Bent functions

The associated codes of almost perfect nonlinear (APN) functions have been widely studied. In this paper, we consider more generally the codes associated with functions that have differential uniformity at least 4. We emphasize, for such a function F , the role of codewords of weight 3 and 4 and of some cosets of its associated code C F . We give some properties on codes associated with differential uniformity exactly 4. We obtain lower bounds and upper bounds for the numbers of codewords of weight less than 5 of the codes C F . We show that the nonlinearity of F decreases when these numbers increase. We obtain a precise expression to compute these numbers, when F is a plateaued or a differentially two-valued function. As an application, we propose a method to construct differentially 4-uniform functions, with a large number of 2-to-1 derivatives, from APN functions.

Crooked permutations were introduced twenty years ago to construct interesting objects in graph theory. These functions, over F 2 n with odd n, are such that their derivatives have as image set a complement of a hyperplane. The field of applications was extended later, in particular to cryptography. However binary crooked functions are rare. It is still unknown if non quadratic crooked functions do exist. We extend the concept and propose to study the crooked property for any characteristic. A function F , from F p n to itself, satisfies this property if all its derivatives have as image set an affine subspace. We show that the partially-bent vectorial functions and the functions satisfying the crooked property are strongly related. We later focus on the components of these functions, establishing that the existence of linear structures is here decisive. We then propose a symbolic approach to identify the linear structures. We claim that this problem consists in solving a system of linear equations, and can often be seen as a combinatorial problem.

The first and the third authors recently introduced a spectral construction of plateaued and of 5-value spectrum functions. In particular, the design of the latter class requires a specification of integers {W (u) : }, so that the sequence {W (u) : u ∈ F n 2 } is a valid spectrum of a Boolean function (recovered using the inverse Walsh transform). Technically, this is done by allocating a suitable Walsh support . In addition, two dual functions g [i] : S [i] → F2 (with #S [i] = 2 λ i ) are employed to specify the signs through W (u) = 2 n+s i 2

From the last few years, generalized bent functions gain a lot of attention in research as they have many applications in various fields such as combinatorial design, sequence design theory, cryptography, CDMA communication, etc. A deep and broad study of generalized bent functions with their properties is done in literature. Kumar et al.[11] first gave the concept of generalized bent function. Many researchers studied the properties and characterizations of generalized bent functions. In [2] authors introduced the concept of generalized (q-ary) negabent functions and studied some properties of generalized (q-ary) negabent functions. In this paper, we study the generalized (q-ary) bent functions f : Z n q → Z q , where Z q is the ring of integers with mod q, Z n q is the vector space of dimension n over Z q and q ≥ 2 is any positive integer. We discuss several properties of generalized (q-ary) bent functions with respect to their nega-Hadamard transform. We also study the relation between generalized nega-Hadamard transforms and generalized nega-autocorrelations. Furthermore, we prove the necessary and sufficient conditions for the bentness and negabentness of generalized (q-ary) bent function generated by the secondary construction for Z n q , where q = 2 s .

In this paper we construct two classes of translation hyperovals in any Hall plane of even order q* P 16. Two hyperovals constructed in the same Hall plane are equivalent under the action of the automorphism group of that Hall plane iff they are in the same class.

Given an inversive plane we define a graph, the vertices of which are the blocks of the inversive plane, where two blocks are adjacent if they meet in one point. We use this graph to provide the following characterization of the classical inversive planes of odd order. An inversive plane of odd order is classical if, and only if, there is a point at which the residue is an affine plane over a field, and the graph defined above has two connected components.

Crooked permutations were introduced twenty years ago to construct interesting objects in graph theory. These functions, over F 2 n with odd n, are such that their derivatives have as image set a complement of a hyperplane. The field of applications was extended later, in particular to cryptography. However binary crooked functions are rare. It is still unknown if non quadratic crooked functions do exist. We extend the concept and propose to study the crooked property for any characteristic. A function F , from F p n to itself, satisfies this property if all its derivatives have as image set an affine subspace. We show that the partially-bent vectorial functions and the functions satisfying the crooked property are strongly related. We later focus on the components of these functions, establishing that the existence of linear structures is here decisive. We then propose a symbolic approach to identify the linear structures. We claim that this problem consists in solving a system of linear equations, and can often be seen as a combinatorial problem.

