On optimality of key pre-distribution schemes for distributed sensor networks

We derive the optimality results for key pre distribution scheme for distributed sensor networks, and relations between interesting parameters. Namely, given a key-pool of size n we derive the optimal value that is jointly achievable for parameters like, Size optimality: using less memory per node - while supporting large network, Connectivity optimality: possibility of establishing secure communication between any two nodes over short path, and Resiliency optimality: large fraction of network remains working under compromise or node capture. We characterize this relation in graph theoretic framework. Our result shows that the desired graph (a combination of network topology graph on which key-share graph is embedded) must have small clique and independent set and must have high expansion properties, in other words Expander graphs are best suited for forming secure networks.