Schoof–Elkies–Atkin algorithm

The Schoof–Elkies–Atkin algorithm (SEA) is an algorithm used for finding the order of or calculating the number of points on an elliptic curve over a finite field. Its primary application is in elliptic curve cryptography. The algorithm is an extension of Schoof's algorithm by Noam Elkies and A. O. L. Atkin to significantly improve its efficiency (under heuristic assumptions).

The Elkies-Atkin extension to Schoof's algorithm works by restricting the set of primes ${\displaystyle S=\{l_{1},\ldots ,l_{s}\}}$ considered to primes of a certain kind. These came to be called Elkies primes and Atkin primes respectively. A prime ${\displaystyle l}$ is called an Elkies prime if the characteristic equation: ${\displaystyle \phi ^{2}-t\phi +q=0}$ splits over ${\displaystyle \mathbb {F} _{l}}$, 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 ${\displaystyle \Phi _{l}(X,Y)}$ that parametrize pairs of ${\displaystyle l}$-isogenous elliptic curves in terms of their j-invariants (in practice alternative modular polynomials may also be used but for the same purpose).

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