Translations of

MATHEMATICAL MONOGRAPHS

Volume 241

American Mathematical Society

Boolean Functions in Coding Theory and Cryptography

O. A. LogachevA. A. SalnikovV. V. Yashchenko

Boolean Functions in Coding Theory and Cryptography

Translations of

MATHEMATICAL MONOGRAPHS

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American Mathematical SocietyProvidence, Rhode Island

Volume 241

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

10.1090/mmono/241

EDITORIAL COMMITTEE

AMS SubcommitteeRobert 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 IKRIPTOGRAFII

M.: MCNMO, 2004This work was originally published in Russian by Izdatelstvo MCNMO under the

title “Bulevy funkcii v teorii kodirovani i kriptografii” c© 2004. The presenttranslation 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, visitwww.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. Sal′nikov, A. A. (Aleksei Alek-

sandrovich) II. IAshchenko, V. V. III. Title.

QA268.L6413 2011003′.54—dc23

2011035308

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10 9 8 7 6 5 4 3 2 1 17 16 15 14 13 12

Contents

Foreword vii

Preface ix

Notation xi

Chapter 1. Arithmetics of Finite Fields and Polynomials 11.1. Basic Algebra 11.2. Construction of finite fields 191.3. Polynomials over finite fields 28Comments to Chapter 1 35

Chapter 2. Boolean Functions 372.1. Basic concepts and definitions 372.2. Numerical and metric characteristics 442.3. Autocorrelation and crosscorrelation 562.4. Group algebra of Boolean functions 612.5. Cryptographic properties of Boolean functions and mappings 652.6. Covering sequences of Boolean functions 74Comments to Chapter 2 76

Chapter 3. Classifications of Boolean Functions 773.1. Group equivalence of mappings. Polya’s theorem 773.2. Classification of Boolean functions of five variables 833.3. Classification of quadratic Boolean functions 913.4. Classification of homogeneous cubic forms of 8 variables 993.5. RM -equivalence of Boolean functions 101Comments to Chapter 3 104

Chapter 4. Linear Codes over the Field F2 1074.1. Basic properties of linear block codes 1074.2. The decoding problem 1164.3. Cyclic codes 1204.4. Some classes of primitive cyclic codes 131Comments to Chapter 4 136

Chapter 5. Reed–Muller Codes 1395.1. General properties of the Reed–Muller codes 1395.2. Reed’s decoding algorithm 1465.3. First order Reed–Muller codes and connections with other codes 1505.4. 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 3rdorder 160

Comments to Chapter 5 163

Chapter 6. Nonlinearity 1656.1. Nonlinearity as a measure of cryptographic quality 1656.2. Maximum-nonlinear bent functions and their properties 1666.3. Some classes of maximum-nonlinear bent functions 1726.4. Partially maximum-nonlinear (partially bent) functions and their

properties 1776.5. Plateaued functions and partially defined mn-bent functions 1796.6. Hyperbent functions 1886.7. Biorthogonal bases 189Comments to Chapter 6 192

Chapter 7. Correlation Immunity and Resiliency 1957.1. Main definitions and properties 1957.2. The inheritance of properties under restrictions of Boolean functions 2087.3. General methods for constructing correlation-immune functions and

resilient mappings 2147.4. Nonlinearity of correlation-immune and resilient functions 2187.5. Construction of resilient Boolean functions with good cryptographic

properties 2227.6. Covering sequences of correlation-immune and resilient functions 2267.7. Quadratic resilient Boolean functions of maximum order 235Comments to Chapter 7 237

Chapter 8. Codes, Boolean Mappings, and Their Cryptographic Properties 2398.1. Almost perfect nonlinear and almost bent mappings 2398.2. Coding-theoretic approach to the study of APN and AB mappings 2498.3. Cyclic codes and Boolean mappings 2558.4. Avalanche criteria and propagation criteria 2618.5. Construction of Boolean functions satisfying the propagation criterion

of degree k and order t 2658.6. Global avalanche characteristics of Boolean functions 266Comments to Chapter 8 269

Chapter 9. Basics of Cryptanalysis 2719.1. The Berlekamp–Massey algorithm. Linear complexity 2719.2. Principles of the statistical method for cryptanalysis of block ciphers 2819.3. Principles of the correlation cryptanalysis method 2879.4. Principles of the linear cryptanalysis method 2959.5. Principles of the difference (differential) cryptanalysis method 300Comments to Chapter 9 301

