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We consider a distributed multi-user secret sharing (DMUSS) setting in whichthere is a dealer, $n$ storage nodes, and $m$ secrets. Each user demands a$t$-subset of $m$ secrets. Earlier work in this setting dealt with the case of$t=1$; in this work, we consider general $t$. The user downloads shares fromthe storage nodes based on the designed access structure and reconstructs itssecrets. We identify a necessary condition on the access structures to ensureweak secrecy. We also make a connection between access structures for thisproblem and $t$-disjunct matrices. We apply various $t$-disjunct matrixconstructions in this setting and compare their performance in terms of thenumber of storage nodes and communication complexity. We also derive bounds onthe optimal communication complexity of a distributed secret sharing protocol.Finally, we characterize the capacity region of the DMUSS problem when theaccess structure is specified.

On Distributed Multi-User Secret Sharing with Multiple Secrets per User