Equivalence of counting the number of points on elliptic curve over the ring zn and factoring n

Abstract

For composite n, we prove that counting the number of points on elliptic curves over the ring Zn is randomly computationally equivalent to factoring n. That is, we prove that if we can count it, we can easily factor n. Furthermore, we also prove that if we can solve the elliptic curve discrete logarithm problem modulo n, we can easily factor n.