Yannis C. Stamatiou

Random Constraint Satisfaction: A More Accurate Picture

In the last few years there has been a great amount of interest in Random Constraint Satisfaction Problems, both from an experimental and a theoretical point of view. Quite intriguingly, experimental results with various models for generating random CSP instances suggest that the probability of such problems having a solution exhibits a “threshold–like” behavior. In this spirit, some preliminary theoretical work has been done in analyzing these models asymptotically, i.e., as the number of variables grows. In this paper we prove that, contrary to beliefs based on experimental evidence, the models commonly used for generating random CSP instances do not have an asymptotic threshold. In particular, we prove that asymptotically almost all instances they generate are overconstrained, suffering from trivial, local inconsistencies. To complement this result we present an alternative, single–parameter model for generating random CSP instances and prove that, unlike current models, it exhibits non–trivial asymptotic behavior. Moreover, for this new model we derive explicit bounds for the narrow region within which the probability of having a solution changes dramatically.

Approximating the unsatisfiability threshold of random formulas

Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by Kamath et al. to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253

Bounding the unsatisfiability threshold of random 3-SAT

The satisfiability threshold conjecture states that fur a randomly generated formula of m clauses of exactly k literals over n variables, the probability that it is satisfiable, as n tends to infinity, changes abruptly from I to 0, as the ratio I = m/n is

Elliptic Curve Based Zero Knowledge Proofs and their Applicability on Resource Constrained Devices

As the Internet of Things (IOT) arises, the use of low-end devices on a daily basis increases. The wireless nature of communication that these devices provide raises security and privacy issues. For protecting a users privacy, cryptography offers the tool of zero knowledge proofs (ZKP). In this paper, we study well-established ZKP protocols based on the discrete logarithm problem and we adapt them to the Elliptic Curve Cryptography (ECC) setting, which consists an ideal candidate for embedded implementations. Then, we implement the proposed protocols on Wiselib, a generic and open source algorithmic library. For the first time, we present a thorough evaluation of the protocols on two popular hardware platforms equipped with low end microcontrollers (Jennic JN5139, TI MSP430) and 802.15.4 RF transceivers, in terms of code size, execution time, message size and energy requirements. This works results can be used from developers who wish to achieve certain levels of privacy in their applications.

Cryptography and Cryptanalysis Through Computational Intelligence

The past decade has witnessed an increasing interest in the application of Computational Intelligence methods to problems derived from the field of cryptography and cryptanalysis. This phenomenon can be attributed both to the effectiveness of these methods to handle hard problems, and to the major importance of automated techniques in the design and cryptanalysis of cryptosystems. This chapter begins with a brief introduction to cryptography and Computational Intelligence methods. A short survey of the applications of Computational Intelligence to cryptographic problems follows, and our contribution in this field is presented. Specifically, some cryptographic problems are viewed as discrete optimization tasks and Evolutionary Computation methods are utilized to address them. Furthermore, the effectiveness of Artificial Neural Networks to approximate some cryptographic functions is studied. Finally, theoretical issues of Ridge Polynomial Networks and cryptography are presented. The experimental results reported suggest that problem formulation and representation are critical determinants of the performance of Computational Intelligence methods in cryptography. Moreover, since strong cryptosystems should not reveal any patterns of the encrypted messages or their inner structure, it appears that Computational Intelligence methods can constitute a first measure of the cryptosystems’ security.

Coupon Collectors, q-Binomial Coefficients and the Unsatisfiability Threshold

The problem of determining the unsatisfiability threshold for random 3-SAT formulas consists in determining the clause to variable ratio that marks the (experimentally observed) abrupt change from almost surely satisfiable formulas to almost surely unsatisfiable. Up to now, there have been rigorously established increasingly better lower and upper bounds to the actual threshold value. An upper bound of 4.506 was announced by Dubois et al. in 1999 but, to the best of our knowledge, no complete proof has been made available from the authors yet. We consider the problem of bounding the threshold value from above using methods that, we believe, are of interest on their own right. More specifically, we explain how the method of local maximum satisfying truth assignments can be combined withresu lts for coupon collectors probabilities in order to achieve an upper bound for the unsatisfiability threshold less than 4.571. Thus, we improve over the best, with an available complete proof, previous upper bound, which was 4.596. In order to obtain this value, we also establish a bound on the q-binomial coefficients (a generalization of the binomial coefficients) which may be of independent interest.

Monotonicity and inert fugitive search games

In general, graph search can be described in terms of a sequence of searchers’ moves on a graph, able to capture a fugitive resorting on its vertices/edges. Several variations of graph search have been introduced, di!ering on the a bilities of the fugitive as well as of the search. In this paper, we examine the case where the fugitive is inert, i.e., it moves only whenever the search is about to capture it. Mainly, there are two variants for “clearing” an edge during a search: when as liding of as earcher occurs along the edge or when both its endpoints are simultaneously occupied by searchers. These variants define the inert edge search and th ei nert node search respectively. A third search variant, the inert mixed search, is defined when both ways of clearing an edge are possible. As we show, inert search and inert mixed search are equivalent (surprisingly, this is not the case if we discard the inertness property). Moreover, we prove that, in any case, by restricting the searches to only those that always reduce further the fugitive’s possible resorts, doe sn ot give any advantage to the fugitive (this monotonicity property is usually expressed as: “recontamination does not help”). So far, the only monotonicity result on inert search concerns inert node search and our results yield a much simpler proof o ft hat result as well. Furthermore, we define a new graph-theoretic parameter, the proper-treewidth ,i n analogy to the parameter proper-pathwidth ,a nd prove it equivalent to the inert mixed search game. Last, we prove that proper-treewidth, in turn, is equivalent to ak nown graph theoretic parameter related to treewidth.

Efficient generation of secure elliptic curves

In many cryptographic applications it is necessary to generate elliptic curves (ECs) whose order possesses certain properties. The method that is usually employed for the generation of such ECs is the so-called Complex Multiplication method. This method requires the use of the roots of certain class field polynomials defined on a specific parameter called the discriminant. The most commonly used polynomials are the Hilbert and Weber ones. The former can be used to generate directly the EC, but they are characterized by high computational demands. The latter have usually much lower computational requirements, but they do not directly construct the desired EC. This can be achieved if transformations of their roots to the roots of the corresponding (generated by the same discriminant) Hilbert polynomials are provided. In this paper we present a variant of the Complex Multiplication method that generates ECs of cryptographically strong order. Our variant is based on the computation of Weber polynomials. We present in a simple and unifying manner a complete set of transformations of the roots of a Weber polynomial to the roots of its corresponding Hilbert polynomial for all values of the discriminant. In addition, we prove a theoretical estimate of the precision required for the computation of Weber polynomials for all values of the discriminant. We present an extensive experimental assessment of the computational efficiency of the Hilbert and Weber polynomials along with their precision requirements for various discriminant values and we compare them with the theoretical estimates. We further investigate the time efficiency of the new Complex Multiplication variant under different implementations of a crucial step of the variant. Our results can serve as useful guidelines to potential implementers of EC cryptosystems involving generation of ECs of a desirable order on resource limited hardware devices or in systems operating under strict timing response constraints.

Generating prime order elliptic curves: difficulties and efficiency considerations

We consider the generation of prime order elliptic curves (ECs) over a prime field

Locating Information with Uncertainty in Fully Interconnected Networks with Applications to World Wide Web Information Retrieval

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