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Towards Breaking the Exponential Barrier for General Secret Sharing

  • Tianren Liu
  • Vinod Vaikuntanathan
  • Hoeteck Wee
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10820)

Abstract

A secret-sharing scheme for a monotone Boolean (access) function \(F: \{0,1\}^n \rightarrow \{0,1\}\) is a randomized algorithm that on input a secret, outputs n shares \(s_1,\ldots ,s_n\) such that for any \((x_1,\ldots ,x_n) \in \{0,1\}^n\), the collection of shares \( \{ s_i : x_i = 1 \}\) determine the secret if \(F(x_1,\ldots ,x_n)=1\) and reveal nothing about the secret otherwise. The best secret sharing schemes for general monotone functions have shares of size \(\varTheta (2^n)\). It has long been conjectured that one cannot do much better than \(2^{\varOmega (n)}\) share size, and indeed, such a lower bound is known for the restricted class of linear secret-sharing schemes.

In this work, we refute two natural strengthenings of the above conjecture:
  • First, we present secret-sharing schemes for a family of \(2^{2^{n/2}}\) monotone functions over \(\{0,1\}^n\) with sub-exponential share size \(2^{O(\sqrt{n} \log n)}\). This unconditionally refutes the stronger conjecture that circuit size is, within polynomial factors, a lower bound on the share size.

  • Second, we disprove the analogous conjecture for non-monotone functions. Namely, we present “non-monotone secret-sharing schemes” for every access function over \(\{0,1\}^n\) with shares of size \(2^{O(\sqrt{n} \log n)}\).

Our construction draws upon a rich interplay amongst old and new problems in information-theoretic cryptography: from secret-sharing, to multi-party computation, to private information retrieval. Along the way, we also construct the first multi-party conditional disclosure of secrets (CDS) protocols for general functions \(F:\{0,1\}^n \rightarrow \{0,1\}\) with communication complexity \(2^{O(\sqrt{n} \log n)}\).

Notes

Acknowledgments

We thank Yuval Ishai for telling us about Conjecture 1. We thank the anonymous EUROCRYPT 2018 reviewers for their insightful comments.

Supplementary material

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Copyright information

© International Association for Cryptologic Research 2018

Authors and Affiliations

  • Tianren Liu
    • 1
  • Vinod Vaikuntanathan
    • 1
  • Hoeteck Wee
    • 2
  1. 1.MITCambridgeUSA
  2. 2.CNRS and ENSParisFrance

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