Maiorana-McFarland Class Research Articles

Using Pτ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_\ au $$\\end{document} property for designing bent functions provably outside the completed Maiorana–McFarland class

In this article, we identify certain instances of bent functions, constructed using the so-called Pτ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_\ au $$\\end{document} property, that are provably outside the completed Maiorana–McFarland (MM#\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal{M}\\mathcal{M}}^\\#$$\\end{document}) class. This also partially answers an open problem in posed by Kan et al. (IEEE Trans Inf Theory, https://doi.org/10.1109/TIT.2022.3140180, 2022). We show that this design framework (using the Pτ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_\ au $$\\end{document} property), can provide instances of bent functions that are outside the known classes of bent functions, including the classes MM#\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal{M}\\mathcal{M}}^\\#$$\\end{document}, C,D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {C}}},{{\\mathcal {D}}}$$\\end{document} and D0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {D}}}_0$$\\end{document}, where the latter three were introduced by Carlet in the early nineties. We provide two generic methods for identifying such instances, where most notably one of these methods uses permutations that may admit linear structures. For the first time, a set of sufficient conditions for the functions of the form h(y,z)=Tr(yπ(z))+G1(Tr1m(α1y),…,Tr1m(αky))G2(Tr1m(βk+1z),…,Tr1m(βτz))+G3(Tr1m(α1y),…,Tr1m(αky))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h(y,z)=Tr(y\\pi (z)) + G_1(Tr_1^m(\\alpha _1y),\\ldots ,Tr_1^m(\\alpha _ky))G_2(Tr_1^m(\\beta _{k+1}z),\\ldots ,Tr_1^m(\\beta _{\ au }z))+ G_3(Tr_1^m(\\alpha _1y),\\ldots ,Tr_1^m(\\alpha _ky))$$\\end{document} to be bent and outside MM#\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal{M}\\mathcal{M}}^\\#$$\\end{document} is specified without a strong assumption that the components of the permutation π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} do not admit linear structures.

Infinite families of minimal binary codes via Krawtchouk polynomials

Linear codes play a crucial role in various fields of engineering and mathematics, including data storage, communication, cryptography, and combinatorics. Minimal linear codes, a subset of linear codes, are particularly essential for designing effective secret sharing schemes. In this paper, we introduce several classes of minimal binary linear codes by carefully selecting appropriate Boolean functions. These functions belong to a renowned class of Boolean functions, namely, the general Maiorana–McFarland class. We employ a method first proposed by Ding et al. (IEEE Trans Inf Theory 64(10):6536–6545, 2018) to construct minimal codes violating the Ashikhmin–Barg bound (wide minimal codes) by using Krawtchouk polynomials. The lengths, dimensions, and weight distributions of the obtained codes are determined using the Walsh spectrum distribution of the chosen Boolean functions. Our findings demonstrate that a vast majority of the newly constructed codes are wide minimal. Furthermore, our proposed codes exhibit a significantly larger minimum distance, in some cases, compared to some existing similar constructions. Finally, we address this method, based on Krawtchouk polynomials, more generally, and highlight certain generic properties related to it. These general results offer insights into the scope of this approach.

On a Lower Bound for the Number of Bent Functions at the Minimum Distance from a Bent Function in the Maiorana–McFarland Class

On a Lower Bound for the Number of Bent Functions at the Minimum Distance from a Bent Function in the Maiorana–McFarland Class

Further investigations on permutation based constructions of bent functions

Constructing bent functions by composing a Boolean function with a permutation was introduced by Hou and Langevin in 1997. The approach appears simple but heavily depends on the construction of desirable permutations. In this paper, we further study this approach by investigating the exponential sums of certain monomials and permutations. We propose several classes of bent functions from quadratic permutations and permutations with (generalized) Niho exponents, and also a class of bent functions from a generalization of the Maiorana-McFarland class. The relations among the proposed bent functions and the known families of bent function are studied. Numerical results show that our constructions include bent functions that are not contained in the completed Maiorana-McFarland class M#, the class PSap or the class H.

