Abstract: As integrated circuit technology plumbs ever greater depths in the scaling of feature sizes, maintaining the paradigm of deterministic Boolean computation is increasingly challenging. Indeed, mounting concerns over noise and uncertainty in signal values motivate a new approach: the design of stochastic logic, that is to say, digital circuitry that processes signals probabilistically, and so can cope with errors and uncertainty. In this paper, we present a general methodology for synthesizing stochastic logic for the computation of polynomial arithmetic functions, a category that is important for applications such as digital signal processing. The method is based on converting polynomials into a particular mathematical form --- Bernstein polynomials --- and then implementing the computation with stochastic logic. The resulting logic processes serial or parallel streams that are random at the bit level. In the aggregate, the computation becomes accurate, since the results depend only on the precision of the statistics. Experiments show that our method produces circuits that are highly tolerant of errors in the input stream, while the area-delay product of the circuit is comparable to that of deterministic implementations.

Abstract: We introduce in this paper two new, complete propositional languages and study their properties in terms of (1) their support for polytime operations and (2) their ability to represent boolean functions compactly. The new languages are based on a structured version of decomposability--a property that underlies a number of tractable languages. The key characteristic of structured decomposability is its support for a polytime conjoin operation, which is known to be intractable for unstructured decomposability. We show that any CNF can be compiled into formulas in the new languages, whose size is only exponential in the treewidth of the CNF. Our study also reveals that one of the languages we identify is as powerful as OBDDs in terms of answering key inference queries, yet is more succinct than OBDDs.

Abstract: This paper presents a unified and simple treatment of basic questions concern- ing two computational models: multiparty communication complexity and polynomials over GF(2). The key is the use of (known) norms on Boolean functions, which capture their proximity to each of these models (and are closely related to property testers of this proximity). The main contributions are new XOR lemmas. We show that if a Boolean function has correlation at most e 1/2 with either of these models, then the correlation of the parity of its values on m independent instances drops exponentially with m. More specifically: • For polynomials over GF(2) of degree d, the correlation drops to exp m/4 d . No

Abstract: A gate level probabilistic error propagation model is presented which takes as input the Boolean function of the gate, the signal and error probabilities of the gate inputs, and the gate error probability and produces the error probability at the output of the gate. The presented model uses the Boolean difference calculus and can be applied to the problem of calculating the error probability at the primary outputs of a multi-level Boolean circuit with a time complexity which is linear in the number of gates in the circuit. This is done by starting from the primary inputs and moving toward the primary outputs by using a post-order traversal. Experimental results demonstrate the accuracy and efficiency of the proposed approach compared to the other known methods for error calculation in VLSI circuits.

