Paper 2018/333

Breaking the Circuit-Size Barrier in Secret Sharing

Tianren Liu and Vinod Vaikuntanathan


We study secret sharing schemes for general (non-threshold) access structures. A general secret sharing scheme for $n$ parties is associated to a monotone function $\mathsf F:\{0,1\}^n\to\{0,1\}$. In such a scheme, a dealer distributes shares of a secret $s$ among $n$ parties. Any subset of parties $T \subseteq [n]$ should be able to put together their shares and reconstruct the secret $s$ if $\mathsf F(T)=1$, and should have no information about $s$ if $\mathsf F(T)=0$. One of the major long-standing questions in information-theoretic cryptography is to minimize the (total) size of the shares in a secret-sharing scheme for arbitrary monotone functions $\mathsf F$. There is a large gap between lower and upper bounds for secret sharing. The best known scheme for general $\mathsf F$ has shares of size $2^{n-o(n)}$, but the best lower bound is $\Omega(n^2/\log n)$. Indeed, the exponential share size is a direct result of the fact that in all known secret-sharing schemes, the share size grows with the size of a circuit (or formula, or monotone span program) for $\mathsf F$. Indeed, several researchers have suggested the existence of a {\em representation size barrier} which implies that the right answer is closer to the upper bound, namely, $2^{n-o(n)}$. In this work, we overcome this barrier by constructing a secret sharing scheme for any access structure with shares of size $2^{0.994n}$ and a linear secret sharing scheme for any access structure with shares of size $2^{0.999n}$. As a contribution of independent interest, we also construct a secret sharing scheme with shares of size $2^{\tilde{O}(\sqrt{n})}$ for $2^{{n\choose n/2}}$ monotone access structures, out of a total of $2^{{n\choose n/2}\cdot (1+O(\log n/n))}$ of them. Our construction builds on recent works that construct better protocols for the conditional disclosure of secrets (CDS) problem.

Available format(s)
Publication info
Published elsewhere. 50th Annual ACM SIGACT Symposium on the Theory of Computing
secret sharing
Contact author(s)
liutr @ mit edu
2018-04-11: received
Short URL
Creative Commons Attribution


      author = {Tianren Liu and Vinod Vaikuntanathan},
      title = {Breaking the Circuit-Size Barrier in Secret Sharing},
      howpublished = {Cryptology ePrint Archive, Paper 2018/333},
      year = {2018},
      doi = {10.1145/3188745.3188936},
      note = {\url{}},
      url = {}
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