## Abstract

Consider a sender S and a group of n recipients. S holds a secret message m of length l bits and the goal is to allow S to create a secret sharing of m with privacy threshold t among the recipients, by broadcasting a single message c to the recipients. Our goal is to do this with information theoretic security in a model with a simple form of correlated randomness. Namely, for each subset A of recipients of size q, S may share a random key with all recipients in A. (The keys shared with different subsets A must be independent.) We call this Broadcast Secret-Sharing (BSS) with parameters l, n, t and q. Our main question is: how large must c be, as a function of the parameters? We show that ^{n−}_{q}^{t} l is a lower bound, and we show an upper bound of (^{n}^{(}_{q}^{t}_{+}^{+1)}_{t} − t)l, matching the lower bound whenever t = 0, or when q = 1 or n − t. When q = n − t, the size of c is exactly l which is clearly minimal. The protocol demonstrating the upper bound in this case requires S to share a key with every subset of size n − t. We show that this overhead cannot be avoided when c has minimal size. We also show that if access is additionally given to an idealized PRG, the lower bound on ciphertext size becomes ^{n−}_{q}^{t} λ + l − negl(λ) (where λ is the length of the input to the PRG). The upper bound becomes (^{n}^{(}_{q}^{t}_{+}^{+1)}_{t} − t)λ + l. BSS can be applied directly to secret-key threshold encryption. We can also consider a setting where the correlated randomness is generated using computationally secure and non-interactive key exchange, where we assume that each recipient has an (independently generated) public key for this purpose. In this model, any protocol for non-interactive secret sharing becomes an ad hoc threshold encryption (ATE) scheme, which is a threshold encryption scheme with no trusted setup beyond a PKI. Our upper bounds imply new ATE schemes, and our lower bound becomes a lower bound on the ciphertext size in any ATE scheme that uses a key exchange functionality and no other cryptographic primitives.

Original language | English |
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Title of host publication | 2nd Conference on Information-Theoretic Cryptography, ITC 2021 |

Editors | Stefano Tessaro |

Number of pages | 20 |

Publisher | Dagstuhl Publishing |

Publication date | Jul 2021 |

Article number | 10 |

ISBN (Electronic) | 9783959771979 |

DOIs | |

Publication status | Published - Jul 2021 |

Event | 2nd Conference on Information-Theoretic Cryptography, ITC 2021 - Virtual, Bertinoro, Italy Duration: 23 Jul 2021 → 26 Jul 2021 |

### Conference

Conference | 2nd Conference on Information-Theoretic Cryptography, ITC 2021 |
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Country/Territory | Italy |

City | Virtual, Bertinoro |

Period | 23/07/2021 → 26/07/2021 |

Series | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 199 |

ISSN | 1868-8969 |

## Keywords

- Ad-hoc threshold encryption
- Secret-sharing