Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography (Mathematical Sciences Research Institute Publications, Series Number 44)

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. 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