We devise an algorithm, e L, with the following specifications: It takes as input an arbitrary basis B = (bi)i ∈ Zd×d of a Euclidean lattice L; It computes a basis of L which is reduced for a mild modification of the Lenstra-Lenstra-Lovász reduction; It terminates in time O(dβ + dβ) where β = log max ‖bi‖ (for any ε > 0 and ω is a valid exponent for matrix multiplication). This is the first LLL-reducing algorithm with a time complexity that is quasi-linear in β and polynomial in d. The backbone structure of e L is able to mimic the Knuth-Schönhage fast gcd algorithm thanks to a combination of innovative ingredients. First the bit-size of our lattice bases can be decreased via truncations whose validity are backed by recent numerical stability results on the QR matrix factorization. Also we establish a new framework for analyzing unimodular transformation matrices which reduce shifts of reduced bases, this includes bit-size control and new perturbation tools. We illustrate the power of this framework by generating a family of reduction algorithms.