Sparse Hashing for Scalable Approximate Model Counting

Given a Boolean formula $\phi$ over $n$ variables, the problem of model counting is to compute the number of solutions of $\phi$. Model counting is a fundamental problem in computer science with wide-ranging applications. Owing to the \#P-hardness of the problems, Stockmeyer initiated the study of the complexity of approximate counting. Stockmeyer showed that $\log n$ calls to an NP oracle are necessary and sufficient to achieve $(\varepsilon,\delta)$ guarantees. The hashing-based framework proposed by Stockmeyer has been very influential in designing practical counters over the past decade, wherein the SAT solver substitutes the NP oracle calls in practice. It is well known that an NP oracle does not fully capture the behavior of SAT solvers, as SAT solvers are also designed to provide satisfying assignments when a formula is satisfiable, without additional overhead. Accordingly, the notion of SAT oracle has been proposed to capture the behavior of SAT solver wherein given a Boolean formula, an SAT oracle returns a satisfying assignment if the formula is satisfiable or returns unsatisfiable otherwise. Since the practical state-of-the-art approximate counting techniques use SAT solvers, a natural question is whether an SAT oracle is more powerful than an NP oracle in the context of approximate model counting. The primary contribution of this work is to study the relative power of the NP oracle and SAT oracle in the context of approximate model counting. The previous techniques proposed in the context of an NP oracle are weak to provide strong bounds in the context of SAT oracle since, in contrast to an NP oracle that provides only one bit of information, a SAT oracle can provide $n$ bits of information. We therefore develop a new methodology to achieve the main result: a SAT oracle is no more powerful than an NP oracle in the context of approximate model counting.

Article Free Access Share on Exact sampling and approximate counting techniques Author: Mark Huber School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NYView Profile Authors Info & Claims STOC '98: Proceedings of the thirtieth annual ACM symposium on Theory of computingMay 1998 Pages 31–40https://doi.org/10.1145/276698.276709Published:23 May 1998Publication History 25citation666DownloadsMetricsTotal Citations25Total Downloads666Last 12 Months66Last 6 weeks15 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

In this paper, we propose a new hash function based on RC4 and we call it RC4-Hash. This proposed hash function produces variable length hash output from 16 bytes to 64 bytes. Our RC4-Hash has several advantages over many popularly known hash functions. Its efficiency is comparable with widely used known hash function (e.g., SHA-1). Seen in the light of recent attacks on MD4, MD5, SHA-0, SHA-1 and on RIPEMD, there is a serious need to consider other hash function design strategies. We present a concrete hash function design with completely new internal structure. The security analysis of RC4-Hash can be made in the view of the security analysis of RC4 (which is well studied) as well as the attacks on different hash functions. Our hash function is very simple and rules out all possible generic attacks. To the best of our knowledge, the design criteria of our hash function is different from all previously known hash functions. We believe our hash function to be secure and will appreciate security analysis and any other comments.

Recently, with the discoveries in neurobiology, high-dimensional sparse hashing has attracted increasing attention. In contrast with general hashing that generates low-dimensional hash codes, the high-dimensional sparse hashing maps inputs into a higher dimensional space and generates sparse hash codes, achieving superior performance. However, the sparse hashing has not been fully studied in hashing literature yet. For example, how to fully explore the power of sparse coding in cross-modal retrieval tasks; how to discretely solve the binary and sparse constraints so as to avoid the quantization error problem. Motivated by these issues, in this paper, we present an efficient sparse hashing method, i.e., High-dimensional Sparse Cross-modal Hashing, HSCH for short. It not only takes the high-level semantic similarity of data into consideration, but also properly exploits the low-level feature similarity. In specific, we theoretically design a fine-grained similarity with two critical fusion rules. Then we take advantage of sparse codes to embed the fine-grained similarity into the to-be-learnt hash codes. Moreover, an efficient discrete optimization algorithm is proposed to solve the binary and sparse constraints, reducing the quantization error. In light of this, it becomes much more trainable, and the learnt hash codes are more discriminative. More importantly, the retrieval complexity of HSCH is as efficient as general hash methods. Extensive experiments on three widely-used datasets demonstrate the superior performance of HSCH compared with several state-of-the-art cross-modal hashing approaches.

