Permutation polynomials and involutions over the finite field F22m

Abstract

In this paper, we propose several classes of permutation polynomials of the form $\sum _{j=1}^t{({\mathrm{Tr}}_m^{2m}{\left(x\right)}^{k_j}+{\delta }_j)}^{s_j}+L\left(x\right)$ over $F_{2^{2m}}$, where $L\left(x\right)=a{\mathrm{Tr}}_m^{2m}\left(x\right)+bx$, $a\in F_{2^{2m}}$ and $b\in F_{2^m}^{⁎}$. The permutation behavior of the proposed polynomials is investigated by the AGW criterion and determination of the number of solutions to certain equations over $F_{2^{2m}}$. Based on an effective method proposed by Mesnager (2014), we construct several classes of involutions and further obtain some self-dual bent functions by employing three permutations of $F_{2^{2m}}$ satisfying an algebraic property $\left(A_{2m}\right)$. Finally, it is worth pointing out that there exist examples of bent functions we obtained which do not belong to $M{M}^{\#}$.