Welcome to the resource topic for 2021/1304
Title:
An Open Problem on the Bentness of Mesnager’s Functions
Authors: Chunming Tang, Peng Han, Qi Wang, Jun Zhang, 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.
ePrint: https://eprint.iacr.org/2021/1304
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