Algorithmic number theory: lattices, number fields, curves and cryptography

J.P. Buhler, P. Stevenhagen0521808545, 9780521808545

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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Table of contents :
Cover……Page 1
Contents……Page 10
Solving the Pell equation……Page 14
Basic algorithms in number theory……Page 38
Smooth numbers and the quadratic sieve……Page 82
The number field sieve……Page 96
Four primality testing algorithms……Page 114
Lattices……Page 140
Elliptic curves……Page 196
The arithmetic of number rings……Page 222
Smooth numbers: computational number theory and beyond……Page 280
Fast multiplication and its applications……Page 338
Elementary thoughts on discrete logarithms……Page 398
The impact of the number field sieve on the discrete logarithm problem in finite fields……Page 410
Reducing lattice bases to find small-height values of univariate polynomials……Page 434
Computing Arakelov class groups……Page 460
Computational class field theory……Page 510
Protecting communications against forgery……Page 548
Algorithmic theory of zeta functions over finite fields……Page 564
Counting points on varieties over finite fields of small characteristic……Page 592
Congruent number problems and their variants
……Page 626
An introduction to computing modular forms using modular symbols……Page 654