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

Chi Zhang, Mingqian Yao

TL;DR

The paper addresses LVP and CVP in $p$-adic vector spaces and develops a deterministic polynomial-time framework to solve them by constructing orthogonal bases from maximal orders and $p$-radicals. The approach combines $p$-adic lattice theory with residue-field data to reduce LVP/CVP to tractable subproblems on orthogonal components, and it provides a concrete toy example over $K= obreak mathbb{Q}_2( heta)$ to illustrate the method. A key theoretical contribution is a norm characterization on finite-dimensional $ obreak ext{$ obreak V$ over $ obreak mathbb{Q}_p$ and the demonstration that such norms encode complete orthogonal decompositions via invertible matrices in $ ext{GL}_n( obreak mathbb{Q}_p)$. The results have cryptographic implications, suggesting that providing data that reveals orthogonal bases (or their generating matrices) can undermine proposed $p$-adic lattice-based schemes, and motivating designs that avoid leaking such structure while still enabling computable norms.

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 LVP and CVP. In addition, we introduce the characterization of norms on vector spaces over $\mathbb{Q}_p$.

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

TL;DR

The paper addresses LVP and CVP in -adic vector spaces and develops a deterministic polynomial-time framework to solve them by constructing orthogonal bases from maximal orders and -radicals. The approach combines -adic lattice theory with residue-field data to reduce LVP/CVP to tractable subproblems on orthogonal components, and it provides a concrete toy example over to illustrate the method. A key theoretical contribution is a norm characterization on finite-dimensional obreak V obreak mathbb{Q}_p ext{GL}_n( obreak mathbb{Q}_p)p$-adic lattice-based schemes, and motivating designs that avoid leaking such structure while still enabling computable norms.

Abstract

This paper explores computational methods for solving the Longest Vector Problem (LVP) and Closest Vector Problem (CVP) in -adic fields. Leveraging the non-Archimedean property of -adic norms, we propose a polynomial time algorithm to compute orthogonal bases for -adic lattices when the -adic field is given by a minimal polynomial. The method utilizes the structure of maximal orders and -radicals in extension fields of to efficiently construct uniformizers and residue field bases, enabling rapid solutions for LVP and CVP. In addition, we introduce the characterization of norms on vector spaces over .
Paper Structure (12 sections, 9 theorems, 41 equations, 2 algorithms)

This paper contains 12 sections, 9 theorems, 41 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Norm and Orthogonal Basis
  4. Basic Facts about Local Fields
  5. $p$-adic Lattice
  6. LVP and CVP
  7. Computing the Maximal Order
  8. Finding an Orthogonal Basis
  9. Solving the LVP and the CVP with Orthogonal Bases
  10. A Toy Example
  11. Characterization of Norms on Vector Spaces over $\mathbb{Q}_p$
  12. Conclusion

Key Result

Proposition 2.1

Let $V$ be a vector space over $\mathbb{Q}_p$ of finite dimension $n>0$, and let $\left|\cdot\right|$ be a norm on $V$. Then there is a decomposition $V=V_1+V_2+\cdots+V_n$ of $V$ into a direct sum of subspaces $V_i$ of dimension $1$, such that for any $v_i\in V_i$, $i=1,2,\dots,n$.

Theorems & Definitions (21)

  • Proposition 2.1: ref-6
  • Definition 2.2: orthogonal basis ref-6
  • Definition 2.3: $p$-adic lattice ref-3
  • Definition 2.4: orthogonal basis of a $p$-adic lattice ref-3
  • Definition 2.5: ref-2
  • Definition 2.6: ref-2
  • Definition 2.7: ref-2
  • Definition 3.1: order ref-1
  • Definition 3.2: $p$-radical ref-1
  • Theorem 3.3: ref-1
  • ...and 11 more