On $p$-adic Gram-Schmidt Orthogonalization Process

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.