Generic constructions of 5-valued spectra Boolean functions

S. Hodžić, E. Pasalic, W. G. Zhang

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 so-called 5-valued spectra Boolean functions whose Walsh spectra takes the values in $\{0, \pm 2^{\lambda_1}, \pm 2^{\lambda_2}\}$. Moreover, these design methods mainly regards the specification of these functions in their ANF (algebraic normal form) domain. In this article, 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 spectral domain) to a family of so-called {\em 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 4-bent decomposition.

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