### Abstract

In this paper we study the suitability of the Metropolis Algorithm and its generalization for solving the shortest lattice vector problem (SVP). SVP has numerous applications spanning from robotics to computational number theory, viz., polynomial factorization. At the same time, SVP is a notoriously hard problem. Not only it is NP-hard, there is not even any polynomial approximation known for the problem that runs in polynomial time. What one normally uses is the LLL algorithm which, although a polynomial time algorithm, may give solutions which are an exponential factor away from the optimum. In this paper, we have defined an appropriate search space for the problem which we use for implementation of the Metropolis algorithm. We have defined a suitable neighbourhood structure which makes the diameter of the space polynomially bounded, and we ensure that each search point has only polynomially many neighbours. We can use this search space formulation for some other classes of evolutionary algorithms, e.g., for genetic and go-with-the-winner algorithms. We have implemented the Metropolis algorithm and Hasting's generalization of Metropolis algorithm for the SVP. Our results are quite encouraging in all instances when compared with LLL algorithm.

Original language | English |
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Title of host publication | Proceedings of the 2011 11th International Conference on Hybrid Intelligent Systems, HIS 2011 |

Pages | 442-447 |

Number of pages | 6 |

DOIs | |

Publication status | Published - 2011 |

Event | 2011 11th International Conference on Hybrid Intelligent Systems, HIS 2011 - Malacca, Malaysia Duration: 05-12-2011 → 08-12-2011 |

### Conference

Conference | 2011 11th International Conference on Hybrid Intelligent Systems, HIS 2011 |
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Country | Malaysia |

City | Malacca |

Period | 05-12-11 → 08-12-11 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Artificial Intelligence
- Information Systems

### Cite this

*Proceedings of the 2011 11th International Conference on Hybrid Intelligent Systems, HIS 2011*(pp. 442-447). [6122146] https://doi.org/10.1109/HIS.2011.6122146

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*Proceedings of the 2011 11th International Conference on Hybrid Intelligent Systems, HIS 2011.*, 6122146, pp. 442-447, 2011 11th International Conference on Hybrid Intelligent Systems, HIS 2011, Malacca, Malaysia, 05-12-11. https://doi.org/10.1109/HIS.2011.6122146

**Metropolis algorithm for solving Shortest lattice Vector Problem (SVP).** / Ajitha, Shenoy K.B.; Biswas, Somenath; Kurur, Piyush P.

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

TY - GEN

T1 - Metropolis algorithm for solving Shortest lattice Vector Problem (SVP)

AU - Ajitha, Shenoy K.B.

AU - Biswas, Somenath

AU - Kurur, Piyush P.

PY - 2011

Y1 - 2011

N2 - In this paper we study the suitability of the Metropolis Algorithm and its generalization for solving the shortest lattice vector problem (SVP). SVP has numerous applications spanning from robotics to computational number theory, viz., polynomial factorization. At the same time, SVP is a notoriously hard problem. Not only it is NP-hard, there is not even any polynomial approximation known for the problem that runs in polynomial time. What one normally uses is the LLL algorithm which, although a polynomial time algorithm, may give solutions which are an exponential factor away from the optimum. In this paper, we have defined an appropriate search space for the problem which we use for implementation of the Metropolis algorithm. We have defined a suitable neighbourhood structure which makes the diameter of the space polynomially bounded, and we ensure that each search point has only polynomially many neighbours. We can use this search space formulation for some other classes of evolutionary algorithms, e.g., for genetic and go-with-the-winner algorithms. We have implemented the Metropolis algorithm and Hasting's generalization of Metropolis algorithm for the SVP. Our results are quite encouraging in all instances when compared with LLL algorithm.

AB - In this paper we study the suitability of the Metropolis Algorithm and its generalization for solving the shortest lattice vector problem (SVP). SVP has numerous applications spanning from robotics to computational number theory, viz., polynomial factorization. At the same time, SVP is a notoriously hard problem. Not only it is NP-hard, there is not even any polynomial approximation known for the problem that runs in polynomial time. What one normally uses is the LLL algorithm which, although a polynomial time algorithm, may give solutions which are an exponential factor away from the optimum. In this paper, we have defined an appropriate search space for the problem which we use for implementation of the Metropolis algorithm. We have defined a suitable neighbourhood structure which makes the diameter of the space polynomially bounded, and we ensure that each search point has only polynomially many neighbours. We can use this search space formulation for some other classes of evolutionary algorithms, e.g., for genetic and go-with-the-winner algorithms. We have implemented the Metropolis algorithm and Hasting's generalization of Metropolis algorithm for the SVP. Our results are quite encouraging in all instances when compared with LLL algorithm.

UR - http://www.scopus.com/inward/record.url?scp=84856722499&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856722499&partnerID=8YFLogxK

U2 - 10.1109/HIS.2011.6122146

DO - 10.1109/HIS.2011.6122146

M3 - Conference contribution

SN - 9781457721502

SP - 442

EP - 447

BT - Proceedings of the 2011 11th International Conference on Hybrid Intelligent Systems, HIS 2011

ER -