Group structure of elliptic curves over ℤ/Nℤ

Abstract

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

Keywords

• group structure
• elliptic curves
• ecdlp
• 11t71
• 13b25
• 14h52