The process is repeated with the successive input blocks until a ciphertext segment is produced from every plaintext segment. In general, each successive input block is enciphered to produce an output block. The s most significant bits of each output block are exclusive-ORed with the corresponding plaintext segment to form a ciphertext segment. Each ciphertext segment (except the last one) is “fed back” into the previous input block, as described above, to form a new input block. The feedback can be described in terms of the individual bits in the strings as follows: if i 1i2…ib is the jth input block, and c1c2…c is the jth ciphertext segment, then the (j+1) th input blocks is is+1is+2…ib c1c2…c .s OUTPUT BLOCK 1 Select Discard s Bits ( b-s) Bits INPUT BLOCK 1 OUTPUT BLOCK 1 Select Discard s Bits ( b-s) Bits CIPH K INITIALIZATION VECTOR ⊕ PLAINTEXT 1 s Bits CIPHERTEXT 1 s Bits INPUT BLOCK 1 ⊕CIPHERTEXT 1 s Bits PLAINTEXT 1 s Bits ENCRYPTDECRYPT OUTPUT BLOCK n Select Discard s Bits ( b-s) Bits INPUT BLOCK n (b-s) Bits s Bits OUTPUT BLOCK n Select Discard s Bits ( b-s) Bits ⊕ PLAINTEXT n s Bits CIPHERTEXT n s Bits INPUT BLOCK n (b-s) Bits s Bits ⊕CIPHERTEXT n s Bits PLAINTEXT n s Bits OUTPUT BLOCK 2 Select Discard s Bits ( b-s) Bits INPUT BLOCK 2 (b-s) Bits s Bits OUTPUT BLOCK 2 Select Discard s Bits ( b-s) Bits ⊕PLAINTEXT 2 s Bits CIPHERTEXT 2 s Bits INPUT BLOCK 2 (b-s) Bits s Bits ⊕CIPHERTEXT 2 s Bits PLAINTEXT 2 s Bits INITIALIZATION VECTOR CIPH K CIPH K CIPH KCIPH KCIPH K Figure 3: The CFB Mode In CFB decryption, the IV is the first input block, and each successive input block is formed as in CFB encryption, by concatenating the b-s least significant bits of the previous input block with 12 the s most significant bits of the previous ciphertext. The forward cipher function is applied to each input block to produce the output blocks. The s most significant bits of the output blocks are exclusive-ORed with the corresponding ciphertext segments to recover the plaintext segments. In CFB encryption, like CBC encryption, the input block to each forward cipher function (except the first) depends on the result of the previous forward cipher function; therefore, multiple forward cipher operations cannot be performed in parallel. In CFB decryption, the required forward cipher operations can be performed in parallel if the input blocks are first constructed (in series) from the IV and the ciphertext. The CFB mode is illustrated in Figure 3. 6.4 The Output Feedback Mode The Output Feedback (OFB) mode is a confidentiality mode that features the iteration of the forward cipher on an IV to generate a sequence of output blocks that are exclusive-ORed with the plaintext to produce the ciphertext, and vice versa. The OFB mode requires that the IV is a nonce, i.e., the IV must be unique for each execution of the mode under the given key; the generation of such IVs is discussed in Appendix C. The OFB mode is defined as follows: OFB Encryption: I 1 = IV; Ij = Oj -1 for j = 2 … n; Oj = CIPHK(Ij) for j = 1, 2 … n; Cj = Pj ⊕ Oj for j = 1, 2 … n-1; C * = P * ⊕ MSB (O ).n n u n OFB Decryption: I1 = IV; Ij = Oj -1 for j = 2 … n; Oj = CIPHK(Ij) for j = 1, 2 … n; Pj = Cj ⊕ Oj for j = 1, 2 … n-1; P * n = C * n ⊕ MSBu(On). In OFB encryption, the IV is transformed by the forward cipher function to produce the first output block. The first output block is exclusive-ORed with the first plaintext block to produce the first ciphertext block. The forward cipher function is then invoked on the first output block to produce the second output block. The second output block is exclusive-ORed with the second plaintext block to produce the second ciphertext block, and the forward cipher function is invoked on the second output block to produce the third output block. Thus, the successive output blocks are produced from applying the forward cipher function to the previous output blocks, and the output blocks are exclusive-ORed with the corresponding plaintext blocks to produce the ciphertext blocks. For the last block, which may be a partial block of u bits, the most significant u bits of the last output block are used for the exclusive-OR operation; the remaining b-u bits of the last output block are discarded. In OFB decryption, the IV is transformed by the forward cipher function to produce the first 13 output block. The first output block is exclusive-ORed with the first ciphertext block to recover the first plaintext block. The first output block is then transformed by the forward cipher function to produce the second output block. The second output block is exclusive-ORed with the second ciphertext block to produce the second plaintext block, and the second output block is also transformed by the forward cipher function to produce the third output block. Thus, the successive output blocks are produced from applying the forward cipher function to the previous output blocks, and the output blocks are exclusive-ORed with the corresponding ciphertext blocks to recover the plaintext blocks. For the last block, which may be a partial block of u bits, the most significant u bits of the last output block are used for the exclusive-OR operation; the remaining b-u bits of the last output block are discarded. OUTPUT BLOCK 1 INPUT BLOCK 1 OUTPUT BLOCK 1 CIPH K INITIALIZATION VECTOR ⊕PLAINTEXT 1 CIPHERTEXT 1 INPUT BLOCK 1 ⊕CIPHERTEXT 1 PLAINTEXT 1 ENCRYPTDECRYPT OUTPUT BLOCK n INPUT BLOCK n OUTPUT BLOCK n ⊕PLAINTEXT n CIPHERTEXT n INPUT BLOCK n ⊕CIPHERTEXT n PLAINTEXT n OUTPUT BLOCK 2 INPUT BLOCK 2 OUTPUT BLOCK 2 ⊕PLAINTEXT 2 CIPHERTEXT 2 INPUT BLOCK 2 ⊕CIPHERTEXT 2 PLAINTEXT 2 INITIALIZATION VECTOR CIPH K CIPH K CIPH KCIPH KCIPH K Figure 4: The OFB Mode In both OFB encryption and OFB decryption, each forward cipher function (except the first) depends on the results of the previous forward cipher function; therefore, multiple forward cipher functions cannot be performed in parallel. However, if the IV is known, the output blocks can be generated prior to the availability of the plaintext or ciphertext data. The OFB mode requires a unique IV for every message that is ever encrypted under the given key. If, contrary to this requirement, the same IV is used for the encryption of more than one message, then the confidentiality of those messages may be compromised. In particular, if a plaintext block of any of these messages is known, say, the jth plaintext block, then the jth output of the forward cipher function can be determined easily from the jth ciphertext block of the message. This information allows the jth plaintext block of any other message that is encrypted 14 using the same IV to be easily recovered from the jth ciphertext block of that message. Confidentiality may similarly be compromised if any of the input blocks to the forward cipher function for the encryption of a message is designated as the IV for the encryption of another message under the given key. The OFB mode is illustrated in Figure 4. 6.5 The Counter Mode The Counter (CTR) mode is a confidentiality mode that features the application of the forward cipher to a set of input blocks, called counters, to produce a sequence of output blocks that are exclusive-ORed with the plaintext to produce the ciphertext, and vice versa. The sequence of counters must have the property that each block in the sequence is different from every other block. This condition is not restricted to a single message: across all of the messages that are encrypted under the given key, all of the counters must be distinct. In this recommendation, the counters for a given message are denoted T 1, T2, … , T . Methods for generating counters aren discussed in Appendix B. Given a sequence of counters, T1 , T2 , … , T , the CTR mode is n defined as follows: CTR Encryption: Oj = CIPHK(Tj) for j = 1, 2 … n; Cj = Pj ⊕ Oj for j = 1, 2 … n-1; C * = P * ⊕ MSB (O ).n n u n CTR Decryption: Oj = CIPHK(Tj) for j = 1, 2 … n; Pj = Cj ⊕ Oj for j = 1, 2 … n-1; P * n = C * n ⊕ MSBu(On). In CTR encryption, the forward cipher function is invoked on each counter block, and the resulting output blocks are exclusive-ORed with the corresponding plaintext blocks to produce the ciphertext blocks. For the last block, which may be a partial block of u bits, the most significant u bits of the last output block are used for the exclusive-OR operation; the remaining b-u bits of the last output block are discarded. In CTR decryption, the forward cipher function is invoked on each counter block, and the resulting output blocks are exclusive-ORed with the corresponding ciphertext blocks to recover the plaintext blocks. For the last block, which may be a partial block of u bits, the most significant u bits of the last output block are used for the exclusive-OR operation; the remaining b-u bits of the last output block are discarded. In both CTR encryption and CTR decryption, the forward cipher functions can be performed in parallel; similarly, the plaintext block that corresponds to any particular ciphertext block can be recovered independently from the other plaintext blocks if the corresponding counter block can be determined. Moreover, the forward cipher functions can be applied to the counters prior to the availability of the plaintext or ciphertext data. 15 OUTPUT BLOCK 1 INPUT BLOCK 1 OUTPUT BLOCK 1 CIPH K COUNTER 1 ⊕PLAINTEXT 1 CIPHERTEXT 1 INPUT BLOCK 1 ⊕CIPHERTEXT 1 PLAINTEXT 1 ENCRYPTDECRYPT COUNTER 1 OUTPUT BLOCK 2 INPUT BLOCK 2 OUTPUT BLOCK 2 COUNTER 2 ⊕PLAINTEXT 2 CIPHERTEXT 2 INPUT BLOCK 2 ⊕CIPHERTEXT 2 PLAINTEXT 2 COUNTER 2 . . . . .CIPH K CIPH KCIPH K INPUT BLOCK n OUTPUT BLOCK n COUNTER n ⊕PLAINTEXT n CIPH K OUTPUT BLOCK n CIPHERTEXT n INPUT BLOCK n ⊕CIPHERTEXT n COUNTER n . . . . . CIPH K PLAINTEXT n Figure 5: The CTR Mode The CTR mode is illustrated in Figure 5. 16 Appendix A: Padding For the ECB, CBC, and CFB modes, the plaintext must be a sequence of one or more complete data blocks (or, for CFB mode, data segments). In other words, for these three modes, the total number of bits in the plaintext must be a positive multiple of the block (or segment) size. If the data string to be encrypted does not initially satisfy this property, then the formatting of the plaintext must entail an increase in the number of bits. A common way to achieve the necessary increase is to append some extra bits, called padding, to the trailing end of the data string as the last step in the formatting of the plaintext. An example of a padding method is to append a single ‘1’ bit to the data string and then to pad the resulting string by as few ‘0’ bits, possibly none, as are necessary to complete the final block (segment). Other methods may be used; in general, the formatting of the plaintext is outside the scope of this recommendation. For the above padding method, the padding bits can be removed unambiguously, provided the receiver can determine that the message is indeed padded. One way to ensure that the receiver does not mistakenly remove bits from an unpadded message is to require the sender to pad every message, including messages in which the final block (segment) is already complete. For such messages, an entire block (segment) of padding is appended. Alternatively, such messages can be sent without padding if, for every message, the existence of padding can be reliably inferred, e.g., from a message length indicator. 17 Appendix B: Generation of Counter Blocks The specification of the CTR mode requires a unique counter block for each plaintext block that is ever encrypted under a given key, across all messages. If, contrary to this requirement, a counter block is used repeatedly, then the confidentiality of all of the plaintext blocks corresponding to that counter block may be compromised. In particular, if any plaintext block that is encrypted using a given counter block is known, then the output of the forward cipher function can be determined easily from the associated ciphertext block. This output allows any other plaintext blocks that are encrypted using the same counter block to be easily recovered from their associated ciphertext blocks. There are two aspects to satisfying the uniqueness requirement. First, an incrementing function for generating the counter blocks from any initial counter block can ensure that counter blocks do not repeat within a given message. Second, the initial counter blocks, T 1, must be chosen to ensure that counters are unique across all messages that are encrypted under the given key. B.1 The Standard Incrementing Function In general, given the initial counter block for a message, the successive counter blocks are derived by applying an incrementing function. As in the above specifications of the modes, n is the number of blocks in the given plaintext message, and b is the number of bits in the block. The standard incrementing function can apply either to an entire block or to a part of a block. Let m be the number of bits in the specific part of the block to be incremented; thus, m is a positive integer such that m ≤ b. Any string of m bits can be regarded as the binary representation of a non-negative integer x that is strictly less than 2 m . The standard incrementing function takes [x]m and returns [x+1 mod 2 m ] .m For example, let the standard incrementing function apply to the five least significant bits of eight bit blocks, so that b=8 and m=5 (unrealistically small values); let * represent each unknown bit in this example, and let **11110 represent a block to be incremented. The following sequence of blocks results from four applications of the standard incrementing function: * * 1 1 1 1 0 * * * 1 1 1 1 1 * * * 0 0 0 0 0 * * * 0 0 0 0 1 * * * 0 0 0 1 0. Counter blocks in which a given set of m bits are incremented by the standard incrementing function satisfy the uniqueness requirement within the given message provided that n ≤ 2 m . Whether the uniqueness requirement for counter blocks is satisfied across all messages that are encrypted under a given key then depends on the choices of the initial counter blocks for the messages, as discussed in the next section. 18 This recommendation permits the use of any other incrementing function that generates n unique strings of m bits in succession from the allowable initial strings. For example, if the initial string of m bits is not the “zero” string, i.e., if it contains at least one ‘1’ bit, then an incrementing function can be constructed from a linear feedback shift register that is specialized to ensure a sufficiently large period; see Ref. [5] for information about linear feedback shift registers. B.2 Choosing Initial Counter Blocks The initial counter blocks, T1, for each message that is encrypted under the given key must be chosen in a manner than ensures the uniqueness of all the counter blocks across all the messages. Two examples of approaches to choosing the initial counter blocks are given in this section. In the first approach, for a given key, all plaintext messages are encrypted sequentially. Within the messages, the same fixed set of m bits of the counter block is incremented by the standard incrementing function. The initial counter block for the initial plaintext message may be any string of b bits. The initial counter block for any subsequent message can be obtained by applying the standard incrementing function to the fixed set of m bits of the final counter block of the previous message. In effect, all of the plaintext messages that are ever encrypted under the given key are concatenated into a single message; consequently, the total number of plaintext blocks must not exceed 2 m . Procedures should be established to ensure the maintenance of the state of the final counter block of the latest encrypted message, and to ensure the proper sequencing of the messages. A second approach to satisfying the uniqueness property across messages is to assign to each message a unique string of b/2 bits (rounding up, if b is odd), in other words, a message nonce, and to incorporate the message nonce into every counter block for the message. The leading b/2 bits (rounding up, if b is odd) of each counter block would be the message nonce, and the standard incrementing function would be applied to the remaining m bits to provide an index to the counter blocks for the message. Thus, if N is the message nonce for a given message, then the jth counter block is given by T j = N \| [ j] , for j = 1…n. The number of blocks, n, in any message must satisfy n < 2 m . A procedure should be established to ensure the uniqueness of the message nonces. m This recommendation allows other methods and approaches for achieving the uniqueness property. Validation that an implementation of the CTR mode conforms to this recommendation will typically include an examination of the procedures for assuring the uniqueness of counter blocks within messages and across all messages that are encrypted under a given key. 19 Appendix C: Generation of Initialization Vectors The CBC, CFB, and OFB modes require an initialization vector as input, in addition to the plaintext. An IV must be generated for each execution of the encryption operation, and the same IV is necessary for the corresponding execution of the decryption operation. Therefore, the IV, or information that is sufficient to calculate the IV, must be available to each party to the communication. The IV need not be secret, so the IV, or information sufficient to determine the IV, may be transmitted with the ciphertext. For the CBC and CFB modes, the IVs must be unpredictable. In particular, for any given plaintext, it must not be possible to predict the IV that will be associated to the plaintext in advance of the generation of the IV. There are two recommended methods for generating unpredictable IVs. The first method is to apply the forward cipher function, under the same key that is used for the encryption of the plaintext, to a nonce. The nonce must be a data block that is unique to each execution of the encryption operation. For example, the nonce may be a counter, as described in Appendix B, or a message number. The second method is to generate a random data block using a FIPS- approved random number generator. For the OFB mode, the IV need not be unpredictable, but it must be a nonce that is unique to each execution of the encryption operation. For example, the nonce may be a counter, as described in Appendix B, or a message number. If, contrary to this requirement, the same IV is used for the OFB encryption of more than one message, then the confidentiality of those messages may be compromised. In particular, if a plaintext block of any of these messages is known, say, the jth plaintext block, then the jth output of the forward cipher function can be determined easily from the jth ciphertext block of the message. This information allows the jth plaintext block of any other message that is encrypted using the same IV to be easily recovered from the jth ciphertext block of that message. Confidentiality may similarly be compromised if any of the input blocks to the forward cipher function for the OFB encryption of a message is designated as the IV for the encryption of another message under the given key. One consequence of this observation is that IVs for the OFB mode should not be generated by invoking the block cipher on another IV. Validation that an implementation of the CBC, CFB, or OFB mode conforms to this recommendation will typically include an examination of the procedures for assuring the unpredictability or uniqueness of the IV. 20 Appendix D: Error Properties A bit error is the substitution of a ‘ 0’ bit for a ‘1’ bit, or vice versa. This appendix contains a discussion of the effects of bit errors in ciphertext blocks (or segments), counter blocks, and IVs on the modes in this recommendation. Insertion or deletion of bits into ciphertext blocks (or segments) is also discussed. For any confidentiality mode, if there are any bit errors in a single ciphertext block (or segment), then the decryption of that ciphertext block (or segment) will be incorrect, i.e., it will differ from the original plaintext block (or segment). In the CFB, OFB, and CTR modes, the bit error(s) in the decrypted ciphertext block (or segment) occur in the same bit position(s) as in the ciphertext block (or segment); the other bit positions are not affected. In the ECB and CBC modes, a bit error may occur, independently, in any bit position of the decrypted ciphertext block, with an expected error rate of fifty percent, depending on the strength of the underlying block cipher. For the ECB, OFB, and CTR modes, bit errors within a ciphertext block do not affect the decryption of any other blocks. In the CBC mode, any bit positions that contain bit errors in a ciphertext block will also contain bit errors in the decryption of the succeeding ciphertext block; the other bit positions are not affected. In the CFB mode, bit errors in a ciphertext segment affect the decryption of the next b/s (rounded up to the nearest integer) ciphertext segments. A bit error may occur, independently, in any bit position in these decrypted segments, with an expected error rate of fifty percent. Similarly, for the CTR mode, if there is a bit error in a counter block, then a bit error may occur, independently, in any bit position of the decryption of the corresponding ciphertext, with an expected error rate of fifty percent. Bit errors in IVs also affect the decryption process. In the OFB mode, bit errors in the IV affect the decryption of every ciphertext block. In the CFB mode, bit errors in the IV affect, at a minimum, the decryption of the first ciphertext segment, and possibly successive ciphertext segments, depending on the bit position of the rightmost bit error in the IV. (In general, a bit error in the ith most significant bit position affects the decryptions of the first i/s (rounding up) ciphertext segments.) For both the OFB and CFB modes, a bit error may occur, independently, in any bit position of the affected ciphertext blocks (or segments), with an expected error rate of fifty percent. In the CBC mode, if bit errors occur in the IV, then the first ciphertext block will be decrypted incorrectly, and bit errors will occur in exactly the same bit positions as in the IV; the decryptions of the other ciphertext blocks are not affected. Consequently, for the CBC mode, the decryption of the first ciphertext block is vulnerable to the (deliberate) introduction of bit errors in specific bit positions of the IV if the integrity of the IV is not protected. Similarly, for the OFB and CTR modes, the decryption of any ciphertext block is vulnerable to the introduction of specific bit errors into that ciphertext block if its integrity is not protected. The same property also holds for the ciphertext segments in the CFB mode; however, for every ciphertext segment except the last one, the existence of such bit errors may be detected by their randomizing effect on the decryption of the succeeding ciphertext segment. 21 Table D.1 summarizes the effects of bit errors in a ciphertext block or IV on the decryption of the ciphertext for each of the five confidentiality modes. Table D.1e five confidentiality modes. Table D.2: Summary of Effect of Bit Errors on Decryption Mode Effect of Bit Errors in Cj Effect of Bit Errors in the IV ECB RBE in the decryption of Cj Not applicable CBC RBE in the decryption of Cj SBE in the decryption of Cj+1 SBE in the decryption of C1 CFB SBE in the decryption of Cj RBE in the decryption of Cj+1,…,Cj+b/s RBE in the decryption of C1, C2, …, Cj for some j between 1 and b/s OFB SBE in the decryption of Cj RBE in the decryption of C1, C2, …, Cn CTR SBE in the decryption of Cj Not applicable * RBE: random bit errors, i.e., bit errors occur independently in any bit position with an expected probability of ½. SBE: specific bit errors, i.e., bit errors occur in the same bit position(s) as the original bit error(s). * Bit errors in the jth counter block, T j, result in RBE in the decryption of Cj. The deletion or insertion of bits into a ciphertext block (or segment) spoils the synchronization of the block (or segment) boundaries; in effect, bit errors may occur in the bit position of the inserted or deleted bit, and in every subsequent bit position. Therefore, the decryptions of the subsequent ciphertext blocks (or segments) will almost certainly be incorrect until the synchronization is restored. When the 1-bit CFB mode is used, then the synchronization is automatically restored b+1 positions after the inserted or deleted bit. For other values of s in the CFB mode, and for the other confidentiality modes in this recommendation, the synchronization must be restored externally. 22 Appendix E: Modes of Triple DES FIPS Pub 46-3 [FIPS 46-3] specifies the Data Encryption Standard (DES) algorithm and approves its three-fold, compound operation that is specified in ANSI X9.52 [1]: the Triple Data Encryption Algorithm (TDEA). Essentially, the TDEA consists of the application of the forward DES algorithm, i.e, DES encryption, under one key, followed by the application of the inverse DES algorithm, i.e., DES decryption, under a second key, followed by the application of the forward DES algorithm under a third key. The TDEA is often called Triple DES. FIPS Pub 46-3 also approves the seven modes of operation of Triple DES that are specified in ANSI X9.52. Four of those modes are equivalent to modes in this recommendation with the TDEA as the underlying block cipher. In particular, the TECB, TCBC, and TOFB modes in ANSI X9.52 are equivalent to the ECB, CBC, and OFB modes in this recommendation, with the TDEA as the underlying block cipher; the TCFB mode in ANSI X9.52 is equivalent to the CFB mode in this recommendation, with the TDEA as the underlying block cipher, provided that the possible choices of the parameter s (the segment size) are restricted to three values: 1, 8, and 64. The remaining three modes in ANSI X9.52 are TCBC-I, TCFB-P, and TOFB-I; they are mode variants that allow for interleaving or pipelining; this recommendation does not provide analogues of these three modes. The Triple DES modes in ANSI X9.52 should not be used as the underlying block cipher algorithm for the modes in this recommendation. However, the Triple DES algorithm, i.e., TDEA, as described above, may be used as the underlying block cipher algorithm for the six modes in this recommendation. One of the resulting modes of Triple DES is new, i.e., not specified in ANSI X9.52: the CTR mode of the TDEA. 23 Appendix F: Example Vectors for Modes of Operation of the AES In this appendix, three examples are provided for each of the modes in this recommendation with the AES algorithm [2] as the underlying block cipher: one example is given for each of the allowed key sizes (128, 192, and 256 bits). Some intermediate results are presented. For the five confidentiality modes, examples are provided for both encryption and decryption. Examples are provided for 1-bit, 8-bit, and 128 bit CFB. The plaintext for all but two of these examples is equivalent to the following string of hexadecimal characters, formatted into four 128 bit blocks: 6bc1bee22e409f96e93d7e117393172a ae2d8a571e03ac9c9eb76fac45af8e51 30c81c46a35ce411e5fbc1191a0a52ef f69f2445df4f9b17ad2b417be66c3710. For the example of 1-bit CFB, the plaintext is the first 16 bits in the above string; for the example of 8-bit CFB, the plaintext is the first 18 octets in the above string. All strings are presented in hexadecimal notation, except in the example of 1-bit CFB, where the plaintext and ciphertext segments are single bits. F.1 ECB Example Vectors F.1.1 ECB-AES128.Encrypt Key 2b7e151628aed2a6abf7158809cf4f3c Block #1 Plaintext 6bc1bee22e409f96e93d7e117393172a Input Block 6bc1bee22e409f96e93d7e117393172a Output Block 3ad77bb40d7a3660a89ecaf32466ef97 Ciphertext 3ad77bb40d7a3660a89ecaf32466ef97 Block #2 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Input Block ae2d8a571e03ac9c9eb76fac45af8e51 Output Block f5d3d58503b9699de785895a96fdbaaf Ciphertext f5d3d58503b9699de785895a96fdbaaf Block #3 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Input Block 30c81c46a35ce411e5fbc1191a0a52ef Output Block 43b1cd7f598ece23881b00e3ed030688 Ciphertext 43b1cd7f598ece23881b00e3ed030688 Block #4 Plaintext f69f2445df4f9b17ad2b417be66c3710 Input Block f69f2445df4f9b17ad2b417be66c3710 Output Block 7b0c785e27e8ad3f8223207104725dd4 Ciphertext 7b0c785e27e8ad3f8223207104725dd4 F.1.2 ECB-AES128.Decrypt Key 2b7e151628aed2a6abf7158809cf4f3c Block #1 Ciphertext 3ad77bb40d7a3660a89ecaf32466ef97 Input Block 3ad77bb40d7a3660a89ecaf32466ef97 24 Output Block 6bc1bee22e409f96e93d7e117393172a Plaintext 6bc1bee22e409f96e93d7e117393172a Block #2 Ciphertext f5d3d58503b9699de785895a96fdbaaf Input Block f5d3d58503b9699de785895a96fdbaaf Output Block ae2d8a571e03ac9c9eb76fac45af8e51 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Block #3 Ciphertext 43b1cd7f598ece23881b00e3ed030688 Input Block 43b1cd7f598ece23881b00e3ed030688 Output Block 30c81c46a35ce411e5fbc1191a0a52ef Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Block #4 Ciphertext 7b0c785e27e8ad3f8223207104725dd4 Input Block 7b0c785e27e8ad3f8223207104725dd4 Output Block f69f2445df4f9b17ad2b417be66c3710 Plaintext f69f2445df4f9b17ad2b417be66c3710 F.1.3 ECB-AES192.Encrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b Block #1 Plaintext 6bc1bee22e409f96e93d7e117393172a Input Block 6bc1bee22e409f96e93d7e117393172a Output Block bd334f1d6e45f25ff712a214571fa5cc Ciphertext bd334f1d6e45f25ff712a214571fa5cc Block #2 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Input Block ae2d8a571e03ac9c9eb76fac45af8e51 Output Block 974104846d0ad3ad7734ecb3ecee4eef Ciphertext 974104846d0ad3ad7734ecb3ecee4eef Block #3 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Input Block 30c81c46a35ce411e5fbc1191a0a52ef Output Block ef7afd2270e2e60adce0ba2face6444e Ciphertext ef7afd2270e2e60adce0ba2face6444e Block #4 Plaintext f69f2445df4f9b17ad2b417be66c3710 Input Block f69f2445df4f9b17ad2b417be66c3710 Output Block 9a4b41ba738d6c72fb16691603c18e0e Ciphertext 9a4b41ba738d6c72fb16691603c18e0e F.1.4 ECB-AES192.Decrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b Block #1 Ciphertext bd334f1d6e45f25ff712a214571fa5cc Input Block bd334f1d6e45f25ff712a214571fa5cc Output Block 6bc1bee22e409f96e93d7e117393172a Plaintext 6bc1bee22e409f96e93d7e117393172a Block #2 Ciphertext 974104846d0ad3ad7734ecb3ecee4eef Input Block 974104846d0ad3ad7734ecb3ecee4eef Output Block ae2d8a571e03ac9c9eb76fac45af8e51 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 25 Block #3 Ciphertext ef7afd2270e2e60adce0ba2face6444e Input Block ef7afd2270e2e60adce0ba2face6444e Output Block 30c81c46a35ce411e5fbc1191a0a52ef Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Block #4 Ciphertext 9a4b41ba738d6c72fb16691603c18e0e Input Block 9a4b41ba738d6c72fb16691603c18e0e Output Block f69f2445df4f9b17ad2b417be66c3710 Plaintext f69f2445df4f9b17ad2b417be66c3710 F.1.5 ECB-AES256.Encrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 Block #1 Plaintext 6bc1bee22e409f96e93d7e117393172a Input Block 6bc1bee22e409f96e93d7e117393172a Output Block f3eed1bdb5d2a03c064b5a7e3db181f8 Ciphertext f3eed1bdb5d2a03c064b5a7e3db181f8 Block #2 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Input Block ae2d8a571e03ac9c9eb76fac45af8e51 Output Block 591ccb10d410ed26dc5ba74a31362870 Ciphertext 591ccb10d410ed26dc5ba74a31362870 Block #3 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Input Block 30c81c46a35ce411e5fbc1191a0a52ef Output Block b6ed21b99ca6f4f9f153e7b1beafed1d Ciphertext b6ed21b99ca6f4f9f153e7b1beafed1d Block #4 Plaintext f69f2445df4f9b17ad2b417be66c3710 Input Block f69f2445df4f9b17ad2b417be66c3710 Output Block 23304b7a39f9f3ff067d8d8f9e24ecc7 Ciphertext 23304b7a39f9f3ff067d8d8f9e24ecc7 F.1.6 ECB-AES256.Decrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 Block #1 Ciphertext f3eed1bdb5d2a03c064b5a7e3db181f8 Input Block f3eed1bdb5d2a03c064b5a7e3db181f8 Output Block 6bc1bee22e409f96e93d7e117393172a Plaintext 6bc1bee22e409f96e93d7e117393172a Block #2 Ciphertext 591ccb10d410ed26dc5ba74a31362870 Input Block 591ccb10d410ed26dc5ba74a31362870 Output Block ae2d8a571e03ac9c9eb76fac45af8e51 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Block #3 Ciphertext b6ed21b99ca6f4f9f153e7b1beafed1d Input Block b6ed21b99ca6f4f9f153e7b1beafed1d Output Block 30c81c46a35ce411e5fbc1191a0a52ef Plaintext 30c81c46a35ce411e5fbc1191a0a52ef 26 Block #4 Ciphertext 23304b7a39f9f3ff067d8d8f9e24ecc7 Input Block 23304b7a39f9f3ff067d8d8f9e24ecc7 Output Block f69f2445df4f9b17ad2b417be66c3710 Plaintext f69f2445df4f9b17ad2b417be66c3710 F.2 CBC Example Vectors F.2.1 CBC-AES128.Encrypt Key 2b7e151628aed2a6abf7158809cf4f3c IV 000102030405060708090a0b0c0d0e0f Block #1 Plaintext 6bc1bee22e409f96e93d7e117393172a Input Block 6bc0bce12a459991e134741a7f9e1925 Output Block 7649abac8119b246cee98e9b12e9197d Ciphertext 7649abac8119b246cee98e9b12e9197d Block #2 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Input Block d86421fb9f1a1eda505ee1375746972c Output Block 5086cb9b507219ee95db113a917678b2 Ciphertext 5086cb9b507219ee95db113a917678b2 Block #3 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Input Block 604ed7ddf32efdff7020d0238b7c2a5d Output Block 73bed6b8e3c1743b7116e69e22229516 Ciphertext 73bed6b8e3c1743b7116e69e22229516 Block #4 Plaintext f69f2445df4f9b17ad2b417be66c3710 Input Block 8521f2fd3c8eef2cdc3da7e5c44ea206 Output Block 3ff1caa1681fac09120eca307586e1a7 Ciphertext 3ff1caa1681fac09120eca307586e1a7 F.2.2 CBC-AES128.Decrypt Key 2b7e151628aed2a6abf7158809cf4f3c IV 000102030405060708090a0b0c0d0e0f Block #1 Ciphertext 7649abac8119b246cee98e9b12e9197d Input Block 7649abac8119b246cee98e9b12e9197d Output Block 6bc0bce12a459991e134741a7f9e1925 Plaintext 6bc1bee22e409f96e93d7e117393172a Block #2 Ciphertext 5086cb9b507219ee95db113a917678b2 Input Block 5086cb9b507219ee95db113a917678b2 Output Block d86421fb9f1a1eda505ee1375746972c Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Block #3 Ciphertext 73bed6b8e3c1743b7116e69e22229516 Input Block 73bed6b8e3c1743b7116e69e22229516 Output Block 604ed7ddf32efdff7020d0238b7c2a5d Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Block #4 Ciphertext 3ff1caa1681fac09120eca307586e1a7 Input Block 3ff1caa1681fac09120eca307586e1a7 27 Output Block 8521f2fd3c8eef2cdc3da7e5c44ea206 Plaintext f69f2445df4f9b17ad2b417be66c3710 F.2.3 CBC-AES192.Encrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b IV 000102030405060708090a0b0c0d0e0f Block #1 Plaintext 6bc1bee22e409f96e93d7e117393172a Input Block 6bc0bce12a459991e134741a7f9e1925 Output Block 4f021db243bc633d7178183a9fa071e8 Ciphertext 4f021db243bc633d7178183a9fa071e8 Block #2 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Input Block e12f97e55dbfcfa1efcf7796da0fffb9 Output Block b4d9ada9ad7dedf4e5e738763f69145a Ciphertext b4d9ada9ad7dedf4e5e738763f69145a Block #3 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Input Block 8411b1ef0e2109e5001cf96f256346b5 Output Block 571b242012fb7ae07fa9baac3df102e0 Ciphertext 571b242012fb7ae07fa9baac3df102e0 Block #4 Plaintext f69f2445df4f9b17ad2b417be66c3710 Input Block a1840065cdb4e1f7d282fbd7db9d35f0 Output Block 08b0e27988598881d920a9e64f5615cd Ciphertext 08b0e27988598881d920a9e64f5615cd F.2.4 CBC-AES192.Decrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b IV 000102030405060708090a0b0c0d0e0f Block #1 Ciphertext 4f021db243bc633d7178183a9fa071e8 Input Block 4f021db243bc633d7178183a9fa071e8 Output Block 6bc0bce12a459991e134741a7f9e1925 Plaintext 6bc1bee22e409f96e93d7e117393172a Block #2 Ciphertext b4d9ada9ad7dedf4e5e738763f69145a Input Block b4d9ada9ad7dedf4e5e738763f69145a Output Block e12f97e55dbfcfa1efcf7796da0fffb9 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Block #3 Ciphertext 571b242012fb7ae07fa9baac3df102e0 Input Block 571b242012fb7ae07fa9baac3df102e0 Output Block 8411b1ef0e2109e5001cf96f256346b5 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Block #4 Ciphertext 08b0e27988598881d920a9e64f5615cd Input Block 08b0e27988598881d920a9e64f5615cd Output Block a1840065cdb4e1f7d282fbd7db9d35f0 Plaintext f69f2445df4f9b17ad2b417be66c3710 F.2.5 CBC-AES256.Encrypt Key 603deb1015ca71be2b73aef0857d7781 28 1f352c073b6108d72d9810a30914dff4 IV 000102030405060708090a0b0c0d0e0f Block #1 Plaintext 6bc1bee22e409f96e93d7e117393172a Input Block 6bc0bce12a459991e134741a7f9e1925 Output Block f58c4c04d6e5f1ba779eabfb5f7bfbd6 Ciphertext f58c4c04d6e5f1ba779eabfb5f7bfbd6 Block #2 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Input Block 5ba1c653c8e65d26e929c4571ad47587 Output Block 9cfc4e967edb808d679f777bc6702c7d Ciphertext 9cfc4e967edb808d679f777bc6702c7d Block #3 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Input Block ac3452d0dd87649c8264b662dc7a7e92 Output Block 39f23369a9d9bacfa530e26304231461 Ciphertext 39f23369a9d9bacfa530e26304231461 Block #4 Plaintext f69f2445df4f9b17ad2b417be66c3710 Input Block cf6d172c769621d8081ba318e24f2371 Output Block b2eb05e2c39be9fcda6c19078c6a9d1b Ciphertext b2eb05e2c39be9fcda6c19078c6a9d1b F.2.6 CBC-AES256.Decrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 IV 000102030405060708090a0b0c0d0e0f Block #1 Ciphertext f58c4c04d6e5f1ba779eabfb5f7bfbd6 Input Block f58c4c04d6e5f1ba779eabfb5f7bfbd6 Output Block 6bc0bce12a459991e134741a7f9e1925 Plaintext 6bc1bee22e409f96e93d7e117393172a Block #2 Ciphertext 9cfc4e967edb808d679f777bc6702c7d Input Block 9cfc4e967edb808d679f777bc6702c7d Output Block 5ba1c653c8e65d26e929c4571ad47587 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Block #3 Ciphertext 39f23369a9d9bacfa530e26304231461 Input Block 39f23369a9d9bacfa530e26304231461 Output Block ac3452d0dd87649c8264b662dc7a7e92 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Block #4 Ciphertext b2eb05e2c39be9fcda6c19078c6a9d1b Input Block b2eb05e2c39be9fcda6c19078c6a9d1b Output Block cf6d172c769621d8081ba318e24f2371 Plaintext f69f2445df4f9b17ad2b417be66c3710 F.3 CFB Example Vectors F.3.1 CFB1-AES128.Encrypt Key 2b7e151628aed2a6abf7158809cf4f3c 000102030405060708090a0b0c0d0e0f 29 IV Segment #1 Input Block Output Block Plaintext Ciphertext Segment #2 Input Block Output Block Plaintext Ciphertext Segment #3 Input Block Output Block Plaintext Ciphertext Segment #4 Input Block Output Block Plaintext Ciphertext Segment #5 Input Block Output Block Plaintext Ciphertext Segment #6 Input Block Output Block Plaintext Ciphertext Segment #7 Input Block Output Block Plaintext Ciphertext Segment #8 Input Block Output Block Plaintext Ciphertext Segment #9 Input Block Output Block Plaintext Ciphertext Segment #10 Input Block Output Block Plaintext Ciphertext Segment #11 Input Block Output Block Plaintext 000102030405060708090a0b0c0d0e0f 50fe67cc996d32b6da0937e99bafec60 0 0 00020406080a0c0e10121416181a1c1e 19cf576c7596e702f298b35666955c79 1 1 0004080c1014181c2024282c3034383d 59e17759acd02b801fa321ea059e331f 1 1 0008101820283038404850586068707b 71f415b0cc109e8b0faa14ab740c22f4 0 0 00102030405060708090a0b0c0d0e0f6 3fb76d3d1048179964597a0f64d5adad 1 1 0020406080a0c0e10121416181a1c1ed 4c943b4bac54ab974e3e52326d29aaa1 0 0 004080c1014181c2024282c3034383da c94da41eb3d3acf1993a512ab1e8203f 1 0 008101820283038404850586068707b4 e07f5e98778f75dbb2691c3f582c3953 1 0 0102030405060708090a0b0c0d0e0f68 02ef5fc8961efcce8568bc0731262dc7 1 1 020406080a0c0e10121416181a1c1ed1 9f5a30367065efbe914b53698c8716b7 1 0 04080c1014181c2024282c3034383da2 d018cfb81d0580edbff955ed74d382db 0 30 Ciphertext 1 Segment #12 Input Block 08101820283038404850586068707b45 Output Block 81272ab351e08e0b695b94b8164d86f4 Plaintext 0 Ciphertext 1 Segment #13 Input Block 102030405060708090a0b0c0d0e0f68b Output Block 094d33f856483d3fa01ba94f7e5ab3e7 Plaintext 0 Ciphertext 0 Segment #14 Input Block 20406080a0c0e10121416181a1c1ed16 Output Block 609900ad61923c8c102cd8d0d7947a2c Plaintext 0 Ciphertext 0 Segment #15 Input Block 4080c1014181c2024282c3034383da2c Output Block 9e5a154de966ab4db9c88b22a398134e Plaintext 0 Ciphertext 1 Segment #16 Input Block 8101820283038404850586068707b459 Output Block 7fe16252b338bc4de3725c4156dfed20 Plaintext 1 Ciphertext 1 F.3.2 CFB1-AES128.Decrypt Key 2b7e151628aed2a6abf7158809cf4f3c IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block 50fe67cc996d32b6da0937e99bafec60 Ciphertext 0 Plaintext 0 Segment #2 Input Block 00020406080a0c0e10121416181a1c1e Output Block 19cf576c7596e702f298b35666955c79 Ciphertext 1 Plaintext 1 Segment #3 Input Block 0004080c1014181c2024282c3034383d Output Block 59e17759acd02b801fa321ea059e331f Ciphertext 1 Plaintext 1 Segment #4 Input Block 0008101820283038404850586068707b Output Block 71f415b0cc109e8b0faa14ab740c22f4 Ciphertext 0 Plaintext 0 Segment #5 Input Block 00102030405060708090a0b0c0d0e0f6 Output Block 3fb76d3d1048179964597a0f64d5adad 31 Ciphertext Plaintext Segment #6 Input Block Output Block Ciphertext Plaintext Segment #7 Input Block Output Block Ciphertext Plaintext Segment #8 Input Block Output Block Ciphertext Plaintext Segment #9 Input Block Output Block Ciphertext Plaintext Segment #10 Input Block Output Block Ciphertext Plaintext Segment #11 Input Block Output Block Ciphertext Plaintext Segment #12 Input Block Output Block Ciphertext Plaintext Segment #13 Input Block Output Block Ciphertext Plaintext Segment #14 Input Block Output Block Ciphertext Plaintext Segment #15 Input Block Output Block Ciphertext Plaintext Segment #16 Input Block 1 1 0020406080a0c0e10121416181a1c1ed 4c943b4bac54ab974e3e52326d29aaa1 0 0 004080c1014181c2024282c3034383da c94da41eb3d3acf1993a512ab1e8203f 0 1 008101820283038404850586068707b4 e07f5e98778f75dbb2691c3f582c3953 0 1 0102030405060708090a0b0c0d0e0f68 02ef5fc8961efcce8568bc0731262dc7 1 1 020406080a0c0e10121416181a1c1ed1 9f5a30367065efbe914b53698c8716b7 0 1 04080c1014181c2024282c3034383da2 d018cfb81d0580edbff955ed74d382db 1 0 08101820283038404850586068707b45 81272ab351e08e0b695b94b8164d86f4 1 0 102030405060708090a0b0c0d0e0f68b 094d33f856483d3fa01ba94f7e5ab3e7 0 0 20406080a0c0e10121416181a1c1ed16 609900ad61923c8c102cd8d0d7947a2c 0 0 4080c1014181c2024282c3034383da2c 9e5a154de966ab4db9c88b22a398134e 1 0 8101820283038404850586068707b459 32 Output