Remarks on the Schoof-Elkies-Atkin algorithm

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 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 M¨uller’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.

• Type: Journal
• Year: 1998
• Where: MOC
• Authors: L. Dewaghe
• Keywords: Algorithm, MOC 1998, Modulo Small Primes, finite field Fq