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Secret Sharing and Statistical Zero Knowledge

  • Vinod Vaikuntanathan
  • Prashant Nalini Vasudevan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9452)

Abstract

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 \({\mathbf {P}} \subseteq {\mathbf {DSPACE}}(q(n))\).

Keywords

Statistical zero knowledge Secret sharing 

Notes

Acknowledgments

We thank an anonymous ASIACRYPT reviewer for comments that helped improve the presentation of this paper.

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Copyright information

© International Association for Cryptologc Research 2015

Authors and Affiliations

  • Vinod Vaikuntanathan
    • 1
  • Prashant Nalini Vasudevan
    • 1
  1. 1.MIT CSAILCambridgeUSA

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