On the distribution of Atkin and Elkies primes for reductions of   elliptic curves on average
  For an elliptic curve E/Q without complex multiplication we study the distribution of Atkin and Elkies primes l, on average, over all good reductions of E modulo primes p. We show that, under the Generalised Riemann Hypothesis, for almost all primes p there are enough small Elkies primes l to ensure that the Schoof-Elkies-Atkin point-counting algorithm runs in (log p)^(4+o(1)) expected time. 
Igor E. Shparlinski and Andrew V. Sutherland
math.NT