In this paper, we exploit the concept of a particular n × n symmetric matrix of the form F = [F k ] n×n , where k = max(i, j)+1 and F K is the kth Fibonacci number. We investigate some special properties of this new matrix. In addition, we construct the Hadamard exponential form of this new matrix. We compute the spectral norm, determinant, principal minors, some upper and lower bounds of this matrix and its Hadamard exponential. Finally, we prove that its Hadamard inverse is positive definite, and investigate some properties of its Hadamard inverse.

Bent functions, having the highest possible nonlinearity, are among the best candidates for construction of S S S-boxes. One problem with bent functions is the fact that they are hard to find among randomly generated set of Boolean functions already for 6 argument functions. There exist some algorithms that allow for easy generation of bent functions. The major drawback of these algorithms is the fact that they rely on deterministic dependencies and are only able to generate bent functions belonging to one specific class. In our paper we present an efficient generator of random bent functions of more than 4 arguments. Resulting functions are not bounded by constraints described above. The generator operates in algebraic normal form domain (ANF). We also present our result on comparing the performance of S S S-boxes build using our bent function generator versus a standard method of bent function construction. We also give some directions for further research.

Starting with a basis of F 2k 2 , we define some sets in F 2k 2 that are the supports of bent functions of 2k variables. We also establish some results in order to count the number of bent functions we can construct, and we provide a complete classification of all bases of F 2k 2 (for k = 2) providing the same supports of bent functions.

The associated codes of almost perfect nonlinear (APN) functions have been widely studied. In this paper, we consider more generally the codes associated with functions that have differential uniformity at least 4. We emphasize, for such a function F , the role of codewords of weight 3 and 4 and of some cosets of its associated code C F. We give some properties on codes associated with differential uniformity exactly 4. We obtain lower bounds and upper bounds for the numbers of codewords of weight less than 5 of the codes C F. We show that the nonlinearity of F decreases when these numbers increase. We obtain a precise expression to compute these numbers, when F is a plateaued or a differentially two-valued function. As an application, we propose a method to construct differentially 4-uniform functions, with a large number of 2-to-1 derivatives, from APN functions.

We present a method to iteratively construct new bent functions of n+2 variables from a bent function of n variables and its cyclic shift permutations using minterms of n variables and minterms of 2 variables. In addition, we provide the number of bent functions of n+2 variables that we can obtain by applying the method here presented, and finally we compare this method with a previous one introduced by us in 2008 and with the Rothaus and Maiorana-McFarland constructions.

In 2017, Tang et al. have introduced a generic construction for bent functions of the form f (x) = g(x) + h(x), where g is a bent function satisfying some conditions and h is a Boolean function. Recently, Zheng et al. [22] generalized this result to construct large classes of bent vectorial Boolean function from known ones in the form F (x) = G(x)+h(X), where G is a bent vectorial and h a Boolean function. In this paper we further generalize this construction to obtain vectorial bent functions of the form F (x) = G(x)+H(X), where H is also a vectorial Boolean function. This allows us to construct new infinite families of vectorial bent functions, EA-inequivalent to G, which was used in the construction. Most notably, specifying H(x) = h(T r n 1 (u 1 x),. .. , T r n 1 (u t x)), the function h : F t 2 → F 2 t can be chosen arbitrary which gives a relatively large class of different functions for a fixed function G. We also propose a method of constructing vectorial (n, n)-functions having maximal number of bent components.

Whereas the design and properties of bent and plateaued functions have been frequently addressed during the past few decades, there are only a few design methods of so-called 5-valued spectra Boolean functions whose Walsh spectra takes the values in {0, ±2 λ1 , ±2 λ2 }. Moreover, these design methods mainly regards the specification of these functions in their ANF (algebraic normal form) domain. In this article, we give a precise characterization of this class of functions in their spectral domain using the concept of a dual of plateaued functions. Both necessary and sufficient conditions on the Walsh support of these functions are given which then connects their design (in spectral domain) to a family of so-called totally (non-overlap) disjoint spectra plateaued functions. We identify some suitable families of plateaued functions having this property, thus providing some generic methods in the spectral domain. Furthermore, we also provide an extensive analysis of their constructions in the ANF domain and provide several generic design methods. The importance of this class of functions is manifolded, where apart from being suitable for some cryptographic applications we emphasize their property of being constituent functions in the so-called 4-bent decomposition.

