Algorithmic Number Theory

Invited Talks.- Gauss Composition and Generalizations.- Elliptic Curves -- The Crossroads of Theory and Computation.- The Weil and Tate Pairings as Building Blocks for Public Key Cryptosystems.- Using Elliptic Curves of Rank One towards the Undecidability of Hilbert's Tenth Problem over Rings of Algebraic Integers.- On p-adic Point Counting Algorithms for Elliptic Curves over Finite Fields.- Number Theory.- On Arithmetically Equivalent Number Fields of Small Degree.- A Survey of Discriminant Counting.- A Higher-Rank Mersenne Problem.- An Application of Siegel Modular Functions to Kronecker's Limit Formula.- Computational Aspects of NUCOMP.- Efficient Computation of Class Numbers of Real Abelian Number Fields.- An Accelerated Buchmann Algorithm for Regulator Computation in Real Quadratic Fields.- Arithmetic Geometry.- Some Genus 3 Curves with Many Points.- Trinomials ax 7 + bx + c and ax 8 + bx + c with Galois Groups of Order 168 and 8 · 168.- Computations on Modular Jacobian Surfaces.- Integral Points on Punctured Abelian Surfaces.- Genus 2 Curves with (3, 3)-Split Jacobian and Large Automorphism Group.- Transportable Modular Symbols and the Intersection Pairing.- Elliptic Curves and CM.- Action of Modular Correspondences around CM Points.- Curves Dy 2 = x 3 -- x of Odd Analytic Rank.- Comparing Invariants for Class Fields of Imaginary Quadratic Fields.- A Database of Elliptic Curves -- First Report.- Point Counting.- Isogeny Volcanoes and the SEA Algorithm.- Fast Elliptic Curve Point Counting Using Gaussian Normal Basis.- An Extension of Kedlaya's Algorithm to Artin-Schreier Curves in Characteristic 2.- Cryptography.- Implementing the Tate Pairing.- Smooth Orders and Cryptographic Applications.- Chinese Remaindering for Algebraic Numbers in a Hidden Field.-Function Fields.- An Algorithm for Computing Weierstrass Points.- New Optimal Tame Towers of Function Fields over Small Finite Fields.- Periodic Continued Fractions in Elliptic Function Fields.- Discrete Logarithms and Factoring.- Fixed Points and Two-Cycles of the Discrete Logarithm.- Random Cayley Digraphs and the Discrete Logarithm.- The Function Field Sieve Is Quite Special.- MPQS with Three Large Primes.- An Improved Baby Step Giant Step Algorithm for Point Counting of Hyperelliptic Curves over Finite Fields.- Factoring N = pq 2 with the Elliptic Curve Method.- Gröbner Bases.- A New Scheme for Computing with Algebraically Closed Fields.- Complexity.- Additive Complexity and Roots of Polynomials over Number Fields and -adic Fields.