The solution to an open problem on the bentness of Mesnager's functions

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They were introduced by Rothaus in 1976. For their own sake as interesting combinatorial objects, but also because of their relations to coding theory (Reed-Muller codes) and applications in cryptography (design of stream ciphers), they have attracted a lot of research, specially in the last 15 years. The class of bent functions contains a subclass of functions, introduced by Youssef and Gong in 2001, the so-called hyper-bent functions whose properties are still stronger and whose elements are still rarer than bent functions. Bent and hyper-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. This paper is devoted to the constructions of bent and hyper-bent Boolean functions in polynomial forms. We survey and present an overview of the constructions discovered recently. We extensively investigate the link between the bentness property of such functions and some exponential sums (involving Dickson polynomials).

The nonlinearity of sequences is one of the important security measures for stream cipher systems. Recently, in the study of vectorized stream cipher systems, bent functions have been generalized to the alphabet ℤq = ℤ/qℤ and also studied by several authors. This generalization maps the classical definition of binary bent functions to generalized bent functions. This paper propose a new approach for generalized bent functions over ℤ4 and their associated derived boolean functions. We present in this paper a new construction of a family of m-variable quaternary (ℤ4 = ℤ/4ℤ = {0,1,2,3}-valued) bent functions over Galois ring and cyclotomic classes using an intern function defined on a particular partition of the Teichmuller set. Using a particular binary projection map we obtain a family of 2m-variable boolean bent functions and a family of 2m + 1-variable boolean plateaued functions of amplitude 2m + 1 with nonlinearity equal to 4m − 2m.

In this paper, we investigate properties of generalized bent Boolean functions and 2k-bent (i.e., negabent, octabent, hexadecabent, et al.) Boolean functions in a uniform framework. From the Hadamard matrices, Hodzic and Pasalic presented sufficient conditions for generalized bent functions. Using cyclotomic fields and the decomposition of generalized bent functions, we generalize their results, prove that Hodzic and Pasalic's conditions of generalized bent functions are not only sufficient but also necessary, and completely characterize generalized bent functions in terms of their component functions. Furthermore, we present a secondary construction of bent functions or semibent functions from generalized bent functions. Finally, we give the relations of generalized bent functions and 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> -bent functions, demonstrate that 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> -bent functions are actually a special class of generalized bent functions, and completely characterize 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> -bent functions.

Boolean functions are fundamental building blocks in both discrete mathematics and computer science, with applications spanning from cryptography to coding theory. Bent functions, a subset of Boolean functions with maximal nonlinearity, are particularly valuable in cryptographic applications. This study introduces a novel equivalence relation among all Boolean functions and presents an algorithm to generate bent functions based on this relation. We systematically generated a collection of 10,000 bent functions over eight variables, all originating from the same equivalence class, and analyzed their structural complexity through rank determination. Our findings revealed the presence of at least five distinct affine classes of bent functions within this collection. By employing this construction, we devised an algorithm to generate a filter function capable of combining Boolean functions. This filter function can be dynamically adjusted based on a key, offering potential applications in symmetric cipher design, such as enhancing security or improving efficiency.

New quadratic bent functions in polynomial form are constructed in this paper. The constructions give new Boolean bent, generalized Boolean bent and p-ary bent functions. Based on Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -valued quadratic forms, a simple method provides several new constructions of generalized Boolean bent functions. From these generalized Boolean bent functions a method is presented to transform them into Boolean bent and semi-bent functions. Moreover, many new p-ary bent functions can also be obtained by applying similar methods.

We consider an approach to the classification of n-variable Boolean bent functions of the nonlinearity degree 3. We utilize the apparatus of bent rectangles introduced by S. V. Agievich. This apparatus was used for the classification of 8-variable Boolean cubic bent functions. The results of our research allow to construct cubic bent functions that depend on an arbitrary even number of variables; the construction is based on well studied quadratic bent functions.

We establish a new connection between invariant theory and the theory of bent functions. This enables us to construct Boolean functions with a prescribed symmetry group action. Besides the quadratic bent functions the only other previously known homogeneous bent functions are the six variable degree three functions constructed in l16r. We show that these bent functions arise as invariants under an action of the symmetric group on four letters and determine the stabilizer which turns out to be a matrix group of order 10752. We apply the machinery of invariant theory in order to construct homogeneous bent functions of degree three in 8, 10, and 12 variables. This approach gives a great computational advantage over the unstructured search problem and yields Boolean functions which have a concise description in terms of certain designs and graphs. We consider the question of linear equivalence of the constructed bent functions and study the properties of the associated elementary abelian difference sets.

Determining the affine equivalence of Boolean functions has significant applications in circuit and cryptography. Previous methods for determining this require a large amount of computation when Boolean functions are bent functions or when the truth table is sparse. This paper presents a new method to determine the affine equivalence based on matrix algebra. By transforming Boolean function to the corresponding matrix representation, we first propose and prove the congruent standard form of Boolean function. It lays the foundation for the determination of equivalence because affine Boolean functions must have the same standard form. Then we find the generators of orthogonal matrix group and symplectic matrix group, which greatly reduce the search space. The computation complexity of our method is o(2(r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2/2+n</sup> * <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n-r)</sup> )), where n is the number of bit operations, and r is the rank of the matrix, which is the product of Boolean-1 matrix of the test Boolean function and its transposition. The experimental results show that our method is useful when the test Boolean function is no more than 7 bits and r is greater than 2.

