On a theorem of Mestre and Schoof
John E. Cremona1; Andrew V. Sutherland2
1 Mathematics Institute University of Warwick Coventry CV4 7AL UK
2 Massachusetts Institute of Technology Department of Mathematics 77 Massachusetts Avenue Cambridge, MA 02139-4307 USA
Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 353-358.

A well known theorem of Mestre and Schoof implies that the order of an elliptic curve E over a prime field 𝔽 q can be uniquely determined by computing the orders of a few points on E and its quadratic twist, provided that q>229. We extend this result to all finite fields with q>49, and all prime fields with q>29.

Un théorème bien connu de Mestre et Schoof implique que la cardinalité d’une courbe elliptique E définie sur un corps premier 𝔽 q peut être déterminée de manière univoque en calculant les ordres de quelques points sur E et sur sa tordue quadratique, à condition que q>229. Nous étendons ce résultat à tous les corps finis avec q>49, et tous les corps premiers avec q>29.

DOI: 10.5802/jtnb.719

John E. Cremona 1; Andrew V. Sutherland 2

1 Mathematics Institute University of Warwick Coventry CV4 7AL UK
2 Massachusetts Institute of Technology Department of Mathematics 77 Massachusetts Avenue Cambridge, MA 02139-4307 USA 
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John E. Cremona; Andrew V. Sutherland. On a theorem of Mestre and Schoof. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 353-358. doi : 10.5802/jtnb.719. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.719/

[1] René Schoof, Counting points on elliptic curves over finite fields. Journal de Théorie des Nombres de Bordeaux 7 (1995), 219–254. | Numdam | MR | Zbl

[2] Andrew V. Sutherland, Order computations in generic groups. PhD thesis, M.I.T., 2007, available at http://groups.csail.mit.edu/cis/theses/sutherland-phd.pdf. | MR

[3] Lawrence C. Washington, Elliptic curves: Number theory and cryptography, 2nd ed. CRC Press, 2008. | MR | Zbl

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