Abstract: Substitution boxes (S-boxes) are important components in many modern-day symmetric key ciphers. Their study has attracted a great deal of attention over many years. The emergence of a variety of cryptosystem attacks has shown that substitutions must be designed with great care. Some general criteria such as high non-linearity and low autocorrelation have been proposed (providing some protection against attacks such as linear cryptanalysis and differential cryptanalysis). The design of appropriate S-boxes is a difficult task: several criteria must be traded off and the design space is huge. There has been little application of evolutionary search to the development of S-boxes. In this paper we show how a cost function that has found excellent single-output Boolean functions can be generalised to provide improved results for small S-boxes.

Abstract: In this paper, we analyze the algebraic immunity of symmetric Boolean functions. The algebraic immunity is a property which measures the resistance against the algebraic attacks on symmetric ciphers. We identify a set of lowest degree annihilators for symmetric functions and propose an efficient algorithm for computing the algebraic immunity of a symmetric function. The existence of several symmetric functions with maximum algebraic immunity is proven. In this way, we have found a new class of functions which have good implementation properties and maximum algebraic immunity.

Abstract: The paper describes an improved approach to Boolean network optimization using internal don't-cares. The improvements concern the type of don't-cares computed, their scope, and the computation method. Instead of the traditionally used compatible observability don't-cares (CODCs), we introduce and justify the use of complete don't-cares (CDC). To ensure the robustness of the don't-care computation for very large industrial networks, an optional windowing scheme is implemented that computes substantial subsets of the CDCs in reasonable time. Finally, we give a SAT-based don't-care computation algorithm that is more efficient than BDD-based algorithms. Experimental results confirm that these improvements work well in practice. Complete don't-cares allow for a reduction in the number of literals compared to the CODCs. Windowing guarantees robustness, even for very large benchmarks on which previous methods could not be applied. SAT reduces the runtime and enhances robustness, making don't-cares affordable for a variety of other Boolean methods applied to the network.

Abstract: We describe a novel decision procedure for Quantified Boolean Formulas (QBFs) which aims to unleash the hidden potential of quantified reasoning in applications. The Skolem theorem acts like a glue holding several ingredients together: BDD-based representations for boolean functions, search-based QBF decision procedure, and compilation-to-SAT techniques, among the others. To leverage all these techniques at once we show how to evaluate QBFs by symbolically reasoning on a compact representation for the propositional expansion of the skolemized problem. We also report about a first implementation of the procedure, which yields very interesting experimental results.

Abstract: In this paper we present a new decision procedure for the satisfiability of Linear Arithmetic Logic (LAL), i.e. boolean combinations of propositional variables and linear constraints over numerical variables. Our approach is based on the well known integration of a propositional SAT procedure with theory deciders, enhanced in the following ways. First, our procedure relies on an incremental solver for linear arithmetic, that is able to exploit the fact that it is repeatedly called to analyze sequences of increasingly large sets of constraints. Reasoning in the theory of LA interacts with the boolean top level by means of a stack-based interface, that enables the top level to add constraints, set points of backtracking, and backjump, without restarting the procedure from scratch at every call. Sets of inconsistent constraints are found and used to drive backjumping and learning at the boolean level, and theory atoms that are consequences of the current partial assignment are inferred. Second, the solver is layered: a satisfying assignment is constructed by reasoning at different levels of abstractions (logic of equality, real values, and integer solutions). Cheaper, more abstract solvers are called first, and unsatisfiability at higher levels is used to prune the search. In addition, theory reasoning is partitioned in different clusters, and tightly integrated with boolean reasoning. We demonstrate the effectiveness of our approach by means of a thorough experimental evaluation: our approach is competitive with and often superior to several state-of-the-art decision procedures.

