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New Families of ECM Curves for Cunningham Numbers

  • Éric Brier
  • Christophe Clavier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)

Abstract

In this paper we study structures related to torsion of elliptic curves defined over number fields. The aim is to build families of elliptic curves more efficient to help factoring numbers of special form, including numbers from the Cunningham Project. We exhibit a family of curves with rational ℤ/4ℤ×ℤ/4ℤ torsion and positive rank over the field ℚ(ζ 8) and a family of elliptic curves with rational ℤ/6ℤ×ℤ/3ℤ torsion and positive rank over the field ℚ(ζ 3). These families have been used in finding new prime factors for the numbers 2972 + 1 and 21048 + 1. Along the way, we classify and give a parameterization of modular curves for some torsion subgroups.

Keywords

Elliptic Curve Elliptic Curf Cyclic Subgroup Torsion Group Modular Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Éric Brier
    • 1
  • Christophe Clavier
    • 2
    • 3
  1. 1.Ingenico S.A.Guilherand-GrangesFrance
  2. 2.Institut d’Ingénierie Informatique de Limoges (3iL)Limoges
  3. 3.Département de Mathématiques et InformatiqueUniversité de Limoges – XLIMLimoges

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