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

In this paper, we present the compositional inverses of several classes permutation polynomials of the form ${\sum }_{i=1}^k{b}_i{({\mathrm{T}\mathrm{r}}_m^{mn}(x{)}^{t_i}+\delta )}^{s_i}+f_1(x)$, where $1\le i\le k,$ $s_i$ 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) ${\mathrm{T}\mathrm{r}}_m^{mn}(x)\circ f_1(x)=\phi (x)\circ {\mathrm{T}\mathrm{r}}_m^{mn}(x),$ where $\phi (x)$ is a polynomial over ${\mathbb{F}}_{p^m};$ (ii) For any $a\in {\mathbb{F}}_{p^m},$ $f_1(x)$ is injective on ${\mathrm{T}\mathrm{r}}_m^{mn}(a{)}^{-1}\mathrm{.}$