Basis Reduction Problems

Abstract

In the first chapters of this book we studied the shortest vector problem and the closest vector problem both from an algorithmic and computational complexity point of view. In fact, the algorithms presented in Chapter 2 to approximately solve SVP and CVP do somehow more than just finding an approximately shortest lattice vector, or a lattice vector approximately closest to a given target. For example, the LLL algorithm on input a lattice basis B, outputs an equivalent basis B’ such that not only b’1is an approximately shortest lattice vector, but also all other basis vectors b’2 are not too long. Moreover, LLL reduced bases have relatively good geometric properties that make them useful to solve other lattice problems. In particular, we have seen that if an LLL basis is used, then the nearest plane algorithm always finds a lattice vector approximately closest to any input target point. The problem of finding a “good” basis for a given lattice is generically called the basis reduction problem. Unfortunately, there is not a unique, clearly defined notion of what makes a basis good, and several different definitions of reduced basis have been suggested. In this chapter we consider the most important notions of basis reduction, define approximation problems naturally associated to such notions, and study the relation between these and other lattice problems.