## 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 for sharing an ℓ-bit secret in which the share size of each party is roughly max{ℓ, log n} bits, and this is tight even for secrets of 1 b. In these schemes, the number of parties n must be given in advance to the dealer. In this paper, 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 t th party arriving the smallest possible share as a function of t . Our main result is such a scheme for the k-Threshold access structure and 1-bit secrets 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 | English |
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Pages (from-to) | 4179-4190 |

Number of pages | 12 |

Journal | IEEE Transactions on Information Theory |

Volume | 64 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2018 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 1963-2012 IEEE.

### Funding

Manuscript received December 28, 2016; revised September 22, 2017; accepted November 24, 2017. Date of publication December 4, 2017; date of current version May 18, 2018. This work was supported in part by the Israel Science Foundation under Grant 950/16 and in part by the Levzion Fellowship. This paper was presented at the 14th International Conference on Theory of Cryptography (TCC 2016-B).

Funders | Funder number |
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Israel Science Foundation | 950/16 |

## Keywords

- Secret sharing
- evolving access structure
- prefix code
- threshold