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 bit-length of the input vectors entries, without fast integer multiplication. The latter bit-complexity strengthens the connection between LLL and Euclid’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.