An Approach for Computing the Number of Points on Elliptic Curve y2 = x3 + a (mod p) via Explicit Formula for That Number Modulo p

Abstract

We present an efficient approach for determining the cardinality of the set of points on each elliptic curve of the family $\mathcal{E}_p = \{E_a :y^2 = x^3 + a\,(mod\,p),a \in \mathbb{Z}_p^* \}$ by applying the famous Hasse’s bound together with an explicit formula for that cardinality reduced to modulo p which is derived by us. As a by-product it is shown that for fixed p ≡ 1 (mod 6) those cardinalities take exactly six distinct values. The latter permits us to provide a reasoning why the curve E7 employed by Bitcoin system has an optimal resistance among the curves from ℰp against the EC variant of Pohlig-Hellman algorithm when p is the actual modulo used in that system. This approach allows also to give another proof of the known result that in case p ≡ 2 (mod 3) the cardinalities of all curves from ℰp are equal to p + 1.