Two secondary constructions of bent functions without initial conditions

Abstract

In this article, we propose two secondary constructions of bent functions without any conditions on initial bent functions employed by these methods. It is shown that both methods generate bent functions that belong to the generalized Maiorana–McFarland ($\mathcal{K}=\mathcal{GMM}_{n/2+k}$) class of $n$-variable Boolean functions, with $n$ even. The class $\mathcal{K}$ contains functions that can be viewed as a concatenation of $(n/2−k)$-variable (not necessarily distinct) affine functions, which was previously (mainly) used in the design of resilient Boolean functions. Most notably, we show that a subclass of bent functions generated by our first method is provably outside the completed Maiorana–McFarland class ${\mathcal {MM}}^{\#}$. This extremely large class of Boolean functions $\mathcal{K}$, which is shown to properly include the standard Maiorana–McFarland class ${\mathcal {MM}}$, may contain a significant subset of bent functions that are not the members of ${\mathcal {MM}}^{\#}$. In general, the inclusion of these bent functions, that are provably outside ${\mathcal {MM}}^{\#}$, into the completed partial spread class remains unknown.

Publication
Designs, Codes and Cryptography