Abstract: We propose a new method to compare numbers which are encrypted by Homomorphic Encryption (HE). Previously, comparison and min/max functions were evaluated using Boolean functions where input numbers are encrypted bit-wise. However, the bit-wise encryption methods require relatively expensive computations for basic arithmetic operations such as addition and multiplication.

Abstract: Probabilistic spin logic is a recently proposed computing paradigm based on unstable stochastic units called probabilistic bits ( $p$ -bits) that can be correlated to form probabilistic circuits (p-circuits). These p-circuits can be used to solve the problems of optimization, inference, and implement precise Boolean functions in an “inverted” mode, where a given Boolean circuit can operate in reverse to find the input combinations that are consistent with a given output. In this brief, we present a scalable field-programmable gate array implementation of such invertible p-circuits. We implement a “weighted” $p$ -bit that combines stochastic units with localized memory structures. We also present a generalized tile of weighted $p$ -bits to which a large class of problems beyond invertible Boolean logic can be mapped and how invertibility can be applied to interesting problems such as the NP-complete subset sum problem by solving a small instance of this problem in hardware.

Abstract: Probabilistic Boolean network (PBN) is a kind of stochastic logical system in which update functions are randomly selected from a set of candidate Boolean functions according to a prescribed probability distribution at each time step. In this brief, a pinning controller design algorithm is proposed to set stabilize any PBN with probability one. First, an algorithm is given to change the columns of its transition matrix. Then, according to the newly obtained transition matrix, a fraction of nodes can be selected as pinning nodes to inject control inputs to achieve set stabilization. The problem is challenging since the Boolean functions in a PBN are not deterministic but are randomly chosen among several Boolean functions. Furthermore, the structure matrices of the pinning controllers are given by solving some logical matrices equations based on which a pinning controller design algorithm is provided to set stabilize the PBN with probability one. Finally, the theoretical results are validated using several examples.

Abstract: In this paper, a novel method is being proposed to construct a substitution box or Boolean function for block ciphers using Gaussian distribution and linear fractional transform. The substitution box is constructed by employing a linear fractional transform based on Box–Muller transform, polarization decision, and central limit algorithm. The cryptographic strength of the proposed S-boxes is evaluated with standardized tests such as linear approximation probability, unified averaged changed intensity, bit independent criterion, histogram analysis, nonlinearity score, strict avalanche criterion, and differential approximation probability. The results show that the proposed substitution box achieves better cryptographic strength as compared with the state-of-the-art techniques.

Abstract: In this paper, we propose a novel algorithm that solves a generalized version of the Deutsch-Jozsa problem. The proposed algorithm has the potential to classify an oracle U F , that represents an unknown Boolean function on n Boolean variables, to one of 2 n different classes instead of only two classes which are constant and balanced classes in the case of Deutsch-Jozsa algorithm. The proposed algorithm is based on the use of entanglement measure to explore 2 n - 2 additional classes compared to the standard Deutsch-Jozsa algorithm. In addition, the comparison between the proposed quantum algorithm and the classical one is investigated in details. The comparison shows that the proposed algorithm is faster when the number of Boolean variables exceed 14 variables.

Abstract: We introduce a framework for the formal specification and verification of quantum circuits based on the Feynman path integral. Our formalism, built around exponential sums of polynomial functions, provides a structured and natural way of specifying quantum operations, particularly for quantum implementations of classical functions. Verification of circuits over all levels of the Clifford hierarchy with respect to either a specification or reference circuit is enabled by a novel rewrite system for exponential sums with free variables. Our algorithm is further shown to give a polynomial-time decision procedure for checking the equivalence of Clifford group circuits. We evaluate our methods by performing automated verification of optimized Clifford+T circuits with up to 100 qubits and thousands of T gates, as well as the functional verification of quantum algorithms using hundreds of qubits. Our experiments culminate in the automated verification of the Hidden Shift algorithm for a class of Boolean functions in a fraction of the time it has taken recent algorithms to simulate.

