论文标题
Z/NZ上椭圆曲线的组结构
The group structure of elliptic curves over Z/NZ
论文作者
论文摘要
我们表征可能的组$ e（\ mathbb {z}/n \ mathbb {z}）$，是由$ \ mathbb {z}/n \ mathbb {z} $组成的$ e（\ mathbb {f} _p）$的组中的$ e（mathbb {z}/n \ mathbb {z} $，以及$ p $ n $ n $ pprime的组。通过证明$ \ mathbb {z}/p^e \ mathbb {z} $是$ \ mathbb {z}/p^e \ mathbb {z} $ - torsor，在其中展示了生成器。首先，当$ e（\ mathbb {z}/n \ mathbb {z}）$是$ p $ -group时，我们提供了明确而敏锐的界限。第二个结果，当$ n = p^e $是主要功率，而投影的曲线$ e（\ mathbb {f} _p）$具有跟踪，我们为ECDLP提供了同构攻击，它仅通过有限环Arithmetic使用。
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.
