数论和密码学是两个不同的学科，且分属于不同的研究领域，而现代公钥密码体制的创立和应用则将这两个不同的学科紧密地联系在一起。这是因为这些密码体制的 安全性几乎完全基于某些数论问题的难解性。比如极负盛誉的rsa密码体制之所以难以破译，就是因为整数分解问题难以快速解决。《计算数论与现代密码学》首 先从计算理论的观点介绍数论中一些难解性问题，如整数分解问题和离散对数问题(包括椭圆曲线离散对数问题)，然后讨论基于这些难解性问题的现代公钥密码体 制，最后讨论这些难解性问题的量子计算方法以及这些密码体制的量子攻击方法；由于量子计算仅适合于快速解决某些难解性数论问题(并非所有难解性的数论及数 学问题)，因此还讨论了某些量子计算鞭长莫及的数学问题以及基于这些问题的抗量子密码体制。此外，书中还配有大量实例和练习，便于读者学习和掌握。
《计算数论与现代密码学》可作为高等学校计算机、信息安全、电子与通信工程、数学等专业高年级本科生和研究生的教材，也可作为相关领域研究人员的参考书。
- Part I Preliminaries
- 1 Introduction
- 1.1 What is Number Theory?
- 1.2 What is Computation Theory?
- 1.3 What is Computational Number Theory?
- 1.4 Whatis Modern Cryptography?
- 1.5 Bibliographic Notes and Further Reading
- References
- 2 Fundamentals
- 2.1 Basic Algebraic Structures
- 2.2 Divisibility Theory
- 2.3 Arithmetic Functions
- 2.4 Congruence Theory
- 2.5 Primitive Roots
- 2.6 Elliptic Curves
- 2.7 Bibliographic Notes and Further Reading
- References
- Part II Computational Number Theory
- 3 Primality Testing
- 3.1 Basic TIests
- 3.2 Miller-Rabin Test
- 3.3 Elliptic Curve Tests
- 3.4 AKS Test
- 3.5 Bibliographic Notes and Further Reading
- References
- 4 Integer Factorization
- 4.1 Basic Concepts
- 4.2 Trial Divisions Factoring
- 4.3 ρand p - 1 Methods
- 4.4 Elliptic Curve Method
- 4.5 Continued Fraction Method
- 4.6 Quadratic SieVe
- 4.7 Number Field Sieve
- 4.8 Bibliographic Notes and Further Reading
- References
- 5 Discrete Logarithms
- 5.1 Basic Concepts
- 5.2 Baby-Step Giant-StepMethod
- 5.3 Pohlig-Hellman Method
- 5.4 Index Ca1culus
- 5.5 Elliptic Curve Discrete Logarithms
- 5.6 Bibliographic Notes and Further Reading
- References
- Part III Modern Cryptography
- 6 Secret-Key Cryptography
- 6.1 Cryptography and Cryptanalysis
- 6.2 Classic Secret-Key Cryptography
- 6.3 Modern Secret-Key Cryptography
- 6.4 Bibliographic Notes and Further Reading
- References
- 7 Integer Factorization ßased Cryptography
- 7.1 RSA Cryptography
- 7.2 Cryptana1ysis of RSA
- 7.3 Rabin Cryptography
- 7.4 Residuosity Based Cryptography
- 7.5 Zero-KnowledgeProof
- 7.6 Bibliographic Notes and Further Reading
- References
- 8 Discrete Logarithm ßased Cryptography
- 8.1 Diffie-Hellman-Merkle Key-Exchange Protocol
- 8.2 EIGamal Cryptography
- 8.3 Massey-Omura Cryptography
- 8.4 DLP-Based Digital Signatures
- 8.5 Bibliographic Notes and Further Reading
- References
- 9 Elliptic Curve Discrete Logarithm ßased Cryptography
- 9.1 Basic Ideas
- 9.2 Elliptic Curve Diffie-Hellman-Merkle Key Exchange Scheme
- 9.3 Elliptic Curve Massey-Omura Cryptography
- 9.4 Elliptic CurveEIGamal Cryptography
- 9.5 Elliptic CurveRSA Cryptosystem
- 9.6 Menezes-VanstoneElliptic CurveCryptography
- 9.7 Elliptic Curve DSA
- 9.8 Bibliographic Notes and Further Reading
- References
- Part IV Quantum Resistant Cryptography
- 10 Quantum Computational Number Theory
- 10.1 Quantum Algorithms for Order Finding
- 10.2 Quantum Algorithms for Integer Factorization
- 10.3 Quantum Algorithms for Discrete Logarithms
- 10.4 Quantum Algorithms for Elliptic Curve Discrete Logarithms
- 10.5 Bibliographic Notes and Further Reading
- References
- 11 Quantum Resistant Cryptography
- 11.1 Coding-Based Cryptography
- 11.2 Lattice-Based Cryptography
- 11.3 Quantum Cryptography
- 11.4 DNA Biological Cryptography
- 11.5 Bibliographic Notes and Further Reading
- References
- Index