In this article, heuristic methods of hill climbing for cryptographic Boolean functions satisfying the required properties of balance, nonlinearity, autocorrelation, and other stability indicators are considered. A technique for estimating the computational efficiency of gradient search methods, based on the construction of selective (empirical) distribution functions characterizing the probability of the formation of Boolean functions with indices of stability not lower than required, is proposed. As an indicator of computational efficiency, an average number of attempts is proposed to be performed using a heuristic method to form a cryptographic Boolean function with the required properties. Comparative assessments of the effectiveness of the heuristic methods are considered. The results of investigations of the cryptographic properties of the formed Boolean functions in comparison with the best known assessments are given. On the basis of the conducted research, it can be concluded that the functions constructed in accordance with the developed method have high persistence indexes and exceed the known functions by these indicators.

In this article, heuristic methods of hill climbing for cryptographic Boolean functions satisfying the required properties of balance, nonlinearity, autocorrelation, and other stability indicators are considered. A technique for estimating the computational efficiency of gradient search methods, based on the construction of selective (empirical) distribution functions characterizing the probability of the formation of Boolean functions with indices of stability not lower than required, is proposed. As an indicator of computational efficiency, an average number of attempts is proposed to be performed using a heuristic method to form a cryptographic Boolean function with the required properties. Comparative assessments of the effectiveness of the heuristic methods are considered. The results of investigations of the cryptographic properties of the formed Boolean functions in comparison with the best known assessments are given. On the basis of the conducted research, it can be conclud...

WHAT IS BOOLEAN FUNCTION?  Let B={0,1} and В n = {0, 1} n. Every function f : В n → В is called Boolean function of n variables. В n = { f |f : В n → В }, |В n | = 2 2 n  Let f 1 , f 2 , …, f m ∈ В n. Mapping F : В n → В m defined by the rule F(x) = (f 1 (х), f 2 (х), …, f m (х)), is called vectorial Boolean function and f 1 , f 2 , …, f m are its coordinate functions.

In this paper, we generalize some existing results on Boolean functions to the q-ary functions defined over Zq, where q≥ 2 is an integer, and obtain some new characterization of q-ary functions based on spectral analysis. We provide a relationship between Walsh-...

From the last few years, generalized bent functions gain a lot of attention in research as they have many applications in various fields such as combinatorial design, sequence design theory, cryptography, CDMA communication, etc. A deep and broad study of generalized bent functions with their properties is done in literature. Kumar et al.[11] first gave the concept of generalized bent function. Many researchers studied the properties and characterizations of generalized bent functions. In [2] authors introduced the concept of generalized (q-ary) negabent functions and studied some properties of generalized (q-ary) negabent functions. In this paper, we study the generalized (q-ary) bent functions f : Z n q → Z q , where Z q is the ring of integers with mod q, Z n q is the vector space of dimension n over Z q and q ≥ 2 is any positive integer. We discuss several properties of generalized (q-ary) bent functions with respect to their nega-Hadamard transform. We also study the relation between generalized nega-Hadamard transforms and generalized nega-autocorrelations. Furthermore, we prove the necessary and sufficient conditions for the bentness and negabentness of generalized (q-ary) bent function generated by the secondary construction for Z n q , where q = 2 s .

In several combinatorial optimization problems arising in cryptography and design theory, the admissible solutions must often satisfy a balancedness constraint, such as being represented by bitstrings with a fixed number of ones. For this reason, several works in the literature tackling these optimization problems with Genetic Algorithms (GA) introduced new balanced crossover operators which ensure that the offspring has the same balancedness characteristics of the parents. However, the use of such operators has never been thoroughly motivated, except for some generic considerations about search space reduction. In this paper, we undertake a rigorous statistical investigation on the effect of balanced and unbalanced crossover operators against three optimization problems from the area of cryptography and coding theory: nonlinear balanced Boolean functions, binary Orthogonal Arrays (OA) and bent functions. In particular, we consider three different balanced crossover operators (each with two variants: "left-to-right" and "shuffled"), two of which have never been published before, and compare their performances with classic one-point crossover. We are able to confirm that the balanced crossover operators performs better than all three balanced crossover operators. Furthermore, in two out of three crossovers, the "left-to-right" version performs better than the "shuffled" version.

