Investigations on bent and negabent functions via the nega-Hadamard transform

Abstract

Parker, et al. considered a new style of discrete Fourier transform, called nega-Hadamard transform. We prove several results regarding its behaviior on combinations of Boolean functions and use this theorry to derive several results on negabentness (that is, flat nega-spactrum) of concatenations, and partially symmetric functions. We derive the uppoer bound (n/2) for the algebraic dgree of a negabent function on n variables.. Further, a characterization of of bent-negabent functions is obtained within a subclass if the Maiorana-McFarland set. We develop a technique to construct bent-negabent Boolean functions by using a complete mappig polynomials. Using this technique, we demonstrate for each l > 2, there exist bent-negabent funcitions on n = 12l variable with algebraic degree n/4 + 1 = 3l + 1. It is also demonstrated that there exist bent nega-bent 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.