## Abstract

Secret sharing schemes allow a dealer to distribute a secret piece of information among several parties such that only qualified subsets of parties can reconstruct the secret. The collection of qualified subsets is called an access structure. The best known example is the k-threshold access structure, where the qualified subsets are those of size at least k. When k = 2 and there are n parties, there are schemes where the size of the share each party gets is roughly log n bits, and this is tight even for secrets of 1 bit. In these schemes, the number of parties n must be given in advance to the dealer. In this work we consider the case where the set of parties is not known in advance and could potentially be infinite. Our goal is to give the tth party arriving the smallest possible share as a function of t. Our main result is such a scheme for the k-threshold access structure where the share size of party t is (k − 1) ・ log t + poly(k) ・ o(log t). For k = 2 we observe an equivalence to prefix codes and present matching upper and lower bounds of the form log t + log log t + log log log t + O(1). Finally, we show that for any access structure there exists such a secret sharing scheme with shares of size 2^{t}^{−1}.

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 | 485-514 |

Number of pages | 30 |

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

Funding Information:I. Komargodski, et al.—Research supported in part by grants from the Israel Science Foundation grant no. 1255/12, BSF and from the I-CORE Program of the Planning and Budgeting Committee and the Israel Science Foundation (grant no. 4/11). Moni Naor is the incumbent of the Judith Kleeman Professorial Chair. Ilan Komargodski is supported in part by a Levzion fellowship.

Publisher Copyright:

© International Association for Cryptologic Research 2016.