Abstract: PREFACE. ACKNOWLEDGMENTS. LIST OF FIGURES. LIST OF TABLES. ACRONYMS.1. LOGIC FUNCTIONS. 1.1 Discrete Functions. 1.2 Tabular Representations of Discrete Functions. 1.3 Functional Expressions. 1.4 Decision Diagrams for Discrete Functions. 1.5 Spectral Representations of Logic Functions. 1.6 Fixed-polarity Reed-Muller Expressions of Logic.Functions. 1.7 Kronecker Expressions of Logic Functions. 1.8 Circuit Implementation of Logic Functions. 2. SPECTRAL TRANSFORMS FOR LOGIC FUNCTIONS. 2.1 Algebraic Structures for Spectral Transforms. 2.2 Fourier Series. 2.3 Bases for Systems of Boolean Functions. 2.4 Walsh Related Transforms. 2.5 Bases for Systems of Multiple-Valued Functions. 2.6 Properties of DiscreteWalsh andVilenkin-Chrestenson Transforms. 2.7 Autocorrelation and Cross-Correlation Functions. 2.8 Harmonic Analysis over an Arbitrary Finite Abelian Group. 2.9 Fourier Transform on Finite Non-Abelian Groups. 3. CALCULATION OF SPECTRAL TRANSFORMS. 3.1 Calculation of Walsh Spectra. 3.2 Calculation of the Haar Spectrum. 3.3 Calculation of the Vilenkin-Chrestenson Spectrum. 3.4 Calculation of the Generalized Haar Spectrum. 3.5 Calculation of Autocorrelation Functions. 4. SPECTRAL METHODS IN OPTIMIZATION OF DECISION DIAGRAMS. 4.1 Reduction of Sizes of Decision Diagrams. 4.2 Construction of Linearly Transformed Binary Decision Diagrams. 4.3 Construction of Linearly Transformed Planar BDD. 4.4 Spectral Interpretation of Decision Diagrams. 5. ANALYSIS AND OPTIMIZATION OF LOGIC FUNCTIONS. 5.1 Spectral Analysis of Boolean Functions. 5.2 Analysis and Synthesis of Threshold Element Networks. 5.3 Complexity of Logic Functions. 5.4 Serial Decomposition of Systems of Switching Functions. 5.5 Parallel Decomposition of Systems of Switching Functions. 6. SPECTRAL METHODS IN SYNTHESIS OF LOGIC NETWORKS. 6.1 Spectral Methods of Synthesis of Combinatorial Devices. 6.2 Spectral Methods for Synthesis of Incompletely Specified Functions. 6.3 Spectral Methods of Synthesis of Multiple-Valued Functions. 6.4 Spectral Synthesis of Digital Functions and Sequences Generators. 7. SPECTRAL METHODS OF SYNTHESIS OF SEQUENTIAL MACHINES. 7.1 Realization of Finite Automata by Spectral Methods. 7.2 Assignment of States and Inputs for Completely Specified Automata. 7.3 State Assignment for Incompletely Specified Automata. 7.4 Some Special Cases of the Assignment Problem. 8. HARDWARE IMPLEMENTATION OF SPECTRAL METHODS. 8.1 Spectral Methods of Synthesis with ROM. 8.2 Serial Implementation of Spectral Methods. 8.3 Sequential Haar Networks. 8.4 Complexity of Serial Realization by Haar Series. 8.5 Parallel Realization of Spectral Methods of Synthesis. 8.6 Complexity of Parallel Realization. 8.7 Realization by Expansions over Finite Fields. 9. SPECTRAL METHODS OF ANALYSIS AND SYNTHESIS OF RELIABLE DEVICES. 9.1 Spectral Methods for Analysis of Error Correcting Capabilities. 9.2 Spectral Methods for Synthesis of Reliable Digital Devices. 9.3 Correcting Capability of Sequential Machines. 9.4 Synthesis of Fault-Tolerant Automata with Self-Error Correction. 9.5 Comparison of Spectral and Classical Methods. 10. SPECTRAL METHODS FOR TESTING OF DIGITAL SYSTEMS. 10.1 Testing and Diagnosis by Verification of Walsh Coefficients. 10.2 Functional Testing, Error Detection, and Correction by Linear Checks. 10.3 Linear Checks for Processors. 10.4 Linear Checks for Error Detection in Polynomial Computations. 10.5 Construction of Optimal Linear Checks for Polynomial Computations. 10.6 Implementations and Error-Detecting Capabilities of Linear Checks. 10.7 Testing for Numerical Computations. 10.8 Optimal Inequality Checks and Error-Correcting Codes. 10.9 Error Detection in Computer Memories by Linear Checks. 10.10 Location of Errors in ROMs by Two Orthogonal Inequality Checks. 10.11 Detection and Location of Errors in Random-Access Memories. 11. EXAMPLES OF APPLICATIONS AND GENERALIZATIONS OF SPECTRAL METHODS ON LOGIC FUNCTIONS. 11.1 Transforms Designed for Particular Applications. 11.2 Wavelet Transforms. 11.3 Fibonacci Transforms. 11.4 Two-Dimensional Spectral Transforms. 11.5 Application of the Walsh Transform in Broadband Radio. APPENDIX A. REFERENCES. INDEX.

