Translations of MATHEMATICAL MONOGRAPHS

Volume 241

Boolean Functions in Coding Theory and Cryptography

O. A. Logachev A. A. Salnikov V. V. Yashchenko

American Mathematical Society Boolean Functions in Coding Theory and Cryptography

10.1090/mmono/241

Translations of MATHEMATICAL MONOGRAPHS

Volume 241

Boolean Functions in Coding Theory and Cryptography O. A. Logachev A. A. Salnikov V. V. Yashchenko

M THE ATI A CA M L ΤΡΗΤΟΣ ΜΗ N ΕΙΣΙΤΩ S A O

C C I

I American Mathematical Society

R E

E T

ΑΓΕΩΜΕ

Y

M A Providence, Rhode Island

F O 8 U 88 NDED 1 EDITORIAL COMMITTEE AMS Subcommittee Robert D. MacPherson Grigorii A. Margulis James D. Stasheff (Chair) ASL Subcommittee Steffen Lempp (Chair) IMS Subcommittee Mark I. Freidlin (Chair) O. A. Logaqev, A. A. Salnikov, V. V. wenko BULEVY FUNKCII V TEORII KODIROVANI I KRIPTOGRAFII M.: MCNMO, 2004 This work was originally published in Russian by Izdatelstvo MCNMO under the title “Bulevy funkcii v teorii kodirovani i kriptografii” c 2004. The present translation was created under license for the American Mathematical Society and is pub- lished by permission. Translated by Svetla Nikova

2000 Mathematics Subject Classification. Primary 94–02; Secondary 94A60, 94C10.

For additional information and updates on this book, visit www.ams.org/bookpages/mmono-241

Library of Congress Cataloging-in-Publication Data Logachev, Oleg A. [Bulevy funktsii v teori kodirovaniia i kriptologii. English] Boolean functions in coding theory and cryptography / O.A. Logachev, A.A. Salnikov, V.V. Yashchenko ; translated by Svetla Nikova. p. cm. — (Translations of mathematical monographs ; v. 241) Includes bibliographical references and index. ISBN 978-0-8218-4680-3 (alk. paper) 1. Coding theory. 2. Cryptography. 3. Algebra, Boolean. I. Salnikov, A. A. (Aleksei Alek- sandrovich) II. IAshchenko, V. V. III. Title. QA268.L6413 2011 003.54—dc23 2011035308

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312 Contents

Foreword vii

Preface ix

Notation xi

Chapter 1. Arithmetics of Finite Fields and Polynomials 1 1.1. Basic Algebra 1 1.2. Construction of finite fields 19 1.3. Polynomials over finite fields 28 Comments to Chapter 1 35

Chapter 2. Boolean Functions 37 2.1. Basic concepts and definitions 37 2.2. Numerical and metric characteristics 44 2.3. Autocorrelation and crosscorrelation 56 2.4. Group algebra of Boolean functions 61 2.5. Cryptographic properties of Boolean functions and mappings 65 2.6. Covering sequences of Boolean functions 74 Comments to Chapter 2 76

Chapter 3. Classifications of Boolean Functions 77 3.1. Group equivalence of mappings. Polya’s theorem 77 3.2. Classification of Boolean functions of five variables 83 3.3. Classification of quadratic Boolean functions 91 3.4. Classification of homogeneous cubic forms of 8 variables 99 3.5. RM-equivalence of Boolean functions 101 Comments to Chapter 3 104

Chapter 4. Linear Codes over the Field F2 107 4.1. Basic properties of linear block codes 107 4.2. The decoding problem 116 4.3. Cyclic codes 120 4.4. Some classes of primitive cyclic codes 131 Comments to Chapter 4 136

Chapter 5. Reed–Muller Codes 139 5.1. General properties of the Reed–Muller codes 139 5.2. Reed’s decoding algorithm 146 5.3. First order Reed–Muller codes and connections with other codes 150 5.4. Reed–Muller codes of second order and related codes 157

