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| Main Authors: | , |
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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2010.15543 |
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| _version_ | 1866916150663708672 |
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| author | Sala, Massimiliano Taufer, Daniele |
| author_facet | Sala, Massimiliano Taufer, Daniele |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_15543 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The group structure of elliptic curves over Z/NZ Sala, Massimiliano Taufer, Daniele Number Theory Algebraic Geometry 11T71, 13B25, 14H52 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. |
| title | The group structure of elliptic curves over Z/NZ |
| topic | Number Theory Algebraic Geometry 11T71, 13B25, 14H52 |
| url | https://arxiv.org/abs/2010.15543 |