Group structure of elliptic curves over ℤ/Nℤ

Abstract

Abstract We characterize the possible groups E ( Z ∕ N Z ) E\left({\mathbb{Z}}/N{\mathbb{Z}}) arising from elliptic curves over Z ∕ N Z {\mathbb{Z}}/N{\mathbb{Z}} in terms of the groups E ( F p ) E\left({{\mathbb{F}}}_{p}) , with p p varying among the prime divisors of N N . This classification is achieved by showing that the infinity part of any elliptic curves over Z ∕ p e Z {\mathbb{Z}}/{p}^{e}{\mathbb{Z}} is a Z ∕ p e Z {\mathbb{Z}}/{p}^{e}{\mathbb{Z}} -torsor, of which a generator is exhibited. As a first consequence, when E ( Z ∕ N Z ) E\left({\mathbb{Z}}/N{\mathbb{Z}}) is a p p -group, we provide an explicit and sharp bound on its rank. As a second consequence, when N = p e N={p}^{e} is a prime power and the projected curve E ( F p ) E\left({{\mathbb{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.