Block 7fe16252b338bc4de3725c4156dfed20 Ciphertext 1 Plaintext 1 F.3.3 CFB1-AES192.Encrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block a609b38df3b1133dddff2718ba09565e Plaintext 0 Ciphertext 1 Segment #2 Input Block 00020406080a0c0e10121416181a1c1f Output Block a0e2bee6eb1734379bd4908be6a991a0 Plaintext 1 Ciphertext 0 Segment #3 Input Block 0004080c1014181c2024282c3034383e Output Block b1a1766bedec7ee3ba9cd3f34fbed4c6 Plaintext 1 Ciphertext 0 Segment #4 Input Block 0008101820283038404850586068707c Output Block b294ae5f393ae0179e6d3d8c45a7a4b9 Plaintext 0 Ciphertext 1 Segment #5 Input Block 00102030405060708090a0b0c0d0e0f9 Output Block f0f703ff5d0634aa8aee7f1e26aafca3 Plaintext 1 Ciphertext 0 Segment #6 Input Block 0020406080a0c0e10121416181a1c1f2 Output Block 4d67df426abdb8c89e7de9fb3069d8be Plaintext 0 Ciphertext 0 Segment #7 Input Block 004080c1014181c2024282c3034383e4 Output Block 30bc892338dfa10664118b9f4ba348d2 Plaintext 1 Ciphertext 1 Segment #8 Input Block 008101820283038404850586068707c9 Output Block 763ad8c63ed78d66452bb44c8bb7a8c8 Plaintext 1 Ciphertext 1 Segment #9 Input Block 0102030405060708090a0b0c0d0e0f93 Output Block bfc36f5cfbc1306859b48f8fa62a43df Plaintext 1 Ciphertext 0 Segment #10 33 Input Block 020406080a0c0e10121416181a1c1f26 Output Block 16e27adac112a0bf6a69c95cbdf584a3 Plaintext 1 Ciphertext 1 Segment #11 Input Block 04080c1014181c2024282c3034383e4d Output Block 1e9d21c3da3de9186251160045756ce0 Plaintext 0 Ciphertext 0 Segment #12 Input Block 08101820283038404850586068707c9a Output Block b836e0f661b51d8bd38c448e0e5a11bb Plaintext 0 Ciphertext 1 Segment #13 Input Block 102030405060708090a0b0c0d0e0f935 Output Block c5efcdd09dbb92d1faada8f6c9bab052 Plaintext 0 Ciphertext 1 Segment #14 Input Block 20406080a0c0e10121416181a1c1f26b Output Block 7c99710018d88e40bd4ac8f1b2bf4dbb Plaintext 0 Ciphertext 0 Segment #15 Input Block 4080c1014181c2024282c3034383e4d6 Output Block 173bcd8b4dad60ae6646813fdcb81f5b Plaintext 0 Ciphertext 0 Segment #16 Input Block 8101820283038404850586068707c9ac Output Block 09844c6d2272d148d5af1c7bf01bb439 Plaintext 1 Ciphertext 1 F.3.4 CFB1-AES192.Decrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block a609b38df3b1133dddff2718ba09565e Ciphertext 1 Plaintext 0 Segment #2 Input Block 00020406080a0c0e10121416181a1c1f Output Block a0e2bee6eb1734379bd4908be6a991a0 Ciphertext 0 Plaintext 1 Segment #3 Input Block 0004080c1014181c2024282c3034383e Output Block b1a1766bedec7ee3ba9cd3f34fbed4c6 Ciphertext 0 Plaintext 1 34 Segment #4 Input Block Output Block Ciphertext Plaintext Segment #5 Input Block Output Block Ciphertext Plaintext Segment #6 Input Block Output Block Ciphertext Plaintext Segment #7 Input Block Output Block Ciphertext Plaintext Segment #8 Input Block Output Block Ciphertext Plaintext Segment #9 Input Block Output Block Ciphertext Plaintext Segment #10 Input Block Output Block Ciphertext Plaintext Segment #11 Input Block Output Block Ciphertext Plaintext Segment #12 Input Block Output Block Ciphertext Plaintext Segment #13 Input Block Output Block Ciphertext Plaintext Segment #14 Input Block Output Block Ciphertext 0008101820283038404850586068707c b294ae5f393ae0179e6d3d8c45a7a4b9 1 0 00102030405060708090a0b0c0d0e0f9 f0f703ff5d0634aa8aee7f1e26aafca3 0 1 0020406080a0c0e10121416181a1c1f2 4d67df426abdb8c89e7de9fb3069d8be 0 0 004080c1014181c2024282c3034383e4 30bc892338dfa10664118b9f4ba348d2 1 1 008101820283038404850586068707c9 763ad8c63ed78d66452bb44c8bb7a8c8 1 1 0102030405060708090a0b0c0d0e0f93 bfc36f5cfbc1306859b48f8fa62a43df 0 1 020406080a0c0e10121416181a1c1f26 16e27adac112a0bf6a69c95cbdf584a3 1 1 04080c1014181c2024282c3034383e4d 1e9d21c3da3de9186251160045756ce0 0 0 08101820283038404850586068707c9a b836e0f661b51d8bd38c448e0e5a11bb 1 0 102030405060708090a0b0c0d0e0f935 c5efcdd09dbb92d1faada8f6c9bab052 1 0 20406080a0c0e10121416181a1c1f26b 7c99710018d88e40bd4ac8f1b2bf4dbb 0 35 Plaintext 0 Segment #15 Input Block 4080c1014181c2024282c3034383e4d6 Output Block 173bcd8b4dad60ae6646813fdcb81f5b Ciphertext 0 Plaintext 0 Segment #16 Input Block 8101820283038404850586068707c9ac Output Block 09844c6d2272d148d5af1c7bf01bb439 Ciphertext 1 Plaintext 1 F.3.5 CFB1-AES256.Encrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block b7bf3a5df43989dd97f0fa97ebce2f4a Plaintext 0 Ciphertext 1 Segment #2 Input Block 00020406080a0c0e10121416181a1c1f Output Block ee93d380e0f01117fffd78017599514a Plaintext 1 Ciphertext 0 Segment #3 Input Block 0004080c1014181c2024282c3034383e Output Block 857749898b3602aad91e699911de89b0 Plaintext 1 Ciphertext 0 Segment #4 Input Block 0008101820283038404850586068707c Output Block dce81c80810e2ba343a6bb402716b7a8 Plaintext 0 Ciphertext 1 Segment #5 Input Block 00102030405060708090a0b0c0d0e0f9 Output Block e5517bfcdccea00501350a601f754823 Plaintext 1 Ciphertext 0 Segment #6 Input Block 0020406080a0c0e10121416181a1c1f2 Output Block 15799c7f4081a78cc41f29955349c5a0 Plaintext 0 Ciphertext 0 Segment #7 Input Block 004080c1014181c2024282c3034383e4 Output Block 84d246bdb391f6a7979ff5ccb8467262 Plaintext 1 Ciphertext 0 Segment #8 Input Block 008101820283038404850586068707c8 36 Output Block bb9e05db9855a9e7e3837a648dd4c3b0 Plaintext 1 Ciphertext 0 Segment #9 Input Block 0102030405060708090a0b0c0d0e0f90 Output Block a413c5714f70287dfcd943004bf7ac8e Plaintext 1 Ciphertext 0 Segment #10 Input Block 020406080a0c0e10121416181a1c1f20 Output Block a7310abf87610d66edf6c892a84460d5 Plaintext 1 Ciphertext 0 Segment #11 Input Block 04080c1014181c2024282c3034383e40 Output Block 8aec6712d89bd147c83b51d787b11399 Plaintext 0 Ciphertext 1 Segment #12 Input Block 08101820283038404850586068707c81 Output Block 2ff05b620f68134f4ba92deffbfc93b2 Plaintext 0 Ciphertext 0 Segment #13 Input Block 102030405060708090a0b0c0d0e0f902 Output Block 819208afd5284316065a76bead028ad3 Plaintext 0 Ciphertext 1 Segment #14 Input Block 20406080a0c0e10121416181a1c1f205 Output Block 1914ed64b2115167ce2ca4c813da5245 Plaintext 0 Ciphertext 0 Segment #15 Input Block 4080c1014181c2024282c3034383e40a Output Block 638abae8724a954ae9e1e2e119deb6e1 Plaintext 0 Ciphertext 0 Segment #16 Input Block 8101820283038404850586068707c814 Output Block 2b4f488a3f958c52a3f1db2da938360e Plaintext 1 Ciphertext 1 F.3.6 CFB1-AES256.Decrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block b7bf3a5df43989dd97f0fa97ebce2f4a Ciphertext 1 Plaintext 0 37 IV Segment #2 Input Block Output Block Ciphertext Plaintext Segment #3 Input Block Output Block Ciphertext Plaintext Segment #4 Input Block Output Block Ciphertext Plaintext Segment #5 Input Block Output Block Ciphertext Plaintext Segment #6 Input Block Output Block Ciphertext Plaintext Segment #7 Input Block Output Block Ciphertext Plaintext Segment #8 Input Block Output Block Ciphertext Plaintext Segment #9 Input Block Output Block Ciphertext Plaintext Segment #10 Input Block Output Block Ciphertext Plaintext Segment #11 Input Block Output Block Ciphertext Plaintext Segment #12 Input Block Output Block Ciphertext 00020406080a0c0e10121416181a1c1f ee93d380e0f01117fffd78017599514a 0 1 0004080c1014181c2024282c3034383e 857749898b3602aad91e699911de89b0 0 1 0008101820283038404850586068707c dce81c80810e2ba343a6bb402716b7a8 1 0 00102030405060708090a0b0c0d0e0f9 e5517bfcdccea00501350a601f754823 0 1 0020406080a0c0e10121416181a1c1f2 15799c7f4081a78cc41f29955349c5a0 0 0 004080c1014181c2024282c3034383e4 84d246bdb391f6a7979ff5ccb8467262 0 1 008101820283038404850586068707c8 bb9e05db9855a9e7e3837a648dd4c3b0 0 1 0102030405060708090a0b0c0d0e0f90 a413c5714f70287dfcd943004bf7ac8e 0 1 020406080a0c0e10121416181a1c1f20 a7310abf87610d66edf6c892a84460d5 0 1 04080c1014181c2024282c3034383e40 8aec6712d89bd147c83b51d787b11399 1 0 08101820283038404850586068707c81 2ff05b620f68134f4ba92deffbfc93b2 0 38 Plaintext 0 Segment #13 Input Block 102030405060708090a0b0c0d0e0f902 Output Block 819208afd5284316065a76bead028ad3 Ciphertext 1 Plaintext 0 Segment #14 Input Block 20406080a0c0e10121416181a1c1f205 Output Block 1914ed64b2115167ce2ca4c813da5245 Ciphertext 0 Plaintext 0 Segment #15 Input Block 4080c1014181c2024282c3034383e40a Output Block 638abae8724a954ae9e1e2e119deb6e1 Ciphertext 0 Plaintext 0 Segment #16 Input Block 8101820283038404850586068707c814 Output Block 2b4f488a3f958c52a3f1db2da938360e Ciphertext 1 Plaintext 1 F.3.7 CFB8-AES128.Encrypt Key 2b7e151628aed2a6abf7158809cf4f3c IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block 50fe67cc996d32b6da0937e99bafec60 Plaintext 6b Ciphertext 3b Segment #2 Input Block 0102030405060708090a0b0c0d0e0f3b Output Block b8eb865a2b026381abb1d6560ed20f68 Plaintext c1 Ciphertext 79 Segment #3 Input Block 02030405060708090a0b0c0d0e0f3b79 Output Block fce6033b4edce64cbaed3f61ff5b927c Plaintext be Ciphertext 42 Segment #4 Input Block 030405060708090a0b0c0d0e0f3b7942 Output Block ae4e5e7ffe805f7a4395b180004f8ca8 Plaintext e2 Ciphertext 4c Segment #5 Input Block 0405060708090a0b0c0d0e0f3b79424c Output Block b205eb89445b62116f1deb988a81e6dd Plaintext 2e Ciphertext 9c Segment #6 Input Block 05060708090a0b0c0d0e0f3b79424c9c Output Block 4d21d456a5e239064fff4be0c0f85488 39 Plaintext Ciphertext Segment #7 Input Block Output Block Plaintext Ciphertext Segment #8 Input Block Output Block Plaintext Ciphertext Segment #9 Input Block Output Block Plaintext Ciphertext Segment #10 Input Block Output Block Plaintext Ciphertext Segment #11 Input Block Output Block Plaintext Ciphertext Segment #12 Input Block Output Block Plaintext Ciphertext Segment #13 Input Block Output Block Plaintext Ciphertext Segment #14 Input Block Output Block Plaintext Ciphertext Segment #15 Input Block Output Block Plaintext Ciphertext Segment #16 Input Block Output Block Plaintext Ciphertext Segment #17 Input Block 40 0d 060708090a0b0c0d0e0f3b79424c9c0d 4b2f5c3895b9efdc85ee0c5178c7fd33 9f d4 0708090a0b0c0d0e0f3b79424c9c0dd4 a0976d856da260a34104d1a80953db4c 96 36 08090a0b0c0d0e0f3b79424c9c0dd436 53674e5890a2c71b0f6a27a094e5808c e9 ba 090a0b0c0d0e0f3b79424c9c0dd436ba f34cd32ffed495f8bc8adba194eccb7a 3d ce 0a0b0c0d0e0f3b79424c9c0dd436bace e08cf2407d7ed676c9049586f1d48ba6 7e 9e 0b0c0d0e0f3b79424c9c0dd436bace9e 1f5c88a19b6ca28e99c9aeb8982a6dd8 11 0e 0c0d0e0f3b79424c9c0dd436bace9e0e a70e63df781cf395a208bd2365c8779b 73 d4 0d0e0f3b79424c9c0dd436bace9e0ed4 cbcfe8b3bcf9ac202ce18420013319ab 93 58 0e0f3b79424c9c0dd436bace9e0ed458 7d9fac6604b3c8c5b1f8c5a00956cf56 17 6a 0f3b79424c9c0dd436bace9e0ed4586a 65c3fa64bf0343986825c636f4a1efd2 2a 4f 3b79424c9c0dd436bace9e0ed4586a4f 40 Output Block 9cff5e5ff4f554d56c924b9d6a6de21d Plaintext ae Ciphertext 32 Segment #18 Input Block 79424c9c0dd436bace9e0ed4586a4f32 Output Block 946c3dc1584cc18400ecd8c6052c44b1 Plaintext 2d Ciphertext b9 F.3.8 CFB8-AES128.Decrypt Key 2b7e151628aed2a6abf7158809cf4f3c IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block 50fe67cc996d32b6da0937e99bafec60 Ciphertext 3b Plaintext 6b Segment #2 Input Block 0102030405060708090a0b0c0d0e0f3b Output Block b8eb865a2b026381abb1d6560ed20f68 Ciphertext 79 Plaintext c1 Segment #3 Input Block 02030405060708090a0b0c0d0e0f3b79 Output Block fce6033b4edce64cbaed3f61ff5b927c Ciphertext 42 Plaintext be Segment #4 Input Block 030405060708090a0b0c0d0e0f3b7942 Output Block ae4e5e7ffe805f7a4395b180004f8ca8 Ciphertext 4c Plaintext e2 Segment #5 Input Block 0405060708090a0b0c0d0e0f3b79424c Output Block b205eb89445b62116f1deb988a81e6dd Ciphertext 9c Plaintext 2e Segment #6 Input Block 05060708090a0b0c0d0e0f3b79424c9c Output Block 4d21d456a5e239064fff4be0c0f85488 Ciphertext 0d Plaintext 40 Segment #7 Input Block 060708090a0b0c0d0e0f3b79424c9c0d Output Block 4b2f5c3895b9efdc85ee0c5178c7fd33 Ciphertext d4 Plaintext 9f Segment #8 Input Block 0708090a0b0c0d0e0f3b79424c9c0dd4 Output Block a0976d856da260a34104d1a80953db4c Ciphertext 36 Plaintext 96 Segment #9 41 Input Block 08090a0b0c0d0e0f3b79424c9c0dd436 Output Block 53674e5890a2c71b0f6a27a094e5808c Ciphertext ba Plaintext e9 Segment #10 Input Block 090a0b0c0d0e0f3b79424c9c0dd436ba Output Block f34cd32ffed495f8bc8adba194eccb7a Ciphertext ce Plaintext 3d Segment #11 Input Block 0a0b0c0d0e0f3b79424c9c0dd436bace Output Block e08cf2407d7ed676c9049586f1d48ba6 Ciphertext 9e Plaintext 7e Segment #12 Input Block 0b0c0d0e0f3b79424c9c0dd436bace9e Output Block 1f5c88a19b6ca28e99c9aeb8982a6dd8 Ciphertext 0e Plaintext 11 Segment #13 Input Block 0c0d0e0f3b79424c9c0dd436bace9e0e Output Block a70e63df781cf395a208bd2365c8779b Ciphertext d4 Plaintext 73 Segment #14 Input Block 0d0e0f3b79424c9c0dd436bace9e0ed4 Output Block cbcfe8b3bcf9ac202ce18420013319ab Ciphertext 58 Plaintext 93 Segment #15 Input Block 0e0f3b79424c9c0dd436bace9e0ed458 Output Block 7d9fac6604b3c8c5b1f8c5a00956cf56 Ciphertext 6a Plaintext 17 Segment #16 Input Block 0f3b79424c9c0dd436bace9e0ed4586a Output Block 65c3fa64bf0343986825c636f4a1efd2 Ciphertext 4f Plaintext 2a Segment #17 Input Block 3b79424c9c0dd436bace9e0ed4586a4f Output Block 9cff5e5ff4f554d56c924b9d6a6de21d Ciphertext 32 Plaintext ae Segment #18 Input Block 79424c9c0dd436bace9e0ed4586a4f32 Output Block 946c3dc1584cc18400ecd8c6052c44b1 Ciphertext b9 Plaintext 2d F.3.9 CFB8-AES192.Encrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b 42 IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block Output Block Plaintext Ciphertext Segment #2 Input Block Output Block Plaintext Ciphertext Segment #3 Input Block Output Block Plaintext Ciphertext Segment #4 Input Block Output Block Plaintext Ciphertext Segment #5 Input Block Output Block Plaintext Ciphertext Segment #6 Input Block Output Block Plaintext Ciphertext Segment #7 Input Block Output Block Plaintext Ciphertext Segment #8 Input Block Output Block Plaintext Ciphertext Segment #9 Input Block Output Block Plaintext Ciphertext Segment #10 Input Block Output Block Plaintext Ciphertext Segment #11 Input Block Output Block 000102030405060708090a0b0c0d0e0f a609b38df3b1133dddff2718ba09565e 6b cd 0102030405060708090a0b0c0d0e0fcd 63c82e99e7289617c49e6851e082142a c1 a2 02030405060708090a0b0c0d0e0fcda2 ec40a5497264bfb4d6820aaae73f75af be 52 030405060708090a0b0c0d0e0fcda252 fc011a96afe968c32bae6495173a9154 e2 1e 0405060708090a0b0c0d0e0fcda2521e de019e09ac995ba46a42916ef77d8fe5 2e f0 05060708090a0b0c0d0e0fcda2521ef0 e980477efb7f896e07c4a2d527e7b537 40 a9 060708090a0b0c0d0e0fcda2521ef0a9 9a9a77b11709b36e08e9321ae8b1e539 9f 05 0708090a0b0c0d0e0fcda2521ef0a905 5ca1d192a780fbca1471e10588593c7c 96 ca 08090a0b0c0d0e0fcda2521ef0a905ca addb26efd21de4d002474c7748e0bc1d e9 44 090a0b0c0d0e0fcda2521ef0a905ca44 f0c410ad6512c5177a5ee40a60de01b8 3d cd 0a0b0c0d0e0fcda2521ef0a905ca44cd 7bbf71f2b4f5cf68f3c0c1b9235dbd53 43 Plaintext 7e Ciphertext 05 Segment #12 Input Block 0b0c0d0e0fcda2521ef0a905ca44cd05 Output Block 6dafb26e3c63b350811394b382e14d69 Plaintext 11 Ciphertext 7c Segment #13 Input Block 0c0d0e0fcda2521ef0a905ca44cd057c Output Block ccd6e25255a80e9bdbec9fbc26e5fad6 Plaintext 73 Ciphertext bf Segment #14 Input Block 0d0e0fcda2521ef0a905ca44cd057cbf Output Block 9e33550f6d47bda77f4f3108181ab21c Plaintext 93 Ciphertext 0d Segment #15 Input Block 0e0fcda2521ef0a905ca44cd057cbf0d Output Block 50b3eae29a6623fbef6d726dbda675a8 Plaintext 17 Ciphertext 47 Segment #16 Input Block 0fcda2521ef0a905ca44cd057cbf0d47 Output Block 8a2a57d1b9158539ef7ff42b33bf0a4a Plaintext 2a Ciphertext a0 Segment #17 Input Block cda2521ef0a905ca44cd057cbf0d47a0 Output Block c94e9102ac731d2f127b657d810ef5a8 Plaintext ae Ciphertext 67 Segment #18 Input Block a2521ef0a905ca44cd057cbf0d47a067 Output Block a765ed650568fbe386660def5f8d491d Plaintext 2d Ciphertext 8a F.3.10 CFB8-AES192.Decrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block a609b38df3b1133dddff2718ba09565e Ciphertext cd Plaintext 6b Segment #2 Input Block 0102030405060708090a0b0c0d0e0fcd Output Block 63c82e99e7289617c49e6851e082142a Ciphertext a2 Plaintext c1 Segment #3 Input Block 02030405060708090a0b0c0d0e0fcda2 44 Output Block Ciphertext Plaintext Segment #4 Input Block Output Block Ciphertext Plaintext Segment #5 Input Block Output Block Ciphertext Plaintext Segment #6 Input Block Output Block Ciphertext Plaintext Segment #7 Input Block Output Block Ciphertext Plaintext Segment #8 Input Block Output Block Ciphertext Plaintext Segment #9 Input Block Output Block Ciphertext Plaintext Segment #10 Input Block Output Block Ciphertext Plaintext Segment #11 Input Block Output Block Ciphertext Plaintext Segment #12 Input Block Output Block Ciphertext Plaintext Segment #13 Input Block Output Block Ciphertext Plaintext Segment #14 ec40a5497264bfb4d6820aaae73f75af 52 be 030405060708090a0b0c0d0e0fcda252 fc011a96afe968c32bae6495173a9154 1e e2 0405060708090a0b0c0d0e0fcda2521e de019e09ac995ba46a42916ef77d8fe5 f0 2e 05060708090a0b0c0d0e0fcda2521ef0 e980477efb7f896e07c4a2d527e7b537 a9 40 060708090a0b0c0d0e0fcda2521ef0a9 9a9a77b11709b36e08e9321ae8b1e539 05 9f 0708090a0b0c0d0e0fcda2521ef0a905 5ca1d192a780fbca1471e10588593c7c ca 96 08090a0b0c0d0e0fcda2521ef0a905ca addb26efd21de4d002474c7748e0bc1d 44 e9 090a0b0c0d0e0fcda2521ef0a905ca44 f0c410ad6512c5177a5ee40a60de01b8 cd 3d 0a0b0c0d0e0fcda2521ef0a905ca44cd 7bbf71f2b4f5cf68f3c0c1b9235dbd53 05 7e 0b0c0d0e0fcda2521ef0a905ca44cd05 6dafb26e3c63b350811394b382e14d69 7c 11 0c0d0e0fcda2521ef0a905ca44cd057c ccd6e25255a80e9bdbec9fbc26e5fad6 bf 73 45 Input Block 0d0e0fcda2521ef0a905ca44cd057cbf Output Block 9e33550f6d47bda77f4f3108181ab21c Ciphertext 0d Plaintext 93 Segment #15 Input Block 0e0fcda2521ef0a905ca44cd057cbf0d Output Block 50b3eae29a6623fbef6d726dbda675a8 Ciphertext 47 Plaintext 17 Segment #16 Input Block 0fcda2521ef0a905ca44cd057cbf0d47 Output Block 8a2a57d1b9158539ef7ff42b33bf0a4a Ciphertext a0 Plaintext 2a Segment #17 Input Block cda2521ef0a905ca44cd057cbf0d47a0 Output Block c94e9102ac731d2f127b657d810ef5a8 Ciphertext 67 Plaintext ae Segment #18 Input Block a2521ef0a905ca44cd057cbf0d47a067 Output Block a765ed650568fbe386660def5f8d491d Ciphertext 8a Plaintext 2d F.3.11 CFB8-AES256.Encrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block b7bf3a5df43989dd97f0fa97ebce2f4a Plaintext 6b Ciphertext dc Segment #2 Input Block 0102030405060708090a0b0c0d0e0fdc Output Block ded5faadb1068af80e774684b9f84870 Plaintext c1 Ciphertext 1f Segment #3 Input Block 02030405060708090a0b0c0d0e0fdc1f Output Block a41e327e5273366ce9403cdbdb92c1cc Plaintext be Ciphertext 1a Segment #4 Input Block 030405060708090a0b0c0d0e0fdc1f1a Output Block 67938ae7d34df4ec2c0aec33eb98318f Plaintext e2 Ciphertext 85 Segment #5 Input Block 0405060708090a0b0c0d0e0fdc1f1a85 Output Block 0e8f2e31efff615d3c93946609808c37 Plaintext 2e 46 Ciphertext Segment #6 Input Block Output Block Plaintext Ciphertext Segment #7 Input Block Output Block Plaintext Ciphertext Segment #8 Input Block Output Block Plaintext Ciphertext Segment #9 Input Block Output Block Plaintext Ciphertext Segment #10 Input Block Output Block Plaintext Ciphertext Segment #11 Input Block Output Block Plaintext Ciphertext Segment #12 Input Block Output Block Plaintext Ciphertext Segment #13 Input Block Output Block Plaintext Ciphertext Segment #14 Input Block Output Block Plaintext Ciphertext Segment #15 Input Block Output Block Plaintext Ciphertext Segment #16 Input Block Output Block 20 05060708090a0b0c0d0e0fdc1f1a8520 e648bb37a95c94c72784162a79dfe306 40 a6 060708090a0b0c0d0e0fdc1f1a8520a6 d278f3147290fc5dd0b7d2e82764a1fd 9f 4d 0708090a0b0c0d0e0fdc1f1a8520a64d 2388d255a3e8a8059675e3a7de19dceb 96 b5 08090a0b0c0d0e0fdc1f1a8520a64db5 b6b8008f6c6dc2d6144641ed2023f0f5 e9 5f 090a0b0c0d0e0fdc1f1a8520a64db55f f18f88a7aa3e3a6167dd93fb1137713a 3d cc 0a0b0c0d0e0fdc1f1a8520a64db55fcc f46c5e67bff7c070b26c0318c52d0ccd 7e 8a 0b0c0d0e0fdc1f1a8520a64db55fcc8a d4dceae622f8f21d27375d8c2c5f9fba 11 c5 0c0d0e0fdc1f1a8520a64db55fcc8ac5 27e9e0d0a016709cd3ae0b5a9a242e31 73 54 0d0e0fdc1f1a8520a64db55fcc8ac554 17f69d50ce64ba0d085de70b9030bbb2 93 84 0e0fdc1f1a8520a64db55fcc8ac55484 59106ee400d18e104337669628c33cdd 17 4e 0fdc1f1a8520a64db55fcc8ac554844e a29c6ac87e2245ec0796772c1f5312a8 47 Plaintext 2a Ciphertext 88 Segment #17 Input Block dc1f1a8520a64db55fcc8ac554844e88 Output Block 397b98fa2ec0ff8cc0cd821909551c9e Plaintext ae Ciphertext 97 Segment #18 Input Block 1f1a8520a64db55fcc8ac554844e8897 Output Block 2d2d6fe9aef72f7b914b623a9c7abd54 Plaintext 2d Ciphertext 00 F.3.12 CFB8-AES256.Decrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block b7bf3a5df43989dd97f0fa97ebce2f4a Ciphertext dc Plaintext 6b Segment #2 Input Block 0102030405060708090a0b0c0d0e0fdc Output Block ded5faadb1068af80e774684b9f84870 Ciphertext 1f Plaintext c1 Segment #3 Input Block 02030405060708090a0b0c0d0e0fdc1f Output Block a41e327e5273366ce9403cdbdb92c1cc Ciphertext 1a Plaintext be Segment #4 Input Block 030405060708090a0b0c0d0e0fdc1f1a Output Block 67938ae7d34df4ec2c0aec33eb98318f Ciphertext 85 Plaintext e2 Segment #5 Input Block 0405060708090a0b0c0d0e0fdc1f1a85 Output Block 0e8f2e31efff615d3c93946609808c37 Ciphertext 20 Plaintext 2e Segment #6 Input Block 05060708090a0b0c0d0e0fdc1f1a8520 Output Block e648bb37a95c94c72784162a79dfe306 Ciphertext a6 Plaintext 40 Segment #7 Input Block 060708090a0b0c0d0e0fdc1f1a8520a6 Output Block d278f3147290fc5dd0b7d2e82764a1fd Ciphertext 4d Plaintext 9f Segment #8 48 Input Block Output Block Ciphertext Plaintext Segment #9 Input Block Output Block Ciphertext Plaintext Segment #10 Input Block Output Block Ciphertext Plaintext Segment #11 Input Block Output Block Ciphertext Plaintext Segment #12 Input Block Output Block Ciphertext Plaintext Segment #13 Input Block Output Block Ciphertext Plaintext Segment #14 Input Block Output Block Ciphertext Plaintext Segment #15 Input Block Output Block Ciphertext Plaintext Segment #16 Input Block Output Block Ciphertext Plaintext Segment #17 Input Block Output Block Ciphertext Plaintext Segment #18 Input Block Output Block Ciphertext Plaintext 0708090a0b0c0d0e0fdc1f1a8520a64d 2388d255a3e8a8059675e3a7de19dceb b5 96 08090a0b0c0d0e0fdc1f1a8520a64db5 b6b8008f6c6dc2d6144641ed2023f0f5 5f e9 090a0b0c0d0e0fdc1f1a8520a64db55f f18f88a7aa3e3a6167dd93fb1137713a cc 3d 0a0b0c0d0e0fdc1f1a8520a64db55fcc f46c5e67bff7c070b26c0318c52d0ccd 8a 7e 0b0c0d0e0fdc1f1a8520a64db55fcc8a d4dceae622f8f21d27375d8c2c5f9fba c5 11 0c0d0e0fdc1f1a8520a64db55fcc8ac5 27e9e0d0a016709cd3ae0b5a9a242e31 54 73 0d0e0fdc1f1a8520a64db55fcc8ac554 17f69d50ce64ba0d085de70b9030bbb2 84 93 0e0fdc1f1a8520a64db55fcc8ac55484 59106ee400d18e104337669628c33cdd 4e 17 0fdc1f1a8520a64db55fcc8ac554844e a29c6ac87e2245ec0796772c1f5312a8 88 2a dc1f1a8520a64db55fcc8ac554844e88 397b98fa2ec0ff8cc0cd821909551c9e 97 ae 1f1a8520a64db55fcc8ac554844e8897 2d2d6fe9aef72f7b914b623a9c7abd54 00 2d 49 F.3.13 CFB128-AES128.Encrypt Key 2b7e151628aed2a6abf7158809cf4f3c IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block 50fe67cc996d32b6da0937e99bafec60 Plaintext 6bc1bee22e409f96e93d7e117393172a Ciphertext 3b3fd92eb72dad20333449f8e83cfb4a Segment #2 Input Block 3b3fd92eb72dad20333449f8e83cfb4a Output Block 668bcf60beb005a35354a201dab36bda Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Ciphertext c8a64537a0b3a93fcde3cdad9f1ce58b Segment #3 Input Block c8a64537a0b3a93fcde3cdad9f1ce58b Output Block 16bd032100975551547b4de89daea630 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Ciphertext 26751f67a3cbb140b1808cf187a4f4df Segment #4 Input Block 26751f67a3cbb140b1808cf187a4f4df Output Block 36d42170a312871947ef8714799bc5f6 Plaintext f69f2445df4f9b17ad2b417be66c3710 Ciphertext c04b05357c5d1c0eeac4c66f9ff7f2e6 F.3.14 CFB128-AES128.Decrypt Key 2b7e151628aed2a6abf7158809cf4f3c IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block 50fe67cc996d32b6da0937e99bafec60 Ciphertext 3b3fd92eb72dad20333449f8e83cfb4a Plaintext 6bc1bee22e409f96e93d7e117393172a Segment #2 Input Block 3b3fd92eb72dad20333449f8e83cfb4a Output Block 668bcf60beb005a35354a201dab36bda Ciphertext c8a64537a0b3a93fcde3cdad9f1ce58b Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Segment #3 Input Block c8a64537a0b3a93fcde3cdad9f1ce58b Output Block 16bd032100975551547b4de89daea630 Ciphertext 26751f67a3cbb140b1808cf187a4f4df Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Segment #4 Input Block 26751f67a3cbb140b1808cf187a4f4df Output Block 36d42170a312871947ef8714799bc5f6 Ciphertext c04b05357c5d1c0eeac4c66f9ff7f2e6 Plaintext f69f2445df4f9b17ad2b417be66c3710 F.3.15 CFB128-AES192.Encrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b 000102030405060708090a0b0c0d0e0f Segment #1 50 IV Input Block 000102030405060708090a0b0c0d0e0f Output Block a609b38df3b1133dddff2718ba09565e Plaintext 6bc1bee22e409f96e93d7e117393172a Ciphertext cdc80d6fddf18cab34c25909c99a4174 Segment #2 Input Block cdc80d6fddf18cab34c25909c99a4174 Output Block c9e3f5289f149abd08ad44dc52b2b32b Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Ciphertext 67ce7f7f81173621961a2b70171d3d7a Segment #3 Input Block 67ce7f7f81173621961a2b70171d3d7a Output Block 1ed6965b76c76ca02d1dcef404f09626 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Ciphertext 2e1e8a1dd59b88b1c8e60fed1efac4c9 Segment #4 Input Block 2e1e8a1dd59b88b1c8e60fed1efac4c9 Output Block 36c0bbd976ccd4b7ef85cec1be273eef Plaintext f69f2445df4f9b17ad2b417be66c3710 Ciphertext c05f9f9ca9834fa042ae8fba584b09ff F.3.16 CFB128-AES192.Decrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block a609b38df3b1133dddff2718ba09565e Ciphertext cdc80d6fddf18cab34c25909c99a4174 Plaintext 6bc1bee22e409f96e93d7e117393172a Segment #2 Input Block cdc80d6fddf18cab34c25909c99a4174 Output Block c9e3f5289f149abd08ad44dc52b2b32b Ciphertext 67ce7f7f81173621961a2b70171d3d7a Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Segment #3 Input Block 67ce7f7f81173621961a2b70171d3d7a Output Block 1ed6965b76c76ca02d1dcef404f09626 Ciphertext 2e1e8a1dd59b88b1c8e60fed1efac4c9 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Segment #4 Input Block 2e1e8a1dd59b88b1c8e60fed1efac4c9 Output Block 36c0bbd976ccd4b7ef85cec1be273eef Ciphertext c05f9f9ca9834fa042ae8fba584b09ff Plaintext f69f2445df4f9b17ad2b417be66c3710 F.3.17 CFB128-AES256.Encrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block b7bf3a5df43989dd97f0fa97ebce2f4a Plaintext 6bc1bee22e409f96e93d7e117393172a Ciphertext dc7e84bfda79164b7ecd8486985d3860 51 Segment #2 Input Block dc7e84bfda79164b7ecd8486985d3860 Output Block 97d26743252b1d54aca653cf744ace2a Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Ciphertext 39ffed143b28b1c832113c6331e5407b Segment #3 Input Block 39ffed143b28b1c832113c6331e5407b Output Block efd80f62b6b9af8344c511b13c70b016 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Ciphertext df10132415e54b92a13ed0a8267ae2f9 Segment #4 Input Block df10132415e54b92a13ed0a8267ae2f9 Output Block 833ca131c5f655ef8d1a2346b3ddd361 Plaintext f69f2445df4f9b17ad2b417be66c3710 Ciphertext 75a385741ab9cef82031623d55b1e471 F.3.18 CFB128-AES256.Decrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 IV 000102030405060708090a0b0c0d0e0f Segment #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block b7bf3a5df43989dd97f0fa97ebce2f4a Ciphertext dc7e84bfda79164b7ecd8486985d3860 Plaintext 6bc1bee22e409f96e93d7e117393172a Segment #2 Input Block dc7e84bfda79164b7ecd8486985d3860 Output Block 97d26743252b1d54aca653cf744ace2a Ciphertext 39ffed143b28b1c832113c6331e5407b Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Segment #3 Input Block 39ffed143b28b1c832113c6331e5407b Output Block efd80f62b6b9af8344c511b13c70b016 Ciphertext df10132415e54b92a13ed0a8267ae2f9 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Segment #4 Input Block df10132415e54b92a13ed0a8267ae2f9 Output Block 833ca131c5f655ef8d1a2346b3ddd361 Ciphertext 75a385741ab9cef82031623d55b1e471 Plaintext f69f2445df4f9b17ad2b417be66c3710 F.4 OFB Example Vectors F.4.1 OFB-AES128.Encrypt Key 2b7e151628aed2a6abf7158809cf4f3c IV 000102030405060708090a0b0c0d0e0f Block #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block 50fe67cc996d32b6da0937e99bafec60 Plaintext 6bc1bee22e409f96e93d7e117393172a Ciphertext 3b3fd92eb72dad20333449f8e83cfb4a Block #2 Input Block 50fe67cc996d32b6da0937e99bafec60 52 Output Block d9a4dada0892239f6b8b3d7680e15674 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Ciphertext 7789508d16918f03f53c52dac54ed825 Block #3 Input Block d9a4dada0892239f6b8b3d7680e15674 Output Block a78819583f0308e7a6bf36b1386abf23 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Ciphertext 9740051e9c5fecf64344f7a82260edcc Block #4 Input Block a78819583f0308e7a6bf36b1386abf23 Output Block c6d3416d29165c6fcb8e51a227ba994e Plaintext f69f2445df4f9b17ad2b417be66c3710 Ciphertext 304c6528f659c77866a510d9c1d6ae5e F.4.2 OFB-AES128.Decrypt Key 2b7e151628aed2a6abf7158809cf4f3c IV 000102030405060708090a0b0c0d0e0f Block #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block 50fe67cc996d32b6da0937e99bafec60 Ciphertext 3b3fd92eb72dad20333449f8e83cfb4a Plaintext 6bc1bee22e409f96e93d7e117393172a Block #2 Input Block 50fe67cc996d32b6da0937e99bafec60 Output Block d9a4dada0892239f6b8b3d7680e15674 Ciphertext 7789508d16918f03f53c52dac54ed825 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Block #3 Input Block d9a4dada0892239f6b8b3d7680e15674 Output Block a78819583f0308e7a6bf36b1386abf23 Ciphertext 9740051e9c5fecf64344f7a82260edcc Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Block #4 Input Block a78819583f0308e7a6bf36b1386abf23 Output Block c6d3416d29165c6fcb8e51a227ba994e Ciphertext 304c6528f659c77866a510d9c1d6ae5e Plaintext f69f2445df4f9b17ad2b417be66c3710 F.4.3 OFB-AES192.Encrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b IV 000102030405060708090a0b0c0d0e0f Block #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block a609b38df3b1133dddff2718ba09565e Plaintext 6bc1bee22e409f96e93d7e117393172a Ciphertext cdc80d6fddf18cab34c25909c99a4174 Block #2 Input Block a609b38df3b1133dddff2718ba09565e Output Block 52ef01da52602fe0975f78ac84bf8a50 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Ciphertext fcc28b8d4c63837c09e81700c1100401 Block #3 Input Block 52ef01da52602fe0975f78ac84bf8a50 53 Output Block bd5286ac63aabd7eb067ac54b553f71d Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Ciphertext 8d9a9aeac0f6596f559c6d4daf59a5f2 Block #4 Input Block bd5286ac63aabd7eb067ac54b553f71d Output Block 9b00044d8885f729318713303fc0fe3a Plaintext f69f2445df4f9b17ad2b417be66c3710 Ciphertext 6d9f200857ca6c3e9cac524bd9acc92a F.4.4 OFB-AES192.Decrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b IV 000102030405060708090a0b0c0d0e0f Block #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block a609b38df3b1133dddff2718ba09565e Ciphertext cdc80d6fddf18cab34c25909c99a4174 Plaintext 6bc1bee22e409f96e93d7e117393172a Block #2 Input Block a609b38df3b1133dddff2718ba09565e Output Block 52ef01da52602fe0975f78ac84bf8a50 Ciphertext fcc28b8d4c63837c09e81700c1100401 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Block #3 Input Block 52ef01da52602fe0975f78ac84bf8a50 Output Block bd5286ac63aabd7eb067ac54b553f71d Ciphertext 8d9a9aeac0f6596f559c6d4daf59a5f2 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Block #4 Input Block bd5286ac63aabd7eb067ac54b553f71d Output Block 9b00044d8885f729318713303fc0fe3a Ciphertext 6d9f200857ca6c3e9cac524bd9acc92a Plaintext f69f2445df4f9b17ad2b417be66c3710 F.4.5 OFB-AES256.Encrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 IV 000102030405060708090a0b0c0d0e0f Block #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block b7bf3a5df43989dd97f0fa97ebce2f4a Plaintext 6bc1bee22e409f96e93d7e117393172a Ciphertext dc7e84bfda79164b7ecd8486985d3860 Block #2 Input Block b7bf3a5df43989dd97f0fa97ebce2f4a Output Block e1c656305ed1a7a6563805746fe03edc Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Ciphertext 4febdc6740d20b3ac88f6ad82a4fb08d Block #3 Input Block e1c656305ed1a7a6563805746fe03edc Output Block 41635be625b48afc1666dd42a09d96e7 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Ciphertext 71ab47a086e86eedf39d1c5bba97c408 Block #4 54 Input Block 41635be625b48afc1666dd42a09d96e7 Output Block f7b93058b8bce0fffea41bf0012cd394 Plaintext f69f2445df4f9b17ad2b417be66c3710 Ciphertext 0126141d67f37be8538f5a8be740e484 F.4.6 OFB-AES256.Decrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 IV 000102030405060708090a0b0c0d0e0f Block #1 Input Block 000102030405060708090a0b0c0d0e0f Output Block b7bf3a5df43989dd97f0fa97ebce2f4a Ciphertext dc7e84bfda79164b7ecd8486985d3860 Plaintext 6bc1bee22e409f96e93d7e117393172a Block #2 Input Block b7bf3a5df43989dd97f0fa97ebce2f4a Output Block e1c656305ed1a7a6563805746fe03edc Ciphertext 4febdc6740d20b3ac88f6ad82a4fb08d Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Block #3 Input Block e1c656305ed1a7a6563805746fe03edc Output Block 41635be625b48afc1666dd42a09d96e7 Ciphertext 71ab47a086e86eedf39d1c5bba97c408 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Block #4 Input Block 41635be625b48afc1666dd42a09d96e7 Output Block f7b93058b8bce0fffea41bf0012cd394 Ciphertext 0126141d67f37be8538f5a8be740e484 Plaintext f69f2445df4f9b17ad2b417be66c3710 F.5 CTR Example Vectors F.5.1 CTR-AES128.Encrypt Key 2b7e151628aed2a6abf7158809cf4f3c Init. Counter f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff Block #1 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff Output Block ec8cdf7398607cb0f2d21675ea9ea1e4 Plaintext 6bc1bee22e409f96e93d7e117393172a Ciphertext 874d6191b620e3261bef6864990db6ce Block #2 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff00 Output Block 362b7c3c6773516318a077d7fc5073ae Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Ciphertext 9806f66b7970fdff8617187bb9fffdff Block #3 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff01 Output Block 6a2cc3787889374fbeb4c81b17ba6c44 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Ciphertext 5ae4df3edbd5d35e5b4f09020db03eab Block #4 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff02 Output Block e89c399ff0f198c6d40a31db156cabfe 55 Plaintext f69f2445df4f9b17ad2b417be66c3710 Ciphertext 1e031dda2fbe03d1792170a0f3009cee F.5.2 CTR-AES128.Decrypt Key 2b7e151628aed2a6abf7158809cf4f3c Init. Counter f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff Block #1 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff Output Block ec8cdf7398607cb0f2d21675ea9ea1e4 Ciphertext 874d6191b620e3261bef6864990db6ce Plaintext 6bc1bee22e409f96e93d7e117393172a Block #2 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff00 Output Block 362b7c3c6773516318a077d7fc5073ae Ciphertext 9806f66b7970fdff8617187bb9fffdff Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Block #3 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff01 Output Block 6a2cc3787889374fbeb4c81b17ba6c44 Ciphertext 5ae4df3edbd5d35e5b4f09020db03eab Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Block #4 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff02 Output Block e89c399ff0f198c6d40a31db156cabfe Ciphertext 1e031dda2fbe03d1792170a0f3009cee Plaintext f69f2445df4f9b17ad2b417be66c3710 F.5.3 CTR-AES192.Encrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b Init. Counter f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff Block #1 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff Output Block 717d2dc639128334a6167a488ded7921 Plaintext 6bc1bee22e409f96e93d7e117393172a Ciphertext 1abc932417521ca24f2b0459fe7e6e0b Block #2 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff00 Output Block a72eb3bb14a556734b7bad6ab16100c5 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Ciphertext 090339ec0aa6faefd5ccc2c6f4ce8e94 Block #3 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff01 Output Block 2efeae2d72b722613446dc7f4c2af918 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Ciphertext 1e36b26bd1ebc670d1bd1d665620abf7 Block #4 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff02 Output Block b9e783b30dd7924ff7bc9b97beaa8740 Plaintext f69f2445df4f9b17ad2b417be66c3710 Ciphertext 4f78a7f6d29809585a97daec58c6b050 56 F.5.4 CTR-AES192.Decrypt Key 8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b Init. Counter f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff Block #1 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff Output Block 717d2dc639128334a6167a488ded7921 Ciphertext 1abc932417521ca24f2b0459fe7e6e0b Plaintext 6bc1bee22e409f96e93d7e117393172a Block #2 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff00 Output Block a72eb3bb14a556734b7bad6ab16100c5 Ciphertext 090339ec0aa6faefd5ccc2c6f4ce8e94 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Block #3 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff01 Output Block 2efeae2d72b722613446dc7f4c2af918 Ciphertext 1e36b26bd1ebc670d1bd1d665620abf7 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Block #4 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff02 Output Block b9e783b30dd7924ff7bc9b97beaa8740 Ciphertext 4f78a7f6d29809585a97daec58c6b050 Plaintext f69f2445df4f9b17ad2b417be66c3710 F.5.5 CTR-AES256.Encrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 Init. Counter f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff Block #1 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff Output Block 0bdf7df1591716335e9a8b15c860c502 Plaintext 6bc1bee22e409f96e93d7e117393172a Ciphertext 601ec313775789a5b7a7f504bbf3d228 Block #2 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff00 Output Block 5a6e699d536119065433863c8f657b94 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Ciphertext f443e3ca4d62b59aca84e990cacaf5c5 Block #3 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff01 Output Block 1bc12c9c01610d5d0d8bd6a3378eca62 Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Ciphertext 2b0930daa23de94ce87017ba2d84988d Block #4 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff02 Output Block 2956e1c8693536b1bee99c73a31576b6 Plaintext f69f2445df4f9b17ad2b417be66c3710 Ciphertext dfc9c58db67aada613c2dd08457941a6 F.5.6 CTR-AES256.Decrypt Key 603deb1015ca71be2b73aef0857d7781 1f352c073b6108d72d9810a30914dff4 Init. Counter f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff 57 Block #1 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff Output Block 0bdf7df1591716335e9a8b15c860c502 Ciphertext 601ec313775789a5b7a7f504bbf3d228 Plaintext 6bc1bee22e409f96e93d7e117393172a Block #2 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff00 Output Block 5a6e699d536119065433863c8f657b94 Ciphertext f443e3ca4d62b59aca84e990cacaf5c5 Plaintext ae2d8a571e03ac9c9eb76fac45af8e51 Block #3 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff01 Output Block 1bc12c9c01610d5d0d8bd6a3378eca62 Ciphertext 2b0930daa23de94ce87017ba2d84988d Plaintext 30c81c46a35ce411e5fbc1191a0a52ef Block #4 Input Block f0f1f2f3f4f5f6f7f8f9fafbfcfdff02 Output Block 2956e1c8693536b1bee99c73a31576b6 Ciphertext dfc9c58db67aada613c2dd08457941a6 Plaintext f69f2445df4f9b17ad2b417be66c3710 58 Appendix G: References [1] American National Standard for Financial Services X9.52-1998, “Triple Data Encryption Algorithm Modes of Operation.” American Bankers Association, Washington, D.C., July 29, 1998. [2] FIPS Publication 197, “ Advanced Encryption Standard (AES). ” U.S. DoC/NIST, November 26, 2001. [3] FIPS Publication 46-3, “Data Encryption Standard (DES).” U.S. DoC/NIST, October 25, 1999. [4] FIPS Publication 81, “DES Modes of Operation.” U.S. DoC/NIST, December 1980. [5] A. Menezes, P. van Oorschot, and S. Vanstone, “Handbook of Applied Cryptography. ” CRC Press, New York, 1997. 59
