**An Open Problem on the Bentness of Mesnager’s Functions**

*Chunming Tang and Peng Han and Qi Wang and Jun Zhang and Yanfeng Qi*

**Abstract: **Let $n=2m$. In the present paper, we study the binomial Boolean functions of the form $$f_{a,b}(x) = \mathrm{Tr}_1^{n}(a x^{2^m-1 }) +\mathrm{Tr}_1^{2}(bx^{\frac{2^n-1}{3} }), $$
where $m$ is an even positive integer, $a\in \mathbb{F}_{2^n}^*$ and $b\in \mathbb{F}_4^*$.
We show that $ f_{a,b}$ is a bent function if the Kloosterman sum
$$K_{m}\left(a^{2^m+1}\right)=1+ \sum_{x\in \mathbb{F}_{2^m}^*} (-1)^{\mathrm{Tr}_1^{m}(a^{2^m+1} x+ \frac{1}{x})}$$
equals $4$, thus settling an open problem of Mesnager. The proof employs tools including computing Walsh coefficients of Boolean functions via multiplicative
characters, divisibility properties of Gauss sums, and graph theory.

**Category / Keywords: **foundations / Boolean function, bent function, Walsh transform, Gauss Sum, directed graph

**Date: **received 27 Sep 2021

**Contact author: **tangchunmingmath at 163 com

**Available format(s): **PDF | BibTeX Citation

**Version: **20210928:182913 (All versions of this report)

**Short URL: **ia.cr/2021/1304

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