Abstract: We present a synthesis framework to map logic networks into quantum circuits for quantum computing. The synthesis framework is based on lookup-table (LUT) networks, which play a key role in conventional logic synthesis. Establishing a connection between LUTs in an LUT network and reversible single-target gates in a reversible network allows us to bridge conventional logic synthesis with logic synthesis for quantum computing, despite several fundamental differences. We call our synthesis framework LUT-based hierarchical reversible logic synthesis (LHRS). Input to LHRS is a classical logic network representing an arbitrary Boolean combinational operation; output is a quantum network (realized in terms of Clifford+ T gates). The framework allows one to account for qubit count requirements imposed by the overlying quantum algorithm or target quantum computing hardware. In a fast first step, an initial network is derived that only consists of single-target gates and already completely determines the number of qubits in the final quantum network. Different methods are then used to map each single-target gate into Clifford+ T gates, while aiming at optimally using available resources. We demonstrate the versatility of our method by conducting a design space exploration using different parameters on a set of large combinational benchmarks. On the same benchmarks, we show that our approach can advance over the state-of-the-art hierarchical reversible logic synthesis algorithms.

Abstract: Boolean networks model finite discrete dynamical systems with complex behaviours. The state of each component is determined by a Boolean function of the state of (a subset of) the components of the network. This paper addresses the synthesis of these Boolean functions from constraints on their domain and emerging dynamical properties of the resulting network. The dynamical properties relate to the existence and absence of trajectories between partially observed configurations, and to the stable behaviours (fixpoints and cyclic attractors). The synthesis is expressed as a Boolean satisfiability problem relying on Answer-Set Programming with a parametrized complexity, and leads to a complete non-redundant characterization of the set of solutions. Considered constraints are particularly suited to address the synthesis of models of cellular differentiation processes, as illustrated on a case study. The scalability of the approach is demonstrated on random networks with scale-free structures up to 100 to 1,000 nodes depending on the type of constraints.

Abstract: Due to the outsourcing of semiconductor design and manufacturing, a number of threats have emerged in recent years, and they are overproduction of integrated circuits (ICs), illegal sale of defective ICs, and piracy of intellectual properties (IPs). Logic locking is one method to enable trust in this complex IC design and manufacturing processes, where a design is obfuscated by inserting a lock to modify the underlying functionality so that an adversary cannot make a chip to function properly. A locked chip will only work properly once it is activated by programming with a secret key into its tamper-proof memory. Over the years, researchers have proposed different locking mechanisms primarily to prevent Boolean satisfiability (SAT)-based attacks, and successfully preserve the security of a locked design. However, an untrusted foundry, the adversary, can use many other effective means to find out the secret key. In this paper, we present a novel oracle-less and topology-guided attack denoted as TGA. The attack relies on identifying repeated functions for determining the value of a key bit. The proposed attack does not require any data from an unlocked chip, and eliminates the need for an oracle. The attack is based on self-referencing, i.e., it compares the internal netlist to find the key. The proposed graph search algorithm efficiently finds a duplicate function of the locked part of the circuit. Our proposed attack correctly estimate a key bit very efficiently, and it only takes few seconds to determine the key bit. We also present a solution to thwart TGA and make logic locking secure.

Abstract: Quantum-dot cellular automata (QCA) has attracted computer scientists as new emerging nanotechnology for replacement the current CMOS technology because it has unique characteristics such as high frequency, extremely small feature size and low power consumption. The main building blocks in QCA are the majority gate and inverter so any Boolean function can be represented using these gates. Many important circuits were the target for implemented in this technology in an optimal form, such as random-access memory (RAM) cell. QCA-RAM cells were introduced in literature with different forms but most of them are not optimized enough. This paper aims to demonstrate QCA inherent capabilities that can facilitate the design of many important gates such as the XOR gate and multiplexer (MUX) without following any Boolean function to get an optimum design in terms of complexity and delay.,In this paper, a novel structure of QCA-MUX in an optimal form will be used to design two unique structures of a RAM cell. The proposed RAM cells are the lowest cost required compared with different counterparts. The presented RAM cells used a new approach that follows the new suggested block diagram. The presented circuits are simulated and tested with QCADesigner and QCAPro tools.,The comparison of the proposed circuits with the previously reported in the literature show noticeable improvements in speed, area, and the number of cells. The cost function analysis results for the proposed RAM cells show significant improvement compared to older circuits.,A novel structure of QCA-MUX in an optimal form will be used to design two unique structures of a RAM cell.

