Sequences of irreducible polynomials over odd prime fields via elliptic curve endomorphisms

TL;DR

This paper introduces a method to generate sequences of irreducible polynomials over odd prime fields using elliptic curve endomorphisms, expanding tools for finite field polynomial construction.

Contribution

It presents a novel construction technique for irreducible polynomials over odd prime fields leveraging elliptic curve endomorphisms.

Findings

Constructs sequences of irreducible polynomials over odd prime fields.

Analyzes the properties of the polynomial transforms.

Provides conditions for irreducibility based on elliptic curve endomorphisms.

Abstract

In this paper we present and analyse a construction of irreducible polynomials over odd prime fields via the transforms which take any polynomial $f \in \mathbf{F}_p[x]$ of positive degree $n$ to $\left(\frac{x}{k} \right)^n \cdot f(k(x+x^{-1}))$, for some specific values of the odd prime $p$ and $k \in \mathbf{F}_p$.