Properties Of Boolean Functions Research Articles

From multivalued to Boolean functions: Preservation of soft nested canalization

Nested canalization (NC) is a property of Boolean functions which has been recently extended to multivalued functions. We study the effect of the Van Ham mapping (from multivalued to Boolean functions) on this property. We introduce the class of softly nested canalizing (SNC) multivalued functions, and prove that the Van Ham mapping sends SNC multivalued functions to NC Boolean functions. Since NC multivalued functions are SNC, this preservation property holds for NC multivalued functions as well. We also study the relevance of SNC functions in the context of gene regulatory network modelling.

Investigations on <i>c</i>-Bent<sub>4</sub> Functions via the Unitary Transform and <i>c</i>-Correlation Functions

In this paper, some properties of Boolean functions via the unitary transform and c-correlation functions are presented. Based on the unitary transform, we present two classes of secondary constructions for c-bent4 functions. Also, by using the c-correlation functions, a direct link between c-autocorrelation function and the unitary transform of Boolean functions is provided, and the relationship among c-crosscorrelation functions of arbitrary four Boolean functions can be obtained.

Stability of Boolean Function Classes with Respect to Clones of Linear Functions

We consider classes of Boolean functions stable under compositions both from the right and from the left with clones. Motivated by the question how many properties of Boolean functions can be defined by means of linear equations, we focus on stability under compositions with the clone of linear idempotent functions. It follows from a result by Sparks that there are countably many such linearly definable classes of Boolean functions. In this paper, we refine this result by completely describing these classes. This work is tightly related with the theory of function minors, stable classes, clonoids, and hereditary classes, topics that have been widely investigated in recent years by several authors including Maurice Pouzet and his coauthors.

Improved cryptographic properties of Boolean functions obtained from the neighbourhood of Patterson-Wiedemann functions

Improved cryptographic properties of Boolean functions obtained from the neighbourhood of Patterson-Wiedemann functions

On cryptographic properties of cubic and splitting Boolean functions

AbstractThe weight, balancedness and nonlinearity are important properties of Boolean functions, but they can be difficult to determine in general. In this paper, we study how to compute them for two classes of functions where these problems are more tractable. In particular, we study functions of degree three and the so-called “splitting” functions. The latter are functions that can be written as the sum of two functions defined over disjoint sets of variables. We show how, for splitting functions, studying these properties reduces to the study of simpler functions. We provide then a procedure to compute the weight of a cubic Boolean function. We show computationally that, for a cubic Boolean function with limited number of terms, this procedure is on average significantly more efficient than some other methods.

Properties of Boolean functions with the extremal number of prime implicants

Properties of Boolean functions with the extremal number of prime implicants

Properties of Boolean Functions with Extremal Number of Prime Implicants

Properties of Boolean Functions with Extremal Number of Prime Implicants

The Walsh Transform of a Class of Boolean Functions

The Walsh transform is an important tool to investigate cryptographic properties of Boolean functions. This paper is devoted to study the Walsh transform of a class of Boolean functions defined as [see formula in PDF], by making use of the known conclusions of Walsh transform and the properties of trace function, and the conclusion is obtained by generalizing an existing result.

Certificate complexity and symmetry of nested canalizing functions

Boolean nested canalizing functions (NCFs) have important applications in molecular regulatory networks, engineering and computer science. In this paper, we study their certificate complexity. For both Boolean values $b\in\{0,1\}$, we obtain a formula for $b$-certificate complexity and consequently, we develop a direct proof of the certificate complexity formula of an NCF. Symmetry is another interesting property of Boolean functions and we significantly simplify the proofs of some recent theorems about partial symmetry of NCFs. We also describe the algebraic normal form of $s$-symmetric NCFs. We obtain the general formula of the cardinality of the set of $n$-variable $s$-symmetric Boolean NCFs for $s=1,\dots,n$. In particular, we enumerate the strongly asymmetric Boolean NCFs.

Constructions of balanced Boolean functions on even number of variables with maximum absolute value in autocorrelation spectra [formula omitted

The autocorrelation properties of Boolean functions are closely related to the Shannon’s concept of diffusion and can be accompanied with other cryptographic criteria (such as high nonlinearity and algebraic degree) for ensuring an overall robustness to various cryptanalytic methods. In a series of recent articles [14,9,15], the design methods of n-variable balanced Boolean functions n is strictly even) with small absolute indicator Δf < 2n/2 have been considered. Whereas the two first articles managed to solve this problem for relatively large n⩾46, a recent approach [15] has introduced a generic design framework achieving Δf < 2n/2 for even n⩾22. Based on a suitable modification of the method of Rothaus, used to construct new bent functions from known ones, we provide a generic iterative framework for designing balanced functions satisfying the condition Δf < 2n/2 and having overall good cryptographic properties for any even n⩾12. Even though the problem of specifying functions having Δf < 2n/2 for smaller n has been considered in [14,9,15] using various search algorithms, our method for the first time provides relatively simple iterative framework for variable spaces of more practical interest. Moreover, our approach can be efficiently applied to certain classes of initial functions (derived from partial spread bent functions) for deriving balanced functions with Δf < 2n/2 for relatively large n, namely for n⩾48 satisfying n≡0 mod 4 and n⩾54 with n≡2 mod 4. In the latter case, our nonlinearity bound is better than the one presented in [14].

