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.
The Reed-Muller-Fourier spectrum of p-valued (p prime, p > 2) Maiorana-McFarland bent functions is studied. It is shown that this spectrum exhibits a unique structure, which allows characterizing ...these functions and to distinguish them from other p-valued bent functions.
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.
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.
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
#
. Moreover, no examples of MNBC functions containing component functions provably outside
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
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.
Bent functions at the minimum distance
from a given bent function of
variables belonging to the Maiorana–McFarland class
are investigated. We provide a criterion for a function obtained using the ...addition of the indicator of an
-dimensional affine subspace to a given bent function from
to be a bent function as well. In other words, all bent functions at the minimum distance from a Maiorana–McFarland bent function are characterized. It is shown that the lower bound
for the number of bent functions at the minimum distance from
is not attained if the permutation used for constructing
is not an APN function. It is proved that for any prime
there exist functions in
for which this lower bound is accurate. Examples of such bent functions are found. It is also established that the permutations of EA-equivalent functions in
are affinely equivalent if the second derivatives of at least one of the permutations are not identically zero.
In the mid 1960s, Rothaus proposed the so-called "most general form" of constructing new bent functions by using three (initial) bent functions whose sum is again bent. In this paper, we utilize a ...special case of Rothaus construction when two of these three bent functions differ by a suitably chosen characteristic function of an n/2-dimensional subspace. This simplification allows us to treat the induced bent conditions more easily, also implying the possibility to specify the initial functions in the partial spread class and most notably to identify several instances of the so-called non-normal bent functions. Affine inequivalent bent functions within this class are then identified using a suitable selection of initial bent functions within the partial spread class (stemming from the complete Desarguesian spread). It is also shown that when the initial bent functions belong to the class D, then, under certain conditions, the constructed functions provably do not belong to the completed Maiorana-McFarland class. We conjecture that our method potentially generates an infinite class of non-normal bent functions (all tested ten-variable functions are non-normal but unfortunately they are weakly normal) though there are no efficient computational tools for confirming this.
The differential-linear connectivity table (DLCT) of a vectorial Boolean function was recently introduced by Bar-On et al. at EUROCRYPT'19, whose value at a point is related to the autocorrelation ...value of its component functions. Further, in INDOCRYPT'19, we proposed a new construction method for vectorial Boolean functions with very low differential-linear uniformity using Maiorana–McFarland bent functions. The difficulty of that construction method was to identify the permutations and the sub-functions that satisfy the conditions to attain good cryptographic properties. In this paper we discover novel techniques to construct such sub-functions to generate vectorial Boolean functions with substantially improved cryptographic properties. Our proposed methods are based on ideas from combinatorics as well as finite fields. In particular, we construct the sub-functions to generate (4t,t−1)-function, t≥5, in a different manner than our Indocrypt'19 paper. Further our new methods help in obtaining sub-functions to generate balanced (4t+2,t−1)-function and (2k,k)-function with very good nonlinearity and very low differential-linear uniformity, that were never demonstrated earlier.
