Notes on the LVP and CVP in $p$-adic Fields

TL;DR

This paper presents a polynomial time algorithm for solving the LVP and CVP in p-adic fields by exploiting their non-Archimedean properties and algebraic structures.

Contribution

It introduces a novel polynomial time method for computing orthogonal bases in p-adic lattices using maximal orders and p-radicals.

Findings

Efficient algorithms for LVP and CVP in p-adic fields.

Characterization of norms on vector spaces over dic fields.

Rapid construction of uniformizers and residue field bases.

Abstract

This paper explores computational methods for solving the Longest Vector Problem (LVP) and Closest Vector Problem (CVP) in $p$-adic fields. Leveraging the non-Archimedean property of $p$-adic norms, we propose a polynomial time algorithm to compute orthogonal bases for $p$-adic lattices when the $p$-adic field is given by a minimal polynomial. The method utilizes the structure of maximal orders and $p$-radicals in extension fields of $\mathbb{Q}_{p}$ to efficiently construct uniformizers and residue field bases, enabling rapid solutions for the LVP and CVP. In addition, we introduce the characterization of norms on vector spaces over $\mathbb{Q}_p$.