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==Improvements to Schoof's algorithm==
In the 1990s, [[Noam Elkies]], followed by [[A. O. L. Atkin]], devised improvements to Schoof's basic algorithm by restricting the set of primes <math>S = \{l_1, \ldots, l_s\}</math> considered before to primes of a certain kind. These came to be called Elkies primes and Atkin primes respectively. A prime <math>l</math> is called an Elkies prime if the characteristic equation: <math>\phi^2-t\phi+ q = 0</math> splits over <math>\mathbb{F}_l</math>, while an Atkin prime is a prime that is not an Elkies prime. Atkin showed how to combine information obtained from the Atkin primes with the information obtained from Elkies primes to produce an efficient algorithm, which came to be known as the [[Schoof–Elkies–Atkin algorithm]]. The first problem to address is to determine whether a given prime is Elkies or Atkin. In order to do so, we make use of modular polynomials, which come from the study of [[modular forms]] and an interpretation of [[Elliptic curve#Elliptic curves over the complex numbers|elliptic curves over the complex numbers]] as lattices. Once we have determined which case we are in, instead of using [[division polynomials]], we proceed by working modulo the modular polynomials <math>f_l</math> which have a lower degree than the corresponding division polynomial <math>\psi_l</math> (degree <math>O(l)</math> rather than <math>O(l^2)</math>). Under the heuristic assumption that approximately half of the primes up to <math>log q</math> are Elkies primes, this yields an algorithm that is more efficient than Schoof's, with complexity <math>O(\log^6 q)</math> using naive arithmetic, and <math>\tilde{O}(\log^4 q)</math> using fast arithmetic. It should be noted that while this heuristic assumption is known to be hold for most elliptic curves, it is not known to hold in every case (even under the [[Generalized Riemann Hypothesis|GRH]]).
==Implementations==
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