Finding ECM-friendly curves through a study of Galois properties
This work addresses the efficiency of integer factorization algorithms, which is crucial for cryptography and number theory, but it appears incremental as it builds on existing curve families like Montgomery or Edwards curves.
The paper tackled the problem of improving the success probability of the elliptic curve method (ECM) for integer factorization by proving divisibility properties of elliptic curve cardinalities modulo primes, leading to the discovery of new families of curves with good division properties.
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.