An Open Problem on the Bentness of Mesnager's Functions

2021/09/28 by Tang, Chunming, Han, Peng, Wang, Qi +2
#FOS: Computer and information sciences #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.2109.13421

Abstract

Let n=2m. In the present paper, we study the binomial Boolean functions of the form f_a,b(x) = Tr₁ⁿ(a x^2m-1 ) +Tr₁²(bx^(2n-1)/(3) ), where m is an even positive integer, a∈ \mathbbF_2ⁿ^* and b∈ \mathbbF₄^*. We show that f_a,b is a bent function if the Kloosterman sum K_m(a^2m+1)=1+ ∑_x∈ \mathbbF_2^m^* (-1)^Tr₁^m(a^2m+1 x+ (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.

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