Floating-point LLL Revisited (Extended Abstract)

Phong Nguyen and Damien Stehlé

Abstract: The Lenstra-Lenstra-Lovász 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 vectors of norm less than B in an n-dimensional space, 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.

Here is a 55-dimensional lattice which makes NTL's LLL_FP (with delta=0.99) loop forever.
This is a 35-dimensional lattice for which NTL's LLL_FP (with delta=0.99) saves the calculations by using extended precision.

An implementation of the algorithm is available in fplll.

An LLL Algorithm with Quadratic Complexity (Full Version)

Phong Nguyen and Damien Stehlé

Download: pdf.