ols2007v2-pages-21-34	The new ext4 ﬁlesystem: current status and future plans Avantika Mathur, Mingming Cao, Suparna Bhattacharya IBM Linux Technology Center [email protected], [email protected], [email protected] Andreas Dilger, Alex Tomas Cluster Filesystem Inc. [email protected], [email protected] Laurent Vivier Bull S.A.S. [email protected] Abstract Ext3 has been the most widely used general Linux R⃝ ﬁlesystem for many years. In keeping with increasing disk capacities and state-of-the-art feature requirements, the next generation of the ext3 ﬁlesystem, ext4, was cre- ated last year. This new ﬁlesystem incorporates scalabil- ity and performance enhancements for supporting large ﬁlesystems, while maintaining reliability and stability. Ext4 will be suitable for a larger variety of workloads and is expected to replace ext3 as the “Linux ﬁlesys- tem.” In this paper we will ﬁrst discuss the reasons for start- ing the ext4 ﬁlesystem, then explore the enhanced ca- pabilities currently available and planned for ext4, dis- cuss methods for migrating between ext3 and ext4, and ﬁnally compare ext4 and other ﬁlesystem performance on three classic ﬁlesystem benchmarks. 1 Introduction Ext3 has been a very popular Linux ﬁlesystem due to its reliability, rich feature set, relatively good performance, and strong compatibility between versions. The conser- vative design of ext3 has given it the reputation of being stable and robust, but has also limited its ability to scale and perform well on large conﬁgurations. With the pressure of increasing capabilities of new hard- ware and online resizing support in ext3, the require- ment to address ext3 scalability and performance is more urgent than ever. One of the outstanding limits faced by ext3 today is the 16 TB maximum ﬁlesystem size. Enterprise workloads are already approaching this limit, and with disk capacities doubling every year and 1 TB hard disks easily available in stores, it will soon be hit by desktop users as well. To address this limit, in August 2006, we posted a series of patches introducing two key features to ext3: larger ﬁlesystem capacity and extents mapping. The patches unavoidably change the on-disk format and break for- wards compatibility. In order to maintain the stability of ext3 for its massive user base, we decided to branch to ext4 from ext3. The primary goal of this new ﬁlesystem is to address scalability, performance, and reliability issues faced by ext3. A common question is why not use XFS or start an entirely new ﬁlesystem from scratch? We want to give the large number of ext3 users the opportunity to easily upgrade their ﬁlesystem, as was done from ext2 to ext3. Also, there has been considerable investment in the capabilities, robustness, and reliability of ext3 and e2fsck. Ext4 developers can take advantage of this pre- vious work, and focus on adding advanced features and delivering a new scalable enterprise-ready ﬁlesystem in a short time frame. Thus, ext4 was born. The new ﬁlesystem has been in mainline Linux since version 2.6.19. As of the writing of this paper, the ﬁlesystem is marked as developmen- tal, titled ext4dev, explicitly warning users that it is not ready for production use. Currently, extents and 48-bit block numbers are included in ext4, but there are many new ﬁlesystem features in the roadmap that will be dis- cussed throughout this paper. The current ext4 develop- ment git tree is hosted at git://git.kernel.org/ • 21 • 22 • The new ext4 ﬁlesystem: current status and future plans pub/scm/linux/kernel/git/tytso/ext4. Up- to-date ext4 patches and feature discussions can be found at the ext4 wiki page, http://ext4.wiki. kernel.org. Some of the features in progress could possibly continue to change the on-disk layout. Ext4 will be converted from development mode to stable mode once the layout has been ﬁnalized. At that time, ext4 will be available for general use by all users in need of a more scalable and modern version of ext3. In the following three sec- tions we will discuss new capabilities currently included in or planned for ext4 in the areas of scalability, frag- mentation, and reliability. 2 Scalability enhancements The ﬁrst goal of ext4 was to become a more scalable ﬁlesystem. In this section we will discuss the scalability features that will be available in ext4. 2.1 Large ﬁlesystem The current 16 TB ﬁlesystem size limit is caused by the 32-bit block number in ext3. To enlarge the ﬁlesystem limit, the straightforward method is to increase the num- ber of bits used to represent block numbers and then ﬁx all references to data and metadata blocks. Previously, there was an extents[3] patch for ext3 with the capacity to support 48-bit physical block numbers. In ext4, instead of just extending the block numbers to 64-bits based on the current ext3 indirect block map- ping, the ext4 developers decided to use extents map- ping with 48-bit block numbers. This both increases ﬁlesystem capacity and improves large ﬁle efﬁciency. With 48-bit block numbers, ext4 can support a maxi- mum ﬁlesystem size up to 2 (48+12) = 260 bytes (1 EB) with 4 KB block size. After changing the data block numbers to 48-bit, the next step was to correct the references to meta- data blocks correspondingly. Metadata is present in the superblock, the group descriptors, and the jour- nal. New ﬁelds have been added at the end of the superblock structure to store the most signiﬁcant 32 bits for block-counter variables, s_free_blocks_count, s_blocks_count, and s_r_blocks_count, extending them to 64 bits. Similarly, we introduced new 32-bit ﬁelds at the end of the block group descriptor structure to store the most signiﬁcant bits of 64-bit values for bitmaps and inode table pointers. Since the addresses of modiﬁed blocks in the ﬁlesys- tem are logged in the journal, the journaling block layer (JBD) is also required to support at least 48-bit block ad- dresses. Therefore, JBD was branched to JBD2 to sup- port more than 32-bit block numbers at the same time ext4 was forked. Although currently only ext4 is using JBD2, it can provide general journaling support for both 32-bit and 64-bit ﬁlesystems. One may question why we chose 48-bit rather than full 64-bit support. The 1 EB limit will be sufﬁcient for many years. Long before this limit is hit there will be reliability issues that need to be addressed. At current speeds, a 1 EB ﬁlesystem would take 119 years to ﬁnish one full e2fsck, and 65536 times that for a 2 64 blocks (64 ZB) ﬁlesystem. Overcoming these kind of reliability is- sues is the priority of ext4 developers before addressing full 64-bit support and is discussed later in the paper. 2.1.1 Future work After extending the limit created by 32-bit block num- bers, the ﬁlesystem capacity is still restricted by the number of block groups in the ﬁlesystem. In ext3, for safety concerns all block group descriptors copies are kept in the ﬁrst block group. With the new uninitial- ized block group feature discussed in section 4.1 the new block group descriptor size is 64 bytes. Given the default 128 MB(2 27 bytes) block group size, ext4 can have at most 2 27/64 = 221 block groups. This limits the entire ﬁlesystem size to 2 21 ∗227 = 248 bytes or 256TB. The solution to this problem is to use the metablock group feature (META_BG), which is already in ext3 for all 2.6 releases. With the META_BG feature, ext4 ﬁlesystems are partitioned into many metablock groups. Each metablock group is a cluster of block groups whose group descriptor structures can be stored in a sin- gle disk block. For ext4 ﬁlesystems with 4 KB block size, a single metablock group partition includes 64 block groups, or 8 GB of disk space. The metablock group feature moves the location of the group descrip- tors from the congested ﬁrst block group of the whole ﬁlesystem into the ﬁrst group of each metablock group itself. The backups are in the second and last group of each metablock group. This increases the 2 21 maximum 2007 Linux Symposium, V olume Two • 23 047 3195 logical block #physical block # lengthuninitialized extent flag ext4_extent structure ext4_extent_header eh_magic eh_entries eh_max eh_depth eh_generation ext4_extent_idx ei_block ei_leaf ei_leaf_hi ei_unused Figure 1: Ext4 extents, header and index structures block groups limit to the hard limit 2 32, allowing support for the full 1 EB ﬁlesystem. 2.2 Extents The ext3 ﬁlesystem uses an indirect block mapping scheme providing one-to-one mapping from logical blocks to disk blocks. This scheme is very efﬁcient for sparse or small ﬁles, but has high overhead for larger ﬁles, performing poorly especially on large ﬁle delete and truncate operations [3]. As mentioned earlier, extents mapping is included in ext4. This approach efﬁciently maps logical to physical blocks for large contiguous ﬁles. An extent is a single descriptor which represents a range of contiguous phys- ical blocks. Figure 1 shows the extents structure. As we discussed in previously, the physical block ﬁeld in an extents structure takes 48 bits. A single extent can represent 215 contiguous blocks, or 128 MB, with 4 KB block size. The MSB of the extent length is used to ﬂag uninitialized extents, used for the preallocation feature discussed in Section 3.1. Four extents can be stored in the ext4 inode structure directly. This is generally sufﬁcient to represent small or contiguous ﬁles. For very large, highly fragmented, or sparse ﬁles, more extents are needed. In this case a constant depth extent tree is used to store the extents map of a ﬁle. Figure 2 shows the layout of the extents tree. The root of this tree is stored in the ext4 inode structure and extents are stored in the leaf nodes of the tree. Each node in the tree starts with an extent header (Fig- ure 1), which contains the number of valid entries in i_block . . . eh_header root node header extent index extent index . . . node header extent . . . extent node header extent . . . extent disk blocksext4_inode index node leaf nodes Figure 2: Ext4 extent tree layout the node, the capacity of entries the node can store, the depth of the tree, and a magic number. The magic num- ber can be used to differentiate between different ver- sions of extents, as new enhancements are made to the feature, such as increasing to 64-bit block numbers. The extent header and magic number also add much- needed robustness to the on-disk structure of the data ﬁles. For very small ﬁlesystems, the block-mapped ﬁles implicitly depended on the fact that random corruption of an indirect block would be easily detectable, because the number of valid ﬁlesystem blocks is a small sub- set of a random 32-bit integer. With growing ﬁlesystem sizes, random corruption in an indirect block is by itself indistinguishable from valid block numbers. In addition to the simple magic number stored in the extent header, the tree structure of the extent tree can be veriﬁed at runtime or by e2fsck in several ways. The ext4_extent_header has some internal con- sistency (eh_entries and eh_max) that also depends on the ﬁlesystem block size. eh_depth decreases from the root toward the leaves. The ext4_extent entries in a leaf block must have increasing ee_block numbers, and must not overlap their neighbors with ee_len. Similarly, the ext4_extent_idx also needs increasing ei_block values, and the range of blocks that an index covers can be veri- ﬁed against the actual range of blocks in the extent leaf. Currently, extents mapping is enabled in ext4 with the extents mount option. After the ﬁlesystem is mounted, 24 • The new ext4 ﬁlesystem: current status and future plans any new ﬁles will be created with extent mapping. The beneﬁts of extent maps are reﬂected in the performance evaluation Section 7. 2.2.1 Future work Extents are not very efﬁcient for representing sparse or highly fragmented ﬁles. For highly fragmented ﬁles, we could introduce a new type of extent, a block-mapped extent. A different magic number, stored in the extent header, distinguishes the new type of leaf block, which contains a list of allocated block numbers similar to an ext3 indirect block. This would give us the increased ro- bustness of the extent format, with the block allocation ﬂexibility of the block-mapped format. In order to improve the robustness of the on-disk data, there is a proposal to create an “extent tail” in the extent blocks, in addition to the extent header. The extent tail would contain the inode number and generation of the inode that has allocated the block, and a checksum of the extent block itself (though not the data). The check- sum would detect internal corruption, and could also de- tect misplaced writes if the block number is included therein. The inode number could be used to detect corruption that causes the tree to reference the wrong block (whether by higher-level corruption, or misplaced writes). The inode number could also be used to recon- struct the data of a corrupted inode or assemble a deleted ﬁle, and also help in doing reverse-mapping of blocks for defragmentation among other things. 2.3 Large ﬁles In Linux, ﬁle size is calculated based on the i_blocks counter value. However, the unit is in sectors (512 bytes), rather than in the ﬁlesystem block size (4096 bytes by default). Since ext4’s i_blocks is a 32-bit vari- able in the inode structure, this limits the maximum ﬁle size in ext4 to 2 32 ∗512 bytes = 2 41 bytes = 2 TB. This is a scalability limit that ext3 has planned to break for a while. The solution for ext4 is quite straightforward. The ﬁrst part is simply changing the i_blocks units in the ext4 inode to ﬁlesystem blocks. An ROCOMPAT fea- ture ﬂag HUGE_FILE is added in ext4 to signify that the i_blocks ﬁeld in some inodes is in units of ﬁlesys- tem block size. Those inodes are marked with a ﬂag EXT4_HUGE_FILE_FL, to allow existing inodes to keep i_blocks in 512-byte units without requiring a full ﬁlesystem conversion. In addition, the i_blocks variable is extended to 48 bits by using some of the reserved in- ode ﬁelds. We still have the limitation of 32 bit logi- cal block numbers with the current extent format, which limits the ﬁle size to 16TB. With the ﬂexible extents for- mat in the future (see Section 2.2.1), we may remove that limit and fully use the 48-bit i_blocks to enlarge the ﬁle size even more. 2.4 Large number of ﬁles Some applications already create billions of ﬁles today, and even ask for support for trillions of ﬁles. In theory, the ext4 ﬁlesystem can support billions of ﬁles with 32- bit inode numbers. However, in practice, it cannot scale to this limit. This is because ext4, following ext3, still allocates inode tables statically. Thus, the maximum number of inodes has to be ﬁxed at ﬁlesystem creation time. To avoid running out of inodes later, users often choose a very large number of inodes up-front. The con- sequence is unnecessary disk space has to be allocated to store unused inode structures. The wasted space be- comes more of an issue in ext4 with the larger default inode. This also makes the management and repair of large ﬁlesystems more difﬁcult than it should be. The uninitialized group feature (Section 4.1) addresses this issue to some extent, but the problem still exists with aged ﬁlesystems in which the used and unused inodes can be mixed and spread across the whole ﬁlesystem. Ext3 and ext4 developers have been thinking about sup- porting dynamic inode allocation for a while [9, 3]. There are three general considerations about the dy- namic inode table allocation: • Performance: We need an efﬁcient way to translate inode number to the block where the inode struc- ture is stored. • Robustness: e2fsck should be able to locate inode table blocks scattered across the ﬁlesystem, in the case the ﬁlesystem is corrupted. • Compatibility: We need to handle the possible in- ode number collision issue with 64-bit inode num- bers on 32-bit systems, due to overﬂow. These three requirements make the design challenging. 2007 Linux Symposium, V olume Two • 25 4-bit offset 15-bit relative block # 32-bit block group # 63 50 18 3 0 Figure 3: 64-bit inode layout With dynamic inode tables, the blocks storing the inode structure are no longer at a ﬁxed location. One way to efﬁciently map the inode number to the block storing the corresponding inode structure, is encoding the block number into the inode number directly, similar to what is done in XFS. This implies the use of 64-bit inode num- bers. The low four to ﬁve bits of the inode number store the offset bits within the inode table block. The rest store the 32-bit block group number as well as 15-bit relative block number within the group, shown in Fig- ure 3. Then, a cluster of contiguous inode table blocks (ITBC) can be allocated on demand. A bitmap at the head of the ITBC would be used to keep track of the free and used inodes, allowing fast inode allocation and deallocation. In the case where the ﬁlesystem is corrupted, the ma- jority of inode tables could be located by checking the directory entries. To further address the reliability con- cern, a magic number could be stored at the head of the ITBC, to help e2fsck to recognize this metadata block. Relocating inodes becomes tricky with this block- number-in-inode-number proposal. If the ﬁlesystem is resized or defragmented, we may have to change the lo- cation of the inode blocks, which would require chang- ing all references to that inode number. The proposal to address this concern is to have a per-group “inode exception map” that translates an old block/inode num- ber into a new block number where the relocated inode structure is actually stored. The map will usually be empty, unless the inode was moved. One concern with the 64-bit inode number is the possi- ble inode number collision with 32-bit applications, as applications might still be using 32-bit stat() to access inode numbers and could break. Investigation is under- way to see how common this case is, and whether most applications are currently ﬁxed to use the 64-bit stat64(). One way to address this concern is to generate 32-bit inode numbers on 32-bit platforms. Seventeen bits is enough to represent block group numbers on 32-bit ar- chitectures, and we could limit the inode table blocks to the ﬁrst 2 10 blocks of a block group to construct the 32-bit inode number. This way user applications will be ensured of getting unique inode numbers on 32-bit plat- forms. For 32-bit applications running on 64-bit plat- forms, we hope they are ﬁxed by the time ext4 is in pro- duction, and this only starts to be an issue for ﬁlesystems over 1TB in size. In summary, dynamic inode allocation and 64-bit inode numbers are needed to support large numbers of ﬁles in ext4. The beneﬁts are obvious, but the changes to the on-disk format may be intrusive. The design details are still under discussion. 2.5 Directory scalability The maximum number of subdirectories contained in a single directory in ext3 is 32,000. To address directory scalability, this limit will be eliminated in ext4 providing unlimited sub-directory support. In order to better support large directories with many en- tries, the directory indexing feature[6] will be turned on by default in ext4. By default in ext3, directory entries are still stored in a linked list, which is very inefﬁcient for directories with large numbers of entries. The di- rectory indexing feature addresses this scalability issue by storing directory entries in a constant depth HTree data structure, which is a specialized BTree-like struc- ture using 32-bit hashes. The fast lookup time of the HTree signiﬁcantly improves performance on large di- rectories. For directories with more than 10,000 ﬁles, improvements were often by a factor of 50 to 100 [3]. 2.5.1 Future work While the HTree implementation allowed the ext2 direc- tory format to be improved from linear to a tree search compatibly, there are also limitations to this approach. The HTree implementation has a limit of 510 * 511 4 KB directory leaf blocks (approximately 25M 24-byte ﬁlenames) that can be indexed with a 2-level tree. It would be possible to change the code to allow a 3-level HTree. There is also currently a 2 GB ﬁle size limit on directories, because the code for using the high 32-bits for i_size on directories was not implemented when the 2 GB limit was ﬁxed for regular ﬁles. Because the hashing used to ﬁnd ﬁlenames in indexed directories is essentially random compared to the lin- ear order in which inodes are allocated, we end up do- ing random seeks around the disk when accessing many 26 • The new ext4 ﬁlesystem: current status and future plans inodes in a large directory. We need to have readdir in hash-index order because directory entries might be moved during the split of a directory leaf block, so to satisfy POSIX requirements we can only safely walk the directory in hash order. To address this problem, there is a proposal to put the whole inode into the directory instead of just a directory entry that references a separate inode. This avoids the need to seek to the inode when doing a readdir, because the whole inode has been read into memory already in the readdir step. If the blocks that make up the directory are efﬁciently allocated, then reading the directory also does not require any seeking. This would also allow dynamic inode allocation, with the directory as the “container” of the inode table. The inode numbers would be generated in a similar manner as previously discussed (Section 2.4), so that the block that an inode resides in can be located directly from the inode number itself. Hard linked ﬁles imply that the same block is allocated to multiple directories at the same time, but this can be reconciled by the link count in the inode itself. We also need to store one or more ﬁle names in the in- ode itself, and this can be done by means of an extended attribute that uses the directory inode number as the EA name. We can then return the name(s) associated with that inode for a single directory immediately when do- ing readdir, and skip any other name(s) for the inode that belong to hard links in another directory. For efﬁcient name-to-inode lookup in the directory, we would still use a secondary tree similar to the current ext3 HTree (though it would need an entry per name instead of per directory block). But because the directory entries (the inodes themselves) do not get moved as the directory grows, we can just use disk block or directory offset or- der for readdir. 2.6 Large inode and fast extended attributes Ext3 supports different inode sizes. The inode size can be set to any power-of-two larger than 128 bytes size up to the ﬁlesystem block size by using the mke2fs -I [inode size] option at format time. The default inode structure size is 128 bytes, which is already crowded with data and has little space for new ﬁelds. In ext4, the default inode structure size will be 256 bytes. Fixed Fields 127 255 0 Fast Extended Attributes Ext4 Large Inode Original 128-bit Inode i_extra_isize i_pad1 i_ctime_extra i_mtime_extra i_atime_extra i_crtime i_crtime_extra i_version_hi Figure 4: Layout of the large inode In order to avoid duplicating a lot of code in the kernel and e2fsck, the large inodes keep the same ﬁxed layout for the ﬁrst 128-bytes, as shown in Figure 4. The rest of the inode is split into two parts: a ﬁxed-ﬁeld section that allows addition of ﬁelds common to all inodes, such as nanosecond timestamps (Section 5), and a section for fast extended attributes (EAs) that consumes the rest of the inode. The ﬁxed-ﬁeld part of the inode is dynamically sized, based on what ﬁelds the current kernel knows about. The size of this area is stored in each inode in the i_extra_isize ﬁeld, which is the ﬁrst ﬁeld beyond the original 128-byte inode. The superblock also contains two ﬁelds, s_min_extra_isize and i_want_extra_isize, which allow down-level kernels to allocate a larger i_extra_isize than it would otherwise do. The s_min_extra_isize is the guaranteed mini- mum amount of ﬁxed-ﬁeld space in each inode. s_want_extra_isize is the desired amount of ﬁxed-ﬁeld space for new inode, but there is no guarantee that this much space will be available in every inode. A ROCOMPAT feature ﬂag EXTRA_ISIZE indicates whether these superblock ﬁelds are valid. The ext4 code will soon also be able to expand i_extra_isize dynamically as needed to cover the ﬁxed ﬁelds, so long as there is space available to store the fast EAs or migrate them to an external EA block. The remaining large inode space may be used for storing EA data inside the inode. Since the EAs are already in memory after the inode is read from disk, this avoids a costly seek to external EA block. This can greatly 2007 Linux Symposium, V olume Two • 27 improve the performance of applications that are using EAs, sometimes by a factor of 3–7 [4]. An external EA block is still available in addition to the fast EA space, which allows storing up to 4 KB of EAs for each ﬁle. The support for fast EAs in large inodes has been avail- able in Linux kernels since 2.6.12, though it is rarely used because many people do not know of this capabil- ity at mke2fs time. Since ext4 will have larger inodes, this feature will be enabled by default. There have also been discussions about breaking the 4 KB EA limit, in order to store larger or more EAs. It is likely that larger single EAs will be stored in their own inode (to allow arbitrary-sized EAs) and it may also be that for many EAs they will be stored in a directory-like structure, possibly leveraging the same code as regular ext4 directories and storing small values inline. 3 Block allocation enhancements Increased ﬁlesystem throughput is the premier goal for all modern ﬁlesystems. In order to meet this goal, de- velopers are constantly attempting to reduce ﬁlesystem fragmentation. High fragmentation rates cause greater disk access time affecting overall throughput, and in- creased metadata overhead causing less efﬁcient map- ping. There is an array of new features in line for ext4, which take advantage of the existing extents mapping and are aimed at reducing ﬁlesystem fragmentation by improv- ing block allocation techniques. 3.1 Persistent preallocation Some applications, like databases and streaming media servers, beneﬁt from the ability to preallocate blocks for a ﬁle up-front (typically extending the size of the ﬁle in the process), without having to initialize those blocks with valid data or zeros. Preallocation helps ensure con- tiguous allocation as far as possible for a ﬁle (irrespec- tive of when and in what order data actually gets writ- ten) and guaranteed space allocation for writes within the preallocated size. It is useful when an application has some foreknowledge of how much space the ﬁle will require. The ﬁlesystem internally interprets the preallo- cated but not yet initialized portions of the ﬁle as zero- ﬁlled blocks. This avoids exposing stale data for each block until it is explicitly initialized through a subse- quent write. Preallocation must be persistent across re- boots, unlike ext3 and ext4 block reservations [3]. For applications involving purely sequential writes, it is possible to distinguish between initialized and uninitial- ized portions of the ﬁle. This can be done by maintain- ing a single high water mark value representing the size of the initialized portion. However, for databases and other applications where random writes into the preal- located blocks can occur in any order, this is not sufﬁ- cient. The ﬁlesystem needs to be able to identify ranges of uninitialized blocks in the middle of the ﬁle. There- fore, some extent based ﬁlesystems, like XFS, and now ext4, provide support for marking allocated but unini- tialized extents associated with a given ﬁle. Ext4 implements this by using the MSB of the extent length ﬁeld to indicate whether a given extent contains uninitialized data, as shown in Figure 1. During reads, an uninitialized extent is treated just like a hole, so that the VFS returns zero-ﬁlled blocks. Upon writes, the ex- tent must be split into initialized and uninitialized ex- tents, merging the initialized portion with an adjacent initialized extent if contiguous. Until now, XFS, the other Linux ﬁlesystem that imple- ments preallocation, provided an ioctl interface to ap- plications. With more ﬁlesystems, including ext4, now providing this feature, a common system-call interface for fallocate and an associated inode operation have been introduced. This allows ﬁlesystem-speciﬁc imple- mentations of preallocation to be exploited by applica- tions using the posix_fallocate API. 3.2 Delayed and multiple block allocation The block allocator in ext3 allocates one block at a time during the write operation, which is inefﬁcient for larger I/O. Since block allocation requests are passed through the VFS layer one at a time, the underlying ext3 ﬁlesys- tem cannot foresee and cluster future requests. This also increases the possibility of ﬁle fragmentation. Delayed allocation is a well-known technique in which block allocations are postponed to page ﬂush time, rather than during the write() operation [3]. This method provides the opportunity to combine many block allo- cation requests into a single request, reducing possible 28 • The new ext4 ﬁlesystem: current status and future plans fragmentation and saving CPU cycles. Delayed alloca- tion also avoids unnecessary block allocation for short- lived ﬁles. Ext4 delayed allocation patches have been imple- mented, but there is work underway to move this sup- port to the VFS layer, so multiple ﬁlesystems can beneﬁt from the feature. With delayed allocation support, multiple block alloca- tion for buffered I/O is now possible. An entire extent, containing multiple contiguous blocks, is allocated at once rather than one block at a time. This eliminates multiple calls to ext4_get_blocks and ext4_new_blocks and reduces CPU utilization. Ext4 multiple block allocation builds per-block group free extents information based on the on-disk block bitmap. It uses this information to guide the search for free extents to satisfy an allocation request. This free extent information is generated at ﬁlesystem mount time and stored in memory using a buddy structure. The performance beneﬁts of delayed allocation alone are very obvious, and can be seen in Section 7. In a previous study [3], we have seen about 30% improved throughput and 50% reduction in CPU usage with the combined two features. Overall, delayed and multi- ple block allocation can signiﬁcantly improve ﬁlesystem performance on large I/O. There are two other features in progress that are built on top of delayed and multiple block allocation, trying to further reduce fragmentation: • In-core Preallocation: Using the in-core free ex- tents information, a more powerful in-core block preallocation/reservation can be built. This further improves block placement and reduces fragmenta- tion with concurrent write workloads. An inode can have a number of preallocated chunks, indexed by the logical blocks. This improvement can help HPC applications when a number of nodes write to one huge ﬁle at very different offsets. • Locality Groups: Currently, allocation policy deci- sions for individual ﬁle are made independently. If the allocator had knowledge of ﬁle relationship, it could intelligently place related ﬁles close together, greatly beneﬁting read performance. The locality groups feature clusters related ﬁles together by a given attribute, such as SID or a combination of SID and parent directory. At the deferred page- ﬂush time, dirty pages are written out by groups, instead of by individual ﬁles. The number of non- allocated blocks are tracked at the group-level, and upon ﬂush time, the allocator can try to preallocate enough space for the entire group. This space is shared by the ﬁles in the group for their individual block allocation. In this way, related ﬁles are place tightly together. In summary, ext4 will have a powerful block allocation scheme that can efﬁciently handle large block I/O and reduce ﬁlesystem fragmentation with small ﬁles under multi-threaded workloads. 3.3 Online defragmentation Though the features discussed in this section improve block allocation to avoid fragmentation in the ﬁrst place, with age, the ﬁlesystem can still become quite fragmented. The ext4 online defragmentation tool, e4defrag, has been developed to address this. This tool can defragment individual ﬁles or the entire ﬁlesystem. For each ﬁle, the tool creates a temporary inode and al- locates contiguous extents to the temporary inode using multiple block allocation. It then copies the original ﬁle data to the page cache and ﬂushes the dirty pages to the temporary inode’s blocks. Finally, it migrates the block pointers from the temporary inode to the original inode. 4 Reliability enhancements Reliability is very important to ext3 and is one of the reasons for its vast popularity. In keeping with this reputation, ext4 developers are putting much effort into maintaining the reliability of the ﬁlesystem. While it is relatively easy for any ﬁlesystem designer to make their ﬁelds 64-bits in size, it is much more difﬁcult to make such large amounts of space actually usable in the real world. Despite the use of journaling and RAID, there are invari- ably corruptions to the disk ﬁlesystem. The ﬁrst line of defense is detecting and avoiding problems proactively by a combination of robust metadata design, internal re- dundancy at various levels, and built-in integrity check- ing using checksums. The fallback will always be doing 2007 Linux Symposium, V olume Two • 29 integrity checking (fsck) to both detect and correct prob- lems that will happen anyway. One of the primary concerns with all ﬁlesystems is the speed at which a ﬁlesystem can be validated and recov- ered after corruption. With reasonably high-end RAID storage, a full fsck of a 2TB ext3 ﬁlesystem can take between 2 to 4 hours for a relatively “clean” ﬁlesystem. This process can degrade sharply to many days if there are large numbers of shared ﬁlesystem blocks that need expensive extra passes to correct. Some features, like extents, have already added to the robustness of the ext4 metadata as previously described. Many more related changes are either complete, in progress, or being designed in order to ensure that ext4 will be usable at scales that will become practical in the future. 4.1 Unused inode count and fast e2fsck In e2fsck, the checking of inodes in pass 1 is by far the most time consuming part of the operation. This re- quires reading all of the large inode tables from disk, scanning them for valid, invalid, or unused inodes, and then verifying and updating the block and inode alloca- tion bitmaps. The uninitialized groups and inode table high watermark feature allows much of the lengthy pass 1 scanning to be safely skipped. This can dramatically reduce the total time taken by e2fsck by 2 to 20 times, depending on how full the ﬁlesystem is. This feature can be enabled at mke2fs time or using tune2fs via the -O uninit_groupsoption. With this feature, the kernel stores the number of un- used inodes at the end of each block group’s inode table. As a result, e2fsck can skip both reading these blocks from disk, and scanning them for in-use inodes. In or- der to ensure that the unused inode count is safe to use by e2fsck, the group descriptor has a CRC16 checksum added to it that allows validation of all ﬁelds therein. Since typical ext3 ﬁlesystems use only in the neighbor- hood of 1% to 10% of their inodes, and the inode alloca- tion policy keeps a majority of those inodes at the start of the inode table, this can avoid processing a large ma- jority of the inodes and speed up the pass 1 processing. The kernel does not currently increase the unused inodes count, when ﬁles are deleted. This counter is updated on every e2fsck run, so in the case where a block group had 0 0.5 10 15 20 25 30 35 0 500 1000 1500 2000 2500 3000 3500 4000 4500 fsck time vs. Inode Count ext3: 0 files ext3 100k files ext3 2.1M files ext4 100kfiles ext4 2.1M files Total Inode Count (millions) fsck time (sec) Figure 5: e2fsck performance improvement with unini- tialized block groups many inodes deleted, e2fsck will be more efﬁcient in the next run. Figure 5 shows that e2fsck time on ext3 grows linearly with the total number of inodes in ﬁlesystem, regardless of how many are used. On ext3, e2fsck takes the same amount of time with zero used ﬁles as with 2.1 million used ﬁles. In ext4, with the unused inode high water- mark feature, the e2fsck time is only dependent on the number of used inodes. As we can see, fsck of an ext4 ﬁlesystem with 100 000 used ﬁles takes a fraction of the time ext3 takes. In addition to the unused inodes count, it is possible for mke2fs and e2fsck to mark a group’s block or in- ode bitmap as uninitialized, so that the kernel does not need to read them from disk when ﬁrst allocating from the group. Similarly, e2fsck does not need to read these bitmaps from disk, though this does not play a major role in performance improvements. What is more sig- niﬁcant is that mke2fs will not write out the bitmaps or inode tables at format time if the mke2fs -O lazy_bgfea- ture is given. Writing out the inode tables can take a signiﬁcant amount of time, and has been known to cause problems for large ﬁlesystems due to the amount of dirty pages this generates in a short time. 4.2 Checksumming Adding metadata checksumming into ext4 will allow it to more easily detect corruption, and behave appropri- ately instead of blindly trusting the data it gets from 30 • The new ext4 ﬁlesystem: current status and future plans disk. The group descriptors already have a checksum added, per the previous section. The next immediate tar- get for checksumming is the journal, because it has such a high density of important metadata and is constantly being written to, so has a higher chance of wearing out the platters or seeing other random corruption. Adding checksumming to the ext4 journal is nearly complete [7]. In ext3 and ext4, each journal transac- tion has a header block and commit block. During nor- mal journal operation the commit block is not sent to the disk until the transaction header and all metadata blocks which make up that transaction have been writ- ten to disk [8]. The next transaction needs to wait for the previous commit block to hit to disk before it can start to modify the ﬁlesystem. With this two-phase commit, if the commit block has the same transaction number as the header block, it should indicate that the transaction can be replayed at recovery time. If they don’t match, the journal recovery is ended. However, there are several scenarios where this can go wrong and lead to ﬁlesystem corruption. With journal checksumming, the journal code computes a CRC32 over all of the blocks in the transaction (in- cluding the header), and the checksum is written to the commit block of the transaction. If the checksum does not match at journal recovery time, it indicates that one or more metadata blocks in the transaction are corrupted or were not written to disk. Then the transaction (along with later ones) is discarded as if the computer had crashed slightly earlier and not written a commit block at all. Since the journal checksum in the commit block allows detection of blocks that were not written into the journal, as an added bonus there is no longer a need for having a two-phase commit for each transaction. The commit block can be written at the same time as the rest of the blocks in the transaction. This can actually speed up the ﬁlesystem operation noticeably (as much as 20% [7]), instead of the journal checksum being an overhead. There are also some long-term plans to add check- summing to the extent tail, the allocation bitmaps, the inodes, and possibly also directories. This can be done efﬁciently once we have journal checksumming in place. Rather than computing the checksum of ﬁlesys- tem metadata each time it is changed (which has high overhead for often-modiﬁed structures), we can write the metadata to the checksummed journal and still be conﬁdent that it is valid and correct at recovery time. The blocks can have metadata-speciﬁc checksums com- puted a single time when they are written into the ﬁlesystem. 5 Other new features New features are continuously being added to ext4. Two features expected to be seen in ext4 are nanosecond timestamps and inode versioning. These two features provide precision when dealing with ﬁle access times and tracking changes to ﬁles. Ext3 has second resolution timestamps, but with today’s high-speed processors, this is not sufﬁcient to record multiple changes to a ﬁle within a second. In ext4, since we use a larger inode, there is room to support nanosec- ond resolution timestamps. High 32-bit ﬁelds for the atime, mtime and ctime timestamps, and also a new cr- time timestamp documenting ﬁle creation time, will be added to the ext4 inode (Figure 4). 30 bits are sufﬁcient to represent the nanosecond ﬁeld, and the remaining 2 bits are used to extend the epoch by 272 years. The NFSv4 clients need the ability to detect updates to a ﬁle made at the server end, in order to keep the client side cache up to date. Even with nanosecond support for ctime, the timestamp is not necessarily updated at the nanosecond level. The ext4 inode versioning feature addresses this issue by providing a global 64-bit counter in each inode. This counter is incremented whenever the ﬁle is changed. By comparing values of the counter, one can see whether the ﬁle has been updated. The counter is reset on ﬁle creation, and overﬂows are unimportant, be- cause only equality is being tested. The i_version ﬁeld already present in the 128-bit inode is used for the low 32 bits, and a high 32-bit ﬁeld is added to the large ext4 inode. 6 Migration tool Ext3 developers worked to maintain backwards compat- ibility between ext2 and ext3, a characteristic users ap- preciate and depend on. While ext4 attempts to retain compatibility with ext3 as much as possible, some of the incompatible on-disk layout changes are unavoid- able. Even with these changes, users can still easily upgrade their ext3 ﬁlesystem to ext4, like it is possible 2007 Linux Symposium, V olume Two • 31 from ext2 to ex3. There are methods available for users to try new ext4 features immediately, or migrate their entire ﬁlesystem to ext4 without requiring back-up and restore. 6.1 Upgrading from ext3 to ext4 There is a simple upgrade solution for ext3 users to start using extents and some ext4 features without requiring a full backup or migration. By mounting an existing ext3 ﬁlesystem as ext4 (with extents enabled), any new ﬁles are created using extents, while old ﬁles are still indi- rect block mapped and interpreted as such. A ﬂag in the inode differentiates between the two formats, allowing both to coexist in one ext4 ﬁlesystem. All new ext4 fea- tures based on extents, such as preallocation and mul- tiple block allocation, are available to the new extents ﬁles immediately. A tool will also be available to perform a system-wide ﬁlesystem migration from ext3 to ext4. This migration tool performs two functions: migrating from indirect to extents mapping, and enlarging the inode to 256 bytes. • Extents migration: The ﬁrst step can be performed online and uses the defragmentation tool. During the defragmentation process, ﬁles are changed to extents mapping. In this way, the ﬁles are being converted to extents and defragmented at the same time. • Inode migration: Enlarging the inode structure size must be done ofﬂine. In this case, data is backed up, and the entire ﬁlesystem is scanned and con- verted to extents mapping and large inodes. For users who are not yet ready to move to ext4, but may want to in the future, it is possible to prepare their ext3 ﬁlesystem to avoid ofﬂine migration later. If an ext3 ﬁlesystem is formatted with a larger inode struc- ture, 256 bytes or more, the fast extended attribute fea- ture (Section 2.6) which is the default in ext4, can be used instantly. When the user later wants to upgrade to ext4, then other ext4 features using the larger inode size, such as nanosecond timestamps, can also be used without requiring any ofﬂine migration. 6.2 Downgrading from ext4 to ext3 Though not as straightforward as ext3 to ext4, there is a path for any user who may want to downgrade from ext4 back to ext3. In this case the user would remount the ﬁlesystem with the noextents mount option, copy all ﬁles to temporary ﬁles and rename those ﬁles over the original ﬁle. After all ﬁles have been converted back to indirect block mapping format, the INCOM- PAT_EXTENTS ﬂag must be cleared using tune2fs, and the ﬁlesystem can be re-mounted as ext3. 