Low-Complexity Secret Sharing Schemes Using Correlated Random Variables and Rate-Limited Public Communication

Abstract

We consider secret sharing where a dealer wants to share a secret with several participants such that predefined subsets of participants can reconstruct the secret and all other subsets of participants cannot learn any information about the secret. To this end, the dealer and the participants have access to samples of correlated random variables and a one-way (from the dealer to the participants), authenticated, public, and rate-limited communication channel. For this problem, we propose the first constructive and low-complexity coding scheme able to handle arbitrary access structures. Our construction relies on a vector quantization coupled with distribution approximations with polar codes 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. Our result is also optimal in the special case of rate-unlimited public communication when all the participants are needed to reconstruct the secret.