Notes on the LVP and CVP in $p$-adic Fields
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913057186250752 |
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| author | Zhang, Chi Yao, Mingqian |
| author_facet | Zhang, Chi Yao, Mingqian |
| contents | 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 $\mathbb{Q}_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24207 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Notes on the LVP and CVP in $p$-adic Fields Zhang, Chi Yao, Mingqian Number Theory Primary 11F85, Secondary 94A60 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 $\mathbb{Q}_p$. |
| title | Notes on the LVP and CVP in $p$-adic Fields |
| topic | Number Theory Primary 11F85, Secondary 94A60 |
| url | https://arxiv.org/abs/2512.24207 |