## Abstract

Since the seminal work of Garg et al. (FOCS’13) in which they proposed the first candidate construction for indistinguishability obfuscation (iO for short), iO has become a central cryptographic primitive with numerous applications. The security of the proposed construction of Garg et al. and its variants are proved based on multi-linear maps (Garg et al. Eurocrypt’13) and their idealized model called the graded encoding model (Brakerski and Rothblum TCC’14 and Barak et al. Eurocrypt’14). Whether or not iO could be based on standard and well-studied hardness assumptions has remain an elusive open question. In this work we prove lower bounds on the assumptions that imply iO in a black-box way, based on computational assumptions. Note that any lower bound for iO needs to somehow rely on computational assumptions, because if P = NP then statistically secure iO does exist. Our results are twofold:1. There is no fully black-box construction of iO from (exponentially secure) collision-resistant hash functions unless the polynomial hierarchy collapses. Our lower bound extends to (separate iO from) any primitive implied by a random oracle in a black-box way.2. Let P be any primitive that exists relative to random trapdoor permutations, the generic group model for any finite abelian group, or degree-O(1) graded encoding model for any finite ring. We show that achieving a black-box construction of iO from P is as hard as basing public-key cryptography on one-way functions. In particular, for any such primitive P we present a constructive procedure that takes any black-box construction of iO from P and turns it into a construction of semantically secure public-key encryption form any one-way functions. Our separations hold even if the construction of iO from P is semi-black-box (Reingold, Trevisan, and Vadhan, TCC’04) and the security reduction could access the adversary in a non-black-box way.

Original language | English |
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Title of host publication | Theory of Cryptography - 13th International Conference, TCC 2016-A, Proceedings |

Editors | Eyal Kushilevitz, Tal Malkin |

Publisher | Springer Verlag |

Pages | 49-66 |

Number of pages | 18 |

ISBN (Print) | 9783662490952 |

DOIs | |

State | Published - 2016 |

Externally published | Yes |

Event | 13th International Conference on Theory of Cryptography, TCC 2016 - Tel Aviv, Israel Duration: 10 Jan 2016 → 13 Jan 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 | 9562 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 13th International Conference on Theory of Cryptography, TCC 2016 |
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Country/Territory | Israel |

City | Tel Aviv |

Period | 10/01/16 → 13/01/16 |

### Funding

Funders | Funder number |
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Microsoft Faculty Fellowship | |

National Science Foundation | CCF-1214844, CCF-1350939, 0939718, 1111781, CNS-1217821, 0845811 |

Air Force Office of Scientific Research | FA9550-15-1-0262 |

Defense Advanced Research Projects Agency | |

Simons Foundation | CNS-1523467 |

Microsoft | |

Air Force Research Laboratory | FA8750-11-2-0211, FA8750-11-C-0080 |

## Keywords

- Black-box separations
- Indistinguishability obfuscation