In this letter, we present a construction of bent functions which generalizes a work of Zhang et al. in 2016. Based on that, we obtain a cubic bent function in 10 variables and prove that, it has no ...affine derivative and does not belong to the completed Maiorana-McFarland class, which is opposite to all 6/8-variable cubic bent functions as they are inside the completed Maiorana-McFarland class. This is the first time a theoretical proof is given to show that the cubic bent functions in 10 variables can be outside the completed Maiorana-McFarland class. Before that, only a sporadic example with such properties was known by computer search. We also show that our function is EA-inequivalent to that sporadic one.
In this article, we provide the first systematic analysis of bent functions <inline-formula> <tex-math notation="LaTeX">f </tex-math></inline-formula> on <inline-formula> <tex-math ...notation="LaTeX">\mathbb {F}_{2}^{n} </tex-math></inline-formula> in the Maiorana-McFarland class <inline-formula> <tex-math notation="LaTeX">\mathcal {M} </tex-math></inline-formula> regarding the origin and cardinality of their <inline-formula> <tex-math notation="LaTeX">\mathcal {M} </tex-math></inline-formula> -subspaces, i.e., vector subspaces such that for any two elements <inline-formula> <tex-math notation="LaTeX">a,b </tex-math></inline-formula> from this subspace, the second-order derivative <inline-formula> <tex-math notation="LaTeX">D_{a}D_{b}f </tex-math></inline-formula> is the zero function on <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{2}^{n} </tex-math></inline-formula>. By imposing restrictions on permutations <inline-formula> <tex-math notation="LaTeX">\pi </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{2}^{n/2} </tex-math></inline-formula>, we specify the conditions so that Maiorana-McFarland bent functions <inline-formula> <tex-math notation="LaTeX">f(x,y)=x\cdot \pi (y) + h(y) </tex-math></inline-formula> admit a unique <inline-formula> <tex-math notation="LaTeX">\mathcal {M} </tex-math></inline-formula>-subspace of dimension <inline-formula> <tex-math notation="LaTeX">n/2 </tex-math></inline-formula>. On the other hand, we show that permutations <inline-formula> <tex-math notation="LaTeX">\pi </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">\mathcal {M} </tex-math></inline-formula>-subspaces of a fixed dimension is invariant under equivalence. Additionally, we give several generic methods of specifying permutations <inline-formula> <tex-math notation="LaTeX">\pi </tex-math></inline-formula> so that <inline-formula> <tex-math notation="LaTeX">f\in \mathcal {M} </tex-math></inline-formula> admits a unique <inline-formula> <tex-math notation="LaTeX">\mathcal {M} </tex-math></inline-formula>-subspace. Most notably, using the knowledge about <inline-formula> <tex-math notation="LaTeX">\mathcal {M} </tex-math></inline-formula>-subspaces, we show that using the bent 4-concatenation of four suitably chosen Maiorana-McFarland bent functions on <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{2}^{n-2} </tex-math></inline-formula>, one can in a generic manner generate bent functions on <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{2}^{n} </tex-math></inline-formula> outside the completed Maiorana-McFarland class <inline-formula> <tex-math notation="LaTeX">\mathcal {M}^{\#} </tex-math></inline-formula> for any even <inline-formula> <tex-math notation="LaTeX">n\geq 8 </tex-math></inline-formula>. Remarkably, with our construction methods, it is possible to obtain inequivalent bent functions on <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{2}^{8} </tex-math></inline-formula> not stemming from the two primary classes, the partial spread class <inline-formula> <tex-math notation="LaTeX">\mathcal {PS} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\mathcal {M} </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">\mathcal {PS} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\mathcal {M} </tex-math></inline-formula>, whereas their total number on <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{2}^{8} </tex-math></inline-formula> is approximately 2106.
In this survey, we revisit the Rothaus paper and the chapter of Dillon’s thesis dedicated to bent functions, and we describe the main results obtained on these functions during these last 40 years. ...We also cover more briefly super-classes of Boolean functions, vectorial bent functions and bent functions in odd characteristic.
In early nineties Carlet (1994) introduced two new classes of bent functions, both derived from the Maiorana–McFarland (M) class, and named them C and D class, respectively. Apart from a subclass of ...D, denoted by D0 by Carlet, which is provably outside two main (completed) primary classes of bent functions, little is known about their efficient constructions. More importantly, both classes may easily remain in the underlying M class which has already been remarked in Mandal et al. (2016). Assuming the possibility of specifying a bent function f that belongs to one of these two classes (apart from D0), the most important issue is then to determine whether f is still contained in the known primary classes or lies outside their completed versions. In this article, we further elaborate on the analysis of the set of sufficient conditions given in Zhang et al. (2017) concerning the specification of bent functions in C and D which are provably outside M. It is shown that these conditions, related to bent functions in class D, can be relaxed so that even those permutations whose component functions admit linear structures still can be used in the design. It is also shown that monomial permutations of the form x2r+1 have inverses which are never quadratic for n>4, which gives rise to an infinite class of bent functions in C but outside M. Similarly, using a relaxed set of sufficient conditions for bent functions in D and outside M, one explicit infinite class of such bent functions is identified. We also extend the inclusion property of certain subclasses of bent functions in C and D, as addressed initially in Carlet (1994), Mandal et al. (2016) that are ultimately within the completed M class. Most notably, we specify other generic and explicit subclasses of D, denoted by Dk⋆ for k∈{1,…,n−2}, whose members are bent functions provably outside the completed M class.
