Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography

Mathematical Sciences Research Institute

doi:10.1017/9781139049801

Algorithmic Number Theory

Lattices, Number Fields, Curves and Cryptography

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Bernstein, Daniel J. Heninger, Nadia Lou, Paul and Valenta, Luke 2017. Post-Quantum Cryptography. Vol. 10346, Issue. , p. 311.

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Edited by J. P. Buhler, Reed College, Oregon, P. Stevenhagen, Universiteit Leiden
Mathematical Sciences Research Institute

Publisher:: Cambridge University Press
Online publication date:: May 2025
Print publication year:: 2008
Online ISBN:: 9781139049801
DOI:: https://doi.org/10.1017/9781139049801

Subjects:: Mathematics, Algorithmics, Complexity, Computer Algebra, Computational Geometry, Real and Complex Analysis, Computer Science, Number Theory
Series:: Mathematical Sciences Research Institute Publications (44)

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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, 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.

Review of the hardback:'… can be warmly recommended to anyone interested in the fascinating area of computational number theory.'

Source: EMS Newsletter

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Algorithmic Number Theory

Lattices, Number Fields, Curves and Cryptography

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Contents

Frontmatter
pp i-vi

Contents
pp vii-viii

Preface
pp ix-x

Solving the Pell equation
pp 1-24

Basic algorithms in number theory
pp 25-68

Smooth numbers and the quadratic sieve
pp 69-82

The number field sieve
pp 83-100

Four primality testing algorithms
pp 101-126

Lattices
pp 127-182

Elliptic curves
pp 183-208

The arithmetic of number rings
pp 209-266

Smooth numbers: computational number theory and beyond
pp 267-324

Fast multiplication and its applications
pp 325-384

Elementary thoughts on discrete logarithms
pp 385-396

The impact of the number field sieve on the discrete logarithm problem in finite fields
pp 397-420

Reducing lattice bases to find small-height values of univariate polynomials
pp 421-446

Computing Arakelov class groups
pp 447-496

Computational class field theory
pp 497-534

Protecting communications against forgery
pp 535-550

Algorithmic theory of zeta functions over finite fields
pp 551-578

Counting points onvarieties over finite fields of small characteristic
pp 579-612

Congruent number problems and their variants
pp 613-640

An introduction to computing modular forms using modular symbols
pp 641-652

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Algorithmic Number Theory

Lattices, Number Fields, Curves and Cryptography

Book description

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