Elliptic Curves: A Computational  Approach

Susanne Schmitt; Horst G. Zimmer

Published by De Gruyter 2003

Elliptic Curves

A Computational Approach

Susanne Schmitt and Horst G. Zimmer
With contributions by: Attila Pethö

Volume 31 in the series De Gruyter Studies in Mathematics

https://doi.org/10.1515/9783110198010

Overview
Contents

About this book

The basics of the theory of elliptic curves should be known to everybody, be he (or she) a mathematician or a computer scientist. Especially everybody concerned with cryptography should know the elements of this theory.

The purpose of the present textbook is to give an elementary introduction to elliptic curves. Since this branch of number theory is particularly accessible to computer-assisted calculations, the authors make use of it by approaching the theory under a computational point of view. Specifically, the computer-algebra package SIMATH can be applied on several occasions. However, the book can be read also by those not interested in any computations. Of course, the theory of elliptic curves is very comprehensive and becomes correspondingly sophisticated. That is why the authors made a choice of the topics treated. Topics covered include the determination of torsion groups, computations regarding the Mordell-Weil group, height calculations, S-integral points. The contents is kept as elementary as possible. In this way it becomes obvious in which respect the book differs from the numerous textbooks on elliptic curves nowadays available.

Author / Editor information

Dr. Susanne Schmitt works at the Max Planck Institute for Computer Sciences at Saarbrücken, Germany.

Horst-Günter Zimmer is Professor at the Mathematics Department of the University of the Saarland at Saarbrücken, Germany.

Reviews

"This is a most welcome addition to the existing literature on elliptic curves."
Ch. Baxa in: Monatshefte für Mathematik 4/2004

"[...] [the authors'] book will function well both as a textbook for students and as a reference book for researchers. [...] it nicely complements the more theoretical books [...], while providing a path for both students and researchers into the computational theory of elliptic curves." Mathematical Reviews

Topics

Frontmatter Requires Authentication Unlicensed Licensed Download PDF	i
Contents Requires Authentication Unlicensed Licensed Download PDF	vii
Chapter 1. Elliptic curves Requires Authentication Unlicensed Licensed Download PDF	1
Chapter 2. Elliptic curves over the complex numbers Requires Authentication Unlicensed Licensed Download PDF	33
Chapter 3. Elliptic curves over finite fields Requires Authentication Unlicensed Licensed Download PDF	63
Chapter 4. Elliptic curves over local fields Requires Authentication Unlicensed Licensed Download PDF	87
Chapter 5. The Mordell-Weil theorem and heights Requires Authentication Unlicensed Licensed Download PDF	103
Chapter 6. Torsion group Requires Authentication Unlicensed Licensed Download PDF	147
Chapter 7. The rank Requires Authentication Unlicensed Licensed Download PDF	198
Chapter 8. Basis Requires Authentication Unlicensed Licensed Download PDF	242
Chapter 9. S-integral points Requires Authentication Unlicensed Licensed Download PDF	263
Appendix A. Algorithmic theory of diophantine equations Requires Authentication Unlicensed Licensed Download PDF	294
Appendix B. Multiquadratic number fields Requires Authentication Unlicensed Licensed Download PDF	316
Backmatter Requires Authentication Unlicensed Licensed Download PDF	351

(Deutsch)

Language: English
Publisher: De Gruyter
Pages
- Front matter: 9
- Main content: 367
Keywords: Elliptische Kurve; Algorithmus; Kryptographie

eBook

Published: August 22, 2008
ISBN: 9783110198010

Hardcover

Published: November 19, 2003
ISBN: 9783110168082