Secret-sharing schemes for general and uniform access structures

Benny Applebaum, Amos Beimel, Oriol Farràs, Oded Nir, Naty Peter

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

11 Scopus citations

Abstract

A secret-sharing scheme allows some authorized sets of parties to reconstruct a secret; the collection of authorized sets is called the access structure. For over 30 years, it was known that any (monotone) collection of authorized sets can be realized by a secret-sharing scheme whose shares are of size 2n-o(n) and until recently no better scheme was known. In a recent breakthrough, Liu and Vaikuntanathan (STOC 2018) have reduced the share size to O(20.994n. Our first contribution is improving the exponent of secret sharing down to 0.892. For the special case of linear secret-sharing schemes, we get an exponent of 0.942 (compared to 0.999 of Liu and Vaikuntanathan). Motivated by the construction of Liu and Vaikuntanathan, we study secret-sharing schemes for uniform access structures. An access structure is k-uniform if all sets of size larger than k are authorized, all sets of size smaller than k are unauthorized, and each set of size k can be either authorized or unauthorized. The construction of Liu and Vaikuntanathan starts from protocols for conditional disclosure of secrets, constructs secret-sharing schemes for uniform access structures from them, and combines these schemes in order to obtain secret-sharing schemes for general access structures. Our second contribution in this paper is constructions of secret-sharing schemes for uniform access structures. We achieve the following results: A secret-sharing scheme for k-uniform access structures for large secrets in which the share size is O(k2)times the size of the secret.A linear secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is (Formula presented) (where h is the binary entropy function). By counting arguments, this construction is optimal (up to polynomial factors).A secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is (Formula presented). Our third contribution is a construction of ad-hoc PSM protocols, i.e., PSM protocols in which only a subset of the parties will compute a function on their inputs. This result is based on ideas we used in the construction of secret-sharing schemes for k-uniform access structures for a binary secret.

Original languageEnglish
Title of host publicationAdvances in Cryptology – EUROCRYPT 2019 - 38th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings
EditorsYuval Ishai, Vincent Rijmen
PublisherSpringer Verlag
Pages441-471
Number of pages31
ISBN (Print)9783030176587
DOIs
StatePublished - 1 Jan 2019
Event38th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Eurocrypt 2019 - Darmstadt, Germany
Duration: 19 May 201923 May 2019

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11478 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference38th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Eurocrypt 2019
Country/TerritoryGermany
CityDarmstadt
Period19/05/1923/05/19

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