The Capacity Region of Distributed Multi-User Secret Sharing Under Perfect Secrecy

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.