On the Distribution of Atkin and Elkies Primes
Number Theory
2014-06-20 v2
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].
Keywords
Cite
@article{arxiv.1112.3390,
title = {On the Distribution of Atkin and Elkies Primes},
author = {Igor E. Shparlinski and Andrew V. Sutherland},
journal= {arXiv preprint arXiv:1112.3390},
year = {2014}
}
Comments
17 pages, minor edits