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 2O(n) 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 2O(n) time. We also give an algorithm for solving the exact Closest Vector Problem in 2O(n) 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 2O(...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Daniel Dadush, Chris Peikert, Santosh Vempala
• Keywords: Algorithm, CORR 2010, Education, Shortest Vector Problem, Vector Problem