Bent functions are optimal combinatorial objects. Since their introduction, substantial efforts have been directed toward their study in the last three decades. A complete classification of bent ...functions is elusive and looks hopeless today, therefore, not only their characterization, but also their generation are challenging problems. This paper is devoted to the construction of bent functions. First, we provide several new effective constructions of bent functions, self-dual bent functions, and antiself-dual bent functions. Second, we provide seven new infinite families of bent functions by explicitly calculating their dual.
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 of vectorial bent-negabent functions and show that in general for a vectorial bent-negabent function <inline-formula> <tex-math notation="LaTeX">F\colon {\mathbb {F}} _{2}^{2m} \rightarrow {\mathbb {F}} _{2}^{k} </tex-math></inline-formula> we necessarily have that <inline-formula> <tex-math notation="LaTeX">k \leq m-1 </tex-math></inline-formula>. We specify a class of vectorial bent-negabent functions of maximal output dimension <inline-formula> <tex-math notation="LaTeX">m-1 </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">F\colon {\mathbb {F}} _{2}^{2m} \rightarrow {\mathbb {F}} _{2}^{k} </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">m \leq k \leq 2m </tex-math></inline-formula>, and identify some families of these functions reaching this upper bound.
A class of bent functions which contains bent functions with various properties like regular, weakly regular and not weakly regular bent functions in even and in odd dimension, is analyzed. It is ...shown that this class includes the Maiorana–McFarland class as a special case. Known classes and examples of bent functions in odd characteristic are examined for their relation to this class. In the second part, normality for bent functions in odd characteristic is analyzed. It turns out that differently to Boolean bent functions, many – also quadratic – bent functions in odd characteristic and even dimension are not normal. It is shown that regular Coulter–Matthews bent functions are normal.
Two new classes of bent functions derived from the Maiorana–McFarland (M) class, so-called C and D, were introduced by Carlet (1993) almost three decades ago. In Zhang (2020) sufficient conditions ...for specifying bent functions in C and D which are outside the completed M class, denoted by M#, were given. Furthermore in Pasalic et al. (2021) the notion of vectorial bent functions which are weakly or strongly outsideM#, referring respectively to the case whether some or all nonzero linear combinations (called components) of its coordinate functions are in class C (or D) but provably outside M#, was introduced. In this article we continue the work of finding new instances of vectorial bent functions weakly/strongly outside M# using a different approach. Namely, a generic method for the construction of vectorial bent (n,t)-functions of the form F(x,y)=G(x,y)+H(x,y), n=2m,t|m, was recently proposed in Bapić (2021), where G is a given bent (n,t)-function satisfying certain properties and H is an arbitrary (t,t)-function having certain form. We introduce a new superclass of bent functions SC which contains the classes D0 and C and whose members are provably outside M#. Most notably, using indicators of the form 1L⊥(x,y)+δ0(x) to define members of this class leads for the first time to modifications of the M class performed on sets rather than on affine subspaces. We also show that for suitable choices of H, the function F is a vectorial bent function weakly/strongly outside the class M#. In this context, a new concept of being almost strongly outside M# is introduced and some families of vectorial bent functions with this property are given. Furthermore, we provide two new families of vectorial bent functions strongly outside M# (considered to be an intrinsically hard problem) whose output dimension is greater than 2, thus giving first examples of such functions in the literature.
In Pasalic et al. (2016) a construction allowing for high levels of modification was presented. It can be used to construct several important combinatorial structures, among them are examples of ...complete permutations. Here the method is used to construct infinite classes of generalised complete permutations F(x), where both F(x) and F(x)+hD(x) are permutations, not just F(x)+x. In the article three families of the function hD(x) are considered: hD(x) being a function multiplying the vector x with a vector D, a permutation matrix D, or a linear mapping matrix D. Most existing results related to complete permutations use the finite field notation, while in this article we are developing permutations based on the vector space structure. The case where D is a binary vector needs to be emphasised. In this case the permutation F(x) is defined in such a way that F(x)+DTx remains a permutation for any of the 2n−3 vectors D as defined in Eq. (5). Let S be the set of all such vectors. We present for arbitrary n>4 construction of an S-complete permutation F(x), where |S|=2n−3. Additionally, we prove that each of these permutations F has such a corresponding linear subspace L that the pair (F,L) satisfies the (C)-property and can be used to construct a huge infinite class of bent functions in Carlet’s C class. In Mandal et al. (2016) it was proven that finding such pairs (F,L) is a difficult problem.
Two new classes of bent functions derived from the Maiorana–McFarland (M) class, so-called C and D, were introduced by Carlet (1994) two decades ago. The difficulty of satisfying their defining ...conditions was emphasized in Mandal et al. (2016). In a recent work Zhang et al. (2017) a set of efficient sufficient conditions for specifying bent functions in C and D which are outside the completed M class, denoted by M#, was given. A natural follow up question is whether there is a possibility of extending this approach to the vectorial case. We introduce the property of vectorial bent functions that we call weakly or strongly outsideM#, referring respectively to the case whether some or all nonzero linear combinations (called components) of its coordinate functions are in class C (or D) but provably outside M#. For the first time, quite different to a straightforward vectorial extension of the Maiorana–McFarland class and the class of Dillon PSap, we show the existence of several classes of vectorial bent functions whose component functions come from different classes of bent functions, mainly from M and D, and in many cases being weakly outside M#. We also address a difficult problem of specifying vectorial bent functions whose all components are in class C but provably outside M#, thus being strongly outside M#. Even though we could only specify a class of such functions whose dimension of bent vector space is only two, thus F:F2n→F22, this is the very first evidence of their existence.
