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
$\lceil {{ n}\over { 2}} \rceil $
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
$\ell \geq 2$
, there exist bent–negabent functions on
$n = 12\ell $
variables with algebraic degree
$ {{ n}\over { 4}}+1 = 3\ell + 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.