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公钥密码学的数学基础（英文版）


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公钥密码学的数学基础（英文版）
  • 书号：9787030474308
    作者：王小云 等
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  • 装帧：平装
    开本：B5
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    语种：en
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Mathematical Foundations of Public Key Crytogtaphy.In Mathcmatical Foundutions of Public Kcy Cryptography. the authors prcscnt the rcsults of more t11an 20 ycars of rcscarch and Lcaching cxpcricnce to hcip studcnts bridge the gap benveen math theory and crypto practice The book provides a theoretical structure of fundamcntal numbcr thcory und algcbra knowicdge supporting puhlic-kcy cryptography.Rather than simply combining number theory and modern algebra: this textbook ftaturcs the intcrclisciplinary charactcristics of cryptography-rcvcaling the intcgrations of mathcmaLical thcorics and puhlic-kcy cryptographic applications incorporating the complexity theory of algoritluns throughoiit, it introduces the basic number theoretic ancl algcbraic algorithms and thor compicxitics to prcwicic a prcliminary uncicrstanding of the applications of mathcmatical thcorics in cryptographic algorithms.SuppLying a seamless assimiLation of cryptography and mathematics. the book incluclcs covcrage of' cicmcntary numbcr thcory; algcbraic structure and attributcs of' group, ring, and fiLid; cryptography-rclatcd computing compicxity and basic algo- rithms, as weLl as lattice and fundamental methods of Lattice cryptanalysis.The tcxt consists of 11 chaptcrs. Basic thcory ancl tools of' cicmcntary numbcr thcory, such as congrucnccs, primitwe roots, rcsiduc classcs. and continucd fractions, are covered in Chapters l-6 The basic concepts of abstract aLgebra are introduced in Chaptcrs 7-9, whcre thrce basic algcbraic structurcs of'groups, rings. and ficids und thcir propcrtics arc cxplaincd.Chapter 10 is about computational complexities of several related mathematical algorithms, mcl harcl probicms such as intcgcr fzictorization md discrcte logurithm Chaptcr il prcscnts the hasics of lattice thcory and the lattice hasis rcduction algorithm-the LLL algorithm and its application in the cryptanalysis ofthe RSA algo-rithm.
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目录

  • Contents
    Preface to Mathematics Monograph Series v
    Foreword vii
    Preface ix
    Acknowledgments xiii
    CHAPTER 1 . Divisibility of Integers 1
    1.1 THE CONCEPT OF DIVISIBILITY 1
    1.2 THE GREATEST COMMON DIVISOR AND THE LEAST COMMON MULTIPLE 6
    1.3 THE EUCLIDEAN ALGORITHM 12
    1.4 SOLVING LINEAR DIOPHANTINE EQUATIONS 16
    1.5 PRIME FACTORIZATION OF INTEGERS 18
    CHAPTER 2 . Congruences 27
    2.1 CONGRUENCES 27
    2.2 RESIDUE CLASSES AND SYSTEMS OF RESIDUES 31
    2.3 EULER/S THEOREM 37
    2.4 WILSON'S THEOREM 40
    CHAP7FER 3 . Congruence Equations 47
    3.1 BASIC CONCEPTS OF CONGRUENCES OF HIGH DEGREES 47
    3.2 LINEAR CONGRUENCES 50
    3.3 SYSTEMS OF LINEAR CONGRUENCE EQUATIONS AND THE CHINESE REMAINDER THEOREM 52
    3.4 GENERAL CONGRUENCE EQUATIONS 55
    3.5 QUADRATIC RESIDUES 57
    3.6 THE LEGENDRE SYMBOL AND THE JACOBI SYMBOL 61
    CHAPTER 4 . Exponents and Primitive Roots 71
    4.1 EXPONENTS AND THEIR PROPERTIES 71
    4.2 PRIMITIVE ROOTS AND THEIR PROPERTIES 76
    4.3 INDICES, CONSTRUCTION OF REDUCED SYSTEM OF RESIDUES 79
    4.4 NTH POWER RESIDUES 85
    CHAPTER 5 . Some Elementary Results for Prime Distribution 91
    5.1 INTRODUCTION TO THE BASIC PROPERTIES OF PRIMES AND THE MAIN RESULTS OF PRIME NUMBER DISTRIBUTION 91
    5.2 PROOF OF THE EULER PRODUCT FORMULA 95
    5.3 PROOF OF A WEAKER VERSION OF THE PRIME NUMBER THEOREM 97
    5.4 EQUIVALENT STATEMENTS OF THE PRIME NUMBER THEOREM 105
    CHAPTER 6 . Simple Continued Fractions 109
    6.1 SIMPLE CONTINUED FRACTIONS AND THEIR BASIC PROPERTIES 109
    6.2 SIMPLE CONTINUED FRACTION REPRESENTATIONS OF REAL NUMBERS 113
    6.3 APPLICATION OF CONTINUED FRACTION IN CRYPTOGRAPHY-ATTACK TO RSA WITH SMALL DECRYPTION EXPONENTS 118
    CHAPTER 7 . Basic Concepts 121
    7.1 MAPS 121
    7.2 ALGEBRAIC OPERATIONS 125
    7.3 HOMOMORPHISMS AND ISOMORPHISMS BET\NEEN SETS WITH OPERATIONS 128
    7.4 EQUIVALENCE RELATIONS AND PARTITIONS 129
    CHAPTER 8 . GroupTheory 133
    8.1 DEFINITIONS 133
    8.2 YCLIC GROUPS 135
    8.3 SUBGROUPS AND COSETS 137
    8.4 FUNDAMENTAL HOMOMORPHISM THEOREM 141
    8.5 CONCRETE EXAMPLES OF FINITE GROUPS 146
    CHAPTER 9 . Ringsand Fields 151
    9.1 DEFINITION OF A RING 151
    9.2 INTEGRAL DOMAINS, FIELDS, AND DIVISION RINGS 154
    9.3 SUBRINGS, IDEALS, AND RING HOMOMORPHISMS 159
    9.4 CHINESE REMAINDER THEOREM 165
    9.5 EUCLIDEAN RINGS 168
    9.6 FINITE FIELDS 170
    9.7 FIELD OF FRACTIONS 172
    CHAPTER 10 . Some Mathematical Problems in Public Key Cryptography 177
    10.1 TIME ESTIMATION AND COMPLEXITY OF ALGORITHMS 177
    10.2 INTEGER FACTORIZATION PROBLEM 184
    10.3 PRIMALITY TESTS 185
    10.4 THE RSA PROBLEM AND THE STRONG RSA PROBLEM 188
    10.5 QUADRATIC RESIDUES 189
    10.6 THE DISCRETE LOGARITHM PROBLEM 192
    CHAPTER 11 . Basics of Lattices 195
    11.1 BASIC CONCEPTS 195
    11.2 SHORTEST VECTOR PROBLEM 196
    11.3 LATTICE BASIS REDUCTION ALGORITHM 197
    11.4 APPLICATIONS OF LLL ALGORITHM 200
    References 207
    Further Reading 209
    Index 211
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