Norm orthogonal bases and invariants of p-adic lattices

Abstract

In 2018, the Longest Vector Problem (LVP) and the Closest Vector Problem (CVP) in p-adic lattices were introduced. These problems are closely linked to the orthogonalization process. In this paper, we first prove that every p-adic lattice has an orthogonal basis respect to any given norm, whereas lattices in Euclidean spaces lack such bases in general. It is an improvement on Weil's result. Then, we prove that the sorted norm sequence of orthogonal basis of a p-adic lattice is unique and give definitions to the successive maxima and the escape distance, as the p-adic analogues of the successive minima and the covering radius in Euclidean lattices. Finally, we present deterministic polynomial time algorithms designed for the orthogonalization process, addressing both the LVP and the CVP with the help of an orthogonal basis of the whole vector space.