Algorithmic Number Theory

1. Front Matter

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2. Invited Talks

   1. Computing Zeta Functions via p-Adic Cohomology

      • Kiran S. Kedlaya

      Pages 1-17

   2. Using Primitive Subgroups to Do More with Fewer Bits

      • K. Rubin, A. Silverberg

      Pages 18-41

   3. Elliptic Curves of Large Rank and Small Conductor

      • Noam D. Elkies, Mark Watkins

      Pages 42-56

3. Contributed Papers

   1. Binary GCD Like Algorithms for Some Complex Quadratic Rings

      • Saurabh Agarwal, Gudmund Skovbjerg Frandsen

      Pages 57-71

   2. On the Complexity of Computing Units in a Number Field

      • V. Arvind, Piyush P. Kurur

      Pages 72-86

   3. Implementing the Arithmetic of C _3,4 Curves

      • Abdolali Basiri, Andreas Enge, Jean-Charles Faugère, Nicolas Gürel

      Pages 87-101

   4. Pseudocubes and Primality Testing

      • P. Berrizbeitia, S. Müller, H. C. Williams

      Pages 102-116

   5. Elliptic Curves with a Given Number of Points

      • Reinier Bröker, Peter Stevenhagen

      Pages 117-131

   6. Rational Divisors in Rational Divisor Classes

      • N. Bruin, E. V. Flynn

      Pages 132-139

   7. Conjectures about Discriminants of Hecke Algebras of Prime Level

      • Frank Calegari, William A. Stein

      Pages 140-152

   8. Montgomery Scalar Multiplication for Genus 2 Curves

      • Sylvain Duquesne

      Pages 153-168

   9. Improved Weil and Tate Pairings for Elliptic and Hyperelliptic Curves

      • Kirsten Eisenträger, Kristin Lauter, Peter L. Montgomery

      Pages 169-183

   10. Elliptic Curves x ³ + y ³ = k of High Rank

       • Noam D. Elkies, Nicholas F. Rogers

       Pages 184-193

   11. Proving the Primality of Very Large Numbers with fastECPP

       • J. Franke, T. Kleinjung, F. Morain, T. Wirth

       Pages 194-207

   12. A Low-Memory Parallel Version of Matsuo, Chao, and Tsujii’s Algorithm

       • Pierrick Gaudry, Éric Schost

       Pages 208-222

   13. Function Field Sieve in Characteristic Three

       • R. Granger, A. J. Holt, D. Page, N. P. Smart, F. Vercauteren

       Pages 223-234

   14. A Comparison of CEILIDH and XTR

       • R. Granger, D. Page, M. Stam

       Pages 235-249

   15. Stable Models of Elliptic Curves, Ring Class Fields, and Complex Multiplication

       • Jordi Guàrdia, Eugenia Torres, Montserrat Vela

       Pages 250-262

   16. An Algorithm for Computing Isomorphisms of Algebraic Function Fields

       • F. Hess

       Pages 263-271

The sixth Algorithmic Number Theory Symposium was held at the University of Vermont, in Burlington, from 13–18 June 2004. The organization was a joint e?ort of number theorists from around the world. There were four invited talks at ANTS VI, by Dan Bernstein of the Univ- sity of Illinois at Chicago, Kiran Kedlaya of MIT, Alice Silverberg of Ohio State University, and Mark Watkins of Pennsylvania State University. Thirty cont- buted talks were presented, and a poster session was held. This volume contains the written versions of the contributed talks and three of the four invited talks. (Not included is the talk by Dan Bernstein.) ANTS in Burlington is the sixth in a series that began with ANTS I in 1994 at Cornell University, Ithaca, New York, USA and continued at Universit´eB- deaux I, Bordeaux, France (1996), Reed College, Portland, Oregon, USA (1998), the University of Leiden, Leiden, The Netherlands (2000), and the University of Sydney, Sydney, Australia (2002). The proceedings have been published as volumes 877, 1122, 1423, 1838, and 2369 of Springer-Verlag’s Lecture Notes in Computer Science series. The organizers of the 2004 ANTS conference express their special gratitude and thanks to John Cannon and Joe Buhler for invaluable behind-the-scenes advice.

• Farey sequence
• Prime
• algebraic geometry
• algebraic number fields
• algebraic number theory
• algorithmic number theory
• algorithms
• complexity
• computational number theory
• elliptic curve computations
• factoring
• finite field computations
• number theory
• statistics
• zeta functions
• algorithm analysis and problem complexity

• Department of Computer Science and Engineering, University of South Carolina , Columbia, USA

  Duncan Buell

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