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Erschienen in:
2018 | OriginalPaper | Buchkapitel
Towards Breaking the Exponential Barrier for General Secret Sharing
verfasst von : Tianren Liu, Vinod Vaikuntanathan, Hoeteck Wee
Erschienen in: Advances in Cryptology – EUROCRYPT 2018
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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:
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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.
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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 \(2^{O(\sqrt{n} \log n)}\).