Iterative constructions of irreducible polynomials from isogenies

TL;DR

This paper explores how certain rational transformations, linked to elliptic curve isogenies, can generate infinite families of irreducible polynomials over finite fields, with algorithms based on complex multiplication theory.

Contribution

It introduces a novel interpretation of polynomial transforms via elliptic curve isogenies and develops algorithms to produce many such transforms with positive density of irreducible polynomials.

Findings

Transformations produce infinite irreducible polynomial families

Algorithms generate numerous rational fractions with desired properties

Positive density of irreducible polynomials achieved

Abstract

Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance, in odd characteristic, this is the case for the rational fraction $S=(x^2+1)/(2x)$, known as the $R$-transform, and for a positive density of all irreducible polynomials $f$. We interpret these transforms in terms of isogenies of elliptic curves. Using complex multiplication theory, we devise algorithms to generate a large number of other rational fractions $S$, each of which yields infinite families of irreducible polynomials for a positive density of starting irreducible polynomials $f$.