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: Proceedings of the 38th Annual ACM Symposium on Theory of Computing
Pages447-456
Number of pages10
Volume2006
StatePublished - 2006
Event38th Annual ACM Symposium on Theory of Computing, STOC'06 - Seattle, WA, United States
Duration: May 21 2006May 23 2006

Other

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

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Hardness

Keywords

  • Embedding
  • Hardness of Approximation
  • Lattices
  • Norms

ASJC Scopus subject areas

  • Software

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 (Vol. 2006, pp. 447-456)

Lattice problems and norm embeddings. / Regev, Oded; Rosen, Ricky.

STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing. Vol. 2006 2006. p. 447-456.

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

Regev, O & Rosen, R 2006, Lattice problems and norm embeddings. in STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing. vol. 2006, pp. 447-456, 38th Annual ACM Symposium on Theory of Computing, STOC'06, Seattle, WA, United States, 5/21/06.
Regev O, Rosen R. Lattice problems and norm embeddings. In STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing. Vol. 2006. 2006. p. 447-456
Regev, Oded ; Rosen, Ricky. / Lattice problems and norm embeddings. STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing. Vol. 2006 2006. pp. 447-456
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