7 Performance evaluation We have conducted a performance evaluation of ext4, as compared to ext3 and XFS, on three well-known ﬁlesys- tem benchmarks. Ext4 was tested with extents and de- layed allocation enabled. The benchmarks in this anal- ysis were chosen to show the impact of new changes in ext4. The three benchmarks chosen were: Flexible Filesystem Benchmark (FFSB) [1], Postmark [5], and IOzone [2]. FFSB, conﬁgured with a large ﬁle work- load, was used to test the extents feature in ext4. Post- mark was chosen to see performance of ext4 on small ﬁle workloads. Finally, we used IOzone to evaluate overall ext4 ﬁlesystem performance. The tests were all run on the 2.6.21-rc4 kernel with de- layed allocation patches. For ext3 and ext4 tests, the ﬁlesystem was mounted in writeback mode, and ap- propriate extents and delayed allocation mount options were set for ext4. Default mount options were used for XFS testing. FFSB and IOzone benchmarks were run on the same 4-CPU 2.8 Ghz Intel(R) Xeon(tm) System with 2 GB of RAM, on a 68GB ultra320 SCSI disk (10000 rpm). Postmark was run on a 4-CPU 700 MHz Pentium(R) III system with 4 GB of RAM on a 9 GB SCSI disk (7200 rpm). Full test results including raw data are available at the ext4 wiki page, http://ext4.wiki.kernel. org. 7.1 FFSB comparison FFSB is a powerful ﬁlesystem benchmarking tool, that can be tuned to simulate very speciﬁc workloads. We have tested multithreaded creation of large ﬁles. The test 32 • The new ext4 ﬁlesystem: current status and future plans /c01/c01/c01/c01/c01/c01/c01/c02/c03/c04/c05/c01/c01/c01/c01/c02/c03/c04/c06 /c01/c01/c01/c01/c03/c07/c08/c01/c01/c01/c01/c01/c01/c01 /c09 /c0a /c0b/c09 /c0b/c0a /c0c/c09 /c0c/c0a /c05/c09 /c05/c0a /c06/c09 /c06/c0a /c0a/c09 /c0d/c0d/c0e/c0f/c01/c10/c01/c11/c12/c13/c14/c02/c01/c0d/c15/c16/c02/c01/c17/c13/c02/c12/c04/c15/c18/c19 /c1a/c1b/c13/c18/c1c/c14/c1b/c1d/c1c/c04/c01 /c1e/c0f/c1f/c08/c02/c20 /c21/c01/c17/c22/c23/c01/c1c/c08/c12/c14/c02 Figure 6: FFSB sequential write comparison /c01/c02/c03/c04 /c01/c02/c03/c05 /c06/c07/c08 /c09 /c09/c0a/c0b /c0c /c0c/c0a/c0b /c0d /c0d/c0a/c0b /c04 /c04/c0a/c0b /c05 /c05/c0a/c0b /c0b /c0b/c0a/c0b /c0e /c0e/c0a/c0b /c0f /c0f/c0a/c0b /c10/c11/c12/c03/c13/c14/c15/c16 /c17/c18/c14/c19 /c1a/c15/c1b/c03/c18 /c1c/c1d/c15/c11/c1e/c1f/c1d/c20/c1e/c03/c21/c22/c23/c24/c25/c12/c26 Figure 7: Postmark read write comparison runs 4 threads, which combined create 24 1-GB ﬁles, and stress the sequential write operation. The results, shown in Figure 6, indicate about 35% im- provement in throughput and 40% decrease in CPU uti- lization in ext4 as compared to ext3. This performance improvement shows a diminishing gap between ext4 and XFS on sequential writes. As expected, the results ver- ify extents and delayed allocation improve performance on large contiguous ﬁle creation. 7.2 Postmark comparison Postmark is a well-known benchmark simulating a mail server performing many single-threaded transactions on small to medium ﬁles. The graph in Figure 7 shows about 30% throughput gain with with ext4. Similar per- cent improvements in CPU utilization are seen, because metadata is much more compact with extents. The write throughput is higher than read throughput because ev- erything is being written to memory. /c01/c02/c03/c04/c05/c06/c05/c07 /c01/c02/c03/c04/c05 /c06/c05/c08/c09/c06/c05/c07 /c06/c05/c08/c09 /c06/c08/c0a/c09/c0b/c0c/c0d /c01/c02/c03/c04/c05 /c06/c08/c0a/c09/c0b/c0c/c0d /c06/c05/c08/c09 /c0e /c0f/c0e/c0e/c0e/c0e /c10/c0e/c0e/c0e/c0e /c11/c0e/c0e/c0e/c0e /c12/c0e/c0e/c0e/c0e /c13/c0e/c0e/c0e/c0e /c14/c0e/c0e/c0e/c0e /c15/c0e/c0e/c0e/c0e /c16/c0e/c0e/c0e/c0e /c17/c0e/c0e/c0e/c0e /c18/c19/c1a/c0b/c0a/c05 /c1b/c1c/c04/c11 /c1b/c1c/c04/c12 /c1d/c1e/c1f /c20/c21/c02/c0b/c22/c23/c21/c24/c22/c04/c0d/c25/c26/c27/c28/c29/c2a Figure 8: IOzone results: throughput of transactions on 512 MB ﬁles These results show that, aside from the obvious perfor- mance gain on large contiguous ﬁles, ext4 is also a good choice on smaller ﬁle workloads. 7.3 IOzone comparison For the IOzone benchmark testing, the system was booted with only 64 M of memory to really stress disk I/O. The tests were performed with 8 MB record sizes on various ﬁle sizes. Write, rewrite, read, reread, random write, and random read operations were tested. Figure 8 shows throughput results for 512 MB sized ﬁles. Over- all, there is great improvement between ext3 and ext4, especially on rewrite, random-write and reread opera- tions. In this test, XFS still has better read performance, while ext4 has shown higher throughput on write opera- tions. 8 Conclusion As we have discussed, the new ext4 ﬁlesystem brings many new features and enhancements to ext3, making it a good choice for a variety of workloads. A tremendous amount of work has gone into bringing ext4 to Linux, with a busy roadmap ahead to ﬁnalize ext4 for produc- tion use. What was once essentially a simple ﬁlesystem has become an enterprise-ready solution, with a good balance of scalability, reliability, performance and sta- bility. Soon, the ext3 user community will have the op- tion to upgrade their ﬁlesystem and take advantage of the newest generation of the ext family. 2007 Linux Symposium, V olume Two • 33 Acknowledgements The authors would like to extend their thanks to Jean- Noël Cordenner and Valérie Clément, for their help on performance testing and analysis, and development and support of ext4. We would also like to give special thanks to Andrew Morton for supporting ext4, and helping to bring ext4 to mainline Linux. We also owe thanks to all ext4 devel- opers who work hard to make the ﬁlesystem better, es- pecially: Ted T’so, Stephen Tweedie, Badari Pulavarty, Dave Kleikamp, Eric Sandeen, Amit Arora, Aneesh Veetil, and Takashi Sato. Finally thank you to all ext3 users who have put their faith in the ﬁlesystem, and inspire us to strive to make ext4 better. Legal Statement Copyright c⃝ 2007 IBM. This work represents the view of the authors and does not necessarily represent the view of IBM. IBM and the IBM logo are trademarks or registered trade- marks of International Business Machines Corporation in the United States and/or other countries. Lustre is a trademark of Cluster File Systems, Inc. Linux is a registered trademark of Linus Torvalds in the United States, other countries, or both. Other company, product, and service names may be trade- marks or service marks of others. References in this publication to IBM products or services do not imply that IBM intends to make them available in all countries in which IBM operates. This document is provided “AS IS,” with no express or im- plied warranties. Use the information in this document at your own risk. References [1] Ffsb project on sourceforge. Technical report. http://sourceforge.net/projects/ffsb. [2] Iozone. Technical report. http://www.iozone.org. [3] Mingming Cao, Theodore Y . Ts’o, Badari Pulavarty, Suparna Bhattacharya, Andreas Dilger, and Alex Tomas. State of the art: Where we are with the ext3 ﬁlesystem. In Ottawa Linux Symposium, 2005. [4] Jonathan Corbet. Which ﬁlesystem for samba4? Technical report. http://lwn.net/Articles/112566/. [5] Jeffrey Katcher. Postmark a new ﬁlesystem benchmark. Technical report, Network Appliances, 2002. [6] Daniel Phillips. A directory index for ext2. In 5th Annual Linux Showcase and Conference, pages 173–182, 2001. [7] Vijayan Prabhakaran, Lakshmi N. Bairavasundaram, Nitin Agrawal, Haryadi S. Gunawi, Andrea C. Arpaci-Dusseau, and Remzi H. Arpaci Dusseau. Iron ﬁle systems. In SOSP’05, pages 206–220, 2005. [8] Stephen Tweedie. Ext3 journalling ﬁlesystem. In Ottawa Linux Symposium, 2000. [9] Stephen Tweedie and Theodore Y Ts’o. Planned extensions to the linux ext2/3 ﬁlesystem. In USENIX Annual Technical Conference, pages 235–244, 2002. 34 • The new ext4 ﬁlesystem: current status and future plans Proceedings of the Linux Symposium V olume Two June 27th–30th, 2007 Ottawa, Ontario Canada Conference Organizers Andrew J. Hutton,Steamballoon, Inc., Linux Symposium, Thin Lines Mountaineering C. Craig Ross,Linux Symposium Review Committee Andrew J. Hutton,Steamballoon, Inc., Linux Symposium, Thin Lines Mountaineering Dirk Hohndel,Intel Martin Bligh,Google Gerrit Huizenga,IBM Dave Jones,Red Hat, Inc. C. Craig Ross,Linux Symposium Proceedings Formatting Team John W. Lockhart,Red Hat, Inc. Gurhan Ozen,Red Hat, Inc. John Feeney,Red Hat, Inc. Len DiMaggio,Red Hat, Inc. John Poelstra,Red Hat, Inc.
paxos-simple	Paxos Made Simple Leslie Lamport 01 Nov 2001 Abstract The Paxos algorithm, when presented in plain English, is very simple. Contents 1 Introduction 1 2 The Consensus Algorithm 1 2.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 Choosing a Value . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 Learning a Chosen Value . . . . . . . . . . . . . . . . . . . . . 6 2.4 Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 The Implementation . . . . . . . . . . . . . . . . . . . . . . . 7 3 Implementing a State Machine 8 References 11 1 Introduction The Paxos algorithm for implementing a fault-tolerant distributed system has been regarded as diﬃcult to understand, perhaps because the original presentation was Greek to many readers [5]. In fact, it is among the sim- plest and most obvious of distributed algorithms. At its heart is a consensus algorithm—the “synod” algorithm of [5]. The next section shows that this consensus algorithm follows almost unavoidably from the properties we want it to satisfy. The last section explains the complete Paxos algorithm, which is obtained by the straightforward application of consensus to the state ma- chine approach for building a distributed system—an approach that should be well-known, since it is the subject of what is probably the most often-cited article on the theory of distributed systems [4]. 2 The Consensus Algorithm 2.1 The Problem Assume a collection of processes that can propose values. A consensus al- gorithm ensures that a single one among the proposed values is chosen. If no value is proposed, then no value should be chosen. If a value has been chosen, then processes should be able to learn the chosen value. The safety requirements for consensus are: • Only a value that has been proposed may be chosen, • Only a single value is chosen, and • A process never learns that a value has been chosen unless it actually has been. We won’t try to specify precise liveness requirements. However, the goal is to ensure that some proposed value is eventually chosen and, if a value has been chosen, then a process can eventually learn the value. We let the three roles in the consensus algorithm be performed by three classes of agents:proposers, acceptors, andlearners. In an implementation, a single process may act as more than one agent, but the mapping from agents to processes does not concern us here. Assume that agents can communicate with one another by sending mes- sages. We use the customary asynchronous, non-Byzantine model, in which: 1 • Agents operate at arbitrary speed, may fail by stopping, and may restart. Since all agents may fail after a value is chosen and then restart, a solution is impossible unless some information can be re- membered by an agent that has failed and restarted. • Messages can take arbitrarily long to be delivered, can be duplicated, and can be lost, but they are not corrupted. 2.2 Choosing a Value The easiest way to choose a value is to have a single acceptor agent. A pro- poser sends a proposal to the acceptor, who chooses the ﬁrst proposed value that it receives. Although simple, this solution is unsatisfactory because the failure of the acceptor makes any further progress impossible. So, let’s try another way of choosing a value. Instead of a single acceptor, let’s use multiple acceptor agents. A proposer sends a proposed value to a set of acceptors. An acceptor mayaccept the proposed value. The value is chosen when a large enough set of acceptors have accepted it. How large is large enough? To ensure that only a single value is chosen, we can let a large enough set consist of any majority of the agents. Because any two majorities have at least one acceptor in common, this works if an acceptor can accept at most one value. (There is an obvious generalization of a majority that has been observed in numerous papers, apparently starting with [3].) In the absence of failure or message loss, we want a value to be chosen even if only one value is proposed by a single proposer. This suggests the requirement: P1. An acceptor must accept the ﬁrst proposal that it receives. But this requirement raises a problem. Several values could be proposed by diﬀerent proposers at about the same time, leading to a situation in which every acceptor has accepted a value, but no single value is accepted by a majority of them. Even with just two proposed values, if each is accepted by about half the acceptors, failure of a single acceptor could make it impossible to learn which of the values was chosen. P1 and the requirement that a value is chosen only when it is accepted by a majority of acceptors imply that an acceptor must be allowed to accept more than one proposal. We keep track of the diﬀerent proposals that an acceptor may accept by assigning a (natural) number to each proposal, so a proposal consists of a proposal number and a value. To prevent confusion, we require that diﬀerent proposals have diﬀerent numbers. How this is 2 achieved depends on the implementation, so for now we just assume it. A value is chosen when a single proposal with that value has been accepted by a majority of the acceptors. In that case, we say that the proposal (as well as its value) has been chosen. We can allow multiple proposals to be chosen, but we must guarantee that all chosen proposals have the same value. By induction on the proposal number, it suﬃces to guarantee: P2. If a proposal with valuev is chosen, then every higher-numbered pro- posal that is chosen has valuev. Since numbers are totally ordered, condition P2 guarantees the crucial safety property that only a single value is chosen. To be chosen, a proposal must be accepted by at least one acceptor. So, we can satisfy P2 by satisfying: P2a. If a proposal with valuev is chosen, then every higher-numbered pro- posal accepted by any acceptor has valuev. We still maintain P1 to ensure that some proposal is chosen. Because com- munication is asynchronous, a proposal could be chosen with some particu- lar acceptorc never having received any proposal. Suppose a new proposer “wakes up” and issues a higher-numbered proposal with a diﬀerent value. P1 requires c to accept this proposal, violating P2a. Maintaining both P1 and P2a requires strengthening P2a to: P2b. If a proposal with valuev is chosen, then every higher-numbered pro- posal issued by any proposer has valuev. Since a proposal must be issued by a proposer before it can be accepted by an acceptor, P2b implies P2a, which in turn impliesP2. To discover how to satisfy P2b, let’s consider how we would prove that it holds. We would assume that some proposal with numberm and value v is chosen and show that any proposal issued with numbern > m also has value v. We would make the proof easier by using induction onn, so we can prove that proposal numbern has valuev under the additional assumption that every proposal issued with a number inm . .(n − 1) has value v, where i . .j denotes the set of numbers fromi through j . For the proposal numberedm to be chosen, there must be some setC consisting of a majority of acceptors such that every acceptor inC accepted it. Combining this with the induction assumption, the hypothesis thatm is chosen implies: 3 Every acceptor in C has accepted a proposal with number in m . .(n − 1), and every proposal with number inm . .(n − 1) accepted by any acceptor has valuev. Since any setS consisting of a majority of acceptors contains at least one member ofC , we can conclude that a proposal numberedn has valuev by ensuring that the following invariant is maintained: P2c. For anyv and n, if a proposal with valuev and numbern is issued, then there is a setS consisting of a majority of acceptors such that either (a) no acceptor inS has accepted any proposal numbered less than n, or (b)v is the value of the highest-numbered proposal among all proposals numbered less thann accepted by the acceptors inS. We can therefore satisfy P2b by maintaining the invariance of P2c. To maintain the invariance of P2c, a proposer that wants to issue a pro- posal numbered n must learn the highest-numbered proposal with number less than n, if any, that has been or will be accepted by each acceptor in some majority of acceptors. Learning about proposals already accepted is easy enough; predicting future acceptances is hard. Instead of trying to pre- dict the future, the proposer controls it by extracting a promise that there won’t be any such acceptances. In other words, the proposer requests that the acceptors not accept any more proposals numbered less thann. This leads to the following algorithm for issuing proposals. 1. A proposer chooses a new proposal numbern and sends a request to each member of some set of acceptors, asking it to respond with: (a) A promise never again to accept a proposal numbered less than n, and (b) The proposal with the highest number less thann that it has accepted, if any. I will call such a request aprepare request with numbern. 2. If the proposer receives the requested responses from a majority of the acceptors, then it can issue a proposal with numbern and value v, where v is the value of the highest-numbered proposal among the responses, or is any value selected by the proposer if the responders reported no proposals. 4 A proposer issues a proposal by sending, to some set of acceptors, a request that the proposal be accepted. (This need not be the same set of acceptors that responded to the initial requests.) Let’s call this anaccept request. This describes a proposer’s algorithm. What about an acceptor? It can receive two kinds of requests from proposers:prepare requests and accept requests. An acceptor can ignore any request without compromising safety. So, we need to say only when it is allowed to respond to a request. It can always respond to aprepare request. It can respond to anaccept request, accepting the proposal, iﬀ it has not promised not to. In other words: P1a. An acceptor can accept a proposal numberedn iﬀ it has not responded to aprepare request having a number greater thann. Observe that P1a subsumes P1. We now have a complete algorithm for choosing a value that satisﬁes the required safety properties—assuming unique proposal numbers. The ﬁnal algorithm is obtained by making one small optimization. Suppose an acceptor receives aprepare request numberedn, but it has already responded to aprepare request numbered greater thann, thereby promising not to accept any new proposal numberedn. There is then no reason for the acceptor to respond to the newprepare request, since it will not accept the proposal numberedn that the proposer wants to issue. So we have the acceptor ignore such aprepare request. We also have it ignore a prepare request for a proposal it has already accepted. With this optimization, an acceptor needs to remember only the highest- numbered proposal that it has ever accepted and the number of the highest- numbered prepare request to which it has responded. Because P2c must be kept invariant regardless of failures, an acceptor must remember this information even if it fails and then restarts. Note that the proposer can always abandon a proposal and forget all about it—as long as it never tries to issue another proposal with the same number. Putting the actions of the proposer and acceptor together, we see that the algorithm operates in the following two phases. Phase 1. (a) A proposer selects a proposal numbern and sends aprepare request with numbern to a majority of acceptors. (b) If an acceptor receives aprepare request with numbern greater than that of anyprepare request to which it has already responded, then it responds to the request with a promise not to accept any more proposals numbered less thann and with the highest-numbered pro- posal (if any) that it has accepted. 5 Phase 2. (a) If the proposer receives a response to itsprepare requests (numbered n) from a majority of acceptors, then it sends anaccept request to each of those acceptors for a proposal numberedn with a value v, wherev is the value of the highest-numbered proposal among the responses, or is any value if the responses reported no proposals. (b) If an acceptor receives anaccept request for a proposal numbered n, it accepts the proposal unless it has already responded to aprepare request having a number greater thann. A proposer can make multiple proposals, so long as it follows the algorithm for each one. It can abandon a proposal in the middle of the protocol at any time. (Correctness is maintained, even though requests and/or responses for the proposal may arrive at their destinations long after the proposal was abandoned.) It is probably a good idea to abandon a proposal if some proposer has begun trying to issue a higher-numbered one. Therefore, if an acceptor ignores aprepare or accept request because it has already received a prepare request with a higher number, then it should probably inform the proposer, who should then abandon its proposal. This is a performance optimization that does not aﬀect correctness. 2.3 Learning a Chosen Value To learn that a value has been chosen, a learner must ﬁnd out that a pro- posal has been accepted by a majority of acceptors. The obvious algorithm is to have each acceptor, whenever it accepts a proposal, respond to all learners, sending them the proposal. This allows learners to ﬁnd out about a chosen value as soon as possible, but it requires each acceptor to respond to each learner—a number of responses equal to the product of the number of acceptors and the number of learners. The assumption of non-Byzantine failures makes it easy for one learner to ﬁnd out from another learner that a value has been accepted. We can have the acceptors respond with their acceptances to a distinguished learner, which in turn informs the other learners when a value has been chosen. This approach requires an extra round for all the learners to discover the chosen value. It is also less reliable, since the distinguished learner could fail. But it requires a number of responses equal only to the sum of the number of acceptors and the number of learners. More generally, the acceptors could respond with their acceptances to some set of distinguished learners, each of which can then inform all the learners when a value has been chosen. Using a larger set of distinguished 6 learners provides greater reliability at the cost of greater communication complexity. Because of message loss, a value could be chosen with no learner ever ﬁnding out. The learner could ask the acceptors what proposals they have accepted, but failure of an acceptor could make it impossible to know whether or not a majority had accepted a particular proposal. In that case, learners will ﬁnd out what value is chosen only when a new proposal is chosen. If a learner needs to know whether a value has been chosen, it can have a proposer issue a proposal, using the algorithm described above. 2.4 Progress It’s easy to construct a scenario in which two proposers each keep issuing a sequence of proposals with increasing numbers, none of which are ever chosen. Proposer p completes phase 1 for a proposal numbern1. Another proposer q then completes phase 1 for a proposal numbern2 > n1. Proposer p’s phase 2accept requests for a proposal numberedn1 are ignored because the acceptors have all promised not to accept any new proposal numbered less than n2. So, proposer p then begins and completes phase 1 for a new proposal number n3 > n2, causing the second phase 2accept requests of proposer q to be ignored. And so on. To guarantee progress, a distinguished proposer must be selected as the only one to try issuing proposals. If the distinguished proposer can com- municate successfully with a majority of acceptors, and if it uses a proposal with number greater than any already used, then it will succeed in issuing a proposal that is accepted. By abandoning a proposal and trying again if it learns about some request with a higher proposal number, the distinguished proposer will eventually choose a high enough proposal number. If enough of the system (proposer, acceptors, and communication net- work) is working properly, liveness can therefore be achieved by electing a single distinguished proposer. The famous result of Fischer, Lynch, and Pat- terson [1] implies that a reliable algorithm for electing a proposer must use either randomness or real time—for example, by using timeouts. However, safety is ensured regardless of the success or failure of the election. 2.5 The Implementation The Paxos algorithm [5] assumes a network of processes. In its consensus algorithm, each process plays the role of proposer, acceptor, and learner. The algorithm chooses a leader, which plays the roles of the distinguished 7 proposer and the distinguished learner. The Paxos consensus algorithm is precisely the one described above, where requests and responses are sent as ordinary messages. (Response messages are tagged with the corresponding proposal number to prevent confusion.) Stable storage, preserved during failures, is used to maintain the information that the acceptor must re- member. An acceptor records its intended response in stable storage before actually sending the response. All that remains is to describe the mechanism for guaranteeing that no two proposals are ever issued with the same number. Diﬀerent proposers choose their numbers from disjoint sets of numbers, so two diﬀerent pro- posers never issue a proposal with the same number. Each proposer remem- bers (in stable storage) the highest-numbered proposal it has tried to issue, and begins phase 1 with a higher proposal number than any it has already used. 3 Implementing a State Machine A simple way to implement a distributed system is as a collection of clients that issue commands to a central server. The server can be described as a deterministic state machine that performs client commands in some se- quence. The state machine has a current state; it performs a step by taking as input a command and producing an output and a new state. For ex- ample, the clients of a distributed banking system might be tellers, and the state-machine state might consist of the account balances of all users. A withdrawal would be performed by executing a state machine command that decreases an account’s balance if and only if the balance is greater than the amount withdrawn, producing as output the old and new balances. An implementation that uses a single central server fails if that server fails. We therefore instead use a collection of servers, each one independently implementing the state machine. Because the state machine is deterministic, all the servers will produce the same sequences of states and outputs if they all execute the same sequence of commands. A client issuing a command can then use the output generated for it by any server. To guarantee that all servers execute the same sequence of state machine commands, we implement a sequence of separate instances of the Paxos consensus algorithm, the value chosen by theith instance being theith state machine command in the sequence. Each server plays all the roles (proposer, acceptor, and learner) in each instance of the algorithm. For now, I assume that the set of servers is ﬁxed, so all instances of the consensus algorithm 8 use the same sets of agents. In normal operation, a single server is elected to be the leader, which acts as the distinguished proposer (the only one that tries to issue proposals) in all instances of the consensus algorithm. Clients send commands to the leader, who decides where in the sequence each command should appear. If the leader decides that a certain client command should be the 135th command, it tries to have that command chosen as the value of the 135th instance of the consensus algorithm. It will usually succeed. It might fail because of failures, or because another server also believes itself to be the leader and has a diﬀerent idea of what the 135th command should be. But the consensus algorithm ensures that at most one command can be chosen as the 135th one. Key to the eﬃciency of this approach is that, in the Paxos consensus algorithm, the value to be proposed is not chosen until phase 2. Recall that, after completing phase 1 of the proposer’s algorithm, either the value to be proposed is determined or else the proposer is free to propose any value. I will now describe how the Paxos state machine implementation works during normal operation. Later, I will discuss what can go wrong. I consider what happens when the previous leader has just failed and a new leader has been selected. (System startup is a special case in which no commands have yet been proposed.) The new leader, being a learner in all instances of the consensus algo- rithm, should know most of the commands that have already been chosen. Suppose it knows commands 1–134, 138, and 139—that is, the values cho- sen in instances 1–134, 138, and 139 of the consensus algorithm. (We will see later how such a gap in the command sequence could arise.) It then executes phase 1 of instances 135–137 and of all instances greater than 139. (I describe below how this is done.) Suppose that the outcome of these ex- ecutions determine the value to be proposed in instances 135 and 140, but leaves the proposed value unconstrained in all other instances. The leader then executes phase 2 for instances 135 and 140, thereby choosing commands 135 and 140. The leader, as well as any other server that learns all the commands the leader knows, can now execute commands 1–135. However, it can’t execute commands 138–140, which it also knows, because commands 136 and 137 have yet to be chosen. The leader could take the next two commands requested by clients to be commands 136 and 137. Instead, we let it ﬁll the gap immediately by proposing, as commands 136 and 137, a special “no- op” command that leaves the state unchanged. (It does this by executing phase 2 of instances 136 and 137 of the consensus algorithm.) Once these 9 no-op commands have been chosen, commands 138–140 can be executed. Commands 1–140 have now been chosen. The leader has also completed phase 1 for all instances greater than 140 of the consensus algorithm, and it is free to propose any value in phase 2 of those instances. It assigns command number 141 to the next command requested by a client, proposing it as the value in phase 2 of instance 141 of the consensus algorithm. It proposes the next client command it receives as command 142, and so on. The leader can propose command 142 before it learns that its proposed command 141 has been chosen. It’s possible for all the messages it sent in proposing command 141 to be lost, and for command 142 to be chosen before any other server has learned what the leader proposed as command 141. When the leader fails to receive the expected response to its phase 2 messages in instance 141, it will retransmit those messages. If all goes well, its proposed command will be chosen. However, it could fail ﬁrst, leaving a gap in the sequence of chosen commands. In general, suppose a leader can get α commands ahead—that is, it can propose commandsi + 1 through i +α after commands 1 throughi are chosen. A gap of up toα−1 commands could then arise. A newly chosen leader executes phase 1 for inﬁnitely many instances of the consensus algorithm—in the scenario above, for instances 135–137 and all instances greater than 139. Using the same proposal number for all instances, it can do this by sending a single reasonably short message to the other servers. In phase 1, an acceptor responds with more than a simple OK only if it has already received a phase 2 message from some proposer. (In the scenario, this was the case only for instances 135 and 140.) Thus, a server (acting as acceptor) can respond for all instances with a single reasonably short message. Executing these inﬁnitely many instances of phase 1 therefore poses no problem. Since failure of the leader and election of a new one should be rare events, the eﬀective cost of executing a state machine command—that is, of achieving consensus on the command/value—is the cost of executing only phase 2 of the consensus algorithm. It can be shown that phase 2 of the Paxos consensus algorithm has the minimum possible cost of any algorithm for reaching agreement in the presence of faults [2]. Hence, the Paxos algo- rithm is essentially optimal. This discussion of the normal operation of the system assumes that there is always a single leader, except for a brief period between the failure of the current leader and the election of a new one. In abnormal circumstances, the leader election might fail. If no server is acting as leader, then no new commands will be proposed. If multiple servers think they are leaders, then 10 they can all propose values in the same instance of the consensus algo- rithm, which could prevent any value from being chosen. However, safety is preserved—two diﬀerent servers will never disagree on the value chosen as the ith state machine command. Election of a single leader is needed only to ensure progress. If the set of servers can change, then there must be some way of deter- mining what servers implement what instances of the consensus algorithm. The easiest way to do this is through the state machine itself. The current set of servers can be made part of the state and can be changed with ordi- nary state-machine commands. We can allow a leader to getα commands ahead by letting the set of servers that execute instancei + α of the con- sensus algorithm be speciﬁed by the state after execution of theith state machine command. This permits a simple implementation of an arbitrarily sophisticated reconﬁguration algorithm. References [1] Michael J. Fischer, Nancy Lynch, and Michael S. Paterson. Impossibility of distributed consensus with one faulty process.Journal of the ACM, 32(2):374–382, April 1985. [2] Idit Keidar and Sergio Rajsbaum. On the cost of fault-tolerant consensus when there are no faults—a tutorial. TechnicalReport MIT-LCS-TR-821, Laboratory for Computer Science, Massachusetts Institute Technology, Cambridge, MA, 02139, May 2001. also published in SIGACT News 32(2) (June 2001). [3] Leslie Lamport. The implementation of reliable distributed multiprocess systems. Computer Networks, 2:95–114, 1978. [4] Leslie Lamport. Time, clocks, and the ordering of events in a distributed system. Communications of the ACM, 21(7):558–565, July 1978. [5] Leslie Lamport. The part-time parliament.ACM Transactions on Com- puter Systems, 16(2):133–169, May 1998. 11
pca	A T utorial on Principal Component Analysis Jonathon Shlens∗ Systems Neurobiology Laboratory , Salk Insitute for Biological Studies La Jolla, CA 92037 and Institute for Nonlinear Science, University of California, San Die go La Jolla, CA 92093-0402 (Dated: December 10, 2005; Version 2) Principal component analysis (PCA) is a mainstay of modern d ata analysis - a black box that is widely used but poorly understood. The goal of this paper i s to dispel the magic behind this black box. This tutorial focuses on building a solid intuiti on for how and why principal component analysis works; furthermore, it crystallizes this knowled ge by deriving from simple intuitions, the mathematics behind PCA . This tutorial does not shy away from explaining the ideas in formally , nor does it shy away from the mathematics. The hope is that by a ddressing both aspects, readers of all levels will be able to gain a better understanding of PCA as well as the when, the how and the why of applying this technique. I. INTRODUCTION Principal component analysis ( PCA) has been called one of the most valuable results from applied linear al- gebra. PCA is used abundantly in all forms of analysis - from neuroscience to computer graphics - because it is a simple, non-parametric method of extracting relevant in- formation from confusing data sets. With minimal addi- tional eﬀort PCA provides a roadmap for how to reduce a complex data set to a lower dimension to reveal the sometimes hidden, simpliﬁed structure that often under- lie it. The goal of this tutorial is to provide both an intuitive feel for PCA, and a thorough discussion of this topic. W e will begin with a simple example and provide an intu- itive explanation of the goal of PCA. W e will continue by adding mathematical rigor to place it within the frame- work of linear algebra to provide an explicit solution. W e will see how and why PCA is intimately related to the mathematical technique of singular value decomposition (SVD). This understanding will lead us to a prescription for how to apply PCA in the real world. W e will discuss both the assumptions behind this technique as well as possible extensions to overcome these limitations. The discussion and explanations in this paper are infor- mal in the spirit of a tutorial. The goal of this paper is to educate. Occasionally , rigorous mathematical proofs are necessary although relegated to the Appendix. Although not as vital to the tutorial, the proofs are presented for the adventurous reader who desires a more complete un- derstanding of the math. The only assumption is that the reader has a working knowledge of linear algebra. Please feel free to contact me with any suggestions, corrections or comments. ∗Electronic address: [email protected] II. MOTIV A TION: A TOY EXAMPLE Here is the perspective: we are an experimenter. W e are trying to understand some phenomenon by measur- ing various quantities (e.g. spectra, voltages, velocities, etc.) in our system. Unfortunately , we can not ﬁgure out what is happening because the data appears clouded, un- clear and even redundant. This is not a trivial problem, but rather a fundamental obstacle in empirical science. Examples abound from complex systems such as neu- roscience, photometry , meteorology and oceanography - the number of variables to measure can be unwieldy and at times even deceptive, because the underlying relation- ships can often be quite simple. T ake for example a simple toy problem from physics diagrammed in Figure 1. Pretend we are studying the motion of the physicist’s ideal spring. This system con- sists of a ball of mass m attached to a massless, friction- less spring. The ball is released a small distance away from equilibrium (i.e. the spring is stretched). Because the spring is “ideal,” it oscillates indeﬁnitely along the x-axis about its equilibrium at a set frequency . This is a standard problem in physics in which the mo- tion along the x direction is solved by an explicit function of time. In other words, the underlying dynamics can be expressed as a function of a single variable x. However, being ignorant experimenters we do not know any of this. W e do not know which, let alone how many , axes and dimensions are important to measure. Thus, we decide to measure the ball’s position in a three-dimensional space (since we live in a three dimen- sional world). Speciﬁcally , we place three movie cameras around our system of interest. At 200 Hz each movie camera records an image indicating a two dimensional position of the ball (a projection). Unfortunately , be- cause of our ignorance, we do not even know what are the real “ x”, “ y” and “ z” axes, so we choose three cam- era axes {⃗a, ⃗b, ⃗c} at some arbitrary angles with respect to the system. The angles between our measurements 2 FIG. 1 A diagram of the toy example. might not even be 90 o! Now, we record with the cameras for several minutes. The big question remains: how do we get from this data set to a simple equation of x? W e know a-priori that if we were smart experimenters, we would have just measured the position along the x- axis with one camera. But this is not what happens in the real world. W e often do not know which measurements best reﬂect the dynamics of our system in question. F ur- thermore, we sometimes record more dimensions than we actually need! Also, we have to deal with that pesky , real-world prob- lem of noise. In the toy example this means that we need to deal with air, imperfect cameras or even friction in a less-than-ideal spring. Noise contaminates our data set only serving to obfuscate the dynamics further. This toy example is the challenge experimenters face everyday. W e will refer to this example as we delve further into ab- stract concepts. Hopefully , by the end of this paper we will have a good understanding of how to systematically extract x using principal component analysis. III. FRAMEWORK: CHANGE OF BASIS The goal of principal component analysis is to compute the most meaningful basis to re-express a noisy data set. The hope is that this new basis will ﬁlter out the noise and reveal hidden structure. In the example of the spring, the explicit goal of PCA is to determine: “the dynamics are along the x-axis.” In other words, the goal of PCA is to determine that ˆx - the unit basis vector along the x-axis - is the important dimension. Determining this fact allows an experimenter to discern which dynamics are important, which are just redundant and which are just noise. A. A Naive Basis With a more precise deﬁnition of our goal, we need a more precise deﬁnition of our data as well. W e treat every time sample (or experimental trial) as an individual sample in our data set. At each time sample we record a set of data consisting of multiple measurements (e.g. voltage, position, etc.). In our data set, at one point in time, camera A records a corresponding ball position (xA, yA). One sample or trial can then be expressed as a 6 dimensional column vector ⃗X =        xA yA xB yB xC yC        where each camera contributes a 2-dimensional projec- tion of the ball’s position to the entire vector ⃗X. If we record the ball’s position for 10 minutes at 120 Hz, then we have recorded 10 × 60 × 120 = 72000 of these vectors. With this concrete example, let us recast this problem in abstract terms. Each sample ⃗X is an m-dimensional vector, where m is the number of measurement types. Equivalently , every sample is a vector that lies in an m- dimensional vector space spanned by some orthonormal basis. F rom linear algebra we know that all measurement vectors form a linear combination of this set of unit length basis vectors. What is this orthonormal basis? This question is usually a tacit assumption often over- looked. Pretend we gathered our toy example data above, but only looked at camera A. What is an orthonormal ba- sis for ( xA, yA)? A naive choice would be {(1, 0), (0, 1)}, but why select this basis over {( √ 2 2 , √ 2 2 ), ( − √ 2 2 , − √ 2 2 )} or any other arbitrary rotation? The reason is that the naive basis reﬂects the method we gathered the data. Pre- tend we record the position (2 , 2). W e did not record 2 √ 2 in the ( √ 2 2 , √ 2 2 ) direction and 0 in the perpindicu- lar direction. Rather, we recorded the position (2 , 2) on our camera meaning 2 units up and 2 units to the left in our camera window. Thus our naive basis reﬂects the method we measured our data. How do we express this naive basis in linear algebra? In the two dimensional case, {(1, 0), (0, 1)} can be recast as individual row vectors. A matrix constructed out of these row vectors is the 2 × 2 identity matrix I. W e can generalize this to the m-dimensional case by constructing an m × m identity matrix B =      b1 b2 . . . bm      =      1 0 · · ·0 0 1 · · ·0 . . . . . . . . . . . . 