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Part Ⅰ Preliminaries 3

1 Introduction 3

1.1 What is Number Theory? 3

1.2 What is Computation Theory? 9

1.3 What is Computational Number Theory? 15

1.4 What is Modern Cryptography? 29

1.5 Bibliographic Notes and Further Reading 32

References 32

2 Fundamentals 35

2.1 Basic Algebraic Structures 35

2.2 Divisibility Theory 46

2.3 Arithmetic Functions 75

2.4 Congruence Theory 89

2.5 Primitive Roots 131

2.6 Elliptic Curves 141

2.7 Bibliographic Notes and Further Reading 154

References 155

Part Ⅱ Computational Number Theory 159

3 Primality Testing 159

3.1 BasicTests 159

3.2 Miller-Rabin Test 168

3.3 Elliptic Curve Tests 173

3.4 AKS Test 178

3.5 Bibliographic Notes and Further Reading 187

References 188

4 Integer Factorization 191

4.1 Basic Concepts 191

4.2 Trial Divisions Factoring 194

4.3 ρ and P-1 Methods 198

4.4 Elliptic Curve Method 205

4.5 Continued Fraction Method 209

4.6 Quadratic Sieve 214

4.7 Number Field Sieve 219

4.8 Bibliographic Notes and Further Reading 231

References 232

5 Discrete Logarithms 235

5.1 Basic Concepts 235

5.2 Baby-Step Giant-Step Method 237

5.3 Pohlig-Hellman Method 240

5.4 Index Calculus 246

5.5 Elliptic Curve Discrete Logarithms 251

5.6 Bibliographic Notes and Further Reading 260

References 261

Part Ⅲ Modern Cryptography 265

6 Secret-Key Cryptography 265

6.1 Cryptography and Cryptanalysis 265

6.2 Classic Secret-Key Cryptography 277

6.3 Modern Secret-Key Cryptography 285

6.4 Bibliographic Notes and Further Reading 291

References 291

7 Integer Factorization Based Cryptography 293

7.1 RSA Cryptography 293

7.2 Cryptanalysis of RSA 302

7.3 Rabin Cryptography 319

7.4 Residuosity Based Cryptography 326

7.5 Zero-Knowledge Proof 331

7.6 Bibliographic Notes and Further Reading 335

References 335

8 Discrete Logarithm Based Cryptography 337

8.1 Diffie-Hellman--Merkle Key-Exchange Protocol 337

8.2 E1Gamal Cryptography 342

8.3 Massey-Omura Cryptography 344

8.4 DLP-Based Digital Signatures 348

8.5 Bibliographic Notes and Further Reading 351

References 351

9 Elliptic Curve Discrete Logarithm Based Cryptography 353

9.1 Basic Ideas 353

9.2 Elliptic Curve Diffie-Hellman-Merkle Key Exchange Scheme 356

9.3 Elliptic Curve Massey-Omura Cryptography 360

9.4 Elliptic Curve ElGamal Cryptography 365

9.5 Elliptic Curve RSA Cryptosystem 370

9.6 Menezes-Vanstone Elliptic Curve Cryptography 371

9.7 Elliptic Curve DSA 373

9.8 Bibliographic Notes and Further Reading 374

References 375

Part Ⅳ Quantum Resistant Cryptography 379

10 Quantum Computational Number Theory 379

10.1 Quantum Algorithms for Order Finding 379

10.2 Quantum Algorithms for Integer Factorization 385

10.3 Quantum Algorithms for Discrete Logarithms 390

10.4 Quantum Algorithms for Elliptic Curve Discrete Logarithms 393

10.5 Bibliographic Notes and Further Reading 397

References 397

11 Quantum Resistant Cryptography 401

11.1 Coding-Based Cryptography 401

11.2 Lattice-Based Cryptography 403

11.3 Quantum Cryptography 404

11.4 DNA Biological Cryptography 406

11.5 Bibliographic Notes and Further Reading 409

References 410

Index 413