The Capacity Region of Distributed Multi-User Secret Sharing under The Perfect Privacy Condition
The Capacity Region of Distributed Multi-User Secret Sharing under The Perfect Privacy Condition
复制标题
完美隐私条件下分布式多用户秘密共享的容量域
DOI：
--
复制
发表时间：
2023
期刊：
影响因子：
--
通讯作者：
Wei Kang
中科院分区：
文献类型：
--
作者：
Jia;Nan Liu;Wei Kang
研究了完全隐私条件下的分布式多用户秘密共享问题。在DMUSS问题中，部署了多个秘密消息，并将共享分流到存储节点。此外，由于每个秘密消息的解码集合只有一个集合，并且在完全隐私条件下，这种集合也是所有其他秘密消息的合谋集合，因此访问结构是极其不完整的。秘密消息速率被定义为由份额大小归一化的秘密消息的大小。当给定访问结构时，我们刻画了DMUSS问题的容量区域，该访问结构被定义为所有可达到的速率元组的集合。在可实现方案中，我们假设所有份额是相互独立的，然后根据每个秘密消息的译码集合只有一个集合的事实来设计译码函数。然后证明了完全隐私条件等价于一些由不同的不定和零点组成的矩阵的满秩性。如果字段大小大于秘密消息的数量，则确实存在这样的解决方案。最后，利用非合谋份额的大小之和表示秘密的大小是上界的匹配逆，刻画了完全保密条件下DMUSS问题的容量域。
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.