Generating secret in a network

Abstract

This monograph studies the theory of information through the multiuser secret key agreement problem. A general notion of mutual dependence is established for the secrecy capacity, as a natural generalization of Shannon's mutual information to the multivariate case. Under linear-type source models, this capacity can be achieved practically by linear network codes. In addition to being an unusual application of the network coding solution to a secrecy problem, it gives secrecy capacity an interpretation of network information flow and partition connectivity, further confirming the intuitive meaning of secrecy capacity as mutual dependence. New identities in submodular function optimization and matroid theory are discovered in proving these results. A framework is also developed to view matroids as graphs, allowing certain theory on graphs to generalize to matroids. In order to study cooperation schemes in a network, a general channel model with multiple inputs is formulated. Single-letter secrecy capacity upper bounds are derived using the Shearer-type lemma. Lower bounds are obtained with a new cooperation scheme called the mixed source emulation. In the same way that mixed strategies may surpass pure strategies in zero-sum games, mixed source emulation outperforms the conventional pure source emulation approach in terms of the achievable key rate. Necessary and sufficient conditions are derived for tightness of these secrecy bounds, which shows that secrecy capacity can be characterized for a larger class of channels than the broadcast-type channels considered in previous work. The mixed source emulation scheme is also shown to be unnecessary for some channels while insufficient for others. The possibility of a better cooperative scheme becomes apparent, but a general scheme remains to be found.