### Abstract

Let y ^{2} = x ^{3} + ax + b be an elliptic curve over F; _{p}, p being a prime number greater than 3, and consider a; b ∈ [1, p]. In this paper, we study elliptic curve isomorphisms, with a view towards reduction in the size of elliptic curves coefficients. We first consider reducing the ratio a/b. We then apply these considerations to determine the number of elliptic curve isomorphism classes. Later we work on both coefficients. We introduce the number M.p/as the lower bound of all M ∈ ℕ such that each isomorphism class has a representative with max (a,b) < M. Using results from the theory of uniform distributions, we prove upper and lower bounds of the form c _{1}p1/2 < M(p) < c _{2}p ^{3/4} with explicit constants c _{1}, c _{2} > 0.

Original language | English |
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Pages (from-to) | 89-100 |

Number of pages | 12 |

Journal | Journal of Mathematical Cryptology |

Volume | 5 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jun 1 2011 |

### Keywords

- Cryptography
- Elliptic curves
- Exponential sums
- Uniform distribution

### ASJC Scopus subject areas

- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

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## Cite this

*Journal of Mathematical Cryptology*,

*5*(1), 89-100. https://doi.org/10.1515/JMC.2011.007