The Capacity Region of Distributed Multi-User Secret Sharing under The Perfect Privacy Condition

TL;DR

This paper characterizes the maximum achievable secret sharing rates in a distributed multi-user setting under perfect privacy, using a novel approach based on matrix rank conditions and independence assumptions.

Contribution

It provides the first complete capacity region characterization for DMUSS under perfect privacy, introducing a new decoding scheme and matrix rank analysis.

Findings

Capacity region is characterized by full rank matrix conditions.

Achievable scheme assumes mutually independent shares.

Capacity bound matches the sum of non-colluding shares.

Abstract

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.