Elliptic curves over finite fields and the computation of square roots mod 𝑝

Abstract

In this paper we present a deterministic algorithm to compute the number of F q {{\mathbf {F}}_q} -points of an elliptic curve that is defined over a finite field F q {{\mathbf {F}}_q} and which is given by a Weierstrass equation. The algorithm takes O ( log 9 q ) O({\log ^9}q) elementary operations. As an application we give an algorithm to compute square roots mod p \bmod p . For fixed x ∈ Z x \in {\mathbf {Z}} , it takes O ( log 9 p ) O({\log ^9}p) elementary operations to compute x mod p \sqrt x \bmod p .