Showing papers on "Boolean function published in 2022"

A New Approach to Pinning Control of Boolean Networks

Abstract: Boolean networks (BNs) are discrete-time systems, where nodes are interconnected (here, we call such connection rule among nodes as a network structure), and the dynamics of each gene node is determined by logical functions. In this article, we propose a new approach on pinning control design for global stabilization of BNs based on BNs’ network structure, named as network-structure-based distributed pinning control. Compared with the existing literature, the design of pinning control is not based on the state transition matrix of BNs. Hence, the computational complexity in this article is reduced from $O(2^{2n})$ to $O(n^2+n2^K)$, where $n$ is the number of nodes and $K\leq n$ is the largest number of in-neighbors of nodes. In addition, without using the state transition matrix, global state information is no longer needed; the design of pinning control is just based on neighbors’ local information, which is easier to implement. The proposed method is well demonstrated by several biological networks with different sizes. The results are shown to be simple and concise, while the traditional pinning control cannot be applied for BNs with such a large dimension.

Sufficient Reasons for Classifier Decisions in the Presence of Domain Constraints

Abstract: Recent work has unveiled a theory for reasoning about the decisions made by binary classifiers: a classifier describes a Boolean function, and the reasons behind an instance being classified as positive are the prime-implicants of the function that are satisfied by the instance. One drawback of these works is that they do not explicitly treat scenarios where the underlying data is known to be constrained, e.g., certain combinations of features may not exist, may not be observable, or may be required to be disregarded. We propose a more general theory, also based on prime-implicants, tailored to taking constraints into account. The main idea is to view classifiers as describing partial Boolean functions that are undefined on instances that do not satisfy the constraints. We prove that this simple idea results in more parsimonious reasons. That is, not taking constraints into account (e.g., ignoring, or taking them as negative instances) results in reasons that are subsumed by reasons that do take constraints into account. We illustrate this improved succinctness on synthetic classifiers and classifiers learnt from real data.

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

Abstract: 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.

Stabilizing Large-Scale Probabilistic Boolean Networks by Pinning Control

Abstract: This article aims to stabilize probabilistic Boolean networks (PBNs) via a novel pinning control strategy. In a PBN, the state evolution of each gene switches among a collection of candidate Boolean functions with preassigned probability distributions, which govern the activation frequency of each Boolean function. Due to the existence of stochasticity, the mode-independent pinning controller might be disabled. Thus, both mode-independent and mode-dependent pinning controller are required here. Moreover, a criterion is derived to determine whether mode-independent controllers are applicable while the pinned nodes are given. It is worth pointing out that this pinning control is based on the $n\times n$ network structure rather than $2^{n} \times 2^{n}$ state transition matrix. Therefore, compared with the existing results, this pinning control strategy is more practicable and has the ability to handle large-scale networks, especially sparsely connected networks. To demonstrate the effectiveness of the designed control scheme, a PBN that describes the mammalian cell-cycle encountering a mutated phenotype is discussed as a simulation.

Observability Criteria for Boolean Networks

Abstract: This article investigates observability of Boolean control networks (BCNs) and probabilistic Boolean networks (PBNs). First, weak observability of BCNs is discussed via the nonaugmented approach. The obtained result is then applied to determine (asymptotic) observability of PBNs. Finally, complexity of algorithms based on new criteria is analyzed. Compared with existing ones, time and space complexity do not get worse, even are improved under some mild conditions.

Artificial Intelligence for the Design of Symmetric Cryptographic Primitives

Abstract: This chapter provides a general overview of AI methods used to support the design of cryptographic primitives and protocols. After giving a brief introduction to the basic concepts underlying the field of cryptography, we review the most researched use cases concerning the use of AI techniques and models to design cryptographic primitives, focusing mainly on Boolean functions, S-boxes and pseudorandom number generators. We then point out two interesting directions for further research on the design of cryptographic primitives where AI methods could be applied in the future.

