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Improved Algorithms for the Shortest Vector Problem and the Closest Vector Problem in the Infinity Norm

Authors Divesh Aggarwal, Priyanka Mukhopadhyay



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Divesh Aggarwal
  • Centre for Quantum Technologies and School of Computing, National University of Singapore
Priyanka Mukhopadhyay
  • Centre for Quantum Technologies, National University of Singapore

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Divesh Aggarwal and Priyanka Mukhopadhyay. Improved Algorithms for the Shortest Vector Problem and the Closest Vector Problem in the Infinity Norm. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 35:1-35:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ISAAC.2018.35

Abstract

Ajtai, Kumar and Sivakumar [Ajtai et al., 2001] gave the first 2^O(n) algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. The algorithm starts with N in 2^O(n) randomly chosen vectors in the lattice and employs a sieving procedure to iteratively obtain shorter vectors in the lattice, and eventually obtaining the shortest non-zero vector. The running time of the sieving procedure is quadratic in N. Subsequent works [Arvind and Joglekar, 2008; Blömer and Naewe, 2009] generalized the algorithm to other norms. We study this problem for the special but important case of the l_infty norm. We give a new sieving procedure that runs in time linear in N, thereby improving the running time of the algorithm for SVP in the l_infty norm. As in [Ajtai et al., 2002; Blömer and Naewe, 2009], we also extend this algorithm to obtain significantly faster algorithms for approximate versions of the shortest vector problem and the closest vector problem (CVP) in the l_infty norm. We also show that the heuristic sieving algorithms of Nguyen and Vidick [Nguyen and Vidick, 2008] and Wang et al. [Wang et al., 2011] can also be analyzed in the l_infty norm. The main technical contribution in this part is to calculate the expected volume of intersection of a unit ball centred at origin and another ball of a different radius centred at a uniformly random point on the boundary of the unit ball. This might be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Lattice
  • Shortest Vector Problem
  • Closest Vector Problem
  • l_infty norm

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References

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