Shortest Vector Problem

Abstract

In this chapter we study the hardness of approximating the shortest vector problem (SVP). Recall that in SVP one is given a matrix \( B \in {\mathbb{Q}^{{m \times n}}} \), and the goal is to find the shortest nonzero vector in the lattice generated by B . In Chapter 3 we have already studied another important algorithmic problem on lattices: the closest vector problem (CVP). In CVP, in addition to the lattice basis \( B \in {\mathbb{Q}^{{m \times n}}} \), one is given a target vector \( t \in {\mathbb{Q}^m} \), and the goal is to find the lattice point in L(B) closest to t. In Chapter 3 we showed that the NP-hardness of CVP can be easily established by reduction from subset sum (Theorem 3.1), and even approximating CVP within any constant or “almost polynomial” factors is hard for NP. We also observed that the reduction from subset sum to CVP can be easily adapted to prove that SVP in the l ∞ norm is NP- hard (Theorem 3.2). Unfortunately, that simple reduction does not work for any other norm. In this chapter, we investigate the computational complexity of SVP in any l p norm other than l ∞ , with special attention to the Euclidean norm l 2 . In the rest of this chapter the l 2 norm is assumed, unless explicitly stated otherwise.