Abstract: In this paper, we first introduce a class of quartic Boolean functions. And then, the construction of weightwise perfectly balanced Boolean functions on $ 2^m $ variables are given by modifying the support of the quartic functions, where $ m $ is a positive integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.

Abstract: Symmetric Searchable Encryption (SSE) schemes facilitate searching over encrypted data, and have been extensively explored to improve function, efficiency or security. There are, however, additional functions that we need to consider in a real-world setting. For example, forward and backward privacy are required to adequately secure newly added documents and deleted documents in Dynamic SSE (DSSE) schemes, and support boolean search (that allows users to search over encrypted data using basic boolean operations) to achieve improved efficiency and retrieval accuracy. Therefore, in this article we first construct the Verifiable Boolean Search over encrypted data (VBS), and then improve VBS to achieve Forward and Backward privacy (VBS-FB). Finally, we formally prove the security of our proposed schemes, and evaluate their performance using real-world datasets.

Abstract: Abstract Motivation From a systematic perspective, it is crucial to infer and analyze gene regulatory network (GRN) from high-throughput single-cell RNA sequencing data. However, most existing GRN inference methods mainly focus on the network topology, only few of them consider how to explicitly describe the updated logic rules of regulation in GRNs to obtain their dynamics. Moreover, some inference methods also fail to deal with the over-fitting problem caused by the noise in time series data. Results In this article, we propose a novel embedded Boolean threshold network method called LogBTF, which effectively infers GRN by integrating regularized logistic regression and Boolean threshold function. First, the continuous gene expression values are converted into Boolean values and the elastic net regression model is adopted to fit the binarized time series data. Then, the estimated regression coefficients are applied to represent the unknown Boolean threshold function of the candidate Boolean threshold network as the dynamical equations. To overcome the multi-collinearity and over-fitting problems, a new and effective approach is designed to optimize the network topology by adding a perturbation design matrix to the input data and thereafter setting sufficiently small elements of the output coefficient vector to zeros. In addition, the cross-validation procedure is implemented into the Boolean threshold network model framework to strengthen the inference capability. Finally, extensive experiments on one simulated Boolean value dataset, dozens of simulation datasets, and three real single-cell RNA sequencing datasets demonstrate that the LogBTF method can infer GRNs from time series data more accurately than some other alternative methods for GRN inference. Availability and implementation The source data and code are available at https://github.com/zpliulab/LogBTF.

Abstract: In this paper we study functions on the Boolean hypercube that have the property that after applying certain random restrictions, the restricted function is correlated to a linear function with non-negligible probability. If the given function is correlated with a linear function then this property clearly holds. Furthermore, the property also holds for low-degree functions as low-degree functions become a constant function under a random restriction with a non-negligible probability. We show that this essentially is the only possible reason. More specifically, we show that the function must be correlated to a product of a linear function and a low-degree function. One of the main motivations of studying this question comes from the recent work of the authors towards understanding approximability of satisfiable Constraint Satisfaction Problems. Towards proving our structural theorem, we analyze a 2-query direct product test for the table F: [n] qn → {0,1}qn where q∈ (0,1). We show that, for every constant ε>0, if the test passes with probability ε>0, then there is a global function g: [n]→ {0,1} such that for at least δ(ε) fraction of sets, the global function g agrees with the given table on all except α(ε) many locations. The novelty of this result lies in the fact that α(ε) is independent of the set sizes. Prior to our work, such a conclusion (in fact, a stronger conclusion with α = 0) was shown by Dinur, Filmus, and Harsha albeit when the test accepts with probability 1−ε for a small constant ε>0. The setting of parameters in our direct product tests is fundamentally different compared to the previous results and hence our analysis involves new techniques, including the use of the small-set expansion property of graphs defined on multi-slices. As one application of our structural result, we give a 4-query linearity test under the p-biased distribution. More specifically, for any p∈ (1/3,2/3), we give a test that queries a given function f: {0,1}n → {0,1} at 4 locations, where the marginal distribution of each query is µp⊗ n. The test has perfect completeness and soundness 1/2+ε – in other words, for every constant ε>0, if the test passes with probability at least 1/2+ε, then the function f is correlated to a linear function under the µp⊗ n measure. This qualitatively improves the results on the linearity testing under the p-biased distribution from the previous work where the authors studied the test with soundness 1−ε, for ε close to 0.