An improved hybrid genetic algorithm to construct balanced Boolean function with optimal cryptographic properties

Boolean functions are used as nonlinear filter functions and combiner functions in several stream ciphers. The security of these stream ciphers largely depends upon cryptographic properties of Boolean functions. Finding a balanced Boolean function with optimal cryptographic properties is an open research problem in the cryptographic community. Since the number of n-variable Boolean functions is $$2^{2^n}$$ , it is not computationally feasible to search the entire space of such functions for cryptographically significant functions when $$n \ge 6$$ . In general, the construction of Boolean functions with optimal or near optimal cryptographically significant properties is formulated as combinatorial optimization problems. In this paper, we apply the Genetic algorithm with the integration of a local search procedure called hybrid GA (HGA) searching for the Boolean functions with high nonlinearity and low autocorrelation. The performance of the Genetic algorithm depends upon the tuning parameters such as crossover mechanism, mutation probability, selection criteria, and the choice of fitness/cost function. To achieve an optimal trade-off among two cryptographic properties, the choice of the cost function plays an important role for finding an optimal solution. So our main focus is to construct balanced Boolean functions using HGA with a new cost function and compare it with existing cost functions. Our experimental results shows that the HGA is more efficient than GA and the new cost function is more efficient than existing cost functions for reaching the optimal solutions.

Fast Algebraic Immunity of Boolean Functions and LCD Codes

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure a Boolean function's resistance to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. Some of the binary LCD codes presented in this paper are optimal. These binary LCD codes have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems.

On the Δ-equivalence of Boolean functions

AbstractA new equivalence relation on the set of Boolean functions is introduced: functions are declared to beΔ-equivalent if their autocorrelation functions are equal. It turns out that this classification agrees well with the cryptographic properties of Boolean functions: for functions belonging to the sameΔ-equivalence class a number of their cryptographic characteristics do coincide. For example, all bent-functions (of a fixed number of variables) make up one class.

Hidden complexity in Life-like rules.

An alternative way to study the rules of life-like cellular automata is presented. The proposed perspective studies some multifractal and informational properties of Boolean functions behind these rules. Results from this approach challenge the traditional argument about the simplicity of Lifelike rules.

Boolean functions as points on the hypersphere in the Euclidean space

Abstract A new approach to the study of algebraic, combinatorial, and cryptographic properties of Boolean functions is proposed. New relations between functions have been revealed by consideration of an injective mapping of the set of Boolean functions onto the sphere in a Euclidean space. Moreover, under this mapping some classes of functions have extremely regular localizations on the sphere. We introduce the concept of curvature of a Boolean function, which characterizes its proximity (in some sense) to maximally nonlinear functions.

A Graph Theoretic Approach to Construct Desired Cryptographic Boolean Functions

In this paper, we present four product operations to construct cryptographic boolean functions from smaller ones with predictable Walsh spectrum. A lot of cryptographic properties of boolean functions can be presented by their Walsh spectrum. In our method, we use the product of Cayley graphs to present new boolean functions with desired Walsh spectrum and investigate their non-linearity, algebraic and correlation immunity.

Primitive permutation groups satisfying the small orbit property and a problem of Bourgain and Kalai

We classify the primitive but not affine permutation subgroups of S_n with the following property: There exists some unbounded subset X of size at most hbox {clog}n whose orbit is polynomially bounded (in terms of n). This class of subgroups arises naturally in the study of threshold behavior of G-symmetric properties of Boolean functions. Our result answers a problem of Bourgain and Kalai. Contrary to a suggestion in their paper, we give an example for a primitive permutation group with this property, which contains {A_{m}} as a section for m rightarrow infty .

Maximal k-Intersecting Families of Subsets and Boolean Functions

A family of subsets of an n-element set is k-intersecting if the intersection of every k subsets in the family is nonempty. A family is maximalk-intersecting if no subset can be added to the family without violating the k-intersection property. There is a one-to-one correspondence between the families of subsets and Boolean functions defined as follows: To each family of subsets, assign the Boolean function whose unit tuples are the characteristic vectors of the subsets.We show that a family of subsets is maximal 2-intersecting if and only if the corresponding Boolean function is monotone and selfdual. Asymptotics for the number of such families is obtained. Some properties of Boolean functions corresponding to k-intersecting families are established fork > 2.

Optimal annihilator immunity with suboptimal bound of other properties

In this paper we present constructions of Boolean functions based on the basic theory of annihilator immunity and bent functions. These constructions provide functions with the maximum possible annihilator immunity, and the nonlinearity and resiliency of the functions can also be calculated with specific cases. Here we have also described certain properties of Boolean functions that are necessary for resistance against algebraic and fast algebraic attacks. Annihilator immunity, nonlinearity and resiliency are important properties of a Boolean function to resist these attacks. There was no systematic attempt made to construct Boolean functions with maximum annihilator immunity that have good bound of nonlinearity and resiliency. We have constructed two different classes of desired Boolean functions through our newly developed method based on multiobjective optimization method(NSGA-II). In the first class, we have constructed bent functions(optimal nonlinearity) with sub-optimal bound of annihilator immunity while in other, Boolean functions having optimal bound of annihilator immunity with good bound of nonlinearity were developed. Our problem was multiobjective(nonlinearity, annihilator immunity and resiliency), hence a heuristic multiobjective optimization technique(NSGA-II) was used.

On the Nonexistence of $q$ -Bent Boolean Functions

We continue the study of the properties of Boolean functions as reflected in The properties of a recently defined transform. For each non-constant Boolean function $q$ , the $q$ - transform of a Boolean function $f$ is related to the Hamming distances from $f$ to the functions obtainable from $q$ by nonsingular linear change of basis. Many properties that can be characterized by the Walsh–Hadamard transform have (for each $q$ ) analogues that can be characterized by the $q$ -transform. In this paper, we study one such property, bentness. We show that if $q$ is balanced and not affine, then there is no function that is both bent and $q$ -bent.