It is known that the Williamson construction for Hadamard matrices can be generalized to constructions using sums of tensor products. This paper describes a specic construction using real monomial representations of Cliord algebras, and its connection with graphs of amicability and anti-amicability. It is proven that this construction works for all such representations where the order of the matrices is a power of 2. Some related results are given for small dimensions.

The quasi-Clifford algebras as described by Gastineau-Hills in 1980 and 1982, should be better known, and have only recently been rediscovered. These algebras and their representation theory provide effective tools to address the following problem arising from a plug-in construction for Hadamard matrices: Given λ, a pattern of amicability / anti-amicability, with λ j,k = λ k,j = ±1, find a set of n monomial {−1, 0, 1} matrices D of minimal order such that

The real monomial representations of Clifford algebras give rise to two sequences of bent functions. For each of these sequences, the corresponding Cayley graphs are strongly regular graphs, and the corresponding sequences of strongly regular graph parameters coincide. Even so, the corresponding graphs in the two sequences are not isomorphic, except in the first 3 cases. The proof of this non-isomorphism is a simple consequence of a theorem of Radon.

This paper examines a pair of bent functions on Z 2m 2 and their relationship to a necessary condition for the existence of an automorphism of an edge-coloured graph whose colours are defined by the properties of a canonical basis for the real representation of the Clifford algebra R m,m. Some other necessary conditions are also briefly examined. Note that the existence of such an automorphism automatically implies that the subgraphs ∆ m [−1] and ∆ m [1] are isomorphic. Considering that it is known that ∆ m [−1] is a strongly regular graph, a more general question can be asked concerning such graphs.

In several combinatorial optimization problems arising in cryptography and design theory, the admissible solutions must often satisfy a balancedness constraint, such as being represented by bitstrings with a fixed number of ones. For this reason, several works in the literature tackling these optimization problems with Genetic Algorithms (GA) introduced new balanced crossover operators which ensure that the offspring has the same balancedness characteristics of the parents. However, the use of such operators has never been thoroughly motivated, except for some generic considerations about search space reduction. In this paper, we undertake a rigorous statistical investigation on the effect of balanced and unbalanced crossover operators against three optimization problems from the area of cryptography and coding theory: nonlinear balanced Boolean functions, binary Orthogonal Arrays (OA) and bent functions. In particular, we consider three different balanced crossover operators (each with two variants: "left-to-right" and "shuffled"), two of which have never been published before, and compare their performances with classic one-point crossover. We are able to confirm that the balanced crossover operators performs better than all three balanced crossover operators. Furthermore, in two out of three crossovers, the "left-to-right" version performs better than the "shuffled" version.

In this thesis hyperovals and ovals are considered in the projective plane PG(2,q), q = 2m even. Traditionally these objects are studied algebraically via o-polynomials. In our work, a different approach is used by means of g-functions. These functions also provide a natural description of Niho bent functions. Using g-functions, all ovals and Niho bent functions are listed up to equivalency for dimensions m ≤ 6

WHAT IS BOOLEAN FUNCTION?  Let B={0,1} and В n = {0, 1} n. Every function f : В n → В is called Boolean function of n variables. В n = { f |f : В n → В }, |В n | = 2 2 n  Let f 1 , f 2 , …, f m ∈ В n. Mapping F : В n → В m defined by the rule F(x) = (f 1 (х), f 2 (х), …, f m (х)), is called vectorial Boolean function and f 1 , f 2 , …, f m are its coordinate functions.

Substitution boxes are the main nonlinear component of block ciphers. The security of these ciphers against linear, differential, or side-channel attacks is dependent on the design of such component and their intrinsic properties. There are several methods that aim to cryptographically define, generate, or search for strong substitution boxes. The application of combinatorial optimization algorithms is one of the most useful methodologies in this research area. In this article, we present a novel hybrid method based on the Leaders and Followers and hill-climbing over Hamming Weight Classes metaheuristics, coupled with a new trade-off fitness function that generates 8-bit bijective substitution boxes with good resisting properties towards classical cryptanalysis and side-channel attacks by power consumption. We address the best Pareto optimal solutions for the multi-objective optimization of non-linearity and confusion coefficient variance.