Abstract: The problem of testing monotonicity for Boolean functions on the hypergrid, f:[n]d → {0,1} is a classic topic in property testing. When n=2, the domain is the hypercube. For the hypercube case, a breakthrough result of Khot-Minzer-Safra (FOCS 2015) gave a non-adaptive, one-sided tester making O(ε−2√d) queries. Up to polylog d and ε factors, this bound matches the Ω(√d)-query non-adaptive lower bound (Chen-De-Servedio-Tan (STOC 2015), Chen-Waingarten-Xie (STOC 2017)). For any n > 2, the optimal non-adaptive complexity was unknown. A previous result of the authors achieves a O(d5/6)-query upper bound (SODA 2020), quite far from the √d bound for the hypercube. In this paper, we resolve the non-adaptive complexity of monotonicity testing for all constant n, up to poly(ε−1logd) factors. Specifically, we give a non-adaptive, one-sided monotonicity tester making O(ε−2n√d) queries. From a technical standpoint, we prove new directed isoperimetric theorems over the hypergrid [n]d. These results generalize the celebrated directed Talagrand inequalities that were only known for the hypercube.

Abstract: Complete complementary codes (CCCs) have important applications in communication, radar, and information security. In modern communication and radar systems, certain spectrum is reserved or prohibited from transmission, which leads to the so-called spectrally null constrained (SNC) problem. Compared with vast works on conventional CCCs, relatively little is known about SNC-CCCs. One objective of this paper is to derive several constructions of CCCs with more flexible settings from the graph of extended Boolean functions. This generalizes earlier achievements on CCCs from recent literature. Another objective of this paper is to employ the graphs of extended Boolean functions and polynomial representation of sequences to construct SNC-CCCs.

Abstract: In this paper, we propose a new method for drawing a cryptographically strong substitution box using the Lorenz system and quantum genetic algorithm techniques. We used the chaotic function to generate an initial random sequence of bits and the quantum crossover to provide a new and improved substitution box with increased non-linearity. The aim of the proposed algorithm was to generate a strong and secure substitution box that can be utilized in symmetric cryptosystems. The use of chaotic Boolean functions, genetic algorithm techniques, and the quantum crossover helped achieve this goal, and statistical tests further confirmed the randomness and efficiency of the generated substitution box. The results of the statistical test suite showed that the substitution box produced by the proposed algorithm is a “pass” in terms of randomness and has strong cryptographic properties. The tests include a frequency (monobit) test, a frequency test within a block, a linear complexity test, an approximate entropy test, and a cumulative sums test among others. The p-values obtained in the tests indicate that the randomness of the generated substitution box meets the requirements of a cryptographically secure substitution box.

Abstract: Abstract Boolean functions are mathematical objects used in diverse domains and have been actively researched for several decades already. One domain where Boolean functions play an important role is cryptography. There, the plethora of settings one should consider and cryptographic properties that need to be fulfilled makes the search for new Boolean functions still a very active domain. There are several options to construct appropriate Boolean functions: algebraic constructions, random search, and metaheuristics. In this work, we concentrate on metaheuristic approaches and examine the related works appearing in the last 25 years. To the best of our knowledge, this is the first survey work on this topic. Additionally, we provide a new taxonomy of related works and discuss the results obtained. Finally, we finish this survey with potential future research directions.

Abstract: The Boolean functions satisfying secure properties on the restricted sets of inputs are studied recently due to their importance in the framework of the FLIP stream cipher. However, finding Boolean functions with optimal cryptographic properties is an open research problem in the cryptographic community. This paper presents an Improved Genetic Algorithm (IGA) with the directed changes that keep the weightwise balancedness of Boolean functions. A cross-protection strategy is proposed to ensure that the offspring has the same weightwise balancedness characteristics of the parents while implementing crossover. Then, a large number of weightwise (almost) perfectly balanced (W(A)PB) functions with a good nonlinearity profile are obtained based on IGA. Finally, we make comparisons between our constructions and relevant works. The comparisons show that IGA has a significant advantage for reaching the W(A)PB functions with high weightwise nonlinearity. Moreover, it is the first time to obtain the 8-variable WPB functions with the weightwise nonlinearity of 28 in the restricted sets of inputs with Hamming weight of 4, and list the statistical indicators of the weightwise nonlinearity for W(A)PB functions for input size n = 9, 10.

