Threshold Secret Sharing Requires a Linear-Size Alphabet

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.