Graph-theoretic design and analysis of key predistribution schemes

Key predistribution schemes for resource-constrained networks are methods for allocating symmetric keys to de-vices in such a way as to provide an efficient trade-off between key storage, connectivity and resilience. While there have been many suggested constructions for key predistribution schemes, a general understanding of the design prin-ciples on which to base such constructions is somewhat lacking. Indeed even the tools from which to develop such an understanding are currently limited, which results in many relatively ad hoc proposals in the research literature. It has been suggested that a large edge-expansion coefficient in the key graph is desirable for efficient key predistri-bution schemes. However, attempts to create key predistribution schemes from known expander graph constructions have only provided an extreme in the trade-off between connectivity and resilience: namely, they provide perfect resilience at the expense of substantially lower connectivity than can be achieved with the same key storage. Our contribution is two-fold. First, we prove that many existing key predistribution schemes produce key graphs with good expansion. This provides further support and justification for their use, and confirms the validity of expan-sion as a sound design principle. Second, we propose the use of incidence graphs and concurrence graphs as tools to represent, design and analyse key predistribution schemes. We show that these tools can lead to helpful insights and new constructions.

Graph-theoretic design and analysis of key predistribution schemes

Key predistribution schemes for resource-constrained networks are methods for allocating symmetric keys to devices in such a way as to provide an efficient trade-off between key storage, connectivity and resilience. While there have been many suggested constructions for key predistribution schemes, a general understanding of the design principles on which to base such constructions is somewhat lacking. Indeed even the tools from which to develop such an understanding are currently limited, which results in many relatively ad hoc proposals in the research literature. It has been suggested that a large edge-expansion coefficient in the key graph is desirable for efficient key predistribution schemes. However, attempts to create key predistribution schemes from known expander graph constructions have only provided an extreme in the trade-off between connectivity and resilience: namely, they provide perfect resilience at the expense of substantially lower connectivity than can be achieved with the same key storage. Our contribution is two-fold. First, we prove that many existing key predistribution schemes produce key graphs with good expansion. This provides further support and justification for their use, and confirms the validity of expansion as a sound design principle. Second, we propose the use of incidence graphs and concurrence graphs as tools to represent, design and analyse key predistribution schemes. We show that these tools can lead to helpful insights and new constructions

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group