Floating-Point LLL Revisited

Abstract

The Lenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL or L3) is a very popular tool in public-key cryptanalysis and in many other fields. Given an integer d-dimensional lattice basis with vectors of norm less than B in an n-dimensional space, L3 outputs a so-called L3-reduced basis in polynomial time O(d 5 n log3 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 L3 is almost never used in practice. Instead, one applies floating-point variants of L3, where the long-integer arithmetic required by Gram-Schmidt orthogonalisation (central in L3) is replaced by floating-point arithmetic. Unfortunately, this is known to be unstable in the worst-case: the usual floating-point L3 is not even guaranteed to terminate, and the output basis may not be L3-reduced at all. In this article, we introduce the L2 algorithm, a new and natural floating-point variant of L3 which provably outputs L3-reduced bases in polynomial time O(d 4 n (d + log B) log B). This is the first L3 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.