## Abstract

Combining the efficient cross-polytope locality-sensitive hash family of Terasawa and Tanaka with the heuristic lattice sieve algorithm of Micciancio and Voulgaris, we show how to obtain heuristic and practical speedups for solving the shortest vector problem (SVP) on both arbitrary and ideal lattices. In both cases, the asymptotic time complexity for solving SVP in dimension n is 2^{0.298n+o(n)}. For any lattice, hashes can be computed in polynomial time, which makes our CPSieve algorithm much more practical than the SphereSieve of Laarhoven and de Weger, while the better asymptotic complexities imply that this algorithm will outperform the GaussSieve of Micciancio and Voulgaris and the HashSieve of Laarhoven in moderate dimensions as well. We performed tests to show this improvement in practice. For ideal lattices, by observing that the hash of a shifted vector is a shift of the hash value of the original vector and constructing rerandomization matrices which preserve this property, we obtain not only a linear decrease in the space complexity, but also a linear speedup of the overall algorithm. We demonstrate the practicability of our cross-polytope ideal lattice sieve ICPSieve by applying the algorithm to cyclotomic ideal lattices from the ideal SVP challenge and to lattices which appear in the cryptanalysis of NTRU.

Original language | English |
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Title of host publication | Progress in Cryptology – AFRICACRYPT 2016 |

Subtitle of host publication | 8th International Conference on Cryptology in Africa, Fes, Morocco, April 13-15, 2016, Proceedings |

Editors | D. Pointcheval, A. Nitaj, T. Rachidi |

Place of Publication | Dordrecht |

Publisher | Springer |

Pages | 3-23 |

Number of pages | 21 |

ISBN (Electronic) | 978-3-319-31517-1 |

ISBN (Print) | 978-3-319-31516-4 |

DOIs | |

Publication status | Published - 1 Jan 2016 |

Event | 8th International Conference on the Theory and Application of Cryptographic Techniques in Africa (Africacrypt 2016) - Fes, Morocco Duration: 13 Apr 2016 → 15 Apr 2016 Conference number: 8 |

### 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 | 9646 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 8th International Conference on the Theory and Application of Cryptographic Techniques in Africa (Africacrypt 2016) |
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Abbreviated title | Africacrypt 2016 |

Country | Morocco |

City | Fes |

Period | 13/04/16 → 15/04/16 |

## Keywords

- (Ideal) lattices
- Locality-sensitive hashing
- Shortest vector problem
- Sieving algorithms