### Abstract

Let p > 1 be any fixed real. We show that assuming NP ⊂ RP, it is hard to approximate the Shortest Vector Problem (SVP) in l
_{p} norm within an arbitrarily large constant factor. Under the stronger assumption NP ⊂ RTIME(2
^{poly((lop n)}), we show that the problem is hard to approximate within factor 2(log n)
^{1/2-ε} where n is the dimension of the lattice and ε > 0 is an arbitrarily small constant. This greatly improves all previous results in
_{p} norms with 1 < p < ∞. The best results so far gave only a constant factor hardness, namely, 2
^{1/p} - ε by Micciancio [27] and
^{1/ε} in high l
_{p} norms by Knot [20]. We first give a new (randomized) reduction from Closest Vector Problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH Codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product, a new variant of tensor product that we introduce. This enables us to boost the hardness factor to 2(log n)
^{1/2-ε}.

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

Title of host publication | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

Pages | 126-135 |

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 |
---|---|

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. 126-135)

**Hardness of approximating the shortest vector problem in lattices.** / Khot, Subhash.

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

*Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS.*pp. 126-135, Proceedings - 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2004, Rome, Italy, 10/17/04.

}

TY - GEN

T1 - Hardness of approximating the shortest vector problem in lattices

AU - Khot, Subhash

PY - 2004

Y1 - 2004

N2 - Let p > 1 be any fixed real. We show that assuming NP ⊂ RP, it is hard to approximate the Shortest Vector Problem (SVP) in l p norm within an arbitrarily large constant factor. Under the stronger assumption NP ⊂ RTIME(2 poly((lop n)), we show that the problem is hard to approximate within factor 2(log n) 1/2-ε where n is the dimension of the lattice and ε > 0 is an arbitrarily small constant. This greatly improves all previous results in p norms with 1 < p < ∞. The best results so far gave only a constant factor hardness, namely, 2 1/p - ε by Micciancio [27] and 1/ε in high l p norms by Knot [20]. We first give a new (randomized) reduction from Closest Vector Problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH Codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product, a new variant of tensor product that we introduce. This enables us to boost the hardness factor to 2(log n) 1/2-ε.

AB - Let p > 1 be any fixed real. We show that assuming NP ⊂ RP, it is hard to approximate the Shortest Vector Problem (SVP) in l p norm within an arbitrarily large constant factor. Under the stronger assumption NP ⊂ RTIME(2 poly((lop n)), we show that the problem is hard to approximate within factor 2(log n) 1/2-ε where n is the dimension of the lattice and ε > 0 is an arbitrarily small constant. This greatly improves all previous results in p norms with 1 < p < ∞. The best results so far gave only a constant factor hardness, namely, 2 1/p - ε by Micciancio [27] and 1/ε in high l p norms by Knot [20]. We first give a new (randomized) reduction from Closest Vector Problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH Codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product, a new variant of tensor product that we introduce. This enables us to boost the hardness factor to 2(log n) 1/2-ε.

UR - http://www.scopus.com/inward/record.url?scp=17744386194&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17744386194&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:17744386194

SP - 126

EP - 135

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

ER -