现代密码学与椭圆曲线：初学者指引（影印版）

样章

作者：

Thomas R. Shemanske

定价：

135.00 元

版面字数：

438千字

开本：

特殊

装帧形式：

精装

版次：

最新版次
印刷时间：

2025年

ISBN：

978-7-04-063235-4

物料号：

63235-00

出版时间：

2025-02-07

读者对象：

学术著作

一级分类：

自然科学

二级分类：

数学与统计

三级分类：

代数几何学

暂无

目录
- 前辅文
- 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: Zn 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. ElGamal 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
  - §8.1. Elliptic Curves and Factoring
  - §8.2. Elliptic Curves and Cryptography
  - §8.3. Remarks on a Post-Quantum Cryptographic World
- Appendix A. Deeper Results and Concluding Thoughts
  - §A.1. The Congruent Number Problem and Tunnell's Solution
  - §A.2. A Digression on Functions of a Complex Variable
  - §A.3. Return to the Birch and Swinnerton-Dyer Conjecture
  - §A.4. Elliptic Curves over C
- Appendix B. Answers to Selected Exercises
  - §B.1. Chapter 2
  - §B.2. Chapter 3
  - §B.3. Chapter 4
  - §B.4. Chapter 5
  - §B.5. Chapter 6
  - §B.6. Chapter 7
- Bibliography
- Index