The Gauß lattice basis reduction algorithm succeeds with any norm

Abstract

We generalize Gauß' definition of lattice basis reduction to an arbitrary norm and analyse the generalized version of the Gauß lattice basis reduction algorithm. We can prove that the worst-case bound established in [5] for the number of iterations of the Gauß algorithm in the euclidean norm, which is known to be the best possible in that case, holds for any norm. We prove for any norm that the norm of two consecutive vectors in the algorithm at every but the first and the last iteration decreases at least by a factor 2. We lift this result to a bound for the number of iterations of log_1+√2 (B/λ₂) + 1, where B denotes the maximum of the norms of the two input vectors and λ₂ denotes the second succesive minimum in the given norm. Furthermore we give two algorithms for the maximum norm ∥ ·∥ _∞ and the sum norm ∥ · ∥₁ that determine the integral reduction coefficient for every iteration in the Gauß algorithm in O(n log n) arithmetic operations, where n is the dimension of the given vector space.