The Closest Vector Problem (CVP) is a computational problem on lattices closely related to SVP. (See Shortest Vector Problem.) Given a lattice L and a target point ~x, CVP asks to find the lattice point closest to the target. As for SVP, CVP can be defined with respect to any norm, but the Euclidean norm is the most common (see the entry lattice for a definition). A more relaxed version of the ...

The closest vector problem for general lattices is NP-hard. However, we can efficiently find the closest lattice points for some special lattices, such as root lattices (An, Dn and some En). In this paper, we discuss the closest vector problem on more general lattices than root lattices.

In this work, we give provable sieving algorithms for the Shortest Vector Problem (SVP) and Closest (CVP) on lattices in ?p norm (1?p??). The running time obtain is better than existing algorithms. We a new linear procedure that works all main idea to divide space into hypercubes such each vector can be mapped efficiently sub-region. achieve complexity of 22.751n+o(n), which much less 23.849n+o...

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...

We give a deterministic algorithm for solving the (1+ε) approximate Closest Vector Problem (CVP) on any n dimensional lattice and any norm in 2(1 + 1/ε) time and 2 poly(n) space. Our algorithm builds on the lattice point enumeration techniques of Micciancio and Voulgaris (STOC 2010) and Dadush, Peikert and Vempala (FOCS 2011), and gives an elegant, deterministic alternative to the “AKS Sieve” b...

In this paper we consider the problem of finding a vector that can be written as a nonnegative, integer and linear combination of given 0-1 vectors, the generators, such that the l1-distance between this vector and a given target vector is minimized. We prove that this closest vector problem is NP-hard to approximate within an additive error of (ln 2− ǫ)d ≈ (0, 693 − ǫ) d for all ǫ > 0 where d ...

In this note we give a polynomial time algorithm for solving the closest vector problem in class of zonotopal lattices. The Voronoi cell lattice is zonotope, i.e. projection regular cube. Examples lattices include Voronoi's first kind and tensor products root type A. combinatorial structure can be described by matroids/totally unimodular matrices. We observe that linear algebra version minimum ...

We present deterministic polynomially space bounded algorithms for the closest vector problem for all lp-norms, 1 < p < ∞, and all polyhedral norms, in particular for the l1norm and the l∞-norm. For all lp-norms with 1 < p < ∞ the running time of the algorithm is p · log2(r)n, where r is an upper bound on the size of the coefficients of the target vector and the lattice basis and n is the dimen...

Recall that in the closest vector problem we are given a lattice and a target vector (which is usually not in the lattice) and we are supposed to find the lattice point that is closest to the target point. More precisely, one can consider three variants of the CVP, depending on whether we have to actually find the closest vector, find its distance, or only decide if it is closer than some given...