  We consider a distributed secret sharing system that consists of a dealer,
$n$ storage nodes, and $m$ users. Each user is given access to a certain subset
of storage nodes, where it can download the stored data. The dealer wants to
securely convey a specific secret $s_j$ to user $j$ via storage nodes, for
$j=1,2,...,m$. More specifically, two secrecy conditions are considered in this
multi-user context. The weak secrecy condition is that each user does not get
any information about the individual secrets of other users, while the perfect
secrecy condition implies that a user does not get any information about the
collection of all other users' secrets. In this system, the dealer encodes
secrets into several secret shares and loads them into the storage nodes. Given
a certain number of storage nodes we find the maximum number of users that can
be served in such a system and construct schemes that achieve this with perfect
secrecy. Lower bounds on the minimum communication complexity and the storage
overhead are characterized given any $n$ and $m$. We construct distributed
secret sharing protocols, under certain conditions on the system parameters,
that attain the lower bound on the communication complexity while providing
perfect secrecy. Furthermore, we construct protocols, again under certain
conditions, that simultaneously attain the lower bounds on the communication
complexity and the storage overhead while providing weak secrecy, thereby
demonstrating schemes that are optimal in terms of both the parameters. It is
shown how to modify the proposed protocols in order to construct schemes with
balanced storage load and communication complexity.
