Paper 2020/345

Bent functions stemming from Maiorana-McFarland class being provably outside its completed version

Fengrong Zhangand, Nastja Cepak, Enes Pasalic, and Yongzhuang Wei

Abstract

In early nineties Carlet [1] introduced two new classes of bent functions, both derived from the Maiorana-McFarland ($\mathcal{M}$) class, and named them $\cC$ and $ \cD$ class, respectively. Apart from a subclass of $\cD$, denoted by $\cD_0$ by Carlet, which is provably outside two main (completed) primary classes of bent functions, little is known about their efficient constructions. More importantly, both classes may easily remain in the underlying $\mathcal{M}$ class which has already been remarked in [21]. Assuming the possibility of specifying a bent function $f$ that belongs to one of these two classes (apart from $\cD_0$), the most important issue is then to determine whether $f$ is still contained in the known primary classes or lies outside their completed versions. In this article, we further elaborate on the analysis of the set of sufficient conditions given in \cite{OutsideMM} concerning the specification of bent functions in $\cC$ and $ \cD$ which are provably outside $\cM$. It is shown that these conditions, related to bent functions in class $\cD$, can be relaxed so that even those permutations whose component functions admit linear structures still can be used in the design. It is also shown that monomial permutations of the form $x^{2^r+1}$ have inverses which are never quadratic for $n >4$, which gives rise to an infinite class of bent functions in $\cC$ but outside $\cM$. Similarly, using a relaxed set of sufficient conditions for bent functions in $\cD$ and outside $\cM$, one explicit infinite class of such bent functions is identified. We also extend the inclusion property of certain subclasses of bent functions in $ \cC$ and $ \cD$, as addressed initially in [1,21], that are ultimately within the completed $\mathcal{M}$ class. Most notably, we specify {\em another generic and explicit subclass} of $\cD$, which we call $\cD_2^\star$, whose members are bent functions provably outside the completed $\mathcal{M}$ class.