TY - GEN

T1 - Lattice problems and norm embeddings

AU - Regev, Oded

AU - Rosen, Ricky

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

KW - Embedding

KW - Hardness of Approximation

KW - Lattices

KW - Norms

UR - http://www.scopus.com/inward/record.url?scp=33748120317&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748120317&partnerID=8YFLogxK

U2 - 10.1145/1132516.1132581

DO - 10.1145/1132516.1132581

M3 - Conference contribution

AN - SCOPUS:33748120317

SN - 1595931341

SN - 9781595931344

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 447

EP - 456

BT - STOC'06

PB - Association for Computing Machinery (ACM)

T2 - 38th Annual ACM Symposium on Theory of Computing, STOC'06

Y2 - 21 May 2006 through 23 May 2006

ER -