Improved algorithms for the Shortest Vector Problem and the Closest Vector Problem in the infinity norm

Divesh Aggarwal; Priyanka Mukhopadhyay

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arXiv · 1801.02358

Improved algorithms for the Shortest Vector Problem and the Closest Vector Problem in the infinity norm

Divesh Aggarwal·Priyanka Mukhopadhyay

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Abstract

Blomer and Naewe[BN09] modified the randomized sieving algorithm of Ajtai, Kumar and Sivakumar[AKS01] to solve the shortest vector problem (SVP). The algorithm starts with $N = 2^{O(n)}$ randomly chosen vectors in the lattice and employs a sieving procedure to iteratively obtain shorter vectors in the lattice. The running time of the sieving procedure is quadratic in $N$. We study this problem for the special but important case of the $\ell_\infty$ norm. We give a new sieving procedure that runs in time linear in $N$, thereby significantly improving the running time of the algorithm for SVP in the $\ell_\infty$ norm. As in [AKS02,BN09], 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 $\ell_\infty$ norm. We also show that the heuristic sieving algorithms of Nguyen and Vidick[NV08] and Wang et al.[WLTB11] can also be analyzed in the $\ell_{\infty}$ 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.

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Divesh Aggarwal, Priyanka Mukhopadhyay. 2018-01-08. Improved algorithms for the Shortest Vector Problem and the Closest Vector Problem in the infinity norm. https://arxiv.org/abs/1801.02358

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