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Fast heuristic algorithms for computing relations in the class group of a quadratic order, with applications to isogeny evaluation

Part of: Algebraic number theory: global fields Abelian varieties and schemes Computational number theory

Published online by Cambridge University Press:  26 August 2016

Jean-François Biasse ,
Claus Fieker  and
Michael J. Jacobson Jr
Jean-François Biasse
Affiliation:
University of South Florida, USA email biasse@usf.edu
Claus Fieker
Affiliation:
University of Kaiserslautern, Germany email fieker@mathematik.uni-kl.de
Michael J. Jacobson Jr
Affiliation:
University of Calgary, Canada email jacobs@ucalgary.ca

Abstract

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In this paper, we present novel algorithms for finding small relations and ideal factorizations in the ideal class group of an order in an imaginary quadratic field, where both the norms of the prime ideals and the size of the coefficients involved are bounded. We show how our methods can be used to improve the computation of large-degree isogenies and endomorphism rings of elliptic curves defined over finite fields. For these problems, we obtain improved heuristic complexity results in almost all cases and significantly improved performance in practice. The speed-up is especially high in situations where the ideal class group can be computed in advance.

MSC classification

Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11Y16: Algorithms; complexity 11R11: Quadratic extensions 14K02: Isogeny

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