#44:
Algorithmic Number Theory
**Up:** MSRI Publications

# MSRI Publications – Volume 44

## Algorithmic Number Theory

Lattices, Number Fields, Curves and Cryptography

### Edited by
Joseph P. Buhler and
Peter Stevenhagen

### Contents

Front matter (front page, copyright page)

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Table of Contents

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Preface, ix-x

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Solving the Pell equation
by **Hendrik W. Lenstra, Jr.**, 1-23

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Basic algorithms in number theory
by **Joe Buhler and Stan Wagon**, 25-68

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Smooth numbers and the quadratic sieve
by **Carl Pomerance**, 69-81

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The number field sieve
by **Peter Stevenhagen**, 83-100

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Four primality testing algorithms
by **René Schoof**, 101-126

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Lattices
by **Hendrik W. Lenstra, Jr.**, 127-181

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Elliptic curves
by **Bjorn Poonen**, 183-207

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The arithmetic of number rings
by **Peter Stevenhagen**, 209-266

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Smooth numbers: computational number theory and beyond
by **Andrew Granville**, 267-323

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Fast multiplication and its applications
by **Daniel J. Bernstein**, 325-384

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Elementary thoughts on discrete logarithms
by **Carl Pomerance**, 385-396

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The impact of the number field sieve on the discrete logarithm problem in finite fields
by **Oliver Schirokauer**, 397-420

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Reducing lattice bases to find small-height values of univariate polynomials
by **Daniel J. Bernstein**, 421-446

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Computing Arakelov class groups
by **René Schoof**, 447-495

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Computational class field theory
by **Henri Cohen and Peter Stevenhagen**, 497-534

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Protecting communications against forgery
by **Daniel J. Bernstein**, 535-549

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Algorithmic theory of zeta functions over finite fields
by **Daqing Wan**, 551-578

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Counting points on varieties over finite fields of small characteristic
by **Alan G. B. Lauder and Daqing Wan**, 579-612

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Congruent number problems and their variants
by **Jaap Top and Noriko Yui**, 613-639

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An introduction to computing modular forms using modular symbols
by **William A. Stein**, 641-652

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