Simultaneously generating multiple keys in many to one networks

The problem of simultaneously establishing multiple keys, one for each user in a set of users, is considered with possible assist from a group of dedicated helpers. For the case in which all users are required to generate keys, we develop a scheme that is sum rate optimal. For the case with dedicated helpers, we develop an achievable scheme and derive an outer bound. We identify conditions under which the developed scheme achieves the full capacity region and conditions under which it is sum rate optimal. We then specialize the study to a pairwise independent network model, for which we convert the key generation problem to a single-source multi-commodity flow over a network problem. Coupling results from graph theory, we fully characterize the capacity region for the general case of generating multiple keys with multiple helpers under the PIN model.