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Minimal codes are characterized by the property that none of the codewords is covered by some other linearly independent codeword. We first show that the use of a bent function g in the so-called direct sum of Boolean functions h (x , y) = f (x) + g (y) , where f is arbitrary, induces minimal codes. This approach gives an infinite class of minimal codes of length 2 n and dimension n + 1 (assuming that h : F 2 n → F 2 ), whose weight distribution is exactly specified for certain choices of f. To increase the dimension of these codes with respect to their length, we introduce the concept of non-covering permutations (referring to the property of minimality) used to construct a bent function g in s variables, which allows us to employ a suitable subspace of derivatives of g and generate minimal codes of dimension s + s / 2 + 1 instead. Their exact weight distribution is also determined. In the second part of this article, we first provide an efficient method (with easily satisfied initial conditions) of generating minimal [ 2 n , n + 1 ] linear codes that cross the so-called Ashikhmin–Barg bound. This method is further extended for the purpose of generating minimal codes of larger dimension n + s / 2 + 2 , through the use of suitable derivatives along with the employment of non-covering permutations. To the best of our knowledge, the latter method is the most general framework for designing binary minimal linear codes that violate the Ashikhmin–Barg bound. More precisely, for a suitable choice of derivatives of h (x , y) = f (x) + g (y) , where g is a bent function and f satisfies certain minimality requirements, for any fixed f, one can derive a huge class of non-equivalent wide binary linear codes of the same length by varying the permutation ϕ when specifying the bent function g (y 1 , y 2) = ϕ (y 2) · y 1 in the Maiorana–McFarland class. The weight distribution is given explicitly for any (suitable) f when ϕ is an almost bent permutation. [ABSTRACT FROM AUTHOR]

We show that the direct sum, under more relaxed conditions compared to those of Polujan and Pott (2020), can generate bent functions provably outside the completed Maiorana-McFarland class (MM #). We also show that the indirect sum method of generating bent functions, by imposing certain conditions (which are completely absent if only the bentness of the resulting function is required) on the initial bent functions, can be employed in the design of bent functions outside MM #. Furthermore, applying this method to suitably chosen bent functions we construct several generic classes of homogeneous cubic bent functions (considered as a difficult problem) that might possess additional properties (namely without affine derivatives and/or outside MM #). Our results significantly improve upon the best known instances of this type of bent functions given by Polujan and Pott (2020), and additionally we provide a solution to an open problem presented in their paper. [ABSTRACT FROM AUTHOR]

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 D 0 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 D 0), 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 x 2 r + 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 D k ⋆ for k ∈ { 1 , ... , n − 2 } , whose members are bent functions provably outside the completed M class. [ABSTRACT FROM AUTHOR]

The design of plateaued functions over $GF(2)^{n}$ , also known as 3-valued Walsh spectra functions (taking the values from the set $\{0, \pm 2^{\lceil ({n+s}/{2}) \rceil }\}$), has been commonly approached by specifying a suitable algebraic normal form which then induces this particular Walsh spectral characterization. In this paper, we consider the reversed design method which specifies these functions in the spectral domain by specifying a suitable allocation of the nonzero spectral values and their signs. We analyze the properties of trivial and nontrivial plateaued functions (as affine inequivalent distinct subclasses), which are distinguished by their Walsh support $S_{f}$ (the subset of $GF(2)^{n}$ having the nonzero spectral values) in terms of whether it is an affine subspace or not. The former class exactly corresponds to partially bent functions and admits linear structures, whereas the latter class may contain functions without linear structures. A simple sufficient condition on $S_{f}$ , which ensures the nonexistence of linear structures, is derived and some generic design methods of nontrivial plateaued functions without linear structures are given. The extended affine equivalence of plateaued functions is also addressed using the concept of dual of plateaued functions. Furthermore, we solve the problem of specifying disjoint spectra (non)trivial plateaued functions of maximal cardinality whose concatenation can be used to construct bent functions in a generic manner. This approach may lead to new classes of bent functions due to large variety of possibilities to select underlying duals that define these disjoint spectra plateaued functions. An additional method of specifying affine inequivalent plateaued functions, obtained by applying a nonlinear transform to their input domain, is also given. [ABSTRACT FROM AUTHOR]

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 \mathcal {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. [ABSTRACT FROM AUTHOR]

In this paper, we first present a novel secondary construction of bent functions (building new bent functions from two already defined ones). Furthermore, the algebraic degree and algebraic immunity of the constructed functions are analysed. Finally, we apply the construction using as initial functions some specific bent functions and then specify sufficient conditions for the resulting bent functions not to be contained in the completed Maiorana-McFarland class. In the second part of the paper, we present a corrigendum of 'Constructions of bent-negabent functions and their relation to the completed Maiorana-McFarland Class' (IEEE Trans Inf Theory 61(3):1496-1506, 2015). [ABSTRACT FROM AUTHOR]

The problem of constructing bent–negabent functions that do not belong to the completed Maiorana–McFarland class emerges implicitly through a series of construction methods proposed recently. These approaches manage to optimize the algebraic degree of bent–negabent functions, but all of the constructed bent–negabent functions belong to the completed Maiorana–McFarland class. In this paper, we use the indirect sum construction (proposed by Carlet in 2004) for constructing the bent–negabent functions that are not provably contained in this class, which is the first significant attempt in this direction. To achieve this, we first provide a class of bent functions with certain desirable properties that does not belong to this class and demonstrate the existence of the class members. Then, embedding these functions in the framework of the indirect sum construction, we are able to specify sufficient conditions for bent–negabent functions not being contained in the completed Maiorana–McFarland class. [ABSTRACT FROM PUBLISHER]

ABSTRACT In this paper, we concentrate on the design of 1-resilient Boolean functions with desirable cryptographic properties. Firstly, we put forward a novel secondary construction to obtain 1-resilient functions. Next, we present the relationships between the properties of these constructed 1-resilient functions and that of the initial functions. Based on the construction and a class of bent functions on n variables, we can obtain a class of ( n + 3)-variable 1-resilient non-separable cryptographic functions with a high algebraic immunity, whose nonlinearity is equal to the bent concatenation bound 2 n + 2 − 2( n + 2)/2. Furthermore, we propose a set of 1-resilient non-separable functions on odd number of variables with an optimal algebraic degree, a high algebraic immunity, and a high nonlinearity. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]