The compositional inverses of three classes of permutation polynomials over finite fields

TL;DR

This paper computes the compositional inverses of three specific classes of permutation polynomials over finite fields, extending recent methods and providing explicit inverse formulas for these classes.

Contribution

It introduces explicit formulas for the compositional inverses of three new classes of permutation polynomials over finite fields, inspired by P. Yuan's local method.

Findings

Derived inverse formulas for permutation polynomials of form $ax^q+bx+(x^q-x)^k$ over $_{q^2}$.

Computed inverses for polynomials $f(x)=-x+x^{(q^2+1)/2}+x^{(q^3+q)/2}$ over $_{q^3}$.

Established inverse formulas for polynomials $A^{m}(x)+L(x)$ over $_{q^n}$ with specific properties.

Abstract

Recently, P. Yuan presented a local method to find permutation polynomials and their compositional inverses over finite fields. The work of P. Yuan inspires us to compute the compositional inverses of three classes of the permutation polynomials: (a) the permutation polynomials of the form $ax^q+bx+(x^q-x)^k$ over $\mathbb{F}_{q^2},$ where $a+b \in \mathbb{F}_q^*$ or $a^q=b;$ (b) the permutation polynomials of the forms $f(x)=-x+x^{(q^2+1)/2}+x^{(q^3+q)/2} $ and $f(x)+x$ over $\mathbb{F}_{q^3};$ (c) the permutation polynomial of the form $A^{m}(x)+L(x)$ over $\mathbb{F}_{q^n},$ where ${\rm Im}(A(x))$ is a vector space with dimension $1$ over $\mathbb{F}_{q}$ and $L(x)$ is not a linearized permutation polynomial.