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Yan, Song Y. : Number Theory for Computing ... ( Ausgestellt von 1.8.2000 bis 7.8.2000 )

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Jul 27, 2000, 3:00:00 AM7/27/00
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Autor : Yan, Song Y.
Titel : Number Theory for Computing
Dokumenttyp : Autorenwerk
ISBN : 3540654720
Erscheinungsjahr : 2000
Verlag : Springer , Berlin
Standort : 0100 (Praesenzbibliothek ; steht unter : YAN )
Signatur : 1.2000A211


Archiviert am : 27.7.2000
Neuerwerbungsregal : von 1.8.2000 bis 7.8.2000


Inhaltsverzeichnis:

Song Y. Yan
Number Theory for Computing
Springer , Berlin , 2000 , 381 S.
ISBN 3540654720

Table of Contents

1. Elementary Number Theory 1
1.1 Introduction 1
1.1.1 What is Number Theory? 1
1.1.2 Algebraic Preliminaries 12
1.2 Theory of Divisibility 20
1.2.1 Basic Properties of Divisibility 20
1.2.2 Fundamental Theorem of Arithmetic 24
1.2.3 Mersenne Primes and Fermat Numbers 27
1.2.4 Euclid's Algorithm 32
1.2.5 Continued Fractions 36
1.3 Diophantine Equations 41
1.3.1 Basic Concepts of Diophantine Equations 41
1.3.2 Linear Diophantine Equations 42
1.3.3 Pell's Equations 45
1.4 Arithmetic Functions 50
1.4.1 Multiplicative Functions 50
1.4.2 Functions T(n), sigma(n) and s(n) 51
1.4.3 Perfect, Amicable and Sociable Numbers 54
1.4.4 Functions phi(n), lambda(n) and my(n) 61
1.5 Distribution of Prime Numbers 64
1.5.1 Prime Distribution Function Pi(x) 65
1.5.2 Approximations of Pi(x) by x/lnx 67
1.5.3 Approximations of Pi(x) by Li(x) 73
1.5.4 The Riemann zeta-Function zeta(s) 74
1.5.5 The nth Prime 83
1.5.6 Distribution of Twin Primes 86
1.5.7 The Arithmetic Progression of Primes 89
1.6 Theory of Congruences 90
1.6.1 Basic Properties of Congruences 90
1.6.2 Modular Arithmetic 94
1.6.3 Linear Congruences 96
1.6.4 The Chinese Remainder Theorem 101
1.6.5 High-Order Congruences 104
1.6.6 Legendre and Jacobi Symbols 107
1.6.7 Orders and Primitive Roots 115
1.6.8 Indices and kth Power Residues 120
1.7 Arithmetic of Elliptic Curves 124
1.7.1 Basic Concepts of Elliptic Curves 125
1.7.2 Geometric Composition Laws of Elliptic Curves 128
1.7.3 Algebraic Computation Laws for Elliptic Curves 129
1.7.4 Group Laws on Elliptic Curves 133
1.7.5 Number of Points on Elliptic Curves 134
1.8 Bibliographic Notes and Further Reading 135

2. Algorithmic Number Theory 139
2.1 Introduction 139
2.1.1 What is Algorithmic Number Theory? 139
2.1.2 Effective Computability 142
2.1.3 Computational Complexity 146
2.1.4 Complexity of Number-Theoretic Algorithms 153
2.1.5 Fast Modular Exponentiations 159
2.1.6 Fast Group Operations on Elliptic Curves 163
2.2 Algorithms for Primality Testing 167
2.2.1 Deterministic and Rigorous Primality Tests 167
2.2.2 Fermat's Pseudoprimality Test 170
2.2.3 Strong Pseudoprimality Test 173
2.2.4 Lucas Pseudoprimality Test 179
2.2.5 Elliptic Curve Test 186
2.2.6 Historical Notes on Primality Testing 190
2.3 Algorithms for Integer Factorization 192
2.3.1 Complexity of Integer Factorization 192
2.3.2 Trial Division and Fermat Method 196
2.3.3 Legendre's Congruence 198
2.3.4 Continued FRACtion Method (CFRAC) 201
2.3.5 Quadratic and Number Field Sieves (QS/NFS) 204
2.3.6 Polland's "rho" and "p-1" Methods 208
2.3.7 Lenstra's Elliptic Curve Method (ECM) 215
2.4 Algorithms for Discrete Logarithms 218
2.4.1 Shanks' Baby-Step Giant-Stop Algorithm 219
2.4.2 Silver Pohlig-Hellman Algorithm 222
2.4.3 Subexponential Algorithms 226
2.4.4 Algorithm for the Root Finding Problem 227
2.5 Quantum Number-Theoretic Algorithms 230
2.5.1 Quantum Information and Computation 230
2.5.2 Quantum Computability and Complexity 235
2.5.3 Quantum Algorithm for Integer Factorization 236
2.5.4 Quantum Algorithms for Discrete Logarithms 241
2.6 Miscellaneous Algorithms in Number Theory 243
2.6.1 Algorithms for Computing Pi(x) 243
2.6.2 Algorithms for Generating Amicable Pairs 249
2.6.3 Algorithms for Verifying Goldbach's Conjecture 252
2.6.4 Algorithm for Finding Odd Perfect Numbers 255
2.7 Bibliographic Notes and Further Reading 257

3. Applied Number Theory 259
3.1 Why Applied Number Theory? 259
3.2 Computer Systems Design 261
3.2.1 Representing Numbers in Residue Number Systems 261
3.2.2 Fast Computations in Residue Number Systems 264
3.2.3 Residue Computers 269
3.2.4 Complementary Arithmetic 269
3.2.5 Hashing Functions 273
3.2.6 Error Detection and Correction Methods 277
3.2.7 Random Number Generation 282
3.3 Cryptography and Information Security 287
3.3.1 Introduction 288
3.3.2 Secret-Key Cryptography 289
3.3.3 Data/Advanced Encryption Standard (DES/AES) 299
3.3.4 Public-Key Cryptography 303
3.3.5 Discrete Logarithm Based Cryptosystems 309
3.3.6 RSA Public-Key Cryptosystem 313
3.3.7 Quadratic Residuosity Cryptosystems 326
3.3.8 Elliptic Curve Public-Key Cryptosystems 332
3.3.9 Digital Signatures 336
3.3.10 Digital Signature Algorithm/Standard (DSA/DSS) 342
3.3.11 Database Security 344
3.3.12 Secret Sharing 348
3.3.13 Internet/Web Security and Electronic Commerce 352
3.3.14 Steganography 356
3.3.15 Quantum Cryptography 358
3.4 Bibliographic Notes and Further Reading 359

Bibliography 363
Index 375

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