A complete characterization of ${\mathcal {D}}_0 \cap {\mathcal {M}}^\#$ and a general framework for specifying bent functions in ${\mathcal {C}}$ outside ${\mathcal {M}}^\#$

Abstract

In this paper we characterize the intersection of the completed Maiorana–McFarland class of bent functions ${\mathcal {M}}^{\#}$ and Carlet’s ${\mathcal {D}}_0$ class of bent functions. As a consequence of this characterization, we prove that when the degree of a permutation$\pi$ is greater than 2 the Boolean function $f(x,y)=x \cdot \pi (y) + \delta _0(x)$, with $f:{\mathbb {F}}_2^n \times {\mathbb {F}}_2^n \rightarrow {\mathbb {F}}_2$, is always outside ${\mathcal {M}}^{\#}$ class. This also refines the sufficient condition of Carlet, which claims that if $\pi$ is not affine on some hyperplane then $f \not \in {\mathcal {M}}^\#$. More precisely, this condition is also necessary when $\deg (\pi )=2$, but it is not needed in the case $\deg (\pi )>2$. We also specify a rather general framework for specifying bent functions in Carlet’s ${\mathcal {C}}$ class which are provably outside the class ${\mathcal {M}}^{\#}$. Finally, using ranks of bent functions, we investigate the intersection of the class ${\mathcal {C}}$ and the subclass of the partial spread class $\mathcal {PS}_{ap}$ and prove that the probability that an n variable function in $\mathcal {PS}_{ap}$ is also in ${\mathcal {C}}$ approaches zero as n increases.