This paper describes the study of cryptographic hash functions, one of the most important classes of primitives used in recent techniques in cryptography. The main aim is the development of recent crypt analysis hash function. We present different approaches to defining security properties more formally and present basic attack on hash function. We recall Merkle-Damgard security properties of iterated hash function. The Main aim of this paper is the development of recent techniques applicable to crypt Analysis hash function, mainly from SHA family. Recent proposed attacks an MD5 & SHA motivate a new hash function design. It is designed not only to have higher security but also to be faster than SHA-256. The performance of the new hash function is at least 30% better than that of SHA-256 in software. And it is secure against any known cryptographic attacks on hash functions.

Two approximate algorithms for model counting

Given a set of constraints F and a weight function W over the assignments, the problem of MaxSAT is to compute a maximum weighted solution of F. MaxSAT is a fundamental problem with applications in numerous areas. The success of MaxSAT solvers has prompted researchers in AI and formal methods communities to develop algorithms that can use MaxSAT solver as oracle. One such problem that stands to benefit from advances in MaxSAT solving is discrete integration. Recently, Ermon et al. achieved a significant breakthrough by reducing the problem of integration to polynomially many queries to an optimization oracle where $F$ is conjuncted with randomly chosen XOR constraints. Unlike approximate model counting, where hashing-based approaches have been able to achieve scalability as well as rigorous formal guarantees, the practical evaluation of Ermon et al's approach, called WISH, often sacrifice theoretical guarantees, largely due to lack of existing MaxSAT solvers with native XOR support. The primary contribution of this paper is a new MaxSAT solver, GaussMaxHS, with built-in XOR support. The architecture of GaussMaxHS is inspired by CryptoMiniSAT, which has been the workhorse of hashing-based approximate model counting techniques. The resulting solver, GaussMaxHS, outperforms MaxHS over 9628 benchmarks arising from spin glass models and network reliability domains. In particular, with a timeout of 5000 seconds, MaxHS could solve only 5473 benchmarks while GaussMaxHS could solve 6120 benchmarks.

Recently, formal complexity-theoretic treatment of cryptographic hash functions was suggested. Two primitives of Collision-free hash functions and Universal one-way hash function families have been defined. The primitives have numerous applications in secure information compression, since their security implies that finding collisions is computationally hard. Most notably, Naor and Yung have shown that the most secure signature scheme can be reduced to the existence of universal one-way hash (this, in turn, gives the first trapdoor-less provably secure signature scheme).In this work, we first present reductions from various one-way function families to universal one-way hash functions. Our reductions are general and quite efficient and show how to base universal one-way hash functions on any of the known concrete candidates for one-way functions. We then show equivalences among various definitions of hardness for collision-free hash functions.

We propose a simple and efficient hash function based on programmable elementary cellular automata. Cryptographic hash functions are important building blocks for many cryptographic protocols such as authentication and integrity verification. They have recently brought an exceptional research interest, especially after the increasing number of attacks against the widely used functions as MD5, SHA-1 and RIPEMD, causing a crucial need to consider new hash functions design and conception strategies. The proposed hash function is built using elementary cellular automata that are very suitable for cryptographic applications, due to their chaotic and complex behavior derived from simple rules interaction. The function is evaluated using several statistical tests, while obtained results demonstrate very admissible cryptographic proprieties such as confusion, diffusion capability and high sensitivity to input changes. Furthermore, the hashing scheme can be easily implemented through software or hardware, and provides very competitive running performances.

A common approach for the construction of cryptographic hash functions is to design the algorithm based on an existing symmetric encryption primitive. While there has been extensive research on the design of block cipher-based hash functions, little has been done on the study of design and security of stream cipher-based hash functions (SCH). In this paper we discuss the general construction of stream cipherbased hash functions, devoting special attention to one of the function's crucial components: the message injection function. We define two types of message injection functions, which may be appended to the keystream generator (e.g. a stream cipher) to build an SCH. Based on these constructions, we evaluate the security of simple SCHs whose stream cipher function consists of a LFSR-based filter generator. We see this as an initial step in the more formal study of the security of hash function constructions based on stream ciphers.

