Distinct Difference Configurations: Multihop Paths and Key Predistribution in Sensor Networks

Abstract

A distinct difference configuration is a set of points in <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"><tex Notation="TeX">$\BBZ ^{2}$</tex></formula></emphasis> with the property that the vectors ( <emphasis emphasistype="boldital" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">difference vectors</emphasis> ) 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 <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"> <tex Notation="TeX">$k$</tex></formula></emphasis> -hop coverage of a distinct difference configuration to be the number of distinct vectors that can be expressed as the sum of <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"> <tex Notation="TeX">$k$</tex></formula></emphasis> 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 <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"> <tex Notation="TeX">$k$</tex></formula></emphasis> -hop coverage of a distinct difference configuration with <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"> <tex Notation="TeX">$m$</tex></formula></emphasis> points, and exploit a connection with <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"><tex Notation="TeX">$B_{h}$</tex> </formula></emphasis> sequences to construct configurations with maximal <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"><tex Notation="TeX">$k$</tex> </formula></emphasis> -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.