**Approaches To Counting Points On Elliptic Curves**

There are several approaches to the problem. Beginning with the naive approach, we trace the developments up to Schoof's definitive work on the subject, while also listing the improvements to Schoof's algorithm made by Elkies (1990) and Atkin (1992).

Several algorithms make use of the fact that groups of the form are subject to an important theorem due to Hasse, that bounds the number of points to be considered.

Hasse's theorem Let *E* be an elliptic curve over the finite field . Then the order of satisfies

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