Distinct difference configurations: multihop paths and key predistribution in sensor networks

A distinct difference configuration is a set of points in Z2 with the property that the vectors (difference vectors) connecting any two of the points are all distinct. Many specific examples of these configurations have been previously studied: the class of distinct difference configurations includes both Costas arrays and sonar sequences, for example. Motivated by an application of these structures in key predistribution for wireless sensor networks, we define the k-hop coverage of a distinct difference configuration to be the number of distinct vectors that can be expressed as the sum of k or fewer difference vectors. This is an important parameter when distinct difference configurations are used in the wireless sensor application, as this parameter describes the density of nodes that can be reached by a short secure path in the network. We provide upper and lower bounds for the k-hop coverage of a distinct difference configuration with m points, and exploit a connection with Bh sequences to construct configurations with maximal k-hop coverage. We also construct distinct difference configurations that enable all small vectors to be expressed as the sum of two of the difference vectors of the configuration, an important task for local secure connectivity in the application

Two-dimensional patterns with distinct differences; constructions, bounds, and maximal anticodes

A two-dimensional (2-D) grid with dots is called a configuration with distinct differences if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various 2-D shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid

PIR schemes with small download complexity and low storage requirements

Shah, Rashmi and Ramchandran recently considered a model for Private Information Retrieval (PIR) where a user wishes to retrieve one of several Ä-bit messages from a set of n non-colluding servers. Their security model is information-theoretic. Their paper is the first to consider a model for PIR in which the database is not necessarily replicated, so allowing distributed storage techniques to be used. Shah et al. show that at least Ä+1 bits must be downloaded from servers, and describe a scheme with linear total storage (in R) that downloads between 2R and 3R bits. For any positive e, we provide a construction with the same storage property, that requires at most (1 + e)R bits to be downloaded; moreover one variant of our scheme only requires each server to store a bounded number of bits (in the sense of being bounded by a function that is independent of R). We also provide variants of a scheme of Shah et al which downloads exactly R +1 bits and has quadratic total storage. Finally, we simplify and generalise a lower bound due to Shah et al. on the download complexity of a PIR scheme. In a natural model, we show that an n-server PIR scheme requires at least nR/(n - 1) download bits in many cases, and provide a scheme that meets this bound. This paper provides various bounds on the download complexity of a PIR scheme, generalising those of Shah et al.\ to the case when the number $n$ of servers is bounded, and providing links with classical techniques due to Chor et al. The paper also provides a range of constructions for PIR schemes that are either simpler or perform better than previously known schemes. These constructions include explicit schemes that achieve the best asymptotic download complexity of Sun and Jafar with significantly lower upload complexity, and general techniques for constructing a scheme with good worst case download complexity from a scheme with good download complexity on average

PIR schemes with small download complexity and low storage requirements

Shah, Rashmi and Ramchandran recently considered a model for Private Information Retrieval (PIR) where a user wishes to retrieve one of several R-bit messages from a set of n non-colluding servers. Their security model is information-theoretic. Their paper is the first to consider a model for PIR in which the database is not necessarily replicated, so allowing distributed storage techniques to be used. They concentrate on minimising the total number of bits downloaded from the servers. Shah et al. provide a construction of a scheme that requires just R + 1 bits to be downloaded from servers, but requires an exponential (in R) number of servers. We provide an improved scheme that requires a linear number of servers. Shah et al. construct a scheme with linear total storage (in R) that needs at least 2R bits to be downloaded. For any positive �, we provide a construction with the same storage property, that requires at most (1 + �)R bits to be downloaded; moreover one variant of our scheme only requires each server to store a bounded number of bits (in the sense of being bounded by a function that is independent of R). Finally, we simplify and generalise a lower bound due to Shah et al. on the download complexity of such a PIR scheme. In a natural model, we show that an n-server PIR scheme requires at least nR/(n−1) download bits, and provide a scheme that meets this bound

