Vol. 1, No. 1, 2013

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Improved techniques for computing the ideal class group and a system of fundamental units in number fields

Jean-François Biasse and Claus Fieker

Vol. 1 (2013), No. 1, 113–133
Abstract

We describe improvements to the subexponential methods for computing the ideal class group, the regulator and a system of fundamental units in number fields under the generalized Riemann hypothesis. We use sieving techniques adapted from the number field sieve algorithm to derive relations between elements of the ideal class group, and $p$-adic approximations to manage the loss of precision during the computation of units. These improvements are particularly efficient for number fields of small degree for which a speedup of an order of magnitude is achieved with respect to the standard methods.

Keywords
number fields, ideal class group, regulator, units, index calculus, subexponentiality
Mathematical Subject Classification 2000
Primary: 54C40, 14E20
Secondary: 46E25, 20C20
Milestones
Published: 14 November 2013
Authors
 Jean-François Biasse Department of Mathematics and Statistics University of Calgary 2500 University Drive NW Calgary, AB T2N 1N4 Canada Claus Fieker Fachbereich Mathematik Universität Kaiserslautern Postfach 3049 D-67653 Kaiserslautern Germany