### Abstract

We show that, unless NP⊆DTIME(2 ^{Poly log(n)}), the closest vector problem with pre-processing, for ℓ _{p} norm for any p ≥ 1, is hard to approximate within a factor of (log n) ^{1/p-ε} for any ε > 0. This improves the previous best factor of 3 ^{1/p} - ε due to Regev [19]. Our results also imply that under the same complexity assumption, the nearest codeword problem with pre-processing is hard to approximate within a factor of (log n) ^{1-ε} for any ε > 0.

Original language | English (US) |
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Title of host publication | Proceedings - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005 |

Pages | 216-225 |

Number of pages | 10 |

Volume | 2005 |

DOIs | |

State | Published - 2005 |

Event | 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005 - Pittsburgh, PA, United States Duration: Oct 23 2005 → Oct 25 2005 |

### Other

Other | 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005 |
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Country | United States |

City | Pittsburgh, PA |

Period | 10/23/05 → 10/25/05 |

### Fingerprint

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Proceedings - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005*(Vol. 2005, pp. 216-225). [1530716] https://doi.org/10.1109/SFCS.2005.40

**Hardness of approximating the closest vector problem with pre-processing.** / Alekhnovich, Mikhail; Khot, Subhash; Kindler, Guy; Vishnoi, Nisheeth K.

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

*Proceedings - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005.*vol. 2005, 1530716, pp. 216-225, 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005, Pittsburgh, PA, United States, 10/23/05. https://doi.org/10.1109/SFCS.2005.40

}

TY - GEN

T1 - Hardness of approximating the closest vector problem with pre-processing

AU - Alekhnovich, Mikhail

AU - Khot, Subhash

AU - Kindler, Guy

AU - Vishnoi, Nisheeth K.

PY - 2005

Y1 - 2005

N2 - We show that, unless NP⊆DTIME(2 Poly log(n)), the closest vector problem with pre-processing, for ℓ p norm for any p ≥ 1, is hard to approximate within a factor of (log n) 1/p-ε for any ε > 0. This improves the previous best factor of 3 1/p - ε due to Regev [19]. Our results also imply that under the same complexity assumption, the nearest codeword problem with pre-processing is hard to approximate within a factor of (log n) 1-ε for any ε > 0.

AB - We show that, unless NP⊆DTIME(2 Poly log(n)), the closest vector problem with pre-processing, for ℓ p norm for any p ≥ 1, is hard to approximate within a factor of (log n) 1/p-ε for any ε > 0. This improves the previous best factor of 3 1/p - ε due to Regev [19]. Our results also imply that under the same complexity assumption, the nearest codeword problem with pre-processing is hard to approximate within a factor of (log n) 1-ε for any ε > 0.

UR - http://www.scopus.com/inward/record.url?scp=33748617126&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748617126&partnerID=8YFLogxK

U2 - 10.1109/SFCS.2005.40

DO - 10.1109/SFCS.2005.40

M3 - Conference contribution

SN - 0769524680

SN - 9780769524689

VL - 2005

SP - 216

EP - 225

BT - Proceedings - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005

ER -