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TR16-131 | 21st August 2016 18:18

Threshold Secret Sharing Requires a Linear Size Alphabet


Authors: Andrej Bogdanov, Siyao Guo, Ilan Komargodski
Publication: 21st August 2016 22:44
Downloads: 3464


We prove that for every $n$ and $1 < t < n$ any $t$-out-of-$n$ threshold secret sharing scheme for one-bit secrets requires share size $\log(t + 1)$. Our bound is tight when $t = n - 1$ and $n$ is a prime power. In 1990 Kilian and Nisan proved the incomparable bound $\log(n - t + 2)$. Taken together, the two bounds imply that the share size of Shamir's secret sharing scheme (Comm. ACM '79) is optimal up to an additive constant even for one-bit secrets for the whole range of parameters $1 < t < n$.
More generally, we show that for all $1 < s < r < n$, any ramp secret sharing scheme with secrecy threshold $s$ and reconstruction threshold $r$ requires share size $\log((r + 1)/(r - s))$.
As part of our analysis we formulate a simple game-theoretic relaxation of secret sharing for arbitrary access structures. We prove the optimality of our analysis for threshold secret sharing with respect to this method and point out a general limitation.

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