v vi CONTENTS

5.5. Classification of Boolean functions and Reed–Muller codes of the 3rd order 160 Comments to Chapter 5 163 Chapter 6. Nonlinearity 165 6.1. Nonlinearity as a measure of cryptographic quality 165 6.2. Maximum-nonlinear bent functions and their properties 166 6.3. Some classes of maximum-nonlinear bent functions 172 6.4. Partially maximum-nonlinear (partially bent) functions and their properties 177 6.5. Plateaued functions and partially defined mn-bent functions 179 6.6. Hyperbent functions 188 6.7. Biorthogonal bases 189 Comments to Chapter 6 192 Chapter 7. Correlation Immunity and Resiliency 195 7.1. Main definitions and properties 195 7.2. The inheritance of properties under restrictions of Boolean functions 208 7.3. General methods for constructing correlation-immune functions and resilient mappings 214 7.4. Nonlinearity of correlation-immune and resilient functions 218 7.5. Construction of resilient Boolean functions with good cryptographic properties 222 7.6. Covering sequences of correlation-immune and resilient functions 226 7.7. Quadratic resilient Boolean functions of maximum order 235 Comments to Chapter 7 237 Chapter 8. Codes, Boolean Mappings, and Their Cryptographic Properties 239 8.1. Almost perfect nonlinear and almost bent mappings 239 8.2. Coding-theoretic approach to the study of APN and AB mappings 249 8.3. Cyclic codes and Boolean mappings 255 8.4. Avalanche criteria and propagation criteria 261 8.5. Construction of Boolean functions satisfying the propagation criterion of degree k and order t 265 8.6. Global avalanche characteristics of Boolean functions 266 Comments to Chapter 8 269 Chapter 9. Basics of Cryptanalysis 271 9.1. The Berlekamp–Massey algorithm. Linear complexity 271 9.2. Principles of the statistical method for cryptanalysis of block ciphers 281 9.3. Principles of the correlation cryptanalysis method 287 9.4. Principles of the linear cryptanalysis method 295 9.5. Principles of the difference (differential) cryptanalysis method 300 Comments to Chapter 9 301

Bibliography 305 Index 329 Foreword

For the last 10 years there have been practically no books in Russian which have the word “cryptography” in the title. Nowadays many people already know that cryptography is the science which studies ciphers, and that only cryptography gives the most reliable tools for ensuring the security of information technology. However, there are not many specialists in this area, because in order to fully understand cryptography it is necessary to have knowledge in many scientific branches such as mathematics, physics, communication theory, and cybernetics. Thus, at present, cryptography (the theoretical branch of cryptology) becomes a university science. A detailed discussion of this issue has been held during the two conferences at Moscow State University (MGU): “Moscow University and development of cryptography in Russia” (October 17–18, 2002) and “Mathematics and information technology security” (October 23–24, 3003). Institute for Problems of Information Security, a new division of MGU, pub- lishes a series of fundamental books on scientific and methodological problems of information security, including those parts of cryptology that are already included in the university mathematical curriculum. The book by O. A. Logachev, A. A. Salnikov, and V. V. Yashchenko “Boolean functions in coding theory and cryptology” belongs to this series. It is written by mathematicians-cryptographers for mathematicians and presents in a systematic way certain results in one branch of cryptology: application of Boolean functions in the analysis and design of ciphers. The book is recommended to readers with basic university knowledge, namely students and graduate students in mathematics, research mathematicians, and cryptographers.

Rector of MGU, Academician V. A. Sadovnichii February, 2004

vii

Preface

The notion of Boolean function was introduced in the second half of the 19th century in connection with investigations in mathematical logic and foundations of mathematics. Boolean functions are named after George Boole (1815–1864), an English mathematician, one of the founders of mathematical logic. In the first half of the 20th century Boolean functions attain fundamental importance in the foundations of mathematics. However, for a long time Boolean functions have not been used in applications. This situation changed drastically in the middle of the 20th century, when the intensive development of communication technology, instrument-building, and computer technology required the creation of an adequate mathematical apparatus. In this period, applied parts of mathematics such as the theory of finite functional systems, information theory, coding theory, and finally mathematical cryptography have been developed. The practice showed the fruitfulness of the application of Boolean functions to the problems of analysis and synthesis of discrete devices for processing and transformation of information. The concept of cryptography that has been established in the scientific liter- ature includes a range of scientific areas, each of them having its own subject of investigations and using specific mathematical techniques. Some researchers do abstract investigations “with cryptographic motifs” in the area of computational complexity theory; others are busy constructing and analyzing algorithms for par- ticular cryptographic systems. In many cryptographic areas, Boolean function tech- niques are often used while formulating and solving various problems. This applies mainly to traditional cryptographic systems with a secret key. The title of the book “Boolean functions in coding theory and cryptography” reflects the relation between many cryptographic problems and encoding and decoding problems for Reed–Muller codes. In this book, for the first time in Russian, we present cryptographic aspects using Boolean functions techniques. The only exceptions are questions related to complexity theory and solving systems of Boolean functions. In this book both classical and recent results are presented. To understand the material, university courses of linear algebra, group theory, finite fields theory and polynomials, combinatorics and discrete mathematics will suffice. A knowledge of basics of probability theory is also assumed. The book is based on courses given by the authors in MGU for students of Mechanics–Mathematics and Computational Mathematics and Cybernetics Depart- ments who major in “Information security”. Recent results obtained by the authors in the framework of the scheduled work of the MGU Laboratory on Mathematical Problems of Cryptography are also included in the book. The book consists of nine chapters.

