Using P 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 (MM#) 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 theP tau property), can provide instances of bent functions that are outside the known classes of bent functions, including the classes MM#,C,D and (0)(D), where the latter three were introduced by Carlet in the early nineties. We pro-vide 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)1(alpha(k)y),...,Tr-1(m)(alpha ky))G(2)(Trm(1)(beta k+1z),...,Trm(1)(beta tau z))+G(3)(Tr-1(m)(alpha(1)y),...,Tr-1(m)(alpha(k)y)) to be bent and outside MM# is specified without a strong assumption that the components of the permutation pi do not admit linear structures.