Algorithmic Number Theory Lattices, Number Fields, Curves and Cryptography
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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.
List price: $129.00
Copyright year: 2008
Publisher: Cambridge University Press
Publication date: 10/20/2008
Size: 6.25" wide x 9.50" long x 1.50" tall
|Solving Pell's equation|
|Basic algorithms in number theory|
|The arithmetic of number rings|
|Fast multiplication and applications|
|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|