On permutation quadrinomials from Niho exponents in characteristic two

TL;DR

This paper confirms that specific coefficient conditions for a certain class of permutation quadrinomials over characteristic two fields are both necessary and sufficient, completing a recent characterization in the literature.

Contribution

It proves the necessity of coefficient conditions for permutation quadrinomials with Niho exponents, extending previous sufficiency results.

Findings

Established the necessity of coefficient conditions for permutation quadrinomials.

Completed the characterization of permutation quadrinomials with Niho exponents.

Provided a full criterion for permutation behavior in the studied class.

Abstract

In a recent paper Zheng et al. characterized the coefficients of $ f(x) = x + a_1x^{s_1(2m-1)+1} + a_2x^{s_2(2m-1)+1} + a_3x^{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.