Constructions of (vectorial) bent functions 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 (1993) almost three decades ago. In Zhang (2020) sufficient conditions for specifying bent functions in $\mathcal{C}$ and $\mathcal{D}$ which are outside the completed $\mathcal{M}$ class, denoted by $\mathcal{M}^{\#}$, were given. Furthermore in Pasalic et al. (2021) the notion of vectorial bent functions which are 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 $\mathcal{M}^{\#}$, was introduced. In this article we continue the work of finding new instances of vectorial bent functions weakly/strongly outside $\mathcal{M}^{\#}$ using a different approach. Namely, a generic method for the construction of vectorial bent $(n,t)$-functions of the form $F(x,y)=G(x,y)+H(x,y)$, $n=2m$, $t$ divides $m$, was recently proposed in Bapić (2021), where $G$ is a given bent $(n,t)-$function satisfying certain properties and $H$ is an arbitrary $(t,t)$-function having certain form. We introduce a new superclass of bent functions $\mathcal{SC}$ which contains the classes $\mathcal{D}_0$ and $\mathcal{C}$ whose members are provably outside $\mathcal{M}^{\#}$. Most notably, using indicators of the form $\mathbf{1}_{L^{\perp}}(x,y)+\delta_0(x)$ to define members of this class leads for the first time to modifications of the $\mathcal{M}$ class performed on sets rather than on affine subspaces. We also show that for suitable choices of $H$, the function $F$ is a vectorial bent function weakly/strongly outside the class $\mathcal{M}^{\#}$. In this context, a new concept of being almost strongly outside $\mathcal{M}^{\#}$ is introduced and some families of vectorial bent functions with this property are given. Furthermore, we provide two new families of vectorial bent functions strongly outside $\mathcal{M}^{\#}$ (considered to be an intrinsically hard problem) whose output dimension is greater than $2$, thus giving first examples of such functions in the literature.