Using $$P_\tau $$ property for designing bent functions provably outside the completed Maiorana–McFarland class

In this article, we identify certain instances of bent functions, constructed using the so-called \(P_\tau \) property, that are provably outside the completed Maiorana–McFarland (\({\mathcal{M}\mathcal{M}}^\#\)) class. This also partially answers an open problem in posed by Kan et al. (IEEE Trans Inf Theory, https://doi.org/10.1109/TIT.2022.3140180, 2022). We show that this design framework (using the \(P_\tau \) property), can provide instances of bent functions that are outside the known classes of bent functions, including the classes \({\mathcal{M}\mathcal{M}}^\#\), \({{\mathcal {C}}},{{\mathcal {D}}}\) and \({{\mathcal {D}}}_0\), where the latter three were introduced by Carlet in the early nineties. We provide two generic methods for identifying such instances, where most notably one of these methods uses permutations that may admit linear structures. For the first time, a set of sufficient conditions for the functions of the form \(h(y,z)=Tr(y\pi (z)) + G_1(Tr_1^m(\alpha _1y),\ldots ,Tr_1^m(\alpha _ky))G_2(Tr_1^m(\beta _{k+1}z),\ldots ,Tr_1^m(\beta _{\tau }z))+ G_3(Tr_1^m(\alpha _1y),\ldots ,Tr_1^m(\alpha _ky))\) to be bent and outside \({\mathcal{M}\mathcal{M}}^\#\) is specified without a strong assumption that the components of the permutation \(\pi \) do not admit linear structures.