Bibliography 305

Index 329

Foreword

For the last 10 years there have been practically no books in Russian which havethe word “cryptography” in the title. Nowadays many people already know thatcryptography is the science which studies ciphers, and that only cryptography givesthe most reliable tools for ensuring the security of information technology. However,there are not many specialists in this area, because in order to fully understandcryptography it is necessary to have knowledge in many scientific branches such asmathematics, physics, communication theory, and cybernetics. Thus, at present,cryptography (the theoretical branch of cryptology) becomes a university science. Adetailed discussion of this issue has been held during the two conferences at MoscowState University (MGU): “Moscow University and development of cryptographyin Russia” (October 17–18, 2002) and “Mathematics and information technologysecurity” (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 ofinformation security, including those parts of cryptology that are already includedin the university mathematical curriculum.

The book by O.A. Logachev, A.A. Salnikov, and V.V. Yashchenko “Booleanfunctions in coding theory and cryptology” belongs to this series. It is written bymathematicians-cryptographers for mathematicians and presents in a systematicway certain results in one branch of cryptology: application of Boolean functionsin the analysis and design of ciphers.

The book is recommended to readers with basic university knowledge, namelystudents and graduate students in mathematics, research mathematicians, andcryptographers.

Rector of MGU, Academician V.A. Sadovnichii

February, 2004

vii

Preface

The notion of Boolean function was introduced in the second half of the 19thcentury in connection with investigations in mathematical logic and foundationsof mathematics. Boolean functions are named after George Boole (1815–1864),an English mathematician, one of the founders of mathematical logic. In the firsthalf of the 20th century Boolean functions attain fundamental importance in thefoundations of mathematics. However, for a long time Boolean functions have notbeen used in applications.

This situation changed drastically in the middle of the 20th century, whenthe intensive development of communication technology, instrument-building, andcomputer technology required the creation of an adequate mathematical apparatus.In this period, applied parts of mathematics such as the theory of finite functionalsystems, information theory, coding theory, and finally mathematical cryptographyhave been developed. The practice showed the fruitfulness of the application ofBoolean functions to the problems of analysis and synthesis of discrete devices forprocessing 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 ofinvestigations and using specific mathematical techniques. Some researchers doabstract investigations “with cryptographic motifs” in the area of computationalcomplexity 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 appliesmainly to traditional cryptographic systems with a secret key. The title of thebook “Boolean functions in coding theory and cryptography” reflects the relationbetween many cryptographic problems and encoding and decoding problems forReed–Muller codes.

In this book, for the first time in Russian, we present cryptographic aspectsusing Boolean functions techniques. The only exceptions are questions related tocomplexity theory and solving systems of Boolean functions. In this book bothclassical 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 willsuffice. A knowledge of basics of probability theory is also assumed.

The book is based on courses given by the authors in MGU for students ofMechanics–Mathematics and Computational Mathematics and Cybernetics Depart-ments who major in “Information security”. Recent results obtained by the authorsin the framework of the scheduled work of the MGU Laboratory on MathematicalProblems of Cryptography are also included in the book.

The book consists of nine chapters.

ix

x PREFACE

Chapter 1 is preliminary. It contains basic notions and results of algebra usedin the book. In Chapter 2, basic notions and theorems of Boolean function theoryare proved. In Chapter 3, problems of Boolean function classification under differ-ent groups of transformations are considered. Chapter 4 presents basics of codingtheory. In Chapter 5, properties of Boolean functions are considered from the pointof view of coding theory. In Chapter 6, properties of maximum-nonlinear functionsare studied. Chapter 7 investigates the correlation immunity property of a func-tion. In Chapter 8, various cryptographic characteristics of Boolean functions andmappings 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 basisfor 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 inChapter 1 (which turns out to be a definition). The mathematical expressions andfigures have similar but independent numbering.

The authors will accept with gratitude any comments on the book. They couldbe sent to the internet site http://www.cryptography.ru.

