The counting function for Elkies primes

Abstract

Let E be an elliptic curve over a finite field $F_q$ where q is a prime power. The Schoof–Elkies–Atkin (SEA) algorithm is a standard method for counting the number of $F_q$-points on E. The asymptotic complexity of the SEA algorithm depends on the distribution of the so-called Elkies primes.

Assuming GRH, we prove that the least Elkies prime is bounded by ${(2\log 4q)}^2$ when $q\ge {10}^9$. Previously, Satoh and Galbraith established an upper bound of $O\left({(\log q)}^{2+\epsilon }\right)$. Let $N_E\left(X\right)$ denote the number of Elkies primes less than X. Assuming GRH, we also show$N_E\left(X\right)=\frac{\pi \left(X\right)}{2}+O\left(\frac{\sqrt{X}{(\log qX)}^2}{\log X}\right).$