Computing the order of points on an elliptic curve modulo N is as difficult as factoring N

Given a square-free integer N, the group of points on an elliptic curve over the ring ZN is defined in the natural way. We prove that computing the order of points on elliptic curves over ZN is as difficult as factoring N, in the sense of randomly polynomial time reduction. Therefore, cryptosystems based on the difficulty of computing the order of points on elliptic curves over the ring ZN will be at least as robust as those based on the difficulty of factoring N.