# 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.

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