## Abstract

A witness encryption (WE) scheme can take any NP statement as a public-key and use it to encrypt a message. If the statement is true then it is possible to decrypt the message given a corresponding witness, but if the statement is false then the message is computationally hidden. Ideally, the encryption procedure should run in polynomial time, but it is also meaningful to define a weaker notion, which we call non-trivially exponentially efficient WE (XWE), where the encryption run-time is only required to be much smaller than the trivial 2 ^{m} bound for NP relations with witness size m. We show how to construct such XWE schemes for all of NP with encryption run-time 2 ^{m} ^{/} ^{2} under the sub-exponential learning with errors (LWE) assumption. For NP relations that can be verified in NC^{1} (e.g., SAT) we can also construct such XWE schemes under the sub-exponential Decisional Bilinear Diffie-Hellman (DBDH) assumption. Although we find the result surprising, it follows via a very simple connection to attribute-based encryption. We also show how to upgrade the above results to get non-trivially exponentially efficient indistinguishability obfuscation for null circuits (niO), which guarantees that the obfuscations of any two circuits that always output 0 are indistinguishable. In particular, under the LWE assumptions we get a XniO scheme where the obfuscation time is 2 ^{n} ^{/} ^{2} for all circuits with input size n. It is known that in the case of indistinguishability obfuscation (iO) for all circuits, non-trivially efficient XiO schemes imply fully efficient iO schemes (Lin et al., PKC ’16) but it remains as a fascinating open problem whether any such connection exists for WE or niO. Lastly, we explore a potential approach toward constructing fully efficient WE and niO schemes via multi-input ABE.

Original language | American English |
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Title of host publication | Security and Cryptography for Networks - 11th International Conference, SCN 2018, Proceedings |

Editors | Dario Catalano, Roberto De Prisco |

Publisher | Springer Verlag |

Pages | 425-441 |

Number of pages | 17 |

ISBN (Print) | 9783319981123 |

DOIs | |

State | Published - 2018 |

Externally published | Yes |

Event | 11th International Conference on Security and Cryptography for Networks, SCN 2018 - Amalfi, Italy Duration: 5 Sep 2018 → 7 Sep 2018 |

### 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 | 11035 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 11th International Conference on Security and Cryptography for Networks, SCN 2018 |
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Country/Territory | Italy |

City | Amalfi |

Period | 5/09/18 → 7/09/18 |

### Bibliographical note

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