Algorithmic Number Theory

Chapter 1 Gauss Composition and Generalizations

Chapter 2 Elliptic Curves — The Crossroads of Theory and Computation

Chapter 3 The Weil and Tate Pairings as Building Blocks for Public Key Cryptosystems

Chapter 4 Using Elliptic Curves of Rank One towards the Undecidability of Hilbert’s Tenth Problem over Rings of Algebraic Integers

Chapter 5 On p -adic Point Counting Algorithms for Elliptic Curves over Finite Fields

Chapter 6 On Arithmetically Equivalent Number Fields of Small Degree

Chapter 7 A Survey of Discriminant Counting

Chapter 8 A Higher-Rank Mersenne Problem

Chapter 9 An Application of Siegel Modular Functions to Kronecker’s Limit Formula

Chapter 10 Computational Aspects of NUCOMP

Chapter 11 Efficient Computation of Class Numbers of Real Abelian Number Fields

Chapter 12 An Accelerated Buchmann Algorithm for Regulator Computation in Real Quadratic Fields

Chapter 13 Some Genus 3 Curves with Many Points

Chapter 14 Trinomials ax 7 + bx + c and ax 8 + bx + c with Galois Groups of Order 168 and 8 · 168

Chapter 15 Computations on Modular Jacobian Surfaces

Chapter 16 Integral Points on Punctured Abelian Surfaces

Chapter 17 Genus 2 Curves with (3, 3)-Split Jacobian and Large Automorphism Group

Chapter 18 Transportable Modular Symbols and the Intersection Pairing

Chapter 19 Action of Modular Correspondences around CM Points

Chapter 20 Curves Dy 2 = x 3 — x of Odd Analytic Rank

Chapter 21 Comparing Invariants for Class Fields of Imaginary Quadratic Fields

Chapter 22 A Database of Elliptic Curves — First Report

Chapter 23 Isogeny Volcanoes and the SEA Algorithm

Chapter 24 Fast Elliptic Curve Point Counting Using Gaussian Normal Basis

Chapter 25 An Extension of Kedlaya’s Algorithm to Artin-Schreier Curves in Characteristic 2

Chapter 26 Implementing the Tate Pairing

Chapter 27 Smooth Orders and Cryptographic Applications

Chapter 28 Chinese Remaindering for Algebraic Numbers in a Hidden Field

Chapter 29 An Algorithm for Computing Weierstrass Points

Chapter 30 New Optimal Tame Towers of Function Fields over Small Finite Fields

Chapter 31 Periodic Continued Fractions in Elliptic Function Fields

Chapter 32 Fixed Points and Two-Cycles of the Discrete Logarithm

Chapter 33 Random Cayley Digraphs and the Discrete Logarithm

Chapter 34 The Function Field Sieve Is Quite Special

Chapter 35 MPQS with Three Large Primes

Chapter 36 An Improved Baby Step Giant Step Algorithm for Point Counting of Hyperelliptic Curves over Finite Fields

Chapter 37 Factoring N = pq 2 with the Elliptic Curve Method

Chapter 38 A New Scheme for Computing with Algebraically Closed Fields

Chapter 39 Additive Complexity and Roots of Polynomials over Number Fields and \( \mathfrak{p} \) -adic Fields