Factoring Integers with Elliptic Curves

Abstract

This paper 19 devoted to the deacnption and analysis of a new algonthm to factor positive mtegers It depends on the use of elliptic curves The new m et b öd α obtained from Pollird's p-1-method (Proc Cambridge Philos Soc 76 (187-1), 521-528) by replacing the multiplicative group by tbe group of points on a random elliptic curve 1t u conjectured thit the algonthm determmes a non-tnvial divuor of a composite number n m expected time at most K(p)(logn)2, where p is the least pnme dmding n and K u a function for which logK(t)=v'(2+o(l))logiloElogz for t->oo In the worst case, when n la the product of two pnmes of the same order of magnitude, this is exp((l+o(l))\/lognloglogn) (for n->oo) There are several other factonng algonthms of which the conjectural expected running time is given by the latter formula However, these algonthms have a running time (hat is basically lodependent of the size of the pnme factors of n, whereas the new elliptic curve method is subatantially faster for amall p