  In his famous book ``Basic Number Theory", Weil proved several theorems about
the existence of norm-orthogonal bases in finite-dimensional vector spaces and
lattices over local fields. In this paper, we transform Weil's proofs into
algorithms for finding out various norm-orthogonal bases. These algorithms are
closely related to the recently introduced closest vector problem (CVP) in
$p$-adic lattices and they have applications in cryptography based on $p$-adic
lattices.