Abstract: The nonlinearity profile of a Boolean function (i.e., the sequence of its minimum Hamming distances nlr(f) to all functions of degrees at most r, for r ges 1) is a cryptographic criterion whose role against attacks on stream and block ciphers has been illustrated by many papers. It plays also a role in coding theory, since it is related to the covering radii of Reed-Muller codes. We introduce a method for lower-bounding its values and we deduce bounds on the second-order nonlinearity for several classes of cryptographic Boolean functions, including the Welch and the multiplicative inverse functions (used in the S-boxes of the Advanced Encryption Standard (AES)). In the case of this last infinite class of functions, we are able to bound the whole profile, and we do it in an efficient way when the number of variables is not too small. This allows showing the good behavior of this function with respect to this criterion as well.

Abstract: In a breakthrough result, Razborov (2003) gave optimal lower bounds on the communication complexity of every function f of the form f(x,y)=D(|x AND y|) for some D:{0,1,...,n}->{0,1}, in the bounded-error quantum model with and without prior entanglement. This was proved by the multidimensional discrepancy method. We give an entirely different proof of Razborov's result, using the original, one-dimensional discrepancy method. This refutes the commonly held intuition (Razborov 2003) that the original discrepancy method fails for functions such as DISJOINTNESS. More importantly, our communication lower bounds hold for a much broader class of functions for which no methods were available. Namely, fix an arbitrary function f:{0,1}n/4->{0,1} and let A be the Boolean matrix whose columns are each an application of f to some subset of the variables x1,x2,...,xn. We prove that the communication complexity of A in the bounded-error quantum model with and without prior entanglement is Omega(d), where d is the approximate degree of f. From this result, Razborov's lower bounds follow easily. Our result also establishes a large new class of total Boolean functions whose quantum communication complexity (regardless of prior entanglement) is at best polynomially smaller than their classical complexity. Our proof method is a novel combination of two ingredients. The first is a certain equivalence of approximation and orthogonality in Euclidean n-space, which follows by linear-programming duality. The second is a new construction of suitably structured matrices with low spectral norm, the pattern matrices, which we realize using matrix analysis and the Fourier transform over (Z2)n. The method of this paper has recently inspired important progress in multiparty communication complexity.

Abstract: In the 2nd Annual FOCS (1961), C. K. Chow proved that every Boolean threshold function is uniquely determined by its degree-0 and degree-1 Fourier coefficients. These numbers became known as the Chow Parameters. Providing an algorithmic version of Chow's theorem --- i.e., efficiently constructing a representation of a threshold function given its Chow Parameters --- has remained open ever since. This problem has received significant study in the fields of circuit complexity, game theory and the design of voting systems, and learning theory. In this paper we effectively solve the problem, giving a randomized PTAS with the following behavior: Theorem: Given the Chow Parameters of a Boolean threshold function f over n bits and any constant e > 0, the algorithm runs in time O(n2 log2 n) and with high probability outputs a representation of a threshold function f' which is e-close to f. Along the way we prove several new results of independent interest about Boolean threshold functions. In addition to various structural results, these include the following new algorithmic results in learning theory (where threshold functions are usually called "halfspaces"): An ~O(n2)-time uniform distribution algorithm for learning halfspaces to constant accuracy in the "Restricted Focus of Attention" (RFA) model of Ben-David et al. [3]. This answers the main open question of [6]. An O(n2)-time agnostic-type learning algorithm for halfspaces under the uniform distribution. This contrasts with recent results of Guruswami and Raghavendra [21] who show that the learning problem we solve is NP-hard under general distributions. As a special case of the latter result we obtain the fastest known algorithm for learning halfspaces to constant accuracy in the uniform distribution PAC learning model. For constant e our algorithm runs in time ~O(n2), which substantially improves on previous bounds and nearly matches the Ω(n2) bits of training data that any successful learning algorithm must use.

