## Abstract

Secret-sharing schemes are an important tool in cryptography that is used in the construction of many secure protocols. However, the shares' size in the best known secret-sharing schemes realizing general access structures is exponential in the number of parties in the access structure, making them impractical. On the other hand, the best lower bound known for sharing of an ℓ-bit secret with respect to an access structure with n parties is Ω(ℓn/log n) (Csirmaz, EUROCRYPT 94). No major progress on closing this gap has been obtained in the last decade. Faced by our lack of understanding of the share complexity of secret sharing schemes, we investigate a weaker notion of privacy in secrets sharing schemes where each unauthorized set can never rule out any secret (rather than not learn any "probabilistic" information on the secret). Such schemes were used previously to prove lower bounds on the shares' size of perfect secret-sharing schemes. Our main results is somewhat surprising upper-bounds on the shares' size in weakly-private schemes. - For every access structure, we construct a scheme for sharing an ℓ-bit secret with (ℓ + c)-bit shares, where c is a constant depending on the access structure (alas, c can be exponential in n). Thus, our schemes become more efficient as ℓ - the secret size - grows. For example, for the above mentioned access structure of Csirmaz, we construct a scheme with shares' size ℓ + n log n. - We construct efficient weakly-private schemes for threshold access structures for sharing a one bit secret. Most impressively, for the 2-out-of-n threshold access structure, we construct a scheme with 2-bit shares (compared to Ω(log n) in any perfect secret sharing scheme).

Original language | English |
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Title of host publication | Theory of Cryptography - 4th Theory of Cryptography Conference, TCC 2007, Proceedings |

Publisher | Springer Verlag |

Pages | 253-272 |

Number of pages | 20 |

ISBN (Print) | 9783540709350 |

DOIs | |

State | Published - 1 Jan 2007 |

Event | 4th Theory of Cryptography Conference, TCC 2OO7 - Amsterdam, Netherlands Duration: 21 Feb 2007 → 24 Feb 2007 |

### 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 | 4392 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 4th Theory of Cryptography Conference, TCC 2OO7 |
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Country/Territory | Netherlands |

City | Amsterdam |

Period | 21/02/07 → 24/02/07 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)