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].

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.

In this paper, we show that the maximum number of bent component functions of a vectorial function F : GF(2)n→ GF(2)nis 2n- 2n/2. We also show that it is very easy to construct such functions. However, it is a much more challenging task to find such functions in polynomial form F ∈ GF(2n)[x], where F has only a few terms. The only known power functions having such a large number of bent components are xd, where d = 2n/2+ 1. In this paper, we show that the binomials Fi(x) = x2i(x + x(2n/2)) also have such a large number of bent components, and these binomials are inequivalent to the monomials x(2n/2+1) if 0n/2+1). We also determine the complete Walsh spectrum of our functions when n/2 is odd and gcd(i, n/2) = 1.

Bent functions can be classified into regular bent functions, weakly regular but not regular bent functions, and non-weakly regular bent functions. Regular and weakly regular bent functions always appear in pairs, since their duals are also bent functions. In general, this does not apply to non-weakly regular bent functions. However, the first known construction of non-weakly regular bent functions by Çeşmelioğlu et al. yields bent functions for which the dual is also bent. In this paper, the first construction of non-weakly regular bent functions for which the dual is not bent is presented. We call such functions non-dual-bent functions. Until now, only sporadic examples found via computer search were known. We then show that with the direct sum of bent functions and with the construction by Çeşmelioğlu et al., one can obtain infinitely many non-dual-bent functions once one example of a non-dual-bent function is known.

We show that bent functions $f$ from ${\mathbb F}_{p^m}\times{\mathbb F}_{p^m}$ to ${\mathbb F}_p$, which are constant or affine on the elements of a given spread of ${\mathbb F}_{p^m}\times{\mathbb F}_{p^m}$, either arise from partial spread bent functions, or they are Boolean and a generalization of Dillon's class $H$. For spreads of a presemifield $S$, we show that a bent function of the second class corresponds to an o-polynomial of a presemifield in the Knuth orbit of $S$. In contrast to the finite fields case, we have to consider pairs of (pre)semifields in a Knuth orbit. We give a canonical example of an o-polynomial for commutative presemifields (which also defines a hyperoval on the semifield plane) and show that the corresponding bent functions belong to the completed Maiorana--McFarland class. Using Albert's twisted fields and Kantor's family of presemifields, we explicitly present examples of such bent functions.

We present necessary and sufficient conditions for a Boolean function to be a negabent function for both an even and an odd number of variables, which demonstrates the relationship between negabent functions and bent functions. By using these necessary and sufficient conditions for Boolean functions to be negabent, we obtain that the nega spectrum of a negabent function has at most four values. We determine the nega spectrum distribution of negabent functions. Further, we provide a method to construct bent-negabent functions innvariables (neven) of algebraic degree ranging from 2 to [(n)/2], which implies that the maximum algebraic degree of ann-variable bent-negabent function is equal to [(n)/2]. Thus, we answer two open problems proposed by Parker and Pott and by Stănică et al.

In 2008, Cusick et al conjectured that certain elementary symmetric Boolean functions of the form σ(2t+1)l-1, (2t) are the only nonlinear balanced ones, wheret,lare any positive integers, and σn,d=⊕1≤( i1)d≤ n xi1xi2⋯xid) for positive integersn, 1 ≤d≤n. In this paper, by analyzing the weight of σn,(2t) and σn, d, we prove that wt σn,dn-1holds in most cases, and so does the conjecture. According to the remainder modulo 4, we also consider the weight of σn, dfrom two aspects: n ≠ 3(mod 4) and n not ≡ 3(mod 4). In particular, our results not only cover the most known results, but also contain some new cases. Thus, we can reduce the conjecture to few remaining cases. We do not fully solve the conjecture, but we also consider the weight of σn, (2t+2s) and also give some experimental results on it.

New infinite classes of almost bent and almost perfect nonlinear polynomials are constructed. It is shown that they are affine inequivalent to any sum of a power function and an affine function

A new almost-perfect nonlinear function (APN) on F(2/sup 10/) which is not equivalent to any of the previously known APN mappings is constructed. This is the first example of an APN mapping which is not equivalent to a power mapping.

Shows that almost perfect autocorrelation sequences are the same as certain cyclic divisible difference sets. The authors apply known theorems on divisible difference sets to simplify and strengthen results by J. Wolfmann [1992]. In particular, they answer two questions which were posed by Wolfman.>