On inverse of permutation polynomials of small degree over finite fields, II

TL;DR

This paper studies the inverse functions of specific permutation polynomials over finite fields, providing explicit formulas for certain cases and general results for polynomials of degrees 6, 7, and 8.

Contribution

It offers explicit inverse expressions for particular permutation polynomials and generalizes the inverse computation for all degree 6, 7, and 8 permutation polynomials over finite fields.

Findings

Explicit inverses for $x(x^3 -a)^2$ and $x(x^2 -a)^3$ over $F_{7^n}$

Inverse formulas for all degree 6, 7, and 8 permutation polynomials over finite fields

Enhanced understanding of permutation polynomial inverses in finite field theory

Abstract

We investigate the permutation property of polynomials of the form $x^{r}(x^{s} -a)^{t}$, and give the expressions of their inverses. In particular, explicit expressions of inverses of permutation polynomials $x(x^3 -a)^2$ and $x(x^2 -a)^3$ on $\mathbb{F}_{7^n}$ are presented. Then, using some known results, we obtain the inverses of all permutation polynomials of degree $6, 7, 8$ over finite fields.