Abstract: The chapter is devoted to programmable logic controllers (PLC). We start from the classification of PLC, their architecture and cycle of operation. Next, the main laws of Boolean algebra are shown. We show the connection between the Boolean algebra and basic logic functions used in programming of PLC. Next, different programming languages used for PLC are shown. The last part is devoted to examples of programming for different Boolean functions and simple control algorithms. All programs are written using the Ladder Diagram language.

Abstract: Very recently, Carlet, Meaux and Rotella have studied the main cryptographic features of Boolean functions when, for a given number n of variables, the input to these functions is restricted to some subset E of $\mathbb {F}_{2}^{n}$. Their study includes the particular case when E equals the set of vectors of fixed Hamming weight, which is important in the robustness of the Boolean function involved in the FLIP stream cipher. In this paper we focus on the nonlinearity of Boolean functions with restricted input and present new results related to the analysis of this nonlinearity improving the upper bound given by Carlet et al.

Abstract: We propose a novel paradigm for solving Inductive Logic Programming (ILP) problems via deep recurrent neural networks. This proposed ILP solver is designed based on differentiable implementation of the deduction via forward chaining. In contrast to the majority of past methods, instead of searching through the space of possible first-order logic rules by using some restrictive rule templates, we directly learn the symbolic logical predicate rules by introducing a novel differentiable Neural Logic (dNL) network. The proposed dNL network is able to learn and represent Boolean functions efficiently and in an explicit manner. We show that the proposed dNL-ILP solver supports desirable features such as recursion and predicate invention. Further, we investigate the performance of the proposed ILP solver in classification tasks involving benchmark relational datasets. In particular, we show that our proposed method outperforms the state of the art ILP solvers in classification tasks for Mutagenesis, Cora and IMDB datasets.

Abstract: We construct strongly walk-regular graphs as coset graphs of the duals of three-weight codes over $$\mathbb {F}_q.$$ The columns of the check matrix of the code form a triple sum set, a natural generalization of partial difference sets. Many infinite families of such graphs are constructed from cyclic codes, Boolean functions, and trace codes over fields and rings. Classification in short code lengths is made for $$q=2,3,4$$ .

Abstract: Dependency quantified Boolean formulas (DQBF) is a logic admitting existential quantification over Boolean functions, which allows us to elegantly state synthesis problems in verification such as the search for invariants, programs, or winning regions of games. In this paper, we lift the clausal abstraction algorithm for quantified Boolean formulas (QBF) to DQBF. Clausal abstraction for QBF is an abstraction refinement algorithm that operates on a sequence of abstractions that represent the different quantifier levels. For DQBF we need to generalize this principle to partial orders of abstractions. The two challenges to overcome are: (1) Clauses may contain literals with incomparable dependencies, which we address by the recently proposed proof rule called Fork Extension, and (2) existential variables may have spurious dependencies, which we prevent by tracking consistency requirements during the execution. Our implementation \(\textsc {dCAQE}\) solves significantly more formulas than the existing DQBF algorithms.

Abstract: The main cryptographic features of Boolean functions when the input is restricted to some subset of ${\mathbb {F}_{2}^{n}}$ are studied recently because of the innovative stream cipher FLIP Meaux et al. (2016). In this paper, we propose a large family of Boolean functions which are (almost) balanced on every set of vectors in ${\mathbb {F}_{2}^{n}}\setminus \{\mathbf {0},\mathbf {1}\}$ with constant Hamming weight (the so-called weightwise (almost) perfectly balanced, W(A)PB). We show that these W(A)PB functions have optimal algebraic immunity on ${\mathbb {F}_{2}^{n}}$ and good algebraic immunity on some subsets of vectors in ${\mathbb {F}_{2}^{n}}$, especially on the subsets of vectors with constant Hamming weight. This is the first time that W(A)PB functions with good local algebraic immunities are presented. Moreover, we discuss the nonlinearity and weightwise nonlinearity of these functions.

Abstract: The multiplicative complexity of a Boolean function is the minimum number of two-input AND gates that are necessary and sufficient to implement the function over the basis (AND, XOR, NOT). Finding the multiplicative complexity of a given function is computationally intractable, even for functions with small number of inputs. Turan et al. [1] showed that n-variable Boolean functions can be implemented with at most n-1 AND gates for n ≤ 5. A counting argument can be used to show that, for n ≥ 7, there exist n-variable Boolean functions with multiplicative complexity of at least n. In this work, we propose a method to find the multiplicative complexity of Boolean functions by analyzing circuits with a particular number of AND gates and utilizing the affine equivalence of functions. We use this method to study the multiplicative complexity of 6-variable Boolean functions, and calculate the multiplicative complexities of all 150357 affine equivalence classes. We show that any 6-variable Boolean function can be implemented using at most 6 AND gates. Additionally, we exhibit specific 6-variable Boolean functions which have multiplicative complexity 6.

