Efficient Robust Secret Sharing from Expander Graphs

Threshold secret sharing is a protocol that allows a dealer to share a secret among $n$ players so that any coalition of $t$ players learns nothing about the secret, but any $t+1$ players can reconstruct the secret in its entirety. Robust secret sharing (RSS) provides the additional guarantee that even if $t$ malicious players mangle their shares, they cannot cause the honest players to reconstruct an incorrect secret. When $t \frac{n}{2}$, RSS is known to be impossible, but for $\frac{n}{3} < t < \frac{n}{2}$ much less is known. When $\frac{n}{3} < t < \frac{n}{2}$ previous RSS protocols could either achieve optimal share size with inefficient (exponential time) reconstruction procedures, or sub-optimal share size with polynomial time reconstruction. In this work, we construct a simple RSS protocol for $t = \left( \frac{1}{2} - \epsilon\right)n$ that achieves logarithmic overhead in terms of share size and simultaneously allows efficient reconstruction. Our shares size increases by an additive term of $O(\kappa + \log n)$, and reconstruction succeeds except with probability at most $2^{-\kappa}$. This provides a partial solution to a problem posed by Cevallos et al. in Eurocrypt 2012. Namely, when $t = \left( \frac{1}{2} - O(1) \right)n$ we show that the share size in RSS schemes do not require an overhead that is linear in $n$. Previous efficient RSS protocols like that of Rabin and Ben-Or (STOC \u2789) and Cevallos et al. (Eurocrypt \u2712) use MACs to allow each player to check the shares of each other player in the protocol. These checks provide robustness, but require significant overhead in share size. Our construction identifies the $n$ players as nodes in an expander graph, each player only checks its neighbors in the expander graph. When $t = \left\{ \frac{1}{2} - O(1) \right\}n$, the concurrent, independent work of Cramer et al. (Eurocrypt \u2715) shows how to achieve shares that \emph{decrease} with the number of players using completely different techniques