Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography

by Edited by J.P. Buhler , P. Stevenhagen

• ISBN13: 9780521808545
• ISBN10: 0521808545
• Format: Hardcover
• Copyright: 2008-10-20
• Publisher: Cambridge University Press

Summary

Number theory is one of the oldest and most appealing areas of mathematics. Computation has always played a role in number theory, a role which has increased dramatically in the last 20 or 30 years, both because of the advent of modern computers, and because of the discovery of surprising and powerful algorithms. As a consequence, algorithmic number theory has gradually emerged as an important and distinct field with connections to computer science and cryptography as well as other areas of mathematics. This text provides a comprehensive introduction to algorithmic number theory for beginning graduate students, written by the leading experts in the field. It includes several articles that cover the essential topics in this area, such as the fundamental algorithms of elementary number theory, lattice basis reduction, elliptic curves, algebraic number fields, and methods for factoring and primality proving. In addition, there are contributions pointing in broader directions, including cryptography, computational class field theory, zeta functions and L-series, discrete logarithm algorithms, and quantum computing.

Table of Contents

Solving Pell's equation

Basic algorithms in number theory

Elliptic curves

The arithmetic of number rings

Fast multiplication and applications

Primality testing

Smooth numbers: computational number theory and beyond

Smooth numbers and the quadratic sieve

The number field sieve

Elementary thoughts on discrete logarithms

The impact of the number field sieve on the discrete logarithm problem in finite fields

Lattices

Reducing lattices to find small-height values of univariate polynomials

Protecting communications against forgery

Computing Arakelov class groups

Computational class field theory

Zeta functions over finite fields

Counting points on varieties over finite fields

How to get your hands on modular forms using modular symbols

Congruent number problems in dimension one and two

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