Abstract: Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the recent FLIP stream cipher (Meaux et al.: in Lecture Notes in Computer Science, vol. 9665, pp. 311–343, Springer, Berlin, 2016). In this paper, we propose a large class of weightwise perfectly balanced (WPB) functions, which is 2-rotation symmetric. This new class of WPB functions is not extended affinely equivalent to the known constructions. We also discuss the weightwise nonlinearity profile of these functions, and present general lower bounds on k-weightwise nonlinearity, where k is a power of 2. Moreover, we exhibit a subclass of the family. By a recursive lower bound, we show that these subclass of WPB functions have very high weightwise nonlinearity profile.

Abstract: In this paper, the robust invariant set (RIS) of Boolean (control) networks with disturbances is investigated. First, for a given fixed point, consider a special set called immediate neighborhoods of the fixed point; then a discrete derivative of Boolean functions at the fixed point is used to analyze the robust invariance, based on which a sufficient condition is obtained. Second, for more general sets, the robust output control invariant set (ROCIS) of Boolean control networks (BCNs) is investigated by semitensor product (STP) of matrices. Then, under a given output feedback controller, we obtain a necessary and sufficient condition to check whether a given set is robust control invariant set (RCIS). Furthermore, output feedback controllers are designed to make a set to be a RCIS. Finally, the proposed methods are illustrated by a reduced model of the lac operon in E. coli.

Abstract: We present a constructive method to create quantum circuits that implement oracles $|x\rangle|y\rangle|0\rangle^k \mapsto |x\rangle|y \oplus f(x)\rangle|0\rangle^k$ for $n$-variable Boolean functions $f$ with low $T$-count. In our method $f$ is given as a 2-regular Boolean logic network over the gate basis $\{\land, \oplus, 1\}$. Our construction leads to circuits with a $T$-count that is at most four times the number of AND nodes in the network. In addition, we propose a SAT-based method that allows us to trade qubits for $T$ gates, and explore the space/complexity trade-off of quantum circuits. Our constructive method suggests a new upper bound for the number of $T$ gates and ancilla qubits based on the multiplicative complexity $c_\land(f)$ of the oracle function $f$, which is the minimum number of AND gates that is required to realize $f$ over the gate basis $\{\land, \oplus, 1\}$. There exists a quantum circuit computing $f$ with at most $4 c_\land(f)$ $T$ gates using $k = c_\land(f)$ ancillae. Results known for the multiplicative complexity of Boolean functions can be transferred. We verify our method by comparing it to different state-of-the-art compilers. Finally, we present our synthesis results for Boolean functions used in quantum cryptoanalysis.

Abstract: Figure A1 describes an r-stage Fibonacci nonlinear feedback shift regisger (NFSR). Here small squares represent binary storage devices, also called bits, whose contents are labelled as Y1, Y2, . . . , Yr from left to right. They together form the state of the NFSR, denoted by Y = [Y1 Y2 · · ·Yr] . The nonlinear Boolean function h in the rectangle is called the feedback function of the NFSR. If the feedback function h is degenerated to a linear Boolean function, then the NFSR is reduced to a linear feedback shift register (LFSR). The content Y1 is the output of the NFSR. The NFSR is nonsingular if and only if its feedback function h is nonsingular, i.e., h(Y1, Y2, . . . , Yr) = Y1 ⊕ h̃(Y2, . . . , Yr) [1]. The function hc(Y1, Y2, . . . , Yr+1) = Yr+1 ⊕ h(Y1, Y2, . . . Yr) is called the characteristic function of the NFSR. The NFSR can be described by the nonlinear system:  Y1(t+ 1) = Y2(t), .. Yr−1(t+ 1) = Yr(t), Yr(t+ 1) = h(Y1(t), Y2(t), . . . , Yr(t)). (A1)

