Algorithmic Number Theory: 6th International Symposium, ANTS-VI, Burlington, VT, USA, June 13-18, 2004, Proceedings (Lecture Notes in Computer Science, 3076)

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. Table of Contents Invited Talks Computing Zeta Functions via p-Adic Cohomology Using Primitive Subgroups to Do More with Fewer Bits Elliptic Curves of Large Rank and Small Conductor Contributed Papers Binary GCD Like Algorithms for Some Complex Quadratic Rings On the Complexity of Computing Units in a Number Field Implementing the Arithmetic of C[sub(,4)] Curves Pseudocubes and Primality Testing Elliptic Curves with a Given Number of Points Rational Divisors in Rational Divisor Classes Conjectures about Discriminants of Hecke Algebras of Prime Level Montgomery Scalar Multiplication for Genus 2 Curves Improved Weil and Tate Pairings for Elliptic and Hyperelliptic Curves Elliptic Curves x[sup()] + y[sup()] = k of High Rank Proving the Primality of Very Large Numbers with fastECPP A Low-Memory Parallel Version of Matsuo, Chao, and Tsujii's Algorithm Function Field Sieve in Characteristic Three A Comparison of CEILIDH and XTR Stable Models of Elliptic Curves, Ring Class Fields, and Complex Multiplication An Algorithm for Computing Isomorphisms of Algebraic Function Fields A Method to Solve Cyclotomic Norm Equations [equation omitted] Imaginary Cyclic Quartic Fields with Large Minus Class Numbers Nonic 3-adic Fields Montgomery Addition for Genus Two Curves Numerical Evaluation at Negative Integers of the Dedekind Zeta Functions of Totally Real Cubic Number Fields Salem Numbers of Trace –2 and Traces of Totally Positive Algebraic Integers Low-Dimensional Lattice Basis Reduction Revisited Computing Order Statistics in the Farey Sequence The Discrete Logarithm in Logarithmic l-Class Groups and Its Applications in K-theory Point Counting on Genus 3 Non Hyperelliptic Curves Algorithmic Aspects of Cubic Function Fields A Binary Recursive Gcd Algorithm Lagrange Resolvents Constructed from Stark Units Cryptanalysis of a Divisor Class Group Based Public-Key Cryptosystem Author Index

This book constitutes the refereed proceedings of the 6th International Algorithmic Number Theory Symposium, ANTS 2004, held in Burlington, VT, USA, in June 2004.

The 30 revised full papers presented together with 3 invited papers were carefully reviewed and selected for inclusion in the book. Among the topics addressed are zeta functions, elliptic curves, hyperelliptic curves, GCD algorithms, number field computations, complexity, primality testing, Weil and Tate pairings, cryptographic algorithms, function field sieve, algebraic function field mapping, quartic fields, cubic number fields, lattices, discrete logarithms, and public key cryptosystems.

We survey some recent applications of p-adic cohomology to machine computation of zeta functions of algebraic varieties over finite fields of small characteristic, and suggest some new avenues for further exploration.