ix xPREFACE

Chapter 1 is preliminary. It contains basic notions and results of algebra used in the book. In Chapter 2, basic notions and theorems of Boolean function theory are proved. In Chapter 3, problems of Boolean function classification under differ- ent groups of transformations are considered. Chapter 4 presents basics of coding theory. In Chapter 5, properties of Boolean functions are considered from the point of view of coding theory. In Chapter 6, properties of maximum-nonlinear functions are studied. Chapter 7 investigates the correlation immunity property of a func- tion. In Chapter 8, various cryptographic characteristics of Boolean functions and mappings are considered in detail. Chapter 9 contains elements of cryptanalysis. To avoid making the book too large, some of the results are presented as prob- lems. Some of the problems included in the book are still open; they may be a basis for future research. All items in the text are numbered consecutively within chapters: definitions, theorems, examples, etc. Thus, for example, Definition 1.121 refers to item 121 in Chapter 1 (which turns out to be a definition). The mathematical expressions and figures have similar but independent numbering. The authors will accept with gratitude any comments on the book. They could be sent to the internet site http://www.cryptography.ru. The authors express their gratitude to Mikhail Vladimirovich Stepanov for his support during the work on the book. Notation

N — the set of natural numbers (1, 2, 3,...); Z — the ring of integers (...,−2, −1, 0, 1, 2,...); Zn — the ring of residues modulo n ∈ N; R — the field of real numbers; Q — the field of rational numbers; C — the field of complex numbers; #A — cardinality of a set A; A × B — Cartesian product of sets A and B; An — nth Cartesian power of a set A (n ∈ N); P{·} — probability of the event in the brackets; E[·] — mathematical expectation of the random variable in the brackets; BA — set of all maps from a set A toasetB; Πn — set of minimal representatives of cyclotomic classes; ϕ−1(b) — complete preimage of b ∈ B under a map ϕ ∈ BA; F — finite field; n Fq — finite field of q elements (q = p , p is a prime number, n ∈ N); F∗ F q — multiplicative group of invertible elements of the field q; Fq[x,y,...,z] — ring of polynomials in variables x,y,...,z over the field Fq; Trqm/q(α) — relative trace of an element α ∈ Fqm over the field Fq; Trm(α) — absolute trace of an element α ∈ Fpm over the field Fp (p is prime); deg P (x,y,...,z) — degree of a polynomial P (x,y,...,z); V — linear vector space; Vn,q — vector space of columns of height n ∈ N with coordinates in the field Fq; Vn — vector space of columns of height n ∈ N with coordinates in the field F2 (n-dimensional Boolean space);  — partial ordering relation of vectors from Vn; dim V — dimension of a vector space V ;  — operator for matrix transposition; v =(v(1),...,v(n)) — column vector in n-dimensional vector space in coordinate notation (in a fixed basis); wt(v) — Hamming weight of a vector v; dist(v, u)=wt(v − u) — Hamming distance between vectors v and u; dist(A, B)=minv∈A,u∈B {dist(v, u)} — Hamming distance between sets of vec- tors A ⊆ V and B ⊆ V ; ⊕ — coordinatewise addition of vectors of the same dimension over the field F2;

xi xii NOTATION

Sn — symmetric (permutation) group of order n ∈ N; SV — symmetric permutation group acting on elements of a space V ; Nn — group of translations acting on the vector space Vn,q of dimension n ∈ N;

Dn — Jevons group acting on the vector space Vn,q of dimension n ∈ N; A — group generated by a set A; GL(V ) — full linear group acting on a vector space V ; GA(V ) — full affine group acting on a vector space V ; r-vector — vector of dimension r ∈ N; r-subset — subset of cardinality r ∈ N; (m × n) matrix — matrix with m ∈ N rows and n ∈ N columns; rank M —rankofamatrixM; det M — determinant of a square matrix M; ⊗ — tensor product; a | b — a divides b (a, b ∈ Z); x — largest integer less than or equal to x ∈ R; x — smallest integer greater than or equal to x ∈ R; ∅ —emptyset; gcd — greatest common divisor; lcm√ — least common multiple; i = −1 ∈ C — imaginary unit; k ∈ Z Tn = exp 2πi n k n —groupofnth roots of unity; T = {x ∈ C ||x| =1} — multiplicative group of complex numbers of absolute value 1;

Fn — set of all Boolean functions of n ∈ N variables; Fn(S) — set of partially defined Boolean functions of 0.5n ∈ N variables with defining set S ⊆ Vn; Fn,m — set of all Boolean functions from Vn to Vm (n, m ∈ N); f(x) exp f(x)=(−1) — function on Vn with values in {−1, 1}; Ln — set of all linear Boolean functions of n ∈ N variables; Ln,m — set of all linear Boolean mappings from Vn to Vm (n, m ∈ N); An — set of all affine Boolean functions of n ∈ N variables; An,m — set of all affine Boolean mappings from Vn to Vm (n, m ∈ N); Bn — set of all Boolean bent functions (maximum-nonlinear functions) of n ∈ N variables;