Abstract: We present a practical FPGA-based accelerator for solving Boolean Satisfiability problems (SAT). Unlike previous efforts for hardware accelerated SAT solving, our design focuses on accelerating the most time consuming part of the SAT solver --- Boolean Constraint Propagation (BCP), leaving the choices of heuristics such as branching order, restarting policy, and learning and backtracking to software. Our novel approach uses an application-specific architecture instead of an instance-specific one to avoid time-consuming FPGA synthesis for each SAT instance. By avoiding global signal wires and carefully pipelining the design, our BCP accelerator is able to achieve much higher clock frequency than that of previous work. In addition, it can load SAT instances in milliseconds, can handle SAT instances with tens of thousands of variables and clauses using a single FPGA, and can easily scale to handle more clauses by using multiple FPGAs. Our evaluation on a cycle-accurate simulator shows that the FPGA co-processor can achieve 3.7--38.6x speedup on BCP compared to state-of-the-art software SAT solvers.

Abstract: In the last years synthesis of reversible logic functions has emerged as an important research area. Other fields such as low-power design, optical computing and quantum computing benefit directly from achieved improvements. Recently, several approaches for exact synthesis of Toffoli networks have been proposed. They all use Boolean satisfiability to solve the underlying synthesis problem. In this paper a new exact synthesis approach based on Quantified Boolean Formula (QBF) satisfiability - a generalization of Boolean satisfiability - is presented. Besides the application of QBF solvers, we propose Binary Decision Diagrams to solve the quantified problem formulation. This allows to easily support different gate libraries during synthesis. In addition, all minimal networks are found in a single step and the best one with respect to quantum costs can be chosen. Experimental results confirm that the new technique is faster than the best previously known approach and leads to cheaper realizations in terms of quantum costs.

Abstract: Non-Linear Feedback Shift Registers (NLFSRs) have been proposed as an alternative to Linear Feedback Shift Registers (LFSRs) for generating pseudo-random sequences for stream ciphers. In this paper, we introduce (n, k)-NLFSRs which can be considered a generalization of the Galois type of LFSR. In an (n, k)-NLFSR, the feedback can be taken from any of the n bits, and the next state functions can be any Boolean function of up to k variables. Our motivation for considering this type NLFSRs is that their Galois configuration makes it possible to compute each next state function in parallel, thus increasing the speed of output sequence generation. Thus, for stream cipher application where the encryption speed is important, (n, k)-NLFSRs may be a better alternative than the traditional Fibonacci ones. We derive a number of properties of (n, k)-NLFSRs. First, we demonstrate that they are capable of generating output sequences with good statistical properties which cannot be generated by the Fibonacci type of NLFSRs. Second, we show that the period of the output sequence of an (n, k)-NLFSR is not necessarily equal to the length of the largest cycle of its states. Third, we compute the period of an (n, k)-NLFSR constructed from several parallel NLFSRs whose outputs are XOR-ed and show how to maximize this period. We also present an algorithm for estimating the length of cycles of states of (n, k)-NLFSRs which uses Binary Decision Diagrams for representing the set of states and the transition relation on this set.

Abstract: A new algorithm for obtaining efficient architectures composed of threshold gates that implement arbitrary Boolean functions is introduced. The method reduces the complexity of a given target function by splitting the function according to the variable with the highest influence. The procedure is iteratively applied until a set of threshold functions is obtained, leading to reduced depth architectures, in which the obtained threshold functions form the nodes and a and or or function is the output of the architecture. The algorithm is tested on a large set of benchmark functions and the results compared to previous existing solutions, showing a considerable reduction on the number of gates and levels of the obtained architectures. An extension of the method for partially defined functions is also introduced and the generalization ability of the method is analyzed.

Abstract: Emerging research device technologies might first appear in special applications that can extend conventional general-purpose processors along one of several axes. These applications could optimize the performance of future workloads such as recognition, mining, and synthesis by using the unique nonlinear output characteristics associated with the emerging research devices. However, a new device technology might emerge that, by way of first supplementing conventionally scaled CMOS, could eventually offer a highly scalable new information- processing paradigm.

