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Secret Sharing and Statistical Zero Knowledge
Authors: Vinod Vaikuntanathan, Prashant Nalini VasudevanAbstract:
We show a general connection between various types of statistical zero-knowledge (SZK) proof systems and (unconditionally secure) secret sharing schemes. Viewed through the SZK lens, we obtain several new results on secret-sharing: Characterizations: We obtain an almost-characterization of access structures for which there are secret-sharing schemes with an efficient sharing algorithm (but not necessarily efficient reconstruction). In particular, we show that for every language L \in \SZKL (the class of languages that have statistical zero knowledge proofs with log-space verifiers and simulators), a (monotonized) access structure associated with L has such a secret-sharing scheme. Conversely, we show that such secret-sharing schemes can only exist for languages in \SZK. Constructions: We show new constructions of secret-sharing schemes with efficient sharing and reconstruction for access structures that are in \P, but are not known to be in \NC, namely Bounded-Degree Graph Isomorphism and constant-dimensional lattice problems. In particular, this gives us the first combinatorial access structure that is conjectured to be outside \NC but has an efficient secret-sharing scheme. Previous such constructions (Beimel and Ishai; CCC 2001) were algebraic and number-theoretic in nature. Limitations: We show that universally-efficient secret-sharing schemes, where the complexity of computing the shares is a polynomial independent of the complexity of deciding the access structure, cannot exist for all (monotone languages in) \P, unless there is a polynomial q such that \P \subseteq \DSPACE(q(n)).
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