Compact elliptic curve representations
(2011) Journal of Mathematical Cryptology — Vol. 5, n° 1, p. 89-100 (2011)
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- Let y <sup>2</sup> = x <sup>3</sup> + ax + b be an elliptic curve over F; <inf>p</inf>, 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 <inf>1</inf>p1/2 < M(p) < c <inf>2</inf>p <sup>3/4</sup> with explicit constants c <inf>1</inf>, c <inf>2</inf> > 0. © de Gruyter 2011.
Citations
Ciet, M., Quisquater, J.-J., & Sica, F. (2011). Compact elliptic curve representations. Journal of Mathematical Cryptology, 5(1), 89-100. https://doi.org/10.1515/JMC.2011.007 (Original work published 2011)