Bn(S) — set of partially defined bent functions of n ∈ N variables with defining set S ⊆ Vn; S ∈ N n — set of all symmetric Boolean functions of n variables;   (j) (j) x, y = j x y — scalar product of vectors x and y; · (1) (1) (n) (n)  x y =(x y ,...,x y ) — product of vectors x and y; W (α)= (−1)f(x)⊕x,α — Walsh–Hadamard transformation of a Boolean f x∈Vn function f ∈Fn (α ∈ Vn); s W (a,s)= (−1)f(x)⊕Tr(ax ) — extended Walsh–Hadamard transforma- f x∈F2n tion of a Boolean function f ∈Fn (α ∈ Vn, s ∈ Πn); NOTATION xiii D − f(x)⊕x,α Wf (α)= x∈D( 1) — partial Walsh–Hadamard transformation of a ∈F ⊆ ∈ Boolean function f n with respect to the set D Vn (α Vn); − x,α ∈F Wf (α)= x∈V f(x)( 1) — Fourier transform of a Boolean function f n ∈n (α Vn); D − x,α Wf (α)= x∈D f(x)( 1) — partial Fourier transform of a Boolean function f ∈Fn with respect to a set D ⊆ Vn (α ∈ Vn); Nf — nonlinearity of a Boolean function f ∈Fn; GNf — generalized nonlinearity of a Boolean function f ∈Fn; ⊂F ∈ DuΦ — derivative of a Boolean mapping Φ n,m in the direction u Vn; − f(x⊕α)⊕f(x) ∈F Δf (α)= x∈V ( 1) — autocorrelation of a Boolean function f n n ∈ with shift α Vn; | | 2 Δf =maxα∈Vn Δf (α) , σf = α∈Vn Δf (α) — numerical measures of the global α= 0 α= 0 avalanche characteristics of a Boolean function f ∈Fn; ill(F ) — linearity index of a Boolean mapping F ∈Fn,m; f — dual of a bent function f ∈Bn; JG(f) — moment group of a function f in a group G; LF — subspace of linearity (subspace of linear structures) of a Boolean mapping F ∈Fn,m; PCF — set of directions (vectors) for which a mapping F ∈Fn,m satisfies the propagation criterion; μl(F ) — maximum element from the difference table of a Boolean mapping F ∈Fn,m; RM(r, n) — binary Reed–Muller code of order r ∈ N and length 2n (n ∈ N); RM ∗(r, n) — binary punctured Reed–Muller code of order r ∈ N and length 2n −1 (n ∈ N); Cj(f) — number of code words in the code C that are at distance j from f (0  j  rC ); Aut(C) — automorphism group of a code C; C⊥ — dual code of a code C; dC — minimum distance of a code C; kC — dimension of a linear code C; rC — covering radius of a code C; RC —rateofacodeC; WC (x, y) — weight function of a code C; [n, k, d]-code — linear code of length n ∈ N,dimensionk ∈ N, and minimum distance d ∈ N; Nmax(n, m) — maximum possible nonlinearity of an m-resilient Boolean function on Fn; ρ(x,r) — ball centered at x ∈ Vn of radius r ∈{0, 1,...,n}; μ —M¨obius function; δ —Diracδ-function; IM — indicator function of a set M; E —identity(n × n)matrix; n 11[n] ∈ N Hn = 1 −1 — Sylvester–Hadamard matrix of order n ;[n]isKronecker (tensor) power; xiv NOTATION

NWf — number of binary vectors for which the Walsh–Hadamard coefficients of a function f are nonzero; NΔf — number of binary vectors for which the values of the autocorrelation function f are nonzero; a conJ — operation of fixing part of variables of a set of functions; it is given by a set of indices J = {j1,...,jl},1≤ j1 < ···

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Index

(n, r)-forms, 103 partial, 173 RM-equivalence, 101 bent mapping, 243 almost, 247 adder, 273 bias, 295 Advanced Encryption Standard, AES, 283 binary operation, 1 algebra over a field, 18 associative, 1 algebraic degree block cipher, 281 of a function, 42 key, 281 of a mapping, 249 Boole, George, ix algebraic system, 1 Boolean function algorithm covering sequence of, 74 deciphering, 281 level of, 74 decoding, 118 nontrivial, 74 enciphering, 281 degeneration structure of, 103 Euclidean, 13 derivative of, 55 Matsui 1, 296 numerical normal form of, 50 Matsui 2, 296 weight of, 45 almost equivalent mappings, 105, 238 boomerang method, 303 array bound of a code Bose–Chaudhuri–Hocquenghem (BCH), standard, 118 133 orthogonal, 206 Elias’, 112 attack on the key, 283 Hamming’s, 111 automorphism Singleton’s, 111 Frobenius, 30 sphere-packing, 111 internal, 6 branching, 70 of a field over another field, 30 linear, 71 of a group, 4 Burnside lemma, 80 avalanche criterion, 261 strict, 261 canonical factorization of a polynomial, 15 strong of order t, 262 center of a group, 7 average character complexity, 285 additive, 25 reliability, 285 canonical, 25 distinguishing, 5 ball, 111 multiplicative, 26 basis nontrivial, 5 biorthogonal, 189 of a group, 4 canonical, 17 trivial, 5 normal, 25 characteristic of a vector space, 17 difference, 300 polynomial, 25 function, 239 bent set, 173 global avalanche, 266 bent function, 166 linear, 295

