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

TL;DR

This paper presents an efficient algorithm for constructing elliptic curves over prime fields with a number of points equal to Fibonacci primes, leveraging their unique arithmetic properties.

Contribution

It introduces a novel variant of the CM-construction algorithm tailored for Fibonacci primes, reducing the expected time complexity.

Findings

Algorithm successfully constructs elliptic curves with Fibonacci prime order.

Expected time complexity is lower than (\u00f8^3).

The method exploits Fibonacci primes' arithmetic properties for efficiency.

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.