@inproceedings{6e9e939fa4dc44a9b470623f612a5477,
title = "Threshold secret sharing requires a linear size alphabet",
abstract = "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{\textquoteright}s secret sharing scheme (Comm. ACM {\textquoteright}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.",
author = "Andrej Bogdanov and Siyao Guo and Ilan Komargodski",
note = "Publisher Copyright: {\textcopyright} International Association for Cryptologic Research 2016.; 14th International Conference on Theory of Cryptography, TCC 2016-B ; Conference date: 31-10-2016 Through 03-11-2016",
year = "2016",
doi = "10.1007/978-3-662-53644-5_18",
language = "الإنجليزيّة",
isbn = "9783662536438",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer Verlag",
pages = "471--484",
editor = "Adam Smith and Martin Hirt",
booktitle = "Theory of Cryptography - 14th International Conference, TCC 2016-B, Proceedings",
address = "ألمانيا",
}