计算数论与现代密码学

part i preliminaries1 introduction1.1 what is number theory1.2 what is computation theory1.3 what is computational number theory1.4 what is modern cryptography1.5 bibliographic notes and further readingreferences2 fundamentals2.1 basic algebraic structures2.2 divisibility theory2.3 arithmetic functions2.4 congruence theory2.5 primitive roots2.6 elliptic curves2.7 bibliographic notes and further readingreferencespart ii computational number theory3 primality testing3.1 basic tests3.2 miller-rabin test3.3 elliptic curve tests3.4 aks test3.5 bibliographic notes and further readingreferences4 integer factorization4.1 basic concepts4.2 trial divisions factoring4.3 p and p - 1 methods4.4 elliptic curve method4.5 continued fraction method4.6 quadratic sieve4.7 number field sieve4.8 bibliographic notes and further readingreferences5 discrete logarithms5.1 basic concepts5.2 baby-step giant-step method5.3 pohlig-hellman method5.4 index calculus5.5 elliptic curve discrete logarithms5.6 bibliographic notes and further readingreferencespart iii modern cryptography6secret-key cryptography6.1 cryptography and cryptanalysis6.2 classic secret-key cryptography6.3 modern secret-key cryptography6.4 bibliographic notes and further readingreferences7 integer factorization based cryptography7.1 rsa cryptography7.2 cryptanalysis of rsa7.3 rabin cryptography7.4 residuosity based cryptography7.5 zero-knowledge proof7.6 bibliographic notes and further readingreferences8 discrete logarithm based cryptography8.1 diffie-heuman-merkle key-exchange protocol8.2 e1gamal cryptography8.3 massey-omura cryptography8.4 dlp-based digital signatures8.5 bibliographic notes and further readingreferences9 elliptic curve discrete logarithm based cryptography9.1 basic ideas9.2 elliptic curve diffie-hellman-merkle key exchange scheme9.3 elliptic curve massey-omura cryptography9.4 elliptic curve eigamal cryptography9.5 elliptic curve rsa cryptosystem9.6 menezes-vanstone elliptic curve cryptography9.7 elliptic curve dsa9.8 bibliographic notes and further readingreferencespart iv quantum resistant cryptography10 quantum computational number theory10.1 quantum algorithms for order finding10.2 quantum algorithms for integer factorization10.3 quantum algorithms for discrete logarithms10.4 quantum algorithms for elliptic curve discretelogarithms10.5 bibliographic notes and further readingreferences11 quantum resistant cryptography11.1 coding-based cryptography11.2 lattice-based cryptography11.3 quantum cryptography11.4 dna biological cryptography11.5 bibliographic notes and further readingreferencesindex

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。