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 $F(x,y)=G(x,y)+H(x,y)$, $n=2m$, $t|m$, 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.

Publication
Discrete Applied Mathematics
Amar Bapić
Amar Bapić
Young Researcher
dr. Enes Pasalic
dr. Enes Pasalic
Full Professor, Head of the center

Related