On the Product of Small Elkies Primes

Given an elliptic curve E over a finite field Fq of q elements, we say that an odd prime l ∤ q is an Elkies prime for E if tE − 4q is a quadratic residue modulo l, where tE = q+1−#E(Fq) and #E(Fq) is the number of Fq-rational points on E. These primes are used in the presently most efficient algorithm to compute #E(Fq). In particular, the bound Lq(E) such that the product of all Elkies primes for E up to Lq(E) exceeds 4q 1/2 is a crucial parameter of this algorithm. We show that there are infinitely many pairs (p,E) of primes p and curves E over Fp with Lp(E) ≥ c log p log log log p for some absolute constant c > 0, while a naive heuristic estimate suggests that Lp(E) ∼ log p. This complements recent results of Galbraith and Satoh (2002), conditional under the Generalised Riemann Hypothesis, and of Shparlinski and Sutherland (2012), unconditional for almost all pairs (p,E).