329 330 INDEX

of a field, 11 simplex, 117, 132 polynomial systematic, 111 of an LRS, 272 uniformly packed, 254 of a register, 273 weight function of, 114 check polynomial, 123 weight spectrum of, 114 check symbols, 111 with maximum distance, 111 cipher code dimension, 107 A5, 70 code distance, 107 stream, 70, 287 dual, 250 symmetric, 65 external, 250 cipher algorithm code rate, 107 DES, 283 code word, 107 GOST 28147-89, 283 coefficient cipher standard Fourier, 46 DES, 283 Walsh–Hadamard, 46 GOST 28147-89, 283 coefficients ciphertext, 281 spectral, 46 block, 281 communication channel intermediate, 282 discrete, 108 class quantum-cryptographic, 203 cyclotomic, 35, 54 completion of a class, 172 equivalence, 2 complexity Maiorana–McFarland, 173 linear, 276 of affine functions, 43 average of statistical classification of maximum-nonlinear functions procedure, 285, 286 M, 173 confusion, 65 complete, 172 conjugate set, 6 code constant, 12 [n, k, d], 107 constructive enumeration problem, 88 automorphism group of, 110 coordinates of a vector, 17, 37 binary Golay, 135, 136 correlation complementary, 123 attack, 294 completely regular, 254 decoding, 152 constructive distance of, 134 coset cyclic, 120 leader, 118 nonzeros, 123 of a code, 118 primitive, 122 of a subgroup, 3 with two zeros, 259 covering radius of a code, 107 zeros, 123 covering sequence dual, 109 perfect, 234 equidistant, 117 simple, 228 generator matrix of, 109 crosscorelation, 58 Hadamard, 190 cryptanalysis Hamming’s, 116 linear, 295 Kerdock, 159 method, 281 linear statistical, 281 block, 107 determined by a mapping, 252 decision area, 284 maximum length, 117 decoder minimum distance of, 107 complete, 119 parity-check matrix of, 109 incomplete, 119 perfect, 117 decoding Hamming code, 117 Preparata, 160 deep hole, 166 primitive BCH, 134 delay device, 273 narrow-sense, 134 Delsarte’s inequality, 255 punctured, 142 dependence Reed–Muller, 139 essential, 38 set of code words of, 109 quasi-linear, 223 INDEX 331 derivative of decomposition, 20 of a Boolean function, 55 prime, 11 of a polynomial, 16 flag of subsets, 69 deviation, 295 form difference table, 239 algebraic normal (ANF), 41 diffusion, 65 alternating, 92 dimension of a space, 18 associated, 92 Dirac δ-function, 46 symplectic, 92 discrepancy bits, 277 Fourier transform, 114 distance function between Boolean functions, 45 d-optimal, 203 from a Boolean function to a set, 49 d-resilient, 203 Hamming, 44 affine, 43 distance of uniqueness, 284 argument of, 38 distributed computations, 203 distribution of random variables, 196 balanced distributivity, 7 with respect to a matrix, 266 divisor of an element of a ring, 9 Boolean, 37 domain, 7 (c0,c1)-regular, 44 dualbases,24 G-invariant, 79 c-regular, 44 element balanced, 45 of a ring bent, 166 prime, 9 correlation-immune, 198 generator of a cyclic group, 2 functionally separable, 42 of a code, 107 maximum-nonlinear, 166 of a field maximum-nonlinear for a subspace, primitive, 23 178 of a ring nondegenerate, 102 reversible, 9 partial, 181 of infinite order, 3 regular, 44 elements conjugate, 6 weakly nondegenerate, 232 equivalent, 2 correlation-immune, 67 of a field in a given direction, 201 conjugate, 29 cryptographic (discrete), 65 of a ring dual, 168 associates, 9 to a plateaued function, 180 congruent modulo an ideal, 8 dual to a partially defined mn-bent Elias bound, 112 function, 182 endomorphism of a group, 4 Euler’s, 4 entropy of a random variable, 196 given as a linear branching, 71 conditional, 196 group-theoretic classification of, 80 enumerator, 81 hyperbent, 189 EPC(k, 0), 264 linear, 43 EPC(k, t), 264 linearly dependent on a variable, 42 epimorphism, 4 M¨obius, 33 equivalence relation, 2 nonlinearity of, 50 equivalent codes, 110 nonlinearly dependent on a variable, 42 ergodic theory, 65 partially defined d-resilient, 217 EWHT, 188 plateaued, 180 exponent of a group, 4 quadratic, 92 extension degree, 17 resilient, 67 extension of a field, 10 self-dual, 168 of finite degree, 17 symmetric, 44 fast correlation attack, 294 functions field, 7 G-equivalent, 79 finite, 19 algebraically independent, 66 332 INDEX generator matrix in the systematic form, information symbols, 111 111 intersection of codes, 127 generator polynomial, 122 invariant of a group, 88 Gilbert–Varshamov bound, 112 complete, 88 global avalanche characteristic, 169 inverse element, 1 absolute index, 266 isomorphic vector spaces, 17 sum of squares, 266 isomorphism, 4 GOST 28147-89, 283 iteration cipher, 282 greatest common divisor of polynomials, 13 Green’s scheme, 152 Jensen’s inequality, 114 group, 1 Jevons group, 85 abelian, 1 center of, 7 kernel commutative, 1 of a bilinear form, 159 complete affine, 86 of a homomorphism, 6 cyclic, 2 of a ring homomorphism, 9 finite, 2 of a symplectic matrix, 159 Galois, 30 key schedule, 283 general linear, 85 Kravchuk polynomials, 116, 227 infinite, 2 large set of orthogonal arrays, 207 isomorphism, 4 least common multiple of polynomials, 14 of affine transformations, 86 length of inverted variables, 84 of a code of linear transformations, 85 primitive, 122 of permutations of variables, 84 of a register, 273 of residue classes, 2 linear of roots of unity, 3 combination, 17 of shifts, 84 complexity, 275 group action on a set of functions, 78 cryptanalysis method, 295 Group Special Mobile, GSM, 70 feedback shift register (LFSR), 272 Hamming bound, 111 recursive sequence (LRS), 272 Hamming code, 116 space, 16 homomorphism, 4 span, 275, 276 of rings, 9 structure, 67 hyperbent function, 189 translator, 67 linearity subspace of a mapping, 68 ideal Lloyd polynomial, 255 minimal, 28 locators of a vector, 255 of a ring maximal, 9 MacWilliams identity, 115 prime, 9 mapping principal, 8 (n, k, d)-resilient, 203, 205 two-sided, 8 almost perfect nonlinear, 245 idempotent, 125 associated with a function, 70 primitive, 128 balanced, 66 proper, 27, 125 branched, 70 identity element of a group, 1 branching, 70 image complete, 261 branching, 70 defined by a polynomial, 16 of a group homomorphism, 4 linearity index, 70 impossible differentials, 302 perfect nonlinear, 243 independent random variables, 196 plateaued, 247 index polynomial, 250 of q modulo n,34 resilient, 67, 203 of a subgroup, 3 material, 283 of linearity, 70 volume of, 283 information matrix mutual, 197 Hadamard, 167 INDEX 333

