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.

• Introduction aimed at beginning graduate students • Extensive coverage by a large body of experts

### Contents

1. Solving Pell's equation Hendrik Lenstra; 2. Basic algorithms in number theory Joe Buhler and Stan Wagon; 3. Elliptic curves Bjorn Poonen; 4. The arithmetic of number rings Peter Stevenhagen; 5. Fast multiplication and applications Dan Bernstein; 6. Primality testing Rene Schoof; 7. Smooth numbers: computational number theory and beyond Andrew Granville; 8. Smooth numbers and the quadratic sieve Carl Pomerance; 9. The number field sieve Peter Stevenhagen; 10. Elementary thoughts on discrete logarithms Carl Pomerance; 11. The impact of the number field sieve on the discrete logarithm problem in finite fields Oliver Schirokauer; 12. Lattices Hendrik Lenstra; 13. Reducing lattices to find small-height values of univariate polynomials Dan Bernstein; 14. Protecting communications against forgery Dan Bernstein; 15. Computing Arakelov class groups Rene Schoof; 16. Computational class field theory Henri Cohen and Peter Stevenhagen; 17. Zeta functions over finite fields Daqing Wan; 18. Counting points on varieties over finite fields Alan Lauder and Daqing Wan; 19. How to get your hands on modular forms using modular symbols William Stein; 20. Congruent number problems in dimension one and two Jaap Top and Noriko Yui.

### Review

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