### 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 |
---|---|

Pages (from-to) | 89-100 |

Number of pages | 12 |

Journal | Journal of Mathematical Cryptology |

Volume | 5 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jun 2011 |

Externally published | Yes |

### Fingerprint

### Keywords

- Cryptography
- Elliptic curves
- Exponential sums
- Uniform distribution

### ASJC Scopus subject areas

- Computer Science Applications
- Applied Mathematics
- Computational Mathematics

### Cite this

*Journal of Mathematical Cryptology*,

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

**Compact elliptic curve representations.** / Ciet, Mathieu; Quisquater, Jean Jacques; Sica, Francesco.

Research output: Contribution to journal › Article

*Journal of Mathematical Cryptology*, vol. 5, no. 1, pp. 89-100. https://doi.org/10.1515/JMC.2011.007

}

TY - JOUR

T1 - Compact elliptic curve representations

AU - Ciet, Mathieu

AU - Quisquater, Jean Jacques

AU - Sica, Francesco

PY - 2011/6

Y1 - 2011/6

N2 - 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 1p1/2 <M(p) 2p 3/4 with explicit constants c 1, c 2 > 0.

AB - 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 1p1/2 <M(p) 2p 3/4 with explicit constants c 1, c 2 > 0.

KW - Cryptography

KW - Elliptic curves

KW - Exponential sums

KW - Uniform distribution

UR - http://www.scopus.com/inward/record.url?scp=84858422907&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858422907&partnerID=8YFLogxK

U2 - 10.1515/JMC.2011.007

DO - 10.1515/JMC.2011.007

M3 - Article

VL - 5

SP - 89

EP - 100

JO - Journal of Mathematical Cryptology

JF - Journal of Mathematical Cryptology

SN - 1862-2976

IS - 1

ER -