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with <math>A,B\in \mathbb{F}_{q}</math>. The set of points defined over <math>\mathbb{F}_{q}</math> consists of the solutions <math>(a,b)\in\mathbb{F}_{q}^2</math> satisfying the curve equation and a [[point at infinity]] <math>O</math>. Using the [[Elliptic_curve#The_group_law|group law]] on elliptic curves restricted to this set one see that this set <math>E(\mathbb{F}_{q})</math> forms an [[abelian group]], with <math>O</math> acting as the zero element.
In order to count points on an elliptic curve, we compute the cardinality of <math>E(\mathbb{F}_{q})</math>.
Schoof's approach to computing the cardinality <math>\sharp E(\mathbb{F}_{q})</math> makes use of [[Hasse's theorem on elliptic curves]] along with the [[Chinese remainder theorem]] and [[division polynomials]].
==Hasse's Theorem==
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