On the Distribution of Atkin and Elkies Primes

Given an elliptic curve E over a finite field F_q of q elements, we say that an odd prime ℓ {does not divide} q is an Elkies prime for E if t_E² - 4q is a square modulo ℓ, where (Formula presented.) is the number of F_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_q, provided that L ≥ (log q)^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_p) prime, where p varies uniformly over primes in a given interval [x, 2x].