
Revised: May 3, 2019
Published: September 7, 2020
Abstract: [Plain Text Version]
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 1979) 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.