SuperBall: A New Approach for MILP Modelings of Boolean Functions

Abstract: Mixed Integer Linear Programming (MILP) solver has become one of the most powerful tools of searching for cryptographic characteristics. It has great significance to study the influencing factors of the efficiency of MILP models. For this goal, different types of MILP models should be constructed and carefully studied. As Boolean functions are the fundamental cryptographic components, in this paper, we study the descriptive models of Boolean functions. Here, a descriptive model of a Boolean function refers to a set of integer linear inequalities, where the set of the binary solutions to these inequalities is exactly the support of this Boolean function. Previously, it is hard to construct various types of descriptive models for study, one important reason is that only a few kinds of inequalities can be generated. On seeing this, a new approach, called SuperBall, is proposed to generate inequalities. The SuperBall approach is based on the method of undetermined coefficients, and it could generate almost all kinds of inequalities by appending appropriate constraints. Besides, the Sasaki-Todo Algorithm is also improved to construct the descriptive models from a set of candidate inequalities by considering both their sizes and strengths, while the strengths of descriptive models have not been considered in the previous works. As applications, we constructed several types of descriptive models for the Sboxes of Liliput, SKINNY-128, and AES. The experimental results first prove that the diversity of the inequalities generated by the SuperBall approach is good. More importantly, the results show that the strengths of descriptive model do affect the efficiencies, and although there is not a type of descriptive model having the best efficiency in all experiments, we did find a specific type of descriptive model which has the minimal size and relatively large strength, and the descriptive models of this type have better efficiencies in most of our experiments.

Generic Constructions of (Boolean and Vectorial) Bent Functions and Their Consequences

Abstract: This article is devoted to Boolean and vectorial bent functions and their duals. Our ultimate objective is to increase such functions’ corpus by designing new ones covering many previous bent functions’ constructions. To this end, we provide several new infinite families of bent functions, including idempotent bent functions of any algebraic degree, bent functions in univariate trace form, and self-dual bent functions. Those bent functions are of great theoretical and practical interest because of their special structures and relationship with self-dual codes. In particular, many well-known bent functions are special cases of our bent functions. Moreover, we extend our results to vectorial bent functions and obtain three new infinite classes of vectorial bent functions of any possible degree by determining the explicit duals of three classes of well-known bent functions.

Two-Dimensional Golay Complementary Array Pairs/Sets With Bounded Row and Column Sequence PAPRs

Abstract: The one-dimensional (1-D) Golay complementary set (GCS) has many well-known properties and has been widely employed in communications engineering. The concept of 1-D GCS can be extended to the two-dimensional (2-D) Golay complementary array set (GCAS) where the 2-D aperiodic autocorrelations of constituent arrays sum to zero except for the 2-D zero shift. The 2-D GCAS includes the 2-D Golay complementary array pair (GCAP) as a special case when the set size is 2. In this paper, 2-D generalized Boolean functions are introduced and constructions of 2-D GCAPs, 2-D GCASs, and 2-D Golay complementary array mates based on generalized Boolean functions are proposed. Explicit expressions of 2-D Boolean functions for 2-D GCAPs and 2-D GCASs are given. Therefore, they are all direct constructions without the aid of other existing 1-D or 2-D sequences. Moreover, the peak-to-average power ratio (PAPR) properties of the column sequences and row sequences of the constructed 2-D GCAPs and 2-D GCASs are investigated and the PAPRs are proved to be upper bounded.

Card-Minimal Protocols for Three-Input Functions with Standard Playing Cards

Abstract: A protocol realizing a secure computation using a deck of physical cards is called a card-based cryptographic protocol. Since Niemi and Renvall first proposed a few protocols using a commercially available deck of playing cards in 1999, several protocols for the two-input AND and XOR functions have been proposed. By combining these existing protocols, one can construct a protocol for any Boolean function using a standard deck of playing cards. However, the minimal numbers of cards needed for Boolean functions having more than two inputs have not been revealed so much. Recently, Koyama et al. developed a card-minimal three-input AND protocol. In this study, by extending Koyama’s AND protocol, we construct a card-minimal protocol for the three-input majority function. Furthermore, carrying the idea behind these protocols further, we provide a generic card-minimal three-input protocol, which covers many important three-input Boolean functions.

Minimum complexity drives regulatory logic in Boolean models of living systems

Abstract: The properties of random Boolean networks have been investigated extensively as models of regulation in biological systems. However, the Boolean functions (BFs) specifying the associated logical update rules should not be expected to be random. In this contribution, we focus on biologically meaningful types of BFs, and perform a systematic study of their preponderance in a compilation of 2,687 functions extracted from published models. A surprising feature is that most of these BFs have odd "bias", that is they produce "on" outputs for a total number of input combinations that is odd. Upon further analysis, we are able to explain this observation, along with the enrichment of read-once functions (RoFs) and its nested canalyzing functions (NCFs) subset, in terms of 2 complexity measures: Boolean complexity based on string lengths in formal logic, which is yet unexplored in biological contexts, and the so-called average sensitivity. RoFs minimize Boolean complexity and all such functions have odd bias. Furthermore, NCFs minimize not only the Boolean complexity but also the average sensitivity. These results reveal the importance of minimum complexity in the regulatory logic of biological networks.

