Common Information and Secret Key Capacity

Abstract: We study the generation of a secret key of maximum rate by a pair of terminals observing correlated sources and with the means to communicate over a noiseless public com- munication channel. Our main result establishes a structural equivalence between the generation of a maximum rate secret key and the generation of a common randomness that renders the observations of the two terminals conditionally independent. The minimum rate of such common randomness, termed interactive common information, is related to Wyner's notion of common information, and serves to characterize the minimum rate of interactive public communication required to generate an optimum rate secret key. This characterization yields a single-letter expression for the aforementioned communication rate when the number of rounds of interaction are bounded. An application of our results shows that interaction does not reduce this rate for binary symmetric sources. Further, we provide an example for which interaction does reduce the minimum rate of communication. Also, certain invariance properties of common information quantities are established that may be of independent interest.