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 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.
The Walsh-Hadamard transform is a powerful tool to investigate interference-resist capabilities and cryptographic of functions which have a wide array of applications in coding theory and ...cryptography. It is interesting to find functions with few Walsh-Hadamard transform values (spectral amplitudes) and to determine their distributions. In this letter, a new modified generalized Maiorana-McFarland (MGMM) construction is presented. A collection of MGMM classes of functions of few spectral amplitudes can be obtained by using the proposed construction. The constructed functions have determined spectral amplitude distributions. As a class of these MGMM functions, generalized 3-ary functions of two nonzero spectral amplitudes 3^{n/2} and 3^{n/2+1} are then exploited to construct spreading sequences for even n. Moreover, an efficient assignment is presented to provide that the smallest distance between pairs of sequences of correlation 3^{n/2+1} is 3, which implies that the spreading sequences based on these functions in our assignment have better interference-resist capability than the spreading sequences based on ternary semi-bent functions.
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 F : F 2 m 2 → F k 2 we necessarily have that k ≤ 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 : F 2 m 2 → F k 2 , where m ≤ k ≤ 2 m , and identify some families of these functions reaching this upper bound.
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.
A function <inline-formula> <tex-math notation="LaTeX">F: \mathbb {F}_{2}^{n}\rightarrow \mathbb {F} _{2}^{n} </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">n=2m ...</tex-math></inline-formula>, can have at most <inline-formula> <tex-math notation="LaTeX">2^{n}-2^{m} </tex-math></inline-formula> bent component functions. Trivial examples are vectorial bent functions from <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{2}^{n} </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{2}^{m} </tex-math></inline-formula>, seen as functions on <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{2}^{n} </tex-math></inline-formula>. The first nontrivial example is given in univariate form as <inline-formula> <tex-math notation="LaTeX">x^{2^{r}} {\rm Tr^{n}_{m}}(x), 1\le r < m </tex-math></inline-formula> (Pott et al. 2018), a few more examples of similar shape are given by Mesnager et al. 2019, and finally it has been shown that the quadratic function <inline-formula> <tex-math notation="LaTeX">F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x)) </tex-math></inline-formula>, has <inline-formula> <tex-math notation="LaTeX">2^{n}-2^{m} </tex-math></inline-formula> bent components if and only if <inline-formula> <tex-math notation="LaTeX">\Lambda </tex-math></inline-formula> is a linearized permutation polynomial of <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{2^{m}}x </tex-math></inline-formula> (Anbar et al. 2021). In the first part of this article, an upper bound for the nonlinearity of plateaued functions with <inline-formula> <tex-math notation="LaTeX">2^{n}-2^{m} </tex-math></inline-formula> bent components is shown, which is attained by the example <inline-formula> <tex-math notation="LaTeX">x^{2^{r}} {\rm Tr^{n}_{m}}(x) </tex-math></inline-formula>. We then analyse in detail nonlinearity and differential spectrum of the class of functions <inline-formula> <tex-math notation="LaTeX">F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x)) </tex-math></inline-formula>, which, as will be seen, requires the study of the functions <inline-formula> <tex-math notation="LaTeX">x^{2^{r}}\Lambda (x) </tex-math></inline-formula>. In the last part we demonstrate that this class belongs to a larger class of functions with <inline-formula> <tex-math notation="LaTeX">2^{n}-2^{m} </tex-math></inline-formula> Maiorana-McFarland bent components, which also contains nonquadratic and non-plateaued functions.
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.
Minimal linear codes have important applications in secret sharing schemes and secure multi-party computation, etc. In this paper, we study the minimality of a kind of linear codes over GF(p) from ...Maiorana-McFarland functions. We first obtain a new sufficient condition for this kind of linear codes to be minimal without analyzing the weights of its codewords, which is a generalization of some works given by Ding et al. in 2015. Using this condition, it is easy to verify that such minimal linear codes satisfy wminwmax≤p−1p for any prime p, where wmin and wmax denote the minimum and maximum nonzero weights in a code, respectively. Then, by selecting the subsets of GF(p)s, we present two new infinite families of minimal linear codes with wminwmax≤p−1p for any prime p. In addition, the weight distributions of the presented linear codes are determined in terms of Krawtchouk polynomials or partial spreads.
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.
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.