Abstract: A filter for impulsive noise removal is presented here. The problem of impulsive noise elimination is closely connected with the problem of maximal preservation of image edges. To avoid smoothing of the image during filtering, all noisy pixels must be detected. We consider here an approach, which is based on threshold Boolean filtering, where the binary slices of an image, obtained by the threshold decomposition, are processed by the impulse-detecting Boolean functions proposed. These functions provide a possibility of single-pass filtering, because they detect and replace impulses at the same time.

Abstract: In this paper, we show that one-qubit polynomial time computations are as powerful as NC^1 circuits. More generally, we define syntactic models for quantum and stochastic branching programs of bounded width and prove upper and lower bounds on their power. We show that any NC^1 language can be accepted exactly by a width-2 quantum branching program of polynomial length, in contrast to the classical case where width 5 is necessary unless NC^1=ACC. This separates width-2 quantum programs from width-2 doubly stochastic programs as we show the latter cannot compute the middle bit of multiplication. Finally, we show that bounded-width quantum and stochastic programs can be simulated by classical programs of larger but bounded width, and thus are in NC^1. For read-once quantum branching programs (QBPs), we give a symmetric Boolean function which is computable by a read-once QBP with O(logn) width, but not by a deterministic read-once BP with o(n) width, or by a classical randomized read-once BP with o(n) width which is ''stable'' in the sense that its transitions depend on the value of the queried variable but do not vary from step to step. Finally, we present a general lower bound on the width of read-once QBPs, showing that our O(logn) upper bound for this symmetric function is almost tight.

Abstract: We present an OBDD-based Computer Algebra system for relational algebra, called RelView. After a short introduction to the OBDD-implementation of relations and the system, we exhibit its application by presenting two typical examples.

Abstract: We present a new and efficient algorithm to accurately polygonize an implicit surface generated by multiple Boolean operations with globally deformed primitives. Our algorithm is special in the sense that it can be applied to objects with both an implicit and a parametric representation, such as superquadrics, supershapes, and Dupin cyclides. The input is a constructive solid geometry tree (CSG tree) that contains the Boolean operations, the parameters of the primitives, and the global deformations. At each node of the CSG tree, the implicit formulations of the subtrees are used to quickly determine the parts to be transmitted to the parent node, while the primitives' parametric definition are used to refine an intermediary mesh around the intersection curves. The output is both an implicit equation and a mesh representing its solution. For the resulting object, an implicit equation with guaranteed differential properties is obtained by simple combinations of the primitives' implicit equations using R-functions. Depending on the chosen R-function, this equation is continuous and can be differentiable everywhere. The primitives' parametric representations are used to directly polygonize the resulting surface by generating vertices that belong exactly to the zero-set of the resulting implicit equation. The proposed approach has many potential applications, ranging from mechanical engineering to shape recognition and data compression. Examples of complex objects are presented and commented on to show the potential of our approach for shape modeling.

Abstract: In this paper, we propose an exact algorithm that maximizes the sharing of partial terms in multiple constant multiplication (MCM) operations. We model this problem as a Boolean network that covers all possible partial terms which may be used to generate the set of coefficients in the MCM instance. The PIs to this network are shifted versions of the MCM input. An AND gate represents an adder or a subtracter, i.e., an AND gate generates a new partial term. All partial terms that have the same numerical value are ORed together. There is a single output which is an /spl and/ over all the coefficients in the MCM. We cast this problem into a 0-1 integer linear programming (ILP) problem by requiring that the output is asserted while minimizing the total number of AND gates that evaluate to one. A SAT-based solver is used to obtain the exact solution. We argue that for many real problems the size of the problem is within the capabilities of current SAT solvers. We present results using binary, CSD and MSD representations. Two main conclusions can be drawn from the results. One is that, in many cases, existing heuristics perform well, computing the best solution, or one close to it. The other is that the flexibility of the MSD representation does not have a significant impact in the solution obtained.

