Secondary Constructions of Plateaued Boolean Functions Through Addition of Indicators

Recently, the design of s-plateaued functions over Fn2, also known as 3-valued Walsh spectra functions (taking the values from the set {0,±2n+s/2 }), has been considered using an algorithmic approach (for example, IEEE Trans. Inf. Theory, vol. 70, no. 2, pp. 1408–1421, 2024) for the purpose of specifying balanced plateaued functions of maximal algebraic degree (called optimal). In this article, we identify these optimal plateaued functions within the Generalized Maiorana-McFarland (GMM) class and provide sufficient conditions that these functions do not admit nonzero linear structures. Moreover, we apply a similar technique to the one C. Carlet used for modifying bent functions in the Maiorana-McFarland (M) class (deriving the so-calledCandDclasses), and we obtain new classes of plateaued functions from theGMMclass. However, when the function ϕ in the definition off(x, y)=x·ϕ(y)+δ0(x) is injective, then we cannot derive the subclassD0as in the case of bent functions. We show that this is still possible when ϕ is not injective even though the sufficient conditions become more complicated compared to the bent case. Moreover, to ensure plateauedness within theCclass, we were forced to impose certain conditions on the dual of a plateaued function, which is harder to handle compared to the bent case since the dual is defined on a subset/subspace of the ambient space. In general, using indicator functions of the form 1E1(x)1E2(y), where the subspacesE1⊆ Fk2andE2⊆ Fn−k2are suitably chosen, we show that it is possible to modify the functiong(x, y)=x· ϕ(y) ∈GMMnkwhile preserving plateauedness, so thatf(x, y)=x· ϕ(y) + 1E1(x)1E2(y) is outside theGMMnkclass.