Direct Construction of Optimal Z-Complementary Code Sets With Even Lengths by Using Generalized Boolean Functions

Abstract: The Z-complementary code set (ZCCS) is well-known for being used in multicarrier code-division multiple access (MC-CDMA) systems to provide interference-free communication in a quasi-synchronous environment. Based on the existing literature, the direct constructions of optimal ZCCSs are limited to their lengths. This letter proposes a direct construction of optimal ZCCSs for all possible even lengths using generalized Boolean functions. The maximum column sequence peak-to-mean envelope power ratio (PMEPR) of the proposed ZCCSs is upper-bounded by two which can benefit in managing PMEPR over a ZCCS-based MC-CDMA system.

Structure-based approach to identifying small sets of driver nodes in biological networks

Abstract: In network control theory, driving all the nodes in the Feedback Vertex Set (FVS) by node-state override forces the network into one of its attractors (long-term dynamic behaviors). The FVS is often composed of more nodes than can be realistically manipulated in a system; for example, only up to three nodes can be controlled in intracellular networks, while their FVS may contain more than 10 nodes. Thus, we developed an approach to rank subsets of the FVS on Boolean models of intracellular networks using topological, dynamics-independent measures. We investigated the use of seven topological prediction measures sorted into three categories-centrality measures, propagation measures, and cycle-based measures. Using each measure, every subset was ranked and then evaluated against two dynamics-based metrics that measure the ability of interventions to drive the system toward or away from its attractors: To Control and Away Control. After examining an array of biological networks, we found that the FVS subsets that ranked in the top according to the propagation metrics can most effectively control the network. This result was independently corroborated on a second array of different Boolean models of biological networks. Consequently, overriding the entire FVS is not required to drive a biological network to one of its attractors, and this method provides a way to reliably identify effective FVS subsets without the knowledge of the network dynamics.

Algebras of Complemented Subsets

Abstract: AbstractComplemented subsets were introduced by Bishop, in order to avoid complementation in terms of negation. In his two approaches to measure theory Bishop used two sets of operations on complemented subsets. Here we study these two algebras and we introduce the notion of Bishop algebra as an abstraction of their common structure. We translate constructively the classical bijection between subsets and boolean-valued functions by establishing a bijection between the proper classes of complemented subsets and of strongly extensional, boolean-valued, partial functions. Avoiding negatively defined concepts, most of our results are within minimal logic.KeywordsBishop setscomplemented subsetspartial functions

Studying Special Operators for the Application of Evolutionary Algorithms in the Seek of Optimal Boolean Functions for Cryptography

Abstract: The role of Boolean functions in modern cryptography has triggered the necessity of developing methods to construct them with adequate properties, such as balancedness and high non-linearity—making them more resistant to a variety of cryptanalytic attacks. Research into the construction of weight-wise perfectly balanced Boolean functions using Evolutionary Algorithms is scarce but encouraging (e.g., [1]). In this work, we first investigate the effect on an evolutionary algorithm’s performance when relying solely on the penalty function, as opposed to the solution repairment method. Second, we focus on the effect of problem-specific crossover operators (e.g., those used on [2]), and particularly proposing a novel one free of solution repairs to preserve balancedness. The results obtained suggest that an adequate penalty factor and the use of specifically designed evolutionary operators is sufficient to find Boolean functions with weight-wise perfect balancedness and high non-linearity, as desired.

Short complete diagnostic tests for circuits with one additional input in the standard basis

Abstract: We prove that each monotone (antimonotone) Boolean function in n variables can be modeled by a logic circuit with one additional input in the basis “conjunction, disjunction, negation” allowing a complete diagnostic test with length no more than n + 2 (no more than n + 1, respectively) relative to constant faults of type 1 at outputs of logic gates.

