Norm Orthogonal Bases and Invariants of $p$-adic Lattices

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 and give definition to the successive maxima and the escape distance, as the $p$-adic analogues of the successive minima and the covering radius in Euclidean lattices. Then, we present deterministic polynomial time algorithms to perform the orthogonalization process, solve the LVP and solve the CVP with an orthogonal basis of the whole vector space. Finally, we conclude that orthogonalization and the CVP are polynomially equivalent.