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.

Rotation-symmetric (RS) bent functions, which are invariant under the action of the cyclic group, have attracted significant attention over the past three decades due to their importance in cryptographic applications. For the last three decades of research on these objects, most known RS bent functions have been obtained by applying extended-affine equivalence to specific Maiorana-McFarland bent functions, in a way that ensures the resulting function retains invariance under the cyclic group action. Due to the intrinsic difficulty of characterizing RS bent functions that do not belong to the completed Maiorana-McFarland classM#, there has been no evidence for the existence of such functions until now. In this paper, we provide, for the first time, a solution to this problem. First, we perform a computational classification of RS cubic bent functions in ten variables under extended-affine equivalence and demonstrate that one of the resulting classes is outside theM#class. Next, we prove that an infinite family of RS bent functions on Fn2of maximum algebraic degreen/2 (Su, Adv. Math. Commun. 13(2): 253–265, 2019) does not belong toM#, for alln≥ 8. Finally, we show that a family of quartic RS bent functions on Fn2(Carlet, Gao, and Liu, J. Comb. Theory Ser. A 127: 161–175, 2014), does not belong to theM#class for infinitely manyn.

Abstract In Pasalic et al. (IEEE Trans Inf Theory 69:2702–2712, 2023), and in Anbar and Meidl (Cryptogr Commun 10:235–249, 2018), two different vectorial negabent and vectorial bent-negabent concepts are introduced, which leads to seemingly contradictory results. One of the main motivations for this article is to clarify the differences and similarities between these two concepts. Moreover, the negabent concept is extended to generalized Boolean functions from $${\mathbb {F}}_2^n$$ F 2 n to the cyclic group $${\mathbb {Z}}_{2^k}$$ Z 2 k . It is shown how to obtain nega- $${\mathbb {Z}}_{2^k}$$ Z 2 k -bent functions from $${\mathbb {Z}}_{2^k}$$ Z 2 k -bent functions, or equivalently, corresponding non-splitting relative difference sets from the splitting relative difference sets. This generalizes the shifting results for Boolean bent and negabent functions. We finally point to constructions of $${\mathbb {Z}}_8$$ Z 8 -bent functions employing permutations with the $$({\mathcal {A}}_m)$$ ( A m ) property, and more generally we show that the inverse permutation gives rise to $${\mathbb {Z}}_{2^k}$$ Z 2 k -bent functions.

Every Boolean bent function f can be written either as a concatenation f = f1|| f2 of two complementary semi-bent functions f1, f2

Every Boolean bent function$f$can be written either as a concatenation$f=f_{1}\Vert f_{2}$of two complementary semi-bent functions$f_{1}, f_{2}$; or as a concatenation$f=f_{1}\Vert f_{2}\Vert f_{3}\Vert f_{4}$of four Boolean functions$f_{1}, f_{2}, f_{3}, f_{A}$, all of which are simultaneously bent, semi-bent, or 5-valued spectra-functions. In this context, it is essential to ask: When does a bent concatenation$f$(not) belong to the completed Maiorana-McFarland class${\mathcal{M}}^{\# }{?}$In this article, we answer this question completely by providing a full characterization of the structure of$\mathcal{M}$-subspaces for the concatenation of the form$f=f_{1}\Vert f_{2}$and$f=f1\Vert f2\Vert f\mathrm{s}\Vert f4$, which allows us to specify the necessary and sufficient conditions so that$f$is outside$\mathcal{M} \neq$. Based on these conditions, we propose several explicit design methods of specifying bent functions outside$\mathcal{M}^{\#}$in the special case when$f=g\Vert h\Vert g\Vert (h+1)$, where$g$and$h$are bent functions.

In this article, we provide the first systematic analysis of bent functions f on Fn2 in the Maiorana-McFarland class M regarding the origin and cardinality of their M-subspaces, i.e., vector subspaces such that for any two elements a, b from this subspace, the second-order derivative DaDbf is the zero function on Fn2. By imposing restrictions on permutations π of Fn/2 2, we specify the conditions so that Maiorana-McFarland bent functions f(x, y) = x · π(y) + h(y) admit a unique M-subspace of dimension n/2. On the other hand, we show that permutations π 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 M-subspaces of a fixed dimension is invariant under equivalence. Additionally, we give several generic methods of specifying permutations π so that f ∈ M admits a unique M-subspace. Most notably, using the knowledge about M-subspaces, we show that using the bent 4-concatenation of four suitably chosen Maiorana-McFarland bent functions on Fn−2 2, one can in a generic manner generate bent functions on Fn2 outside the completed Maiorana-McFarland class M# for any even n ≥ 8. Remarkably, with our construction methods, it is possible to obtain inequivalent bent functions on F82 not stemming from the two primary classes, the partial spread class PS and 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 PS and M, whereas their total number on F82 is approximately 2106.

Boolean bent functions which at the same time have a flat nega-Hadamard transform are called bent-negabent functions. The known families of these functions mostly stem from the Maiorana-McFarland class of bent functions and their vectorial counterparts have not been considered in the literature. In this article, we introduce the notion ofvectorial bent-negabentfunctions and show that in general for a vectorial bent-negabent function$F\colon {\mathbb {F}} _{2}^{2m} \rightarrow {\mathbb {F}} _{2}^{k}$we necessarily have that$k \leq m-1$. We specify a class of vectorial bent-negabent functions of maximal output dimension$m-1$by using a set of linear complete mappings. On the other hand, we propose several methods (one of which is generic) of specifying vector spaces of nonlinear complete mappings which then induce vectorial bent-negabent functions (whose dimension is not maximal) having a certain number of component functions outside the completed Maiorana-McFarland class. Finally, we derive an upper bound on the maximum number of bent-negabent components for mappings$F\colon {\mathbb {F}} _{2}^{2m} \rightarrow {\mathbb {F}} _{2}^{k}$, where$m \leq k \leq 2m$, and identify some families of these functions reaching this upper bound.