The authors express their gratitude to Mikhail Vladimirovich Stepanov for hissupport 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 to a set B;

Πn — set of minimal representatives of cyclotomic classes;

ϕ−1(b) — complete preimage of b ∈ B under a map ϕ ∈ BA;

F — finite field;

Fq — finite field of q elements (q = pn, p is a prime number, n ∈ N);F∗q — multiplicative group of invertible elements of the field Fq;

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);

degP (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 fieldFq;

Vn — vector space of columns of height n ∈ N with coordinates in the fieldF2 (n-dimensional Boolean space);

— partial ordering relation of vectors from Vn;

dimV — dimension of a vector space V ;

— operator for matrix transposition;

v = (v(1), . . . , v(n)) — column vector in n-dimensional vector space in coordinatenotation (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 thefield 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 dimensionn ∈ 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;

rankM — rank of a matrix M ;

detM — 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;∅ — empty set;

gcd — greatest common divisor;

lcm — least common multiple;

i =√−1 ∈ C — imaginary unit;

Tn =exp2πi kn

∣∣ k ∈ Zn

— group of nth roots of unity;

T = x ∈ C | |x| = 1 — multiplicative group of complex numbers of absolutevalue 1;

Fn — set of all Boolean functions of n ∈ N variables;

Fn(S) — set of partially defined Boolean functions of 0.5n ∈ N variables withdefining set S ⊆ Vn;

Fn,m — set of all Boolean functions from Vn to Vm (n,m ∈ N);

exp f(x) = (−1)f(x) — 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 definingset S ⊆ Vn;

Sn — set of all symmetric Boolean functions of n ∈ N variables;

〈x,y〉 =∑

j x(j)y(j) — scalar product of vectors x and y;

x · y = (x(1)y(1), . . . , x(n)y(n)) — product of vectors x and y;

Wf (α) =∑

x∈Vn(−1)f(x)⊕〈x,α〉 — Walsh–Hadamard transformation of a Boolean

function f ∈ Fn (α ∈ Vn);

Wf (a, s) =∑

x∈F2n(−1)f(x)⊕Tr(axs) — extended Walsh–Hadamard transforma-

tion of a Boolean function f ∈ Fn (α ∈ Vn, s ∈ Πn);

NOTATION xiii

WDf (α) =

∑x∈D(−1)f(x)⊕〈x,α〉 — partial Walsh–Hadamard transformation of a

Boolean function f ∈ Fn with respect to the set D ⊆ Vn (α ∈ Vn);

Wf (α) =∑

x∈Vnf(x)(−1)〈x,α〉 — Fourier transform of a Boolean function f ∈ Fn

(α ∈ Vn);

WDf (α) =

∑x∈D f(x)(−1)〈x,α〉 — 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;

DuΦ — derivative of a Boolean mapping Φ ⊂ Fn,m in the direction u ∈ Vn;

Δf (α) =∑

x∈Vn(−1)f(x⊕α)⊕f(x) — autocorrelation of a Boolean function f ∈ Fn

with shift α ∈ Vn;

Δf = maxα∈Vnα=0

|Δf (α)|, σf =∑

α∈Vnα=0

Δ2f (α) — numerical measures of the global

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 Booleanmapping F ∈ Fn,m;

PCF — set of directions (vectors) for which a mapping F ∈ Fn,m satisfies thepropagation criterion;

μl(F ) — maximum element from the difference table of a Boolean mappingF ∈ 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 — rate of a code C;

WC(x, y) — weight function of a code C;

[n, k, d]-code — linear code of length n ∈ N, dimension k ∈ N, and minimumdistance d ∈ N;

Nmax(n,m) — maximum possible nonlinearity of an m-resilient Boolean functionon Fn;

ρ(x, r) — ball centered at x ∈ Vn of radius r ∈ 0, 1, . . . , n;μ — Mobius function;

δ — Dirac δ-function;

IM — indicator function of a set M ;

En — identity (n× n) matrix;

Hn =(1 11 −1

)[n]— Sylvester–Hadamard matrix of order n ∈ N; [n] is Kronecker

(tensor) power;

xiv NOTATION

NWf — number of binary vectors for which the Walsh–Hadamard coefficientsof a function f are nonzero;

NΔf — number of binary vectors for which the values of the autocorrelationfunction f are nonzero;

conaJ — operation of fixing part of variables of a set of functions; it is givenby a set of indices J = j1, . . . , jl, 1 ≤ j1 < · · · < jl ≤ n, and binaryvectors a = (a(1), . . . , a(l)) ∈ Vl;

pri — projection operator of a Boolean mapping to the ith coordinate;

per(u) — period of a sequence u;

loc(c) — set of locators of a row c;

charu(λ) — characteristic polynomial of a recurrent sequence u.

