Information-Theoretic Key Agreement of Multiple Terminals - Part I: Source Model

This is the first part of a two-part paper on information-theoretically secure secret key agreement. In this part, we study the secrecy problem under the widely studied source model. In the source model the terminals wishing to generate a secret key, as well as the eavesdropper, receive the respective coordinates of a block of independent and identically distributed copies of jointly distributed random variables, after which the terminals are allowed interactive authenticated public communication, at the end of which each terminal should be able to generate the key. We derive a new upper bound on the secrecy capacity that strictly improves the currently best upper bound, due to Renner and Wolf. Further, while the RennerWolf bound is defined only in the case of two terminals, the new upper bound applies to the general multi-terminal case. The technique used for deriving our bound is to find certain properties of functions of joint probability distributions which will imply that they dominate the secrecy capacity, and then prove the bound by a verification argument. We also define a problem of communication for omniscience by a neutral observer and establish the equivalence between this new problem and the problem of secret key agreement. This generalizes an earlier result of Csiszár and Narayan. Finally, we prove a new lower bound on the secrecy capacity in the general multi-terminal case that in the two terminal case is strictly better than what is essentially the currently best known lower bound, namely the maximum of the two one-way secrecy capacities.