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

We give a deterministic O(logn)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 l2-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.

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