Abstract
Let y2 = x3 + 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 c1p1/2 < M(p) < c2p 3/4 with explicit constants c1, c2 > 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 2011 |
Keywords
- Cryptography
- Elliptic curves
- Exponential sums
- Uniform distribution
ASJC Scopus subject areas
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics