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2012
ACM

Deterministic construction of an approximate M-ellipsoid and its applications to derandomizing lattice algorithms

7 years 1 months ago
Deterministic construction of an approximate M-ellipsoid and its applications to derandomizing lattice algorithms
We give a deterministic O(log n)n -time and space algorithm for the Shortest Vector Problem (SVP) of a lattice under any norm, improving on the previous best deterministic nO(n) -time algorithms for general norms. This approaches the 2O(n) -time and space complexity of the randomized sieve based SVP algorithms (Arvind and Joglekar, FSTTCS 2008), first introduced by Ajtai, Kumar and Sivakumar (STOC 2001) for 2-SVP, and the M-ellipsoid covering based SVP algorithm of Dadush et al. (FOCS 2011). Here we continue with the covering approach of Dadush et al., and our main technical contribution is a deterministic approximation of an M-ellipsoid for any convex body. To achieve this we exchange the M-position of a convex body by a related position, known as the minimal mean width position of the polar body. We reduce the task of computing this position to solving a semi-definite program whose objective is a certain Gaussian expectation, which we show can be approximated deterministically.
Daniel Dadush, Santosh Vempala
Added 28 Sep 2012
Updated 28 Sep 2012
Type Journal
Year 2012
Where SODA
Authors Daniel Dadush, Santosh Vempala
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