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

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.