### Abstract

We give a randomized 2^{n+o(n)} -time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic Õ(4^{n})-time and Õ(2^{n})-space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp.2013). In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample 2^{n/2} vectors from the discrete Gaussian distribution at any parameter in 2^{n+o(n)} time and space. (Prior work only solved DGS for very large parameters.) Our SVP result then follows from a natural reduction from SVP to DGS. In addition, we give a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate 2^{n/2} discrete Gaussian samples in just 2^{n/2+o(n)} time and space. Among other things, this implies a 2^{n/2+o(n)} -time and space algorithm for 1.93-approximate decision SVP.

Original language | English (US) |
---|---|

Title of host publication | STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing |

Publisher | Association for Computing Machinery |

Pages | 733-742 |

Number of pages | 10 |

Volume | 14-17-June-2015 |

ISBN (Print) | 9781450335362 |

DOIs | |

State | Published - Jun 14 2015 |

Event | 47th Annual ACM Symposium on Theory of Computing, STOC 2015 - Portland, United States Duration: Jun 14 2015 → Jun 17 2015 |

### Other

Other | 47th Annual ACM Symposium on Theory of Computing, STOC 2015 |
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Country | United States |

City | Portland |

Period | 6/14/15 → 6/17/15 |

### Fingerprint

### Keywords

- Discrete Gaussian
- Lattices
- Shortest Vector Problem

### ASJC Scopus subject areas

- Software

### Cite this

^{n}time via discrete Gaussian sampling. In

*STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing*(Vol. 14-17-June-2015, pp. 733-742). Association for Computing Machinery. https://doi.org/10.1145/2746539.2746606

**Solving the shortest vector problem in 2 ^{n} time via discrete Gaussian sampling.** / Aggarwal, Divesh; Dadush, Daniel; Regev, Oded; Stephens-Davidowitz, Noah.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

^{n}time via discrete Gaussian sampling. in

*STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing.*vol. 14-17-June-2015, Association for Computing Machinery, pp. 733-742, 47th Annual ACM Symposium on Theory of Computing, STOC 2015, Portland, United States, 6/14/15. https://doi.org/10.1145/2746539.2746606

^{n}time via discrete Gaussian sampling. In STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing. Vol. 14-17-June-2015. Association for Computing Machinery. 2015. p. 733-742 https://doi.org/10.1145/2746539.2746606

}

TY - GEN

T1 - Solving the shortest vector problem in 2n time via discrete Gaussian sampling

AU - Aggarwal, Divesh

AU - Dadush, Daniel

AU - Regev, Oded

AU - Stephens-Davidowitz, Noah

PY - 2015/6/14

Y1 - 2015/6/14

N2 - We give a randomized 2n+o(n) -time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic Õ(4n)-time and Õ(2n)-space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp.2013). In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample 2n/2 vectors from the discrete Gaussian distribution at any parameter in 2n+o(n) time and space. (Prior work only solved DGS for very large parameters.) Our SVP result then follows from a natural reduction from SVP to DGS. In addition, we give a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate 2n/2 discrete Gaussian samples in just 2n/2+o(n) time and space. Among other things, this implies a 2n/2+o(n) -time and space algorithm for 1.93-approximate decision SVP.

AB - We give a randomized 2n+o(n) -time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic Õ(4n)-time and Õ(2n)-space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp.2013). In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample 2n/2 vectors from the discrete Gaussian distribution at any parameter in 2n+o(n) time and space. (Prior work only solved DGS for very large parameters.) Our SVP result then follows from a natural reduction from SVP to DGS. In addition, we give a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate 2n/2 discrete Gaussian samples in just 2n/2+o(n) time and space. Among other things, this implies a 2n/2+o(n) -time and space algorithm for 1.93-approximate decision SVP.

KW - Discrete Gaussian

KW - Lattices

KW - Shortest Vector Problem

UR - http://www.scopus.com/inward/record.url?scp=84958771669&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958771669&partnerID=8YFLogxK

U2 - 10.1145/2746539.2746606

DO - 10.1145/2746539.2746606

M3 - Conference contribution

AN - SCOPUS:84958771669

SN - 9781450335362

VL - 14-17-June-2015

SP - 733

EP - 742

BT - STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing

PB - Association for Computing Machinery

ER -