We consider a distributed multi-user secret sharing (DMUSS) setting in which there 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 from the storage nodes based on the designed access structure and reconstructs its secrets. We identify a necessary condition on the access structures to ensure weak secrecy. We also make a connection between access structures for this problem and $t$-disjunct matrices. We apply various $t$-disjunct matrix constructions in this setting and compare their performance in terms of the number of storage nodes and communication complexity. We also derive bounds on the optimal communication complexity of a distributed secret sharing protocol. Finally, we characterize the capacity region of the DMUSS problem when the access structure is specified.

翻译：我们考虑一种分布式多用户秘密共享（DMUSS）场景，其中存在一个分发者、$n$个存储节点和$m$个秘密。每个用户需要$m$个秘密中的一个$t$子集。此前对该场景的研究仅针对$t=1$的情况；本文中将考虑一般$t$的情况。用户根据设计好的访问结构从存储节点下载份额，并重构其所需秘密。我们识别出访问结构需满足的必要条件以确保弱安全性，并建立了该问题中访问结构与$t$-析取矩阵之间的关联。在该场景下，我们应用多种$t$-析取矩阵构造方法，并从存储节点数量和通信复杂度两方面比较其性能。此外，我们推导了分布式秘密共享协议最优通信复杂度的界限。最后，在指定访问结构的情况下，我们刻画了DMUSS问题的容量区域。