The algorithm is no longer guaranteed to be polynomial-time: it may not even terminate. And if ever it terminates, the output basis may not be L 3 -reduced. The main number theory computer packages =-=[7, 21,30]-=- contain heuristic fp-variants of L 3 ? la Schnorr-Euchner [29] suffering from stability problems. On the theoretic side, the fastest provable fp variant of L 3 is Nguyen-Stehl?’s L 2 [23], whose runn