Explicit infinite families of bent functions outside the completed Maiorana–McFarland class

During the last five decades, many different secondary constructions of bent functions were proposed in the literature. Nevertheless, apart from a few works, the question about the class inclusion of bent functions generated using these methods is rarely addressed. Especially, if such a “new” family belongs to the completed Maiorana–McFarland ({{{mathcal {M}}}{{mathcal {M}}}}^#) class then there is no proper contribution to the theory of bent functions. In this article, we provide some fundamental results related to the inclusion in {{{mathcal {M}}}{{mathcal {M}}}}^# and eventually we obtain many infinite families of bent functions that are provably outside {{{mathcal {M}}}{{mathcal {M}}}}^#. The fact that a bent function f is in/outside {{{mathcal {M}}}{{mathcal {M}}}}^# if and only if its dual is in/outside {{{mathcal {M}}}{{mathcal {M}}}}^# is employed in the so-called 4-decomposition of a bent function on {mathbb {F}}_2^n, which was originally considered by Canteaut and Charpin (IEEE Trans Inf Theory 49(8):2004–2019, 2003) in terms of the second-order derivatives and later reformulated in (Hodžić et al. in IEEE Trans Inf Theory 65(11):7554–7565, 2019) in terms of the duals of its restrictions to the cosets of an (n-2)-dimensional subspace V. For each of the three possible cases of this 4-decomposition of a bent function (all four restrictions being bent, semi-bent, or 5-valued spectra functions), we provide generic methods for designing bent functions provably outside {{{mathcal {M}}}{{mathcal {M}}}}^#. For instance, for the elementary case of defining a bent function h(textbf{x},y_1,y_2)=f(textbf{x}) oplus y_1y_2 on {mathbb {F}}_2^{n+2} using a bent function f on {mathbb {F}}_2^n, we show that h is outside {{{mathcal {M}}}{{mathcal {M}}}}^# if and only if f is outside {{{mathcal {M}}}{{mathcal {M}}}}^#. This approach is then generalized to the case when two bent functions are used. More precisely, the concatenation f_1||f_1||f_2||(1oplus f_2) also gives bent functions outside {{{mathcal {M}}}{{mathcal {M}}}}^# if f_1 or f_2 is outside {{{mathcal {M}}}{{mathcal {M}}}}^#. The cases when the four restrictions of a bent function are semi-bent or 5-valued spectra functions are also considered and several design methods of constructing infinite families of bent functions outside {{{mathcal {M}}}{{mathcal {M}}}}^# are provided.

Vectorial Boolean functions with the maximum number of bent components beyond the Nyberg’s bound

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 M#\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {M}}^{\\#}$$\\end{document}. Moreover, no examples of MNBC functions containing component functions provably outside M#\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {M}}^{\\#}$$\\end{document} 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 M#\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {M}}^{\\#}$$\\end{document} 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].

Constructions of (vectorial) bent functions outside the completed Maiorana–McFarland class

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.

A New 10-Variable Cubic Bent Function Outside the Completed Maiorana-McFarland Class

A New 10-Variable Cubic Bent Function Outside the Completed Maiorana-McFarland Class

Permutations without linear structures inducing bent functions outside the completed Maiorana-McFarland class

Permutations without linear structures inducing bent functions outside the completed Maiorana-McFarland class

Analysis of (n, n)-Functions Obtained From the Maiorana-McFarland Class

Pott et al. (2018) showed that $\mathcal {F}(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(x)$ , $n = 2m$ , $r\ge 1$ , is a nontrivial example of a vectorial function with the maximal possible number $2^{n}-2^{m}$ of bent components. Mesnager et al. (2019) generalized this result by showing conditions on $\Lambda (x) = x + \sum _{j=1}^\sigma \alpha _{j}x^{2^{t_{j}}}$ , $\alpha _{j}\in {\mathbb F} _{2^{m}}$ , under which $\mathcal {F}(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$ has the maximal possible number of bent components. We simplify these conditions and further analyse this class of functions. For all related vectorial bent functions $F(x) = {\rm Tr^{n}_{m}}(\gamma \mathcal {F}(x))$ , $\gamma \in {\mathbb F}_{2^{n}}\setminus {\mathbb F} _{2^{m}}$ , which as we will point out belong to the Maiorana-McFarland class, we describe the collection of the solution spaces for the linear equations $\mathcal {D}_{a}F(x) = F(x) + F(x+a) + F(a) = 0$ , which forms a spread of ${\mathbb F}_{2^{n}}$ . Analysing these spreads, we can infer neat conditions for functions $H(x) = (F(x),G(x))$ from ${\mathbb F}_{2^{n}}$ to ${\mathbb F}_{2^{m}}\times {\mathbb F} _{2^{m}}$ to exhibit small differential uniformity (for instance for $\Lambda (x) = x$ and $r=0$ this fact is used in the construction of Carlet’s, Pott-Zhou’s, Taniguchi’s APN-function). For some classes of $H(x)$ we determine differential uniformity and with a method based on Bezout’s theorem nonlinearity.

