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In this paper we present a deterministic algorithm to compute the number of F^-points of an elliptic curve that is defined over a finite field Fv and which is given by a Weierstrass equation. The algorithm takes 0(log9 q) elementary operations. As an application wc give an algorithm to compute square roots mod p. For fixed .i e Z, it takes 0(log9p) elementary operations to compute fx mod p. 1. Introduction. In this paper we present an algorithm to compute the number of F(/-points of an elliptic curve defined over a finite field F , which is given by a Weierstrass equation. We restrict ourselves to the case where the characteristic of F^ is not 2 or 3. The algorithm is deterministic, does not depend on any unproved hypotheses and takes 0(log9 0. If one applies fast multiplication techniques, the algorithm will take 0((|x|1/2log p)6+f) elementary operations for any e > 0. Let £ be an elliptic curve defined over the prime field Fp and let an affine model of it be given by a Weierstrass equation Y2 = X3 + AX + B (A,BeFp). An explicit formula for the number of F^-points on £ is given by

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