Constructing irreducible polynomials recursively with a reverse composition method

TL;DR

This paper introduces a recursive method to construct irreducible polynomials over finite fields using a reverse composition approach, enabling the generation of many polynomials with specific properties for various applications.

Contribution

The paper presents a novel recursive construction technique for minimal polynomials of powers of elements in finite fields, expanding the toolkit for generating irreducible polynomials.

Findings

Allows construction of many irreducible polynomials of the same degree

Construction is performed entirely within the base field _q

Method is applicable when prime factors of k divide q-1

Abstract

We suggest a construction of the minimal polynomial $m_{\beta^k}$ of $\beta^k\in \mathbb F_{q^n}$ over $\mathbb F_q$ from the minimal polynomial $f= m_\beta$ for all positive integers $k$ whose prime factors divide $q-1$. The computations of our construction are carried out in $\mathbb F_q$. The key observation leading to our construction is that for $k \mid q-1$ holds $$m_{\beta^k}(X^k) = \prod_{j=1}^{\frac kt} \zeta_k^{-jn} f (\zeta_k^j X),$$ where $t= \max \{m\mid \gcd(n,k): f (X) = g (X^m), g \in \mathbb F_q[X]\}$ and $\zeta_{k}$ is a primitive $k$-th root of unity in $\mathbb F_q$. The construction allows to construct a large number of irreducible polynomials over $\mathbb F_q$ of the same degree. Since different applications require different properties, this large number allows the selection of the candidates with the desired properties.