Wide minimal binary linear codes from the general Maiorana–McFarland class

Wide minimal binary linear codes from the general Maiorana–McFarland class

Vectorial bent functions weakly/strongly outside the completed Maiorana–McFarland class

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.

Further study on constructing bent functions outside the completed Maiorana–McFarland class

In the mid-sixties, Rothaus introduced the notion of bent function and later presented a secondary construction of bent functions (building new bent functions from already defined ones), called Rothaus’ construction. In Zhang et al. 2017 (‘Constructing bent functions outside the Maiorana–Mcfarland class using a general form of Rothaus,’ IEEE Transactions on Information Theory, 2017, vol. 63, no. 8, pp. 5336–5349.’) provided two constructions of bent functions using a general form of Rothaus and showed that the obtained classes lie outside the completed Maiorana–McFarland ( ) class. In this study, the authors propose two similar methods for constructing bent functions outside the completed class but with significantly simplified sufficient conditions compared to those in Zhang et al. 2017. These simplified conditions do not induce any serious restrictions on the choice of permutations used in the construction apart from a simple requirement on their algebraic degree and the request that the component functions of one permutation do not admit linear structures. This enables us to generate a huge class of bent functions lying outside the completed class. Even more importantly, they prove that the new classes of bent functions are affine inequivalent to the bent functions in Zhang et al. 2017.

A construction of highly nonlinear Boolean functions with optimal algebraic immunity and low hardware implementation cost

In this paper we put forward a method to construct balanced Boolean functions by modifying the Tang–Carlet–Tang functions with the help of functions in Generalized Maiorana–McFarland class. Essential cryptographic properties, for instance, nonlinearity, algebraic immunity and immunity to fast algebraic attacks are considered. A new lower bound on the nonlinearity of the functions that are contained in our construction is obtained. It is slightly better than the known result. Owing to a combinatorial fact, it can be demonstrated that the proposed functions have optimal algebraic immunity. Meanwhile, they also possess high nonlinearity and a moderate behavior against fast algebraic attacks. Apart from theoretical contributions, hardware implementation efficiency on the proposed functions is also analyzed. Since the participation of GMM class component, experimental results imply that functions constructed in this paper have a small implementation size and employ fewer logic gates compared with the construction containing an iterative structure.

On constructions and properties of (n, m)-functions with maximal number of bent components

For any positive integers $n=2k$ and $m$ such that $m\geq k$, in this paper we show the maximal number of bent components of any $(n,m)$-functions is equal to $2^{m}-2^{m-k}$, and for those attaining the equality, their algebraic degree is at most $k$. It is easily seen that all $(n,m)$-functions of the form $G(x)=(F(x),0)$ with $F(x)$ being any vectorial bent $(n,k)$-function, have the maximum number of bent components. Those simple functions $G$ are called trivial in this paper. We show that for a power $(n,n)$-function, it has such large number of bent components if and only if it is trivial under a mild condition. We also consider the $(n,n)$-function of the form $F^{i}(x)=x^{2^{i}}h({\rm Tr}^{n}_{e}(x))$, where $h: \mathbb{F}_{2^{e}} \rightarrow \mathbb{F}_{2^{e}}$, and show that $F^{i}$ has such large number if and only if $e=k$, and $h$ is a permutation over $\mathbb{F}_{2^{k}}$. It proves that all the previously known nontrivial such functions are subclasses of the functions $F^{i}$. Based on the Maiorana-McFarland class, we present constructions of large numbers of $(n,m)$-functions with maximal number of bent components for any integer $m$ in bivariate representation. We also determine the differential spectrum and Walsh spectrum of the constructed functions. It is found that our constructions can also provide new plateaued vectorial functions.

