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。

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