Probabilistic analyses of lattice reduction algorithms

Abstract

The general behavior of lattice reduction algorithms is far from being well understood. Indeed, many experimental observations, regarding the execution of the algorithms and the geometry of their outputs, pose challenging questions, which remain unanswered and lead to natural conjectures yet to be settled. This survey describes complementary approaches which can be adopted for analyzing these algorithms, namely, dedicated modeling, probabilistic methods, and a dynamical systems approach. We explain how a mixed methodology has already proved fruitful for small dimensions p, corresponding to the variety of Euclidean algorithms (p = 1) and to the Gauss algorithm (p = 2). Such small dimensions constitute an important step in the analysis of lattice reduction in any (high) dimension, since the celebrated LLL algorithm, due to Lenstra, Lenstra, and Lovász, precisely involves a sequence of Gauss reduction steps on sublattices of a large lattice.