Compact elliptic curve representations

• Main Authors: Ciet Mathieu, Quisquater Jean-Jacques, Sica Francesco
• Format: Article
• Language: English
• Published: De Gruyter 2011-06-01
• Series: Journal of Mathematical Cryptology
• Subjects: elliptic curves; exponential sums; uniform distribution; cryptography
• Online Access: https://doi.org/10.1515/jmc.2011.007

Let y2 = x3 + ax + b be an elliptic curve over 𝔽p, 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) < c2p3/4 with explicit constants c1, c2 > 0.