Vol. 1, No. 1, 2013

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Finding ECM-friendly curves through a study of Galois properties

Razvan Barbulescu, Joppe W. Bos, Cyril Bouvier, Thorsten Kleinjung and Peter L. Montgomery

Vol. 1 (2013), No. 1, 63–86
Abstract

We prove some divisibility properties of the cardinality of elliptic curve groups modulo primes. These proofs explain the good behavior of certain parameters when using Montgomery or Edwards curves in the setting of the elliptic curve method (ECM) for integer factorization. The ideas behind the proofs help us to find new infinite families of elliptic curves with good division properties increasing the success probability of ECM.

Keywords
elliptic curve method (ECM), Edwards curves, Montgomery curves, torsion properties, Galois groups
Mathematical Subject Classification 2010
Primary: 14H52
Secondary: 11Y05
Milestones
Published: 14 November 2013
Authors
Razvan Barbulescu
Université de Lorraine
LORIA - Bât. A
Équipe CARAMEL
Campus Scientifique, BP 239
54506 Vandoeuvre-lès-Nancy
France
Joppe W. Bos
Microsoft Research
One Microsoft Way
Redmond, WA 98052
United States
Cyril Bouvier
ENS Paris and Université de Lorraine
LORIA - Bât. A
Équipe CARAMEL
Campus Scientifique, BP 239
54506 Vandoeuvre-lès-Nancy
France
Thorsten Kleinjung
EPFL
Laboratory for Cryptologic Algorithms
CH-1015 Lausanne
Switzerland
Peter L. Montgomery
Microsoft Research
One Microsoft Way
Redmond, WA 98052
United States