# Vectorial bent functions weakly/strongly outside the completed Maiorana–McFarland class

### Abstract

Two new classes of bent functions derived from the Maiorana–McFarland ($\mathcal{M}$) class, so-called $\mathcal{C}$ and $\mathcal{D}$, were introduced by Carlet (1994) two decades ago. The difficulty of satisfying their defining conditions was emphasized in Mandal et al. (2016). In a recent work Zhang et al. (2017) a set of efficient sufficient conditions for specifying bent functions in $\mathcal{C}$ and $\mathcal{D}$ which are outside the completed $\mathcal{M}$ class, denoted by $\mathcal{M}^{\#}$, was given. A natural follow up question is whether there is a possibility of extending this approach to the vectorial case. We introduce the property of vectorial bent functions that we call weakly or strongly outside $\mathcal{M}^{\#}$, referring respectively to the case whether some or all nonzero linear combinations (called components) of its coordinate functions are in class $\mathcal{C}$ (or $\mathcal{D}$) but provably outside $\mathcal{M}^{\#}$. For the first time, quite different to a straightforward vectorial extension of the Maiorana–McFarland class and the class of Dillon $\mathcal{PS}_{ap}$, we show the existence of several classes of vectorial bent functions whose component functions come from different classes of bent functions, mainly from $\mathcal{M}$ and $\mathcal{D}$, and in many cases being weakly outside $\mathcal{M}^{\#}$. We also address a difficult problem of specifying vectorial bent functions whose all components are in class $\mathcal{C}$ but provably outside $\mathcal{M}^{\#}$, thus being strongly outside $\mathcal{M}^{\#}$. Even though we could only specify a class of such functions whose dimension of bent vector space is only two, thus $F:GF(2)^n\to GF(2)^2$, this is the very first evidence of their existence.

Publication
Discrete Applied Mathematics