Several classes of permutation polynomials based on the AGW criterion over the finite field F22m

Let $F_q$ denote the finite field with q elements. Permutation polynomials and complete permutation polynomials over finite fields have been widely investigated in recent years due to their applications in cryptography, coding theory and combinatorial design. In this paper, several classes of (complete) permutation polynomials with the form ${({\mathrm{Tr}}_m^n{\left(x\right)}^k+\delta )}^s+L\left(x\right)$ and ${({\mathrm{Tr}}_m^n{\left(x\right)}^{k_1}+{\delta }_1)}^{s_1}+{({\mathrm{Tr}}_m^n{\left(x\right)}^{k_2}+{\delta }_2)}^{s_2}+L\left(x\right)$ are proposed based on the AGW criterion and some techniques in solving equations over the finite field $F_{2^n}$, where $L\left(x\right)=a{\mathrm{Tr}}_m^n\left(x\right)+bx$, $a\in F_{2^n}$ and $b\in F_{2^m}^{⁎}$. We also determine the compositional inverse of these polynomials in some special cases. Besides, Mesnager (2014) [13] proposed a construction of bent functions by finding some triples of permutation polynomials satisfying a particular property named $\left(A_n\right)$. With the help of this approach, several classes of bent functions are presented.