Abstract: A two-terminal interactive distributed source coding problem with alternating messages is studied. The focus is on function computation at both locations with a probability which tends to one as the blocklength tends to infinity. A single-letter characterization of the rate region is provided. It is observed that interaction is useless (in terms of the minimum sum-rate) if the goal is pure source reproduction at one or both locations but the gains can be arbitrarily large for (general) function computation. For doubly symmetric binary sources and any function, interaction is useless with even infinite messages, when computation is desired at only one location, but is useful, when desired at both locations. For independent Bernoulli sources and the Boolean AND function computation at both locations, an interesting achievable infinite-message sum-rate is derived. This sum-rate is expressed, in analytic closed-form, in terms of a two-dimensional definite integral with an infinitesimal rate for each message.

Abstract: This paper presents new methods for restructuring logic networks based on fast Boolean techniques. The basis for these are 1) a cut-based view of a logic network, 2) exploiting the uniqueness and speed of disjoint-support decompositions, 3) a new heuristic for speeding these up, 4) extending these to general decompositions, and 5) limiting local transformations to functions with 16 or less inputs so that fast truth table manipulations can be used in all operations. Boolean methods lessen the structural bias of algebraic methods, while still allowing for high speed and multiple iterations. Experimental results on K-LUT networks show an average additional reduction of 5.4% in LUT count, while preserving delay, compared to heavily optimized versions of the same networks.

Abstract: We present foundational work on standard bases over rings and on Boolean Groebner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems arising from formal verification of digital circuits. In fact, algebraic modelling of formal verification problems is developed on the word-level as well as on the bit-level. The word-level model leads to Groebner basis in the polynomial ring over Z/2n while the bit-level model leads to Boolean Groebner bases. In addition to the theoretical foundations of both approaches, the algorithms have been implemented. Using these implementations we show that special data structures and the exploitation of symmetries make Groebner bases competitive to state-of-the-art tools from formal verification but having the advantage of being systematic and more flexible.

Abstract: This paper presents a decidable characterization of tree languages that can be defined by a boolean combination of Sigma1 formulas. This is a tree extension of the Simon theorem, which says that a string language can be defined by a boolean combination of Sigma1 formulas if and only if its syntactic monoid is J-trivial.

Abstract: The recent algebraic attacks have received a lot of attention in cryptographic literature. The algebraic immunity of a Boolean function quantifies its resistance to the standard algebraic attacks of the pseudorandom generators using it as a nonlinear filtering or combining function. Very few results have been found concerning its relation with the other cryptographic parameters or with the rth-order nonlinearity. As recalled by Carlet at CRYPTO'06, many papers have illustrated the importance of the r th-order nonlinearity profile (which includes the first-order nonlinearity). The role of this parameter relatively to the currently known attacks has been also shown for block ciphers. Recently, two lower bounds involving the algebraic immunity on the rth-order nonlinearity have been shown by Carlet . None of them improves upon the other one in all situations. In this paper, we prove a new lower bound on the rth-order nonlinearity profile of Boolean functions, given their algebraic immunity, that improves significantly upon one of these lower bounds for all orders and upon the other one for low orders.

Abstract: The basic principle in designing digital circuit hovers around reducing the required hardware thus reducing the cost too. To achieve this, we use Boolean expression that helps in obtaining minimum number of terms and does not contain any redundant pairs. The conventional methods for the minimization of the Boolean expressions are K-Map method and the . The minimized expressions are used to design digital circuits. Since K-Map method gets exceedingly complex when the number of the variable exceed six, hence Quine-McCluskey tabulation method scores over this and is widely used .In the following paper we present optimized Quine- McCluskey method that reduces the run time complexity of the algorithm by proposing an efficient algorithm for determination of Prime Implicants.

