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JKL-ECM: an implementation of ECM using Hessian curves

Part of: Arithmetic algebraic geometry Computational number theory Curves

Published online by Cambridge University Press:  26 August 2016

Henriette Heer ,
Gary McGuire  and
Oisín Robinson
Henriette Heer
Affiliation:
Faculty of Mathematics, Technical University of Kaiserslautern, PO Box 3049, D-67653 Kaiserslautern, Germany email heer@rhrk.uni-kl.de
Gary McGuire
Affiliation:
School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland email gary.mcguire@ucd.ie
Oisín Robinson
Affiliation:
School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland email oisin.robinson@ucdconnect.ie

Abstract

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We present JKL-ECM, an implementation of the elliptic curve method of integer factorization which uses certain twisted Hessian curves in a family studied by Jeon, Kim and Lee. This implementation takes advantage of torsion subgroup injection for families of elliptic curves over a quartic number field, in addition to the ‘small parameter’ speedup. We produced thousands of curves with torsion $\mathbb{Z}/6\mathbb{Z}\oplus \mathbb{Z}/6\mathbb{Z}$ and small parameters in twisted Hessian form, which admit curve arithmetic that is ‘almost’ as fast as that of twisted Edwards form. This allows JKL-ECM to compete with GMP-ECM for finding large prime factors. Also, JKL-ECM, based on GMP, accepts integers of arbitrary size. We classify the torsion subgroups of Hessian curves over $\mathbb{Q}$ and further examine torsion properties of the curves described by Jeon, Kim and Lee. In addition, the high-performance curves with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/8\mathbb{Z}$ of Bernstein et al. are completely recovered by the $\mathbb{Z}/4\mathbb{Z}\oplus \mathbb{Z}/8\mathbb{Z}$ family of Jeon, Kim and Lee, and hundreds more curves are produced besides, all with small parameters and base points.

MSC classification

Primary: 11Y05: Factorization
Secondary: 11G05: Elliptic curves over global fields 14H52: Elliptic curves

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