### Abstract

We show that unless NP RTIME (2^{poly(log n)}), for any > 0 there is no polynomial-time algorithm approximating the Shortest Vector Problem (SVP) on n-dimensional lattices inthe l_{p} norm (1 q p<) to within a factor of 2(log n)^{1-}. This improves the previous best factor of 2(logn)^{1/2-} under the same complexity assumption due to Khot. Under the stronger assumption NP RSUBEXP, we obtain a hardness factor of n ^{c/log log n} for some c > 0. Our proof starts with Khot's SVP instances from that are hard to approximate to within some constant. To boost the hardness factor we simply apply the standard tensor product oflattices. The main novel part is in the analysis, where we show that Khot's lattices behave nicely under tensorization. At the heart of the analysis is a certain matrix inequality which was first used in the context of lattices by de Shalit and Parzanchevski.

Original language | English (US) |
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Title of host publication | STOC'07: Proceedings of the 39th Annual ACM Symposium on Theory of Computing |

Pages | 469-477 |

Number of pages | 9 |

DOIs | |

State | Published - 2007 |

Event | STOC'07: 39th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States Duration: Jun 11 2007 → Jun 13 2007 |

### Other

Other | STOC'07: 39th Annual ACM Symposium on Theory of Computing |
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Country | United States |

City | San Diego, CA |

Period | 6/11/07 → 6/13/07 |

### Fingerprint

### Keywords

- Hardness of approximation
- Lattices
- Tensor product

### ASJC Scopus subject areas

- Software

### Cite this

*STOC'07: Proceedings of the 39th Annual ACM Symposium on Theory of Computing*(pp. 469-477) https://doi.org/10.1145/1250790.1250859

**Tensor-based hardness of the shortest vector problem to within almost polynomial factors.** / Haviv, Ishay; Regev, Oded.

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

*STOC'07: Proceedings of the 39th Annual ACM Symposium on Theory of Computing.*pp. 469-477, STOC'07: 39th Annual ACM Symposium on Theory of Computing, San Diego, CA, United States, 6/11/07. https://doi.org/10.1145/1250790.1250859

}

TY - GEN

T1 - Tensor-based hardness of the shortest vector problem to within almost polynomial factors

AU - Haviv, Ishay

AU - Regev, Oded

PY - 2007

Y1 - 2007

N2 - We show that unless NP RTIME (2poly(log n)), for any > 0 there is no polynomial-time algorithm approximating the Shortest Vector Problem (SVP) on n-dimensional lattices inthe lp norm (1 q p<) to within a factor of 2(log n)1-. This improves the previous best factor of 2(logn)1/2- under the same complexity assumption due to Khot. Under the stronger assumption NP RSUBEXP, we obtain a hardness factor of n c/log log n for some c > 0. Our proof starts with Khot's SVP instances from that are hard to approximate to within some constant. To boost the hardness factor we simply apply the standard tensor product oflattices. The main novel part is in the analysis, where we show that Khot's lattices behave nicely under tensorization. At the heart of the analysis is a certain matrix inequality which was first used in the context of lattices by de Shalit and Parzanchevski.

AB - We show that unless NP RTIME (2poly(log n)), for any > 0 there is no polynomial-time algorithm approximating the Shortest Vector Problem (SVP) on n-dimensional lattices inthe lp norm (1 q p<) to within a factor of 2(log n)1-. This improves the previous best factor of 2(logn)1/2- under the same complexity assumption due to Khot. Under the stronger assumption NP RSUBEXP, we obtain a hardness factor of n c/log log n for some c > 0. Our proof starts with Khot's SVP instances from that are hard to approximate to within some constant. To boost the hardness factor we simply apply the standard tensor product oflattices. The main novel part is in the analysis, where we show that Khot's lattices behave nicely under tensorization. At the heart of the analysis is a certain matrix inequality which was first used in the context of lattices by de Shalit and Parzanchevski.

KW - Hardness of approximation

KW - Lattices

KW - Tensor product

UR - http://www.scopus.com/inward/record.url?scp=35448997748&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35448997748&partnerID=8YFLogxK

U2 - 10.1145/1250790.1250859

DO - 10.1145/1250790.1250859

M3 - Conference contribution

SN - 1595936319

SN - 9781595936318

SP - 469

EP - 477

BT - STOC'07: Proceedings of the 39th Annual ACM Symposium on Theory of Computing

ER -