Lattice problems and norm embeddings

Oded Regev, Ricky Rosen

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

Abstract

We present reductions from lattice problems in the ℓ2 norm to the corresponding problems in other norms such as ℓ1, ℓ (and in fact in any other ℓp norm where 1 ≤ p ≤ ∞). We consider lattice problems such as the Shortest Vector Problem, Shortest Independent Vector Problem, Closest Vector Problem and the Closest Vector Problem with Preprocessing. Most reductions are simple and follow from known constructions of embeddings of normed spaces. Among other things, our reductions imply that the Shortest Vector Problem in the ℓ1 norm and the Closest Vector Problem with Preprocessing in the ℓ norm are hard to approximate to within any constant (and beyond). Previously, the former problem was known to be hard to approximate to within 2 - ε, while no hardness result was known for the latter problem.

Original languageEnglish (US)
Title of host publicationSTOC'06
Subtitle of host publicationProceedings of the 38th Annual ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery (ACM)
Pages447-456
Number of pages10
ISBN (Print)1595931341, 9781595931344
DOIs
StatePublished - 2006
Event38th Annual ACM Symposium on Theory of Computing, STOC'06 - Seattle, WA, United States
Duration: May 21 2006May 23 2006

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
Volume2006
ISSN (Print)0737-8017

Other

Other38th Annual ACM Symposium on Theory of Computing, STOC'06
CountryUnited States
CitySeattle, WA
Period5/21/065/23/06

Keywords

  • Embedding
  • Hardness of Approximation
  • Lattices
  • Norms

ASJC Scopus subject areas

  • Software

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  • Cite this

    Regev, O., & Rosen, R. (2006). Lattice problems and norm embeddings. In STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (pp. 447-456). (Proceedings of the Annual ACM Symposium on Theory of Computing; Vol. 2006). Association for Computing Machinery (ACM). https://doi.org/10.1145/1132516.1132581