On an open problem on permutation polynomials from self-reciprocal polynomials

Official URL: https://dx.doi.org/10.1016/j.ffa.2026.102879

In this paper, we address two open problems posed by Martínez, Gupta and Quoos in [24] . We solve both problems completely and therefore generalize [28, Theorems 3.4 and 3.5] . We study polynomials of the form f(x)=x3g(xq−1) over the finite field F2k such that g(x)=h(x)+xu+xv, where h(x)=ax4+bx3+cx2+bx+a is a self-reciprocal polynomial with a,b,c∈F2k and (u,v)∈{(3,−1),(1,−1),(2,−2),(2,−1)}. The studied classes of polynomials either generalize some existing pentanomials and hexanomials or they are not quasi-multiplicative equivalent to any of the known permutation polynomials in the literature. We find necessary and sufficient conditions on a,b,c∈F2k so that f(x) is a permutation polynomial for F22k. Moreover, we show that some known permutation pentanomials are QM equivalent.