On Connectivity Thresholds in the Intersection of Random Key Graphs on Random Geometric Graphs

In a random key graph (RKG) of \(n\) nodes each node is randomly assigned a key ring of \(K_n\) cryptographic keys from a pool of \(P_n\) keys. Two nodes can communicate directly if they have at least one common key in their key rings. We assume that the \(n\) nodes are distributed uniformly in \([0,1]^2.\) In addition to the common key requirement, we require two nodes to also be within \(r_n\) of each other to be able to have a direct edge. Thus we have a random graph in which the RKG is superposed on the familiar random geometric graph (RGG). For such a random graph, we obtain tight bounds on the relation between \(K_n,\) \(P_n\) and \(r_n\) for the graph to be asymptotically almost surely connected.