## Abstract

In this paper, we study several related computational problems for supersingular elliptic curves, their isogeny graphs, and their endomorphism rings. We prove reductions between the problem of path finding in the

We show that conjugacy classes of maximal orders have a representative of polynomial size, and we define a way to represent endomorphism ring generators in a way that allows for efficient evaluation at points on the curve. We relate these problems to the security of the Charles-Goren-Lauter hash function. We provide a collision attack for special but natural parameters of the hash function and prove that for general parameters its preimage and collision resistance are also equivalent to the endomorphism ring computation problem.

*l*-isogeny graph, computing maximal orders isomorphic to the endomorphism ring of a supersingular elliptic curve, and computing the endomorphism ring itself. We also give constructive versions of Deuring's correspondence, which associates to a maximal order in a certain quaternion algebra an isomorphism class of supersingular elliptic curves. The reductions are based on heuristics regarding the distribution of norms of elements in quaternion algebras.We show that conjugacy classes of maximal orders have a representative of polynomial size, and we define a way to represent endomorphism ring generators in a way that allows for efficient evaluation at points on the curve. We relate these problems to the security of the Charles-Goren-Lauter hash function. We provide a collision attack for special but natural parameters of the hash function and prove that for general parameters its preimage and collision resistance are also equivalent to the endomorphism ring computation problem.

Original language | English |
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Title of host publication | Advances in Cryptology – EUROCRYPT 2018 |

Subtitle of host publication | 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tel Aviv, Israel, April 29 - May 3, 2018 Proceedings, Part III |

Editors | Jesper Buus Nielsen, Vincent Rijmen |

Publisher | Springer |

Pages | 329-368 |

Number of pages | 40 |

ISBN (Electronic) | 9783319783727 |

ISBN (Print) | 9783319783710 |

DOIs | |

Publication status | Published - 31 Mar 2018 |

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 |
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Publisher | Springer |

Volume | 10822 |

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 |