Abstract: Linear pseudo-Boolean (LPB) constraints denote inequalities between arithmetic sums of weighted Boolean functions and provide a significant extension of the modeling power of purely propositional constraints They can be used to compactly describe many discrete electronic design automation problems with constraints on linearly combined, weighted Boolean variables, yet also offer efficient search strategies for proving or disproving whether a satisfying solution exists Furthermore, corresponding decision procedures can easily be extended for minimizing or maximizing an LPB objective function, thus providing a core optimization method for many problems in logic and physical synthesis In this paper, we review how recent advances in satisfiability search can be extended for pseudo-Boolean constraints and describe a new LPB solver that is based on generalized constraint propagation and conflict-based learning We present a comparison with other, state-of-the-art LPB solvers which demonstrates the overall efficiency of our method

Abstract: Preface. 1. Introduction. 2. Preliminaries. 2.1. Notation. 2.2. Boolean Functions. 2.3. Decomposition of Boolean Functions. 2.4. Reduced Ordered Binary Decision Diagrams.- 3. Exact node Minimization. 3.1. Branch and Bound Algorithm. 3.2. A*-Based Optimization. 3.3. Summary.- 4. Heuristic node Minimization. 4.1. Efficient Dynamic Minimization. 4.2. Improved Lower Bounds for Dynamic Reordering. 4.3. Efficient Forms of Improved Lower Bounds. 4.4. Combination of Improved Lower Bounds with Classical Bounds. 4.5. Experimental Results. 4.6. Summary.- 5. Path Minimization. 5.1. Minimization of Number of Paths. 5.2. Minimization of Expected Path Length. 5.3. Minimization of Average Path Length. 5.4. Summary.- 6. Relation between SAT and BDDS. 6.1. Davis-Putnam Procedure. 6.2. On the Relation between DP Procedure and BDDs. 6.3. Dynamic Variable Ordering Strategy for DP Procedure. 6.4. Experimental Results. 6.5. Summary.- 7. Final Remarks. References. Index.

Abstract: A new connectionist model, called Switching Neural Network (SNN), for the solution of classification problems is presented. SNN includes a first layer containing a particular kind of A/D converters, called latticizers, that suitably transform input vectors into binary strings. Then, the subsequent two layers of an SNN realize a positive Boolean function that solves in a lattice domain the original classification problem. Every function realized by an SNN can be written in terms of intelligible rules. Training can be performed by adopting a proper method for positive Boolean function reconstruction, called Shadow Clustering (SC). Simulation results obtained on the StatLog benchmark show the good quality of the SNNs trained with SC.

Abstract: In this paper, we study the algebraic immunity of Boolean functions and consider in particular the problem of constructing Boolean functions with a good algebraic immunity. We first give heuristic arguments which seem to indicate that the algebraic immunity of a random Boolean function on n variables is at least lfloorn/2rfloor with a very high probability (while the upper bound is lceiln/2rceil, the "ceiling" of n/2). We give an upper bound, under a reasonable assumption, on the algebraic immunity of Boolean functions constructed through Maiorana-MacFarland construction. At last we give examples of balanced functions with optimal algebraic immunity and a good nonlinearity and of balanced functions with a good algebraic immunity, a good nonlinearity and a good correlation immunity, which can be used for cryptographic purposes

Abstract: Technology mapping based on DAG-covering suffers from the problem of structural bias: the structure of the mapped netlist depends strongly on the subject graph. In this paper we present a new mapper aimed at mitigating structural bias. It is based on a simplified cut-based Boolean matching algorithm, and using the speed afforded by this simplification we explore two ideas to reduce structural bias. The first, called lossless synthesis, leverages recent advances in structure-based combinational equivalence checking to combine the different networks seen during technology independent synthesis into a single network with choices in a scalable manner. We show how cut based mapping extends naturally to handle such networks with choices. The second idea is to combine several library gates into a single gate (called a supergate) in order to make the matching process less local. We show how supergates help address the structural bias problem, and how they fit naturally into the cut-based Boolean matching scheme. An implementation based on these ideas significantly outperforms state-of-the-art mappers in terms of delay, area and run-time on academic and industrial benchmarks.