Abstract: Recently, with the improvement of the requirements for fast and efficient data processing in the era of artificial intelligence, new forms of computing have come into being. Developing memristor devices that can simulate the brain's computing neutral network is particularly important for applications in the field of artificial intelligence. However, there are still some challenges in their biological function simulation and related circuit design. In this work, a memristor based on perovskite rare earth nickelates (RNiO3) is presented with excellent electrical performance, including three orders of magnitude higher current switching ratio and good repeatability, and can achieve bidirectional conductance regulation like weight modulation in bio-synapse. Furthermore, the synaptic like characteristics of the device have been mimicked successfully, such as excitatory postsynaptic current (EPSC), paired pulse facilitation (PPF), classical double pulse spike time-dependent plasticity (classical pair-STDP), triplet spike time-dependent plasticity (triplet-STDP), short-term plasticity (STP), long-term plasticity (LTP), the refractory period phenomenon and learning and forgetting rules. In particular, two synaptic devices and a leaky integrate-and-fire (LIF) neuron device are used to achieve a logic gate circuit to realize "AND", "OR", and "NOT" functions. The device paves the way for the application of high-density circuits in artificial intelligence.

Abstract: Cryptographic S-boxes are vectorial Boolean functions that must fulfill strict criteria to provide security for cryptographic algorithms. There are several existing methods for generating strong cryptographic S-boxes, including stochastic search algorithms. These search algorithms typically generate random candidate Boolean functions (or permutations) that are improved during the search by examining the search space in a specific way. Here, we introduce a new type of stochastic algorithm for generating cryptographic S-boxes. We do not generate and then improve the Boolean function; instead, we build the vector of values incrementally. New values are obtained by randomized search driven by restrictions on the differential spectrum of the generated S-box. In this article, we formulate two new algorithms based on this new approach and study the better one in greater detail. We prove the correctness of the proposed algorithm and evaluate its complexity. The final part contains an experimental evaluation of the method. We show that the algorithm generates S-boxes with better properties than a random search. We believe that our approach can be extended in the future by adopting more advanced stochastic search methods.

Abstract: The FLIP cipher was proposed at Eurocrypt 2016 for the purpose of meliorating the efficiency of fully homomorphic cryptosystems. Weightwise perfectly balanced Boolean functions meet the balancedness requirement of the filter function in FLIP ciphers, and the construction of them has attracted serious attention from researchers. Nevertheless, the literature is still thin. Modifying the supports of functions with a low degree is a general construction technique whose key problem is to find a class of available low-degree functions. We first seek out a class of quadratic functions and then, based on these functions, present the new construction of weightwise perfectly balanced Boolean functions by adopting an iterative approach. It is worth mentioning that the functions we construct have good performance in weightwise nonlinearity. In particular, some p-weight nonlinearities achieve the highest values in the literature for a small number of variables.

Abstract: 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.

Abstract: Исследована проблема существования и единственности полилинейных продолжений некоторых дискретных функций. Доказано, что для любой булевой функции существует соответствующее полилинейное продолжение и оно единственно. Предложен алгоритм нахождения полилинейного продолжения булевой функции и доказана его корректность. На основе предложенного алгоритма найдены явные формы полилинейных продолжений сначала для булевой функции, а затем для произвольной функции, определенной на множестве вершин n-мерного единичного куба, произвольного куба и параллелепипеда, и в каждом конкретном случае доказана единственность соответствующего полилинейного продолжения. In this paper, we study the existence and uniqueness of polylinear continuations of some discrete functions. It is proved that for any Boolean function, there exists a corresponding polylinear continuation and it is unique. An algorithm for finding a polylinear continuation of a Boolean function is proposed and its correctness is proved. Based on the result of the proposed algorithm, explicit forms of polylinear continuations are found first for a Boolean function and then for an arbitrary function defined only at the vertices of an n-dimensional unit cube, an arbitrary cube, and a parallelepiped, and in each particular case the uniqueness of the corresponding polylinear continuations is proved.

Abstract: A logical function can be used to characterize a property of states of a Boolean network (BN), which is considered as an aggregation of states. The dynamics of a set of logical functions are called the dual dynamics of the set. To illustrate the dual dynamics of a given set, which characterizes our concerned properties of a BN, the invariant subspace containing the set of logical functions is proposed, and its properties are investigated. Then, the invariant subspace of Boolean control network (BCN) is also proposed, and its dynamics are obtained. Finally, using outputs as the set of logical functions, the minimum output based dual dynamics is considered and proposed as the minimum realization of BCNs. The minimum realization might have much smaller size, which provides a possible solution to overcome the computational complexity of large scale BNs/BCNs. As an example, the proposed approaches for both BN and BCN are applied to an opinion dynamic network to demonstrate the efficiency of the technique proposed in this article.

Abstract: We prove the covering radius of the third-order Reed-Muller code $\mathrm {RM}(3,7)$ is 20, which was previously known to be between 20 and 23 (inclusive). The covering radius of $\mathrm {RM}(3,7)$ is the maximum third-order nonlinearity among all 7-variable Boolean functions. It was known that there exist 7-variable Boolean functions with third-order nonlinearity 20. We prove the third-order nonlinearity cannot achieve 21. According to the classification of the quotient space of $\mathrm {RM}(6,6)/\mathrm {RM}(3,6)$ , we classify all 7-variable Boolean functions into 66 types. Firstly, we prove 62 types (among 66) cannot have third-order nonlinearity 21; Secondly, we prove that any function in the remaining 4 types can be transformed into a type (6, 10) function, if its third-order nonlinearity is 21; Finally, we transform type (6, 10) functions into a specific form, and prove the functions in that form cannot achieve the third-order nonlinearity 21 (with the assistance of computers). By the way, we prove that the affine transformation group over any finite field can be generated by two elements.

Abstract: We propose a new three-level XOR-AND-XOR form for autosymmetric functions, called XORAX expression. In general, a Boolean function f over n variables is k-autosymmetric if it can be projected onto a smaller function fk, which depends on n-k variables only. We show that XORAX expressions can ease the reversible synthesis of autosymmetric functions, producing compact reversible networks, without inserting additional new input lines. Autosymmetry occurs especially for functions that exhibit a regular structure, as for instance arithmetic functions. For this reason, compact reversible networks for autosymmetric functions might be interesting for quantum computing. Experimental results validate the proposed approach.

Abstract: Boolean functions and their representation through logics, circuits, machine learning classifiers, or binary decision diagrams (BDDs) play a central role in the design and analysis of computing systems. Quantifying the relative impact of variables on the truth value by means of importance values can provide useful insights to steer system design and debugging. In this paper, we introduce a uniform framework for reasoning about such values, relying on a generic notion of importance value functions (IVFs). The class of IVFs is defined by axioms motivated from several notions of importance values introduced in the literature, including Ben-Or and Linial’s influence and Chockler, Halpern, and Kupferman’s notion of responsibility and blame. We establish a connection between IVFs and game-theoretic concepts such as Shapley and Banzhaf values, both of which measure the impact of players on outcomes in cooperative games. Exploiting BDD-based symbolic methods and projected model counting, we devise and evaluate practical computation schemes for IVFs.

Abstract: Secure multi-party computation using a physical deck of cards, often called card-based cryptography, has been extensively studied during the past decade. Card-based protocols to compute various Boolean functions have been developed. As each input bit is typically encoded by two cards, computing an n-variable Boolean function requires at least 2n cards. We are interested in optimal protocols that use exactly 2n cards. In particular, we focus on symmetric functions. In this paper, we formulate the problem of developing 2n-card protocols to compute n-variable symmetric Boolean functions by classifying all such functions into several NPN-equivalence classes. We then summarize existing protocols that can compute some representative functions from these classes, and also solve some open problems in the cases $$n=4$$ , 5, 6, and 7. In particular, we develop a protocol to compute a function kMod3, which determines whether the sum of all inputs is congruent to k modulo 3 ( $$k \in \{0,1,2\}$$ ).

Abstract: We apply Siegenthaler’s construction, along with several techniques, to classify all (n−4)-resilient Boolean functions with n variables, for all values of n ≥ 4, up to the extended variable-permutation equivalence. We show that, up to this equivalence, there are only 761 functions for any n larger than or equal to 10, and for smaller values of n, i.e., for n increasing from 4 to 9, there are 58, 256, 578, 720, 754, and 760 functions, respectively. Furthermore, we classify all 1-resilient 6-variable Boolean functions and show that there are 1 035 596 784 such functions up to the extended variable-permutation equivalence.

