Primality proofs with elliptic curves:
Heuristic on the expected number of curve orders

Abstract. In article [14] we have shown that during the downrun part of the elliptic curve primality proving algorithm, the expected number of curve orders for a given probable prime grows asymptotically as the square root of the limit until we take the negative fundamental discriminants, even if these discriminants are smooth. In this article, based on a heuristic assumption, we are going to show that this behaviour is still expectable if we only use those smooth negative fundamental discriminants which are quadratic residues modulo the tested probable prime number.