Threshold Secret Sharing Requires a Linear-Size Alphabet
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AbstractWe 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.
All Author(s) ListAndrej Bogdanov, Siyao Guo, Ilan Komargodski
Journal nameTheory of Computing
Year2020
Month9
Day7
Volume Number16
PublisherUniversity of Chicago, Department of Computer Science
Article number2
Pages1 - 18
ISSN1557-2862
LanguagesEnglish-United States

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