symplectic, 92, 158 of a shortened row of values of a Matsui function, 189 algorithm 1, 296 periodic sequence, 271 algorithm 2, 296 Peterson–Gorenstein–Zierler decoder, 271 maximum-nonlinear functions piling-up lemma, 299 PS, 177 plaintext, 281 PS+, 176 block, 281 PS−, 176 plateaued function class D, 177 complementary, 185 class D0, 177 of order 2r, 180 method Pless identities, 251 boomerang, 303 Polya’s theorem, 82 of conditional differentials, 302 polynomial, 12 of multiple approximation, 302 characteristic of an element, 30 of partial differentials, 302 constant, 12 rectangle, 303 constant term of, 12 minimal polynomial of a sequence, 274 cyclotomically homogeneous, 54 minimum period of a sequence, 271 cyclotomically reduced, 54 mixing, 65 degree of, 12 mn-bent function dual, 31 partially defined, 181 generator of a cyclic code, 122 mn-function irreducible, 14 partially bent, 178 Kravchuk, 227 multiplicity of a root, 16 leading coefficient of, 12 minimal, 28 natural cryptographic assumption, 298 monic, 12 Neyman–Pearson lemma, 290 primitive, 32 nonlinearity, 67 quadratic, 256 generalized, 188 reducible, 14 nonzeros of a cyclic code, 123 root of, 16 norm, 24 multiple, 16 absolute, 24 simple, 16 normalizer unitary, 12 of a set, 7 Zhegalkin, 41 of an element, 7 pre-period of a sequence, 271 procedure for statistical classification, 283 operator product fixing some of the variables, 73 Kronecker, 151 projection, 72 of elements of a group, 2 taking a Boolean derivative, 73 scalar, 26 optimal Bayes procedure, 287 of vectors, 45 orbit index, 77 propagation criteria, 67, 201, 261 order of degree k and order t, 264 lexicographic, 38 extended, 264 of a group, 2 propagation matrix, 264 of a polynomial, 31 property of an element of a group, 3 reducible, 72 partial, 41 secondary, 73 orthogonality equations, 47 quotient group, 6 pair of variables quotient ring, 9 covering, 226 quasi-linear, 223 rectangle method, 303 Parseval’s equation, 48 Reed’s decoding algorithm, 146 partial spreads, 177 reflectivity, 2 PC(k, t), 264 reliability period, 271 of an algorithm, 285 of a polynomial, 31 representative of a cyclotomic class, 54 of a sequence, 271 residue class, 8 334 INDEX resilient, 203 summand Rijndael, 283 in ANF, 42 ring, 7 in Zhegalkin polynomial, 42 commutative, 7 weight, 42 division, 7 linear, 42 domain, 7 support of an element, 206 irreducible, 27 symmetry, 2 of polynomials over a field, 12 syndrome vector, 118 principal ideal domain, 9 tabular method, 38 reducible, 27 trace, 23, 53 with identity, 7 absolute, 23 root of unity, 34 relative, 54 primitive, 34 trace equvalence, 53 Rothaus criterion, 169 transform round, 282 fast Hadamard, 151 subkey, 282 Fourier, 46 transformation, 282 M¨obius, 41 row operations, 110 Walsh–Hadamard, 46 extended, 188 incomplete, 181 SAC(t), 262 transitivity, 2 self-information of an event, 195 triangle inequality, 45 set trigger, 273 difference, 169 truth table, 206 simple Hadamard, 169 type of a permutation, 77 generating a subgroup, 3 of a code ultimately periodic sequence, 271 characteristic, 250 unknown, 12 generating, 255 Shannon’s principles, 66 variable, 12 shift operator, 272 covering, 226 essential, 38 Siegenthaler inequality, 202 fictitious, 38 Singleton bound, 111 adding, 39 skew field, 7 deleting, 39 space linear, 223 r-nonlinearity of, 69 nonessential, 38 branching, 70 variable of a function, 38 vector, 37 vector, 16, 37 stabilizer of a function, 79 r-covered by a code, 107 stable subspace, 170 preceding, 41 statistical classisfication, 283 strictly, 41 statistical cryptanalysis method, 281 vector space, 16 stream cipher, 65 isomorphism of, 17 subalgebra, 18 subfield, 10 weight proper, 10 Hamming, 41 subfunction, 39 of a function, 81 subgroup, 3 of an equivalence class, 81 generated by a set, 3 word error probability, 120 generated by an element, 3 zero element nontrivial, 3 of a ring, 7 normal, 6 of a group, 2 subkey, 282 zero tail expansion, 276 subring, 8 zerodivisors, 7 sum zeros of a cyclic code, 123 of codes, 127 of elements of a group, 2 Selected Titles in This Series

