### Abstract

Finding a short element g of a number field, given the ideal generated by g, is a classic problem in computational algebraic number theory. Solving this problem recovers the private key in cryptosystems introduced by Gentry, Smart–Vercauteren, Gentry–Halevi, Garg– Gentry–Halevi, et al. Work over the last few years has shown that for some number fields this problem has a surprisingly low post-quantum security level. This paper shows, and experimentally verifies, that for some number fields this problem has a surprisingly low pre-quantum security level.

Original language | English |
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Title of host publication | Advances in Cryptology – EUROCRYPT 2017 |

Subtitle of host publication | 36th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Paris, France, April 30 – May 4, 2017, Proceedings, Part I |

Editors | J.-S. Coron, J.B. Nielsen |

Place of Publication | Dordrecht |

Publisher | Springer |

Pages | 27-59 |

Number of pages | 33 |

ISBN (Electronic) | 978-3-319-56620-7 |

ISBN (Print) | 978-3-319-56619-1 |

DOIs | |

Publication status | Published - 2017 |

Event | 36th Annual International Conference on the Theory and Applications of Cryptographic Techniques (Eurocrypt 2017) - Paris, France Duration: 30 Apr 2017 → 4 May 2017 Conference number: 36 |

### 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 | 10210 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 36th Annual International Conference on the Theory and Applications of Cryptographic Techniques (Eurocrypt 2017) |
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Abbreviated title | EUROCRYPT 2017 |

Country | France |

City | Paris |

Period | 30/04/17 → 4/05/17 |

### Keywords

- Gentry
- Ideal lattices
- Lattice-based cryptography
- Multiquadratic fields
- Public-key encryption
- Smart–Vercauteren
- Soliloquy
- Units

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## Cite this

Bauch, J., Bernstein, D. J., de Valence, H., Lange, T., & van Vredendaal, C. (2017). Short generators without quantum computers: the case of multiquadratics. In J-S. Coron, & J. B. Nielsen (Eds.),

*Advances in Cryptology – EUROCRYPT 2017: 36th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Paris, France, April 30 – May 4, 2017, Proceedings, Part I*(pp. 27-59). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10210 LNCS). Springer. https://doi.org/10.1007/978-3-319-56620-7_2