Shortest Vector Problem

The Shortest Vector Problem (SVP) is the most famous and widely studied computational problem on lattices. Given a lattice L (typically represented by a basis), SVP asks to find the shortest nonzero vector in L. The problem can be defined with respect to any norm, but the Euclidean norm is the most common (see the entry lattice for a definition). A variant of SVP (commonly studied in computational complexity theory) only asks to compute the length (denoted λ(L)) of the shortest nonzero vector in L, without necessarily finding the vector. SVP has been studied by mathematicians (in the equivalent language of quadratic forms) since the 19th century because of its connection to many problems in number theory. One of the earliest references to SVP in the computer science literature is [7], where the problem is conjectured to be NP-hard. A cornerstone result about SVP isMinkowski’s first theorem, which states that the shortest nonzero vector in any n-dimentional lattice has length at most γn det(L) , where γn is an absolute constant (approximately equal to √ n) that depends only of the dimension n, and det(L) is the determinant of the lattice (see the entry lattice for a definition). The upper bound provided by Minkowski’s theorem is tight, i.e., there are lattices such that the shortest nonzero vector has length γn det(L) . However, general lattices may contain vectors much shorter than that. Moreover, Minkowski’s theorem only proves that short vectors exist, i.e., it does not give an efficient algorithmic procedure to find such vectors. An algorithm to find the shortest nonzero vector in 2-dimensional lattices was already known to Gauss in the 19th century, but no general methods to efficiently find (approximately) shortest vectors in n-dimentional lattices were known