### Abstract

We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of √n lie in NP intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk, Goldreich and Goldwasser, and Aharonov and Regev. Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier transform over the lattice. This technique might be useful elsewhere - we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP) improving on a previous result of Regev. An interesting fact is that our result emerged from a "dequantization" of our previous quantum result in. This route to proving purely classical results might be beneficial elsewhere.

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

Pages | 362-371 |

Number of pages | 10 |

State | Published - 2004 |

Event | Proceedings - 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2004 - Rome, Italy Duration: Oct 17 2004 → Oct 19 2004 |

### Other

Other | Proceedings - 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2004 |
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Country | Italy |

City | Rome |

Period | 10/17/04 → 10/19/04 |

### Fingerprint

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS*(pp. 362-371)

**Lattice problems in N P ∩ coNP.** / Aharonov, Dorit; Regev, Oded.

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

*Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS.*pp. 362-371, Proceedings - 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2004, Rome, Italy, 10/17/04.

}

TY - GEN

T1 - Lattice problems in N P ∩ coNP

AU - Aharonov, Dorit

AU - Regev, Oded

PY - 2004

Y1 - 2004

N2 - We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of √n lie in NP intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk, Goldreich and Goldwasser, and Aharonov and Regev. Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier transform over the lattice. This technique might be useful elsewhere - we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP) improving on a previous result of Regev. An interesting fact is that our result emerged from a "dequantization" of our previous quantum result in. This route to proving purely classical results might be beneficial elsewhere.

AB - We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of √n lie in NP intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk, Goldreich and Goldwasser, and Aharonov and Regev. Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier transform over the lattice. This technique might be useful elsewhere - we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP) improving on a previous result of Regev. An interesting fact is that our result emerged from a "dequantization" of our previous quantum result in. This route to proving purely classical results might be beneficial elsewhere.

UR - http://www.scopus.com/inward/record.url?scp=17744382015&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17744382015&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:17744382015

SP - 362

EP - 371

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

ER -