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

TL;DR

This paper presents an extension of a method to generate irreducible polynomials over odd prime fields using elliptic curve endomorphisms, specifically through $Q_k$ and $ ilde{Q}_k$ transforms linked to elliptic curve isogenies.

Contribution

It introduces a novel iterative approach employing elliptic curve isogenies to construct irreducible polynomials over finite fields of odd characteristic.

Findings

Successful construction of irreducible polynomials via elliptic curve transforms

Extension of previous methods to new families of transforms

Potential applications in finite field cryptography and coding theory

Abstract

In this paper we extend a previous investigation by us regarding an iterative construction of irreducible polynomials over finite fields of odd characteristic. In particular, we show how it is possible to iteratively construct irreducible polynomials by means of two families of transforms, which we call the $Q_k$ and $\hat{Q}_k$-transforms, related to certain degree two isogenies over elliptic curves, which split the multiplication-by-$2$ map.