Abstract We study combinatorial properties of plateaued functions $$F :\mathbb {F}_p^n \rightarrow \mathbb {F}_p^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mo>→</mml:mo> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> <mml:mi>m</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> . All quadratic functions, bent functions and most known APN functions are plateaued, so many cryptographic primitives rely on plateaued functions as building blocks. The main focus of our study is the interplay of the Walsh transform and linearity of a plateaued function, its differential properties, and their value distributions, i.e., the sizes of image and preimage sets. In particular, we study the special case of “almost balanced” plateaued functions, which only have two nonzero preimage set sizes, generalising, for instance, all monomial functions. We achieve several direct connections and (non)existence conditions for these functions, showing in particular that plateaued d -to-1 functions (and thus plateaued monomials) only exist for a very select choice of d , and we derive for all these functions their linearity as well as bounds on their differential uniformity. We also specifically study the Walsh transform of plateaued APN functions and their relation to their value distribution.

Bent functions are Boolean functions in an even number of variables that are indicators of Hadamard difference sets in elementary abelian 2-groups. A bent functionfinmvariables is said to be normal if it is constant on an affine space of dimensionm/2. In this paper, we demonstrate that all bent functions in m = 8 variables, whose exact count, determined by Langevin and Leander (Des. Codes Cryptogr. 59(1–3): 193–205, 2011), is approximately 2106, share a common algebraic property: every 8-variable bent function is normal, up to the addition of a linear function. With this result, we complete the analysis of the normality of bent functions for the last unresolved case,m= 8. It is already known that all bent functions inmvariables are normal form≤ 6, while form≥ 10, there exist bent functions that cannot be made normal by adding linear functions. Consequently, we provide a complete solution to an open problem by Charpin (J. Complex. 20(2-3): 245-265, 2004).

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.

Non-normal Boolean bent functions are one of the least understood classes of bent functions, and only a few difficult-to-find examples of such functions are known. In this paper, we consider the following two open problems on the normality of bent functions: 1. Do non-normal bent functions in 8 variables and degree 4 exist? 2. Do non-normal bent functions in the PS − ∖ PS a p class exist? We solve both of these problems by finding among the known PS bent functions in n = 8 variables a non-normal bent function in the PS − ∖ PS a p class.

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

Almost perfect nonlinear (APN) functions yield the best possible resistance to differential attacks when used as a substitution box in the design of a block cipher. Constructing APN functions is a non-trivial problem and so far there is only one sporadic example that is not equivalent to either a monomial or a quadratic function. This is the Brinkmann-Leander-Edel-Pott function that is defined in 6 variables. In this paper, we consider in detail an original construction approach of this function suggested by Suder, which is based on the application of antiderivatives in conjunction with fast point spaces. We generalize this approach to higher dimensions and show that it does not yield any cubic APN function in 7 variables.

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.

Abstract Recently, several interesting constructions of vectorial Boolean functions with the maximum number of bent components (MNBC functions, for short) were proposed. However, many of them have component functions from the completed Maiorana-McFarland class $${\mathcal {M}}^{\#}$$ M # . Moreover, no examples of MNBC functions containing component functions provably outside $${\mathcal {M}}^{\#}$$ M # are known. In this paper, we classify all MNBC functions in six variables. Based on the analysis of the obtained equivalence classes, we propose several infinite families of MNBC functions with component functions outside the $${\mathcal {M}}^{\#}$$ M # class. In particular, two of our new constructions are solutions to the open problem [Bapić et al (eds) Proceedings of the twelfth international workshop on coding and cryptography, 2022, Item 1., p. 9].

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.

There are two construction methods of designs from$(n,m)$-bent functions, known as translation and addition designs. In this article we analyze, which equivalence relation for Boolean bent functions, i.e.$(n,1)$-bent functions, and vectorial bent functions, i.e.$(n,m)$-bent functions with$2\le m\le n/2$, is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs. First, we observe that similar to the Boolean bent functions, extended-affine equivalence of vectorial$(n,m)$-bent functions and isomorphism of addition designs are the same concepts for all even$n$and$m\le n/2$. Further, we show that extended-affine inequivalent Boolean bent functions in$n$variables, whose translation designs are isomorphic, exist for all$n\ge 6$. This implies, that isomorphism of translation designs for Boolean bent functions is a coarser equivalence relation than extended-affine equivalence. However, we do not observe the same phenomenon for vectorial bent functions in a small number of variables. We classify and enumerate all vectorial bent functions in six variables and show, that in contrast to the Boolean case, one cannot exhibit isomorphic translation designs from extended-affine inequivalent vectorial$(6,m)$-bent functions with$m\in \{ 2,3 \}$.

Abstract In this paper we prove that in opposite to the cases of 6 and 8 variables, the Maiorana-McFarland construction does not describe the whole class of cubic bent functions in n variables for all $$n\ge 10$$ n ≥ 10 . Moreover, we show that for almost all values of n, these functions can simultaneously be homogeneous and have no affine derivatives.

For a function $f$ from $\mathbb {F}_{2}^{n}$ to $\mathbb {F}_{2}^{n}$ , the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of $f$ can be regarded as a measure of the distance between $f$ and the set of almost perfect nonlinear functions. In some cases, the number of vanishing flats serves as an “intermediate” concept between differential uniformity and differential spectrum, which contains more information than differential uniformity, however less than the differential spectrum. Secondly, the set of vanishing flats forms a combinatorial configuration called partial quadruple system, since it conveys a detailed structural information about $f$ . We initiate this study by considering the number of vanishing flats and the partial quadruple systems associated with monomials and Dembowski-Ostrom polynomials. In addition, we present an application of vanishing flats to the partition of a vector space into disjoint equidimensional affine spaces. We conclude the paper with several further questions and challenges.