The arithmetic of elliptic curves
The arithmetic of elliptic curves
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DOI：
10.1007/978-0-387-09494-6
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发表时间：
1986
期刊：
影响因子：
--
通讯作者：
J. Silverman
中科院分区：
文献类型：
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作者：
J. Silverman
椭圆曲线理论以其悠久的历史和研究方法的多样性而独树一帜。这本书通过使用基本的代数数论和代数几何，在其现代公式中处理椭圆曲线的算术理论。该书以必要的代数几何结果的简要讨论开始，并继续阐述椭圆曲线的几何，椭圆曲线的形式群，以及有限域上的椭圆曲线，复数，局部域和全局域。其中包括给出有理点群有限生成的Mordell-Weil定理的证明和关于整点有限的Siegel定理。在第二版椭圆曲线算法中，有一个新的章节名为椭圆曲线的算法方面，重点介绍了具有密码应用的有限域上的算法。其中包括Lenstra的因式分解算法、Schoof的点计数算法、Miller的计算Tate和Weil对的算法，以及对椭圆曲线密码术方面的描述。还有一个关于Szpiro猜想和ABC的新部分，以及对最近发展和许多新练习的扩大和更新的描述。
The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel's theorem on finiteness of integral points.For this second edition of The Arithmetic of Elliptic Curves, there is a new chapter entitled Algorithmic Aspects of Elliptic Curves, with an emphasis on algorithms over finite fields which have cryptographic applications. These include Lenstra's factorization algorithm, Schoof's point counting algorithm, Miller's algorithm to compute the Tate and Weil pairings, and a description of aspects of elliptic curve cryptography. There is also a new section on Szpiro's conjecture and ABC, as well as expanded and updated accounts of recent developments and numerous new exercises.