  We consider a secret-sharing model 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. Our study generalizes Shamir's secret-sharing model in three
directions. First, we allow a joint design of the protocols for the creation of
the shares and the distribution of the shares, instead of constraining the
model to independent designs. Second, instead of assuming that the participants
and the dealer have access to information-theoretically secure channels at no
cost, we assume that they have access to a public channel and correlated
randomness. Third, motivated by a wireless network setting where the correlated
randomness is obtained from channel gain measurements, we explore a setting
where the dealer is an entity made of multiple sub-dealers. Our main results
are inner and outer regions for the achievable secret rates that the dealer and
the participants can obtain in this model. To this end, we develop two new
achievability techniques, a first one to successively handle reliability and
security constraints in a distributed setting, and a second one to reduce a
multi-dealer setting to multiple single-user dealer settings. Our results yield
the capacity region for threshold access structures when the correlated
randomness corresponds to pairwise secret keys shared between each sub-dealer
and each participant, and the capacity for the all-or-nothing access structure
in the presence of a single dealer and arbitrarily correlated randomness.
