Design and Analysis of Bent Functions Using M-Subspaces

In this article, we provide the first systematic analysis of bent functions
$f$
on
$\mathbb {F}_{2}^{n}$
in the Maiorana-McFarland class
$\mathcal {M}$
regarding the origin and cardinality of their
$\mathcal {M}$
-subspaces, i.e., vector subspaces such that for any two elements
$a,b$
from this subspace, the second-order derivative
$D_{a}D_{b}f$
is the zero function on
$\mathbb {F}_{2}^{n}$
. By imposing restrictions on permutations
$\pi $
of
$\mathbb {F}_{2}^{n/2}$
, we specify the conditions so that Maiorana-McFarland bent functions
$f(x,y)=x\cdot \pi (y) + h(y)$
admit a unique
$\mathcal {M}$
-subspace of dimension
$n/2$
. On the other hand, we show that permutations
$\pi $
with linear structures give rise to Maiorana-McFarland bent functions that do not have this property. In this way, we contribute to the classification of Maiorana-McFarland bent functions, since the number of
$\mathcal {M}$
-subspaces of a fixed dimension is invariant under equivalence. Additionally, we give several generic methods of specifying permutations
$\pi $
so that
$f\in \mathcal {M}$
admits a unique
$\mathcal {M}$
-subspace. Most notably, using the knowledge about
$\mathcal {M}$
-subspaces, we show that using the bent 4-concatenation of four suitably chosen Maiorana-McFarland bent functions on
$\mathbb {F}_{2}^{n-2}$
, one can in a generic manner generate bent functions on
$\mathbb {F}_{2}^{n}$
outside the completed Maiorana-McFarland class
$\mathcal {M}^{\#}$
for any even
$n\geq 8$
. Remarkably, with our construction methods, it is possible to obtain inequivalent bent functions on
$\mathbb {F}_{2}^{8}$
not stemming from the two primary classes, the partial spread class
$\mathcal {PS}$
and
$\mathcal {M}$
. In this way, we contribute to a better understanding of the origin of bent functions in eight variables, since only a small fraction of about 276 bent functions stems from
$\mathcal {PS}$
and
$\mathcal {M}$
, whereas their total number on
$\mathbb {F}_{2}^{8}$
is approximately 2106.