Class Of Boolean Functions Research Articles

On one-sided testing affine subspaces

We study the query complexity of one-sided ϵ-testing the class of Boolean functions f:Fn→{0,1} that describe affine subspaces and Boolean functions that describe axis-parallel affine subspaces, where F is any finite field. We give polynomial-time ϵ-testers that ask O˜(1/ϵ) queries. This improves the query complexity O˜(|F|/ϵ) in [14]. The almost optimality of the algorithms follows from the lower bound of Ω(1/ϵ) for the query complexity proved by Bshouty and Goldreich [3].We then show that any one-sided ϵ-tester with proximity parameter ϵ<1/|F|d for the class of Boolean functions that describe (n−d)-dimensional affine subspaces and Boolean functions that describe axis-parallel (n−d)-dimensional affine subspaces must make at least Ω(1/ϵ+|F|d−1log⁡n) and Ω(1/ϵ+|F|d−1n) queries, respectively. This improves the lower bound Ω(log⁡n/log⁡log⁡n) that is proved in [14] for F=GF(2). We also give testers for those classes with query complexity that almost match the lower bounds.1

Bent Functions and Strongly Regular Graphs

The family of bent functions is a known class of Boolean functions, which have a great importance in cryptography. The Cayley graph defined on \(\mathbb{Z}_{2}^{n}\) by the support of a bent function is a strongly regular graph \(srg(v,k,\lambda,\mu)\), with \(\lambda=\mu\). In this paper we list the parameters of such Cayley graphs. Moreover, a condition is given on \((n,m)\)-bent functions \(F=(f_1,\ldots,f_m)\), involving the support of their components \(f_i\), and their \(n\)-ary symmetric differences.

Biologically meaningful regulatory logic enhances the convergence rate in Boolean networks and bushiness of their state transition graph.

Boolean models of gene regulatory networks (GRNs) have gained widespread traction as they can easily recapitulate cellular phenotypes via their attractor states. Their overall dynamics are embodied in a state transition graph (STG). Indeed, two Boolean networks (BNs) with the same network structure and attractors can have drastically different STGs depending on the type of Boolean functions (BFs) employed. Our objective here is to systematically delineate the effects of different classes of BFs on the structural features of the STG of reconstructed Boolean GRNs while keeping network structure and biological attractors fixed, and explore the characteristics of BFs that drive those features. Using $10$ reconstructed Boolean GRNs, we generate ensembles that differ in BFs and compute from their STGs the dynamics' rate of contraction or 'bushiness' and rate of 'convergence', quantified with measures inspired from cellular automata (CA) that are based on the garden-of-Eden (GoE) states. We find that biologically meaningful BFs lead to higher STG 'bushiness' and 'convergence' than random ones. Obtaining such 'global' measures gets computationally expensive with larger network sizes, stressing the need for feasible proxies. So we adapt Wuensche's $Z$-parameter in CA to BFs in BNs and provide four natural variants, which, along with the average sensitivity of BFs computed at the network level, comprise our descriptors of local dynamics and we find some of them to be good proxies for bushiness. Finally, we provide an excellent proxy for the 'convergence' based on computing transient lengths originating at random states rather than GoE states.

Infinite families of minimal binary codes via Krawtchouk polynomials

Linear codes play a crucial role in various fields of engineering and mathematics, including data storage, communication, cryptography, and combinatorics. Minimal linear codes, a subset of linear codes, are particularly essential for designing effective secret sharing schemes. In this paper, we introduce several classes of minimal binary linear codes by carefully selecting appropriate Boolean functions. These functions belong to a renowned class of Boolean functions, namely, the general Maiorana–McFarland class. We employ a method first proposed by Ding et al. (IEEE Trans Inf Theory 64(10):6536–6545, 2018) to construct minimal codes violating the Ashikhmin–Barg bound (wide minimal codes) by using Krawtchouk polynomials. The lengths, dimensions, and weight distributions of the obtained codes are determined using the Walsh spectrum distribution of the chosen Boolean functions. Our findings demonstrate that a vast majority of the newly constructed codes are wide minimal. Furthermore, our proposed codes exhibit a significantly larger minimum distance, in some cases, compared to some existing similar constructions. Finally, we address this method, based on Krawtchouk polynomials, more generally, and highlight certain generic properties related to it. These general results offer insights into the scope of this approach.

