Abstract. Schoof’s algorithm computes the number m of points on an elliptic curve E defined over a finite field Fq. Schoof determines m modulo small primes ℓ using the characteristic equation of the Frobenius of E and polynomials of degree O(ℓ 2). With the works of Elkies and Atkin, we have just to compute, when ℓ is a “good ” prime, an eigenvalue of the Frobenius using polynomials of degree O(ℓ). In this article, we compute the complexity of Müller’s algorithm, which is the best known method for determining one eigenvalue and we improve the final step in some cases. Finally, when ℓ is “bad”, we describe how to have polynomials of small degree and how to perform computations, in Schoof’s algorithm, on x-values only. 1.

A prohibitive barrier faced by elliptic curve users is the difficulty of computing the curves' cardinalities. Despite recent theoretical breakthroughs, point counting still remains very cumbersome and intensively time consuming.