schoof

Schoof is a surname of Dutch origin. In mathematics the name is most prominently associated with Schoof's algorithm for counting points on elliptic curves over finite fields, a foundational result in computational number theory.

Schoof's algorithm, introduced by René Schoof, determines the number of F_q-rational points on an elliptic curve E defined over a finite field F_q. The problem is to compute |E(F_q)|, the order of the group of points. The method analyzes the action of the Frobenius endomorphism on the ℓ-torsion subgroup E[ℓ] for small primes ℓ not equal to the characteristic of F_q. By computing congruences for the trace of Frobenius modulo each ℓ and combining them via the Chinese remainder theorem, the exact point count can be recovered.

The original approach was later refined and accelerated by Elkies and Atkin, yielding the Schoof–Elkies–Atkin (SEA) algorithm. SEA uses isogenies of small degree and modular polynomials to improve efficiency, making practical point counting feasible for a broad range of finite fields. This development has had substantial impact in computational number theory and cryptography, where knowing the exact group order of an elliptic curve is important for selecting secure parameters and validating curve properties.

Beyond mathematics, Schoof serves as a surname for various individuals. In mathematical literature, the algorithm bearing the name Schoof is the most notable association with the term.