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Index

(n, r)-forms, 103

RM -equivalence, 101

adder, 273

Advanced Encryption Standard, AES, 283

algebra over a field, 18

algebraic degree

of a function, 42

of a mapping, 249

algebraic system, 1

algorithm

deciphering, 281

decoding, 118

enciphering, 281

Euclidean, 13

Matsui 1, 296

Matsui 2, 296

almost equivalent mappings, 105, 238

array

of a code

standard, 118

orthogonal, 206

attack on the key, 283

automorphism

Frobenius, 30

internal, 6

of a field over another field, 30

of a group, 4

avalanche criterion, 261

strict, 261

strong of order t, 262

average

complexity, 285

reliability, 285

ball, 111

basis

biorthogonal, 189

canonical, 17

normal, 25

of a vector space, 17

polynomial, 25

bent set, 173

bent function, 166

partial, 173

bent mapping, 243

almost, 247

bias, 295

binary operation, 1associative, 1

block cipher, 281

key, 281

Boole, George, ix

Boolean functioncovering sequence of, 74

level of, 74

nontrivial, 74

degeneration structure of, 103

derivative of, 55

numerical normal form of, 50weight of, 45

boomerang method, 303

bound

Bose–Chaudhuri–Hocquenghem (BCH),133

Elias’, 112Hamming’s, 111

Singleton’s, 111

sphere-packing, 111

branching, 70

linear, 71

Burnside lemma, 80

canonical factorization of a polynomial, 15

center of a group, 7character

additive, 25

canonical, 25

distinguishing, 5

multiplicative, 26nontrivial, 5

of a group, 4

trivial, 5

characteristic

difference, 300

function, 239global avalanche, 266

linear, 295

329

330 INDEX

of a field, 11

polynomial

of an LRS, 272

of a register, 273

check polynomial, 123

check symbols, 111

cipher

A5, 70

stream, 70, 287

symmetric, 65

cipher algorithm

DES, 283

GOST 28147-89, 283

cipher standard

DES, 283

GOST 28147-89, 283

ciphertext, 281

block, 281

intermediate, 282

class

cyclotomic, 35, 54

equivalence, 2

Maiorana–McFarland, 173

of affine functions, 43

of maximum-nonlinear functions

M, 173

complete, 172

code

[n, k, d], 107

automorphism group of, 110

binary Golay, 135, 136

complementary, 123

completely regular, 254

constructive distance of, 134

cyclic, 120

nonzeros, 123

primitive, 122

with two zeros, 259

zeros, 123

dual, 109

equidistant, 117

generator matrix of, 109

Hadamard, 190

Hamming’s, 116

Kerdock, 159

linear

block, 107

determined by a mapping, 252

maximum length, 117

minimum distance of, 107

parity-check matrix of, 109

perfect, 117

Preparata, 160

primitive BCH, 134

narrow-sense, 134

punctured, 142

Reed–Muller, 139

set of code words of, 109

simplex, 117, 132

systematic, 111

uniformly packed, 254

weight function of, 114

weight spectrum of, 114

with maximum distance, 111

code dimension, 107

code distance, 107

dual, 250

external, 250

code rate, 107

code word, 107

coefficient

Fourier, 46

Walsh–Hadamard, 46

coefficients

spectral, 46

communication channel

discrete, 108

quantum-cryptographic, 203

completion of a class, 172

complexity

linear, 276

average of statistical classificationprocedure, 285, 286

confusion, 65

conjugate set, 6

constant, 12

constructive enumeration problem, 88

coordinates of a vector, 17, 37

correlation

attack, 294

decoding, 152

coset

leader, 118

of a code, 118

of a subgroup, 3

covering radius of a code, 107

covering sequence

perfect, 234

simple, 228

crosscorelation, 58

cryptanalysis

linear, 295

method, 281

statistical, 281

decision area, 284

decoder

complete, 119

incomplete, 119

decoding Hamming code, 117

deep hole, 166

delay device, 273

Delsarte’s inequality, 255

dependence

essential, 38

quasi-linear, 223

INDEX 331

derivativeof a Boolean function, 55of a polynomial, 16

deviation, 295difference table, 239diffusion, 65dimension of a space, 18

Dirac δ-function, 46discrepancy bits, 277distance

between Boolean functions, 45from a Boolean function to a set, 49Hamming, 44

distance of uniqueness, 284distributed computations, 203distribution of random variables, 196distributivity, 7divisor of an element of a ring, 9domain, 7dual bases, 24

