## Abstract

A computational secret-sharing scheme is a method that enables a dealer, that has a secret, to distribute this secret among a set of parties such that a “qualified” subset of parties can efficiently reconstruct the secret while any “unqualified” subset of parties cannot efficiently learn anything about the secret. The collection of “qualified” subsets is defined by a monotone Boolean function.It has been a major open problem to understand which (monotone) functions can be realized by a computational secret-sharing scheme. Yao suggested a method for secret-sharing for any function that has a polynomial-size monotone circuit (a class which is strictly smaller than the class of monotone functions in P). Around 1990 Rudich raised the possibility of obtaining secret-sharing for all monotone functions in NP: In order to reconstruct the secret a set of parties must be “qualified” and provide a witness attesting to this fact.Recently, Garg et al. [14] put forward the concept of witness encryption, where the goal is to encrypt a message relative to a statement x ∈ L for a language L ∈ NP such that anyone holding a witness to the statement can decrypt the message, however, if x ∉ L, then it is computationally hard to decrypt. Garg et al. showed how to construct several cryptographic primitives from witness encryption and gave a candidate construction.One can show that computational secret-sharing implies witness encryption for the same language. Our main result is the converse: we give a construction of a computational secret-sharing scheme for any monotone function in NP assuming witness encryption for NP and one-way functions. As a consequence we get a completeness theorem for secret- sharing: computational secret-sharing scheme for any single monotone NP-complete function implies a computational secret-sharing scheme for every monotone function in NP.

Original language | English |
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Title of host publication | Advances in Cryptology - ASIACRYPT 2014 - 20th International Conference on the Theory and Application of Cryptology and Information Security, Proceedings, Part II |

Editors | Palash Sarkar, Tetsu Iwata |

Publisher | Springer Verlag |

Pages | 254-273 |

Number of pages | 20 |

ISBN (Electronic) | 9783662456071 |

DOIs | |

State | Published - 2014 |

Externally published | Yes |

Event | 20th International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2014 - Kaoshiung, Taiwan, Province of China Duration: 7 Dec 2014 → 11 Dec 2014 |

### 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 | 8874 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 20th International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2014 |
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Country/Territory | Taiwan, Province of China |

City | Kaoshiung |

Period | 7/12/14 → 11/12/14 |