On Boolean Functions Which Are Bent and Negabent

Abstract

Bent functions $f:\mathbb{F}_2^m\rightarrow \mathbb{F}_2$ achieve largest distance to all linear functions. Equivalently, their spectrum with respect to the Hadamard-Walsh transform is flat (i.e. all spectral values have the same absolute value). That is equivalent to saying that the function f has optimum periodic autocorrelation properties. Negaperiodic correlation properties of f are related to another unitary transform called the nega-Hadamard transform. A function is called negabent if the spectrum under the nega-Hadamard transform is flat. In this paper, we consider functions f which are simultaneously bent and negabent, i.e. which have optimum periodic and negaperiodic properties. Several constructions and classifications are presented.