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

Includes index

Includes bibliography references

Solving the Pell equation / Hendrik W. Lenstra, Jr. Basic algorithms in number theory / Stan Wagon Smooth numbers and the quadratic sieve / Carl Pomerance The number field sieve / Peter Stevenhagen Four primality testing algorithms / Rene Schoof Lattices / Hendrik W. Lenstra, Jr. Elliptic curves / Bjorn Poonen The arithmetic of number rings / Peter Stevenhagen Smooth numbers: computational number theory and beyond / Andrew Granville Fast multiplication and its applications / Daniel J. Bernstein Elementary thoughts on discrete logarithms / Carl Pomerance The impact of the number field sieve on the discrete logarithm problem in finite fields / Oliver Schirokauer Reducing lattice bases to find small-height values of univariate polynomials / Daniel J. Bernstein Computing Arakelov class groups / Rene Schoof Computational class field theory / Peter Stevenhagen Contents note continued: Protecting communications against forgery / Daniel J. Bernstein Algorithmic theory of zeta functions over finite fields / Daqing Wan Counting points on varieties over finite fields of small characteristic / Daqing Wan Congruent number problems and their variants / Noriko Yui. An introduction to computing modular forms using modular symbols / William A. Stein.

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, and 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.