Lattice Basis Reduction in Infinity Norm

In the high-tech world of today, the demand for security is constantly rising. That is why identifying hard computational problems for cryptographical use has become a very important task. It is crucial to find computational problems, which complexity could provide a basis for the security of the cryptosystems. However, there are only very few hard computational problems that are useful for cryptography. One of these, which holds great importance, is finding the shortest basis of a lattice, also called lattice basis reduction. The purpose of this paper is to provide an overview of the lattice basis reduction algorithms and to give an insight into the lattice reduction theory. The most important concepts, Gauss, LLL and BKZ reduction, were initially created to work with the Euclidean norm. Here however, the accent falls on the generalisation of the original algorithms to an arbitrary norm, and more precicely to the infinity norm. All three concepts in their three versions are explained in detail. An analysis of the complexity of the algorithms with respect to l∞-norm is provided.