Enumerative Algorithms for the Shortest and Closest Lattice Vector Problems in Any Norm via M-Ellipsoid Coverings

We give an algorithm for solving the exact Shortest Vector Problem in n-dimensional lattices, in any norm, in deterministic 2 time (and space), given poly(n)-sized advice that depends only on the norm. In many norms of interest, including all `p norms, the advice is efficiently and deterministically computable, and in general we give a randomized algorithm to compute it in expected 2 time. We also give an algorithm for solving the exact Closest Vector Problem in 2 time and space, when the target point is within any constant factor of the minimum distance of the lattice. Our approach may be seen as a derandomization of ‘sieve’ algorithms for exact SVP and CVP (Ajtai, Kumar, and Sivakumar; STOC 2001 and CCC 2002), and uses as a crucial subroutine the recent deterministic algorithm of Micciancio and Voulgaris (STOC 2010) for lattice problems in the `2 norm. Our main technique is to reduce the enumeration of lattice points in an arbitrary convex body K to enumeration in 2 copies of an M-ellipsoid of K, a classical concept in asymptotic convex geometry. Building on the techniques of Klartag (Geometric and Functional Analysis, 2006), we also give an expected 2-time algorithm to compute an M-ellipsoid covering of any convex body, which may be of independent interest.

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