Cryptology ePrint Archive: Report 2021/1304

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

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

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