Abstract:
The Lenstra-Lenstra-Lovasz lattice basis reduction algorithm (LLL or L^3)
is a very popular tool in public-key cryptanalysis and in many other fields.
Given an integer d-dimensional lattice basis
with n-dimensional vectors of norm less than B,
L^3 outputs a so-called L^3-reduced basis
in polynomial time $O(d^5 n \log^3 B)$,
using arithmetic operations on integers of bit-length $O(d \log B)$.
This worst-case complexity is problematic for lattices arising in cryptanalysis
where $d$ or/and $\log B$ are often large.
As a result, the original L^3 is almost never used in practice.
Instead, one applies floating-point variants of L^3,
where the long-integer arithmetic required by Gram-Schmidt orthogonalisation
(central in L^3) is replaced by floating-point arithmetic. Unfortunately,
this is known to be unstable in the worst-case: the usual floating-point L^3
is not even guaranteed to terminate, and the output basis may not be
L^3-reduced at all. In this article, we introduce the L^2 algorithm,
a new and natural floating-point variant of L^3 which provably outputs
L^3-reduced bases in polynomial time $O(d^4 n (d+\log B) \log B)$.
This is the first L^3 algorithm whose running time (without fast integer arithmetic) provably grows only
quadratically with respect to $\log B$,
like the well-known Euclidean and Gaussian algorithms, which it generalizes.

Bibtex:
@article{NgSt09,
title={An {LLL} Algorithm with Quadratic Complexity},
author={P. Q. Nguyen and D. Stehl\'e},
journal={SIAM J. of Computing},
volume={ 39},
number= 3,
pages = {874–-903},
year=2009
}