The optimal LLL algorithm is still polynomial in fixed dimension

AbstractIn this paper, we consider the open problem of the complexity of the LLL algorithm in the case when the approximation parameter of the algorithm has as its extreme value 1. This case is of interest because the output is then the strongest Lovász-reduced basis. Experiments reported by Lagarias and Odlyzko (J. ACM 32(1) (1985) 229) seem to show that the algorithm remains polynomial in average. However, no bound better than a naive exponential order one is established for the worst-case complexity of the optimal LLL algorithm, even for fixed small dimension (higher than 2). Here, we prove that, for any fixed dimension n, the number of iterations of the LLL algorithm is linear with respect to the size of the input. It is easy to deduce from Vallée (J. Algorithms 12 (1991) 556) that the linear order is optimal. Moreover in 3 dimensions, we give a tight bound for the maximum number of iterations and we characterize precisely the output basis. Our bound also improves the known one for the usual (non-optimal) LLL algorithm