By Topic

Hardness of approximating the shortest vector problem in lattices

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$31 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

1 Author(s)
Khot, S. ; Inst. for Adv. Study, Princeton, NJ, USA

Let p > 1 be any fixed real. We show that assuming NP


RP, it is hard to approximate the shortest vector problem (SVP) in lp norm within an arbitrarily large constant factor. Under the stronger assumption NP

RTIME(2poly(log n)), we show that the problem is hard to approximate within factor 2(log n)1/2 - ε where n is the dimension of the lattice and ε> 0 is an arbitrarily small constant. This greatly improves all previous results in lp norms with 1 < p < ∞. The best results so far gave only a constant factor hardness, namely, 21/p - ε by Micciancio and p1 - ε in high lp 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(log n)1/2-ε.

Published in:

Foundations of Computer Science, 2004. Proceedings. 45th Annual IEEE Symposium on

Date of Conference:

17-19 Oct. 2004