Elliptic Curves of Fibonacci order over $\mathbb{F}_p$

Abstract

We will describe an algorithm to construct an elliptic curve $E_{f_q}$ over some prime field $\mathbb{F}_p$ such that such that $|E_{f_q}(\mathbb{F}_p)| = f_q$, where $f_q$ is a probable Fibonacci prime for some prime index $q$. The algorithm is a variant of the efficient CM-construction by Br$\ddot{o}$ker and Stevenhagen, which is well suited for Fibonacci primes due to their arithmetic properties. The time complexity of our algorithm is expected to be lower than $\widetilde{O}(\log^3({f_q}))$. The construction process is a series of algorithms, where each is a test for primality.