## Abstract

A secret-sharing scheme allows to distribute a secret s among n parties such that only some predefined “authorized” sets of parties can reconstruct the secret, and all other “unauthorized” sets learn nothing about s. The collection of authorized/unauthorized sets can be captured by a monotone function f: { 0, 1 } ^{n}→ { 0, 1 }. In this paper, we focus on monotone functions that all their min-terms are sets of size a, and on their duals – monotone functions whose max-terms are of size b. We refer to these classes as (a, n)-upslices and (b, n)-downslices, and note that these natural families correspond to monotone a-regular DNFs and monotone (n- b) -regular CNFs. We derive the following results. 1.(General downslices) Every downslice can be realized with total share size of 1. 5 ^{n}^{+}^{o}^{(}^{n}^{)}< 2 ^{0.585}^{n}. Since every monotone function can be cheaply decomposed into n downslices, we obtain a similar result for general access structures improving the previously known 2 ^{0.637}^{n}^{+}^{o}^{(}^{n}^{)} complexity of Applebaum, Beimel, Nir and Peter (STOC 2020). We also achieve a minor improvement in the exponent of linear secrets sharing schemes.2.(Random mixture of upslices) Following Beimel and Farràs (TCC 2020) who studied the complexity of random DNFs with constant-size terms, we consider the following general distribution F over monotone DNFs: For each width value a∈ [ n], uniformly sample k_{a} monotone terms of size a, where k= (k_{1}, …, k_{n}) is an arbitrary vector of non-negative integers. We show that, except with exponentially small probability, F can be realized with share size of 2 ^{0.5}^{n}^{+}^{o}^{(}^{n}^{)} and can be linearly realized with an exponent strictly smaller than 2/3. Our proof also provides a candidate distribution for “exponentially-hard” access structure. We use our results to explore connections between several seemingly unrelated questions about the complexity of secret-sharing schemes such as worst-case vs. average-case, linear vs. non-linear and primal vs. dual access structures. We prove that, in at least one of these settings, there is a significant gap in secret-sharing complexity.

Original language | English |
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Title of host publication | Advances in Cryptology – CRYPTO 2021 - 41st Annual International Cryptology Conference, CRYPTO 2021, Proceedings |

Editors | Tal Malkin, Chris Peikert |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 627-655 |

Number of pages | 29 |

ISBN (Print) | 9783030842512 |

DOIs | |

State | Published - 2021 |

Event | 41st Annual International Cryptology Conference, CRYPTO 2021 - Virtual, Online Duration: 16 Aug 2021 → 20 Aug 2021 |

### 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 | 12827 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 41st Annual International Cryptology Conference, CRYPTO 2021 |
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City | Virtual, Online |

Period | 16/08/21 → 20/08/21 |

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