Progress on LLL and lattice reduction

Abstract

We review variants and extensions of the LLL-algorithm of Lenstra, Lenstra Lovász, extensions to quadratic indefinite forms and to faster and stronger reduction algorithms. The LLL-algorithm with Householder orthogonalisation in floating-point arithmetic is very efficient and highly accurate. We review approximations of the shortest lattice vector by feasible lattice reduction, in particular by block reduction, primal–dual reduction and random sampling reduction. Segment reduction performs LLL-reduction in high dimension, mostly working with a few local coordinates.