Group structure of elliptic curves over Z/NZ

We characterize the possible groups E(Z/NZ)arising from elliptic curves over Z/NZ in terms of the groups E(F-p), with p varying among the prime divisors of N. This classification is achieved by showing that the infinity part of any elliptic curves over Z/p(e)Z is a Z/p(e)Z-torsor, of which a generator is exhibited. As a first consequence, when E(Z/NZ) is a p -group, we provide an explicit and sharp bound on its rank. As a second consequence, when N=p(e) is a prime power and the projected curve E(F-p) has trace one, we provide an isomorphism attack to the elliptic curve discrete logarithm problem, which works only by means of finite ring arithmetic.