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Algorithmic Number Theory Lattices, Number Fields, Curves and Cryptography

ISBN-10: 0521808545

ISBN-13: 9780521808545

Edition: 2008

Authors: J. P. Buhler, Peter Stevenhagen, P. Stevenhagen, P. Stevenhagen

List price: $129.00
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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.
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Book details

List price: $129.00
Copyright year: 2008
Publisher: Cambridge University Press
Publication date: 10/20/2008
Binding: Hardcover
Pages: 664
Size: 6.25" wide x 9.50" long x 1.50" tall
Weight: 2.244
Language: English

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
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