Remarks on the Schoof-Elkies-Atkin algorithm
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- by L. Dewaghe PDF
- Math. Comp. 67 (1998), 1247-1252 Request permission
Abstract:
Schoof’s algorithm computes the number $m$ of points on an elliptic curve $E$ defined over a finite field ${\Bbb F}_q$. Schoof determines $m$ modulo small primes $\ell$ using the characteristic equation of the Frobenius of $E$ and polynomials of degree $O(\ell ^2)$. With the works of Elkies and Atkin, we have just to compute, when $\ell$ is a “good" prime, an eigenvalue of the Frobenius using polynomials of degree $O(\ell )$. 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 $\ell$ is “bad", we describe how to have polynomials of small degree and how to perform computations, in Schoof’s algorithm, on $x$-values only.References
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Additional Information
- L. Dewaghe
- Affiliation: Université de Lille I, UFR de Mathématiques, 59655 Villeneuve d’Ascq cedex, France
- Email: dewaghe@gat.univ-lille1.fr
- Received by editor(s): May 11, 1996
- Received by editor(s) in revised form: October 2, 1996, and February 19, 1997
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp. 67 (1998), 1247-1252
- MSC (1991): Primary 14H52, 14K02, 11Y16
- DOI: https://doi.org/10.1090/S0025-5718-98-00962-4
- MathSciNet review: 1468941