Probabilistic analysis of Gauss’ algorithm in a realistic model

The set {bi}i=1 is called basis of L, and is represented by a matrix B having as columns the bi’s. Each matrix of the form BP , with P ∈ GLn(Z), represents a basis of L. A lattice can have bases with arbitrarily long and skew vectors. See figure 1. By lattice basis reduction we mean solving the following problem: given a lattice basis formed by long (and probably skew) vectors, compute a basis formed by short (and quite orthogonal) vectors. This problem plays a primary rôle in many areas of computational mathematics and computer science: for instance, modern cryptanalysis [10], computer algebra [16], integer linear programming [9] and number theory [3]. In the two-dimensional case, there exists an algorithm due to Lagrange and Gauss which computes a minimal basis (i.e., a basis formed by shortest possible vectors) in linear time: it is a generalization of the Euclidean algorithm. This