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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
ISBN：
978-7-04-063235-4
版面字数：
438.00千字
开本：
特殊
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
出版时间：
2025-02-07
物料号：
63235-00
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
  • 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

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