Quaternary cryptographic bent functions and their binary projection

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).

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.

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.

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.

On the Bent Boolean Functions That are Symmetric

Binary bent functions have a strictly specified number of non-zero values. In the same way, ternary bent functions satisfy certain requirements on the elements of their value vectors. These requirements can be used to specify six classes of ternary bent functions. Classes are mutually related by encoding of function values. Functions within a class are mutually related by permutation of elements in their function vectors. Given a basic ternary bent function, other functions in the same class can be constructed by permutation matrices having a block structure similar to that of the factor matrices appearing in the Good-Thomas decomposition of Cooley-Tukey Fast Fourier transform and related algorithms. Conversion of function vectors into matrices or equivalent matrix-valued vectors of smaller length leads to the redistribution of space complexity of related manipulation algorithms. In this matrix representations, construction of other bent functions from a given known bent function is performed by using either properties of bent functions or by manipulation of such representations in terms of FFT-like permutation matrices of dimensions smaller than the length of the function vector of the initial bent functions.

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 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.

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.

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

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.

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.

$$\mathbb {Z}$$-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.

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They have attracted a lot of research for four decades because of their own sake as interesting combinatorial objects, and also because of their relations to coding theory, sequences and their applications in cryptography and other domains such as design theory. In this paper we investigate explicit constructions of bent functions which are linear on elements of spreads. After presenting an overview on this topic, we study bent functions which are linear on elements of presemifield spreads and give explicit descriptions of such functions for known commutative presemifields. A direct connection between bent functions which are linear on elements of the Desarguesian spread and oval polynomials over finite fields was proved by Carlet and the second author. Very recently, further nice extensions have been made by Carlet in another context. We introduce oval polynomials for semifields which are dual to symplectic semifields. In particular, it is shown that from a linear oval polynomial for a semifield one can get an oval polynomial for transposed semifield.

Bent Boolean functions exist only for an even number of variables, moreover, they are unbalanced. Therefore, they are used in coding theory and in many areas of computer science. General form of bent functions is still unknown. One way of representing Boolean functions is with a reduced ordered binary decision diagram (ROBDD). The strength of ROBDDs is that they can represent Boolean functions data with a high level of redundancy in a compact form, as long as the data is encoded in such a way that the redundancy is exposed. This paper investigates characteristics of bent functions with focus on their ROBDD parameters. Decision diagram experimental framework has been used for implementation of a program for calculation of the ROBDD parameters. The results presented in this paper are intended to be used to create methods for the construction of bent functions using a ROBDD as a data structure from which the bent functions can be discovered.