@inbook{9f503ebb51984fa7887d9fc4a25d1669,
title = "Floating-point LLL: theoretical and practical aspects",
abstract = "The text-book LLL algorithm can be sped up considerably by replacing the underlying rational arithmetic used for the Gram–Schmidt orthogonalisation by floating-point approximations. We review how this modification has been and is currently implemented, both in theory and in practice. Using floating-point approximations seems to be natural for LLL even from the theoretical point of view: it is the key to reach a bit-complexity which is quadratic with respect to the bitlength of the input vectors entries, without fast integer multiplication. The latter bit-complexity strengthens the connection between LLL and Euclid{\textquoteright}s gcd algorithm. On the practical side, the LLL implementer may weaken the provable variants in order to further improve their efficiency: we emphasise on these techniques. We also consider the practical behaviour of the floating-point LLL algorithms, in particular their output distribution, their running-time and their numerical behaviour. After 25 years of implementation, many questions motivated by the practical side of LLL remain open.",
author = "Damien Stehl{\'e}",
year = "2009",
doi = "10.1007/978-3-642-02295-1_5",
language = "English",
isbn = "9783642022944",
series = "Information security and cryptography",
publisher = "Springer, Springer Nature",
pages = "179--213",
editor = "Nguyen, {Phong Q.} and Brigitte Vallee",
booktitle = "The LLL algorithm",
address = "United States",
}