elementof a ring

prime, 9generator of a cyclic group, 2

of a code, 107of a field

primitive, 23of a ring

reversible, 9of infinite order, 3

elementsconjugate, 6equivalent, 2of a field

conjugate, 29of a ring

associates, 9congruent modulo an ideal, 8

Elias bound, 112endomorphism of a group, 4entropy of a random variable, 196

conditional, 196enumerator, 81EPC(k, 0), 264EPC(k, t), 264epimorphism, 4equivalence relation, 2

equivalent codes, 110ergodic theory, 65EWHT, 188exponent of a group, 4extension degree, 17extension of a field, 10

of finite degree, 17

fast correlation attack, 294field, 7

finite, 19

of decomposition, 20

prime, 11

flag of subsets, 69

form

algebraic normal (ANF), 41

alternating, 92

associated, 92

symplectic, 92

Fourier transform, 114

function

d-optimal, 203

d-resilient, 203

affine, 43

argument of, 38

balanced

with respect to a matrix, 266

Boolean, 37

(c0, c1)-regular, 44

G-invariant, 79

c-regular, 44

balanced, 45

bent, 166

correlation-immune, 198

functionally separable, 42

maximum-nonlinear, 166

maximum-nonlinear for a subspace,178

nondegenerate, 102

partial, 181

regular, 44

weakly nondegenerate, 232

correlation-immune, 67

in a given direction, 201

cryptographic (discrete), 65

dual, 168

to a plateaued function, 180

dual to a partially defined mn-bentfunction, 182

Euler’s, 4

given as a linear branching, 71

group-theoretic classification of, 80

hyperbent, 189

linear, 43

linearly dependent on a variable, 42

Mobius, 33

nonlinearity of, 50

nonlinearly dependent on a variable, 42

partially defined d-resilient, 217

plateaued, 180

quadratic, 92

resilient, 67

self-dual, 168

symmetric, 44

functions

G-equivalent, 79

algebraically independent, 66

332 INDEX

generator matrix in the systematic form,111

generator polynomial, 122Gilbert–Varshamov bound, 112global avalanche characteristic, 169

absolute index, 266sum of squares, 266

GOST 28147-89, 283greatest common divisor of polynomials, 13Green’s scheme, 152group, 1

abelian, 1center of, 7commutative, 1complete affine, 86cyclic, 2finite, 2Galois, 30general linear, 85infinite, 2isomorphism, 4of affine transformations, 86of inverted variables, 84of linear transformations, 85of permutations of variables, 84of residue classes, 2of roots of unity, 3of shifts, 84

group action on a set of functions, 78Group Special Mobile, GSM, 70

Hamming bound, 111Hamming code, 116homomorphism, 4

of rings, 9hyperbent function, 189

idealminimal, 28of a ring

maximal, 9prime, 9principal, 8two-sided, 8

idempotent, 125primitive, 128proper, 27, 125

identity element of a group, 1image

branching, 70of a group homomorphism, 4

impossible differentials, 302independent random variables, 196index

of q modulo n, 34of a subgroup, 3of linearity, 70

informationmutual, 197

information symbols, 111

intersection of codes, 127

invariant of a group, 88complete, 88

inverse element, 1

isomorphic vector spaces, 17

isomorphism, 4iteration cipher, 282

Jensen’s inequality, 114

Jevons group, 85

kernel

of a bilinear form, 159

of a homomorphism, 6

of a ring homomorphism, 9of a symplectic matrix, 159

key schedule, 283

Kravchuk polynomials, 116, 227

large set of orthogonal arrays, 207

least common multiple of polynomials, 14

length

of a codeprimitive, 122

of a register, 273

linear

combination, 17complexity, 275

cryptanalysis method, 295

feedback shift register (LFSR), 272recursive sequence (LRS), 272

space, 16

span, 275, 276

structure, 67translator, 67

linearity subspace of a mapping, 68

Lloyd polynomial, 255locators of a vector, 255

MacWilliams identity, 115

mapping

(n, k, d)-resilient, 203, 205almost perfect nonlinear, 245

associated with a function, 70

balanced, 66

branched, 70branching, 70

complete, 261

defined by a polynomial, 16linearity index, 70

perfect nonlinear, 243

plateaued, 247

polynomial, 250resilient, 67, 203

material, 283

volume of, 283matrix

Hadamard, 167

INDEX 333

symplectic, 92, 158

Matsuialgorithm 1, 296

algorithm 2, 296maximum-nonlinear functions

PS, 177PS+, 176PS−, 176

class D, 177class D0, 177

methodboomerang, 303

of conditional differentials, 302of multiple approximation, 302of partial differentials, 302

