Finding ECM-friendly curves through a study of Galois properties
Razvan Barbulescu (INRIA Nancy - Grand Est / LORIA), Joppe W. Bos, (LACAL), Cyril Bouvier (INRIA Nancy - Grand Est / LORIA), Thorsten Kleinjung, (LACAL), Peter L. Montgomery
TL;DR
This paper investigates divisibility properties of elliptic curves modulo primes, explaining why Montgomery and Edwards curves perform well in ECM, and introduces new elliptic curve families with improved properties for factorization.
Contribution
It provides new divisibility results and constructs novel elliptic curve families that enhance the effectiveness of the elliptic curve method for integer factorization.
Findings
Divisibility properties of elliptic curve cardinalities established
Explanation of good parameters for Montgomery and Edwards curves in ECM
New elliptic curve families with improved division properties
Abstract
In this paper we prove some divisibility properties of the cardinality of elliptic curves 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 of the proofs help us to find new families of elliptic curves with good division properties which increase the success probability of ECM.
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Taxonomy
TopicsCryptography and Residue Arithmetic · Algebraic Geometry and Number Theory · Polynomial and algebraic computation