Constructing new superclasses of bent functions from known ones
Date:
Contributed talk at the 6th International Workshop on Boolean Functions and their Applications (BFA).
Abstract: Some recent research articles addressed the specification of indicators leading to two classes of bent functions, denoted $\mathcal{C}$ and $\mathcal{D}$, derived from the Maiorana-McFarland ($\mathcal{M}$) class by C. Carlet in 1994. Many of these explicitly specified bent functions that belong to $\mathcal{C}$ or $\mathcal{D}$ are provably outside the completed $\mathcal{M}$ class. Nevertheless, these modifications are performed on affine subspaces whereas modifying bent functions on suitable sets may provide us with further classes of bent functions which { are provably outside the completed $\mathcal{M}$ class}. In this article, we exactly specify new families of bent functions by adding together indicators well suited for the $\mathcal{C}$ and $\mathcal{D}$ class, thus essentially modifying bent functions in $\mathcal{M}$ on suitable sets instead of subspaces. Apart from the desirable property of being outside the completed $\mathcal{M}$ class, these bent functions can be potentially used for constructing vectorial bent functions whose components (possibly not all) share the same property, which is an interesting research challenge. The question whether these functions are simultaneously outside the completed $\mathcal{M}$ and $\mathcal{PS}^+$ classes is also addressed.