Secret Sharing and Statistical Zero Knowledge

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 \(L \in {{\mathbf {SZK}}_{\mathbf {L}}}\) (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 \({\mathbf {SZK}}\). Constructions: We show new constructions of secret-sharing schemes with both efficient sharing and efficient reconstruction for access structures associated with languages that are in \({\mathbf {P}}\), but are not known to be in \({\mathbf {NC}}\), 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 \({\mathbf {NC}}\) but has an efficient secret-sharing scheme. Previous such constructions (Beimel and Ishai; CCC 2001) were algebraic and number-theoretic in nature. Limitations: We also 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) \(\mathbf {P}\), unless there is a polynomial q such that \({\mathbf {P}} \subseteq {\mathbf {DSPACE}}(q(n))\).