Boolean function classes with high nonlinearity

In 2020, Liu (Q. Liu, The lower bounds on the second-order nonlinearity of three classes of Boolean functions, Advances in Mathematics of Communications, AIMS (2021), Doi: 10.3934/amc.2020136) studied the second-order nonlinearity of three classes of Boolean functions, which have high first-order nonlinearity. Liu gave bounds on the first-order nonlinearity of these classes based on some experimental results using Magma. This article provides theoretical proof for computing the lower bounds on the first-order nonlinearity of two classes of Boolean functions out of three. Our bounds are identical to the bounds obtained by Liu experimentally.

Union-closed sets and Horn Boolean functions

A family F of sets is union-closed if the union of any two sets from F belongs to F. The union-closed sets conjecture states that if F is a finite union-closed family of finite sets, then there is an element that belongs to at least half of the sets in F. The conjecture has several equivalent formulations in terms of other combinatorial structures such as lattices and graphs. In its whole generality the conjecture remains wide open, but it was verified for various important classes of lattices, such as lower semimodular lattices, and graphs, such as chordal bipartite graphs. In the present paper we develop a Boolean approach to the conjecture and verify it for several classes of Boolean functions, such as submodular functions and double Horn functions.

The Zhegalkin Polynomial of Multiseat Sole Sufficient Operator

Among functionally complete sets of Boolean functions, sole sufficient operators are of particular interest. They have a wide range of applicability and are not limited to the two-seat case. In this paper, the conditions, imposed on the Zhegalkin polynomial coefficients, are formulated. The conditions are necessary and sufficient for the polynomial to correspond to a sole sufficient operator. The polynomial representation of constant-preserving Boolean functions is considered. It is shown that the properties of monotone and linearity do not require special consideration in describing a sole sufficient operator. The concept of a dual remainder polynomial is introduced. The value of it allows one to determine the self-duality of a Boolean function. It is proved that the preserving 0 and 1 or preserving neither 0 nor 1 Boolean function is self-dual if and only if the dual remainder of its corresponding Zhegalkin polynomial is equal to 0 for any sets of function variable values. Based on this fact, a system of leading coefficients is obtained. The solution of the system made it possible to formulate the criterion for the self-duality of the Boolean function represented by the Zhegalkin polynomial. It imposes necessary and sufficient conditions on the polynomial coefficients. Thus, it is shown that Zhegalkin polynomials are a rather convenient tool for studying precomplete classes of Boolean functions.

Some classes of easily testable circuits in the Zhegalkin basis

Abstract We identify the classes of Boolean functions that may be implemented by easily testable circuits in the Zhegalkin basis for constant type-1 faults on outputs of gates. An upper estimate for the length of a complete fault detection test for three-place functions is 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.

Quantum algorithm for ESOP minimization

The Exclusive-OR Sum-of-Product (ESOP) minimization problem has long been of interest to the research community because of its importance in classical logic design (including low-power design and design for test), reversible logic synthesis, and knowledge discovery, among other applications. However, no exact minimal minimization method has been presented for more than seven variables on arbitrary functions. This paper presents a novel quantum-classical hybrid algorithm for the exact minimal ESOP minimization of incompletely specified Boolean functions. This algorithm constructs oracles from sets of constraints and leverages the quantum speedup offered by Grover's algorithm to find solutions to these oracles, thereby improving over classical algorithms. Improved encoding of ESOP expressions results in substantially fewer decision variables compared to many existing algorithms for many classes of Boolean functions. This paper also extends the idea of exact minimal ESOP minimization to additionally minimize the cost of realizing an ESOP expression as a quantum circuit. To the extent of the authors' knowledge, such a method has never been published. This algorithm was tested on completely and incompletely specified Boolean functions via quantum simulation.

Quantum rotational cryptanalysis for preimage recovery of round-reduced Keccak

The Exclusive-OR Sum-of-Product (ESOP) minimization problem has long been of interest to the research community because of its importance in classical logic design (including low-power design and design for test), reversible logic synthesis, and knowledge discovery, among other applications. However, no exact minimal minimization method has been presented for more than seven variables on arbitrary functions. This paper presents a novel quantum-classical hybrid algorithm for the exact minimal ESOP minimization of incompletely specified Boolean functions. This algorithm constructs oracles from sets of constraints and leverages the quantum speedup offered by Grover's algorithm to find solutions to these oracles, thereby improving over classical algorithms. Improved encoding of ESOP expressions results in substantially fewer decision variables compared to many existing algorithms for many classes of Boolean functions. This paper also extends the idea of exact minimal ESOP minimization to additionally minimize the cost of realizing an ESOP expression as a quantum circuit. To the extent of the authors' knowledge, such a method has never been published. This algorithm was tested on completely and incompletely specified Boolean functions via quantum simulation.