Weightwise almost perfectly balanced functions: secondary constructions for all n and better weightwise nonlinearities

Abstract: AbstractThe design of FLIP stream cipher presented at Eurocrypt 2016 motivates the study of Boolean function with good cryptographic criteria when restricted to subsets of \(\mathbb {F}_2^n\). Since the security of FLIP relies on properties of functions restricted to subsets of constant Hamming weight, called slices, several studies investigate functions with good properties on the slices, i.e. weightwise properties. A major challenge is to build functions balanced on each slice, from which we get the notion of Weightwise Almost Perfectly Balanced (WAPB) functions. Although various constructions of WAPB functions have been exhibited since 2017, building WAPB functions with high weightwise nonlinearities remains a difficult task. Lower bounds on the weightwise nonlinearities of WAPB functions are known for very few families, and exact values were computed only for functions in at most 16 variables.In this article, we introduce and study two new secondary constructions of WAPB functions. This new strategy allows us to bound the weightwise nonlinearities from those of the parent functions enabling us to produce WAPB functions with high weightwise nonlinearities. As a practical application, we build several novel WAPB functions in up to 16 variables by taking parent functions from two different known families. Moreover, combining these outputs, we also produce the 16-variable WAPB function with the highest weightwise nonlinearities known so far. KeywordsFLIP cipherBoolean functionsWeightwise (almost) perfectly balanced functionWeightwise nonlinearity

Relation between spectra of Narain CFTs and properties of associated boolean functions

Abstract: A bstract Recently, the construction of Narain CFT from a certain class of quantum error correcting codes has been discovered. In particular, the spectral gap of Narain CFT corresponds to the binary distance of the code, not the genuine Hamming distance. In this paper, we show that the binary distance is identical to the so-called EPC distance of the boolean function uniquely associated with the quantum code. Therefore, seeking Narain CFT with large spectral gap can be addressed by getting a boolean function with high EPC distance. Furthermore, this problem can be undertaken by finding lower Peak-to-Average Power ratio (PAR) with respect to the binary truth table of the boolean function. Though this is neither sufficient nor necessary condition for high EPC distance, we construct some examples of relatively high EPC distances referring to the constructions for lower PAR. We also see that codes with high distance are related to induced graphs with low independence numbers.

Novel Quantum Algorithms to Minimize Switching Functions Based on Graph Partitions

Abstract: After Google reported its realization of quantum supremacy, Solving the classical problems with quantum computing is becoming a valuable research topic. Switching function minimization is an important problem in Electronic Design Automation (EDA) and logic synthesis, most of the solutions are based on heuristic algorithms with a classical computer, it is a good practice to solve this problem with a quantum processer. In this paper, we introduce a new hybrid classic quantum algorithm using Grover’s algorithm and symmetric functions to minimize small Disjoint Sum of Product (DSOP) and Sum of Product (SOP) for Boolean switching functions. Our method is based on graph partitions for arbitrary graphs to regular graphs, which can be solved by a Grover-based quantum searching algorithm we proposed. The Oracle for this quantum algorithm is built from Boolean symmetric functions and implemented with Lattice diagrams. It is shown analytically and verified by simulations on a quantum simulator that our methods can find all solutions to these problems.

Xor-And-Inverter Graphs for Quantum Compilation

Abstract: Abstract Quantum compilation is the task of translating a high-level description of a quantum algorithm into a sequence of low-level quantum operations. We propose and motivate the use of Xor-And-Inverter Graphs (XAG) to specify Boolean functions for quantum compilation. We present three different XAG-based compilation algorithms to synthesize quantum circuits in the Clifford + T library, hence targeting fault-tolerant quantum computing. The algorithms are designed to minimize relevant cost functions, such as the number of qubits, the T -count, and the T -depth, while allowing the flexibility of exploring different solutions. We present novel resource estimation results for relevant cryptographic and arithmetic benchmarks. The achieved results show a significant reduction in both T -count and T -depth when compared with the state-of-the-art.

Test Suite Generation for Boolean Conditions with Equivalence Class Partitioning

Abstract: Boolean test input generation is the process of finding sets of values for variables of a logical expression such that a given coverage criterion is achieved. This paper presents a formal framework in which evaluating an expression produces a tree structure, and where a coverage criterion is expressed as equivalence classes induced by a particular transformation over these trees. It then defines many well-known coverage criteria as particular cases of this framework. The paper describes an algorithm to generate test suites by a reduction through a graph problem; this algorithm works in the same way regardless of the criterion considered. An experimental evaluation of this technique shows that it produces test suites that are in many cases smaller than existing tools.