241 O.A.Logachev,A.A.Salnikov,andV.V.Yashchenko, Boolean Functions in Coding Theory and Cryptography, 2012 240 Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito, Number Theory 2, 2011 239 I. Ya. Novikov, V. Yu. Protasov, and M. A. Skopina, Wavelet Theory, 2011 238 Leonid L. Vaksman, Quantum Bounded Symmetric Domains, 2010 237 Hitoshi Moriyoshi and Toshikazu Natsume, Operator Algebras and Geometry, 2008 236 Anatoly A. Goldberg, Iossif V. Ostrovskii, and Iossif V. Ostrovskii, Value Distribution of Meromorphic Functions, 2008 235 Mikio Furuta, Index Theorem. 1, 2007 234 G. A. Chechkin, G. A. Chechkin, A. L. Piatnitski, A. L. Piatnitski, A. S. Shamaev, and A. S. Shamaev, Homogenization, 2007 233 A. Ya. Helemskii, Lectures and Exercises on Functional Analysis, 2006 232 O. N. Vasilenko, Number-Theoretic Algorithms in Cryptography, 2007 231 Kiyosi Itˆo, Essentials of Stochastic Processes, 2006 230 Akira Kono and Dai Tamaki, Generalized Cohomology, 2006 229 Yu. N. Lin kov, Lectures in Mathematical Statistics, 2005 228 D. Zhelobenko, Principal Structures and Methods of Representation Theory, 2006 227 Takahiro Kawai and Yoshitsugu Takei, Algebraic Analysis of Singular Perturbation Theory, 2005 226 V. M. Manuilov and E. V. Troitsky, Hilbert C∗-Modules, 2005 225 S. M. Natanzon and S. M. Natanzon, Moduli of Riemann Surfaces, Real Algebraic Curves, and Their Superanalogs, 2004 224 Ichiro Shigekawa, Stochastic Analysis, 2004 223 Masatoshi Noumi, Painlev´e Equations through Symmetry, 2004 222 G. G. Magaril-Il’yaev and V. M. Tikhomirov, Convex Analysis: Theory and Applications, 2003 221 Katsuei Kenmotsu, Surfaces with Constant Mean Curvature, 2003 220 I. M. Gelfand, S. G. Gindikin, and M. I. Graev, Selected Topics in Integral Geometry, 2003 219 S. V. Kerov, Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis, 2003 218 Kenji Ueno, Algebraic Geometry 3, 2003 217 Masaki Kashiwara, D-modules and Microlocal Calculus, 2003 216 G. V. Badalyan, Quasipower Series and Quasianalytic Classes of Functions, 2002 215 Tatsuo Kimura, Introduction to Prehomogeneous Vector Spaces, 2002 214 L. S.ˇ Grinblat, Algebras of Sets and Combinatorics, 2002 213 V. N. Sachkov and V. E. Tarakanov, Combinatorics of Nonnegative Matrices, 2002 212 A. V. Melnikov, S. N. Volkov, and M. L. Nechaev, Mathematics of Financial Obligations, 2002 211 Takeo Ohsawa, Analysis of Several Complex Variables, 2002 210 Toshitake Kohno, Conformal Field Theory and Topology, 2002 209 Yasumasa Nishiura, Far-from-Equilibrium Dynamics, 2002 208 Yukio Matsumoto, An Introduction to Morse Theory, 2002 207 Ken’ichi Ohshika, Discrete Groups, 2002 206 Yuji Shimizu and Kenji Ueno, Advances in Moduli Theory, 2002 205 Seiki Nishikawa, Variational Problems in Geometry, 2002 204 A. M. Vinogradov, Cohomological Analysis of Partial Differential Equations and Secondary Calculus, 2001 203 Te Sun Han and Kingo Kobayashi, Mathematics of Information and Coding, 2002 202 V. P. Maslov and G. A. Omel’yanov, Geometric Asymptotics for Nonlinear PDE. I, 2001 SELECTED TITLES IN THIS SERIES