Tight bounds on sensitivity and block sensitivity of some classes of transitive functions

Nisan and Szegedy [18] conjectured that block sensitivity is at most polynomial in sensitivity for any Boolean function. Until a recent breakthrough of Huang [15], the conjecture had been wide open in the general case, and was proved only for a few special classes of Boolean functions. Huang's result [15] implies that block sensitivity is at most the 4th power of sensitivity for any Boolean function. It remains open if a tighter relationship between sensitivity and block sensitivity holds for arbitrary Boolean functions; the largest known gap between these measures is quadratic [20,23,9,12,4,10].We prove tighter bounds showing that block sensitivity is at most 3rd power, and in some cases at most square of sensitivity for subclasses of transitive functions, defined by various properties of their DNF (or CNF) representation. Our results improve and extend previous results regarding transitive functions. We obtain these results by proving tight (up to constant factors) lower bounds on the smallest possible sensitivity of functions in these classes.In another line of research, the smallest possible block sensitivity of transitive functions has been closely examined. Our results yield tight (up to constant factors) lower bounds on the block sensitivity of the classes we consider.

A Novel Method of Computing Quadratic Weil Sum

Let χ be an additive character of F q and let L(x) be a linearized polynomial over F qn. In this paper, we consider the Weil sums of quadratic functions, namely, . Quadratic Weil sum is a critical topic in number theory and has applications in sequence design, coding, and cryptography. The computation of a quadratic Weil sum is complicated and sometimes impractical. This paper introduces a novel and generic way of computing quadratic Weil sums, which allows us to deal with the computation more efficiently compared with previous methods. As a result, we give a formula for computing explicitly in the case aij ∈ Fq . We also incorporate our results to obtain the condition for a class of Boolean functions to be bent, which are highly non-linear and widely applied in cryptography.

Resiliency and Nonlinearity Profiles of Some Cryptographic Functions

Boolean functions are important in terms of their cryptographic and combinatorial properties for different kinds of cryptosystems. The nonlinearity and resiliency of cryptographic functions are crucial criteria with respect to protection of ciphers from affine approximation and correlation attacks. In this article, some constructions of disjoint spectra Boolean that function by concatenating the functions on a lesser number of variables are provided. The nonlinearity and resiliency profiles of the constructed functions are obtained. From the profiles of the constructed functions, it is observed that the nonlinearity of these functions is greater than or equal to the nonlinearity of some existing functions. Furthermore, in the security analysis of cryptosystems, 4th order nonlinearity of Boolean functions play a crucial role. It provides protection against various higher order approximation attacks. The lower bounds on 4th order nonlinearity of some classes of Boolean functions having degree 5 are provided. The lower bounds of two classes of functions have form Tr1n(λxd) for all x∈F2n,λ∈F2n*, where (i) d=2i+2j+2k+2ℓ+1, where i,j,k,ℓ are integers such that i&gt;j&gt;k&gt;ℓ≥1 and n&gt;2i, and (ii) d=24ℓ+23ℓ+22ℓ+2ℓ+1, where ℓ&gt;0 is an integer with property gcd(ℓ,n)=1, n&gt;8 are provided. The obtained lower bounds are compared with some existing results available in the literature.

What perceptron neural networks are (not) good for?

Perceptron Neural Networks (PNNs) are essential components of intelligent systems because they produce efficient solutions to problems of overwhelming complexity for conventional computing methods.Many papers show that PNNs can approximate a wide variety of functions, but comparatively, very few discuss their limitations and the scope of this paper. To this aim, we define two classes of Boolean functions – sensitive and robust –, and prove that an exponentially large set of sensitive functions are exponentially difficult to compute by multi-layer PNNs (hence incomputable by single-layer PNNs). A comparatively large set of functions in the second one, but not all, are computable by single-layer PNNs.Finally, we used polynomial threshold PNNs to compute all Boolean functions with quantum annealing and present in detail a QUBO computation on the D-Wave Advantage.These results confirm that the successes of PNNs, or lack of them, are in part determined by properties of the learned data sets and suggest that sensitive functions may not be (efficiently) computed by PNNs.

