Probabilistic behaviour of lattice reduction algorithms
Abstract
The general behavior of lattice reduction algorithms is far from beingwell understood. Indeed, many experimental observations, regarding the executionof the algorithms and the geometry of their outputs, pose challenging questions,which remain unanswered and lead to natural conjectures yet to be settled. This surveydescribes complementary approaches which can be adopted for analyzing thesealgorithms, namely, dedicated modeling, probabilistic methods, and a dynamicalsystems approach.We explain how a mixed methodology has already proved fruitfulfor small dimensions p, corresponding to the variety of Euclidean algorithms(p D 1) and to the Gauss algorithm (p D 2). Such small dimensions constitute animportant step in the analysis of lattice reduction in any (high) dimension, since thecelebrated LLL algorithm, due to Lenstra, Lenstra, and Lov´asz, precisely involves asequence of Gauss reduction steps on sublattices of a large lattice.
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