Random Robust Secret Sharing with Perfect Privacy and its Applications

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Secret Sharing schemes allow a dealer to distribute $n$ shares $s_1,\ldots, s_n$ of a secret $s$ so that any $t$ shares suffice to reconstruct the secret, while any $t-1$ shares reveal no information about $s$. In fact, schemes such as Shamir Secret Sharing satisfy a stronger guarantee called $(t\!-\!1)$-perfect privacy, meaning that for any subset $S \subseteq [n]$ with $|S| \le t-1$, the joint distribution $(s_i)_{i \in S}$ is uniformly distributed over its domain. This strong guarantee is es