Compact elliptic curve representations

Let y D x C ax C b be an elliptic curve over Fp , p being a prime number greater than 3, and consider a; b 2 Œ1; 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 2 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 c1p < M.p/ < c2p with explicit constants c1; c2 > 0.