A sieve algorithm for the shortest lattice vector problem
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
The Two Faces of Lattices in Cryptology
CaLC '01 Revised Papers from the International Conference on Cryptography and Lattices
Fast Algorithms for Logconcave Functions: Sampling, Rounding, Integration and Optimization
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Simulated Annealing for Convex Optimization
Mathematics of Operations Research
Worst-Case to Average-Case Reductions Based on Gaussian Measures
SIAM Journal on Computing
Robust Stochastic Approximation Approach to Stochastic Programming
SIAM Journal on Optimization
Proceedings of the forty-second ACM symposium on Theory of computing
Enumerative Lattice Algorithms in any Norm Via M-ellipsoid Coverings
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
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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.