On the Distribution of Atkin and Elkies Primes

TL;DR

This paper investigates the distribution of Atkin and Elkies primes for elliptic curves over finite fields, showing they occur with similar frequency on average, and applies this to develop a fast algorithm for generating elliptic curves with prime order.

Contribution

It proves that Atkin and Elkies primes are asymptotically equally distributed on average, and introduces an efficient method to generate elliptic curves with prime number of points.

Findings

Atkin and Elkies primes are equally distributed on average for large q.

The distribution result holds for primes less than (log q)^e.

A new fast algorithm for generating elliptic curves with prime order is proposed.

Abstract

Given an elliptic curve E over a finite field F_q of q elements, we say that an odd prime ell not dividing q is an Elkies prime for E if t_E^2 - 4q is a square modulo ell, where t_E = q+1 - #E(F_q) and #E(F_q) is the number of F_q-rational points on E; otherwise ell is called an Atkin prime. We show that there are asymptotically the same number of Atkin and Elkies primes ell < L on average over all curves E over F_q, provided that L >= (log q)^e for any fixed e > 0 and a sufficiently large q. We use this result to design and analyse a fast algorithm to generate random elliptic curves with #E(F_p) prime, where p varies uniformly over primes in a given interval [x,2x].