An LLL Algorithm with Quadratic Complexity

The Lenstra–Lenstra–Lovász lattice basis reduction algorithm (called LLL or L3) is a fundamental tool in computational number theory and theoretical computer science, which can be viewed as an efficient algorithmic version of Hermite’s inequality on Hermite’s constant. Given an integer d-dimensional lattice basis with vectors of Euclidean norm less than B in an ndimensional space, the L3 algorithm outputs a reduced basis in O(d3n logB ·M(d logB)) bit operations, where M(k) denotes the time required to multiply k-bit integers. This worst-case complexity is problematic for applications where d or/and logB are often large. As a result, the original L3 algorithm is almost never used in practice, except in tiny dimension. Instead, one applies floating-point variants where the long-integer arithmetic required by Gram–Schmidt orthogonalization is replaced by floating-point arithmetic. Unfortunately, this is known to be unstable in the worst case: the usual floating-point L3 algorithm 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 floatingpoint variant of the L3 algorithm which provably outputs L3-reduced bases in polynomial time O(d2n(d+logB) logB ·M(d)). This is the first L3 algorithm whose running time (without fast integer arithmetic) provably grows only quadratically with respect to logB, like Euclid’s gcd algorithm and Lagrange’s two-dimensional algorithm.