Constructions of (vectorial) bent functions outside the completed Maiorana–McFarland class

Abstract

Two new classes of bent functions derived from the Maiorana–McFarland (M) class, so-called C and D, were introduced by Carlet (1993) almost three decades ago. In Zhang (2020) sufficient conditions for specifying bent functions in C and D which are outside the completed M class, denoted by M#, were given. Furthermore in Pasalic et al. (2021) the notion of vectorial bent functions which are weakly or strongly outsideM#, referring respectively to the case whether some or all nonzero linear combinations (called components) of its coordinate functions are in class C (or D) but provably outside M#, was introduced. In this article we continue the work of finding new instances of vectorial bent functions weakly/strongly outside 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 SC which contains the classes D0 and C and whose members are provably outside M#. Most notably, using indicators of the form 1L⊥(x,y)+δ0(x) to define members of this class leads for the first time to modifications of the 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 M#. In this context, a new concept of being almost strongly outside 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 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.