Paper 2026/1476

Constructing new permutation polynomials by the AGW Criterion

Abstract

In this paper, we propose several classes of permutation polynomials having the form $\sum\limits_i(x^{2^m}+x+\delta)^{s_i}+ax$ for $i=1$ or $i=2$, where the exponent $s_i$ satisfies $s_i\equiv 2^j\pmod{2^m+1}$ or $s_i \equiv 2^j\pmod{2^m-1}$ for some different integers $j$, $a\in\mathbb{F}_{2^m}^*$ and $\delta\in \mathbb{F}_{2^{2m}}$. More precisely, by applying the AGW criterion and determining the number of solutions to certain equations over $\mathbb{F}_{2^{2m}}$, several classes of permutation polynomials of the form $(x^{2^m}+x+\delta)^s+ax$ over $\mathbb{F}_{2^{2m}}$ are presented. In addition, we construct some classes of permutation polynomials of the form $(x^{2^m}+x+\delta)^{s_1}+(x^{2^m}+x+\delta)^{s_2}+ax$ over $\mathbb{F}_{2^{2m}}$. Our results generalize some known constructions of permutation polynomials. Finally, we demonstrate that the permutation polynomials proposed in this paper are not quasi-multiplicative equivalent to known ones.