Generic Constructions of Five-Valued Spectra Boolean Functions

Abstract

Whereas the design and properties of bent and plateaued functions have been frequently addressed during the past few decades, there are only a few design methods of the so-called five-valued spectra Boolean functions whose Walsh spectra take the values in $\{0, \pm 2^{\lambda _{1}}, \pm 2^{\lambda _{2}}\}$ . Moreover, these design methods mainly regard the specification of these functions in their algebraic normal form (ANF) domain. In this paper, we give a precise characterization of this class of functions in their spectral domain using the concept of a dual of plateaued functions. Both necessary and sufficient conditions on the Walsh support of these functions are given, which then connects their design (in the spectral domain) to a family of the so-called totally (non-overlap) disjoint spectra plateaued functions . We identify some suitable families of plateaued functions having this property, thus providing some generic methods in the spectral domain. Furthermore, we also provide an extensive analysis of their constructions in the ANF domain and provide several generic design methods. The importance of this class of functions is manifolded, where apart from being suitable for some cryptographic applications, we emphasize their property of being constituent functions in the so-called four-bent decomposition.