Abstract: We prove a $k^{-\Omega\left(\log \left(\varepsilon_{2}-\varepsilon_{1}\right)\right)}$ lower bound for adap- tively testing whether a Boolean function is $\varepsilon_{1}$-close to or $\varepsilon_{2}-$ far from k-juntas. Our results provide the first superpolynomial separation between tolerant and non-tolerant testing for a natural property of boolean functions under the adaptive setting. Furthermore, our techniques generalize to show that adaptively testing whether a function is $\varepsilon_{1}$-close to a k-junta or $\varepsilon_{2}$-far from $(k+o(k))$-juntas cannot be done with poly $(k,\left(\varepsilon_{2}-\varepsilon_{1}\right)^{-1})$ queries. This is in contrast to an algorithm by Iyer, Tal and Whitmeyer [CCC 2021] which uses poly $(k,\left(\varepsilon_{2}-\varepsilon_{1}\right)^{-1})$ queries to test whether a function is $\varepsilon_{1}$-close to a k-junta or $\varepsilon_{2}$-far from $O(k /\left(\varepsilon_{2}-\varepsilon_{1}\right)^{2})$-juntas

Abstract: In this paper, we study systems of Boolean equations over a network, where each node in the network possesses only one Boolean equation from the system. The Boolean equation assigned at any particular node is a private equation known to this node only, and the aim of the paper is to develop distributed algorithms that allow all the nodes to obtain solutions to the network Boolean equations without exchanging their local Boolean equations. First, we observe that the Boolean equations can be locally lifted to a system of linear algebraic equations under a basis of Boolean vectors, which is distributedly solvable using existing distributed linear equation algorithms as a subroutine. Next, we construct a distributed Boolean equation solver by the nodes solving the lifted linear network equation for a number of randomly selected initial values, and then converting the algebraic solutions into solutions to the original Boolean equations by a novel Boolean vector search algorithm. We prove that for solvable Boolean equations, when the initial values of the nodes for the distributed linear equation solving step are i.i.d selected according to a uniform distribution in a high-dimensional cube, such an algorithm returns the exact solution set of the Boolean equations at each node with high probability. We also present an algorithm for distributed verification of the satisfiability of Boolean equations when the solvability is not known beforehand, and prove its correctness. Finally, we show that by utilizing linear equation solvers with differential privacy to replace the in-network computing routines, the distributed Boolean equation algorithms can be made differentially private.

Abstract: In recent years, hardware systems have significantly grown in complexity. Due to the increasing complexity, there is a need to continuously improve the quality of the hardware design process. This leads designers to strive for more efficient data structures and algorithms operating on them to guarantee the correct be-havior of such systems through verification techniques like model checking and meet time-to-market constraints. A Binary Decision Diagram (BDD) is a suitable data structure as it provides a canon-ical compact representation of Boolean functions, given variable ordering, and efficient algorithms for manipulating them. How-ever, reduced ordered BDDs also have challenges: There is a large memory consumption for the BDD construction of some complex practical functions and the use of realizations in the form of BDD packages strongly depends on the application. To address these issues, this paper presents a novel multi-core package called Engineer Decision Diagrams Yourself (EDDY) with dynamic memory management and reduced fragmentation. Exper-iments on BDD benchmarks of both combinational circuits and model checking show that using EDDY leads to a significantly performance boost compared to state-of-the-art packages.

Abstract: Many logic synthesis methods are based on the optimization of reduced order Binary Decision Diagrams (BDDs). The complexity of a BDD greatly depends on the chosen order of variables. Most of the methods find optimal variable ordering focused primarily on some subset of BDD parameters, for example, the size, the number of paths, the average path length, etc. In some cases, there is a need for using several parameters for the evaluation of overall BDD complexity. Therefore, this paper proposes a BDD complexity evaluation using the relative arithmetic width parameter. This parameter is calculated as the arithmetic mean of the BDD relative widths over all its levels. The relative arithmetic width expresses the numerical value of the relative shape of the BDD. Experimental results with optimal variable ordering demonstrate the efficiency of the approach.