The compositional inverses of permutation polynomials from trace functions over finite fields

TL;DR

This paper derives the compositional inverses of certain permutation polynomials over finite fields, specifically those involving trace functions and polynomial functions satisfying particular algebraic conditions.

Contribution

It provides explicit formulas for the inverses of a broad class of permutation polynomials involving trace functions and polynomial conditions over finite fields.

Findings

Explicit inverse formulas for permutation polynomials involving trace functions.

Conditions under which the permutation polynomials are invertible.

Application of the results to construct permutation polynomials with known inverses.

Abstract

In this paper, we present the compositional inverses of several classes permutation polynomials of the form $\sum_{i=1}^kb_i\left({\rm Tr}_m^{mn}(x)^{t_i}+\delta\right)^{s_i}+f_1(x)$, where $1\leq i \leq k,$ $s_i$ are positive integers, $b_i \in \mathbb{F}_{p^m},$ $p$ is a prime and $f_1(x)$ is a polynomial over $\mathbb{F}_{p^{mn}}$ satisfying the following conditions: (i) ${\rm Tr}_m^{mn}(x) \circ f_1(x)=\varphi(x) \circ {\rm Tr}_m^{mn}(x),$ where $\varphi(x)$ is a polynomial over $\mathbb{F}_{p^m};$ (ii) For any $a \in \mathbb{F}_{p^m},$ $f_1(x)$ is injective on ${\rm Tr}_m^{mn}(a)^{-1}.$