Investigations on Bent and Negabent Functions via the Nega-Hadamard Transform

Parker considered a new type of discrete Fourier transform, called nega-Hadamard transform. We prove several results regarding its behavior on combinations of Boolean functions and use this theory to derive several results on negabentness (that is, flat nega-spectrum) of concatenations, and partially symmetric functions. We derive the upper bound ⌈n/2⌉ for the algebraic degree of a negabent function on n variables. Further, a characterization of bent-negabent functions is obtained within a subclass of the Maiorana-McFarland set. We develop a technique to construct bent-negabent Boolean functions by using complete mapping polynomials. Using this technique, we demonstrate that for each ℓ ≥ 2, there exist bent-negabent functions on n = 12 ℓ variables with algebraic degree n /4 + 1 = 3 ℓ + 1. It is also demonstrated that there exist bent-negabent functions on eight variables with algebraic degrees 2, 3, and 4. Simple proofs of several previously known facts are obtained as immediate consequences of our work.