Abstract: We study the questions concerning the properties and capabilities of computational bithreshold real-weighted neural-like units. We give and justify the two sufficient conditions ensuring the possibility of separation of two sets in n-dimensional vector space by means of one bithreshold neuron. Our approach is based on application of convex and affine hulls of sets and is feasible in the case when one of the two sets is a compact and the second one is finite. We also correct and refine some previous results concerning bithreshold separability. Then the hardness of the learning bithreshold neurons is considered. We examine the complexity of the problem of checking whether the given Boolean function of n variables can be realizable by single bithreshold unit. Our main result is that the problem of verifying the bithreshold separability is NP-complete. The same is true for neural networks consisting of such computational units. We propose some continuous modifications of the bithreshold activation function to smooth away these difficulties and to make possible the application of modern paradigms and learning techniques for such networks.

Abstract: Motivation Inferring gene regulatory networks from gene expression time series data is important for gaining insights into the complex processes of cell life. A popular approach is to infer Boolean networks. However, it is still a pressing open problem to infer accurate Boolean networks from experimental data that are typically short and noisy. Results To address the problem, we propose a Boolean network inference algorithm which is able to infer accurate Boolean network topology and dynamics from short and noisy time series data. The main idea is that, for each target gene, we use an And/Or tree ensemble algorithm to select prime implicants of which each is a conjunction of a set of input genes. The selected prime implicants are important features for predicting the states of the target gene. Using these important features we then infer the Boolean function of the target gene. Finally, the Boolean functions of all target genes are combined as a Boolean network. Using the data generated from artificial and real-world gene regulatory networks, we show that our algorithm can infer more accurate Boolean network topology and dynamics from short and noisy time series data than other algorithms. Our algorithm enables us to gain better insights into complex regulatory mechanisms of cell life. Availability and implementation Package ATEN is freely available at https://github.com/ningshi/ATEN. Supplementary information Supplementary data are available at Bioinformatics online.

Abstract: The problem of tolerant junta testing is a natural and challenging problem which asks if the property of a function having some specified correlation with a k-Junta is testable. In this paper we give an affirmative answer to this question: There is an algorithm which given distance parameters c, d, and oracle access to a Boolean function f on the hypercube, has query complexity exp(k).poly(1/(c-d)) and distinguishes between the following cases: 1. The distance of f from any k-junta is at least c; 2. There is a k-junta g which has distance at most d from f. This is the first non-trivial tester (i.e., query complexity is independent of the ambient dimension n) which works for all c and d (bounded by 0.5). The best previously known results by Blais et~al., required c to be at least 16d. In fact, with the same query complexity, we accomplish the stronger goal of identifying the most correlated k-junta, up to permutations of the coordinates. We can further improve the query complexity to poly(k/(c-d)) for the (weaker) task of distinguishing between the following cases: 1. The distance of f from any k'-junta is at least c. 2. There is a k-junta g which is at a distance at most d from f. Here k'=poly(k/(c-d)). Our main tools are Fourier analysis based algorithms that simulate oracle access to influential coordinates of functions.

Abstract: Given a Boolean formula $F(\mathrm{X},\ \mathrm{Y})$, where X is a vector of outputs and Y is a vector of inputs, the Boolean functional synthesis problem requires us to compute a Skolem function vector $\Psi(\mathrm{Y})$ such that $F(\Psi(\mathrm{Y}),\ \mathrm{Y})$ holds whenever $\exists \mathrm{X}F(\mathrm{X},\ \mathrm{Y})$ holds. In this paper, we investigate the relation between the representation of the specification $F(\mathrm{X},\ \mathrm{Y})$ and the complexity of synthesis. We introduce a new normal form for Boolean formulas, called SynNNF, that guarantees polynomial-time synthesis and also polynomial-time existential quantification for some order of quantification of variables. We show that several normal forms studied in the knowledge compilation literature are subsumed by SynNNF, although SynNNF can be super-polynomially more succinct than them. Motivated by these results, we propose an algorithm to convert a specification in CNF to SynNNF, with the intent of solving the Boolean functional synthesis problem. Experiments with a prototype implementation show that this approach solves several benchmarks beyond the reach of state-of-the-art tools.

Abstract: We consider the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0, 1}n. Chen, Liu, Servedio, Sheng and Xie [35] showed that the distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that makes O(k2)/ϵ queries. In this paper, we give a simple two-sided error adaptive algorithm that makes O{k/ϵ) queries.