Key Generation Algorithms for Pairwise Independent Networks Based on Graphical Models

We consider two secret key generation problems under a pairwise independent network model, and propose low complexity key generation schemes in a framework that connects our problems to network flow problems in graphs. Our schemes have two components: 1) local key generation and 2) global key propagation. In the local key generation, we use point-to-point source coding with side information to establish pairwise keys, from which we construct a graph with the capacity of each edge being the key rate of the corresponding point-to-point local key. In the global key propagation, depending on the particular problem, secret keys are delivered to users in the network using various network flow algorithms. In particular, in the first problem in which one is required to generate a group key for a group of users in the network, we propose a network coding-based global key propagation approach. This approach has a low complexity and has a better performance than the existing approach. In the second problem, in which one is required to generate multiple keys simultaneously for different pairs of users, we propose a multicommodity flow-based global key propagation approach. We show that the proposed approach is optimal for the case of generating two keys. For the general case of generating more than two keys, we show that the sum rate of the proposed scheme is larger than an upper bound characterized in this paper divided by a constant.