现代密码学与椭圆曲线：初学者指引(影印版)

本书为低年级本科生提 供了现代数学的一些全景， 通过开发和呈现所需工具， 帮助理解有限域上椭圆曲线 的算术及其在现代密码学中 的应用。这种渐近式的引入 也为教会学生如何通过将数 学作为一种探索来产生或发 现证明做出了重大努力，同 时，它为研究椭圆曲线密码 学（ECC）的实践和实现提 供了必要的数学基础。 本书引入并发展了抽象 代数、数论、仿射几何与射 影几何的要素，并解释了它 们之间的相互作用。代数和 几何结合起来，通过单位圆 上的有理点来表征同余数， 椭圆曲线上点集的群法则则 源于贝祖定理提供的几何直 觉以及射影空间的构造。整 数模素数的单位群结构解释 了RSA加密、Pollard因数分 解方法、Diffie–Hellman密 钥交换和ElGamal加密，而 有限域上椭圆曲线的点群激 发了Lenstra椭圆曲线因子 分解方法和ECC。 本书唯一的实际先修课 程是一元微积分课程；其他 必要的数学主题将随着学习 过程逐步引入。大量的练习 进一步引导学习探索。

Preface Introduction Chapter 1. Three Motivating Problems §1.1.Fermat's Last Theorem §1.2.The Congruent Number Problem §1.3.Cryptography Chapter 2. Back to the Beginning §2.1.The Unit Circle: Real vs. Rational Points §2.2.Parametrizing the Rational Points on the Unit Circle §2.3.Finding all Pythagorean Triples §2.4.Looking for Underlying Structure: Geometry vs. Algebra §2.5.More about Points on Curves §2.6.Gathering Some Insight about Plane Curves §2.7.Additional Exercises Chapter 3. Some Elementary Number Theory §3.1.The Integers §3.2.Some Basic Properties of the Integers §3.3.Euclid's Algorithm §3.4.A First Pass at Modular Arithmetic §3.5.Elementary Cryptography:Caesar Cipher §3.6.Affine Ciphers and Linear Congruences §3.7.Systems of Congruences Chapter 4. A Second View of Modular Arithmetic: 7§n and Un §4.1.Groups and Rings §4.2.Fractions and the Notion of an Equivalence Relation §4.3.Modular Arithmetic §4.4.A Few More Comments on the Euler Totient Function §4.5.An Application to Factoring Chapter 5. Public-Key Cryptography and RSA §5.1.A Brief Overview of Cryptographic Systems §5.2 . RSA §5.3.Hash Functions §5.4.Breaking Cryptosystems and Practical RSA Security Considerations Chapter 6. A Little More Algebra §6.1.Towards a Classification of Groups §6.2.Cayley Tables §6.3.A Couple of Non-abelian Groups §6.4.Cyclic Groups and Direct Products §6.5.Fundamental Theorem of Finite Abelian Groups §6.6.Primitive Roots §6.7.Diffie-Hellman Key Exchange §6.8.E1Gamal Encryption Chapter 7. Curves in Affine and Projective Space §7.1.Affine and Projective Space §7.2.Curves in the Affine and Projective Plane §7.3.Rational Points on Curves §7.4.The Group Law for Points on an Elliptic Curve §7.5.A Formula for the Group Law on an Elliptic Curve §7.6.The Number of Points on an Elliptic Curve Chapter 8. Applications of Elliptic Curves