On the Distribution of Atkin and Elkies Primes

Abstract

Given an elliptic curve E over a finite field F[subscript q] of q elements, we say that an odd prime ℓ∤q is an Elkies prime for E if t[superscript 2][subscript E]−4q is a square modulo ℓ, where t[subscript E]=q+1−#E(F[subscript q]) and #E(F[subscript q]) is the number of F[subscript q]-rational points on E; otherwise, ℓ is called an Atkin prime. We show that there are asymptotically the same number of Atkin and Elkies primes ℓ<L on average over all curves E over F[subscript q], provided that L≥(log q)[superscript ε] for any fixed ε>0 and a sufficiently large q. We use this result to design and analyze a fast algorithm to generate random elliptic curves with #E(F[subscript p]) prime, where p varies uniformly over primes in a given interval [x,2x].