Counting the number of points on elliptic curves over finite fields of characteristic greater than three

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We present a variant of an algorithm of Oliver Atkin for counting the number of points on an elliptic curve over a finite field. We describe an implementation of this algorithm for prime fields. We report on the use of this implementation to count the number of points on a curve over \(\mathbb{F}\) p, where p is a 375-digit prime.