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.