An Open Problem on the Bentness of Mesnager's Functions
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Let n=2m. In the present paper, we study the binomial Boolean functions of the form f_a,b(x) = Tr_1^n(a x^2^m-1 ) +Tr_1^2(bx^2^n-1/3), where m is an even positive integer, a∈𝔽_2^n^* and b∈𝔽_4^*. We show that f_a,b is a bent function if the Kloosterman sum K_m(a^2^m+1)=1+ ∑_x∈𝔽_2^m^* (-1)^Tr_1^m(a^2^m+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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