A fast variant of the Gaussian reduction algorithm

We propose a fast variant of the Gaussian algorithm for the reduction of two– dimensional lattices for the l1−, l2− and l∞−norm. The algorithm runs in at most O(nM(B) log B) bit operations for the l∞−norm and in O(n log n M(B) log B) bit operations for the l1− and l2−norm on input vectors a, b ∈ ZZ with norm at most 2 where M(B) is a time bound for B-bit integer multiplication. This generalizes Schönhages monotone Algorithm [Sch91] to the centered case and to various norms.