Elliptic Curve Method for Integer Factorization on Parallel Architectures

The elliptic curve method (ECM) for integer factorization is an algorithm that uses the algebraic structure of the set of points of an elliptic curve for factoring integers. The running time of ECM depends on the size of the smallest prime divisor of the number to be factored. One of its main applications is the co-factorization step in the number field sieve algorithm that is used for assessing the security of the RSA cryptosystem. The principal goal emphasized in this proposal is the efficient implementation of ECM on highly parallel low-cost devices, like graphics cards. This requires theoretical and practical study of parallel algorithms for elliptic curve and finite field arithmetic.