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John E. Cremona; Andrew V. Sutherland
On a theorem of Mestre and Schoof
Journal de théorie des nombres de Bordeaux, 22 no. 2 (2010), p. 353-358, doi: 10.5802/jtnb.719
Article PDF | Reviews MR 2769066 | Zbl 1223.11072

Résumé - Abstract

A well known theorem of Mestre and Schoof implies that the order of an elliptic curve $E$ over a prime field $\mathbb{F}_q$ 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 $q>29$.

Bibliography

[1] René Schoof, Counting points on elliptic curves over finite fields. Journal de Théorie des Nombres de Bordeaux 7 (1995), 219–254. Cedram |  MR 1413578 |  Zbl 0852.11073
[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 2717420
[3] Lawrence C. Washington, Elliptic curves: Number theory and cryptography, 2nd ed. CRC Press, 2008.  MR 2404461 |  Zbl 1200.11043