Remarks on Particular Properties of Ternary Bent Functions and Construction Algorithms

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).

As in the binary case, ternary bent functions are defined as most non-linear ternary functions meaning that they are at the largest possible Hamming distance from affine ternary functions. It is therefore interesting to observe that some ternary bent functions can be constructed as various combinations of linear ternary functions. By starting from the combination of linear functions corresponding to the basic ternary bent functions the construction of different other ternary bent functions can be performed by the application of different combinations of FFT-like permutation matrices for ternary functions.

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.

This paper aims at the identification of classes of bent functions in order to allow their construction without searching or sieving. In order to reach this aim, we studied first the relationship between bent functions and complexity classes defined by the Specific Normal Forms of all Boolean functions. As result of this exploration we found classes of bent functions which are embedded in different complexity classes defined by the Specific Normal Form. In the second step to reach our global aim, we utilized the found classes of bent functions in order to express bent functions in terms of derivative operations of the Boolean Differential Calculus. In detail, we studied bent functions of two and four variables. This exploration leads finally to Boolean differential equations that will allow the direct calculation of all bent functions of two and four variables. A given generalization allows to calculate subsets of bent functions for each even number of Boolean variables.

О свойствах бент-функций, построенных по некоторой бентфункции с помощью подпространств

Bent functions are a class of discrete functions which exhibit the highest degree of nonlinearity. As such bent functions form an essential part of cryptographic systems. Original concept of bent functions defined in GF(2) can be extended to multiple-valued case. Multiple-valued bent functions are defined in therms of properties of their Vilenkin-Chrestenson spectra. Decision diagrams are a method of compact representation of discrete functions. Special types of decision diagrams have been introduced for various types of discrete functions. In this paper we demonstrate how Vilenkin-Chrestenson decision diagrams can be used for efficient representation of multiple-valued 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.

The question of establishing EA-inequivalence among the classes of bent functions remains in general an open problem. The EA-inequivalence is also relevant in classifying bent functions within the same class. This paper is an attempt to investigate these questions for the Maiorana–McFarland ($$\mathcal{M}$$) class and so-called class $$\mathcal{H}$$ of bent functions. For cubic bent functions of the form $$Tr_1^t(xy^{2^i+1})$$ in $$\mathcal{M}$$, the necessary and sufficient conditions related to EA-equivalence are derived. It is also shown that in most of the cases, at least over finite fields of relatively small order, bent functions within $$\mathcal{H}$$ are EA-inequivalent.

In this paper, we obtain a new class of $p$-ary binomial bent functions which are determined by Kloosterman sums. The bentness of another three classes of functions is characterized by some exponential sums and some results in \cite{Linian2013} are generalized. Furthermore we obtain, in some special cases, some bent functions are determined by Kloosterman sums.

<p style='text-indent:20px;'>This paper first gives out basic background and some definitions and propositions for Fourier matrix and bent function. Secondly we construct an standard orthogonal basis by the eigenvectors of the corresponding Fourier matrix. At last the diagonalization work of Fourier matrix is completed and some theorems about them are proved.

Recently in Çeşmelioğlu, Meidl (Adv. Math. Commun.,18, 2024), the study of EA-equivalence and CCZ-equivalence for functions from {mathbb {V}}_n^{(p)} to the cyclic group {mathbb {Z}}_{p^k} has been initiated, where {mathbb {V}}_n^{(p)} denotes an n-dimensional vector space over {mathbb {F}}_p. Amongst others it has been shown that there exist functions from {mathbb {V}}_n^{(2)} to {mathbb {Z}}_4 which are CCZ-equivalent but not EA-equivalent. We extend these results to larger classes of functions from {mathbb {V}}_n^{(p)} to {mathbb {Z}}_{p^k}. We then discuss constructions of generalized bent functions from {mathbb {V}}_n^{(p)} to {mathbb {Z}}_{p^k}, p odd or p=2 and n is even, which correspond to large affine spaces of bent functions. In particular we employ versions of the direct sum, the semi-direct sum and of a recent secondary bent function construction in Wang et. al., (IEEE Trans. Inform. Theory69, 2023), to generate large affine spaces of bent functions. Finally we present a solution for constructing generalized bent functions from {mathbb {V}}_n^{(2)} to {mathbb {Z}}_{2^k}, n odd, from arbitrary generalized bent functions from {mathbb {V}}_{n-1}^{(2)} to {mathbb {Z}}_{2^{k-1}}.

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

Recently, the concept of (normal) bent partitions, which are partitions of elementary abelian groups having similar properties to spreads, has been introduced by Anbar and Meidl. A large number of bent partitions, so-called generalized semifield spreads, can be obtained from semifields with certain properties. A strongly related concept, namely Latin square partial difference set packings (LP-packings) in finite abelian groups, has also been introduced recently by Jedwab and Li. The examples for LP-packings in an elementary abelian group are obtained from spreads. LP-packings yield bent partitions (not only for elementary abelian groups). In this paper, we first point out that conversely, generalized semifield spreads yield LP-packings. As a result, there is a huge amount of LP-packings in elementary abelian groups, other than spreads. With some examples from ternary bent functions, we then show that normal bent partitions and LP-packings are not the same concept. Finally, we extend the lifting procedure from spreads to LP-packings in nonelementary abelian groups to a lifting procedure from some generalized spreads to LP-packings in nonelementary abelian groups and in larger elementary abelian groups. This potentially yields bent partitions other than generalized semifield spreads.

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.

The bent functions are the Boolean functions of an even number of variables which are at the maximum possible distance from the set of all affine functions. In this paper, it is shown that each isometric mapping of the set of Boolean functions of n variables to itself preserving the class of bent functions is a combination of an affine transformation of coordinates and a shift by an affine function. It is proved that the affine functions are precisely all Boolean functions which are at the maximum possible distance from the class of bent functions.