An affine point of view on minima finding in integer lattices of lower dimensions

Abstract

We study here algorithms that determine successive minima in integer lattices of lower dimensions (n=2 or n=3). We adopt an affine point of view that leads us to a better understanding of the complexity of Gauss' algorithm and we can exhibit its worst-case input configuration. We then propose for the three dimensional case a new algorithm that constitutes the natural generalisation of Gauss' algorithm. We build in polynomial time a “minimal” basis of the lattice and we also get a new structural result — on hyperacute tetrahedra. Furthermore, our algorithm has a better computational complexity that of the LLL algorithm in the 3-dimensional case. Detailed proofs and a more thorough algorithmic discussion are given in [5].