## Abstract

A collision resistant hash (CRH) function is one that compresses its input, yet it is hard to find a collision, i.e. a x_{1}≠ x_{2} s.t. h(x_{1}) = h(x_{2}). Collision resistant hash functions are one of the more useful cryptographic primitives both in theory and in practice and two prominent applications are in signature schemes and succinct zero-knowledge arguments. In this work we consider a relaxation of the above requirement that we call Multi-CRH: a function where it is hard to find x_{1}, x_{2}, …, x_{k} which are all distinct, yet h(x_{1}) = h(x_{2}) = ⋯ = h(x_{k}). We show that for some of the major applications of CRH functions it is possible to replace them by the weaker notion of a Multi-CRH, albeit at the price of adding interaction: we show a constant-round statistically-hiding commitment scheme with succinct interaction (committing to poly(n) bits requires exchanging O~ (n) bits) that can be opened locally (without revealing the full string). This in turn can be used to provide succinct arguments for any NP statement. We formulate four possible worlds of hashing-related assumptions (in the spirit of Impagliazzo’s worlds). They are (1) Nocrypt, where no one-way functions exist, (2) Unihash, where one-way functions exist, and hence also UOWHFs and signature schemes, but no Multi-CRH functions exist, (3) Minihash, where Multi-CRH functions exist but no CRH functions exist, and (4) Hashomania, where CRH functions exist. We show that these four worlds are distinct in a black-box model: we show a separation of CRH from Multi-CRH and a separation of Multi-CRH from one-way functions.

Original language | American English |
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Title of host publication | Advances in Cryptology - EUROCRYPT 2018 - 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, 2018 Proceedings |

Editors | Jesper Buus Nielsen, Vincent Rijmen |

Publisher | Springer Verlag |

Pages | 162-194 |

Number of pages | 33 |

ISBN (Print) | 9783319783741 |

DOIs | |

State | Published - 2018 |

Externally published | Yes |

Event | 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2018 - Tel Aviv, Israel Duration: 29 Apr 2018 → 3 May 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 | 10821 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2018 |
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Country/Territory | Israel |

City | Tel Aviv |

Period | 29/04/18 → 3/05/18 |

### Bibliographical note

Funding Information:I. Komargodski—Supported in part by a Packard Foundation Fellowship and AFOSR grant FA9550-15-1-0262. Most work done while the author was a Ph.D. student at the Weizmann Institute of Science, supported in part by a grant from the Israel Science Foundation (no. 950/16) and by a Levzion Fellowship. M. Naor and E. Yogev—Supported in part by a grant from the Israel Science Foundation (no. 950/16). Moni Naor is the incumbent of the Judith Kleeman Professorial Chair.

Publisher Copyright:

© 2018, International Association for Cryptologic Research.