On the Product of Small Elkies Primes

TL;DR

This paper investigates the product of small Elkies primes for elliptic curves over finite fields, showing that for infinitely many pairs, this product can be significantly larger than heuristic estimates suggest, impacting point-counting algorithms.

Contribution

It proves the existence of infinitely many elliptic curves over finite fields with a large product of Elkies primes, surpassing heuristic bounds, thus providing new insights into the distribution of Elkies primes.

Findings

Existence of infinitely many pairs with large Elkies prime products

Demonstrates that $L_p(E)$ can grow faster than heuristic estimates

Implications for the efficiency of point-counting algorithms

Abstract

Given an elliptic curve $E$ over a finite field $\F_q$ of $q$ elements, we say that an odd prime $\ell \nmid q$ is an Elkies prime for $E$ if $t_E^2 - 4q$ is a quadratic residue modulo $\ell$, where $t_E = q+1 - #E(\F_q)$ and $#E(\F_q)$ is the number of $\F_q$-rational points on $E$. These primes are used in the presently most efficient algorithm to compute $#E(\F_q)$. In particular, the bound $L_q(E)$ such that the product of all Elkies primes for $E$ up to $L_q(E)$ exceeds $4q^{1/2}$ is a crucial parameter of this algorithm. We show that there are infinitely many pairs $(p, E)$ of primes $p$ and curves $E$ over $\F_p$ with $L_p(E) \ge c \log p \log \log \log p$ for some absolute constant $c>0$, while a naive heuristic estimate suggests that $L_p(E) \sim \log p$. This complements recent results of Galbraith and Satoh (2002), conditional under the Generalised Riemann Hypothesis, and of Shparlinski and Sutherland (2012), unconditional for almost all pairs $(p,E)$.