Paper 2019/231

Secret-Sharing Schemes for General and Uniform Access Structures

Benny Applebaum
Amos Beimel
Oriol Farràs
Oded Nir
Naty Peter
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 and until recently no better scheme was known. In a recent breakthrough, Liu and Vaikuntanathan (STOC 2018) have reduced the share size to . Our first contribution is improving the exponent of secret sharing down to . For the special case of linear secret-sharing schemes, we get an exponent of (compared to 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 -uniform if all sets of size larger than are authorized, all sets of size smaller than are unauthorized, and each set of size 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) A secret-sharing scheme for -uniform access structures for large secrets in which the share size is times the size of the secret. b) A linear secret-sharing scheme for -uniform access structures for a binary secret in which the share size is (where is the binary entropy function). By counting arguments, this construction is optimal (up to polynomial factors). c) A secret-sharing scheme for -uniform access structures for a binary secret in which the share size is . 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 -uniform access structures for a binary secret.

Note: Full version of the EUROCRYPT 2019 paper.

Metadata
Available format(s)
PDF
Category
Foundations
Publication info
A minor revision of an IACR publication in EUROCRYPT 2019
DOI
10.1007/978-3-030-17659-4_15
Keywords
secret sharingconditional disclosure of secrets protocolsprivate simultaneous messages protocols
Contact author(s)
benny applebaum @ gmail com
amos beimel @ gmail com
oriol farras @ urv cat
odednir123 @ gmail com
natypeter @ mail tau ac il
History
2023-10-06: last of 2 revisions
2019-02-28: received
See all versions
Short URL
https://ia.cr/2019/231
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2019/231,
      author = {Benny Applebaum and Amos Beimel and Oriol Farràs and Oded Nir and Naty Peter},
      title = {Secret-Sharing Schemes for General and Uniform Access Structures},
      howpublished = {Cryptology {ePrint} Archive, Paper 2019/231},
      year = {2019},
      doi = {10.1007/978-3-030-17659-4_15},
      url = {https://eprint.iacr.org/2019/231}
}
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