201 Shigeyuki Morita, Geometry of Differential Forms, 2001 200 V. V. Prasolov and V. M. Tikhomirov, Geometry, 2001 199 Shigeyuki Morita, Geometry of Characteristic Classes, 2001 198 V. A. Smirnov, Simplicial and Operad Methods in Algebraic Topology, 2001 197 Kenji Ueno, Algebraic Geometry 2, 2001 196 Yu. N. Lin kov, Asymptotic Statistical Methods for Stochastic Processes, 2001 195 Minoru Wakimoto, Infinite-Dimensional Lie Algebras, 2001 194 Valery B. Nevzorov, Records: Mathematical Theory, 2001 193 Toshio Nishino, Function Theory in Several Complex Variables, 2001 192 Yu.P.SolovyovandE.V.Troitsky, C∗-Algebras and Elliptic Operators in Differential Topology, 2001 191 Shun-ichi Amari and Hiroshi Nagaoka, Methods of Information Geometry, 2000 190 Alexander N. Starkov, Dynamical Systems on Homogeneous Spaces, 2000 189 Mitsuru Ikawa, Hyperbolic Partial Differential Equations and Wave Phenomena, 2000 188 V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, 2000 187 A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, 2000 186 Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito, Number Theory 1, 2000 185 Kenji Ueno, Algebraic Geometry 1, 1999 184 A. V. Melnikov, Financial Markets, 1999 183 Hajime Sato, Algebraic Topology: An Intuitive Approach, 1999 182 A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor’kova, A. V. Samokhin, Yu. N. Torkhov, and A. M. VerbovetskySymmetries and Conservation Laws for Differential Equations of Mathematical Physics, 1999 181 Ya.G.BerkovichandE.M.Zhmud, Characters of Finite Groups. Part 2, 1999 180 A. A. Milyutin and N. P. Osmolovskii, Calculus of Variations and Optimal Control, 1998 179 V. E. Voskresenski˘ı, Algebraic Groups and Their Birational Invariants, 1998 178 Mitsuo Morimoto, Analytic Functionals on the Sphere, 1998 177 Satoru Igari, Real Analysis—With an Introduction to Wavelet Theory, 1998 176 L. M. Lerman and Ya. L. Umanskiy, Four-Dimensional Integrable Hamiltonian Systems with Simple Singular Points (Topological Aspects), 1998 175 S. K. Godunov, Modern Aspects of Linear Algebra, 1998 174 Ya-Zhe Chen and Lan-Cheng Wu, Second Order Elliptic Equations and Elliptic Systems, 1998 173 Yu. A. Davydov, M. A. Lifshits, and N. V. Smorodina, Local Properties of Distributions of Stochastic Functionals, 1998 172 Ya.G.BerkovichandE.M.Zhmud, Characters of Finite Groups. Part 1, 1998 171 E. M. Landis, Second Order Equations of Elliptic and Parabolic Type, 1998 170 Viktor Prasolov and Yuri Solovyev, Elliptic Functions and Elliptic Integrals, 1997 169 S. K. Godunov, Ordinary Differential Equations with Constant Coefficient, 1997 168 Junjiro Noguchi, Introduction to Complex Analysis, 1998 167 Masaya Yamaguti, Masayoshi Hata, and Jun KigamiMathematics of Fractals, 1997 166 Kenji UenoAn Introduction to Algebraic Geometry, 1997 165 V. V. Ishkhanov, B. B. Lure, and D. K. Faddeev, The Embedding Problem in Galois Theory, 1997

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. This book offers a systematic presentation of cryptographic and code- theoretic aspects of the theory of Boolean functions. Both classical and recent results are thoroughly presented. Prerequisites for the book include basic knowledge of linear algebra, group theory, theory of fi nite fi elds, combinatorics, and probability. The book can be used by research mathematicians and graduate students interested in discrete mathematics, coding theory, and cryptography.

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