eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-12-06
35:1
35:13
10.4230/LIPIcs.ISAAC.2018.35
article
Improved Algorithms for the Shortest Vector Problem and the Closest Vector Problem in the Infinity Norm
Aggarwal, Divesh
1
Mukhopadhyay, Priyanka
2
Centre for Quantum Technologies and School of Computing, National University of Singapore
Centre for Quantum Technologies, National University of Singapore
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol123-isaac2018/LIPIcs.ISAAC.2018.35/LIPIcs.ISAAC.2018.35.pdf
Lattice
Shortest Vector Problem
Closest Vector Problem
l_infty norm