We study the problem of distributed multi-user secret sharing (DMUSS), involving a main node, N storage nodes, and K users. Every user has access to the contents of a certain subset of storage nodes and wants to decode an independent secret message. With knowledge of K secret messages, the main node strategically places encoded shares in the storage nodes, ensuring two crucial conditions: (i) each user can recover its own secret message from the storage nodes that it has access to; (ii) each user is unable to acquire any information regarding the collection of $K-1$ secret messages for all the other users. The rate of each user is defined as the size of its secret message normalized by the size of a storage node. We characterize the capacity region of the DMUSS problem, which is the closure of the set of all achievable rate tuples that satisfy the correctness and perfect secrecy conditions. The converse proof relies on a bound from the traditional single-secret sharing regime. In the achievability proof, we firstly design the linear decoding functions, based on the fact that each secret message needs to be recovered from a single set of storage nodes. It turns out that the perfect secrecy condition holds if K matrices, whose entries are extracted from the decoding functions, are full rank. We prove that the decoding functions can be constructed explicitly if the rate tuple satisfies the converse and the field size is not less than K. At last, the encoding functions are obtained by solving the system of linear decoding functions, where some shares are equal to the randomness and the other shares are linear combinations of the secret messages and the randomness.

中文翻译：

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完全保密下的分布式多用户 Secret 共享的容量区域


我们研究了分布式多用户密钥共享 （DMUSS） 问题，涉及一个主节点、N 个存储节点和 K 个用户。每个用户都可以访问某个存储节点子集的内容，并希望解码独立的秘密消息。在知道 K 个秘密消息后，主节点战略性地将编码的共享放置在存储节点中，确保两个关键条件：（i） 每个用户都可以从其有权访问的存储节点恢复自己的秘密消息;（ii） 每个用户都无法获得有关所有其他用户收集 $K-1 美元秘密消息的任何信息。每个用户的速率定义为其秘密消息的大小，由存储节点的大小进行标准化。我们描述了 DMUSS 问题的容量区域，即满足正确性和完全保密条件的所有可实现的速率元组的集合的闭合。相反的证明依赖于传统单一秘密共享制度的束缚。在可实现性证明中，我们首先设计了线性解码函数，基于每个秘密消息都需要从一组存储节点中恢复的事实。事实证明，如果 K 矩阵（其条目是从解码函数中提取的）是满秩的，则完美保密条件成立。我们证明，如果速率元组满足相反的情况并且字段大小不小于 K，则可以显式构造解码函数。最后，通过求解线性解码函数系统得到编码函数，其中一些份额等于随机性，其他份额是秘密消息和随机性的线性组合。