Deterministic Construction of an Approximate M-Ellipsoid and its Application to Derandomizing Lattice Algorithms

We give a deterministic O(log n) algorithm for the Shortest Vector Problem (SVP) of a lattice under any norm, improving on the previous best deterministic bound of n for general norms and nearly matching the bound of 2 for the standard Euclidean norm established by Micciancio and Voulgaris (STOC 2010). Our algorithm can be viewed as a derandomization of the AKS randomized sieve algorithm, which can be used to solve SVP for any norm in 2 time with high probability. We use the technique of covering a convex body by ellipsoids, as introduced for lattice problems in (Dadush et al., FOCS 2011). Our main contribution is a deterministic approximation of an M-ellipsoid of any convex body. We achieve this via a convex programming formulation of the optimal ellipsoid with the objective function being an n-dimensional integral that we show can be approximated deterministically, a technique that appears to be of independent interest. ∗School of Industrial and Systems Engineering, Georgia Tech. [email protected] †School of Computer Science, Georgia Tech. [email protected]