### Abstract

There is surprisingly little consensus on the precise role of the generator g in group-based assumptions such as DDH. Some works consider g to be a fixed part of the group description, while others take it to be random. We study this subtle distinction from a number of angles. In the generic group model, we demonstrate the plausibility of groups in which random-generator DDH (resp. CDH) is hard but fixed-generator DDH (resp. CDH) is easy. We observe that such groups have interesting cryptographic applications.We find that seemingly tight generic lower bounds for the Discrete-Log and CDH problems with preprocessing (Corrigan-Gibbs and Kogan, Eurocrypt 2018) are not tight in the sub-constant success probability regime if the generator is random. We resolve this by proving tight lower bounds for the random generator variants; our results formalize the intuition that using a random generator will reduce the effectiveness of preprocessing attacks.We observe that DDH-like assumptions in which exponents are drawn from low-entropy distributions are particularly sensitive to the fixed- vs. random-generator distinction. Most notably, we discover that the Strong Power DDH assumption of Komargodski and Yogev (Komargodski and Yogev, Eurocrypt 2018) used for non-malleable point obfuscation is in fact false precisely because it requires a fixed generator. In response, we formulate an alternative fixed-generator assumption that suffices for a new construction of non-malleable point obfuscation, and we prove the assumption holds in the generic group model. We also give a generic group proof for the security of fixed-generator, low-entropy DDH (Canetti, Crypto 1997).

Original language | English (US) |
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Title of host publication | Advances in Cryptology – CRYPTO 2019 - 39th Annual International Cryptology Conference, Proceedings |

Editors | Alexandra Boldyreva, Daniele Micciancio |

Publisher | Springer Verlag |

Pages | 801-830 |

Number of pages | 30 |

ISBN (Print) | 9783030269500 |

DOIs | |

State | Published - Jan 1 2019 |

Event | 39th Annual International Cryptology Conference, CRYPTO 2019 - Santa Barbara, United States Duration: Aug 18 2019 → Aug 22 2019 |

### 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 | 11693 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 39th Annual International Cryptology Conference, CRYPTO 2019 |
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Country | United States |

City | Santa Barbara |

Period | 8/18/19 → 8/22/19 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Advances in Cryptology – CRYPTO 2019 - 39th Annual International Cryptology Conference, Proceedings*(pp. 801-830). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11693 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-26951-7_27