Sampling Short Lattice Vectors and the Closest Lattice Vector Problem

We present a 2 O(n) time Turing reduction from the closest lattice vector problem to the shortest lattice vector problem. Our reduction assumes access to a subroutine that solves SVP exactly and a subroutine to sample short vectors from a lattice, and computes a (1+)-approximation to CVP. As a consequence, using the SVP algorithm from 1], we obtain a randomized 2 O(1+ ?1)n algorithm to obtain a (1+)-approximation for the closest lattice vector problem in n dimensions. This improves the existing time bound of O(n!) for CVP (achieved by a deterministic algorithm in 2]). Given an n-dimensional lattice L and a point x 2 R n , the closest lattice vector problem (CVP) is to nd a v 2 L such that the Euclidean norm kx ? vk is minimized. CVP is one of the most fundamental problems concerning lattices and has many applications. The homogeneous version of CVP is the shortest lattice vector problem (SVP) where x = 0 and v is required to be non-zero. In the-approximate version of CVP, it is required to nd a v 0 2 L such that for every v 2 L, kx ? v 0 k kx ? vk. In this paper we give a Turing reduction from CVP to SVP. Our reduction assumes access to two subroutines for variants of SVP: one that solves SVP exactly, and one that can sample short vectors from a lattice (with very weak uniformity guarantees). The reduction solves the (1 +)-approximate version of CVP. Using the SVP algorithm from 1] in place of the subroutines, we obtain a randomized 2 O(1+ ?1)n algorithm to obtain a (1 +)-approximation for CVP in n dimensions. CVP is a well-studied problem from many points of view. For the problem of computing the closest vector exactly, Kannan obtained an n O(n) time deterministic algorithm 10] and the constant in the exponent was improved by Helfrich 9]. Recently, Bll omer obtained an O(n!) time deterministic algorithm to compute the closest vector exactly 2]. For the problem of approximating the closest vector, using the LLL algorithm 12], Babai obtained a (3= p 2) n-approximation algorithm that runs in polynomial time 3]. Using a 2 O(n) algorithm for SVP 1 and the polynomial-time Turing reduction from approximate CVP to SVP given in 10], the present authors obtained a p n=2-approximation algorithm that runs in 2 O(n) time and a 2 n log …