LLL on the Average

Phong Nguyen and Damien Stehlé

Abstract: Despite their popularity, lattice reduction algorithms remain mysterious in many ways. It has been widely reported that they behave much more nicely than what was expected from the worst-case proved bounds, both in terms of the running time and the output quality. In this article, we investigate this puzzling statement by trying to model the average case of lattice reduction algorithms, starting with the celebrated Lenstra-Lenstra-Lovász algorithm (LLL). We discuss what is meant by lattice reduction on the average, and we present extensive experiments on the average case behavior of LLL, in order to give a clearer picture of the differences/similarities between the average and worst cases. Our work is intended to clarify the practical behavior of LLL and to raise theoretical questions on its average behavior.

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Warning: The conference proceedings version contains a calculation mistake. Throughout the paper, the 0.18 constant must be replaced by 0.25.