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Abstract <p> This paper explores computational methods for solving the Longest Vector Problem (LVP) and Closest Vector Problem (CVP) in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-adic fields. Leveraging the non-Archimedean property of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-adic norms, we propose a polynomial time algorithm to compute orthogonal bases for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-adic lattices when the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-adic field is given by a minimal polynomial. The method utilizes the structure of maximal orders and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-radicals in extension fields of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathbb{Q}_{p}\)</EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\mathbb{Q}_p\)</EquationSource> </InlineEquation>. </p>

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Notes on the LVP and CVP in \(p\)-Adic Fields

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Chi Zhang,
  • Mingqian Yao

摘要

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 \(\mathbb{Q}_p\) .