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This article explains Schoof's approach, laying emphasis on the mathematical ideas underlying the structure of the algorithm.
Let <math>E</math> be an [[elliptic curve]] defined over the finite field <math>\mathbb{F}_{q}</math>, where <math>q=p^n</math> for <math>p</math> a prime and <math>n</math> an integer <math>\geq 1</math>. Over a field of characteristic <math>\neq 2, 3</math> an elliptic curve can be given by a (short) Weierstrass equation
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