The Lower Bound of Second-Order Nonlinearity of a Class of Boolean Functions

The Lower Bound of Second-Order Nonlinearity of a Class of Boolean Functions

Exponential separation between quantum and classical ordered binary decision diagrams, reordering method and hierarchies

In this paper, we study quantum Ordered Binary Decision Diagrams(\(\mathrm {OBDD}\)) model; it is a restricted version of read-once quantum branching programs, with respect to “width” complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a new technique (“reordering”) for proving lower bounds and upper bounds for OBDD with an arbitrary order of input variables if we have similar bounds for the natural order. Using this transformation, we construct a total function \(\mathrm {REQ}\) such that the deterministic \(\mathrm {OBDD}\) complexity of it is at least \(2^{\varOmega (n / \log n)}\), and the quantum \(\mathrm {OBDD}\) complexity of it is at most \(O(n^2/\log n)\). It is the biggest known gap for explicit functions not representable by \(\mathrm {OBDD}\)s of a linear width. Another function(shifted equality function) allows us to obtain a gap \(2^{\varOmega (n)}\) vs \(O(n^2)\). Moreover, we prove the bounded error quantum and probabilistic \(\mathrm {OBDD}\) width hierarchies for complexity classes of Boolean functions. Additionally, using “reordering” method we extend a hierarchy for read-k-times Ordered Binary Decision Diagrams (\({\textit{k}}\text {-}\mathrm {OBDD}\)) of polynomial width, for \(k = o(n / \log ^3 n)\). We prove a similar hierarchy for bounded error probabilistic \({\textit{k}}\text {-}\mathrm {OBDD}\)s of polynomial, superpolynomial and subexponential width. The extended abstract of this work was presented on International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8 – 12, 2017 Khadiev and Khadieva (2017)

Nonlinearity of nonbalanced and nearly bent boolean functions based on Galois ring

Designing of nonlinear confusion component of block cipher is one of the most important and inevitable research problem. Nowadays mostly heuristic search schemes were utilized for the construction of these confusion component. To construct, a cryptographically secure confusion component several algebraic structures were utilized. The thirst for new algebraic structure for the construction of these nonlinear confusion component has always been a point of interest. In this communication, we have utilized a maximal cyclic subgroup of unit of Galois ring. The offered algorithm is more general as compared to Galois field. The class of Boolean functions over Galois ring fall in mixed category which are not completely balanced. Boolean functions having higher nonlinearity and others cryptographic aspects added an inevitable significance in the construction of modern block ciphers. The primary idea of this article is to structure non-balanced Boolean functions on n variables, where n is an even integer, sustaining strict avalanche criterion (SAC) and bit independent criterion (BIC). By comparing SAC with available cryptographic Boolean functions, the constructed multivalued Boolean function acquire highest nonlinearity which does not follow the existing nonlinearity bound of Boolean functions. These newly proposed S-boxes consists of n basic Boolean functions which satisfy the balancedness and non-balancedness criterion. Therefore, these S-box structure lies within a less balanced and more bent Boolean function categories.

Learning the Rules of the Game: An Interpretable AI for Learning How to Play

In this article, we present an interpretable artificial intelligence, and its associated machine learning algorithm, that is capable of automatically learning the rules of a game whenever the rules&#x2014;the relationship between a player&#x2019;s current state and their corresponding set of legal moves&#x2014;can be represented as a set of low degree Zhegalkin polynomials, a special class of Boolean functions. This is true for many popular games including <i>Spanish Domin&#x00F3;</i> and the card game <i>President</i>. Our method takes advantage of such low polynomial degree to compute an exact representation of the rules in polynomial time instead of the required exponential time for generic Boolean functions. The rules can also be represented using significantly less storage than in the generic case which, for many games, leads to a representation that is easy to interpret.

A Complete Study of Two Classes of Boolean Functions: Direct Sums of Monomials and Threshold Functions

In this paper, we make a comprehensive study of two classes of Boolean functions whose interest originally comes from hybrid symmetric-FHE encryption (with stream ciphers like FiLIP), but which also present much interest for general stream ciphers. The functions in these two classes are cheap and easy to implement, and they allow the resistance to all classical attacks and to their guess and determine variants as well. We determine exactly all the main cryptographic parameters (algebraic degree, resiliency order, nonlinearity, algebraic immunity) for all functions in these two classes, and we give close bounds for the others (fast algebraic immunity, the dimension of the space of annihilators of minimal degree). This is the first time that this is done for all functions in large classes of cryptographic interest.