Iterative constructions of irreducible polynomials from isogenies

Abstract

Let S be a rational fraction and let f be a polynomial over a finite field. Consider the transform $T\left(f\right)=\mathrm{numerator}\left(f\right(S\left)\right)$. In certain cases, the polynomials f, $T\left(f\right)$, $T\left(T\right(f\left)\right)\dots$ are all irreducible. For instance, in odd characteristic, this is the case for the rational fraction $S=(x^2+1)/\left(2x\right)$, known as the R-transform, and for a positive density of 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 rational fractions S, each of which yields infinite families of irreducible polynomials for a positive density of starting irreducible polynomials f.