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：我们考虑一个分布式多用户秘密共享（DMUSS）场景，其中包含一个分发者、$n$个存储节点和$m$个秘密。每位用户需获取$m$个秘密中的$t$个子集。此前该场景的相关工作仅针对$t=1$的情况，而本文则研究一般化$t$的情形。用户根据设计的访问结构从存储节点下载份额并重构其秘密。我们识别了保证弱安全性所需的访问结构必要条件，并建立了该问题中访问结构与$t$-分离矩阵之间的关联。我们在此场景中应用多种$t$-分离矩阵构造方法，从存储节点数量和通信复杂度角度比较其性能，同时推导了分布式秘密共享协议最优通信复杂度的界。最后，我们刻画了在指定访问结构时DMUSS问题的容量区域。