  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.
