## 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 2^{n-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(2^{0.994n}. 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(k^{2})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 language | English |
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Title of host publication | Advances in Cryptology – EUROCRYPT 2019 - 38th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings |

Editors | Yuval Ishai, Vincent Rijmen |

Pages | 441-471 |

Number of pages | 31 |

DOIs | |

State | Published - 1 Jan 2019 |

Event | 38th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Eurocrypt 2019 - Darmstadt, Germany Duration: 19 May 2019 → 23 May 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11478 LNCS |

### Conference

Conference | 38th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Eurocrypt 2019 |
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Country/Territory | Germany |

City | Darmstadt |

Period | 19/05/19 → 23/05/19 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- General Computer Science