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 $\funnyF_q$, and let $p$ be a prime number. We give a conjectural formula for the probability that the order of the quotient group $E(\funnyF_{q^p})/E(\funnyF_q)$ is prime, and compare it with experimental data. The motivation for this study comes from public key cryptography.