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

Divesh Aggarwal, Daniel Dadush, Oded Regev, Noah Stephens-Davidowitz

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

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.

Original languageEnglish (US)
Title of host publicationSTOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery
Pages733-742
Number of pages10
ISBN (Electronic)9781450335362
DOIs
StatePublished - Jun 14 2015
Event47th Annual ACM Symposium on Theory of Computing, STOC 2015 - Portland, United States
Duration: Jun 14 2015Jun 17 2015

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
Volume14-17-June-2015
ISSN (Print)0737-8017

Other

Other47th Annual ACM Symposium on Theory of Computing, STOC 2015
CountryUnited States
CityPortland
Period6/14/156/17/15

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Keywords

  • Discrete Gaussian
  • Lattices
  • Shortest Vector Problem

ASJC Scopus subject areas

  • Software

Cite this

Aggarwal, D., Dadush, D., Regev, O., & Stephens-Davidowitz, N. (2015). Solving the shortest vector problem in 2n time via discrete Gaussian sampling. In STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing (pp. 733-742). (Proceedings of the Annual ACM Symposium on Theory of Computing; Vol. 14-17-June-2015). Association for Computing Machinery. https://doi.org/10.1145/2746539.2746606