### Abstract

We present reductions from lattice problems in the ℓ_{2} norm to the corresponding problems in other norms such as ℓ_{1}, ℓ_{∞} (and in fact in any other ℓ_{p} 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 ℓ_{1} 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.

Original language | English (US) |
---|---|

Title of host publication | STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing |

Pages | 447-456 |

Number of pages | 10 |

Volume | 2006 |

State | Published - 2006 |

Event | 38th Annual ACM Symposium on Theory of Computing, STOC'06 - Seattle, WA, United States Duration: May 21 2006 → May 23 2006 |

### Other

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

Country | United States |

City | Seattle, WA |

Period | 5/21/06 → 5/23/06 |

### Fingerprint

### Keywords

- Embedding
- Hardness of Approximation
- Lattices
- Norms

### ASJC Scopus subject areas

- Software

### Cite this

*STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing*(Vol. 2006, pp. 447-456)

**Lattice problems and norm embeddings.** / Regev, Oded; Rosen, Ricky.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing.*vol. 2006, pp. 447-456, 38th Annual ACM Symposium on Theory of Computing, STOC'06, Seattle, WA, United States, 5/21/06.

}

TY - GEN

T1 - Lattice problems and norm embeddings

AU - Regev, Oded

AU - Rosen, Ricky

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

M3 - Conference contribution

SN - 1595931341

SN - 9781595931344

VL - 2006

SP - 447

EP - 456

BT - STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing

ER -