Tenth Algorithmic Number Theory Symposium ANTS-X University of California, San Diego July 9 – 13, 2012 Tenth Algorithmic Number Theory Symposium (ANTS-X) July 9 – 13, 2012

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

Jean-François Biasse and Claus Fieker

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 speed-up of an order of magnitude is achieved with respect to the standard methods.

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