Abstract: Chapter 7-Combinatorial Searching 1 71 Zeros and Ones 47 711 Boolean Basics 47 712 Boolean Evaluation 96 Answers to Exercises 134 Index and Glossary 201

Abstract: We introduce a new state discrimination problem in which we are given additional information about the state after the measurement, or more generally, after a quantum memory bound applies. The following special case plays an important role in quantum cryptographic protocols in the bounded storage model: Given a string x encoded in an unknown basis chosen from a set of mutually unbiased bases (MUBs), you may perform any measurement, but then store at most q qubits of quantum information, and an unlimited amount of classical information. Later on, you learn which basis was used. How well can you compute a function f(x) of x, given the initial measurement outcome, the q qubits, and the additional basis information? We first show a lower bound on the success probability for any balanced function, and any number of mutually unbiased bases, beating the naive strategy of simply guessing the basis. We then show that for two bases, any Boolean function f(x) can be computed perfectly if you are allowed to store just a single qubit, independent of the number of possible input strings x. However, we show how to construct three bases, such that you need to store all qubits in order to compute f(x) perfectly. We then investigate how much advantage the additional basis information can give for a Boolean function. To this end, we prove optimal bounds for the success probability for the AND and the XOR function for up to three mutually unbiased bases. Our result shows that the gap in success probability can be maximal: without the basis information, you can never do better than guessing the basis, but with this information, you can compute f(x) perfectly. We also give an example where the extra information does not give any advantage at all.

Abstract: Boolean functions which are simultaneously bent and negabent are studied. Transformations that leave the bent-negabent property invariant are presented. A construction for infinitely many bent-negabent Boolean functions in 2mnvariables (m> 1) and of algebraic degree at most nis described, this being a subclass of the Maiorana---McFarland class of bent functions. Finally it is shown that a bent-negabent function in 2nvariables from the Maiorona---McFarland class has algebraic degree at most ni¾? 1.

Abstract: It is well known that modal satisfiability is PSPACE-complete (Ladner 1977). However, the complexity may decrease if we restrict the set of propositional operators used. Note that there exist an infinite number of propositional operators, since a propositional operator is simply a Boolean function. We completely classify the complexity of modal satisfiability for every finite set of propositional operators, i.e., in contrast to previous work, we classify an infinite number of problems. We show that, depending on the set of propositional operators, modal satisfiability is PSPACE-complete, coNP-complete, or in P. We obtain this trichotomy not only for modal formulas, but also for their more succinct representation using modal circuits. We consider both the uni-modal and the multi-modal case, and study the dual problem of validity as well.

Abstract: Side-channel attacks are an important class of cryptanalytic techniques against cryptographic' implementations and masking is a frequently considered solution to improve the resistance of a cryptographic implementation against side-channel attacks. The security of higher-order Boolean masking schemes in various contexts is analysed. The results presented are 2-fold. First, the definitions of higher-order side-channel attacks with the related security notions are formalised and certain security weaknesses in recently proposed masking schemes are put forward. Second, the conditions upon which a substitution box in a block cipher can be perfectly masked by Boolean values in order to counteract side-channel attacks are investigated. That is, can the leakages' statistical distributions at a masked S-box output (over all possible masks) be independent of the secret key targeted in the attacks? The consequences of this requirement are studied in two commonly considered leakage models, namely the Hamming weight and distance models, and conditions on the substitution boxes are derived. As a result of the analysis, it appears that these conditions are not achievable as they lead to evident cryptanalytic weaknesses. Thus, it is formally confirmed that masking cannot be used as a stand-alone countermeasure and cannot offer provable security against side-channel attacks.

Abstract: Shannon in his 1938 Masterpsilas Thesis demonstrated that any Boolean function can be realized by a switching relay circuit, leading to the development of deterministic digital logic. Here, we replace each classical switch with a probabilistic switch (pswitch). We present algorithms for synthesizing circuits closed with a desired probability, including an algorithm that generates optimal size circuits for any binary fraction. We also introduce a new duality property for series-parallel stochastic switching circuits. Finally, we construct a universal probability generator which maps deterministic inputs to arbitrary probabilistic outputs. Potential applications exist in the analysis and design of stochastic networks in biology and engineering.