We study the distributed multi-user secret sharing (DMUSS) problem under the perfect privacy condition. In a DMUSS problem, multiple secret messages are deployed and the shares are offloaded to the storage nodes. Moreover, the access structure is extremely incomplete, as the decoding collection of each secret message has only one set, and by the perfect privacy condition such collection is also the colluding collection of all other secret messages. The secret message rate is defined as the size of the secret message normalized by the size of a share. We characterize the capacity region of the DMUSS problem when given an access structure, defined as the set of all achievable rate tuples. In the achievable scheme, we assume all shares are mutually independent and then design the decoding function based on the fact that the decoding collection of each secret message has only one set. Then it turns out that the perfect privacy condition is equivalent to the full rank property of some matrices consisting of different indeterminates and zeros. Such a solution does exist if the field size is bigger than the number of secret messages. Finally with a matching converse saying that the size of the secret is upper bounded by the sum of sizes of non-colluding shares, we characterize the capacity region of DMUSS problem under the perfect privacy condition.

翻译：我们研究了完美隐私条件下的分布式多用户秘密共享问题。在该问题中，多个秘密消息被部署，且份额被卸载到存储节点。此外，访问结构极不完整，每个秘密消息的解码集合仅包含一个集合，且根据完美隐私条件，该集合也是所有其他秘密消息的串谋集合。秘密消息速率定义为秘密消息大小与份额大小之比。我们刻画了给定访问结构时分布式多用户秘密共享问题的容量区域，即所有可达速率元组的集合。在可达方案中，我们假设所有份额相互独立，然后基于每个秘密消息的解码集合仅包含一个集合这一事实设计解码函数。进而发现完美隐私条件等价于由不同不定元与零组成的某些矩阵的满秩性质。当域的大小超过秘密消息数量时，这样的解确实存在。最后，通过匹配的逆命题（即秘密大小受非串谋份额大小之和的上界约束），我们刻画了完美隐私条件下分布式多用户秘密共享问题的容量区域。