Classes of bent functions identified by specific normal forms and generated using Boolean differential equations

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).

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.

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.

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.

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.

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}}.

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.

Homogeneous bent functions

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

Inspired by the works of Mesnager (IEEE Trans Inf Theory 60(7):4397–4407, 2014) and Tang et al. (IEEE Trans Inf Theory 63(10):6149–6157, 2017), we study a class of bent functions of the form $$f(x)=g(x)+F(f_1(x),f_2(x),\ldots ,$$ $$f_{\tau }(x))$$ , where g(x) is a function from $$\mathbb {F}_{p^{n}}$$ to $${\mathbb {F}}_{p}$$ , $$f_i(x)=\prod _{j=1}^{\kappa _i}{\text {Tr}}(u_{ij}x)$$ for $$1\le i\le \tau $$ , $$\kappa _i\ge 1$$ and $$F(x_1,\ldots ,x_{\tau })$$ is a reduced polynomial in $${\mathbb {F}}_{p}[x_1,\ldots ,x_{\tau }]$$ . As a consequence, we obtain a generic result on the Walsh transform of f(x) in terms of g(x) and characterize the bentness of f(x) for the cases $$F(x_1,\ldots ,x_{\tau })$$ without and with restrictions respectively, which enables us to generalize some earlier works and derive new bent functions from known ones. In addition, we study the construction of bent functions f(x) when g(x) is not bent for the first time and present a class of bent functions from non-bent Gold functions.

Des que estava cursant els primers anys de la Llicenciatura en Matemtiques he tingut la illusi de fer una tesi doctoral en aquesta disciplina, com deia Gauss, la reina de les cincies. Segurament aquesta illusi ha sorgit de la curiositat que les Matemtiques i la cincia

On the confusion and diffusion properties of Maiorana–McFarland's and extended Maiorana–McFarland's functions

The optimal design of a non-deterministic finite state machine by solving and resolving Boolean equations is shown, taking advantage of the relationships between single Boolean functions, sets of Boolean functions, Boolean equations, and their solutions. The Boolean differential calculus is used to point out the key ideas of the applied resolvability theory.

Background: Using Boolean algebra for differential diagnosis of neurological disorders involves treating neurologic findings and disease entities such as Boolean variables and combining them via OR (∨) or other logical operations. Aim: To demonstrate how Boolean algebra (mathematical logic) and Boolean matrices can be applied to support differential diagnosis in neurology. Results: Each neurologic finding was encoded as a Boolean variable (1 = present, 0 = absent). Boolean variables and Boolean matrices reflect mutual interdependencies of neurologic findings across diseases, enabling reverse inference (i.e., inferring possible diseases from a pattern of findings). Conclusion: Boolean algebra offers a formal, rigorous framework to represent and manipulate differential diagnostic relationships in neurology; it may complement clinical reasoning and computational decision support.