Almost Primality of Group Orders of Elliptic Curves Defined over Small Finite Fields

Abstract

Let E be an elliptic curve defined over a small finite field F_q, and let p be a prime number. We give a conjectural formula for the probabil ity that the order of the quotient group E(F_q^p )/E(F_q) is prime, and compare it with experimental data. The motivation for this study comes from public key cryptography.