計算數(shù)論與現(xiàn)代密碼學

前言

　　The book is about number theory and modem cryptography. More specically， it is about computational number theory and modem public-key cryptography based on number theory. It consists of four parts. The first part， consisting of two chapters， provides some preliminaries. Chapter 1 provides some basic concepts of number theory， computation theory， computational number theory， and modem public-key cryptography based on number theory. In chapter 2， a complete introduction to some basic concepts and results in abstract algebra and elementary number theory is given. 　　The second part is on computational number theory. There are three chapters in this part.Chapter 3 deals with algorithms for primality testing， with an emphasis on the Miller-Rabin test， the elliptic curve test， and the AKS test. Chapter 4 treats with algorithms for integer factorization， including the currently fastest factoring algorithm NFS （Number Field Sieve），and the elliptic curve factoring algorithm ECM （Elliptic Curve Method）. Chapter 5 discusses various modem algorithms for discrete logarithms and for elliptic curve discrete logarithms.It is well-known now that primality testing can be done in polynomial-time on a digital computer， however， integer factorization and discrete logarithms still cannot be performed in polynomial-time. From a computational complexity point of view， primality testing is feasible （tractable， easy） on a digital computer， whereas integer factorization and discrete logarithms are infeasible （intractable， hard， difficult）. Of course， no-one has yet been able to prove that the integer factorization and the discrete logarithm problems must be infeasible on a digital computer. 　　Building on the results in the first two parts， the third part of the book studies the modem cryptographic schemes and protocols whose security relies exactly on the infeasibility of the integer factorization and discrete logarithm problems. There are four chapters in this part.Chapter 6 presents some basic concepts and ideas of secret-key cryptography. Chapter 7 studies the integer factoring based public-key cryptography， including， among others， the most famous and widely used RSA cryptography， the Rabin cryptosystem， the probabilistic encryption and the zero-knowledge proof protocols. Chapter 8 studies the discrete logarithm based cryptography， including the DHM key-exchange protocol （the world's first public-key system）， the E1Gamal cryptosystem， and the US Government's Digital Signature Standard （DSS）， Chapter 9 discusses various cryptographic systems and digital signature schemes based on the infeasibility of the elliptic curve discrete logarithm problem， some of them are just the elliptic curve analogues of the ordinary public-key cryptography such as elliptic curve DHM， elliptic curve E1Gamal， elliptic curve RSA， and elliptic curve DSA/DSS.

內(nèi)容概要

　　數(shù)論和密碼學是兩個不同的學科，且分屬于不同的研究領域，而現(xiàn)代公鑰密碼體制的創(chuàng)立和應用則將這兩個不同的學科緊密地聯(lián)系在一起。這是因為這些密碼體制的安全性幾乎完全基于某些數(shù)論問題的難解性。比如極負盛譽的rsa密碼體制之所以難以破譯，就是因為整數(shù)分解問題難以快速解決?！队嬎銛?shù)論與現(xiàn)代密碼學》首先從計算理論的觀點介紹數(shù)論中一些難解性問題，如整數(shù)分解問題和離散對數(shù)問題(包括橢圓曲線離散對數(shù)問題)，然后討論基于這些難解性問題的現(xiàn)代公鑰密碼體制，最后討論這些難解性問題的量子計算方法以及這些密碼體制的量子攻擊方法；由于量子計算僅適合于快速解決某些難解性數(shù)論問題(并非所有難解性的數(shù)論及數(shù)學問題)，因此還討論了某些量子計算鞭長莫及的數(shù)學問題以及基于這些問題的抗量子密碼體制。此外，書中還配有大量實例和練習，便于讀者學習和掌握。
　　《計算數(shù)論與現(xiàn)代密碼學》可作為高等學校計算機、信息安全、電子與通信工程、數(shù)學等專業(yè)高年級本科生和研究生的教材，也可作為相關領域研究人員的參考書。

作者簡介

　　Song Y. Yan（顏松遠），江西吉安人。獲中國科學院研究生院理學碩士學位，并獲英國約克大學數(shù)學博士學位。長期在國外大學從事計算數(shù)論、計算理論和密碼學等方面的科研與教學工作，在Springer出版專著4部：1.《Number Theory for Computing》第1版（2000年）、第2版（2002年）、波蘭文版（2006年，華沙國家科技出版社）、中文版（2007年，清華大學出版社）；2.《Primality Testing and Integer Factorization in Public-Key Cryptography》第1版（2004年）、第2版（2009年）；3.《Cryptanalytic Attacks on RSA》第1版（2007年）、俄文版（2010年，莫斯科國家科技出版中心）；4.《Quantum Attacks on Public-Key Cryptosystems》第1版（2012年）。

書籍目錄

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 what is 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 tests
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 p and 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-step method
5.3 pohlig-hellman method
5.4 index calculus
5.5 elliptic curve discrete logarithms
5.6 bibliographic notes and further reading
references
part iii modern cryptography
6secret-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 based cryptography
7.1 rsa cryptography
7.2 cryptanalysis of rsa
7.3 rabin cryptography
7.4 residuosity based cryptography
7.5 zero-knowledge proof
7.6 bibliographic notes and further reading
references
8 discrete logarithm based cryptography
8.1 diffie-heuman-merkle key-exchange protocol
8.2 e1gamal 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 based 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 curve eigamal cryptography
9.5 elliptic curve rsa cryptosystem
9.6 menezes-vanstone elliptic curve cryptography
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

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