Abstract. In the last few years there has been a great amount o f i n terest 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 numberofvariables grows. In this paper we p r o ve 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, su ering from trivial, local inconsistencies. To complement this result we present a n 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 d e r i v e explicit bounds for the narrow region within which the probability o f h a ving a solution changes dramatically.

Let be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number such that if the ratio of the number of clauses over the number of variables of strictly exceeds , then is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that F 5.191. 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 is around 4.2. In this work, we define, in terms of the random formula , a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then 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 equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q .

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We lower the upper bound for the threshold for random 3-SAT from 4.6011 to 4.596 through two different approaches, both giving the same result. (Assuming the threshold exists, as is generally believed but still not rigorously shown.) In both approaches, we start with a sum over all truth assignments that appears in an upper bound by Kirousis et al. to the the probability that a random 3-SAT formula is satisfiable. In the first approach, this sum is reformulated as the partition function of a spin system consisting of n sites each of which may assume the values 0 or 1. We then obtain an asymptotic expression for this function that results from the application of an optimization technique from statistical * Research performed while this author was visiting the School of Computer Science of Carleton University in Ottawa, supported by the Greek Ministry of National Economy through a NATO scholarship for conducting postdoctoral studies (contract number 106384/∆OO 1222/2-7-98).

Abstract. We present a variant of the complex multiplication method that generates elliptic curves of cryptographically strong order. Our variant is based on the computation of Weber polynomials that require significantly less time and space resources than their Hilbert counterparts. We investigate the time efficiency and precision requirements for generating off-line Weber polynomials and its comparison to another variant based on the off-line generation of Hilbert polynomials. We also investigate the efficiency of our variant when the computation of Weber polynomials should be made on-line due to limitations in resources (e.g., hardware devices of limited space). We present trade-offs that could be useful to potential implementors of elliptic curve cryptosystems on resource-limited hardware devices.

Elliptic Curve Cryptography (ECC) is an attractive alternative to conventional public key cryptography, such as RSA. ECC is an ideal candidate for implementation on constrained devices where the major computational resources i.e. speed, memory are limited and low-power wireless communication protocols are employed. That is because it attains the same security levels with traditional cryptosystems using smaller parameter sizes. Moreover, in several application areas such as person identification and eVoting, it is frequently required of entities to prove knowledge of some fact without revealing this knowledge. Such proofs of knowledge are called Zero Knowledge Interactive Proofs (ZKIP) and involve interactions between two communicating parties, the Prover and the Verifier. In a ZKIP, the Prover demonstrates the possesion of some information (e.g. authentication information) to the Verifier without disclosing it. In this paper, we focus on the application of ZKIP protocols on resource constrained devices. We study well-established ZKIP protocols based on the discrete logarithm problem and we transform them under the ECC setting. Then, we implement the proposed protocols on Wiselib, a generic and open source algorithmic library. Finally, 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. To the best of our knowledge, this is the first attempt of implementing and evaluating ZKIP protocols with emphasis on low-end devices. This work's results can be used from developers who wish to achieve certain levels of security and privacy in their applications.

We lower the upper bound for the threshold for random 3-SAT from 4.6011 to 4.596 through two different approaches, both giving the same result. (Assuming the threshold exists, as is generally believed but still not rigorously shown.) In both approaches, we start with a sum over all truth assignments that appears in an upper bound by Kirousis et al. to the the probability that a random 3-SAT formula is satisfiable. In the first approach, this sum is reformulated as the partition function of a spin system consisting of n sites each of which may assume the values 0 or 1. We then obtain an asymptotic expression for this function that results from the application of an optimization technique from statistical * Research performed while this author was visiting the School of Computer Science of Carleton University in Ottawa, supported by the Greek Ministry of National Economy through a NATO scholarship for conducting postdoctoral studies (contract number 106384/∆OO 1222/2-7-98).

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