NDLI: Hardness of approximating the shortest vector problem in lattices

Content Provider	IEEE Xplore Digital Library
Author	Khot, S.
Copyright Year	2004
Description	Author affiliation: Inst. for Adv. Study, Princeton, NJ, USA (Khot, S.)
Abstract	Let p > 1 be any fixed real. We show that assuming NP /spl nsube/ RP, it is hard to approximate the shortest vector problem (SVP) in l/sub p/ norm within an arbitrarily large constant factor. Under the stronger assumption NP /spl nsube/ RTIME(2/sup poly(log n)/), we show that the problem is hard to approximate within factor 2/sup log n1/2 - /spl epsi// where n is the dimension of the lattice and /spl epsi/> 0 is an arbitrarily small constant. This greatly improves all previous results in l/sub p/ norms with 1 < p < /spl infin/. The best results so far gave only a constant factor hardness, namely, 2/sup 1/p/ - /spl epsi/ by Micciancio and p/sup 1 - /spl epsi// in high l/sub p/ norms by Khot. We first give a new (randomized) reduction from closest vector problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product, a new variant of tensor product that we introduce. This enables us to boost the hardness factor to 2/sup log n1/2-/spl epsi//.
Sponsorship	IEEE Comput. Soc. Tech. Soc. on Foundations of Comput
Starting Page	126
Ending Page	135
File Size	422498
Page Count	10
File Format	PDF
ISBN	0769522289
ISSN	02725428
DOI	10.1109/FOCS.2004.31
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2004-10-17
Publisher Place	Italy
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Lattices Tensile stress Polynomials Paper technology History Books Gaussian processes Geometry Linear programming Computer science
Content Type	Text
Resource Type	Article

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