“Chinese & Match”, an alternative to Atkin’s “Match and Sort” method used in the SEA algorithm

Abstract:

A classical way to compute the number of points of elliptic curves defined over finite fields from partial data obtained in SEA (Schoof Elkies Atkin) algorithm is a so-called “Match and Sort” method due to Atkin. This method is a “baby step/giant step” way to find the number of points among $C$ candidates with $O(\sqrt {C})$ elliptic curve additions. Observing that the partial information modulo Atkin’s primes is redundant, we propose to take advantage of this redundancy to eliminate the usual elliptic curve algebra in this phase of the SEA computation. This yields an algorithm of similar complexity, but the space needed is smaller than what Atkin’s method requires. In practice, our technique amounts to an acceleration of Atkin’s method, allowing us to count the number of points of an elliptic curve defined over $\mathbb {F}_{2^{1663}}$. As far as we know, this is the largest point-counting computation to date. Furthermore, the algorithm is easily parallelized.