The Capacity Region of Distributed Multi-User Secret Sharing

TL;DR

This paper characterizes the capacity region for distributed multi-user secret sharing, providing a theoretical framework for encoding secrets with privacy constraints in a networked storage system.

Contribution

It introduces the first complete characterization of the capacity region for distributed multi-user secret sharing with privacy and correctness constraints.

Findings

Derived the capacity region for the secret sharing model.

Proposed an achievable scheme using polynomial-based encoding.

Established bounds matching the achievable scheme.

Abstract

In this paper, we study the problem of distributed multi-user secret sharing, including a trusted master node, $N\in \mathbb{N}$ storage nodes, and $K$ users, where each user has access to the contents of a subset of storage nodes. Each user has an independent secret message with certain rate, defined as the size of the message normalized by the size of a storage node. Having access to the secret messages, the trusted master node places encoded shares in the storage nodes, such that (i) each user can recover its own message from the content of the storage nodes that it has access to, (ii) each user cannot gain any information about the message of any other user. We characterize the capacity region of the distributed multi-user secret sharing, defined as the set of all achievable rate tuples, subject to the correctness and privacy constraints. In the achievable scheme, for each user, the master node forms a polynomial with the degree equal to the number of its accessible storage nodes minus one, where the value of this polynomial at certain points are stored as the encoded shares. The message of that user is embedded in some of the coefficients of the polynomial. The remaining coefficients are determined such that the content of each storage node serves as the encoded shares for all users that have access to that storage node.