Tensor-based hardness of the shortest vector problem to within almost polynomial factors

Ishay Haviv, Oded Regev

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

Abstract

We show that unless NP RTIME (2poly(log n)), for any > 0 there is no polynomial-time algorithm approximating the Shortest Vector Problem (SVP) on n-dimensional lattices inthe lp norm (1 q p<) to within a factor of 2(log n)1-. This improves the previous best factor of 2(logn)1/2- under the same complexity assumption due to Khot. Under the stronger assumption NP RSUBEXP, we obtain a hardness factor of n c/log log n for some c > 0. Our proof starts with Khot's SVP instances from that are hard to approximate to within some constant. To boost the hardness factor we simply apply the standard tensor product oflattices. The main novel part is in the analysis, where we show that Khot's lattices behave nicely under tensorization. At the heart of the analysis is a certain matrix inequality which was first used in the context of lattices by de Shalit and Parzanchevski.

Original languageEnglish (US)
Title of host publicationSTOC'07: Proceedings of the 39th Annual ACM Symposium on Theory of Computing
Pages469-477
Number of pages9
DOIs
StatePublished - 2007
EventSTOC'07: 39th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States
Duration: Jun 11 2007Jun 13 2007

Other

OtherSTOC'07: 39th Annual ACM Symposium on Theory of Computing
CountryUnited States
CitySan Diego, CA
Period6/11/076/13/07

Fingerprint

Tensors
Hardness
Polynomials

Keywords

  • Hardness of approximation
  • Lattices
  • Tensor product

ASJC Scopus subject areas

  • Software

Cite this

Haviv, I., & Regev, O. (2007). Tensor-based hardness of the shortest vector problem to within almost polynomial factors. In STOC'07: Proceedings of the 39th Annual ACM Symposium on Theory of Computing (pp. 469-477) https://doi.org/10.1145/1250790.1250859

Tensor-based hardness of the shortest vector problem to within almost polynomial factors. / Haviv, Ishay; Regev, Oded.

STOC'07: Proceedings of the 39th Annual ACM Symposium on Theory of Computing. 2007. p. 469-477.

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

Haviv, I & Regev, O 2007, Tensor-based hardness of the shortest vector problem to within almost polynomial factors. in STOC'07: Proceedings of the 39th Annual ACM Symposium on Theory of Computing. pp. 469-477, STOC'07: 39th Annual ACM Symposium on Theory of Computing, San Diego, CA, United States, 6/11/07. https://doi.org/10.1145/1250790.1250859
Haviv I, Regev O. Tensor-based hardness of the shortest vector problem to within almost polynomial factors. In STOC'07: Proceedings of the 39th Annual ACM Symposium on Theory of Computing. 2007. p. 469-477 https://doi.org/10.1145/1250790.1250859
Haviv, Ishay ; Regev, Oded. / Tensor-based hardness of the shortest vector problem to within almost polynomial factors. STOC'07: Proceedings of the 39th Annual ACM Symposium on Theory of Computing. 2007. pp. 469-477
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