PIR schemes with small download complexity and low storage requirements

Shah, Rashmi and Ramchandran recently considered a model for Private Information Retrieval (PIR) where a user wishes to retrieve one of several Ä-bit messages from a set of n non-colluding servers. Their security model is information-theoretic. Their paper is the first to consider a model for PIR in which the database is not necessarily replicated, so allowing distributed storage techniques to be used. Shah et al. show that at least Ä+1 bits must be downloaded from servers, and describe a scheme with linear total storage (in R) that downloads between 2R and 3R bits. For any positive e, we provide a construction with the same storage property, that requires at most (1 + e)R bits to be downloaded; moreover one variant of our scheme only requires each server to store a bounded number of bits (in the sense of being bounded by a function that is independent of R). We also provide variants of a scheme of Shah et al which downloads exactly R +1 bits and has quadratic total storage. Finally, we simplify and generalise a lower bound due to Shah et al. on the download complexity of a PIR scheme. In a natural model, we show that an n-server PIR scheme requires at least nR/(n - 1) download bits in many cases, and provide a scheme that meets this bound. This paper provides various bounds on the download complexity of a PIR scheme, generalising those of Shah et al.\ to the case when the number $n$ of servers is bounded, and providing links with classical techniques due to Chor et al. The paper also provides a range of constructions for PIR schemes that are either simpler or perform better than previously known schemes. These constructions include explicit schemes that achieve the best asymptotic download complexity of Sun and Jafar with significantly lower upload complexity, and general techniques for constructing a scheme with good worst case download complexity from a scheme with good download complexity on average

A unified approach to combinatorial key predistribution schemes for sensor networks

There have been numerous recent proposals for key predistribution schemes for wireless sensor networks based on various types of combinatorial structures such as designs and codes. Many of these schemes have very similar properties and are analysed in a similar manner. We seek to provide a unified framework to study these kinds of schemes. To do so, we define a new, general class of designs, termed “partially balanced t-designs”, that is sufficiently general that it encompasses almost all of the designs that have been proposed for combinatorial key predistribution schemes. However, this new class of designs still has sufficient structure that we are able to derive general formulas for the metrics of the resulting key predistribution schemes. These metrics can be evaluated for a particular scheme simply by substituting appropriate parameters of the underlying combinatorial structure into our general formulas. We also compare various classes of schemes based on different designs, and point out that some existing proposed schemes are in fact identical, even though their descriptions may seem different. We believe that our general framework should facilitate the analysis of proposals for combinatorial key predistribution schemes and their comparison with existing schemes, and also allow researchers to easily evaluate which scheme or schemes present the best combination of performance metrics for a given application scenario

On the Role of Expander Graphs in Key Predistribution Schemes for Wireless Sensor Networks

Providing security for a wireless sensor network composed of small sensor nodes with limited battery power and memory can be a non-trivial task. A variety of key predistribution schemes have been proposed which allocate symmetric keys to the sensor nodes before deployment. In this paper we examine the role of expander graphs in key predistribution schemes for wireless sensor networks. Roughly speaking, a graph has good expansion if every `small\u27 subset of vertices has a `large\u27 neighbourhood, and intuitively, expansion is a desirable property for graphs of networks. It has been claimed that good expansion in the product graph is necessary for `optimal\u27 networks. We demonstrate flaws in this claim, argue instead that good expansion is desirable in the intersection graph, and discuss how this can be achieved. We then consider key predistribution schemes based on expander graph constructions and compare them to other schemes in the literature. Finally, we propose the use of expansion and other graph-theoretical techniques as metrics for assessing key predistribution schemes and their resulting wireless sensor networks

Modelling harmonic generation measurements in solids

Harmonic generation measurements typically make use of the plane wave result when extracting values for the nonlinearity parameter, beta, from experimental measurements. This approach, however, ignores the effects of diffraction, attenuation, and receiver integration which are common features in a typical experiment. Our aim is to determine the importance of these effects when making measurements of beta over different sample dimensions, or using different input frequencies. We describe a three-dimensional numerical model designed to accurately predict the results of a typical experiment, based on a quasi-linear assumption. An experiment is designed to measure the axial variation of the fundamental and second harmonic amplitude components in an ultrasonic beam, and the results are compared with those predicted by the model. The absolute b values are then extracted from the experimental data using both the simulation and the standard plane wave result. A difference is observed between the values returned by the two methods, which varies with axial range and input frequency. (C) 2013 Elsevier B. V. All rights reserved.</p

Detector Description and Performance for the First Coincidence Observations between LIGO and GEO

For 17 days in August and September 2002, the LIGO and GEO interferometer gravitational wave detectors were operated in coincidence to produce their first data for scientific analysis. Although the detectors were still far from their design sensitivity levels, the data can be used to place better upper limits on the flux of gravitational waves incident on the earth than previous direct measurements. This paper describes the instruments and the data in some detail, as a companion to analysis papers based on the first data.Comment: 41 pages, 9 figures 17 Sept 03: author list amended, minor editorial change