License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/OASIcs.SOSA.2018.12
URN: urn:nbn:de:0030-drops-83062
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8306/
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Aggarwal, Divesh ; Stephens-Davidowitz, Noah

Just Take the Average! An Embarrassingly Simple 2^n-Time Algorithm for SVP (and CVP)

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OASIcs-SOSA-2018-12.pdf (0.6 MB)

Abstract

We show a 2^{n+o(n)}-time (and space) algorithm for the Shortest Vector Problem on lattices (SVP) that works by repeatedly running an embarrassingly simple "pair and average" sieving-like procedure on a list of lattice vectors. This matches the running time (and space) of the current fastest known algorithm, due to Aggarwal, Dadush, Regev, and Stephens-Davidowitz (ADRS, in STOC, 2015), with a far simpler algorithm. Our algorithm is in fact a modification of the ADRS algorithm, with a certain careful rejection sampling step removed.

The correctness of our algorithm follows from a more general "meta-theorem," showing that such rejection sampling steps are unnecessary for a certain class of algorithms and use cases. In particular, this also applies to the related 2^{n + o(n)}-time algorithm for the Closest Vector Problem (CVP), due to Aggarwal, Dadush, and Stephens-Davidowitz (ADS, in FOCS, 2015), yielding a similar embarrassingly simple algorithm for gamma-approximate CVP for any gamma = 1+2^{-o(n/log n)}. (We can also remove the rejection sampling procedure from the 2^{n+o(n)}-time ADS algorithm for exact CVP, but the resulting algorithm is still quite complicated.)

BibTeX - Entry

@InProceedings{aggarwal_et_al:OASIcs:2018:8306,
  author =	{Divesh Aggarwal and Noah Stephens-Davidowitz},
  title =	{{Just Take the Average! An Embarrassingly Simple 2^n-Time Algorithm for SVP (and CVP)}},
  booktitle =	{1st Symposium on Simplicity in Algorithms (SOSA 2018)},
  pages =	{12:1--12:19},
  series =	{OpenAccess Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-064-4},
  ISSN =	{2190-6807},
  year =	{2018},
  volume =	{61},
  editor =	{Raimund Seidel},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8306},
  URN =		{urn:nbn:de:0030-drops-83062},
  doi =		{10.4230/OASIcs.SOSA.2018.12},
  annote =	{Keywords: Lattices, SVP, CVP}
}

Keywords: Lattices, SVP, CVP
Collection: 1st Symposium on Simplicity in Algorithms (SOSA 2018)
Issue Date: 2018
Date of publication: 05.01.2018

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