The group structure of elliptic curves over Z/NZ

TL;DR

This paper classifies the possible groups arising from elliptic curves over Z in terms of the groups E(\mathbb{Z}/N\mathbb {Z}), and provides an isomorphism attack to the ECDLP, which works only by means of finite rings arithmetic rather than involved methods.

Abstract

We characterize the possible groups $E(\mathbb{Z}/N\mathbb{Z})$ arising from elliptic curves over $\mathbb{Z}/N\mathbb{Z}$ in terms of the groups $E(\mathbb{F}_p)$, with $p$ varying among the prime divisors of $N$. This classification is achieved by showing that the infinity part of any elliptic curve over $\mathbb{Z}/p^e\mathbb{Z}$ is a $\mathbb{Z}/p^e\mathbb{Z}$-torsor, of which a generator is exhibited. As a first consequence, when $E(\mathbb{Z}/N\mathbb{Z})$ 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 $E(\mathbb{F}_p)$ has trace one, we provide an isomorphism attack to the ECDLP, which works only by means of finite rings arithmetic.

Theorems & Definitions (22)

• definition 1: Primitivity
• definition 2: Minor ideal
• definition 3: Strong rank
• lemma 1
• definition 4: Projective $n$-space
• definition 5: Elliptic curve over $R$
• lemma 2
• proposition 1: Washington, Corollary 2.32
• lemma 3: ECNTA, Section 4
• proposition 2