# A Mixed Integer Quadratic Formulation for the Shortest Vector Problem

## Abstract

Lattice-based cryptography is based on the hardness of the lattice problems, e.g., the shortest vector problem and the closed vector problem. In fact, these mathematical optimization problems are known to be NP-hard. Our interest is to know how large-scale shortest vector problems can be solved. For this, we provide a mixed integer quadratic programming formulation for the shortest vector problem and propose a technique to restrict the search space of the shortest vector problem. This approach is a potential technique to improve the performance of the state-of-the-art software for mixed integer programming problems. In fact, we observe that this technique improves the numerical performance for TU Darmstadt’s benchmark instances with the dimension up to 49.

## Keywords

Shortest vector problem Integer program Mixed integer program Presolve Optimization-based bound tightening Second-order cone program## Notes

### Acknowledgements

We would like to thank Dr. Masaya Yasuda in Kyushu University for fruitful discussions.

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