Secure Hypergraphs: Privacy from Partial Broadcast

A "partial broadcast channel" enables one processor to send the same message---simultaneously and privately---to a fixed subset of processors. Suppose that a collection of processors are connected by an arbitrary network of partial broadcast channels (a hypergraph). We initiate the study of necessary and sufficient conditions, complexity bounds, and protocols for individual processors to exchange private messages across this network. Private message exchange, in turn, enables the realization of general secure computation primitives. The model (motivated by various environments such as multicast network architectures and group communication in distributed systems) is an intermediate setting between the private channels model and the full information model, both of which have been investigated extensively in the last few years. We assume a computationally unlimited adversary (i.e., the information theoretic notion of security), and our techniques are combinatorial. Both the possibility and the polynomial-time feasibility of private message exchange are investigated.