Secret Sharing Schemes from Correlated Random Variables and Rate-Limited Public Communication

TL;DR

This work proposes the first explicit coding scheme able to handle arbitrary access structures and achieve the best known achievable rates, previously obtained non-constructively for secret-key generation under one-way, rate-limited public communication.

Abstract

A dealer aims to share a secret with participants so that only predefined subsets can reconstruct it, while others learn nothing. The dealer and participants access correlated randomness and communicate over a one-way, public, rate-limited channel. For this problem, we propose the first explicit coding scheme able to handle arbitrary access structures and achieve the best known achievable rates, previously obtained non-constructively. Our construction relies on lossy source coding coupled with distribution approximation to handle the reliability constraints, followed by universal hashing to handle the security constraints. We stress that our coding scheme does not require symmetry or degradation assumptions on the correlated random variables, and does not need a pre-shared secret among the participants and dealer. As a by-product, our construction also yields explicit coding schemes for secret-key generation under one-way, rate-limited public communication that, unlike prior work, achieves the capacity for arbitrary source correlations and do not require a pre-shared secret to ensure strong secrecy.

Figures (1)

• Figure 1: Secret sharing from correlated randomness and rate-limited public communication with $J=3$, $\mathbb{A } \triangleq \{\{1, 2\}, \{2, 3\}, \{1, 2, 3\}\}$, $\mathbb{U}\triangleq 2^{[J]}\backslash \mathbb{A } = \{\{1, 3\}, \{1\}, \{2\}, \{3\}\}$.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Theorem 1
• Theorem 2
• Corollary 1
• proof
• Theorem 3: csiszar2000common
• Theorem 4: csiszar2000common
• Definition 3
• Theorem 5