Closest Vector Problem

Abstract

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 problem (used mostly in complexity theory) only asks to compute the distance of the target from the lattice, without actually finding the closest lattice vector. CVP has been studied in mathematics (in the equivalent language of quadratic forms) since the 19th century. One of the first references to CVP (under the name "Nearest Vector Problem") in the computer science literature is [11], where the problem is shown to be NP-hard to solve exactly. Many applications of the CVP only require finding a lattice vector that is not too far from the target, even if not necessarily the closest. A g-approximation algorithm for CVP finds a lattice vector within distance at most g times the distance of the optimal solution. The best known polynomial time algorithms to solve CVP due to Babai and Kannan [2, 7] are based on lattice reduction, and achieve approximation factors that (in the worst case) are essentially exponential in the dimension of the lattice. In practice, heuristics approaches (e.g., the "embedding technique", see lattice reduction) seem to find relatively good approximations to CVP in a reasonable amount of time when the dimension of the lattice is sufficiently small. CVP is widely regarded, both in theory and in practice, as a considerably harder problem than SVP. CVP is known to be NP-hard to solve approximately within any constant factor or even some slowly increasing (sub-polynomial) function of the dimension n [1, 3]. However, CVP is un-1