On a theorem of Mestre and Schoof

TL;DR

This paper extends a theorem by Mestre and Schoof, showing that the order of an elliptic curve over certain finite fields can be uniquely determined with fewer point computations, broadening its applicability.

Contribution

The authors generalize the Mestre and Schoof theorem to all finite fields with q > 49 and prime fields with q > 29, reducing the field size constraints.

Findings

Order of elliptic curves can be determined with minimal point computations.

The theorem's applicability is extended to smaller finite fields.

Unique determination of elliptic curve order is possible under new conditions.

Abstract

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.