Faster exponential time algorithms for the shortest vector problem

Abstract

We present new faster algorithms for the exact solution of the shortest vector problem in arbitrary lattices. Our main result shows that the shortest vector in any <i>n</i>-dimensional lattice can be found in time 2<sup>3.199<i>n</i></sup> (and space 2<sup>1.325<i>n</i></sup>), or in space 2<sup>1.095<i>n</i></sup> (and still time 2<sup><i>O(n)</i></sup>). This improves the best previously known algorithm by Ajtai, Kumar and Sivakumar [Proceedings of STOC 2001] which was shown by Nguyen and Vidick [J. Math. Crypto. 2(2):181--207] to run in time 2<sup>5.9<i>n</i></sup> and space 2<sup>2.95<i>n</i></sup>. We also present a practical variant of our algorithm which provably uses an amount of space proportional to &#964;<i>n</i>, the "kissing" constant in dimension <i>n</i>. No upper bound on the running time of our second algorithm is currently known, but experimentally the algorithm seems to perform fairly well in practice, with running time 2<sup>0.52<i>n</i></sup>, and space complexity 2<sup>0.2<i>n</i></sup>.

DOI: 10.1137/1.9781611973075.119
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@inproceedings{Voulgaris2009FasterET, title={Faster exponential time algorithms for the shortest vector problem}, author={Panagiotis Voulgaris and Daniele Micciancio}, booktitle={Electronic Colloquium on Computational Complexity}, year={2009} }