Bent functions linear on elements of some classical spreads and presemifields spreads

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 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.

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.

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.

Bent functions are maximally nonlinear Boolean functions. They are important functions introduced by Rothaus and studied firstly by Dillon and next by many researchers for four decades. Since the complete classification of bent functions seems elusive, many researchers turn to design constructions of bent functions. In this paper, we show that linear involutions (which are an important class of permutations) over finite fields give rise to bent functions in bivariate representations. In particular, we exhibit new constructions of bent functions involving binomial linear involutions whose dual functions are directly obtained without computation. The existence of bent functions from involutions heavily relies on solving systems of equations over finite fields.

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.

On the Bent Boolean Functions That are Symmetric

We present a new characterization of semi-bent and bent quadratic functions on finite fields. First, we determine when a GF(2)-linear combination of Gold functions Tr(x2i+1) is semi-bent over GF(2n), n odd, by a polynomial GCD computation. By analyzing this GCD condition, we provide simpler characterizations of semi-bent functions. For example, we deduce that all linear combinations of Gold functions give rise to semi-bent functions over GF(2p) when p belongs to a certain class of primes. Second, we generalize our results to fields GF(pn) where p is an odd prime and n is odd. In that case, we can determine whether a GF(p)-linear combination of Gold functions Tr(xpi+1) is (generalized) semi-bent or bent by a polynomial GCD computation. Similar to the binary case, simple characterizations of these p-ary semi-bent and bent functions are provided.

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 p-ary function f(x) mapping GF(p4k) to GF(p) and given by f(x)=Tr4k(xp3k+p2k-pk+1+x2) is proven to be a weakly regular bent function and the exact value of its Walsh transform coefficients is found. This is the first proven infinite class of nonquadratic generalized bent functions over the fields of an arbitrary odd characteristic. The proof is based on a few new results in the area of exponential sums and polynomials over finite fields that may also be interesting as independent problems.

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.

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

An Erratum has been published for this article in Journal of Combinatorial Designs 14: 82–82, 2006.We give the equivalence between perfect nonlinear functions and appropriate splitting semi‐regular relative difference sets, construct a class of splitting relative difference sets by using Galois rings and bent functions, and prove that there exists a 4‐phase perfect nonlinear function if and only if the number of input variables is at least twice the number of output variables. © 2005 Wiley Periodicals, Inc.

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.