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Verifiable secret sharing (VSS) allows honest parties to ensure consistency of their shares even if a dealer and/or a subset of parties are corrupt. We focus on perfect VSS, i.e., those providing perfect privacy, correctness and commitment with zero error, in the unconditional (information-theoretic) security setting where no assumption on the computational power of the participants is imposed. Our study is motivated by both practical and theoretical considerations. For the practical side, MPC with perfect security is now being implemented. Multi-cloud storage has been implemented by IBM. Modern users rely on smartphones with limited internet connectivity, limited battery power, etc. We focus on such a user outsourcing her data to multi-clouds with the capability to have these multi-clouds participate on her behalf in MPC protocols. We show that in the case of VSS based on the replicated secret sharing scheme, there is no need for that user to be involved in any interaction. In addition, this scheme has an optimal number of rounds. This construction is derived from Maurer's VSS based on the replicated secret sharing scheme. A disadvantage of the replicated scheme is that it generally requires a considerable amount of randomness. We address this issue by showing a VSS scheme based on Shamir's secret sharing, where the dealer does not need any randomness at all.
Keyword:
Secret sharing
Verifiable secret sharing
Round complexity
Randomness complexity
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