Remarks on the Schoof-Elkies-Atkin algorithm

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 ` using the characteristic equation of the Frobenius of E and polynomials of degree O(`2). With the works of Elkies and Atkin, we have just to compute, when ` is a “good” prime, an eigenvalue of the Frobenius using polynomials of degree O(`). 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 ` is “bad”, we describe how to have polynomials of small degree and how to perform computations, in Schoof’s algorithm, on x-values only.