On permutation quadrinomials from Niho exponents in characteristic two

Abstract

In a recent paper Zheng et al. characterized the coefficients of $f(x)=x+a_1{x}^{s_1(2m-1)+1}+a_2{x}^{s_2(2m-1)+1}+a_3{x}^{s_3(2m-1)+1}$ over ${\mathbb{F}}_2^{2m}$ that lead $f(x)$ to be a permutation of ${\mathbb{F}}_2^{2m}$ for $(s_1,s_2,s_3)=(\frac{1}{4},1,\frac{3}{4})$. They left open the question whether those conditions were also necessary. In this paper we give a positive answer to that question.