Vectorial bent functions weakly/strongly 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 (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 C and D which are outside the completed M class, denoted by 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 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#. For the first time, quite different to a straightforward vectorial extension of the Maiorana–McFarland class and the class of Dillon PSap, we show the existence of several classes of vectorial bent functions whose component functions come from different classes of bent functions, mainly from M and D, and in many cases being weakly outside M#. We also address a difficult problem of specifying vectorial bent functions whose all components are in class C but provably outside M#, thus being strongly outside M#. Even though we could only specify a class of such functions whose dimension of bent vector space is only two, thus F:F2n→F22, this is the very first evidence of their existence.