Abstract: This paper describes a Boolean satisfiability checking (SAT)-based unbounded symbolic model-checking algorithm. The conjunctive normal form is used to represent sets of states and transition relation. A logical operation on state sets is implemented as an operation on conjunctive normal form formulas. A satisfy-all procedure is proposed to compute the existential quantification required in obtaining the preimage and fix point. The proposed satisfy-all procedure is implemented by modifying a SAT procedure to generate all the satisfying assignments of the input formula, which is based on new efficient techniques such as line justification to make an assignment covering more search space, excluding clause management, and two-level logic minimization to compress the set of found assignments. In addition, a cache table is introduced into the satisfy-all procedure. It is a difficult problem for a satisfy-all procedure to detect the case that a previous result can be reused. This paper shows that the case can be detected by comparing sets of undetermined variables and clauses. Experimental results show that the proposed algorithm can check more circuits than binary decision diagram-based and previous SAT-based model-checking algorithms.

Abstract: In this paper, we propose a method of finding simple disjoint decompositions in frequent itemset data. The techniques for decomposing Boolean functions have been studied for long time in the area of logic circuit design, and recently, there is a very efficient algorithm to find all possible simple disjoint decompositions for a given Boolean functions based on BDDs (Binary Decision Diagrams). We consider the data model called “sets of combinations” instead of Boolean functions, and present a similar efficient algorithm for finding all possible simple disjoint decompositions for a given set of combinations. Our method will be useful for extracting interesting hidden structures from the frequent itemset data on a transaction database. We show some experimental results for conventional benchmark data.

Abstract: An efficient and compact canonical form is proposed for the Boolean matching problem under permutation and complementation of variables. In addition, an efficient algorithm for computing the proposed canonical form is provided. The efficiency of the algorithm allows it to be applicable to large complex Boolean functions with no limitation on the number of input variables as apposed to previous approaches, which are not capable of handling functions with more than seven inputs. Generalized signatures are used to define and compute the canonical form while symmetry of variables is used to minimize the computational complexity of the algorithm. Experimental results demonstrate the efficiency and applicability of the proposed canonical form.

Abstract: Separation logic is a subset of the quantifier-free first order logic. It has been successfully used in the automated verification of systems that have large (or unbounded) integer-valued state variables, such as pipelined processor designs and timed systems. In this paper, we present a fast decision procedure for separation logic, which combines Boolean satisfiability (SAT) with a graph based incremental negative cycle elimination algorithm. Our solver abstracts a separation logic formula into a Boolean formula by replacing each predicate with a Boolean variable. Transitivity constraints over predicates are detected from the constraint graph and added on a need-to basis. Our solver handles Boolean and theory conflicts uniformly at the Boolean level. The graph based algorithm supports not only incremental theory propagation, but also constant time theory backtracking without using a cumbersome history stack. Experimental results on a large set of benchmarks show that our new decision procedure is scalable, and outperforms existing techniques for this logic.

Abstract: A short introduction to quantum error correction is given, and it is shown that zero-dimensional quantum codes can be represented as self-dual additive codes over GF(4) and also as graphs. We show that graphs representing several such codes with high minimum distance can be described as nested regular graphs having minimum regular vertex degree and containing long cycles. Two graphs correspond to equivalent quantum codes if they are related by a sequence of local complementations. We use this operation to generate orbits of graphs, and thus classify all inequivalent self-dual additive codes over GF(4) of length up to 12, where previously only all codes of length up to 9 were known. We show that these codes can be interpreted as quadratic Boolean functions, and we define non-quadratic quantum codes, corresponding to Boolean functions of higher degree. We look at various cryptographic properties of Boolean functions, in particular the propagation criteria. The new aperiodic propagation criterion (APC) and the APC distance are then defined. We show that the distance of a zero-dimensional quantum code is equal to the APC distance of the corresponding Boolean function. Orbits of Boolean functions with respect to the {I,H,N}^n transform set are generated. We also study the peak-to-average power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove that PAR_IHN of a quadratic Boolean function is related to the size of the maximum independent set over the corresponding orbit of graphs. A construction technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It is finally shown that both PAR_IHN and APC distance can be interpreted as partial entanglement measures.

