利用有限Abel群构建公钥密码系统现在已经成为著名的范例，而代数几何学通过有限域上的Abel簇提供了一些这样的群,特别令人感兴趣的是Abel簇为代数曲线的Jacobi簇的情形。本书中的所有文章都聚焦于有限域上曲线的Jacobi簇的点计数和显式算法这一主题。这些文章的论题包括Schoof的l进点计数算法、Kedlaya 和 Denef-Vercauteren的p进算法、Cab曲线和zeta函数的Jacobi簇的显式算法。
库马尔·默蒂主编的《代数曲线与密码学（英文版）（精）》的文章大部分都适合希望进入这一领域的研究生独立学习，这些文章既介绍了基础性材料，又能引导读者深入到文献中去。密码学的文献看上去是呈指数型增长的，对于一个入门者来说，穿越这片海洋令人望而却步。本书会将读者引向关于这一数学分支的若干新思想的讨论,并给出进一步阅读的简明指引。
本书适合对密码学以及数论和代数几何的应用感兴趣的研究生和研究人员阅读。

Chapter 1 An Overview of Algebraic Curves and Cryptography
V. KUMAR MURTY
1.1 Introduction
1.2 The basic paradigm
1.3 The Diffie-Hellman decision problem
1.4 Constraints on the group
1.5 Abelian varieties over finite fields
1.6 Elliptic curves
1.7 Statistical results
1.8 Abelian varieties of higher dimension
1.9 Outline of contents
Chapter 2 School's Point Counting Algorithm
NICOLAS THERIAULT
2.1 Preliminaries
2.2 Division polynomials
2.3 Schoof's algorithm
2.4 Implementation
2.5 Improvements by Atkin and Elkies
2.6 Computing the modular equations
2.7 Computing Pl, 5 and
2.8 Computing the factor of fe
2.9 Parallelization
Chapter 3 Report on the Denef-Vercauteren/Kedlaya Algorithm
ZUBAIR ASHRAF ALI JUMA AND PRAMATHANATH SASTRY
3.1 Background
3.2 Generalities
3.3 Main strategy
3.4 Monsky-Washnitzer cohomology
3.5 Hyperelliptic curves
3.6 Data structures
3.7 Algorithm for lifting the curve to characteristic zero
3.8 Inversion
3.9 The 2-power Frobenius on K
3.10 The characteristic polynomial of Frobenius
3.11 Multiplication
3.12 Running times
3.13 Parallelization
Chapter 4 An Introduction to Gr5bner Bases
MOHAMMED RADI-BENJELLOUN
4.1 Introduction
4.2 GrSbner bases
Chapter 5 Cab Curves and Arithmetic on Their Jacobians
FARZALI IZADI
5.1 Introduction
5.2 Preliminaries
5.3 The Cab curves
5.4 Addition algorithm for Jacobian group in divisor representation
5.5 Addition algorithm for Jacobian group in ideal representation
Chapter 6 The Zeta Functions of Two Garcia-Stichtenoth Towers
KENNETH W. SHUM 6.1 Introduction
6.2 Background on zeta functions
6.3 The first Garcia-Stichtenoth tower
6.4 The second Garcia-Stichtenoth tower
6.5 Conclusion
Appendix: Counting points over P0 in GS1
Bibliography
Index