## Abstract

The security of most lattice based cryptography relies on the hardness of computing a shortest nonzero vector of lattices. We say that a lattice basis is SV-reduced if it contains a shortest nonzero vector of the lattice. In this paper, we prove that, π∕6 orthogonality between the shortest vector of the basis and the vector space spanned by other vectors of the basis is enough to be SV-reduced under the assumption that a plausible condition C_{n} holds. By using the π∕6 orthogonality under C_{2}, we prove a new complexity bound log_{3}[Formula presented]+1 for Gauss–Lagrange algorithm which clarifies why the currently known complexity is so far fall short to expose the efficiency of the algorithm we experience in practice. Our experiments suggest that our complexity bound of Gauss–Lagrange algorithm is somewhat close to actual efficiency of the algorithm. We also show that LLL(δ) algorithm outputs a SV-reduced basis if δ≥1∕3 for two dimensional lattice. We present an efficient three dimensional SV-reduction algorithm by using the condition C_{3} and π∕6 orthogonality and how to generalize the algorithm for higher dimension.

Original language | English |
---|---|

Pages (from-to) | 323-335 |

Number of pages | 13 |

Journal | Discrete Applied Mathematics |

Volume | 283 |

DOIs | |

State | Published - 15 Sep 2020 |

### Bibliographical note

Funding Information:Hyang-Sook Lee was supported by the National Research Foundation of Korea grant funded by the Korea government (Grant Number: NRF-2018R1A2A1A05079095 ). Seongan Lim was supported by the National Research Foundation of Korea grant funded by the Korea government (Grant Number: NRF-2016R1D1A1B01008562 ). Kyunghwan Song was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (Grant Number: NRF-2019R1A6A1A11051177 ). Ikkwon Yie was supported by the National Research Foundation of Korea grant funded by the Korea government (Grant Number: NRF-2017R1D1A1B03034721 ).

Publisher Copyright:

© 2020 Elsevier B.V.

## Keywords

- Lattice
- Orthogonality of basis
- Shortest vector problem