Abstract: Perfectly synchronous systems immediately react to the inputs of their environment, which may lead to so-called causality cycles between actions and their trigger conditions. Algorithms to analyze the consistency of such cycles usually extend data types by an additional value to explicitly indicate unknown values. In particular, Boolean functions are thereby extended to ternary functions. However, a Boolean function usually has several ternary extensions, and the result of the causality analysis depends on the chosen ternary extension. In this paper, we show that there always is a maximal ternary extension that allows one to solve as many causality problems as possible. Moreover, we elaborate the relationship to hazard elimination in hardware circuits, and finally show how the maximal ternary extension of a Boolean function can be efficiently computed by means of binary decision diagrams.

Abstract: We present a novel framework to identify all the testable and untestable path delay faults (PDFs) in a circuit. The method uses a combination of decision diagrams for manipulating PDFs as well as Boolean functions. The approach benefits from processing partial paths or fanout-free segments in the circuit rather than the entire path. The methodology is modified to identify all testable critical PDFs under the bounded delay fault model. The effectiveness of the proposed framework is demonstrated experimentally. It is observed that the methodology outperforms any existing method for identifying testable PDFs. Its scalability by focusing on critical PDFs is demonstrated by experimenting on very path-intensive benchmarks.

Abstract: In this paper we study functions with low influences on product probability spaces. The analysis of boolean functions with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics. We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known non-linear invariance principles. It has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly ``smoothed''; this extension is essential for our applications to ``noise stability''-type problems. In particular, as applications of the invariance principle we prove two conjectures: the ``Majority Is Stablest'' conjecture from theoretical computer science, which was the original motivation for this work, and the ``It Ain't Over Till It's Over'' conjecture from social choice theory.

Abstract: We introduce two new complexity measures for Boolean functions, or more generally for functions of the form f:S->T. We call these measures sumPI and maxPI. The quantity sumPI has been emerging through a line of research on quantum query complexity lower bounds via the so-called quantum adversary method [Amb02, Amb03, BSS03, Zha04, LM04], culminating in [SS04] with the realization that these many different formulations are in fact equivalent. Given that sumPI turns out to be such a robust invariant of a function, we begin to investigate this quantity in its own right and see that it also has applications to classical complexity theory. As a surprising application we show that sumPI^2(f) is a lower bound on the formula size, and even, up to a constant multiplicative factor, the probabilistic formula size of f. We show that several formula size lower bounds in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93], including a key lemma of [Has98], are in fact special cases of our method. The second quantity we introduce, maxPI(f), is always at least as large as sumPI(f), and is derived from sumPI in such a way that maxPI^2(f) remains a lower bound on formula size. While sumPI(f) is always a lower bound on the quantum query complexity of f, this is not the case in general for maxPI(f). A strong advantage of sumPI(f) is that it has both primal and dual characterizations, and thus it is relatively easy to give both upper and lower bounds on the sumPI complexity of functions. To demonstrate this, we look at a few concrete examples, for three functions: recursive majority of three, a function defined by Ambainis, and the collision problem.

Abstract: Linear pseudo-Boolean optimization (PBO) is a widely used modeling framework in electronic design automation (EDA). Due to significant advances in Boolean satisfiability (SAT), new algorithms for PBO have emerged, which are effective on highly constrained instances. However, these algorithms fail to handle effectively the information provided by the cost function of PBO. This paper addresses the integration of lower bound estimation methods with SAT-related techniques in PBO solvers. Moreover, the paper shows that the utilization of lower bound estimates can dramatically improve the overall performance of PBO solvers for most existing benchmarks from EDA.