0 0 · · ·1      = I where each row is an orthornormal basis vector bi with m components. W e can consider our naive basis as the eﬀective starting point. All of our data has been recorded in this basis and thus it can be trivially expressed as a linear combination of {bi}. 3 B. Change of Basis With this rigor we may now state more precisely what PCA asks: Is there another basis, which is a linear com- bination of the original basis, that best re-expresses our data set? A close reader might have noticed the conspicuous ad- dition of the word linear. Indeed, PCA makes one strin- gent but powerful assumption: linearity. Linearity vastly simpliﬁes the problem by (1) restricting the set of poten- tial bases, and (2) formalizing the implicit assumption of continuity in a data set. 1 With this assumption PCA is now limited to re- expressing the data as a linear combination of its ba- sis vectors. Let X be the original data set, where each column is a single sample (or moment in time) of our data set (i.e. ⃗X). In the toy example X is an m × n matrix where m = 6 and n = 72000. Let Y be another m × n matrix related by a linear transformation P. X is the original recorded data set and Y is a re-representation of that data set. PX = Y (1) Also let us deﬁne the following quantities. 2 • pi are the rows of P • xi are the columns of X (or individual ⃗X). • yi are the columns of Y. Equation 1 represents a change of basis and thus can have many interpretations. 1. P is a matrix that transforms X into Y. 2. Geometrically , P is a rotation and a stretch which again transforms X into Y. 3. The rows of P, {p1, . . . , pm}, are a set of new basis vectors for expressing the columns of X. The latter interpretation is not obvious but can be seen by writing out the explicit dot products of PX. PX =    p1 . . . pm    [ x1 · · ·xn ] Y =    p1 ·x1 · · ·p1 ·xn . . . . . . . . . pm ·x1 · · ·pm ·xn    1 A subtle point it is, but we have already assumed linearity by implicitly stating that the data set even characterizes the dy- namics of the system. In other words, we are already relying o n the superposition principal of linearity to believe that th e data provides an ability to interpolate between individual data points 2 In this section xi and yi are column vectors, but be forewarned. In all other sections xi and yi are row vectors. W e can note the form of each column of Y. yi =    p1 ·xi . . . pm ·xi    W e recognize that each coeﬃcient of yi is a dot-product of xi with the corresponding row in P. In other words, the jth coeﬃcient of yi is a projection on to the jth row of P. This is in fact the very form of an equation where yi is a projection on to the basis of {p1, . . . , pm}. Therefore, the rows of P are indeed a new set of basis vectors for representing of columns of X. C. Questions Remaining By assuming linearity the problem reduces to ﬁnd- ing the appropriate change of basis . The row vectors {p1, . . . , pm} in this transformation will become the principal components of X. Several questions now arise. • What is the best way to “re-express” X? • What is a good choice of basis P? These questions must be answered by next asking our- selves what features we would like Y to exhibit . Evi- dently , additional assumptions beyond linearity are re- quired to arrive at a reasonable result. The selection of these assumptions is the subject of the next section. IV. V ARIANCE AND THE GOAL Now comes the most important question: what does “best express” the data mean? This section will build up an intuitive answer to this question and along the way tack on additional assumptions. The end of this section will conclude with a mathematical goal for deciphering “garbled” data. In a linear system “garbled” can refer to only three potential confounds: noise, rotation and redundancy. Let us deal with each situation individually . A. Noise and Rotation Measurement noise in any data set must be low or else, no matter the analysis technique, no information about a system can be extracted. There exists no absolute scale for noise but rather all noise is measured relative to the measurement. A common measure is the signal-to-noise ratio (SNR), or a ratio of variances σ2, SN R = σ2 signal σ2 noise . (2) A high SNR (≫ 1) indicates high precision data, while a low SNR indicates noise contaminated data. 4 (a) xA yA (b) p* variance along p * 0 45 90 p* angle of p (degrees) SNR FIG. 2 (a) Simulated data of ( xA , y A ) for camera A. The signal and noise variances σ 2 signal and σ 2 noise are graphically represented by the two lines subtending the cloud of data. (b) Rotating these axes ﬁnds an optimal p∗ where the variance and SNR are maximized. The SNR is deﬁned as the ratio of the variance along p∗ and the variance in the perpindicular direction. Pretend we plotted all data from camera A from the spring in Figure 2a. Remembering that the spring travels in a straight line, every individual camera should record motion in a straight line as well. Therefore, any spread deviating from straight-line motion must be noise. The variance due to the signal and noise are indicated by each line in the diagram. The ratio of the two lengths, the SNR, thus measures how “fat” the cloud is - a range of possiblities include a thin line ( SNR ≫ 1), a perfect circle (SNR = 1) or even worse. By positing reasonably good measurements, quantitatively we assume that directions with largest variances in our measurement vector space contain the dynamics of interest. In Figure 2 the di- rection with the largest variance is not ˆ xA = (1 , 0) nor ˆyA = (0 , 1), but the direction along the long axis of the cloud. Thus, by assumption the dynamics of interest ex- ist along directions with largest variance and presumably highest SNR . Our assumption suggests that the basis for which we are searching is not the naive basis (ˆ xA, ˆyA) because these directions do not correspond to the directions of largest variance. Maximizing the variance (and by assumption the SNR ) corresponds to ﬁnding the appropriate rota- tion of the naive basis. This intuition corresponds to ﬁnding the direction p∗ in Figure 2b. How do we ﬁnd p∗? In the 2-dimensional case of Figure 2a, p∗ falls along the direction of the best-ﬁt line for the data cloud. Thus, rotating the naive basis to lie parallel to p∗ would reveal the direction of motion of the spring for the 2-D case. How do we generalize this notion to an arbitrary number of dimensions? Before we approach this question we need to examine this issue from a second perspective. B. Redundancy Figure 2 hints at an additional confounding factor in our data - redundancy . This issue is particularly evident in the example of the spring. In this case multiple sensors FIG. 3 A spectrum of possible redundancies in data from the two separate recordings r1 and r2 (e.g. xA , y B ). The best-ﬁt line r2 = kr1 is indicated by the dashed line. record the same dynamic information. Reconsider Fig- ure 2 and ask whether it was really necessary to record 2 variables? Figure 3 might reﬂect a range of possibile plots between two arbitrary measurement types r1 and r2. Panel (a) depicts two recordings with no apparent relationship. In other words, r1 is entirely uncorrelated with r2. Because one can not predict r1 from r2, one says that r1 and r2 are statistcally independent. This situation could occur by plotting two variables such as (xA, humidity ). On the other extreme, Figure 3c depicts highly corre- lated recordings. This extremity might be achieved by several means: • A plot of ( xA, xB ) if cameras A and B are very nearby . • A plot of ( xA, ˜xA) where xA is in meters and ˜ xA is in inches. Clearly in panel (c) it would be more meaningful to just have recorded a single variable, not both. Why? Because one can calculate r1 from r2 (or vice versa) using the best- ﬁt line. Recording solely one response would express the data more concisely and reduce the number of sensor recordings (2 → 1 variables). Indeed, this is the very idea behind dimensional reduction. C. Covariance Matrix In a 2 variable case it is simple to identify redundant cases by ﬁnding the slope of the best-ﬁt line and judging the quality of the ﬁt. How do we quantify and generalize these notions to arbitrariliy higher dimensions? Consider two sets of measurements with zero means 3 A = {a1, a2, . . . , a n} , B = {b1, b2, . . . , b n} 3 These data sets are in mean deviation form because the means have been subtracted oﬀ or are zero. 5 where the subscript denotes the sample number. The variance of A and B are individually deﬁned as, σ2 A = ⟨aiai⟩i , σ 2 B = ⟨bibi⟩i where the expectation 4 is the average over n variables. The covariance between A and B is a straight-forward generalization. covariance of A and B ≡ σ2 AB = ⟨aibi⟩i The covariance measures the degree of the linear rela- tionship between two variables. A large (small) value in- dicates high (low) redundancy . In the case of Figures 2a and 3c, the covariances are quite large. Some additional facts about the covariance. • σ2 AB ≥ 0 (non-negative). σAB is zero if and only if A and B are entirely uncorrelated. • σ2 AB = σ2 A if A = B. W e can equivalently convert A and B into corresponding row vectors. a = [ a1 a2 . . . a n] b = [ b1 b2 . . . b n] so that we may now express the covariance as a dot prod- uct matrix computation. σ2 ab ≡ 1 n − 1 abT (3) where 1 n−1 is a constant for normalization. 5 Finally , we can generalize from two vectors to an arbi- trary number. W e can rename the row vectors x1 ≡ a, x2 ≡ b and consider additional indexed row vectors x3, . . . , xm. Now we can deﬁne a new m × n matrix X. X =    x1 . . . xm    (4) One interpretation of X is the following. Each row of X corresponds to all measurements of a particular type ( xi). Each column of X corresponds to a set of measurements from one particular trial (this is ⃗X from section 3.1). W e now arrive at a deﬁnition for the covariance matrix CX. CX ≡ 1 n − 1 XXT . (5) 4 ⟨·⟩i denotes the average over values indexed by i. 5 The most straight-forward normalization is 1 n . However, this provides a biased estimation of variance particularly for s mall n. It is beyond the scope of this paper to show that the proper normalization for an unbiased estimator is 1 n−1 . Consider how the matrix form XXT , in that order, com- putes the desired value for the ijth element of CX. Speciﬁcally , the ijth element of CX is the dot product between the vector of the ith measurement type with the vector of the jth measurement type (or substituting xi and xj for a and b into Equation 3). W e can summarize several properties of CX: • CX is a square symmetric m × m matrix. • The diagonal terms of CX are the variance of par- ticular measurement types. • The oﬀ-diagonal terms of CX are the covariance between measurement types. CX captures the correlations between all possible pairs of measurements. The correlation values reﬂect the noise and redundancy in our measurements. • In the diagonal terms, by assumption, large (small) values correspond to interesting dynamics (or noise). • In the oﬀ-diagonal terms large (small) values cor- respond to high (low) redundancy . Pretend we have the option of manipulating CX. W e will suggestively deﬁne our manipulated covariance ma- trix CY. What features do we want to optimize in CY? D. Diagonalize the Covariance Matrix W e can summarize the last two sections by stating that our goals are (1) to minimize redundancy , mea- sured by covariance, and (2) maximize the signal, mea- sured by variance. By deﬁnition covariances must be non-negative, thus the minimal covariance is zero. What would the optimized covariance matrix CY look like? Ev- idently , in an “optimized” matrix all oﬀ-diagonal terms in CY are zero. Thus, CY must be diagonal. There are many methods for diagonalizing CY. It is curious to note that PCA arguably selects the easiest method, perhaps accounting for its widespread applica- tion. PCA assumes that all basis vectors {p1, . . . , pm} are orthonormal (i.e. pi ·pj = δij ). In the language of linear algebra, PCA assumes P is an orthonormal matrix. Sec- ondly PCA assumes the directions with the largest vari- ances the signals and the most “important.” In other words, they are the most principal. Why are these as- sumptions easiest? Envision how PCA works. In our simple example in Figure 2b, P acts as a generalized rotation to align a basis with the maximally variant axis. In multiple dimensions this could be performed by a simple algorithm: 1. Select a normalized direction in m-dimensional space along which the variance in X is maximized. Save this vector as p1. 6 2. Find another direction along which variance is max- imized, however, because of the orthonormality condition, restrict the search to all directions per- pindicular to all previous selected directions. Save this vector as pi 3. Repeat this procedure until m vectors are selected. The resulting ordered set of p’s are the principal compo- nents. In principle this simple algorithm works, however that would bely the true reason why the orthonormality as- sumption is particularly judicious. Namely , the true beneﬁt to this assumption is that it makes the solution amenable to linear algebra . There exist decompositions that can provide eﬃcient, explicit algebraic solutions. Notice what we also gained with the second assump- tion. W e have a method for judging the “importance” of the each prinicipal direction. Namely , the variances as- sociated with each direction pi quantify how “principal” each direction is. W e could thus rank-order each basis vector pi according to the corresponding variances.W e will now pause to review the implications of all the as- sumptions made to arrive at this mathematical goal. E. Summary of Assumptions and Limits This section provides an important context for under- standing when PCA might perform poorly as well as a road map for understanding new extensions to PCA. The Discussion returns to this issue and provides a more lengthy discussion. I. Linearity Linearity frames the problem as a change of basis. Several areas of research have explored how applying a nonlinearity prior to perform- ing PCA could extend this algorithm - this has been termed kernel PCA . II. Mean and variance are suﬃcient statistics . The formalism of suﬃcient statistics captures the notion that the mean and the variance en- tirely describe a probability distribution. The only class of probability distributions that are fully described by the ﬁrst two moments are exponential distributions (e.g. Gaussian, Ex- ponential, etc). In order for this assumption to hold, the prob- ability distribution of xi must be exponen- tially distributed. Deviations from this could invalidate this assumption. 6 On the ﬂip side, 6 A sidebar question: What if xi are not Gaussian distributed? Diagonalizing a covariance matrix might not produce satisf ac- tory results. The most rigorous form of removing redundancy is this assumption formally guarantees that the SNR and the covariance matrix fully charac- terize the noise and redundancies. III. Large variances have important dynamics. This assumption also encompasses the belief that the data has a high SNR. Hence, prin- cipal components with larger associated vari- ances represent interesting dynamics, while those with lower variances represent noise. IV. The principal components are orthogonal. This assumption provides an intuitive sim- pliﬁcation that makes PCA soluble with lin- ear algebra decomposition techniques. These techniques are highlighted in the two follow- ing sections. W e have discussed all aspects of deriving PCA - what remain are the linear algebra solutions. The ﬁrst solution is somewhat straightforward while the second solution involves understanding an important algebraic decompo- sition. V. SOL VING PCA: EIGENVECTORS OF COV ARIANCE W e derive our ﬁrst algebraic solution to PCA using linear algebra. This solution is based on an important property of eigenvector decomposition. Once again, the data set is X, an m × n matrix, where m is the number of measurement types and n is the number of samples. The goal is summarized as follows. Find some orthonormal matrix P where Y = PX such that CY ≡ 1 n−1 YYT is diago- nalized. The rows of P are the principal com- ponents of X. statistical independence. P (y1, y2) = P (y1)P (y2) where P (·) denotes the probability density . The class of algo- rithms that attempt to ﬁnd the y1, , . . . , ym that satisfy this statistical constraint are termed independent component a naly- sis ( ICA). In practice though, quite a lot of real world data are Gaussian distributed - thanks to the Central Limit Theorem - and PCA is usually a robust solution to slight deviations from this assumption. 7 W e begin by rewriting CY in terms of our variable of choice P. CY = 1 n − 1 YYT = 1 n − 1 (PX)(PX)T = 1 n − 1 PXXT PT = 1 n − 1 P(XXT )PT CY = 1 n − 1 PAPT Note that we deﬁned a new matrix A ≡ XXT , where A is symmetric (by Theorem 2 of Appendix A). Our roadmap is to recognize that a symmetric matrix (A) is diagonalized by an orthogonal matrix of its eigen- vectors (by Theorems 3 and 4 from Appendix A). F or a symmetric matrix A Theorem 4 provides: A = EDET (6) where D is a diagonal matrix and E is a matrix of eigen- vectors of A arranged as columns. The matrix A has r ≤ m orthonormal eigenvectors where r is the rank of the matrix. The rank of A is less than m when A is degenerate or all data occupy a sub- space of dimension r ≤ m. Maintaining the constraint of orthogonality , we can remedy this situation by selecting (m − r) additional orthonormal vectors to “ﬁll up” the matrix E. These additional vectors do not eﬀect the ﬁ- nal solution because the variances associated with these directions are zero. Now comes the trick. We select the matrix P to be a matrix where each row pi is an eigenvector of XXT . By this selection, P ≡ ET. Substituting into Equation 6, we ﬁnd A = PT DP. With this relation and Theorem 1 of Appendix A ( P−1 = PT ) we can ﬁnish evaluating CY. CY = 1 n − 1 P APT = 1 n − 1 P(PT DP)PT = 1 n − 1 (PPT )D(PPT ) = 1 n − 1 (PP−1)D(PP−1) CY = 1 n − 1 D It is evident that the choice of P diagonalizes CY. This was the goal for PCA. W e can summarize the results of PCA in the matrices P and CY. • The principal components of X are the eigenvectors of XXT ; or the rows of P. • The ith diagonal value of CY is the variance of X along pi. In practice computing PCA of a data set X entails (1) subtracting oﬀ the mean of each measurement type and (2) computing the eigenvectors of XXT . This solution is encapsulated in demonstration Matlab code included in Appendix B. VI. A MORE GENERAL SOLUTION: SVD This section is the most mathematically involved and can be skipped without much loss of continuity . It is pre- sented solely for completeness. W e derive another alge- braic solution for PCA and in the process, ﬁnd that PCA is closely related to singular value decomposition ( SVD). In fact, the two are so intimately related that the names are often used interchangeably . What we will see though is that SVD is a more general method of understanding change of basis . W e begin by quickly deriving the decomposition. In the following section we interpret the decomposition and in the last section we relate these results to PCA. A. Singular Value Decomposition Let X be an arbitrary n × m matrix7 and XT X be a rank r, square, symmetric n × n matrix. In a seemingly unmotivated fashion, let us deﬁne all of the quantities of interest. • { ˆ v1, ˆ v2, . . . , ˆ vr } is the set of orthonormal m × 1 eigenvectors with associated eigenval- ues {λ1, λ2, . . . , λ r } for the symmetric matrix XT X. (XT X)ˆ vi = λi ˆ vi • σi ≡ √ λi are positive real and termed the singular values. • { ˆ u1, ˆ u2, . . . , ˆ ur } is the set of orthonormal n ×1 vec- tors deﬁned by ˆ ui ≡ 1 σ i Xˆ vi. W e obtain the ﬁnal deﬁnition by referring to Theorem 5 of Appendix A. The ﬁnal deﬁnition includes two new and unexpected properties. • ˆ ui ·ˆ uj = δij • ∥Xˆ vi∥= σi These properties are both proven in Theorem 5. W e now have all of the pieces to construct the decomposition. The 7 Notice that in this section only we are reversing convention from m × n to n × m. The reason for this derivation will become clear in section 6.3. 8 “value” version of singular value decomposition is just a restatement of the third deﬁnition. Xˆ vi = σi ˆ ui (7) This result says a quite a bit. X multiplied by an eigen- vector of XT X is equal to a scalar times another vec- tor. The set of eigenvectors {ˆ v1, ˆ v2, . . . , ˆ vr } and the set of vectors {ˆ u1, ˆ u2, . . . , ˆ ur } are both orthonormal sets or bases in r-dimensional space. W e can summarize this result for all vectors in one matrix multiplication by following the prescribed con- struction in Figure 4. W e start by constructing a new diagonal matrix Σ . Σ ≡           σ˜ 1 . . . 0 σ˜r 0 0 . . . 0           where σ˜ 1 ≥ σ˜ 2 ≥ . . . ≥ σ˜r are the rank-ordered set of sin- gular values. Likewise we construct accompanying or- thogonal matrices V and U. V = [ ˆ v˜ 1 ˆ v˜ 2 . . . ˆ v˜ m] U = [ ˆ u˜ 1 ˆ u˜ 2 . . . ˆ u˜ n] where we have appended an additional ( m−r) and ( n−r) orthonormal vectors to “ﬁll up” the matrices for V and U respectively8. Figure 4 provides a graphical represen- tation of how all of the pieces ﬁt together to form the matrix version of SVD. XV = UΣ (8) where each column of V and U perform the “value” ver- sion of the decomposition (Equation 7). Because V is orthogonal, we can multiply both sides by V−1 = VT to arrive at the ﬁnal form of the decomposition. X = UΣV T (9) Although it was derived without motivation, this decom- position is quite powerful. Equation 9 states that any arbitrary matrix X can be converted to an orthogonal matrix, a diagonal matrix and another orthogonal matrix (or a rotation, a stretch and a second rotation). Making sense of Equation 9 is the subject of the next section. 8 This is the same procedure used to ﬁx the degeneracy in the previous section. B. Interpreting SVD The ﬁnal form of SVD (Equation 9) is a concise but thick statement to understand. Let us instead reinterpret Equation 7. Xa = kb where a and b are column vectors and k is a scalar con- stant. The set {ˆ v1, ˆ v2, . . . , ˆ vm} is analogous to a and the set {ˆ u1, ˆ u2, . . . , ˆ un} is analogous to b. What is unique though is that {ˆ v1, ˆ v2, . . . , ˆ vm} and {ˆ u1, ˆ u2, . . . , ˆ un} are orthonormal sets of vectors which span an m or n dimen- sional space, respectively . In particular, loosely speak- ing these sets appear to span all possible “inputs” ( a) and “outputs” ( b). Can we formalize the view that {ˆ v1, ˆ v2, . . . , ˆ vn} and {ˆ u1, ˆ u2, . . . , ˆ un} span all possible “inputs” and “outputs”? W e can manipulate Equation 9 to make this fuzzy hy- pothesis more precise. X = UΣV T UT X = ΣV T UT X = Z where we have deﬁned Z ≡ ΣV T . Note that the previous columns {ˆ u1, ˆ u2, . . . , ˆ un} are now rows in UT . Compar- ing this equation to Equation 1, {ˆ u1, ˆ u2, . . . , ˆ un} per- form the same role as {ˆ p1, ˆ p2, . . . , ˆ pm}. Hence, UT is a change of basis from X to Z. Just as before, we were transforming column vectors, we can again infer that we are transforming column vectors. The fact that the or- thonormal basis UT (or P) transforms column vectors means that UT is a basis that spans the columns of X. Bases that span the columns are termed the column space of X. The column space formalizes the notion of what are the possible “outputs” of any matrix. There is a funny symmetry to SVD such that we can deﬁne a similar quantity - the row space . XV = ΣU (XV)T = ( ΣU )T VT XT = UT Σ VT XT = Z where we have deﬁned Z ≡ UTΣ . Again the rows of VT (or the columns of V) are an orthonormal basis for transforming XT into Z. Because of the transpose on X, it follows that V is an orthonormal basis spanning the row space of X. The row space likewise formalizes the notion of what are possible “inputs” into an arbitrary matrix. W e are only scratching the surface for understanding the full implications of SVD. F or the purposes of this tu- torial though, we have enough information to understand how PCA will fall within this framework. 9 The “value” form of SVD is expressed in equation 7. Xˆ vi = σ i ˆ ui The mathematical intuition behind the construction of the matrix for m is that we want to express all n “value” equations in just one equation. It is easiest to understand this process graphic ally . Drawing the matrices of equation 7 looks likes the following. We can construct three new matrices V, U and Σ . All singular values are ﬁrst rank-ordered σ ˜ 1 ≥ σ ˜ 2 ≥ . . . ≥ σ ˜r , and the corresponding vectors are indexed in the same rank order. Each pai r of associated vectors ˆ vi and ˆ ui is stacked in the ith column along their respective matrices. The corresponding singula r value σ i is placed along the diagonal (the iith position) of Σ . This generates the equation XV = UΣ , which looks like the following. The matrices V and U are m × m and n × n matrices respectively and Σ is a diagonal matrix with a few non-zero values (represented by the checkerboard) along its diagonal. Solving th is single matrix equation solves all n “value” form equations. FIG. 4 How to construct the matrix form of SVD from the “value” form. C. SVD and PCA With similar computations it is evident that the two methods are intimately related. Let us return to the original m × n data matrix X. W e can deﬁne a new matrix Y as an n × m matrix9. Y ≡ 1 √n − 1 XT where each column of Y has zero mean. The deﬁnition of Y becomes clear by analyzing YT Y. YT Y = ( 1√n − 1 XT ) T ( 1√n − 1 XT ) = 1 n − 1 XT T XT = 1 n − 1 XXT YT Y = CX 9 Y is of the appropriate n × m dimensions laid out in the deriva- tion of section 6.1. This is the reason for the “ﬂipping” of di - mensions in 6.1 and Figure 4. By construction YT Y equals the covariance matrix of X. F rom section 5 we know that the principal compo- nents of X are the eigenvectors of CX. If we calculate the SVD of Y, the columns of matrix V contain the eigen- vectors of YT Y = CX. Therefore, the columns of V are the principal components of X. This second algorithm is encapsulated in Matlab code included in Appendix B. What does this mean? V spans the row space of Y ≡ 1√n−1 XT . Therefore, V must also span the column space of 1√n−1 X. W e can conclude that ﬁnding the principal components10 amounts to ﬁnding an orthonormal basis that spans the column space of X. 10 If the ﬁnal goal is to ﬁnd an orthonormal basis for the coulmn space of X then we can calculate it directly without constructing Y. By symmetry the columns of U produced by the SVD of 1 √n−1 X must also be the principal components. 10 FIG. 5 Data points (black dots) tracking a person on a ferris wheel. The extracted principal components are ( p1, p2) and the phase is ˆθ . VII. DISCUSSION AND CONCLUSIONS A. Quick Summary Performing PCA is quite simple in practice. 1. Organize a data set as an m × n matrix, where m is the number of measurement types and n is the number of trials. 2. Subtract oﬀ the mean for each measurement type or row xi. 3. Calculate the SVD or the eigenvectors of the co- variance. In several ﬁelds of literature, many authors refer to the individual measurement types xi as the sources. The data projected into the principal components Y = PX are termed the signals, because the projected data pre- sumably represent the underlying items of interest. B. Dimensional Reduction One beneﬁt of PCA is that we can examine the vari- ances CY associated with the principle components. Of- ten one ﬁnds that large variances associated with the ﬁrst k < m principal components, and then a precipi- tous drop-oﬀ. One can conclude that most interesting dynamics occur only in the ﬁrst k dimensions. In the example of the spring, k = 1. This process of throwing out the less important axes can help reveal hidden, simpliﬁed dynamics in high dimensional data. This process is aptly named dimensional reduction . C. Limits and Extensions of PCA Both the strength and weakness of PCA is that it is a non-parametric analysis. One only needs to make the FIG. 6 Non-Gaussian distributed data causes PCA to fail. In exponentially distributed data the axes with the largest variance do not correspond to the underlying basis. assumptions outlined in section 4.5 and then calculate the corresponding answer. There are no parameters to tweak and no coeﬃcients to adjust based on user experience - the answer is unique 11 and independent of the user. This same strength can also be viewed as a weakness. If one knows a-priori some features of the structure of a system, then it makes sense to incorporate these assump- tions into a parametric algorithm - or an algorithm with selected parameters. Consider the recorded positions of a person on a fer- ris wheel over time in Figure 5. The probability distri- butions along the axes are approximately Gaussian and thus PCA ﬁnds ( p1, p2), however this answer might not be optimal. The most concise form of dimensional re- duction is to recognize that the phase (or angle along the ferris wheel) contains all dynamic information. Thus, the appropriate parametric algorithm is to ﬁrst convert the data to the appropriately centered polar coordinates and then compute PCA. This prior non-linear transformation is sometimes termed a kernel transformation and the entire parametric algorithm is termed kernel PCA . Other common kernel transformations include F ourier and Gaussian transfor- mations. This procedure is parametric because the user must incorporate prior knowledge of the structure in the selection of the kernel but it is also more optimal in the sense that the structure is more concisely described. Sometimes though the assumptions themselves are too stringent. One might envision situations where the prin- cipal components need not be orthogonal. F urthermore, the distributions along each dimension ( xi) need not be 11 T o be absolutely precise, the principal components are not uniquely deﬁned; only the sub-space is unique. One can alway s ﬂip the direction of the principal components by multiplyin g by −1. In addition, eigenvectors beyond the rank of a matrix (i.e . σ i = 0 for i > rank ) can be selected almost at whim. However, these degrees of freedom do not eﬀect the qualitative featur es of the solution nor a dimensional reduction. 11 Gaussian. F or instance, Figure 6 contains a 2-D expo- nentially distributed data set. The largest variances do not correspond to the meaningful axes; thus PCA fails. This less constrained set of problems is not trivial and only recently has been solved adequately via Independent Component Analysis (ICA) . The formulation is equiva- lent. Find a matrix P where Y = PX such that CY is diagonalized. however it abandons all assumptions except linearity , and attempts to ﬁnd axes that satisfy the most formal form of redundancy reduction - statistical independence . Mathe- matically ICA ﬁnds a basis such that the joint probability distribution can be factorized 12. P (yi, yj) = P (yi)P (yj) for all i and j, i ̸= j. The downside of ICA is that it is a form of nonlinear optimizaiton, making the solu- tion diﬃcult to calculate in practice and potentially not unique. However ICA has been shown a very practical and powerful algorithm for solving a whole new class of problems. W riting this paper has been an extremely instructional experience for me. I hope that this paper helps to demys- tify the motivation and results of PCA, and the under- lying assumptions behind this important analysis tech- nique. Please send me a note if this has been useful to you as it inspires me to keep writing! APPENDIX A: Linear Algebra This section proves a few unapparent theorems in linear algebra, which are crucial to this paper. 1. The inverse of an orthogonal matrix is its transpose. The goal of this proof is to show that if A is an or- thogonal matrix, then A−1 = AT . Let A be an m × n matrix. A = [ a1 a2 . . . an] where ai is the ith column vector. W e now show that AT A = I where I is the identity matrix. Let us examine the ijth element of the matrix AT A. The ijth element of AT A is ( AT A)ij = aiT aj. Remember that the columns of an orthonormal matrix are orthonormal to each other. In other words, the dot product of any two columns is zero. The only exception is 12 Equivalently , in the language of information theory the goal is to ﬁnd a basis P such that the mutual information I(yi, yj) = 0 for i ̸= j. a dot product of one particular column with itself, which equals one. (AT A)ij = ai T aj = { 1 i = j 0 i ̸= j AT A is the exact description of the identity matrix. The deﬁnition of A−1 is A−1A = I. Therefore, because AT A = I, it follows that A−1 = AT . 2. If A is any matrix, the matrices A T A and AAT are both symmetric. Let’s examine the transpose of each in turn. (AAT )T = AT T AT = AAT (AT A)T = AT AT T = AT A The equality of the quantity with its transpose completes this proof. 3. A matrix is symmetric if and only if it is orthogonally diagonalizable. Because this statement is bi-directional, it requires a two-part “if-and-only-if” proof. One needs to prove the forward and the backwards “if-then” cases. Let us start with the forward case. If A is orthogo- nally diagonalizable, then A is a symmetric matrix. By hypothesis, orthogonally diagonalizable means that there exists some E such that A = EDET , where D is a diag- onal matrix and E is some special matrix which diago- nalizes A. Let us compute AT . AT = ( EDET )T = ET T DT ET = EDET = A Evidently , if A is orthogonally diagonalizable, it must also be symmetric. The reverse case is more involved and less clean so it will be left to the reader. In lieu of this, hopefully the “forward” case is suggestive if not somewhat convincing. 4. A symmetric matrix is diagonalized by a matrix of its orthonormal eigenvectors. Restated in math, let A be a square n × n symmet- ric matrix with associated eigenvectors {e1, e2, . . . , en}. Let E = [ e1 e2 . . . en] where the ith column of E is the eigenvector ei. This theorem asserts that there exists a diagonal matrix D where A = EDET . This theorem is an extension of the previous theorem 3. It provides a prescription for how to ﬁnd the matrix E, the “diagonalizer” for a symmetric matrix. It says that the special diagonalizer is in fact a matrix of the original matrix’s eigenvectors. This proof is in two parts. In the ﬁrst part, we see that the any matrix can be orthogonally diagonalized if and only if it that matrix’s eigenvectors are all linearly independent. In the second part of the proof, we see that 12 a symmetric matrix has the special property that all of its eigenvectors are not just linearly independent but also orthogonal, thus completing our proof. In the ﬁrst part of the proof, let A be just some ma- trix, not necessarily symmetric, and let it have indepen- dent eigenvectors (i.e. no degeneracy). F urthermore, let E = [ e1 e2 . . . en] be the matrix of eigenvectors placed in the columns. Let D be a diagonal matrix where the ith eigenvalue is placed in the iith position. W e will now show that AE = ED. W e can examine the columns of the right-hand and left-hand sides of the equation. Left hand side : AE = [ Ae1 Ae2 . . . Aen] Right hand side : ED = [ λ1e1 λ2e2 . . . λ nen] Evidently , if AE = ED then Aei = λiei for all i. This equation is the deﬁnition of the eigenvalue equation. Therefore, it must be that AE = ED. A little rearrange- ment provides A = EDE−1, completing the ﬁrst part the proof. F or the second part of the proof, we show that a sym- metric matrix always has orthogonal eigenvectors. F or some symmetric matrix, let λ1 and λ2 be distinct eigen- values for eigenvectors e1 and e2. λ1e1 ·e2 = ( λ1e1)T e2 = ( Ae1)T e2 = e1 T AT e2 = e1 T Ae2 = e1 T (λ2e2) λ1e1 ·e2 = λ2e1 ·e2 By the last relation we can equate that (λ1 − λ2)e1 ·e2 = 0. Since we have conjectured that the eigenvalues are in fact unique, it must be the case that e1 ·e2 = 0. Therefore, the eigenvectors of a symmetric matrix are orthogonal. Let us back up now to our original postulate that A is a symmetric matrix. By the second part of the proof, we know that the eigenvectors of A are all orthonormal (we choose the eigenvectors to be normalized). This means that E is an orthogonal matrix so by theorem 1, ET = E−1 and we can rewrite the ﬁnal result. A = EDET . Thus, a symmetric matrix is diagonalized by a matrix of its eigenvectors. 5. F or any arbitrary m × n matrix X, the symmetric matrix X T X has a set of orthonor- mal eigenvectors of {ˆ v1, ˆ v2, . . . , ˆ vn} and a set of associated eigenvalues {λ1, λ2, . . . , λ n}. The set of vectors {Xˆ v1, Xˆ v2, . . . , Xˆ vn} then form an orthog- onal basis, where each vector Xˆ v i is of length √ λi. All of these properties arise from the dot product of any two vectors from this set. (Xˆ vi) ·(Xˆ vj) = ( Xˆ vi)T (Xˆ vj) = ˆ vT i XT Xˆ vj = ˆ vT i (λj ˆ vj) = λj ˆ vi ·ˆ vj (Xˆ vi) ·(Xˆ vj) = λj δij The last relation arises because the set of eigenvectors of X is orthogonal resulting in the Kronecker delta. In more simpler terms the last relation states: (Xˆ vi) ·(Xˆ vj) = { λj i = j 0 i ̸= j This equation states that any two vectors in the set are orthogonal. The second property arises from the above equation by realizing that the length squared of each vector is deﬁned as: ∥Xˆ vi∥2 = ( Xˆ vi) ·(Xˆ vi) = λi APPENDIX B: Code This code is written for Matlab 6.5 (Release 13) from Mathworks 13. The code is not computationally ef- ﬁcient but explanatory (terse comments begin with a %). This ﬁrst version follows Section 5 by examining the covariance of the data set. function [signals,PC,V] = pca1(data) % PCA1: Perform PCA using covariance. % data - MxN matrix of input data % (M dimensions, N trials) % signals - MxN matrix of projected data % PC - each column is a PC % V - Mx1 matrix of variances [M,N] = size(data); % subtract off the mean for each dimension mn = mean(data,2); data = data - repmat(mn,1,N); % calculate the covariance matrix covariance = 1 / (N-1) * data * data’; % find the eigenvectors and eigenvalues [PC, V] = eig(covariance); 13 http://www.mathworks.com 13 % extract diagonal of matrix as vector V = diag(V); % sort the variances in decreasing order [junk, rindices] = sort(-1V); V = V(rindices); PC = PC(:,rindices); % project the original data set signals = PC’ data; This second version follows section 6 computing PCA through SVD. function [signals,PC,V] = pca2(data) % PCA2: Perform PCA using SVD. % data - MxN matrix of input data % (M dimensions, N trials) % signals - MxN matrix of projected data % PC - each column is a PC % V - Mx1 matrix of variances [M,N] = size(data); % subtract off the mean for each dimension mn = mean(data,2); data = data - repmat(mn,1,N); % construct the matrix Y Y = data’ / sqrt(N-1); % SVD does it all [u,S,PC] = svd(Y); % calculate the variances S = diag(S); V = S .* S; % project the original data signals = PC’ * data; APPENDIX C: References Bell, Anthony and Sejnowski, T erry . (1997) “The Independent Components of Natural Scenes are Edge Filters.” Vision Research 37(23), 3327-3338. [A paper from my ﬁeld of research that surveys and explores diﬀerent forms of decorrelating data sets. The authors examine the features of PCA and compare it with new ideas in redun- dancy reduction, namely Independent Component Analysis.] Bishop, Christopher. (1996) Neural Networks for Pattern Recognition. Clarendon, Oxford, UK. [A challenging but brilliant text on statistical pattern recog- nition. Although the derivation of PCA is tough in section 8.6 (p.310-319), it does have a great discussion on potential extensions to the method and it puts PCA in context of other methods of dimensional reduction. Also, I want to acknowledge this book for several ideas about the limitations of PCA.] Lay , David. (2000). Linear Algebra and It’s Applica- tions. Addison-W esley , New Y ork. [This is a beautiful text. Chapter 7 in the second edition (p. 441-486) has an exquisite, intuitive derivation and discussion of SVD and PCA. Extremely easy to follow and a must read.] Mitra, Partha and Pesaran, Bijan. (1999) ”Analysis of Dynamic Brain Imaging Data.” Biophysical Journal . 76, 691-708. [A comprehensive and spectacular paper from my ﬁeld of research interest. It is dense but in two sections ”Eigenmode Analysis: SVD” and ”Space-frequency SVD” the authors discuss the beneﬁts of performing a Fourier transform on the data before an SVD.] Will, T odd (1999) ”Introduction to the Sin- gular V alue Decomposition” Davidson College. www.davidson.edu/academic/math/will/svd/index.html [A math professor wrote up a great web tutorial on SVD with tremendous intuitive explanations, graphics and animations. Although it avoids PCA directly , it gives a great intuitive feel for what SVD is doing mathematically . Also, it is the inspiration for my ”spring” example.]
svm-notes-long-08	1 An Idiot’s guide to Support vectormachines (SVMs)R. Berwick, Village Idiot SVMs: A NewGeneration of Learning Algorithms•Pre 1980:–Almost all learning methods learned linear decision surfaces.–Linear learning methods have nice theoretical properties•1980’s–Decision trees and NNs allowed efficient learning of non-linear decision surfaces–Little theoretical basis and all suffer from local minima•1990’s–Efficient learning algorithms for non-linear functions basedon computational learning theory developed–Nice theoretical properties. 2 Key Ideas•Two independent developments within last decade–New efficient separability of non-linear regions that use“kernel functions” : generalization of ‘similarity’ tonew kinds of similarity measures based on dot products–Use of quadratic optimization problem to avoid ‘localminimum’ issues with neural nets–The resulting learning algorithm is an optimizationalgorithm rather than a greedy search Organization•Basic idea of support vector machines: just like 1-layer or multi-layer neural nets–Optimal hyperplane for linearly separablepatterns–Extend to patterns that are not linearlyseparable by transformations of original data tomap into new space – the Kernel function•SVM algorithm for pattern recognition 3 Support Vectors•Support vectors are the data points that lie closestto the decision surface (or hyperplane)•They are the data points most difficult to classify•They have direct bearing on the optimum locationof the decision surface•We can show that the optimal hyperplane stemsfrom the function class with the lowest“capacity”= # of independent features/parameterswe can twiddle [note this is ‘extra’ material notcovered in the lectures… you don’t have to knowthis] Recall from 1-layer nets : Which SeparatingHyperplane? •In general, lots of possiblesolutions for a,b,c (aninfinite number!)•Support Vector Machine(SVM) finds an optimalsolution 4 Support Vector Machine (SVM)Support vectors Maximizemargin •SVMs maximize the margin(Winston terminology: the ‘street’)around the separating hyperplane.•The decision function is fullyspecified by a (usually very small)subset of training samples, thesupport vectors.•This becomes a Quadraticprogramming problem that is easyto solve by standard methods Separation by Hyperplanes•Assume linear separability for now (we will relax thislater)•in 2 dimensions, can separate by a line–in higher dimensions, need hyperplanes 5 General input/output for SVMs just like forneural nets, but for one important addition…Input: set of (input, output) training pair samples; call theinput sample features x1, x2…xn, and the output result y.Typically, there can be lots of input features xi.Output: set of weights w (or wi), one for each feature,whose linear combination predicts the value of y. (So far,just like neural nets…)Important difference: we use the optimization of maximizingthe margin (‘street width’) to reduce the number of weightsthat are nonzero to just a few that correspond to theimportant features that ‘matter’ in deciding the separatingline(hyperplane)…these nonzero weights correspond to thesupport vectors (because they ‘support’ the separatinghyperplane) 2-D Case Find a,b,c, such thatax + by ≥ c for red pointsax + by ≤ (or < ) c for greenpoints. 6 Which Hyperplane to pick?•Lots of possible solutions for a,b,c.•Some methods find a separatinghyperplane, but not the optimal one (e.g.,neural net)•But: Which points should influenceoptimality?–All points?•Linear regression•Neural nets–Or only “difficult points” close todecision boundary•Support vector machines Support Vectors again for linearly separable case•Support vectors are the elements of the training set thatwould change the position of the dividing hyperplane ifremoved.•Support vectors are the critical elements of the training set•The problem of finding the optimal hyper plane is anoptimization problem and can be solved by optimizationtechniques (we use Lagrange multipliers to get thisproblem into a form that can be solved analytically). 7 X X X X X X Support Vectors: Input vectors that just touch the boundary of themargin (street) – circled below, there are 3 of them (or, rather, the‘tips’ of the vectorsw0Tx + b0 = 1 or w0Tx + b0 = –1d X X X X X X Here, we have shown the actual support vectors, v1, v2, v3, instead ofjust the 3 circled points at the tail ends of the support vectors. ddenotes 1/2 of the street ‘width’ dv1v2 v3 8 d+d- DefinitionsDefine the hyperplanes H such that:w•xi+b ≥ +1 when yi =+1 w•xi+b ≤ -1 when yi = –1 d+ = the shortest distance to the closest positive pointd- = the shortest distance to the closest negative pointThe margin (gutter) of a separating hyperplane is d+ + d–. HH1 and H2 are the planes:H1: w•xi+b = +1H2: w•xi+b = –1The points on the planes H1 andH2 are the tips of the SupportVectorsThe plane H0 is the median inbetween, where w•xi+b =0 H1 H2H0 Moving a support vectormoves the decisionboundary Moving the other vectorshas no effect The optimization algorithm to generate the weights proceeds in such away that only the support vectors determine the weights and thus theboundary 9 Defining the separating Hyperplane•Form of equation defining the decision surface separatingthe classes is a hyperplane of the form:wTx + b = 0–w is a weight vector–x is input vector–b is bias•Allows us to writewTx + b ≥ 0 for di = +1wTx + b < 0 for di = –1 Some final definitions•Margin of Separation (d): the separation between thehyperplane and the closest data point for a given weightvector w and bias b.•Optimal Hyperplane (maximal margin): the particularhyperplane for which the margin of separation d ismaximized. 10 Maximizing the margin (aka street width) d+d- We want a classifier (linear separator) with as big a margin as possible. Recall the distance from a point(x0,y0) to a line:Ax+By+c = 0 is: \|Ax0 +By0 +c\|/sqrt(A2+B2), so,The distance between H0 and H1 is then:\|w•x+b\|/\|\|w\|\|=1/\|\|w\|\|, soThe total distance between H1 and H2 is thus: 2/\|\|w\|\| In order to maximize the margin, we thus need to minimize \|\|w\|\|. With the condition that there are no datapoints between H1 and H2:xi•w+b ≥ +1 when yi =+1 xi•w+b ≤ –1 when yi =–1 Can be combined into: yi(xi•w) ≥ 1 H1 H2H0 We now must solve a quadraticprogramming problem•Problem is: minimize \|\|w\|\|, s.t. discrimination boundary isobeyed, i.e., min f(x) s.t. g(x)=0, which we can rewrite as: min f: ½ \|\|w\|\|2 (Note this is a quadratic function)s.t. g: yi(w•xi)–b = 1 or [yi(w•xi)–b] – 1 =0 This is a constrained optimization problemIt can be solved by the Lagrangian multipler methodBecause it is quadratic, the surface is a paraboloid, with just asingle global minimum (thus avoiding a problem we hadwith neural nets!) 11 Example: paraboloid 2+x2+2y2 s.t. x+y=1flatten Intuition: find intersection of two functions f, g ata tangent point (intersection = both constraintssatisfied; tangent = derivative is 0); this will be amin (or max) for f s.t. the contraint g is satisfied Flattened paraboloid f: 2x2+2y2=0 with superimposedconstraint g: x +y = 1 Minimize when the constraint line g (shown in green)is tangent to the inner ellipse contour linez of f (shown in red) –note direction of gradient arrows. 12 flattened paraboloid f: 2+x2+2y2=0 with superimposed constraintg: x +y = 1; at tangent solution p, gradient vectors of f,g areparallel (no possible move to increment f that also keeps you inregion g) Minimize when the constraint line g is tangent to the inner ellipsecontour line of f Two constraints1.Parallel normal constraint (= gradient constrainton f, g s.t. solution is a max, or a min)2.g(x)=0 (solution is on the constraint line as well)We now recast these by combining f, g as the newLagrangian function by introducing new ‘slackvariables’ denoted a or (more usually, denoted αin the literature) 13 Redescribing these conditions•Want to look for solution point p where •Or, combining these two as the Langrangian L &requiring derivative of L be zero: ()()()0fpgpgx!"="= L(x,a)=f(x)!ag(x)"(x,a)=0At a solution p•The the constraint line g and the contour lines of f mustbe tangent•If they are tangent, their gradient vectors(perpendiculars) are parallel•Gradient of g must be 0 – i.e., steepest ascent & soperpendicular to f•Gradient of f must also be in the same direction as g 14 How Langrangian solves constrainedoptimization L(x,a)=f(x)!ag(x) where"(x,a)=0Partial derivatives wrt x recover the parallel normal constraintPartial derivatives wrt λ recover the g(x,y)=0In general, L(x,a)=f(x)+aii!gi(x)In general L(x,a)=f(x)+aii!gi(x) a function of n+m variablesn for the x's, m for the a. Differentiating gives n+m equations, each set to 0. The n eqns differentiated wrt each xi give the gradient conditions; the m eqns differentiated wrt each ai recover the constraints giGradient min of fconstraint condition g In our case, f(x): ½\|\| w\|\|2 ; g(x): yi(w•xi +b)–1=0 so Lagrangian is:min L= ½\|\| w\|\|2 – Σai[yi(w•xi +b)–1] wrt w, bWe expand the last to get the following L form:min L= ½\|\| w\|\|2 – Σaiyi(w•xi +b) +Σai wrt w, b 15 Lagrangian Formulation•So in the SVM problem the Lagrangian is •From the property that the derivatives at min = 0we get: minLP=12w2!aii=1l"yixi#w+b()+aii=1l"s.t. $i,ai%0 where l is the # of training points w=aii=1l!yixi, aii=1l!yi=0!LP!w=w"aiyixi=0i=1l#!LP!b=aiyi=0 so i=1l"What’s with this Lp business?•This indicates that this is the primal form of theoptimization problem•We will actually solve the optimization problemby now solving for the dual of this originalproblem•What is this dual formulation? 16 The Lagrangian Dual Problem: instead of minimizing over w, b,subject to constraints involving a’s, we can maximize over a (thedual variable) subject to the relations obtained previously for wand b w=aii=1l!yixi, aii=1l!yi=0Our solution must satisfy these two relations: By substituting for w and b back in the original eqn we can get rid of thedependence on w and b.Note first that we already now have our answer for what the weights wmust be: they are a linear combination of the training inputs and thetraining outputs, xi and yi and the values of a. We will now solve for thea’s by differentiating the dual problem wrt a, and setting it to zero. Mostof the a’s will turn out to have the value zero. The non-zero a’s willcorrespond to the support vectors Primal problem:minLP=12w2!aii=1l"yixi#w+b()+aii=1l"s.t. $i ai%0 w=aii=1l!yixi, aii=1l!yi=0 Dual problem:maxLD(ai)=aii=1l!"12aiaji=1l!yiyjxi#xj()s.t. aiyi=0i=1l! & ai$0(note that we have removed the dependence on w and b) 17 The Dual problem•Kuhn-Tucker theorem: the solution we find here willbe the same as the solution to the original problem•Q: But why are we doing this???? (why not justsolve the original problem????)•Ans: Because this will let us solve the problem bycomputing the just the inner products of xi, xj (whichwill be very important later on when we want tosolve non-linearly separable classification problems) The Dual Problem Dual problem:maxLD(ai)=aii=1l!"12aiaji=1l!yiyjxi#xj()s.t. aiyi=0i=1l! & ai$0Notice that all we have are the dot products of xi,xjIf we take the derivative wrt a and set it equal to zero,we get the following solution, so we can solve for ai:aiyii=1l!=00"ai"C 18 Now knowing the ai we can find theweights w for the maximal marginseparating hyperplane: w=aii=1l!yixiAnd now, after training and finding the w by this method,given an unknown point u measured on features xi wecan classify it by looking at the sign of: f(x)=wiu+b=(aiyixiiu)+bi=1l!Remember: most of the weights wi, i.e., the a’s, will be zeroOnly the support vectors (on the gutters or margin) will have nonzeroweights or a’s – this reduces the dimensionality of the solution Why should inner product kernels be involved in patternrecognition using SVMs, or at all?– Intuition is that inner products provide some measure of‘similarity’– Inner product in 2D between 2 vectors of unit length returns thecosine of the angle between them = how ‘far apart’ they aree.g. x = [1, 0]T , y = [0, 1]Ti.e. if they are parallel their inner product is 1 (completely similar) xT y = x•y = 1If they are perpendicular (completely unlike) their inner product is0 (so should not contribute to the correct classifier) xT• y = x•y = 0 Inner products, similarity, and SVMs 19 Insight into inner products Consider that we are trying to maximize the form:LD(ai)=aii=1l!"12aiaji=1l!yiyjxi#xj()s.t. aiyi=0i=1l! & ai$0The claim is that this function will be maximized if we give nonzero values to a’s thatcorrespond to the support vectors, ie, those that ‘matter’ in fixing the maximum widthmargin (‘street’). Well, consider what this looks like. Note first from the constraintcondition that all the a’s are positive. Now let’s think about a few cases.Case 1. If two features xi , xj are completely dissimilar, their dot product is 0, and they don’tcontribute to L.Case 2. If two features xi,xj are completely alike, their dot product is 0. There are 2 subcases.Subcase 1: both xi,and xj predict the same output value yi (either +1 or –1). Then yix yj is always 1, and the value of aiajyiyjxixj will be positive. But this would decrease thevalue of L (since it would subtract from the first term sum). So, the algorithm downgradessimilar feature vectors that make the same prediction.Subcase 2: xi,and xj make opposite predictions about the output value yi (ie, one is+1, the other –1), but are otherwise very closely similar: then the product aiajyiyjxix isnegative and we are subtracting it, so this adds to the sum, maximizing it. This is preciselythe examples we are looking for: the critical ones that tell the two classses apart. Insight into inner products, graphically: 2 veryvery similar xi, xj vectors that predict difftclasses tend to maximize the margin width xi xj 20 2 vectors that are similar but predict thesame class are redundant xi xj 2 dissimilar (orthogonal) vectors don’tcount at all xixj 21 But…are we done??? Not Linearly Separable! Find a line that penalizespoints on “the wrong side” 22 x xxxxxx ϕ (o) X F ϕ ϕ (x)ϕ (x) ϕ (x) ϕ (x) ϕ (x)ϕ (x) ϕ (x)ϕ (o) ϕ (o)ϕ (o)ϕ (o)ϕ (o)ϕ (o) oooooo Transformation to separate Non–Linear SVMs a b()()()2xaxbxabxab!!=!++{}2,xxx!•The idea is to gain linearly separation by mapping the data toa higher dimensional space–The following set can’t be separated by a linear function,but can be separated by a quadratic one –So if we map we gain linear separation 23 Problems with linear SVM =-1=+1 What if the decision function is not linear? What transform would separate these? Ans: polar coordinates!Non-linear SVMThe Kernel trick =-1=+1 Imagine a function φ that maps the data into another space: φ=Radial→Η =-1=+1 Remember the function we want to optimize: Ld = ∑ai – ½∑ai ajyiyj (xi•xj) where (xi•xj) is thedot product of the two feature vectors. If we now transform to φ, instead of computing this dot product (xi•xj) we will have to compute (φ (xi)• φ (xj)). But how can we do this? This isexpensive and time consuming (suppose φ is a quartic polynomial… or worse, we don’t know thefunction explicitly. Well, here is the neat thing: If there is a ”kernel function” K such that K(xi,xj) = φ (xi)• φ (xj), then we do not need to knowor compute φ at all!! That is, the kernel function defines inner products in the transformed space.Or, it defines similarity in the transformed space. Radial Ηφ 24 Non-linear SVMsSo, the function we end up optimizing is:Ld = ∑ai – ½∑aiaj yiyjK(xi•xj),Kernel example: The polynomial kernelK(xi,xj) = (xi•xj + 1)p, where p is a tunable parameterNote: Evaluating K only requires one addition and one exponentiationmore than the original dot product Examples for Non Linear SVMs ()(),1pK=!+xyxy(){}222,expK!"="xyxy()(),tanhK!"=#$xyxy1st is polynomial (includes x•x as special case)2nd is radial basis function (gaussians)3rd is sigmoid (neural net activation function) 25 We’ve already seen such nonlineartransforms…•What is it??? •tanh(β0xTxi + β1)•It’s the sigmoidtransform (for neuralnets)•So, SVMs subsumeneural nets! (but w/otheir problems…) Inner Product Kernels Actually works only forsome values of β0 andβ1 tanh(β0xTxi + β1)Two layer neural net The width σ2 isspecified a prioriexp(1/(2σ2)\|\|x-xi\|\|2)Radial-basis function(RBF) Power p is specified apriori by the user(xTxi + 1)pPolynomial learningmachine Usual inner productInner Product KernelK(x,xi), I = 1, 2, …, NType of Support VectorMachine 26 Kernels generalize the notion of ‘innerproduct similarity’Note that one can define kernels over more than justvectors: strings, trees, structures, … in fact, just aboutanythingA very powerful idea: used in comparing DNA, proteinstructure, sentence structures, etc. Examples for Non Linear SVMs 2 –Gaussian Kernel Gaussian Linear 27 Nonlinear rbf kernel Admiral’s delight w/ difft kernelfunctions 28 Overfitting by SVM Every point is a support vector… too much freedom to bend to fit thetraining data – no generalization.In fact, SVMs have an ‘automatic’ way to avoid such issues, but wewon’t cover it here… see the book by Vapnik, 1995. (We add apenalty function for mistakes made after training by over-fitting: recallthat if one over-fits, then one will tend to make errors on new data.This penalty fn can be put into the quadratic programming problemdirectly. You don’t need to know this for this course.)
time-clocks	Operating R. Stockton Gaines Systems Editor Time, Clocks, and the Ordering of Events in a Distributed System Leslie Lamport Massachusetts Computer Associates, Inc. The concept of one event happening before another in a distributed system is examined, and is shown to define a partial ordering of the events. A distributed algorithm is given for synchronizing a system of logical clocks which can be used to totally order the events. The use of the total ordering is illustrated with a method for solving synchronization problems. The algorithm is then specialized for synchronizing physical clocks, and a bound is derived on how far out of synchrony the clocks can become. Key Words and Phrases: distributed systems, computer networks, clock synchronization, multiprocess systems CR Categories: 4.32, 5.29 Introduction The concept of time is fundamental to our way of thinking. It is derived from the more basic concept of the order in which events occur. We say that something happened at 3:15 if it occurred after our clock read 3:15 and before it read 3:16. The concept of the temporal ordering of events pervades our thinking about systems. For example, in an airline reservation system we specify that a request for a reservation should be granted if it is made before the flight is filled. However, we will see that this concept must be carefully reexamined when consid- ering events in a distributed system. General permission to make fair use in teaching or research of all or part of this material is granted to individual readers and to nonprofit libraries acting for them provided that ACM's copyright notice is given and that reference is made to the publication, to its date of issue, and to the fact that reprinting privileges were granted by permission of the Association for Computing Machinery. To otherwise reprint a figure, table, other substantial excerpt, or the entire work requires specific permission as does republication, or systematic or multiple reproduc- tion. This work was supported by the Advanced Research Projects Agency of the Department of Defense and Rome Air Development Center. It was monitored by Rome Air Development Center under contract number F 30602-76-C-0094. Author's address: Computer Science Laboratory, SRI Interna- tional, 333 Ravenswood Ave., Menlo Park CA 94025. © 1978 ACM 0001-0782/78/0700-0558 $00.75 558 A distributed system consists of a collection of distinct processes which are spatially separated, and which com- municate with one another by exchanging messages. A network of interconnected computers, such as the ARPA net, is a distributed system. A single computer can also be viewed as a distributed system in which the central control unit, the memory units, and the input-output channels are separate processes. A system is distributed if the message transmission delay is not negligible com- pared to the time between events in a single process. We will concern ourselves primarily with systems of spatially separated computers. However, many of our remarks will apply more generally. In particular, a mul- tiprocessing system on a single computer involves prob- lems similar to those of a distributed system because of the unpredictable order in which certain events can occur. In a distributed system, it is sometimes impossible to say that one of two events occurred first. The relation "happened before" is therefore only a partial ordering of the events in the system. We have found that problems often arise because people are not fully aware of this fact and its implications. In this paper, we discuss the partial ordering defined by the "happened before" relation, and give a distributed algorithm for extending it to a consistent total ordering of all the events. This algorithm can provide a useful mechanism for implementing a distributed system. We illustrate its use with a simple method for solving syn- chronization problems. Unexpected, anomalous behav- ior can occur if the ordering obtained by this algorithm differs from that perceived by the user. This can be avoided by introducing real, physical clocks. We describe a simple method for synchronizing these clocks, and derive an upper bound on how far out of synchrony they can drift. The Partial Ordering Most people would probably say that an event a happened before an event b if a happened at an earlier time than b. They might justify this definition in terms of physical theories of time. However, if a system is to meet a specification correctly, then that specification must be given in terms of events observable within the system. If the specification is in terms of physical time, then the system must contain real clocks. Even if it does contain real clocks, there is still the problem that such clocks are not perfectly accurate and do not keep precise physical time. We will therefore define the "happened before" relation without using physical clocks. We begin by defining our system more precisely. We assume that the system is composed of a collection of processes. Each process consists of a sequence of events. Depending upon the application, the execution of a subprogram on a computer could be one event, or the execution of a single machine instruction could be one Communications July 1978 of Volume 21 the ACM Number 7 Fig. 1. a, CY ,Y (9 (9 ~o ~ o P4' P3 P2' Pl ~ q7 q6 q5 ql r 4 r 3 r 2 r 1 event. We are assuming that the events of a process form a sequence, where a occurs before b in this sequence if a happens before b. In other words, a single process is defined to be a set of events with an a priori total ordering. This seems to be what is generally meant by a process.~ It would be trivial to extend our definition to allow a process to split into distinct subprocesses, but we will not bother to do so. We assume that sending or receiving a message is an event in a process. We can then define the "happened before" relation, denoted by "---~", as follows. Definition. The relation "---->" on the set of events of a system is the smallest relation satisfying the following three conditions: (1) If a and b are events in the same process, and a comes before b, then a ~ b. (2) If a is the sending of a message by one process and b is the receipt of the same message by another process, then a ~ b. (3) If a ~ b and b ~ c then a ---* c. Two distinct events a and b are said to be concurrent if a ~ b and b -/-* a. We assume that a ~ a for any event a. (Systems in which an event can happen before itself do not seem to be physically meaningful.) This implies that ~ is an irreflexive partial ordering on the set of all events in the system. It is helpful to view this definition in terms of a "space-time diagram" such as Figure 1. The horizontal direction represents space, and the vertical direction represents time--later times being higher than earlier ones. The dots denote events, the vertical lines denote processes, and the wavy lines denote messagesfl It is easy to see that a ~ b means that one can go from a to b in ' The choice of what constitutes an event affects the ordering of events in a process. For example, the receipt of a message might denote the setting of an interrupt bit in a computer, or the execution of a subprogram to handle that interrupt. Since interrupts need not be handled in the order that they occur, this choice will affect the order- ing of a process' message-receiving events. 2 Observe that messages may be received out of order. We allow the sending of several messages to be a single event, but for convenience we will assume that the receipt of a single message does not coincide with the sending or receipt of any other message. 559 Fig. 2. cy c~ (9 (9 ~) O O U -2 - - - q6 -- ;#.i Y _ P3' ~ ~ ~ ~ ~ _~~-~ r3 the diagram by moving forward in time along process and message lines. For example, we have p, --~ r4 in Figure 1. Another way of viewing the definition is to say that a --) b means that it is possible for event a to causally affect event b. Two events are concurrent if neither can causally affect the other. For example, events pa and q:~ of Figure 1 are concurrent. Even though we have drawn the diagram to imply that q3 occurs at an earlier physical time than 1)3, process P cannot know what process Q did at qa until it receives the message at p, (Before event p4, P could at most know what Q was planning to do at q:~.) This definition will appear quite natural to the reader familiar with the invariant space-time formulation of special relativity, as described for example in [1] or the first chapter of [2]. In relativity, the ordering of events is defined in terms of messages that could be sent. However, we have taken the more pragmatic approach of only considering messages that actually are sent. We should be able to determine if a system performed correctly by knowing only those events which did occur, without knowing which events could have occurred. Logical Clocks We now introduce clocks into the system. We begin with an abstract point of view in which a clock is just a way of assigning a number to an event, where the number is thought of as the time at which the event occurred. More precisely, we define a clock Ci for each process Pi to be a function which assigns a number Ci(a) to any event a in that process. The entire system ofclbcks is represented by the function C which assigns to any event b the number C(b), where C(b) = C/(b) ifb is an event in process Pj. For now, we make no assumption about the relation of the numbers Ci(a) to physical time, so we can think of the clocks Ci as logical rather than physical clocks. They may be implemented by counters with no actual timing mechanism. Communications July 1978 of Volume 21 the ACM Number 7 Fig. 3. CY n¢ 8 8 8 c~! ~ ~iLql ~ .r 4 We now consider what it means for such a system of clocks to be correct. We cannot base our definition of correctness on physical time, since that would require introducing clocks which keep physical time. Our defi- nition must be based on the order in which events occur. The strongest reasonable condition is that if an event a occurs before another event b, then a should happen at an earlier time than b. We state this condition more formally as follows. Clock Condition. For any events a, b: if a---> b then C(a) < C(b). Note that we cannot expect the converse condition to hold as well, since that would imply that any two con- current events must occur at the same time. In Figure 1, p2 and p.~ are both concurrent with q3, so this would mean that they both must occur at the same time as q.~, which would contradict the Clock Condition because p2 -----> /93. It is easy to see from our definition of the relation "---~" that the Clock Condition is satisfied if the following two conditions hold. C 1. If a and b are events in process P~, and a comes before b, then Ci(a) < Ci(b). C2. If a is the sending of a message by process Pi and b is the receipt of that message by process Pi, then Ci(a) < Ci(b). Let us consider the clocks in terms of a space-time diagram. We imagine that a process' clock "ticks" through every number, with the ticks occurring between the process' events. For example, if a and b are consec- utive events in process Pi with Ci(a) = 4 and Ci(b) = 7, then clock ticks 5, 6, and 7 occur between the two events. We draw a dashed "tick line" through all the like- numbered ticks of the different processes. The space- time diagram of Figure 1 might then yield the picture in Figure 2. Condition C 1 means that there must be a tick line between any two events on a process line, and 560 condition C2 means that every message line must cross a tick line. From the pictorial meaning of--->, it is easy to see why these two conditions imply the Clock Con- dition. We can consider the tick lines to be the time coordi- nate lines of some Cartesian coordinate system on space- time. We can redraw Figure 2 to straighten these coor- dinate lines, thus obtaining Figure 3. Figure 3 is a valid alternate way of representing the same system of events as Figure 2. Without introducing the concept of physical time into the system (which requires introducing physical clocks), there is no way to decide which of these pictures is a better representation. The reader may find it helpful to visualize a two- dimensional spatial network of processes, which yields a three-dimensional space-time diagram. Processes and messages are still represented by lines, but tick lines become two-dimensional surfaces. Let us now assume that the processes are algorithms, and the events represent certain actions during their execution. We will show how to introduce clocks into the processes which satisfy the Clock Condition. Process Pi's clock is represented by a register Ci, so that C~(a) is the value contained by C~ during the event a. The value of C~ will change between events, so changing Ci does not itself constitute an event. To guarantee that the system of clocks satisfies the Clock Condition, we will insure that it satisfies conditions C 1 and C2. Condition C 1 is simple; the processes need only obey the following implementation rule: IR1. Each process P~ increments Ci between any two successive events. To meet condition C2, we require that each message m contain a timestamp Tm which equals the time at which the message was sent. Upon receiving a message time- stamped Tin, a process must advance its clock to be later than Tin. More precisely, we have the following rule. IR2. (a) If event a is the sending of a message m by process P~, then the message m contains a timestamp Tm= Ci(a). (b) Upon receiving a message m, process Pi sets Ci greater than or equal to its present value and greater than Tin. In IR2(b) we consider the event which represents the receipt of the message m to occur after the setting of C i. (This is just a notational nuisance, and is irrelevant in any actual implementation.) Obviously, IR2 insures that C2 is satisfied. Hence, the simple implementation rules IR l and IR2 imply that the Clock Condition is satisfied, so they guarantee a correct system of logical clocks. Ordering the Events Totally We can use a system of clocks satisfying the Clock Condition to place a total ordering on the set of all system events. We simply order the events by the times Communications July 1978 of Volume 21 the ACM Number 7 at which they occur. To break ties, we use any arbitrary total ordering < of the processes. More precisely, we define a relation ~ as follows: if a is an event in process Pi and b is an event in process Pj, then a ~ b if and only if either (i) Ci{a) < Cj(b) or (ii) El(a) ----" Cj(b) and Pi < Py. It is easy to see that this defines a total ordering, and that the Clock Condition implies that if a ----> b then a ~ b. In other words, the relation ~ is a way of completing the "happened before" partial order- ing to a total ordering, a The ordering ~ depends upon the system of clocks Cz, and is not unique. Different choices of clocks which satisfy the Clock Condition yield different relations ~. Given any total ordering relation ~ which extends --->, there is a system of clocks satisfying the Clock Condition which yields that relation. It is only the partial ordering which is uniquely determined by the system of events. Being able to totally order the events can be very useful in implementing a distributed system. In fact, the reason for implementing a correct system of logical clocks is to obtain such a total ordering. We will illustrate the use of this total ordering of events by solving the following version of the mutual exclusion problem. Con- sider a system composed of a fixed collection of processes which share a single resource. Only one process can use the resource at a time, so the processes must synchronize themselves to avoid conflict. We wish to find an algo- rithm for granting the resource to a process which satis- fies the following three conditions: (I) A process which has been granted the resource must release it before it can be granted to another process. (II) Different requests for the resource must be granted in the order in which they are made. (III) If every process which is granted the resource eventually releases it, then every request is eventually granted. We assume that the resource is initially granted to exactly one process. These are perfectly natural requirements. They pre- cisely specify what it means for a solution to be correct/ Observe how the conditions involve the ordering of events. Condition II says nothing about which of two concurrently issued requests should be granted first. It is important to realize that this is a nontrivial problem. Using a central scheduling process which grants requests in the order they are received will not work, unless additional assumptions are made. To see this, let P0 be the scheduling process. Suppose P1 sends a request to Po and then sends a message to P2. Upon receiving the latter message, Pe sends a request to Po. It is possible for P2's request to reach P0 before Pl's request does. Condi- tion II is then violated if P2's request is granted first. To solve the problem, we implement a system of ;~ The ordering < establishes a priority among the processes. If a "fairer" method is desired, then < can be made a function of the clock value. For example, if Ci(a) = C/b) andj < L then we can let a ~ b ifj < C~(a) mod N --< i, and b ~ a otherwise; where N is the total number of processes. 4 The term "eventually" should be made precise, but that would require too long a diversion from our main topic. 561 clocks with'rules IR 1 and IR2, and use them to define a total ordering ~ of all events. This provides a total ordering of all request and release operations. With this ordering, finding a solution becomes a straightforward exercise. It just involves making sure that each process learns about all other processes' operations. To simplify the problem, we make some assumptions. They are not essential, but they are introduced to avoid distracting implementation details. We assume first of all that for any two processes P/and Pj, the messages sent from Pi to Pi are received in the same order as they are sent. Moreover, we assume that every message is even- tually received. (These assumptions can be avoided by introducing message numbers and message acknowledg- ment protocols.) We also assume that a process can send messages directly to every other process. Each process maintains its own request queue which is never seen by any other process. We assume that the request queues initially contain the single message To:Po requests resource, where Po is the process initially granted the resource and To is less than the initial value of any clock. The algorithm is then defined by the following five rules. For convenience, the actions defined by each rule are assumed to form a single event. 1. To request the resource, process Pi sends the mes- sage TIn:P/requests resource to every other process, and puts that message on its request queue, where T,~ is the timestamp of the message. 2. When process Pj receives the message T,~:P~ re- quests resource, it places it on its request queue and sends a (timestamped) acknowledgment message to P~.'~ 3. To release the resource, process P~ removes any Tm:Pi requests resource message from its request queue and sends a (timestamped) Pi releases resource message to every other process. 4. When process Pj receives a Pi releases resource message, it removes any Tm:P~ requests resource message from its request queue. 5. Process P/is granted the resource when the follow- ing two conditions are satisfied: (i) There is a Tm:Pi requests resource message in its request queue which is ordered before any other request in its queue by the relation ~. (To define the relation "~" for messages, we identify a message with the event of sending it.) (ii) P~ has received a message from every other process time- stamped later than Tin. ~ Note that conditions (i) and (ii) of rule 5 are tested locally by P~. It is easy to verify that the algorithm defined by these rules satisfies conditions I-III. First of all, observe that condition (ii) of rule 5, together with the assumption that messages are received in order, guarantees that P~ has learned about all requests which preceded its current '~ This acknowledgment message need not be sent if Pj has already sent a message to Pi timestamped later than T .... " If P, -< Pi, then Pi need only have received a message timestamped _> T,,, from P/. Communications July 1978 of Volume 21 the ACM Number 7 request. Since rules 3 and 4 are the only ones which delete messages from the request queue, it is then easy to see that condition I holds. Condition II follows from the fact that the total ordering ~ extends the partial ordering ---~. Rule 2 guarantees that after Pi requests the resource, condition (ii) of rule 5 will eventually hold. Rules 3 and 4 imply that if each process which is granted the resource eventually releases it, then condition (i) of rule 5 will eventually hold, thus proving condition III. This is a distributed algorithm. Each process inde- pendently follows these rules, and there is no central synchronizing process or central storage. This approach can be generalized to implement any desired synchroni- zation for such a distributed multiprocess system. The synchronization is specified in terms of a State Machine, consisting of a set C of possible commands, a set S of possible states, and a function e: C× S--~ S. The relation e(C, S) -- S' means that executing the command C with the machine in state S causes the machine state to change to S'. In our example, the set C consists of all the commands Pi requests resource and P~ releases resource, and the state consists of a queue of waiting request commands, where the request at the head of the queue is the currently granted one. Executing a request com- mand adds the request to the tail of the queue, and executing a release command removes a command from 'he queue. 7 Each process independently simulates the execution of the State Machine, using the commands issued by all the processes. Synchronization is achieved because all processes order the commands according to their time- stamps (using the relation ~), so each process uses the same sequence of commands. A process can execute a command timestamped T when it has learned of all commands issued by all other processes with timestamps less than or equal to T. The precise algorithm is straight- forward, and we will not bother to describe it. This method allows one to implement any desired form of multiprocess synchronization in a distributed system. However, the resulting algorithm requires the active participation of all the processes. A process must know all the commands issued by other processes, so that the failure of a single process will make it impossible for any other process to execute State Machine com- mands, thereby halting the system. The problem of failure is a difficult one, and it is beyond the scope of this paper to discuss it in any detail. We will just observe that the entire concept of failure is only meaningful in the context of physical time. Without physical time, there is no way to distinguish a failed process from one which is just pausing between events. A user can tell that a system has "crashed" only because he has been waiting too long for a response. A method which works despite the failure of individual processes or communication lines is described in [3]. 7 If each process does not strictly alternate request and release commands, then executing a release command could delete zero, one, or more than one request from the queue. Anomalous Behavior Our resource scheduling algorithm ordered the re- quests according to the total ordering =*. This permits the following type of "anomalous behavior." Consider a nationwide system of interconnected computers. Suppose a person issues a request A on a computer A, and then telephones a friend in another city to have him issue a request B on a different computer B. It is quite possible for request B to receive a lower timestamp and be ordered before request A. This can happen because the system has no way of knowing that A actually preceded B, since that precedence informatiori is based on messages exter- nal to the system. Let us examine the source of the problem more closely. Let O ° be the set of all system events. Let us introduce a set of events which contains the events in b ° together with all other relevant external events, such as the phone calls in our example. Let ~ denote the "hap- pened before" relation for ~. In our example, we had A B, but A-~ B. It is obvious that no algorithm based entirely upon events in 0 °, and which does not relate those events in any way with the other events in~, can guarantee that request A is ordered before request B. There are two possible ways to avoid such anomalous behavior. The first way is to explicitly introduce into the system the necessary information about the ordering --~. In our example, the person issuing request A could receive the timestamp TA of that request from the system. When issuing request B, his friend could specify that B be given a timestamp later than TA. This gives the user the responsibility for avoiding anomalous behavior. The second approach is to construct a system of clocks which satisfies the following condition. Strong Clock Condition. For any events a, b in O°: ifa --~ b then C(a} < C(b). This is stronger than the ordinary Clock Condition be- cause ~ is a stronger relation than ---~. It is not in general satisfied by our logical clocks. Let us identify ~ with some set of "real" events in physical space-time, and let ~ be the partial ordering of events defined by special relativity. One of the mysteries of the universe is that it is possible to construct a system of physical clocks which, running quite independently of one another, will satisfy the Strong Clock Condition. We can therefore use physical clocks to eliminate anomalous behavior. We now turn our attention to such clocks. Physical Clocks Let us introduce a physical time coordinate into our space-time picture, and let Ci(t) denote the reading of the clock Ci at physical time t. 8 For mathematical con- We will assume a Newtonian space-time. If the relative motion of the clocks or gravitational effects are not negligible, then CM) must be deduced from the actual clock reading by transforming from proper time to the arbitrarily chosen time coordinate. 562 Communications July 1978 of Volume 2 l the ACM Number 7 venience, we assume that the clocks run continuously rather than in discrete "ticks." (A discrete clock can be thought of as a continuous one in which there is an error of up to ½ "tick" in reading it.) More precisely, we assume that Ci(t) is a continuous, differentiable function of t except for isolated jump discontinuities where the clock is reset. Then dCg(t)/dt represents the rate at which the clock is running at time t. In order for the clock Cg to be a true physical clock, it must run at approximately the correct rate. That is, we must have dCi(t)/dt -~ 1 for all t. More precisely, we will assume that the following condition is satisfied: PCI. There exists a constant x << 1 such that for all i: [dCg(t)/dt - 1 [ < x. For typical crystal controlled clocks, x _< 10 -(~. It is not enough for the clocks individually to run at approximately the correct rate. They must be synchro- nized so that Cg(t) = C/(t) for all i,j, and t. More precisely, there must be a sufficiently small constant e so that the following condition holds: PC2. For all i, j: [ Ci(t) - Cy(t)[ < •. If we consider vertical distance in Figure 2 to represent physical time, then PC2 states that the variation in height of a single tick line is less than E. Since two different clocks will never run at exactly the same rate, they will tend to drift further and further apart. We must therefore devise an algorithm to insure that PC2 always holds. First, however, let us examine how small x and • must be to prevent anomalous behav- ior. We must insure that the system 5e of relevant physical events satisfies the Strong Clock Condition. We assume that our clocks satisfy the ordinary Clock Condition, so we need only require that the Strong Clock Condition holds when a and b are events in 0 ° with a 4-> b. Hence, we need only consider events occurring in differ- ent processes. Let # be a number such that if event a occurs at physical time t and event b in another process satisfies a ~ b, then b occurs later than physical time t + bt. In other words,/~ is less than the shortest transmission time for interprocess messages. We can always choose # equal to the shortest distance between processes divided by the speed of light. However, depending upon how messages in ~ are transmitted, # could be significantly larger. To avoid anomalous behavior, we must make sure that for any i, j, and t: Ci(t + #) - CAt) > 0. Combining this with PC I and 2 allows us to relate the required smallness of x and ~ to the value of # as follows. We assume that when a clock is reset, it is always set forward and never back. (Setting it back could cause C I to be violated.) PCI then implies that Cg(t + #) - Cg(t) > (1 - x)#. Using PC2, it is then easy to deduce that Cg(t + #) - C/(t) > 0 if the following inequality holds: E/(I - ~) _< ~. This inequality together with PC 1 and PC2 implies that anomalous behavior is impossible. 563 We now describe our algorithm for insuring that PC2 holds. Let m be a message which is sent at physical time t and received at time t'. We define l, m ~- t t -- I to be the total delay of the message m. This delay will, of course, not be known to the process which receives m. However, we assume that the receiving process knows some mini- mum delay tzm >_ 0 such that ~£m ~ Pro. We call ~,, = I,m -- #m the unpredictable delay of the message. We now specialize rules IRI and 2 for our physical clocks as follows: IR 1'. For each i, if Pi does not receive a message at physical time t, then C/is differentiable at t and dCg(t)/dt >0. IR2'. (a) If Pg sends a message m at physical time t, then m contains a timestamp Tm= C/(t). (b) Upon receiving a message m at time t', process P/ sets C/(t') equal to maximum (Cj(t' - 0), Tm + /Zm). 