rectangle, 303minimal polynomial of a sequence, 274

minimum period of a sequence, 271mixing, 65

mn-bent functionpartially defined, 181

mn-functionpartially bent, 178

multiplicity of a root, 16

natural cryptographic assumption, 298

Neyman–Pearson lemma, 290nonlinearity, 67

generalized, 188nonzeros of a cyclic code, 123

norm, 24absolute, 24

normalizer

of a set, 7of an element, 7

operator

fixing some of the variables, 73projection, 72

taking a Boolean derivative, 73optimal Bayes procedure, 287

orbit index, 77order

lexicographic, 38

of a group, 2of a polynomial, 31

of an element of a group, 3partial, 41

orthogonality equations, 47

pair of variablescovering, 226quasi-linear, 223

Parseval’s equation, 48partial spreads, 177

PC(k, t), 264period, 271

of a polynomial, 31of a sequence, 271

of a shortened row of values of afunction, 189

periodic sequence, 271Peterson–Gorenstein–Zierler decoder, 271piling-up lemma, 299plaintext, 281

block, 281

plateaued functioncomplementary, 185of order 2r, 180

Pless identities, 251Polya’s theorem, 82polynomial, 12

characteristic of an element, 30constant, 12constant term of, 12cyclotomically homogeneous, 54cyclotomically reduced, 54degree of, 12dual, 31generator of a cyclic code, 122irreducible, 14Kravchuk, 227leading coefficient of, 12minimal, 28monic, 12primitive, 32quadratic, 256reducible, 14root of, 16

multiple, 16simple, 16

unitary, 12Zhegalkin, 41

pre-period of a sequence, 271procedure for statistical classification, 283product

Kronecker, 151of elements of a group, 2scalar, 26

of vectors, 45propagation criteria, 67, 201, 261

of degree k and order t, 264extended, 264

propagation matrix, 264property

reducible, 72secondary, 73

quotient group, 6quotient ring, 9

rectangle method, 303Reed’s decoding algorithm, 146reflectivity, 2reliability

of an algorithm, 285representative of a cyclotomic class, 54residue class, 8

334 INDEX

resilient, 203

Rijndael, 283

ring, 7

commutative, 7

division, 7

domain, 7

irreducible, 27

of polynomials over a field, 12

principal ideal domain, 9

reducible, 27

with identity, 7

root of unity, 34

primitive, 34

Rothaus criterion, 169

round, 282

subkey, 282

transformation, 282

row operations, 110

SAC(t), 262

self-information of an event, 195

set

difference, 169

simple Hadamard, 169

generating a subgroup, 3

of a code

characteristic, 250

generating, 255

Shannon’s principles, 66

shift operator, 272

Siegenthaler inequality, 202

Singleton bound, 111

skew field, 7

space

r-nonlinearity of, 69

branching, 70

vector, 37

stabilizer of a function, 79

stable subspace, 170

statistical classisfication, 283

statistical cryptanalysis method, 281

stream cipher, 65

subalgebra, 18

subfield, 10

proper, 10

subfunction, 39

subgroup, 3

generated by a set, 3

generated by an element, 3

nontrivial, 3

normal, 6

subkey, 282

subring, 8

sum

of codes, 127

of elements of a group, 2

summandin ANF, 42in Zhegalkin polynomial, 42

weight, 42linear, 42

support of an element, 206symmetry, 2

syndrome vector, 118

tabular method, 38trace, 23, 53

absolute, 23relative, 54

trace equvalence, 53transform

fast Hadamard, 151Fourier, 46Mobius, 41Walsh–Hadamard, 46

extended, 188incomplete, 181

transitivity, 2triangle inequality, 45trigger, 273truth table, 206

type of a permutation, 77

ultimately periodic sequence, 271unknown, 12

variable, 12covering, 226essential, 38fictitious, 38

adding, 39deleting, 39

linear, 223nonessential, 38

variable of a function, 38vector, 16, 37

r-covered by a code, 107preceding, 41

strictly, 41vector space, 16

isomorphism of, 17

weightHamming, 41of a function, 81of an equivalence class, 81

word error probability, 120

zero elementof a ring, 7

of a group, 2zero tail expansion, 276zerodivisors, 7zeros of a cyclic code, 123

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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.