The property of nonlinearity has high importance for the design of strong substitution boxes. Therefore, the development of new techniques to produce substitution boxes with high values of nonlinearity is essential. Many research papers have shown that optimization algorithms are an efficient technique to obtain good solutions. However, there is no reference in the public literature showing that a heuristic method obtains optimal nonlinearity unless seeded with optimal initial solutions. Moreover, the majority of papers with the best nonlinearity reported for pseudo-random seeding of the algorithm(s) often achieve their results with the help of some cost function(s) over the Walsh–Hadamard spectrum of the substitution. In the sense, we proposed to present, in this paper, a novel external parameter independent cost function for evolving bijective s-boxes of high nonlinearity, which is highly correlated to this property. Several heuristic approaches including GaT (genetic and tree), L...

Substitution boxes are the main nonlinear component of block ciphers. The security of these ciphers against linear, differential, or side-channel attacks is dependent on the design of such component and their intrinsic properties. There are several methods that aim to cryptographically define, generate, or search for strong substitution boxes. The application of combinatorial optimization algorithms is one of the most useful methodologies in this research area. In this article, we present a novel hybrid method based on the Leaders and Followers and hill-climbing over Hamming Weight Classes metaheuristics, coupled with a new trade-off fitness function that generates 8-bit bijective substitution boxes with good resisting properties towards classical cryptanalysis and side-channel attacks by power consumption. We address the best Pareto optimal solutions for the multi-objective optimization of non-linearity and confusion coefficient variance.

Substitution boxes are the main nonlinear component of block ciphers. The security of these ciphers against linear, differential, or side-channel attacks is dependent on the design of such component and their intrinsic properties. There are several methods that aim to cryptographically define, generate, or search for strong substitution boxes. The application of combinatorial optimization algorithms is one of the most useful methodologies in this research area. In this article, we present a novel hybrid method based on the Leaders and Followers and hill-climbing over Hamming Weight Classes metaheuristics, coupled with a new trade-off fitness function that generates 8-bit bijective substitution boxes with good resisting properties towards classical cryptanalysis and side-channel attacks by power consumption. We address the best Pareto optimal solutions for the multi-objective optimization of non-linearity and confusion coefficient variance.

Boolean functions have a prominent role in many real-world applications, which makes them a very active research domain. Throughout the years, various heuristic techniques proved to be an attractive choice for the construction of Boolean functions with different properties. One of the most important properties is nonlinearity, and in particular maximally nonlinear Boolean functions are also called bent functions. In this paper, instead of considering Boolean functions, we experiment with quaternary functions. The corresponding problem is much more difficult and presents an interesting benchmark as well as realworld applications. The results we obtain show that evolutionary metaheuristics, especially genetic programming, succeed in finding quaternary functions with the desired properties. The obtained results in the quaternary domain can also be translated into the binary domain, in which case this approach compares favorably with the state-of-the-art in Boolean optimization. Our techniques are able to find quaternary bent functions for up to 8 inputs, which corresponds to obtaining Boolean bent functions of 16 inputs.

Boolean functions as well as their generalizations, vectorial Boolean functions are extremely active areas of research. Their applications can be found in domains such as error correcting codes, communication, and cryptography. Accordingly, various methods how to obtain Boolean functions are explored where one group belongs to heuristic techniques and more precisely, evolutionary algorithms. In this paper we explore how to evolve (vectorial) Boolean functions with specific properties by utilizing several different algorithms and encodings. As far as we are aware, we are the first to explore the topic of evolution of vectorial Boolean functions where the output dimension is strictly smaller than the input dimension. Our results show that evolutionary algorithms represent a valuable option to produce vectorial Boolean functions where good results are obtained for various sizes. On the other hand, as the number of outputs grow, we can observe that evolutionary algorithms are still able to obtain high quality results but with much more difficulty.

The real monomial representations of Clifford algebras give rise to two sequences of bent functions. For each of these sequences, the corresponding Cayley graphs are strongly regular graphs, and the corresponding sequences of strongly regular graph parameters coincide. Even so, the corresponding graphs in the two sequences are not isomorphic, except in the first 3 cases. The proof of this non-isomorphism is a simple consequence of a theorem of Radon.