We observe a property on Boolean functions which explains how work some secondary constructions recently obtained for Boolean bent functions. It leads to a generalization and to a unification of these constructions. It also permits to design highly nonlinear resilient functions from known ones. This construction does not increase the number of variables, contrary to the known general secondary constructions, and it permits to improve some cryptographic characters of the functions (e.g. their algebraic immunity) while keeping good the other characteristics

On the Bent Boolean Functions That are Symmetric

In this article we study two party Communication Complexity of Boolean bent functions from Maiorana–McFarland class. In particular, we describe connections between Maiorana–McFarland construction of bent functions and operations on matrix form of Boolean functions and show that bent functions of 2n variables from Maiorana–McFarland class have deterministic communication complexity equal n + 1. Finally, we show that not all bent functions have high communication complexity lower bound by giving the example of such function.

Bent functions over the finite fields of an odd characteristic received a lot of attention of late years. In Cesmelioglu et al. (J Comb Theory Ser A 119:420---429, 2012), Cesmelioglu and Meidl (Des Codes Cryptogr 66:231---242, 2013), an efficient method of construction of weakly regular and non-weakly regular bent functions defined over a finite field with odd characteristic is presented. In this paper, we give an adaptation of this method to the ring of integers modulo $$p^m$$pm, where p is an odd prime and m is a positive integer. We emphasize that different results than the results of the finite field case are obtained in every application process. First, we give a method that constructs bent functions using plateaued functions by increasing the dimension. Then, in order to give concrete examples, we compute Walsh spectrum of some specific quadratic functions defined over the ring of integers modulo $$p^m$$pm and apply the construction method on these functions. There are notable differences between the cases when m is odd and even. Also, we explain how to determine weakly regular and non-weakly regular bent functions among the bent functions that are constructed by the method.

$$\mathbb {Z}$$-bent functions, mappings from $$\mathbb {F}_2^n$$ to a subset of $$\mathbb {Z}$$, were introduced by Dobbertin and Leander (Des Codes Cryptogr 49:3–22, 2008) as an attempt to capture the origin of standard bent functions and in particular to understand better a recursive construction framework of bent functions. Nevertheless, many questions have been left open in Dobbertin and Leander (2008) such as efficient construction methods of $$\mathbb {Z}$$-bent functions of different levels, where these levels specify precisely a subset of $$\mathbb {Z}$$ containing both the image values of f and its normalized Fourier coefficients. In this article, using different design techniques, we provide several generic construction methods of $$\mathbb {Z}$$-bent functions of arbitrary levels, thereby solving an open problem posed in Dobbertin and Leander (2008). On the other hand, apart from an independent theoretical interest in these objects, our rigor treatment of the so-called gluing technique reveals that this approach is equivalent to a classical concept of concatenation. More precisely, gluing four suitable n-variables $$\mathbb {Z}$$-bent functions of level one to obtain an $$(n+2)$$-variable bent function directly corresponds to a concatenation of four suitable n-variable Boolean functions. Nevertheless, the recursive framework based on $$\mathbb {Z}$$-bent functions remains to be investigated further in this context.

It is a difficult task to compute the r-th order nonlinearity of a given function with algebraic degree strictly greater than r &gt; 1. Though lower bounds on the second order nonlinearity are known only for a few particular functions, the majority of which are cubic. We investigate lower bounds on the second order nonlinearity of cubic Boolean functions [Formula: see text], where [Formula: see text], dl = 2il + 2jl + 1, m, il and jl are positive integers, n &gt; il &gt; jl. Furthermore, for a class of Boolean functions [Formula: see text] we deduce a tighter lower bound on the second order nonlinearity of the functions, where [Formula: see text], dl = 2ilγ + 2jlγ + 1, il &gt; jl and γ ≠ 1 is a positive integer such that gcd(n,γ) = 1. Lower bounds on the second order nonlinearity of cubic monomial Boolean functions, represented by fμ(x) = Tr(μx2i+2j+1), [Formula: see text], i and j are positive integers such that i &gt; j, were obtained by Gode and Gangopadhvay in 2009. In this paper, we first extend the results of Gode and Gangopadhvay from monomial Boolean functions to Boolean functions with more trace terms. We further generalize and improve the results to a wider range of n. Our bounds are better than those of Gode and Gangopadhvay for monomial functions fμ(x). Especially, our lower bounds on the second order nonlinearity of some Boolean functions F(x) are better than the existing ones.

The paper considers maximally non-linear Boolean functions of even number of variables - bent functions. These functions have a number applications in coding theory and cryptography. For each bent function, the dual to it bent functions. A bent function is called self-dual if it coincides with its dual. Characteristic vectors of self-dual bent functions are eigenvectors of the matrix Sylvester-Hadamard, which has applications in combinatorics, theory signals and quantum computing. The paper proposes a number of new iterative constructions of self-dual bent functions of n variables, in within which the vectors of values of bent functions from n-4 variables. Based on the analysis of sets of functions generated by data constructions, as well as previously known iterative constructions, a new iterative lower bound for the number of self-dual bent functions.