Secret Sharing over a Public Channel from Correlated Random Variables

We consider a model for secret sharing, where a dealer distributes the shares of a secret among a set of participants with the constraint that only predetermined subsets of participants must be able to reconstruct the secret by pooling their shares. Unlike traditional secret sharing models, no secure channels are available between the dealer and the participants. Additionally, we assume that the dealer is an abstract entity made of several sub-dealers able to communicate with the participants through a noiseless and authenticated public channel. In this setting, we assume that the sub-dealers and the participants observe realizations of independently and identically distributed random variables. Our main results are inner and outer bounds on the rate of the secret that the dealer can distribute to the participants via its sub-dealers.