Conflict-Driven Clause Learning (CDCL) SAT solvers can automatically solve very large real-world problems. IntSat is a new technique extending CDCL to Integer Linear Programming (ILP). For some conflicts, IntSat generates no strong enough ILP constraint, and the backjump has to be done based on the conflicting set of bounds. The techniques given in this article precisely analyze how and when that set can be translated into the required new ILP constraint. Moreover, this can be done efficiently. This obviously strengthens learning and significantly improves the performance of IntSat (as confirmed by our experiments). We also briefly discuss extensions and other applications.

The Boolean Satisfiability Problem (SAT) is well known in computational complexity, representing the first decision problem to be proved NP-complete. SAT is also often the subject of work in proof complexity. Besides its theoretical interest, SAT finds a wide range of practical applications. Moreover, SAT solvers have been the subject of remarkable efficiency improvements since the mid-1990s, motivating their widespread use in many practical applications including Bounded and Unbounded Model Checking. The success of SAT solvers has also motivated the development of algorithms for natural extensions of SAT, including Quantified Boolean Formulas (QBF), Pseudo-Boolean constraints (PB), Maximum Satisfiability (MaxSAT) and Satisfiability Modulo Theories (SMT) which also see application in the model-checking context. This chapter first covers the organization of modern conflict-driven clause learning (CDCL) SAT solvers, which are used in the vast majority of practical applications of SAT. It then reviews the techniques shown to be effective in modern SAT solvers.

Model counting is a problem to count different truth assignments for a given propositional formula. Approximate model counting via extension rule is a feasible method. In this paper we introduce the technique of clause reduction and avoiding multiple accesses to enhance the performance of #SAT solver. Using clause reduction can eliminate short clauses and restrict the size of the residual formula. We also optimize the code to avoid getting access to the same variable for several times. We combine these two optimizations with extension rule to achieve ERCRApprox algorithm. In addition, we control the upper bound of the approximate dispersion of ERCRApprox algorithm in a feasible method. Experiment results show that the algorithm increases the efficiency for solving model counting especially when some short clauses exist and the approximate dispersion in each case is always acceptable.

Many real-world problems are encoded into SAT instances and efficiently solved by CDCL (Conflict-Driven Clause Learning) SAT solvers. However, some scenarios require distributed problem solving approaches. Privacy is often the main reason. This motivates the need to solve distributed SAT problems We analyze how this problem can be tacked in an efficient way, and present ABTSAT, a new version of the ABT (Asynchronous Backtracking) algorithm adapted to solve distributed SAT instances. It combines ABT execution with calls to CDCL SAT solvers and clause learning. ABTSAT is sound and complete, properties inherited from ABT, and solves local problems efficiently by using CDCL SAT solvers.

Modern conflict-driven clause-learning (CDCL) Boolean SAT solvers provide efficient automatic analysis of real-world feature models (FM) of systems ranging from cars to operating systems. It is well-known that solver-based analysis of real-world FMs scale very well even though SAT instances obtained from such FMs are large, and the corresponding analysis problems are known to be NP-complete. To better understand why SAT solvers are so effective, we systematically studied many syntactic and semantic characteristics of a representative set of large real-world FMs. We discovered that a key reason why large real-world FMs are easy-to-analyze is that the vast majority of the variables in these models are unrestricted, i.e., the models are satisfiable for both true and false assignments to such variables under the current partial assignment. Given this discovery and our understanding of CDCL SAT solvers, we show that solvers can easily find satisfying assignments for such models without too many backtracks relative to the model size, explaining why solvers scale so well. Further analysis showed that the presence of unrestricted variables in these real-world models can be attributed to their high-degree of variability. Additionally, we experimented with a series of well-known nonbacktracking simplifications that are particularly effective in solving FMs. The remaining variables/clauses after simplifications, called the core, are so few that they are easily solved even with backtracking, further strengthening our conclusions. We explain the connection between our findings and backdoors, an idea posited by theorists to explain the power of SAT solvers. This connection strengthens our hypothesis that SAT-based analysis of FMs is easy. In contrast to our findings, previous research characterizes the difficulty of analyzing randomly-generated FMs in terms of treewidth. Our experiments suggest that the difficulty of analyzing real-world FMs cannot be explained in terms of treewidth.