The compositional inverses of some permutation polynomials of the form $x+γ\operatorname{Tr}_q^{q^2}(h(x))$

TL;DR

We address the problem of obtaining explicit compositional inverses for six permutation polynomials of the form $f(x)=x+\gamma\operatorname{Tr}_q^{q^2}(h(x))$ over $\mathbb{F}_{q^2}$ with $q=2^m$. The approach combines auxiliary results on trace-linear maps and a conjugation-based method using a linearized map $T(x)=\operatorname{Tr}_q^{q^2}(\beta x)+\alpha\operatorname{Tr}_q^{q^2}(\delta x)$ to construct $T^{-1}$ and transfer inversion to the target polynomials. The paper provides explicit inverse expressions for $f_1$ through $f_6$, with case distinctions depending on whether $\gamma\in\mathbb{F}_q$ or $\operatorname{Tr}_q^{q^2}(\gamma)=1$ and the parity of $m$, often employing traces and a $t$ satisfying $3t\equiv1\pmod{2^m-1}$. These closed-form inverses enhance practical use in areas requiring both permutation polynomials and their inverses, such as cryptography and coding theory.

Abstract

Recently, Jiang et al. \cite{JIANG2025102522} obtained several classes of Permutation Polynomial of the form $x+γ\operatorname{Tr}_q^{q^2}(h(x))$ over finite fields $\mathbb{F}_{q^2},q=2^n$. In this paper, we find the compositional inverse of six classes of permutation polynomials of this form.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof