Permutation polynomials from a linearized decomposition

TL;DR

This paper explores a class of permutation polynomials over finite fields constructed via linearized polynomials, linking their properties to polynomial factorization over smaller fields and providing methods for their construction and inversion.

Contribution

It introduces a new approach to constructing permutation polynomials using linearized decompositions and the AGW criterion, connecting permutation properties to polynomial factorization over finite fields.

Findings

Permutation polynomials constructed from linearized polynomials are characterized by coprimality conditions.

The construction method is linked to factorization of x^n - 1 over the base field.

Explicit inverses for certain permutation polynomials are derived.

Abstract

In this paper we discuss the permutational property of polynomials of the form $f(L(x))+k(L(x))\cdot M(x)\in \mathbb F_{q^n}[x]$ over the finite field $\mathbb F_{q^n}$, where $L, M\in \mathbb F_q[x]$ are $q$-linearized polynomials. The restriction $L, M\in \mathbb F_q[x]$ implies a nice correspondence between the pair $(L, M)$ and the pair $(g, h)$ of conventional $q$-associates over $\mathbb F_q$ of degree at most $n-1$. In particular, by using the AGW criterion, permutational properties of our class of polynomials translates to some arithmetic properties of polynomials over $\mathbb F_q$, like coprimality. This relates the problem of constructing PPs of $\mathbb F_{q^n}$ to the problem of factorizing $x^n-1$ in $\mathbb F_q[x]$. We then specialize to the case where $L(x)$ is the trace polynomial from $\mathbb F_{q^n}$ over $\mathbb F_q$, providing results on the construction of permutation and complete permutation polynomials, and their inverses. We further demonstrate that the latter can be extended to more general linearized polynomials of degree $q^{n-1}$.