9 Although the rules are formally specified in terms of the physical time parameter, a process only needs to know its own clock reading and the timestamps of mes- sages it receives. For mathematical convenience, we are assuming that each event occurs at a precise instant of physical time, and different events in the same process occur at different times. These rules are then specializa- tions of rules IR1 and IR2, so our system of clocks satisfies the Clock Condition. The fact that real events have a finite duration causes no difficulty in implement- ing the algorithm. The only real concern in the imple- mentation is making sure that the discrete clock ticks are frequent enough so C 1 is maintained. We now show that this clock synchronizing algorithm can be used to satisfy condition PC2. We assume that the system of processes is described by a directed graph in which an arc from process Pi to process P/represents a communication line over which messages are sent directly from Pi to P/. We say that a message is sent over this arc every T seconds if for any t, Pi sends at least one message to P/between physical times t and t + -r. The diameter of the directed graph is the smallest number d such that for any pair of distinct processes P/, Pk, there is a path from P/to P~ having at most d arcs. In addition to establishing PC2, the following theo- rem bounds the length of time it can take the clocks to become synchronized when the system is first started. THEOREM. Assume a strongly connected graph of processes with diameter d which always obeys rules IR 1' and IR2'. Assume that for any message m, #m --< # for some constant g, and that for all t > to: (a) PC 1 holds. (b) There are constants ~" and ~ such that every ~- seconds a message with an unpredictable delay less than ~ is sent over every arc. Then PC2 is satisfied with • = d(2x~- + ~) for all t > to + Td, where the approximations assume # + ~<< z. The proof of this theorem is surprisingly difficult, and is given in the Appendix. There has been a great deal of work done on the problem of synchronizing physical clocks. We refer the reader to [4] for an intro- :) C/(t' - 0) = lim C,(t' - 181). Communications July 1978 of Volume 21 the ACM Number 7 duction to the subject. The methods described in the literature are useful for estimating the message delays ktm and for adjusting the clock frequencies dCi/dt (for clocks which permit such an adjustment). However, the requirement that clocks are never set backwards seems to distinguish our situation from ones previously studied, and we believe this theorem to be a new result. Conclusion We have seen that the concept of "happening before" defines an invariant partial ordering of the events in a distributed multiprocess system. We described an algo- rithm for extending that partial ordering to a somewhat arbitrary total ordering, and showed how this total or- dering can be used to solve a simple synchronization problem. A future paper will show how this approach can be extended to solve any synchronization problem. The total ordering defined by the algorithm is some- what arbitrary. It can produce anomalous behavior if it disagrees with the ordering perceived by the system's users. This can be prevented by the use of properly synchronized physical clocks. Our theorem showed how closely the clocks can be synchronized. In a distributed system, it is important to realize that the order in which events occur is only a partial ordering. We believe that this idea is useful in understanding any multiprocess system. It should help one to understand the basic problems of multiprocessing independently of the mechanisms used to solve them. Appendix Proof of the Theorem For any i and t, let us define C~ t to be a clock which is set equal to C~ at time t and runs at the same rate as Ci, but is never reset. In other words, Cit(t ') = Ci(t) + [dCz(t)/dtldt (1) for all t' >_ t. Note that Ci(t') >_ Cit(t ') for all t' >__ t. (2) Suppose process P~ at time tl sends a message to process Pz which is received at time t2 with an unpre- dictable delay _< ~, where to <- ta _< t2. Then for all t ___ t2 we have: C~(t) >_ C~(t2) + (1 - x)(t - t2) [by (1) and PCI] > Cfftl) +/~m + (1 -- x)(t -- t2) [by IR2' (b)] = Cl(tl) + (1 - x)(t - tl) - [(t2 - tO - ~m] + x(t2 - t,) >-- Cl(tl) + (1 - x)(t - tl) - 4. Hence, with these assumptions, for all t >_ t2 we have: C~(t) _> Cl(tl) + (1 - x)(t -/1) -- 4" (3) NOW suppose that for i = 1, ..., n we have t, _< t ~, < ti+l, to <-- t~, and that at time t[ process Pi sends a message to process Pi+l which is received at time ti+l with an unpredictable delay less than 4. Then repeated applica- tion of the inequality (3) yields the following result for t >_ tn+l. Ct~t(t) --> Cl(tl') + (1 - ~)(t - tl') - n~. (4) From PC1, IRI' and 2' we deduce that Cl(/l') >" Cl(tl) + (1 -- K)(tl' -- /1). Combining this with (4) and using (2), we get Cn+a(t) > C~(tl) + (1 - x)(t - t~) - n~ (5) for t > tn+l. For any two processes P and P', we can find a sequence of processes P -- Po, P~ ..... Pn+~ = P', n _< d, with communication arcs from each Pi to Pi+~. By hy- pothesis (b) we can find times ti, t[ with t[ - ti <- T and ti+l - t" <_ v, where v = # + 4. Hence, an inequality of the form (5) holds with n <_ d whenever t >_ t~ + d('r + v). For any i, j and any t, tl with tl > to and t ___ t~ + d(z + v) we therefore have: Ci(t) _> Cj(ta) + (1 - x)(t - tx) - d~. (6) Now let m be any message timestamped Tin, and suppose it is sent at time t and received at time t'. We pretend that m has a clock Cm which runs at a constant rate such that C,~(t) = tm and Cm(t') = tm +/~m. Then #0, ___ t' - t implies that dCm/dt _< 1. Rule IR2' (b) simply sets Cj(t') to maximum (Cj(t' - 0), Cm(t')). Hence, clocks are reset only by setting them equal to other clocks. For any time tx >-- to +/~/(1 - ~), let Cx be the clock having the largest value at time t~. Since all clocks run at a rate less than 1 + x, we have for all i and all t >_ tx: Ci(t) _< Cx(tx) + (1 + x)(t - tx). (7) We now consider the following two cases: (i) Cx is the clock Cq of process Pq. (ii) Cx is the clock Cm of a message sent at time ta by process Pq. In case (i), (7) simply becomes El(t) -< Cq(tx) + (1 + x)(t - tx). (8i) In case (ii), since Cm(tx) = Cq(tl) and dCm/dt _ 1, we have Cx(tx) <_ Cq(tl) + (tx - tO. Hence, (7) yields Ci(t) <-~ Cq(/1) + (1 + K)(t -- /1). Since tx Cq(tx -- (8ii) >-- to + ~t/(1 -- X), we get ~/(1 -- K)) <_ Cq(tx) - ~ [by PCI] ___ Cm(tx) -/z [by choice of m] -< Cm(tx) - (tx - tl)#m/Vm [tXm <-- r.t, tx - tl <_ v,,] =Tm [by definition of Cm] = Cq(tl) [by IR2'(a)]. Hence, Cq(tx -/L/(1 - x)) ___ Cq(tl), so tx - tl <-/x/(l - ~) and thus ll ~ to. 564 Communications July 1978 of Volume 21 the ACM Number 7 Letting tl = tx in case (i)~ we can combine (8i) and (8ii) to deduce that for any t, tx with t _> tx _> to + #/ (1 - x) there is a process Pq and a time tl with tx - #/ (1 - i¢) ~ tl "< tx such that for all i: Ci(t) _< Cq(t~) + (1 + x)(t - h). (9) Choosing t and tx with t _ tx + d(r + ~,), we can combine (6) and (9) to conclude that there exists a tl and a process Pq such that for all i: Cq(t 0 + (1 - x)(t - tl) - d~ -< Ci(t) (10) <: Cq(tl) + (1 + tc)(t - t]) Letting t = t~ + d(r + u), we get d(r+p)__.t-tl_<d(r+v)+#/(1-x). Combining this with (10), we get Cq(tl) + (t - tl) - xd(r + p) - d~ -< Ci(t) _< Cq(tl) (11) + (t - tl) + x[d(r + u) +/z/(1 - x)] Using the hypotheses that x << 1 and/t _< ~, << r, we can rewrite (11) as the following approximate inequality. Cq(tl) + (t - tl) - d(tg'r + ~) ~.< Ci(t) (12) ~' Cq(/1) + (l -- tl) -I- dKr. Since this holds for all i, we get [Ci(t) - CAt)[ ~< d(2xr + O, and this holds for all t > to + dr. [] Note that relation (11) of the proof yields an exact upper bound for [Ci(t) - Cj(t) l in case the assumption /~ + ~ << r is invalid. An examination of the proof suggests a simple method for rapidly initializing the clocks, or resynchronizing them if they should go out of synchrony for any reason. Each process sends a message which is relayed to every other process. The procedure can be initiated by any process, and requires less than 2d~ + ~) seconds to effect the synchronization, assuming each of the messages has an unpredictable delay less than ~. Programming J.J. Homing Languages Editor Shallow Binding in Lisp 1.5 Henry G. Baker, Jr. Massachusetts Institute of Technology Shallow binding is a scheme which allows the value of a variable to be accessed in a bounded amount of computation. An elegant model for shallow binding in Lisp 1.5 is presented in which context-switching is an environment tree transformation called rerooting. Rerooting is completely general and reversible, and is optional in the sense that a Lisp 1.5 interpreter will operate correctly whether or not rerooting is in- voked on every context change. Since rerooting leaves assoc [ v, a] invariant, for all variables v and all environments a, the programmer can have access to a rerooting primitive, shallow[], which gives him dynamic control over whether accesses are shallow or deep, and which affects only the speed of execution of a program, not its semantics. In addition, multiple processes can be active in the same environment structure, so long as rerooting is an indivisible operation. Finally, the concept of rerooting is shown to combine the concept of shallow binding in Lisp with Dijkstra's display for Algol and hence is a general model for shallow binding. Key Words and Phrases: Lisp 1.5, environment trees, FUNARG'S, shallow binding, deep binding, multiprogramming, Algol display CR Categories: 4.13, 4.22, 4.32 Acknowledgment. The use of timestamps to order operations, and the concept of anomalous behavior are due to Paul Johnson and Robert Thomas. Received March 1976; revised October 1977 References 1. Schwartz, J.T. Relativity in lllustrations. New York U. Press, New York, 1962. 2. Taylor, E.F., and Wheeler, J.A. Space-Time Physics, W.H. Freeman, San Francisco, 1966. 3. Lamport, L. The implementation of reliable distributed multiprocess systems. To appear in Computer Networks. 4. Ellingson, C, and Kulpinski, R.J. Dissemination of system-time. 1EEE Trans. Comm. Com-23, 5 (May 1973), 605-624. 565 General permission to make fair use in teaching or research of all or part of this material is granted to individual readers and to nonprofit libraries acting for them provided that ACM's copyright notice is given and that reference is made to the publication, to its date of issue, and to the fact that reprinting privileges were granted by permission of the Association for Computing Machinery. To otherwise reprint a figure, table, other substantial excerpt, or the entire work requires specific permission as does republication, or systematic or multiple reproduc- tion. This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Office of Naval Research under contract number N00014-75-C-0522. Author's present address: Computer Science Department, Univer- sity of Rochester, Rochester, NY 14627. © 1978 ACM 0001-0782/78/0700-0565 $00.75 Communications July 1978 of Volume 21 the ACM Number 7
ts_1013760309v010101p	ETSI TS 101 376-3-9V1.1.1(2001-03) Technical Specification GEO-Mobile Radio Interface Specifications; Part 3: Network specifications; Sub-part 9: Security related Network Functions; GMR-1 03.020 ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)2GMR-1 03.020 Reference DTS/SES-001-03020 Keywords GMR, GSM, GSO, inetrface, MES, mobile, MSS, network, radio, satellite, security, S-PCN ETSI 650 Route des Lucioles F-06921 Sophia Antipolis Cedex - FRANCE T e l . :+ 3 349 29 44 20 0 F a x :+ 3 349 36 54 71 6 Siret N° 348 623 562 00017 - NAF 742 C Association à but non lucratif enregistrée à la Sous-Préfecture de Grasse (06) N° 7803/88 Important notice Individual copies of the present document can be downloaded from: http://www.etsi.org The present document may be made available in more than one electronic version or in print. In any case of existing or perceived difference in contents between such versions, the reference version is the Portable Document Format (PDF). In case of dispute, the reference shall be the printing on ETSI printers of the PDF version kept on a specific network drive within ETSI Secretariat. Users of the present document should be aware that the document may be subject to revision or change of status. Information on the current status of this and other ETSI documents is available athttp://www.etsi.org/tb/status/ If you find errors in the present document, send your comment to: [email protected] Copyright Notification No part may be reproduced except as authorized by written permission. The copyright and the foregoing restriction extend to reproduction in all media. © European Telecommunications Standards Institute 2001. All rights reserved. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)3GMR-1 03.020 Contents Intellectual Property Rights ..........................................................................................................................5 Foreword......................................................................................................................................................7 Introduction..................................................................................................................................................8 1 Scope..................................................................................................................................................9 2 References..........................................................................................................................................9 3 Abbreviations ...................................................................................................................................10 4 Security features provided in a GMR-1 PLMN..................................................................................11 4.1 General ..................................................................................................................................................... 11 4.2 Subscriber identity confidentiality.............................................................................................................. 11 4.2.1 Definition ............................................................................................................................................ 11 4.2.2 Purpose................................................................................................................................................ 11 4.2.3 Functional requirements....................................................................................................................... 11 4.3 Subscriber identity authentication .............................................................................................................. 12 4.3.1 Definition ............................................................................................................................................ 12 4.3.2 Purpose................................................................................................................................................ 12 4.3.3 Functional requirements....................................................................................................................... 12 4.3.3.1 Access to services........................................................................................................................... 12 4.3.3.2 Storage of subscriber related information ........................................................................................ 13 4.4 User data confidentiality on physical connections (voice and nonvoice)...................................................... 15 4.4.1 Definition ............................................................................................................................................ 15 4.4.2 Purpose................................................................................................................................................ 15 4.4.3 Functional requirements....................................................................................................................... 15 4.5 Signalling information element confidentiality........................................................................................... 17 4.5.1 Definition ............................................................................................................................................ 17 4.5.2 Purpose................................................................................................................................................ 17 4.5.3 Functional requirements....................................................................................................................... 17 5 Subscriber identity confidentiality.....................................................................................................17 5.1 Overview .................................................................................................................................................. 17 5.2 Identifying a mobile earth station............................................................................................................... 17 5.3 TMSI management procedures .................................................................................................................. 18 5.3.1 User identification during location update within the same GS/MSC/VLR coverage area ...................... 18 5.3.2 User identification during location update between different GS/MSC/VLRs ........................................ 19 5.3.3 User identification during location registration ..................................................................................... 20 5.3.4 User identification with a system malfunction....................................................................................... 20 5.3.4.1 Location update, TMSI lost............................................................................................................. 20 5.3.4.2 Location update between different GS/MSC/VLR coverage areas, old VLR not reachable ............... 21 5.3.4.3 Location update in the same GS/MSC/VLR coverage area, local TMSI unknown ............................ 22 5.3.4.4 Location update between different GS/MSC/VLR coverage areas, loss of information..................... 23 6 Subscriber identity authentication .....................................................................................................24 6.1 General ..................................................................................................................................................... 24 6.2 The authentication procedure..................................................................................................................... 24 6.3 Subscriber authentication key management................................................................................................ 24 6.3.1 General authentication procedure ......................................................................................................... 25 6.3.2 Authentication at location updating in a new VLR, using TMSI............................................................ 26 6.3.3 Authentication at location updating in a new VLR, using IMSI............................................................. 26 6.3.4 Authentication at location updating in a new VLR, using TMSI, TMSI unknown in "old" VLR............. 27 6.3.5 Authentication at location updating in a new VLR, using TMSI, old VLR not reachable........................ 27 6.3.6 Authentication with IMSI if authentication with TMSI fails.................................................................. 28 6.3.7 Reuse of security-related information in failure situations.....................................................................28 7 Confidentiality of signalling information and user information on physical connections ....................28 7.1 General ..................................................................................................................................................... 28 ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)4GMR-1 03.020 7.2 Ciphering.................................................................................................................................................. 29 7.3 Setting the session key............................................................................................................................... 29 7.4 Start of ciphering and deciphering.............................................................................................................. 30 7.5 Frame number tag...................................................................................................................................... 30 7.6 Negotiation of A5-GMR-1......................................................................................................................... 30 7.7 Implementation of bidirectional ciphering.................................................................................................. 31 8 Terminal-to-terminal call privacy......................................................................................................31 8.1 Ciphering requirement............................................................................................................................... 31 8.2 Generation of the common key (Ktt) and start of ciphering and deciphering ............................................... 32 8.2.1 The use of S1 and S2............................................................................................................................ 32 8.2.2 Frame number correction ..................................................................................................................... 32 8.2.3 Change of cipher mode with Kc to cipher mode with Ktt ...................................................................... 32 9 Summary ..........................................................................................................................................34 Annex A (informative): Security issues related to signalling schemes and key management.........35 A.1 Introduction......................................................................................................................................35 A.2 Short description of the scheme.........................................................................................................35 Annex B (informative): Security information to be stored in the GMR-1 system ..........................36 B.1 Introduction......................................................................................................................................36 B.2 Entities and security information.......................................................................................................36 B.2.1 Home location register............................................................................................................................... 36 B.2.2 Visitor location register ............................................................................................................................. 36 B.2.3 Mobile services switching center/gateway station....................................................................................... 36 B.2.4 Mobile earth station................................................................................................................................... 37 B.2.5 Authentication center (AuC)...................................................................................................................... 37 Annex C (normative): External specifications of security related algorithms..............................38 C.1 Scope................................................................................................................................................38 C.2 Specifications for algorithm A5-GMR-1 ...........................................................................................38 C.2.1 Purpose..................................................................................................................................................... 38 C.2.2 Implementation indications........................................................................................................................ 38 C.2.3 External specifications of algorithm A5-GMR-1 ........................................................................................ 40 C.2.4 Internal specification of algorithm A5-GMR-1........................................................................................... 40 C.3 Algorithm A3 ...................................................................................................................................41 C.3.1 Purpose..................................................................................................................................................... 41 C.3.2 Implementation and operational requirements ............................................................................................ 41 C.4 Algorithm A8 ...................................................................................................................................41 C.4.1 Purpose..................................................................................................................................................... 41 C.4.2 Implementation and operational requirements ............................................................................................ 41 Annex D (informative): Generation of session keys for direct terminal-to-terminal calls..............43 History .......................................................................................................................................................44 ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)5GMR-1 03.020 Intellectual Property Rights The information pertaining to essential IPRs is publicly available forETSI members and non-members , and can be found in ETSI SR 000 314:"Intellectual Property Rights (IPRs); Essential, or potentially Essential, IPRs notified to ETSI in respect of ETSI standards", which is available from the ETSI Secretariat. Latest updates are available on the ETSI Web server (http://www.etsi.org/ipr). The attention of ETSI has been drawn to the Intellectual Property Rights (IPRs) listed below which are, or may be, or may become, Essential to the present document. The IPR owner has undertaken to grant irrevocable licences, on fair, reasonable and non-discriminatory terms and conditions under these IPRs pursuant to the ETSI IPR Policy. Further details pertaining to these IPRs can be obtained directly from the IPR owner. The present IPR information has been submitted to ETSI and pursuant to the ETSI IPR Policy, no investigation, including IPR searches, has been carried out by ETSI. No guarantee can be given as to the existence of other IPRs not referenced in ETSI SR 000 314 (or the updates on the ETSI Web server) which are, or may be, or may become, essential to the present document. IPRs: Project Company Title Country of Origin Patent n° Countries Applicable TS 101 376 V1.1.1 Digital Voice Systems Inc US US 5,226,084 US TS 101 376 V1.1.1 Digital Voice Systems Inc US US 5,715,365 US TS 101 376 V1.1.1 Digital Voice Systems Inc US US 5,826,222 US TS 101 376 V1.1.1 Digital Voice Systems Inc US US 5,754,974 US TS 101 376 V1.1.1 Digital Voice Systems Inc US US 5,701,390 US IPR Owner: Digital Voice Systems Inc One Van de Graaff Drive Burlington, MA 01803 USA Contact: John C. Hardwick Tel.: +1 781 270 1030 Fax: +1 781 270 0166 Project Company Title C ountry of Origin Patent n° Countries Applicable TS 101 376 V1.1.1 Ericsson Mobile Communication Improvements in, or in relation to, equalisers GB GB 2 215 567 GB TS 101 376 V1.1.1 Ericsson Mobile Communication Power Booster GB GB 2 251 768 GB TS 101 376 V1.1.1 Ericsson Mobile Communication Receiver Gain GB GB 2 233 846 GB TS 101 376 V1.1.1 Ericsson Mobile Communication Transmitter Power Control for Radio Telephone System GB GB 2 233 517 GB IPR Owner: Ericsson Mobile Communications (UK) Limited The Keytech Centre, Ashwood Way Basingstoke Hampshire RG23 8BG United Kingdom Contact: John Watson Tel.: +44 1256 864 821 ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)6GMR-1 03.020 Project Company Title Country of Origin Patent n° Countries Applicable TS 101 376 V1.1.1 Hughes Network Systems US Pending US IPR Owner: Hughes Network Systems 11717 Exploration Lane Germantown, Maryland 20876 USA Contact: John T. Whelan Tel: +1 301 428 7172 Fax: +1 301 428 2802 Project Company Title Country of Origin Patent n° Countries Applicable TS 101 376 V1.1.1 Lockheed Martin Global Telecommunic. Inc 2.4-to-3 KBPS Rate Adaptation Apparatus for Use in Narrowband Data and Facsimile Communication Systems US US 6,108,348 US TS 101 376 V1.1.1 Lockheed Martin Global Telecommunic. Inc Cellular Spacecraft TDMA Communications System with Call Interrupt Coding System for Maximizing Traffic ThroughputCellular Spacecraft TDMA Communications System with Call Interrupt Coding System for Maximizing Traffic Throughput US US 5,717,686 US TS 101 376 V1.1.1 Lockheed Martin Global Telecommunic. Inc Enhanced Access Burst for Random Access Channels in TDMA Mobile Satellite System US US 5,875,182 TS 101 376 V1.1.1 Lockheed Martin Global Telecommunic. Inc Spacecraft Cellular Communication System US US 5,974,314 US TS 101 376 V1.1.1 Lockheed Martin Global Telecommunic. Inc Spacecraft Cellular Communication System US US 5,974,315 US TS 101 376 V1.1.1 Lockheed Martin Global Telecommunic. Inc Spacecraft Cellular Communication System with Mutual Offset High-argin Forward Control Signals US US 6,072,985 US TS 101 376 V1.1.1 Lockheed Martin Global Telecommunic. Inc Spacecraft Cellular Communication System with Spot Beam Pairing for Reduced Updates US US 6,118,998 US IPR Owner: Lockheed Martin Global Telecommunications, Inc. 900 Forge Road Norristown, PA. 19403 USA Contact: R.F. Franciose Tel.: +1 610 354 2535 Fax: +1 610 354 7244 ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)7GMR-1 03.020 Foreword This Technical Specification (TS) has been produced by ETSI Technical Committee Satellite Earth Stations and Systems (SES). The contents of the present document are subject to continuing work within TC-SES and may change following formal TC-SES approval. Should TC-SES modify the contents of the present document it will then be republished by ETSI with an identifying change of release date and an increase in version number as follows: Version 1.m.n where: - the third digit (n) is incremented when editorial only changes have been incorporated in the specification; - the second digit (m) is incremented for all other types of changes, i.e. technical enhancements, corrections, updates, etc. The present document is part 3, sub-part 9 of a multi-part deliverable covering the GEO-Mobile Radio Interface Specifications, as identified below: Part 1: "General specifications"; Part 2: "Service specifications"; Part 3: "Network specifications"; Sub-part 1: "Network Functions; GMR-1 03.001"; Sub-part 2: "Network Architecture; GMR-1 03.002"; Sub-part 3: "Numbering, Addressing and identification; GMR-1 03.003"; Sub-part 4: "Organization of Subscriber Data; GMR-1 03.008"; Sub-part 5: "Technical realization of Supplementary Services; GMR-1 03.011"; Sub-part 6: "Location Registration and Position Identification Procedures; GMR-1 03.012"; Sub-part 7: "Discontinuous Reception (DRX); GMR-1 03.013"; Sub-part 8: "Support of Dual-Tone Multifrequency Signalling (DTMF); GMR-1 03.014"; Sub-part 9: "Security related Network Functions; GMR-1 03.020"; Sub-part 10: "Functions related to Mobile Earth station (MES) in idle mode; GMR-1 03.022"; Sub-part 11: "Technical realization of the Short Message Service (SMS) Point-to-Point (PP); GMR-1 03.040"; Sub-part 12: "Technical realization of the Short Message Service Cell Broadcast (SMSCB); GMR-1 03.041"; Sub-part 13: "Technical realization of group 3 facsimile using transparent mode of transmission; GMR-1 03.045"; Sub-part 14: Transmission Planning Aspects of the Speech Service in the GMR-1 system; GMR-1 03.050"; Sub-part 15: "Line Identification supplementary service - Stage 2; GMR-1 03.081"; Sub-part 16: "Call Barring (CB) supplementary services - Stage 2; GMR-1 03.088"; Sub-part 17: "Unstructured Supplementary Service Data (USSD) - Stage 2; GMR-1 03.290"; Sub-part 18: "Terminal-to-Terminal Call (TtT); GMR-1 03.296"; ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)8GMR-1 03.020 Sub-part 19: "Optimal Routing technical realization; GMR-1 03.297"; Sub-part 20: "Technical realization of High-Penetration Alerting; GMR-1 03.298"; Sub-part 21: "Position Reporting services; Stage 2 Service description; GMR-1 03.299"; Part 4: "Radio interface protocol specifications"; Part 5: "Radio interface physical layer specifications"; Part 6: "Speech coding specifications"; Part 7: "Terminal adaptor specifications". Introduction GMR stands for GEO (Geostationary Earth Orbit) Mobile Radio interface, which is used for mobile satellite services (MSS) utilizing geostationary satellite(s). GMR is derived from the terrestrial digital cellular standard GSM and supports access to GSM core networks. Due to the differences between terrestrial and satellite channels, some modifications to the GSM standard are necessary. Some GSM specifications are directly applicable, whereas others are applicable with modifications. Similarly, some GSM specifications do not apply, while some GMR specifications have no corresponding GSM specification. Since GMR is derived from GSM, the organization of the GMR specifications closely follows that of GSM. The GMR numbers have been designed to correspond to the GSM numbering system. All GMR specifications are allocated a unique GMR number as follows: GMR-n xx.zyy where: - xx.0yy (z = 0) is used for GMR specifications that have a corresponding GSM specification. In this case, the numbers xx and yy correspond to the GSM numbering scheme. - xx.2yy (z = 2) is used for GMR specifications that do not correspond to a GSM specification. In this case, only the number xx corresponds to the GSM numbering scheme and the number yy is allocated by GMR. - N denotes the first (n = 1) or second (n = 2) family of GMR specifications. A GMR system is defined by the combination of a family of GMR specifications and GSM specifications as follows: • If a GMR specification exists it takes precedence over the corresponding GSM specification (if any). This precedence rule applies to any references in the corresponding GSM specifications. NOTE: Any references to GSM specifications within the GMR specifications are not subject to this precedence rule. For example, a GMR specification may contain specific references to the corresponding GSM specification. • If a GMR specification does not exist, the corresponding GSM specification may or may not apply. The applicability of the GSM specifications is defined in GMR-1 01.201 [2]. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)9GMR-1 03.020 1 Scope The present document specifies the network functions needed to provide the security related service and functions specified in technical specification GSM 02.09 [8]. The use of satellite communications for transmission to mobile subscribers makes public land mobile networks (PLMNs) particularly sensitive to: • misuse of their resources by unauthorized persons using manipulated MESs, who try to impersonate authorized subscribers; • eavesdropping on the various information that is exchanged on the satellite path. It can be seen that PLMNs intrinsically do not provide the same level of protection to their operators and subscribers as the traditional telecommunication networks provide. This fact leads to the need to implement security features in a G M R - 1P L M Ni no r d e rt op r o t e c t : • the access to the mobile services; • any relevant item from being disclosed at the satellite path, mainly in order to ensure the privacy of user-related information. Therefore two levels of protection are assumed: 1) where security features are provided, the level of protection at the satellite path of the corresponding items is as good as the level of protection provided in fixed networks; 2) where no special provision is made, the level of protection at the satellite path is null. All items that are not dealt with in clause 4 are considered to need no protection. The present document draws on GSM 02.09 [8] "Security Aspect" and GSM 03.20 [10] "Security-Related Network Functions" to establish functional requirements as well as detailed procedures for GMR-1 system security. The present document does not address the cryptological algorithms that are needed to provide different security related features. This topic is addressed in annex C. Wherever a cryptological algorithm or mechanism is needed, this is signalled with a reference to annex C. The references refers only to functionalities, and some algorithms may be identical or use common hardware. 2 References The following documents contain provisions which, through reference in this text, constitute provisions of the present document. • References are either specific (identified by date of publication and/or edition number or version number) or non-specific. • For a specific reference, subsequent revisions do not apply. • For a non-specific reference, the latest version applies. [1] GMR-1 01.004 (ETSI TS 101 376-1-1): "GEO-Mobile Radio Interface Specifications; Part 1: General specifications; Sub-part 1: Abbreviations and acronyms; GMR-1 01.004". [2] GMR-1 01.201 (ETSI TS 101 376-1-2): "GEO-Mobile Radio Interface Specifications; Part 1: General specifications; Sub-part 2: Introduction to the GMR-1 Family; GMR-1 01.201". [3] GMR-1 03.003 (ETSI TS 101 376-3-3): "GEO-Mobile Radio Interface Specifications; Part 3: Network specifications; Sub-part 3: Numbering, Addressing and identification; GMR-1 03.003". ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)10GMR-1 03.020 [4] GMR-1 03.296 (ETSI TS 101 376-3-18): "GEO-Mobile Radio Interface Specifications; Part 3: Network specifications; Sub-part 18: Terminal-to-Terminal Call (TtT); GMR-1 03.296". [5] GMR-1 03.297 (ETSI TS 101 376-3-19): "GEO-Mobile Radio Interface Specifications; Part 3: Network specifications; Sub-part 19: Optimal Routing technical realisation; GMR-1 03.297". [6] GMR-1 05.002 (ETSI TS 101 376-5-2): "GEO-Mobile Radio Interface Specifications; Part 5: Radio interface physical layer specifications; Sub-part 2: Multiplexing and Multiple Access; Stage 2 Service Description; GMR-1 05.002". [7] GMR-1 05.010 (ETSI TS 101 376-5-7): "GEO-Mobile Radio Interface Specifications; Part 5: Radio interface physical layer specifications; Sub-part 7: Radio Subsystem Synchronisation; GMR-1 05.010". [8] GSM 02.09 ETSI ETS 300 506: "Digital cellular telecommunications system (Phase 2); Security a s p e c t s( G S M0 2 . 0 9v e r s i o n4 . 4 . 1 ) " . [9] GSM 02.17 (ETSI ETS 300 509): "European digital cellular telecommunications system (Phase 2); Subscriber Identity Module (SIM); Functional characteristics (GSM 02.17 V4.3.3)". [10] GSM 03.20 (ETSI ETS 300 534): "Digital cellular telecommunications system (Phase 2); Security related network functions (GSM 03.20 version 4.4.1)". 3 Abbreviations For the purposes of the present document, the abbreviations given in GMR-1 01.004 [1] and the following apply. A3 Authentication Algorithm, used in security schemes A5-GMR-1 Signalling data and user data encryption algorithm, used in security schemes A8 Session key generating algorithm, used in security schemes Block-1 Ciphering stream used in the direction from network to MES Block-2 Ciphering stream used in the direction from MES to network CKSN Ciphering Key Sequence Number GSC GMR-1 Security Custodian, used in security schemes HLR Home Location Register HPLMN Home Public Land Mobile Network IMEI International Mobile Equipment Identity IMSI International Mobile Station Identity Kc Ciphering Key, used in security schemes Kc[M] Message encrypted with ciphering key Kc, used in security schemes Kc[TMSI] TMSI encrypted with ciphering key Kc, used in security schemes keynr Key number associated with a session key, used in security schemes Ki Individual subscriber authentication key, used in security schemes Ktt Common ciphering key used in mobile-to-mobile calls LAI Location Area Identification lsb least significant bit LU Location Update M Clear text message, used in security schemes MCC Mobile County Code MNC Mobile Network Code msb most significant bit MSC Mobile Switching Center MSCID MSC/VLR Identity n a Size of triplet array, used in security schemes PLMN Public Land Mobile Networks RAND Random Number SBID Spot beam Identity SIM Subscriber Identity Module SRES Signed Response TMSI Temporary Mobile Subscriber Identity ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)11GMR-1 03.020 TMSI o/n Temporary Mobile Subscriber Identity old/new, used in security schemes triplet Set of three numbers: RAND, SRES, and Kc, used in security schemes VLR Visitor Location Register VLR o/n Visitor Location Register old/new 4 Security features provided in a GMR-1 PLMN 4.1 General The following security features are considered: • Subscriber identity (IMSI) confidentiality. • Subscriber identity (IMSI) authentication. • User data confidentiality on physical connections. • Signalling information element confidentiality. • Equipment identity number (IMEI) confidentiality. The implementation of these five security features is mandatory on both the fixed infrastructure side and the mobile earth station (MES) side. This means that all GMR-1 PLMNs and all MESs will be able to support every security feature. For all subscribers, use of these five security features is mandatory. In case of an emergency call that does not require the SIM, all security features are bypassed. Details of the authentication algorithm and the session key algorithm are left to the discretion of the local PLMN service provider. 4.2 Subscriber identity confidentiality 4.2.1 Definition The subscriber identity confidentiality feature is the property that the IMSI is not made available or disclosed to unauthorized individuals, entities, or processes. 4.2.2 Purpose This feature provides for the privacy of the identities of the subscribers who are using GMR-1 PLMN resources (e.g., a traffic channel or any signalling means). It allows for the improvement of all other security features (e.g., user data confidentiality) and provides for the protection against tracing the location of a mobile subscriber by listening to the signalling exchanges on the satellite path. 4.2.3 Functional requirements This feature necessitates the confidentiality of the subscriber identity (IMSI) when it is transferred in signalling messages (see clause 4.5) together with specific measures to preclude the possibility to derive it indirectly from listening to specific information such as addresses at the satellite path. The means used to identify a mobile subscriber on the satellite path consists of a local number called temporary mobile subscriber identity (TMSI). When used, the subscriber identity confidentiality feature will apply for all signalling sequences on the satellite path. However, in the case of location register failure, or in case the MES has no TMSI available, use of IMSI or IMEI is allowed on the satellite path. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)12GMR-1 03.020 4.3 Subscriber identity authentication 4.3.1 Definition IMSI authentication corroborates that the subscriber identity claimed by the user (IMSI or TMSI) is correct. 4.3.2 Purpose The purpose of this authentication security feature is to protect the network against unauthorized use. It enables also the protection of the GMR-1 PLMN subscribers by denying the possibility for intruders to impersonate authorized users. 4.3.3 Functional requirements The authentication of the GMR-1 PLMN subscriber identity may be triggered by the network when the subscriber applies for: • Access to service: set-up of mobile-originated or terminated calls and the activation or deactivation of a supplementary service. • Location update: this refers to a change of subscriber-related information element in the visitor location register (VLR) or home location register (HLR) including location updating involving change of VLR. Confidential information contained in the SIM should never be transmitted over the satellite radio interface. Physical security means shall be provided to preclude the possibility to obtain sufficient information to impersonate or duplicate a subscriber in a GMR-1 PLMN, in particular by deriving sensitive information from the MES equipment. If, on an access request to the GMR-1 PLMN, the subscriber identity authentication procedure fails and this failure is not due to network malfunction, then the access to the GMR-1 PLMN will be denied to the requesting party. 4.3.3.1 Access to services Except in the case of an emergency call, an authentication process is required for all mobile originated and mobile terminated calls. The following list will provide an illustration of the situations requiring an authentication procedure for the originating user: • GMR-1 MES calls PSTN user. • GMR-1 MES calls GMR-1 MES. • GMR-1 MES calls GSM user. • GMR-1 MES calls GSM user roaming in GMR-1 network. • GSM user roaming in GMR-1 network calls PSTN user. • GSM user roaming in GMR-1 network calls GMR-1 MES. • GSM user roaming in GMR-1 network calls GSM PLMN user. • GSM user roaming in GMR-1 network calls another GSM user roaming in GMR-1 network. The following list will provide an illustration of the situations requiring an authentication procedure for the terminating user: • PSTN user calls GMR-1 MES. • GMR-1 MES calls GMR-1 MES. • GSM user calls GMR-1 MES. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)13GMR-1 03.020 • GSM user roaming in GMR-1 network calls GMR-1 MES. • PSTN user calls GSM user roaming in GMR-1 network. • GMR-1 MES calls GSM user roaming in GMR-1 network. • GSM user calls another GSM user roaming in GMR-1 network. • GSM user roaming in GMR-1 network calls another GSM user roaming in GMR-1 network. Connectionless service also requires authentication, but is not now implemented. 4.3.3.2 Storage of subscriber related information In the GMR-1 system, the subscriber related information is always stored in the VLR which provides service for the user's current roaming area. This information is passed from VLR to another VLR as the user moves around in the system. Transfer of the subscriber related information is performed by Location Update procedure, and the later is triggered whenever the MES finds a change of Location Area Identity (LAI) received from the BCCH channel. The LAI is defined to have four different components, i.e.: LAI = MCC + MNC + SBID + MSCID where MCC is defined as mobile country code, MNC is defined mobile network code, SBID is defined as spot beam identity and, MSCID is defined as MSC/VLR identity. Due to a change of mobile user's location, one or more than one component in the LAI might be changed, which eventually will initiate the procedure of a location update. Table 4.1 outlines various reasons for triggering a location update in the GMR-1 system. In the same table, the number of MSC/VLRs involved in the location update procedure is also specified. Table 4.1: Various types of location update in the GMR-1 system Case No. MCC MNC SBID MSCID Number of MSC/VLRs Involved in the LU 1- - - x 2 2- - x - 1 3- - x x 2 4x x - x 2 5x x x x 2 "-": corresponding term without any change during user's motion. "x": a corresponding change takes place due to user motion. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)14GMR-1 03.020 The relationship between location update and a mobile user's motion is summarized in figure 4.1. Beam border Country border Gateway connection with spotbeam Location area border PLMN coverage border n Case number 'n' GS1/MSC1/VLR1 GS3/MSC3/VLR3 HLR1 HLR2 1 2 3 4 5 GS2/MSC2/VLR2 PLMN-1 PLMN-2 Beam-1 Beam-2 Beam-3 Beam-4 Beam-5 Beam-6 Country-1 Country-2 Figure 4.1: Various reasons to trigger location update due to mobile user's motion There are five different location update cases in figure 4.1: Case 1: In the Location Update (LU), the Gateway Station (GS) changes from GS1 to GS2, subscriber related information is passed from VLR1 to VLR2. Case 2: In the LU, the spot beam changes from beam-1 to beam-2, subscriber related information remains in the same VLR (VLR1). Case 3: In the LU, the GS changes from GS1 to GS2, and the spot beam changes from beam-3 to beam-4, subscriber related information is passed from VLR1 to VLR2. Case 4: In the LU, the GS changes from GS2 to GS3, the network changes from PLMN-1 to PLMN-2, and the country changes from country-1 to country-2, subscriber related information is passed from VLR2 to VLR3. Case 5: In the LU, the GS changes from GS2 to GS3, the spot beam changes from beam-5 to beam-6, the network changes from PLMN-1 to PLMN-2, and the country changes from country-1 to country-2, subscriber related information is passed from VLR2 to VLR3. Apart from mobile user's motion, a location update also can be triggered by the procedure of optimal routing (OR). At the start of a mobile-originated call, if the network finds that the optimal routing GS is different from the MES's local GS, the MES is required to perform a location update from the local GS to the optimal routing GS. Upon conclusion of the mobile-originated call, the MES may register back to its original GS by performing another location update from the optimal routing GS to its original GS. See GMR-1 03.297 [5] for details. The location update procedure triggered by the OR is the same as that triggered by the mobile user's motion. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)15GMR-1 03.020 If an MES is registered and has been successfully authenticated, calls are permitted (including continuation). If the MES is not registered or ceases to be registered, a new registration needs to be performed, and the preceding cases apply. 4.4 User data confidentiality on physical connections (voice and nonvoice) 4.4.1 Definition The user data confidentiality feature on physical connections is the property that the user information exchanged on traffic channels is not made available or disclosed to unauthorized individuals, entities or processes. 4.4.2 Purpose The purpose of this feature is to ensure privacy of user information on traffic channels. 4.4.3 Functional requirements Encryption will normally be applied to all voice and nonvoice communications. See table 4.2 for a list of GMR-1 channels that are encrypted. A standard algorithm called A5-GMR-1 will be employed throughout the system. It is permissible for the MES and/or PLMN infrastructure to support more than one algorithm. In this case, the infrastructure is responsible for deciding which algorithm to use (including the possibility not to use encryption, in which case confidentiality is not applied). When necessary, the MES will signal to the network indicating which algorithms it supports. The serving network then selects one of these that it can support (based on an order of priority preset in the network), and signals this to the MES. The selected algorithm is then used by the MES and network. An ON/OFF indicator for encryption is a useful feature but is not required for all situations. There should be a user option to temporarily disable or bypass this indicator if one is provided. In the case that this indicator is turned on, the MES has to check to see if user data confidentiality is switched on. If so, an indicator (e.g., a panel light) will be provided to show to the user. In the event that the MES confidentiality feature is turned off or abruptly transitions from ON to OFF, e.g., during handover, then an indication is given to the user. This ciphering indicator feature may be disabled by the subscriber identity module (SIM). During the establishment of a call, the trigger point for the indicator will be when the called party answers the call at the latest. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)16GMR-1 03.020 Table 4.2: GMR-1 channels which are and are not encrypted Channel Type Channel Name Encryption Used? Reasons for Use or Nonuse of Encryption TCH Channel TCH3 Yes If the call set-up is performed through the TCH channel instead of the SDCCH channel, signalling messages in the TCH channel are not encrypted up to the message "Cipher Mode Command." The "Cipher Mode Acknowledge" is sent encrypted. NOTE: in order to facilitate synchronization, the cipher stream blocks are generated at both ends of the link but are discarded for DKABs. TCH6 Yes Same as TCH3. TCH9 Yes Same as TCH3. BCCH BCCH No Common channel shared by multiple users, encryption is not applied. Channel FCCH No Same as BCCH. AGCH No Same as BCCH. BACH No Same as BCCH. CCCH CBCH No Same as BCCH. Channel GBCH No Same as BCCH. PCH No Same as BCCH. RACH No Same as BCCH. TTCH No To avoid two bursts within the same frame using the same ciphering stream. FACCH/3 Yes When traffic channels are encrypted, FACCHs also use encryption. Both ends of the link generate the same size of encryption bursts suitable for TCH burst, the FACCH burst discards some bits that are not needed. FACCH/6 Yes Same as FACCH/3. DCCH FACCH/9 Yes Same as FACCH/3. Channel SACCH Yes Same as FACCH/3. SDCCH Yes If the call set-up is performed through SDCCH channel instead of TCH channel, signalling messages in the SDCCH channel are not encrypted before and including the message "Cipher Mode Command," but "Cipher Mode Acknowledge" is sent encrypted. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)17GMR-1 03.020 4.5 Signalling information element confidentiality 4.5.1 Definition The signalling information element confidentiality feature is the property that a given piece of signalling information that is exchanged between MESs and Ground Stations is not made available or disclosed to unauthorized individuals, entities, or processes. 4.5.2 Purpose The purpose of the signalling confidentiality feature is to ensure the privacy of user-related signalling elements. 4.5.3 Functional requirements Up to a certain point in time, when the "cipher on" command is given, the call set-up procedure is not ciphered. Information transmitted during this period includes protocol discriminator, connection reference, message type, and TMSI according to the circumstance. When ciphering is turned on, then all information in the signalling channel is ciphered. 5 Subscriber identity confidentiality 5.1 Overview The purpose of this function is to avoid the possibility for an intruder to identify which subscriber is using a given resource on the satellite path (e.g., TCH or signalling resources) by listening to the signalling exchanges on the satellite path. This function allows both a high level of confidentiality for user data and signalling and protection against the tracing of a user's location. The provision of this function implies that the IMSI, or any information allowing a listener to derive the IMSI easily, should not normally be transmitted in clear text in any signalling message on the satellite path. Consequently, to obtain the required level of protection, it is necessary that: • a protected identifying method is normally used instead of the IMSI on the satellite path; and • when the signalling procedures permit it, signalling information elements that convey information about the mobile subscriber identity shall be ciphered for transmission on the satellite path. The identifying method is specified in the following clause. The ciphering of communication over the satellite path is specified in clause 7. 5.2 Identifying a mobile earth station The means used to identify a mobile subscriber on the satellite path is to use TMSI. This TMSI is a local number, having a meaning only in a given location area. Therefore the TMSI shall be accompanied by the LAI (location area identification) to avoid ambiguities. The maximum length and guidance for defining the format of a TMSI are specified i nG M R - 10 3 . 0 0 3[ 3 ] . The TMSI is always allocated by a VLR currently visited by the mobile user. The VLR manages a suitable database to keep the relationship between TMSIs and IMSIs. If the user moves to a new location area under the control of another VLR, both security related information and the user's IMSI follow the user's motion and are passed from the original VLR to the new VLR. The allocation of a new TMSI is always associated with the procedure of location update or initial registration. Meanwhile, the allocation of a new TMSI also triggers the de-allocation of the previous one. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)18GMR-1 03.020 When a new TMSI is allocated to an MES, it is transmitted to the MES in ciphered mode. The MES shall store its current TMSI in a non-volatile memory in the SIM, together with the LAI, so that these data are not lost when the MES is switched off. 5.3 TMSI management procedures The means of TMSI management have the following features: TMSI allocation and deallocation are always associated with mobile user's location update; the way of managing the TMSI is closely dependent on the location update procedure. From a user identification point of view, the signalling procedure is a function of the number of MSC/VLRs involved in the location update. Two situations have been identified: location update within the same MSC/VLR and location update between different MSC/VLRs, as shown in table 4.1. In the following clauses the TMSI management procedures will be described in terms of these two different situations. 5.3.1 User identification during location update within the same GS/MSC/VLR coverage area This procedure is part of the location updating procedure that takes place when the original location area and the new location area depend on the same GS/MSC/VLR. The procedure relative to TMSI management is reduced to a TMSI reallocation (from TMSIo with "o" for "old" to TMSIn with "n" for "new"). Note that the location area identity (LAI) is always associated with a TMSI. Thus, a new LAI will require a new TMSI. However, the designators "o" and "n" are applied to the TMSI, by convention, and not to the LAI in the following figures. The MES sends TMSIo as an identifying field at the beginning of the location updating procedure. The procedure is summarized in figure 5.1. MES Satellite radio path GS/MSC/VLR Ciphering Initiation Allocation of TMSIn LAI, TMSIo Cipher (TMSIn) Acknowledge De-allocation of TMSIn Figure 5.1: User identification during location update in the same GS/MSC/VLR area Signalling functionality: • MES initiates location update procedure, both LAI and TMSIo are transmitted over the satellite link in clear text. • Ciphering Initiation (clause 6). The MES and GS/MSC/VLR agree on means for ciphering signalling information elements, in particular for transmission of TMSIn. • A new TMSI is assigned by the VLR. It is passed to the MES in ciphered mode. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)19GMR-1 03.020 5.3.2 User identification during location update between different GS/MSC/VLRs This procedure is part of the normal location updating procedure, using TMSI and LAI, when the original location area and the new location area depend on different GS/MSC/VLRs. The MES is still registered in VLRo ("o" for old or original) and requests registration in VLRn ("n" for new). LAI and TMSIo are sent by MES as identifying fields during the location updating procedure. Note, that in this normal procedure, the new VLR obtains Location Update information directly from the old VLR. It will be noted later that in some cases, the VLRn shall go directly to the HLR in order to obtain the subscriber related information. The procedure is shown figure 5.2. MES Satellite path GS/MSC/VLRn Ciphering Initiation Allocation of TMSIn LAI, TMSIo Cipher (TMSIn) Acknowledge De-allocation of TMSIn GS/MSC/VLRo HLR TMSIo IMSI Sec. Rel. Inf. Location Update Acknowledge Cancellation Figure 5.2: User identification during location update in different GS/MSC/VLR areas Signalling functionality: • MES initiates location update procedure; both LAI and TMSIo are transmitted over the satellite link in clear text. • The MSC/VLRn needs some information for authentication and ciphering; this information is obtained from MSC/VLRo. • Ciphering initiation. The MES and GS/MSC/VLR agree on means for ciphering signalling information elements, in particular for transmission of TMSIn. • A new TMSI is assigned by the VLRn. It is passed to the MES in ciphered mode. • The VLRn informs the MES's HLR about this location update. • The HLR indicates to VLRo that the MES is now under control of another VLR. The "old" TMSI is free for allocation. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)20GMR-1 03.020 5.3.3 User identification during location registration This situation occurs where an MES requests first time registration, but there is no TMSI available. In this case, the IMSI is used for identification. The IMSI is sent in clear text via the satellite path as part of the registration process. This procedure is shown in figure 5.3. MES Satellite path GS/MSC/VLR HLR Allocation of TMSIn Cipher (TMSIn) Acknowledge IMSI Ciphering Initiation Sec. Rel. Info. Figure 5.3: User identification during MES registration for the first time Signalling functionality: • The MES initiates the registration procedure. The mobile user's IMSI is transmitted over the satellite path in clear text. • Ciphering initiation. The VLR asks for security-related information from its HLR. The MES and GS/MSC/VLR agree on the means for ciphering signalling information elements, in particular for transmission of TMSIn. • A new TMSI is assigned by the VLR. It is passed to the MES in ciphered mode. 5.3.4 User identification with a system malfunction To cope with malfunctioning situations, e.g., arising from a software failure or loss of database, the fixed part of the network can require the user's IMSI without encryption. This procedure is a breach in the provision of the service and should be used only when necessary. When a TMSI is received with an LAI that does not correspond to the current VLR, the IMSI of the MES shall be requested by the VLR in charge of the indicated location area. 5.3.4.1 Location update, TMSI lost This situation occurs where an MES asks for location update but its TMSI is lost. In this case, the IMSI is used for identification. The IMSI is sent in clear text via the satellite path as part of the location update. The same signalling procedure shown in figure 5.3 can be reused in this case. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)21GMR-1 03.020 5.3.4.2 Location update between different GS/MSC/VLR coverage areas, old VLR not reachable This situation arises when the VLR receiving the LAI and TMSIo cannot identify the VLRo. In that case the relation between TMSIo and IMSI is lost, and the identification of the MES in clear is necessary. The procedure is shown in figure 5.4. MES Satellite path GS/MSC/VLRn Allocation of TMSIn LAI, TMSIo Cipher (TMSIn) Acknowledge De-allocation of TMSIn GS/MSC/VLRo HLR Location Update Acknowledge Cancellation Old VLR not reachable Identity Request IMSI Ciphering Initiation Sec. Rel. Info. Figure 5.4: User identification during location update, different MSC/VLRs, old VLR not reachable Signalling functionality: • MES initiates location update procedure, both LAI and TMSIo are transmitted over the satellite link in clear text. • By analysing the LAI contained in the location update request, the MSC/VLRn realizes the VLRo is not reachable. The GS/MSC/VLRn asks the MES to submit its identity. • The mobile user's IMSI is passed to the VLRn over the satellite path in clear text. • Ciphering initiation. The VLRn asks for security related information from its HLR. The MES and GS/MSC/VLRn agree on means for ciphering signalling information elements, in particular for transmission of TMSIn. • A new TMSI is assigned by the VLRn. It is passed to the MES in ciphered mode. • The VLRn informs the MES's HLR about this location update. • The HLR indicates to VLRo that the MES is now under control of another VLR. The "old" TMSI is free for allocation. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)22GMR-1 03.020 5.3.4.3 Location update in the same GS/MSC/VLR coverage area, local TMSI unknown This situation arises when a data loss has occurred in a VLR and when an MES uses an unknown TMSI, e.g., for a communication request or for a location updating request in a location area managed by the same VLR. This procedure is shown in figure 5.5. MES Satellite path GS/MSC/VLRn Allocation of TMSIn LAI, TMSIo Cipher (TMSIn) Acknowledge HLR TMSI is unknown Identity Request IMSI Ciphering Initiation Sec. Rel. Info. Figure 5.5: User identification during location update, same MSC/VLR, local TMSI unknown Signalling functionality: • MES initiates location update procedure, both LAI and TMSIo are transmitted over the satellite link in clear text. • By analysing the LAI, it is decided that the MES has already registered in this VLR, but the VLR cannot find a corresponding record in its database matching the TMSIo. The GS/MSC/VLR asks the MES to submit its identity. • The mobile user's IMSI is passed to the VLR over the satellite path in clear text. • Ciphering initiation. The VLR asks for security related information from its HLR. The MES and GS/MSC/VLR agree on means for ciphering signalling information elements, in particular for transmission of TMSIn. • A new TMSI is assigned by the VLR. It is passed to the MES in ciphered mode. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)23GMR-1 03.020 5.3.4.4 Location update between different GS/MSC/VLR coverage areas, loss of information This situation arises when the VLR in charge of the MES has suffered a loss of data. In that case the relationship between TMSIo and IMSI is lost, and the identification of the MES without encryption is necessary. The procedure is summarized figure 5.6. MES Satellite path GS/MSC/VLRn Allocation of TMSIn LAI, TMSIo Cipher (TMSIn) Acknowledge De-allocation of TMSIn GS/MSC/VLRo HLR Location Update Acknowledge Cancellation Identity Request IMSI Ciphering Initiation TMSIo Unknown Sec. Rel. Info. Figure 5.6: User identification during location update, different MSC/VLRs, loss of information Signalling functionality: • MES initiates location update procedure, both LAI and TMSIo are transmitted over the satellite link in clear text. • Based on the received LAI, the VLRn interrogates the VLRo in order to obtain all security related information and the user's IMSI. But the VLRo cannot recognize this user due to a loss of information. In this case, the GS/MSC/VLRn shall ask the MES to submit its identity. • The mobile user's IMSI is passed to the VLRn over the satellite path in clear text. • Ciphering initiation. The VLRn asks for security related information from its HLR. The MES and GS/MSC/VLRn agree on means for ciphering signalling information elements, in particular for transmission of TMSIn. • A new TMSI is assigned by the VLRn. It is passed to the MES in ciphered mode. • The VLRn informs the MES's HLR about this location update. • The HLR indicates to the VLRo that the MES is now under the control of another VLR. The "old" TMSI is free for allocation. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)24GMR-1 03.020 6 Subscriber identity authentication 6.1 General The definition and operational requirements of subscriber identity authentication were described in clause 4. The authentication procedure will also be used to set the ciphering key (see clause 6). Therefore, it is performed after the subscriber identity (TMSI/IMSI) is known by the network and before the channel is encrypted. It is possible that a call can begin in cipher mode as soon as a traffic channel is allocated. Currently this feature is not supported in the GMR-1. Two network functions are necessary: the authentication procedure itself and the key management within the ground subsystem. 6.2 The authentication procedure The authentication procedure consists of the following exchange between the fixed subsystem and the MES. • The fixed subsystem transmits a non-predictable number RAND to the MES. • The MES computes the signature of RAND, say SRES, using algorithm A3 stored in the SIM and some secret information: the Individual Subscriber Authentication Key, denoted below by Ki also stored in the SIM. • The MES transmits the signature SRES to the fixed subsystem. • The fixed subsystem tests SRES for validity. The general procedure is shown in figure 6.1. Satellite Path SRES A3 Network Side RAND IMSI (Note) Ki = A3 MES Ki YES/NO N O T E : I M S Ii su s e dt or e t r i e v eK i . Figure 6.1: The authentication procedure Authentication algorithm A3 is specified in annex C. 6.3 Subscriber authentication key management The subscriber authentication key Ki is allocated, together with the IMSI, at subscription time and both are contained on the SIM in non-readable form. Ki is stored on the network side in the HLR, in an authentication center (AuC). A PLMN may contain one or more AuCs. An AuC can be physically integrated with other functions, e.g., in an HLR. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)25GMR-1 03.020 6.3.1 General authentication procedure When needed for each MES, the GS/MSC/VLR requests security-related information from the HLR/AuC corresponding to the MES. This information includes an array of "triplets" of corresponding RAND, SRES, and session key, called Kc, to be described in clause 7. The first two numbers in each triplet, RAND and SRES, are obtained by applying algorithm A3 to each RAND and the key Ki, as shown in figure 6.1. The pairs are stored in the VLR as part of the security-related information. The procedure used for updating the vectors RAND/SRES is shown in figure 6.2. GS/MSC/VLR HLR/AuC generate RAND (1..na) A3 Store RAND/ SRES vectors Ki Security-Related Information Request Authentication Vector Response (SRES (1..na), RAND (1..na)) NOTE: The Authentication Vector Response also contains Kc(1..na). For clarity, the Kc vector is not shown in figure 6.2 and the remaining figures in clause 6. For discussion of Kc, see clause 7. Figure 6.2: Procedure for updating the vectors RAND/SRES When an MSC/VLR performs an authentication, including the case of a location updating within the same VLR area, it chooses a RAND value in the array corresponding to the MES. It then tests the answer from the MES by comparing it with the corresponding SRES, as shown in figure 6.3. Satellite Path SRES(j) GS/MSC/VLR SRES(j) = A3 MES Ki yes/no RAND(j) RAND(j) SRES(j) Figure 6.3: General authentication procedure ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)26GMR-1 03.020 6.3.2 Authentication at location updating in a new VLR, using TMSI During location updating in a new VLR (VLRn), the procedure to get pairs for subsequent authentication may differ from that described in the previous clause. In the case where identification is done using TMSI, pairs for authentication as part of security related information are given by the old VLR (VLRo). The old VLR will send only those pairs that have not been used to the new VLR. The procedure is shown in figure 6.4. MES Satellite path GS/MSC/VLRn LAI, TMSIo SRES Location Updating GS/MSC/VLRo HLR RAND TMSIo IMSI RAND(1..na) SRES(1..na) A3 = Ki yes/no Figure 6.4: Authentication during location updating in a new VLR, using TMSI 6.3.3 Authentication at location updating in a new VLR, using IMSI When the IMSI is used for identification, or more generally when the old VLR is not reachable, the procedure described in clause 6.3.2 cannot be used. Instead, pairs of RAND/SRES contained in the security related information are requested directly from the HLR. The procedure is shown in figure 6.5. MES Satellite path GS/MSC/VLRn IMSI SRES Location Updating HLR RAND Security-related information request RAND(1,..na) SRES(1,..na) A3 = Ki yes/no Figure 6.5: Authentication at location updating in a new VLR, using IMSI ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)27GMR-1 03.020 6.3.4 Authentication at location updating in a new VLR, using TMSI, TMSI unknown in "old" VLR This case, where a data loss has occurred in the "old" VLR, is an abnormal one. The procedure is shown in figure 6.6. MES Satellite path GS/MSC/VLRn LAI, TMSIo GS/MSC/VLRo HLR TMSIo Unknown Ki SRES Location Updating RAND A3 = yes/no Identity Request IMSI Security-related information request RAND (1,..,na) SRES (1,..,na) Figure 6.6: Authentication at location updating in a new VLR, using TMSI, TMSI unknown in "old" VLR 6.3.5 Authentication at location updating in a new VLR, using TMSI, old VLR not reachable This case occurs when an old VLR cannot be reached by the new VLR. The procedure is shown in figure 6.7. MES Satellite path GS/MSC/VLRn LAI, TMSIo GS/MSC/VLRo HLR Ki SRES Location Updating RAND A3 = yes/no Identity Request IMSI Security-related information request RAND (1,..,na) SRES (1,..,na) VLR not reachable Figure 6.7: Authentication at location updating in a new VLR, using TMSI, old VLR not reachable ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)28GMR-1 03.020 6.3.6 Authentication with IMSI if authentication with TMSI fails If authentication of an MES that identifies itself with a TMSI is unsuccessful, the network requests the IMSI from the MES and repeats the authentication using the IMSI. Optionally, if authentication using the TMSI fails, the network may reject the access request or location registration request that triggered the authentication. 6.3.7 Reuse of security-related information in failure situations Security-related information consisting of sets of RAND, SRES, and Kc is stored in the VLR and in the HLR. When a VLR has used a set of security-related information to authenticate an MES, it will delete the set of security-related information or mark it as used. When a VLR needs to use security-related information, it will use a set that is not marked as used in preference to a set that is marked as used; if there are no sets that are not marked as used, then the VLR may use a set that is marked as used. It is an operator option to define how many times a set of security-related information may be reused in the VLR; when a set of security-related information has been reused as many times as is permitted by the operator, it will be deleted. If a VLR successfully obtains security-related information from the HLR, it will discard any security-related i n f o r m a t i o nt h a ti sm a r k e da su s e di nt h eV L R . If a VLR receives a request from another VLR for security-related information, it will send only the sets that are not marked as used. If an HLR receives a request for security-related information, it will send any sets that are not marked as used; those sets will then be deleted or marked as used. If there are no sets that are not marked as used, the HLR may, as an operator option, send sets that are marked as used. It is an operator option to define how many times a set of security-related information may be resent by the HLR; when a set of security-related information has been sent as many times as is permitted by the operator, it will be deleted. 7 Confidentiality of signalling information and user information on physical connections 7.1 General As noted in clause 4, some signalling information elements are considered sensitive and shall be protected. To ensure identity confidentiality, the temporary subscriber identity shall be transferred in a protected mode at allocation time and at other times when the signalling procedures permit it. Confidentiality of user information on physical connections concerns the information transmitted on a traffic channel on the MES-GS interface (e.g., for speech). It is not an end-to-end confidentiality service. These needs for a protected mode of transmission are fulfilled with the same mechanism as where the confidentiality function is an Open Systems Interconnection (OSI) layer 1 function. The scheme description that follows assumes that the main part of the signalling information elements is transmitted on DCCH and that the CCCH is only used for the allocation of a DCCH. This clause does not treat the subject of confidentiality in terminal-to-terminal (TtT) connections. See clause 8 for TtT encryption. Four points have to be specified: 1) Ciphering. 2) Setting the session key. 3) Initiation and acknowledgement of start of cipher. 4) Synchronization of the cipher streams at both ends of the link. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)29GMR-1 03.020 7.2 Ciphering Layer 1 data flow (transmitted on DCCH or TCH) is ciphered on a bit-per-bit basis, i.e., the data flow on the satellite path is obtained by the bit-per-bit binary addition of the user data flow and a ciphering bit stream, generated by algorithm A5-GMR-1 using a session key determined as specified in clause 7.3. The session key is denoted below by Kc and is called "Ciphering Key." As its name suggests, the session key is used for the duration of the cipher-ON mode at both ends of the link. Deciphering is performed by exactly the same method. Algorithm A5-GMR-1 is specified in annex C. 7.3 Setting the session key Mutual key setting is the procedure that allows the MES and the network to agree on the key Kc to use in the ciphering and deciphering algorithms A5-GMR-1. A key setting is triggered by the authentication procedure. Key setting may be initiated by the network as often as the network operator wishes. Key setting shall occur on a DCCH not yet encrypted and as soon as the identity of the mobile subscriber (i.e., TMSI or IMSI) is known by the network. The transmission of Kc to the MES is indirect and uses the authentication RAND value; Kc is derived from RAND by using algorithm A8 and the Subscriber Authentication key Ki, as defined in annex C. As a consequence, the procedures for the management of Kc are the authentication procedures described in clause 6. Just as with RAND and SRES (clause 6), there is an array of session keys, Kc (1…na) that are sent out to GSs as the MES changes its location. The Kc values are computed together with the SRES values. The security-related information known as a "triplet," (see clause 6.3.1) consists of RAND, SRES, and Kc. The ciphering key sequence number, keynr, is simply an accounting variable which enables the GS and the MES to cite a specific triplet in the array. All numbers in the array are stored together in the MES and in the network. Kc is stored by the MES until it is updated at the next authentication. It is possible to begin a call using the stored session key from a previous call. In this case, it is not necessary to give the "cipher on" command. This feature is not currently supported in the GMR-1. Session key setting is shown in figure 7.1. MES Satellite path Network side LAI, TMSIo RANDKi A8 A8 Ki RAND RAND Store Kc Store Kc Kc Kc Figure 7.1: Key Setting ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)30GMR-1 03.020 7.4 Start of ciphering and deciphering The MES and the GS shall coordinate the frame number (FN) in which the ciphering and deciphering processes start on DCCH and TCH. On DCCH, this procedure takes place under the control of the network some time after the completion of the authentication procedure (if any) or after Kc has been made available at the GS. The transition from clear text mode to ciphered mode proceeds as shown in figure 7.2. Deciphering starts in the GS, which sends in clear text to the MES a specific message, here called "Start cipher." Both the ciphering and deciphering start on the MES side after the message "Start cipher" has been correctly received by the MES. Finally, ciphering on the GS side starts as soon as a frame or a message from the MES has been correctly deciphered at the GS. MES Satellite path GS/MSC/VLR "Start cipher" Start deciphering Start enciphering Start deciphering and Start enciphering Any correctly deciphered message Figure 7.2: Starting of the ciphering and deciphering processes When a TCH is allocated for user data transmission, the key used is the one set during the preceding DCCH session (Call Set-up). Ciphering and deciphering will start immediately on the first frame of a TCH. 7.5 Frame number tag GMR-1 will use the FN to ensure that the cipher streams appear statistically independent from one frame to the next and that cipher streams do not repeat within a hyperframe. The combination of session key Kc and FN will create a unique internal initialization variable to the A5-GMR-1 algorithm (annex C) that is guaranteed not to repeat within a hyperframe. 7.6 Negotiation of A5-GMR-1 When an MES wishes to establish a connection with the network, the MES will indicate to the network which versions of the A5-GMR-1 algorithm it supports. The network will not provide service to an MES which indicates that it does not support the minimum ciphering algorithm(s) required by GMR-1. The network will compare its ciphering capabilities and preferences, and any special requirements, with those indicated by the MES and act according to the following rules: 1) If the MES and the network have no versions of the A5-GMR-1 algorithm in common and the network is not prepared to use an unciphered connection, then the connection will be released. 2) If the MES and the network have at least one version of the A5-GMR-1 algorithm in common, then the network will select one of the mutually acceptable versions of the A5-GMR-1 algorithm for use on that connection. 3) If the MES and the network have no versions of the A5-GMR-1 algorithm in common and the network is willing to use an unciphered connection, then an unciphered connection will be used. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)31GMR-1 03.020 7.7 Implementation of bidirectional ciphering For implementation purposes, two ciphering streams will be generated on each side of the communication link, one for ciphering and the other for deciphering. As shown in figure 7.3, S1 is the ciphering stream used in the direction from network to MES and S2 is the ciphering stream used in the direction from MES to network. Kc Kc A5-GMR-1 A5-GMR-1 Frame number ⊕ ⊕ ⊕ ⊕ Block-2 Block-1 Block-2 Block-1 ciphering deciphering deciphering ciphering encryption MES side Frame number network side Figure 7.3: Implementation of bidirectional ciphering Both ciphering and deciphering are performed by applying an "XOR" operation between the coded bits of the information stream and a ciphering sequence generated by the A5-GMR-1 algorithm. The forward link and return link directions use two different ciphering sequences: one sequence is used for ciphering in the mobile and deciphering in the GS, denoted as "S2" in figure 7.3; and another one is used for deciphering in the mobile and ciphering in the GS, denoted as "S1" in figure 7.3. 8 Terminal-to-terminal call privacy 8.1 Ciphering requirement The GMR-1 network will be able to support a single–hop call in the ciphering mode. In this mode there are two MESs and the network involved at the same time. It is required that both MESs will be able to communicate directly in the ciphered mode in a single satellite hop, that is without having to work in double hop through the network. In the GMR-1 network, the ciphering in a single-hop terminal-to-terminal call is carried out by modifying existing ciphering technique between MES/network and by establishing a ciphering between MESs. The additional requirements to carry out ciphering in a single-hop call in a GMR-1 network are as follows: 1) Both MESs as well as the network will use the common ciphering algorithm and a common ciphering key during a TtT call. 2) In the GMR-1 single-hop call the HLR and MESs are not involved in the generation of the common ciphering key (Ktt). 3) The common ciphering key (Ktt) will be generated by the network (TCS) and will be transferred to both MESs in normal cipher mode between the network and each MES, using separate session Key Kc. After switching to L-to-L link, one of the MES's functions as "network" for purpose of the definition of S1 and S2, as shown in figure 7.3. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)32GMR-1 03.020 4) The network will be able to transfer the following parameters to each MES during the subsequent TtT channel assignment procedure: - the common ciphering key Ktt (64 bits); - mobile/network designation for the use of S1 and S2 (1 bit); - the frame number correction indication (1 bit). a) In the single-hop call, the ciphering mode of operation begins first with both MESs talking to the GS in normal cipher mode, each with a separate Kc. Then, after the common key (Ktt) is transferred to both MESs, the mode is changed from the ciphered mode with Kc to the ciphered mode with the common key Ktt, but the ciphering algorithm remains unchanged. It remains in this ciphered mode until the end of the call. b) During a TtT call, a direct L-to-L link is established through the satellite to enable the call to take place. The original L-C links are maintained to enable the GS to monitor both sides of the TtT call. 8.2 Generation of the common key (Ktt) and start of ciphering and deciphering Each MES and the network perform the security procedures such as authentication, TMSI allocation and ciphering in the same way as in a normal TtG/GtT call. The network (TCS) will generate sets of common ciphering keys (Ktt) to be employed during calls, as described in annex D. The Ktt generation does not involve the HLR and Ats as in a normal TtG/GtT call. During the subsequent TtT channel assignment procedure, the network will deliver the common ciphering key (Ktt), mobile/network designation for the use of S1 and S2 (see GMR-1 03.296 [4]), and the frame number correction indication. The parameters are transferred to each MES in the ciphered mode with the Ats using separate cipher keys. 8.2.1 The use of S1 and S2 The MES instructed to act as a mobile shall use S1 on its receive side. The MES instructed to act as a network shall use S2 on its receive side. After the call gets transferred to an L-to-L link, the GS will continue to monitor both sides of a TtT call by way of the L-C connections of the satellite. Both L-L and L-C links will exist on the satellite simultaneously to permit TtT calls. Since the role of S1 and S2 reverses in the MES designated as "network," the monitoring function of the GS also reverses the role of S1 and S2 for that terminal. There is no ciphering on the TTCH from GSs to MESs. 8.2.2 Frame number correction Due to the timeslot mapping delay at the satellite L-L switch, the timeslots at which the satellite receives and those at which the satellite transmits may not be in the same frame. The frame number of the receive (RX) timeslots and that of the transmit (TX) timeslots for TtT connection at the satellite may differ, at most, by one. If the RX timeslots are in frame N, the TX timeslots are either in frame N or frame N+1. This frame number slip information is known at the timeslot assignment for a TtT call, and it is sent to both MESs as frame number offset IE (1 bit, whether the current frame number will be decrement or not) in the ASSIGNMENT COMMAND 2 message. The receiving side of both MESs will implement frame number correction based on the frame number offset IE, starting from the SABM/UA exchange with SAPI 2. There is no change in frame number slip issue in the L-C links from MES(o) to GS(t1) and from MES(t) to GS(t2). 8.2.3 Change of cipher mode with Kc to cipher mode with Ktt The change of cipher mode to start the enciphering and deciphering process is schematized in figure 8.1. The cipher mode with key Kc(o) between MES(o) and the network is changed into the cipher mode with key Ktt first. Deciphering with Ktt starts in the GS, which sends in ciphered text with key Kc(o) to the MES(o) a specific message, here called "Start cipher with Ktt." Both the ciphering and deciphering start on the MES(o) side after the message "Start cipher with Ktt" has been correctly received by the MES(o). Ciphering synchronization between MES(o) and the GS is confirmed when a frame or a message from the MES(o) has been correctly deciphered at the GS. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)33GMR-1 03.020 After the cipher mode with key Ktt between MES(o) and the GS has been established successfully, the same procedure used between MES(o) and the GS is performed between MES(t) and the GS. At the same time, an L-to-L link on the satellite is established. When a frame or a message from the MES(t) has been correctly deciphered at the GS, ciphering synchronization between MES(t) and the GS is confirmed. At this stage, both MESs have already "turned on" the cipher operation with key Ktt, both at the transmitters and the receivers. The MES(t) will initiate the verification of ciphering synchronization with MES(o) by sending a message in cipher mode with key Ktt. If the message is correctly received and deciphered by MES(o), MES(o) will respond to MES(t) by sending a message in cipher mode with key Ktt. Ciphering synchronization with Ktt between MES(o) and MES(t) is confirmed when MES(t) successfully deciphers the response message in cipher mode with Ktt from MES(o). MES(o) MES(t)GS/MSC/VLR Cipher Mode with key Kc(o) Cipher Mode with key Kc(t) GSC/TCS Start deciphering with key Ktt with MES(o) Start cipher with Ktt Start deciphering with Ktt and start enciphering with Ktt Any correctly deciphered message Confirm cipher synch. with key Ktt for MES(o) Cipher Mode with key Ktt Start deciphering with key Ktt for MES(t) Start cipher with Ktt Start deciphering with Ktt and start enciphering with Ktt Any correctly deciphered message Confirm cipher synch. with key Ktt for MES(t) Cipher Mode with key Ktt Verify with MES(o) Verify with MES(t) Cipher Mode with key Ktt between MES(o) and MES(t) Figure 8.1: Starting of the enciphering and deciphering process with key ktt ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)34GMR-1 03.020 9S u m m a r y Figure 9.1 shows in a synopsis a normal location updating procedure with all elements pertaining to security functions, i.e., to TMSI management, authentication, and Kc management. MES Satellite path GS/MSC/VLRn LAI, TMSIo SRES GS/MSC/VLRo HLR RAND TMSIo IMS I RAND(1..na) SRES(1..na) Kc(1..na) = Ki Yes / no Kc Location Update Acknowledge Cipher "ON" command Cipher mode complete Allocation of TMSIn Location Update Complete TMSI acknowledge Deallocation of TMSIo Cancellation A3 + A8 NOTE: n a is the size of the array Figure 9.1: Normal location updating procedure ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)35GMR-1 03.020 Annex A (informative): Security issues related to signalling schemes and key management A.1 Introduction The diagram in this annex indicates the security items related to signalling functions and to some of the key management functions. The purpose of the diagram is to give a general overview of signalling, both on the satellite path and in the fixed network. The diagram indicates how and where keys are generated, distributed, stored, and used. The security functions are split between VLR and GS/MSC. A.2 Short description of the scheme In order to provide an illustrative example of an MES performing a location update in the same VLR, the following will show the steps involved. Figure A.1 shows the exchange of security information for this scenario. The MES stays within the area controlled by the VLR. The MES is already registered in this VLR. All information belonging to the MES is stored in the VLR, so no connection with the HLR is necessary. Identification is done by the ciphering key sequence number (CKSN), LAI, and TMSI. For authentication, a new set of RAND, SRES, and Kc is already available in the VLR. MES GS/MSC VLR HLR Loc. Upd. Req. (TMSI, LAIo, keynr) Upd. Loc. Area (TMSI, LAIo, keynr) Authenticate (keynr, RAND)Authentication Request. (keynr, RAND) Authentication Response (SRES) A3 + A8 Ki RAND Kc SRES Authentication Acknowledge (SRES) SRES1= SRES2 ? SRES1/SRES2 Y N Generate new TMSI Update Location (IMSI, MSRN) Forward new TMSI (TMSIn) Store TMSI, LAI, keynr Start ciphering (Kc) Location update accept (IMSI)Location area update acceptStore Kc Cipher mode commond A5-GMR-1 Kc M Kc[M], etc Cipher mode complete (Kc[M]) A5-GMR-1 Kc Kc[M] M, etcLocation update accept (Kc[M,TMSIn]) Store TMSIn, LAI TMSI reallocation complete (Kc[M]) TMSI acknowledgement Send para. From HLR (auth. para) (IMSI) Auth. Para. (IMSI, Kc, RAND, SRES) Update database, remove TMSIo Fetch new values from AuC IMSI, TMSI, Ki, Kc, LAI, keynr TMSI, IMSI, Kc, R, S Kc, R, S ...... IMSI, Kc, R, S K c ,R ,S ...... This symbol denotes an action taken by corresponding entity as shown in the Figure Figure A.1: Location updating in the same VLR ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)36GMR-1 03.020 Annex B (informative): S e c u r i t yi n f o r m a t i o nt ob es t o r e di nt h eG M R - 1s y s t e m B.1 Introduction This annex gives an overview of the security related information and the places where this information is stored in the GMR-1 network. The entities of the GMR-1 network where security information is stored are: • HLR • VLR • MSC • GS • MES • AUC B.2 Entities and security information B.2.1 Home location register If required, sets of Kc, RAND, and SRES coupled to each IMSI are stored in the HLR. B.2.2 Visitor location register Sets of Kc, RAND, and SRES coupled to each IMSI are stored in the VLR. In addition the CKSN, LAI, and TMSI are stored together with the presumed valid Kc. After a new TMSI is generated, both the old and the new TMSI are stored. When the old TMSI is no longer valid, it is removed from the database. B.2.3 Mobile services switching center/gateway station E n c r y p t i o na l g o r i t h mA 5 - G M R - 1i ss t o r e di nt h eM S C / G S . Call related information stored in the MSC includes the ciphering key Kc and CKSN associated with the identity of the mobile engaged in this call. After a new TMSI is generated, both the old and the new TMSI are stored. When the old TMSI is no longer valid, it is removed from the database. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)37GMR-1 03.020 B.2.4 Mobile earth station The MES permanently stores these variables in the SIM: • Authentication algorithm A3. • Encryption algorithm A5-GMR-1. • Ciphering key generating algorithm A8. • Individual subscriber authentication key Ki. The MES generates and stores: • Ciphering key Kc. The MES receives and stores: • Ciphering key sequence number. • TMSI. • LAI. The last four variables are stored in the SIM. B.2.5 Authentication center (AuC) The AuC are implements: • Authentication algorithm(s) A3; • Ciphering key generating algorithm(s) A8. The secret individual authentication keys Ki of each subscriber are stored in an AuC ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)38GMR-1 03.020 Annex C (normative): External specifications of security related algorithms C.1 Scope This annex specifies the cryptological algorithms that are needed to provide the various security features and mechanisms. The following three algorithms are considered in the present document. • Algorithm A3: Authentication algorithm. • Algorithm A5-GMR-1: Ciphering/deciphering algorithm. • Algorithm A8: Ciphering key generator. Algorithm A5-GMR-1 shall be common to all GMR-1 PLMNs and all MESs (in particular, to allow roaming). The external specifications of A5-GMR-1 are defined in clause C.2.3. The internal specifications of algorithm A5-GMR-1 are managed under the responsibility of GMR-1 Security Custodian (GSC); they will be made available in response to an appropriate request. Algorithms A3 and A8 are at each PLMN operator discretion. Only the formats of their inputs and outputs shall be specified. It is also desirable that the processing times of these algorithms remain below a maximum value. Proposals for algorithms A3 and A8 are managed by GSC and available in response to an appropriate request for those PLMN operators who wish to use them. C.2 Specifications for algorithm A5-GMR-1 C.2.1 Purpose Algorithm A5-GMR-1 is used for both data and signalling information elements at the physical layer on the dedicated channels (TCH or DCCH). C.2.2 Implementation indications Algorithm A5-GMR-1 is implemented into both the MES and the GS. The GS side description assumes that one algorithm, A5-GMR-1, is implemented for each physical channel (TCH or DCCH). The ciphering takes place before modulation and after interleaving the deciphering takes place after demodulation symmetrically. Both ciphering and deciphering need algorithm A5-GMR-1 and start at different times. As an indication, recall that, due to the TDMA techniques used in the system, the useful data (also called the plain text) are organized into bursts NT3, NT6, and NT9. See table C.1. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)39GMR-1 03.020 Interface 1: Information bits (d) Interface 2: Information + parity bits (u) Interface 3: Coded bits (c) Interface 4: Interleaved bits ( or ) LLC message Cyclic code +t a i l Channel interleave Convolutional code Intraburst multiplex SACCH bits ′e ′′e Interface 6: Encryptedbits (y) Interface 7: Multiplexed bits (m) Scrambling Intraburst multiplex Interface 5: Scrambled bits (x) Encryption unit Status field bits Encoded bits (e) Interface8: Figure C.1: Channel coding and interleaving organization Note, that 4 status bits, which are part of a 24-bit status field, are multiplexed with the coded bits and become encrypted. The unique word is never encrypted. (Numbers come from GMR-1 05.002 [6]) Table C.1: Cipher stream block sizes for encrypted bursts in GMR-1 GMR-1 Burst Size Number of Coded Bits Number of status bits Size of Cipher Stream Block Needed To Encrypt Traffic Burst TCH3 208 4 208 TCH6/FACCH6 430 4 430 SDCCH 208 0 208 TCH9/FACCH9 658 4 658 NT3 for FACCH 96 8 96 For ciphering TCH3 bursts, for example, algorithm A5-GMR-1 produces a sequence of 208 encipher/decipher bits (here called Block) in each frame, which are combined by a bit-wise modulo 2 addition with the 208-bit plain text block. The first encipher/decipher bit produced by A5-GMR-1 is added to the intraburst multiplexer output as shown in figure C.1. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)40GMR-1 03.020 For each slot deciphering is performed on the MES side with the first block (Block-1) of 208 bits produced by A5-GMR-1, and ciphering is performed with the second block (Block-2). As a consequence, on the network side, Block-1 is used for ciphering and Block-2 for deciphering. Therefore algorithm A5-GMR-1 shall produce two blocks of 208 bits (i.e., Block-1 and Block-2). One of the inputs to the A5-GMR-1 is the 19-bit frame number. Use of the frame number ensures no repetition of a cipher block within a hyperframe, which lasts 3 hours and 28 minutes in GMR-1. For details on frame numbering synchronization, see GMR-1 05.010 [7] "Radio Subsystem Synchronization". Figure C.2 summarizes the implementation described above, with only one ciphering/ deciphering procedure represented (the second one for deciphering/ciphering is symmetrical). MES sideNetwork side Bit wise binary addition channel Frame number Frame number A5-GMR-1 A5-GMR-1Key Kc Key Kc Block-1 Bit wise binary addition Block-1 208 bits cipher block 208 bits cipher block 208 bits plain text in TCH3 208 bits plain text in TCH3 Figure C.2: Deciphering on the MES side C.2.3 External specifications of algorithm A5-GMR-1 The two input parameters (COUNT and Kc) and the output parameters (BLOCK1 and BLOCK2) of algorithm A5-GMR-1 will use the following formats: • Length of Kc: 64 bits • Length of FN: 19 bits • Length of BLOCK1: See table C.1 • Length of BLOCK2: See table C.1 • Length of Ktt: 64 bits Algorithm A5-GMR-1 will produce BLOCK1 and BLOCK2 in less than a TDMA frame duration. NOTE: If the actual length of the ciphering key is less than 64 bits, then it is assumed that the actual ciphering key corresponds to the most significant bits of Kc and that the remaining and less significant bits are set to zero. For signalling and testing purposes, the ciphering key Kc is considered to be 64 unstructured bits. C.2.4 Internal specification of algorithm A5-GMR-1 The internal specification of algorithm A5-GMR-1 is managed under the responsibility of the GSC; it will be made available in response to an appropriate request. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)41GMR-1 03.020 C.3 Algorithm A3 Algorithm A3 is considered as a matter for GMR-1 PLMN operators. Therefore, only external specifications are given. However a proposal for a possible algorithm A3 is managed by the GSC and available upon appropriate request. C.3.1 Purpose As defined here, the purpose of algorithm A3 is to allow authentication of a mobile subscriber's identity. To this end, algorithm A3 shall compute an expected response SRES from a random challenge RAND sent by the network. For this computation, algorithm A3 makes use of the secret authentication key Ki. C.3.2 Implementation and operational requirements On the MES side, algorithm A3 is contained in a SIM. On the network side, it is implemented in the HLR or the AuC. The two input parameters (RAND and Ki) and the output parameter (SRES) of algorithm A3 will use the following formats: • Length of Ki: 128 bits • Length of RAND: 128 bits • Length of SRES: 32 bits • Length of Ktt: 64 bits The runtime of algorithm A3 will be less than 200 msec. C.4 Algorithm A8 A l g o r i t h mA 8i sc o n s i d e r e da sam a t t e rf o rG M R - 1P L M No p e r a t o r sa si sa l g o r i t h mA 3 . A proposal for a possible algorithm A8 is available upon appropriate request. C.4.1 Purpose Algorithm A8 shall compute the ciphering key Kc from the random challenge RAND sent during the authentication procedure, using the authentication key Ki. C.4.2 Implementation and operational requirements On the MES side, algorithm A8 is contained in the SIM, as specified in GSM 02.17 [9]. On the network side, algorithm A8 is colocated with algorithm A3. The two input parameters (RAND and Ki) and the output parameter (Kc) of algorithm A8 will follow the following formats: • Length of Ki: 128 bits • Length of RAND: 128 bits • Length of Kc: 64 bits • Length of Ktt: 64 bits ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)42GMR-1 03.020 Because the maximum length of the actual ciphering key is fixed by the GSC algorithm A8 will produce this actual ciphering key and extend it (if necessary) into a 64-bit word where the non-significant bits are forced to zero. It is assumed that any non-significant bits are the lsbs and that the actual ciphering key is contained in the msbs. For signalling and testing purposes, the ciphering key Kc will be 64 unstructured bits. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)43GMR-1 03.020 Annex D (informative): Generation of session keys for direct terminal-to-terminal calls The GS setting up the TtT call will generate a 64 bit session key, Ktt, using a random number generator. It will keep and number up to 10 session keys at a time for use by MESs wishing to make TtT calls. Session keys for TtT calls will be transferred in encrypted mode only. Storage of session keys Ktt for archival purposes will be possible following guidelines for storage of Kc. ETSI ETSI TS 101 376-3-9 V1.1.1 (2001-03)44GMR-1 03.020 History Document history V1.1.1 March 2001 Publication

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