## Abstract

We prove that for every n and 1 < t < n any t-out-of-n threshold secret sharing scheme for one-bit secrets requires share size log(t + 1). Our bound is tight when t = n − 1 and n is a prime power. In 1990 Kilian and Nisan proved the incomparable bound log(n−t+2). Taken together, the two bounds imply that the share size of Shamir’s secret sharing scheme (Comm. ACM ’79) is optimal up to an additive constant even for one-bit secrets for the whole range of parameters 1 < t < n. More generally, we show that for all 1 < s < r < n, any ramp secret sharing scheme with secrecy threshold s and reconstruction threshold r requires share size log((r + 1)/(r − s)). As part of our analysis we formulate a simple game-theoretic relaxation of secret sharing for arbitrary access structures. We prove the optimality of our analysis for threshold secret sharing with respect to this method and point out a general limitation.

Original language | American English |
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Title of host publication | Theory of Cryptography - 14th International Conference, TCC 2016-B, Proceedings |

Editors | Adam Smith, Martin Hirt |

Publisher | Springer Verlag |

Pages | 471-484 |

Number of pages | 14 |

ISBN (Print) | 9783662536438 |

DOIs | |

State | Published - 2016 |

Externally published | Yes |

Event | 14th International Conference on Theory of Cryptography, TCC 2016-B - Beijing, China Duration: 31 Oct 2016 → 3 Nov 2016 |

### 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 | 9986 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 14th International Conference on Theory of Cryptography, TCC 2016-B |
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Country/Territory | China |

City | Beijing |

Period | 31/10/16 → 3/11/16 |

### Bibliographical note

Publisher Copyright:© International Association for Cryptologic Research 2016.