LLL reducing with the most significant bits

Let B be a basis of a Euclidean lattice, and B an approximation thereof. We give a sufficient condition on the closeness between B and B so that an LLL-reducing transformation U for B remains valid for B. Further, we analyse an efficient reduction algorithm when B is itself a small deformation of an LLL-reduced basis. Applications include speeding-up reduction by keeping only the most significant bits of B, reducing a basis that is only approximately known, and efficiently batching LLL reductions for closely related inputs.