Floating-Point LLL Revisited
نویسندگان
چکیده
The Lenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL or L) 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 outputs a socalled L-reduced basis in polynomial time O(dn log 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 is almost never used in practice. Instead, one applies floating-point variants of L, where the long-integer arithmetic required by Gram-Schmidt orthogonalisation (central in L) is replaced by floating-point arithmetic. Unfortunately, this is known to be unstable in the worst-case: the usual floating-point L is not even guaranteed to terminate, and the output basis may not be L-reduced at all. In this article, we introduce the L algorithm, a new and natural floating-point variant of L which provably outputs Lreduced bases in polynomial time O(dn(d + log B) log B). This is the first L algorithm whose running time (without fast integer arithmetic) provably grows only quadratically with respect to log B, like the wellknown Euclidean and Gaussian algorithms, which it generalizes.
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