Schoof's algorithm

The number of points on a specific curve can be computed with Schoof's algorithm.

Schoof's algorithm stores the values of in a variable for each prime considered.

This is heavily used in Schoof's algorithm for counting points on elliptic curves.

They play a central role in the study of counting points on elliptic curves in Schoof's algorithm.

Elkies primes are the key to improving the asymptotic complexity of Schoof's algorithm.

Schoof's algorithm implementation for with prime .

Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields.

For Elkies primes, this allows one to compute the number of points on modulo more efficiently than in Schoof's algorithm.

Atkin and Morain state "the problem with GK is that Schoof's algorithm seems almost impossible to implement.

The polynomial is a divisor of the corresponding division polynomial used in Schoof's algorithm, and it has significantly lower degree, versus .

G. Musiker: Schoof's Algorithm for Counting Points on .

Given that this computation needs to be carried out for each of the primes, the total complexity of Schoof's algorithm turns out to be .

Central to Schoof's algorithm are the use of division polynomials and Hasse's theorem, along with the Chinese remainder theorem.

In any case Schoof's algorithm is most frequently used in addressing the case since there are more efficient, so called adic algorithms for small characteristic fields.

The Elkies-Atkin extension to Schoof's algorithm works by restricting the set of primes considered to primes of a certain kind.

Atkin, along with Noam Elkies, extended Schoof's algorithm to create the Schoof-Elkies-Atkin algorithm.

The running time of Schoof's Algorithm is polynomial in , with an asymptotic complexity of , where denotes the complexity of multiplication.

The resulting algorithm is probabilistic (of Las Vegas type), and its expected running time is, heuristically, , making it more efficient in practice than Schoof's algorithm.

It is very slow and cumbersome to count all of the points on E using Schoof's algorithm, which is the preferred algorithm for the Goldwasser-Kilian algorithm.

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 modification of Schoof's algorithm to use Elkies and Atkin primes is known as the Schoof-Elkies-Atkin (SEA) algorithm.

Before Schoof's algorithm, approaches to counting points on elliptic curves such as the naive and baby-step giant-step algorithms were, for the most part, tedious and had an exponential running time.

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).

Unlike Schoof's algorithm, the SEA algorithm is typically implemented as a probabilistic algorithm (of the Las Vegas type), so that root-finding and other operations can be performed more efficiently.

Next we need an algorithm to count the number of points on E. Applied to E, this algorithm (Koblitz and others suggest Schoof's algorithm) produces a number m which is the number of points on curve E over F, provided N is prime.