计算数论 第2版 世界图书出版公司2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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1．Elementary Number Theory1

1.1 Introduction1

1.1.1 What is Number Theory?1

1.1.2 Applications of Number Theory13

1.1.3 Algebraic Preliminaries14

1.2 Theory of Divisibility21

1.2.1 Basic Concepts and Properties of Divisibility21

1.2.2 Fundamental Theorem of Arithmetic27

1.2.3 Mersenne Primes and Fermat Numbers33

1.2.4 Euclid's Algorithm40

1.2.5 Continued Fractions44

1.3 Diophantine Equations52

1.3.1 Basic Concepts of Diophantine Equations52

1.3.2 Linear Diophantine Equations54

1.3.3 Pell's Equations57

1.4 Arithmetic Functions63

1.4.1 Multiplicative Functions63

1.4.2 Functions ？（n）,σ（n）and s（n）66

1.4.3 Perfect,Amicable and Sociable Numbers71

1.4.4 Functionsφ（n）,λ（n）andμ（n）79

1.5 Distribution of Prime Numbers85

1.5.1 Prime Distribution Functionπ（x）85

1.5.2 Approximations of π（x）by x/lnx87

1.5.3 Approximations of π（x）by Li（x）94

1.5.4 The Riemannξ-Functionξ（s）95

1.5.5 The nth Prime104

1.5.6 Distribution of Twin Primes106

1.5.7 The Arithmetic Progression of Primes110

1.6 Theory of Congruences111

1.6.1 Basic Concepts and Properties of Congruences111

1.6.2 Modular Arithmetic118

1.6.3 Linear Congruences123

1.6.4 The Chinese Remainder Theorem130

1.6.5 High-Order Congruences133

1.6.6 Legendre and Jacobi Symbols139

1.6.7 Orders and Primitive Roots150

1.6.8 Indices and kth Power Residues155

1.7 Arithmetic of Elliptic Curves160

1.7.1 Basic Concepts of Elliptic Curves160

1.7.2 Geometric Composition Laws of Elliptic Curves163

1.7.3 Algebraic Computation Laws for Elliptic Curves164

1.7.4 Group Laws on Elliptic Curves168

1.7.5 Number of Points on Elliptic Curves169

1.8 Bibliographic Notes and Further Reading171

2．Computational/Algorithmic Number Theory173

2.1 Introduction173

2.1.1 What is Computational/Algorithmic Number Theory?174

2.1.2 Effective Computability177

2.1.3 Computational Complexity181

2.1.4 Complexity of Number-Theoretic Algorithms188

2.1.5 Fast Modular Exponentiations194

2.1.6 Fast Group Operations on Elliptic Curves198

2.2 Algorithms for Primality Testing202

2.2.1 Deterministic and Rigorous Primality Tests202

2.2.2 Fermat's Pseudoprimality Test206

2.2.3 Strong Pseudoprimality Test208

2.2.4 Lucas Pseudoprimality Test215

2.2.5 Elliptic Curve Test222

2.2.6 Historical Notes on Primality Testing225

2.3 Algorithms for Integer Factorization228

2.3.1 Complexity of Integer Factorization228

2.3.2 Trial Division and Fermat Method232

2.3.3 Legendre's Congruence234

2.3.4 Continued FRACtion Method （CFRAC）237

2.3.5 Quadratic and Number Field Sieves（QS/NFS）240

2.3.6 Polland's"rho"and"p-1"Methods244

2.3.7 Lenstra's Elliptic Curve Method （ECM）251

2.4 Algorithms for Discrete Logarithms254

2.4.1 Shanks'Baby-Step Giant-Step Algorithm255

2.4.2 Silver-Pohlig-Hellman Algorithm258

2.4.3 Index Calculus for Discrete Logarithms262

2.4.4 Algorithms for Elliptic Curve Discrete Logarithms266

2.4.5 Algorithm for Root Finding Problem270

2.5 Quantum Number-Theoretic Algorithms273

2.5.1 Quantum Information and Computation273

2.5.2 Quantum Computability and Complexity278

2.5.3 Quantum Algorithm for Integer Factorization279

2.5.4 Quantum Algorithms for Discrete Logarithms285

2.6 Miscellaneous Algorithms in Number Theory287

2.6.1 Algorithms for Computing π（x）287

2.6.2 Algorithms for Generating Amicable Pairs292

2.6.3 Algorithms for Verifying Goldbach's Conjecture295

2.6.4 Algorithm for Finding Odd Perfect Numbers299

2.7 Bibliographic Notes and Further Reading300

3．Applied Number Theory in Computing/Cryptography303

3.1 Why Applied Number Theory?303

3.2 Computer Systems Design305

3.2.1 Representing Numbers in Residue Number Systems305

3.2.2 Fast Computations in Residue Number Systems308

3.2.3 Residue Computers312

3.2.4 Complementary Arithmetic315

3.2.5 Hash Functions317

3.2.6 Error Detection and Correction Methods321

3.2.7 Random Number Generation326

3.3 Cryptography and Information Security332

3.3.1 Introduction332

3.3.2 Secret-Key Cryptography333

3.3.3 Data/Advanced Encryption Standard （DES/AES）344

3.3.4 Public-Key Cryptography348

3.3.5 Discrete Logarithm Based Cryptosystems354

3.3.6 RSA Public-Key Cryptosystem358

3.3.7 Quadratic Residuosity Cryptosystems373

3.3.8 Elliptic Curve Public-Key Cryptosystems379

3.3.9 Digital Signatures385

3.3.10 Digital Signature Standard（DSS）392

3.3.11 Database Security395

3.3.12 Secret Sharing399

3.3.13 Internet/Web Security and Electronic Commerce403

3.3.14 Steganography409

3.3.15 Quantum Cryptography410

3.4 Bibliographic Notes and Further Reading411

Bibliography415

Index429