LLL and PSLQ

Abstract

The single most useful algorithm of computational number theory is the LLL lattice basis reduction algorithm of Lenstra, Lenstra, and Lovasz [1982]. It finds a relatively short vector in an integer lattice. In this chapter we give some examples of how LLL can be used to approach some of the central problems of the book. Appendix B deals, in detail, with the LLL algorithm and the closely related PSLQ algorithm for finding integer relations. In many of our applications LLL can be treated as a “black box”—why it works doesn’t matter. One inputs a lattice and receives as output a candidate short vector that can be verified to have the requisite properties for the particular problem under consideration.