Factoring polynomials of the form $f(x^n)\in \mathbb{F}_q[x]$

TL;DR

This paper presents a fast algorithm for factoring polynomials of the form $f(x^n)$ over finite fields, with specific cases and applications to splitting $x^n-1$ into irreducibles.

Contribution

It introduces a novel, efficient algorithm for factoring such polynomials, extending previous methods to particular cases involving divisibility and gcd conditions.

Findings

Algorithm efficiently factors $f(x^n)$ over finite fields.

Successfully splits $x^n-1$ into irreducible factors for specific $n$ and $q$.

Provides explicit factorization methods under certain divisibility conditions.

Abstract

Let $f(x)\in \mathbb{F}_q[x]$ be an irreducible polynomial of degree $m$ and exponent $e$, and $n$ be a positive integer such that $\nu_p(q-1)\ge \nu_{p}(e)+\nu_p(n)$ for all $p$ prime divisor of $n$. We show a fast algorithm to determine the irreducible factors of $f(x^n)$. We also show the irreducible factors in the case when ${\rm rad}(n)$ divides $q-1$ and ${\rm gcd}(m, n)=1$. Finally, using this algorithm we split $x^n-1$ into irreducible factors, in the case when $n=2^mp^t$ and $q$ is a generator of the group $\mathbb{Z}_{p^2}^*$.