Secret key generation for a pairwise independent network model

We investigate secret key generation for a ldquopair-wise independent networkrdquo model in which every pair of terminals observes correlated sources which are independent of sources observed by all other pairs of terminals. The terminals are then allowed to communicate interactively in multiple rounds over a public noiseless channel of unlimited capacity, with all such communication being observed by all the terminals. The objective is to generate a secret key shared by a given subset of terminals at the largest rate possible. All the terminals cooperate in generating the secret key, with secrecy being required from an eavesdropper which has access to the public interterminal communication. We provide a (single-letter) formula for the secrecy capacity for this model, and show a natural connection between the problem of secret key generation and the combinatorial problem of maximal packing of Steiner trees in an associated multigraph. In particular, we show that the maximum number of Steiner tree packings in the multigraph is always a lower bound for the secrecy capacity. The bound is tight for the case when all the terminals seek to share a secret key; the mentioned connection yields an explicit capacity-achieving algorithm. This algorithm, which can be executed in polynomial time, extracts a group-wide secret key of the optimum rate from the collection of optimum and mutually independent secret keys for pairs of terminals.