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Lattice problems and norm embeddings

Published: 21 May 2006 Publication History

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.

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    cover image ACM Conferences
    STOC '06: Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing
    May 2006
    786 pages
    ISBN:1595931341
    DOI:10.1145/1132516
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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    Published: 21 May 2006

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    Author Tags

    1. embedding
    2. hardness of approximation
    3. lattices
    4. norms

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    May 21 - 23, 2006
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