Simultaneously generating multiple keys and multi-commodity flow in networks

The problem of simultaneously generating multiple independent keys for multiple pairs of users is considered. This problem is motivated by the fact that typically in wireless networks, multiple pairs of users need to establish secret keys for secure communications between these pairs. We propose a secure routing based key distribution approach to establish keys for the terminals. This approach connects the problem at the hand to that of multi-commodity flow problem studied in graph theory. Using the Max Bi-Flow Min Cut Theorem in the graph theory and developing a matching outer-bound, we show that the proposed approach achieves the key capacity region for the case of establishing two keys. For the general case of establishing more than two keys, an upper bound on the achievable sum rate is derived based on the concept of multicut and our proposed approach can achieve a sum rate equals to the upper bound divided by a constant factor.