Several new infinite families of bent functions via second order derivatives

Inspired by a recent work of Tang et al. on constructing bent functions [14, IEEE TIT, 63(1): 6149-6157, 2017], we introduce a property (Pτ) of any Boolean function that its second order derivatives vanish at any direction (ui,uj) for some τ-subset {u1,…,uτ} of $\mathbb {F}_{2^{n}}$ , and then establish a link between this property and the construction of Tang et al. (IEEE Trans. Inf. Theory 63(10), 6149–6157 2017). It enables us to find more bent functions efficiently. We construct (at least) five new infinite families of bent functions from some known functions: the Gold’s bent functions and some quadratic non-monomial bent functions, Leander’s monomial bent functions, Canteaut-Charpin-Kyureghyan’s monomial bent functions, and the Maiorana-McFarland class of bent functions, respectively. Our result generalizes some recent works on bent functions. We also provide the corresponding dual functions in all our constructions except the quadratic non-monomial one. It also turns out that we can get new bent functions outside the Maiorana-McFarland completed class.

Cubic bent functions outside the completed Maiorana-McFarland class

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 nge 10. Moreover, we show that for almost all values of n, these functions can simultaneously be homogeneous and have no affine derivatives.

Generalized bent functions into $\mathbb {Z}_{p^{k}}$ from the partial spread and the Maiorana-McFarland class

Functions f from ${\mathbb {F}_{p}^{n}}$, n = 2m, to $\mathbb {Z}_{{p}^{k}}$ for which the character sum $\mathcal {H}^{k}_{f}(p^{t},u)=\sum\limits _{x\in {\mathbb {F}_{p}^{n}}}\zeta _{p^{k}}^{p^{t}f(x)}\zeta _{p}^{u\cdot x}$ (where $\zeta _{q} = e^{2\pi i/q}$ is a q-th root of unity), has absolute value $p^{m}$ for all $u\in {\mathbb {F}_{p}^{n}}$ and $0\le t\le k-1$, induce relative difference sets in ${\mathbb {F}_{p}^{n}}\times \mathbb {Z}_{{p}^{k}}$ hence are called bent. Functions only necessarily satisfying $|\mathcal {H}^{k}_{f}(1,u)| = p^{m}$ are called generalized bent. We show that with spreads we not only can construct a variety of bent and generalized bent functions, but also can design functions from ${\mathbb {F}_{p}^{n}}$ to $\mathbb {Z}_{{p}^{m}}$ satisfying $|\mathcal {H}_{f}^{m}(p^{t},u)| = p^{m}$ if and only if $t\in T$ for any $T\subset \{0,1\ldots ,m-1\}$. A generalized bent function can also be seen as a Boolean (p-ary) bent function together with a partition of ${\mathbb {F}_{p}^{n}}$ with certain properties. We show that the functions from the completed Maiorana-McFarland class are bent functions, which allow the largest possible partitions.

Design methods for semi-bent functions

Semi-bent functions play an important role in the construction of orthogonal variable spreading factor codes used in code-division multiple-access (CDMA) systems as well as in certain cryptographic applications. In this article we provide several infinite classes of semi-bent functions, where each class is characterized by either a different decomposition of such a function with respect to the Walsh spectra of its subfunctions, or by the method used for its derivation. In particular, we also give the exact number of possibilities of decomposing bent functions, in a subclass of the Maiorana–McFarland class.

Further Results on Generalized Bent Functions and Their Complete Characterization

This paper contributes to increase our knowledge on generalized bent functions (including generalized bent Boolean functions and generalized $p$ -ary bent functions with odd prime $p$ ) by bringing new results on their characterization and construction in arbitrary characteristic. More specifically, we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions. This enables us to completely characterize generalized bent functions and $\mathbb Z_{p^{k}}$ -bent functions by some affine space associated with the generalized bent functions. We also present the relationship between generalized bent Boolean functions with an odd number of variables and generalized bent Boolean functions with an even number of variables. Based on the well-known Maiorana-McFarland class of Boolean functions, we present some infinite classes of generalized bent Boolean functions. In addition, we introduce a class of generalized hyperbent functions that can be seen as generalized Dillon’s $PS$ functions. Finally, we solve an open problem related to the description of the dual function of a weakly regular generalized bent Boolean function with an odd number of variables via the Walsh–Hadamard transform of their component functions, and we generalize these results to the case of odd prime.