Orthogonal Bases for p-adic Lattices

• Orthogonal bases for p-adic lattices are defined by ultrametric norms that enable explicit norm computations and straightforward vector decompositions.
• Their construction leverages reduction modulo uniformizers and residue field theory, leading to polynomial-time algorithmic orthogonalization.
• These bases underpin applications in cryptography, harmonic analysis, and lattice algorithms, with duality properties essential for arithmetic insights.

An orthogonal basis for $p$-adic lattices is foundational across $p$-adic number theory, harmonic analysis, and cryptographic applications. These are bases whose ultrametric structure enables explicit norm computations, efficient solutions to the lattice closest vector and longest vector problems, and tractable invariants such as successive minima. The existence, explicit construction, complexity, and classification of orthogonal bases in finite-dimensional $p$-adic vector spaces (and their lattices) rest on the strong non-Archimedean properties of $p$-adic norms and their connection to maximal orders, residue field theory, and reduction modulo uniformizers.

1. $p$-adic Norms, Lattices, and Orthogonality

Let $p$ be a prime, $\Q_p$ the field of $p$-adic numbers, and $K/\Q_p$ a finite extension of degree $n=ef$ (ramification index $e$, residue field degree $f$). Every $x\in K$ uniquely decomposes as $x=u\,\pi^t$ with $u\in\O_K^\times$ (unit), $\pi$ a uniformizer, and $t\in\Z$. The normalized absolute value is $|x|_p = p^{-t/e}$. This ultrametric satisfies $|x+y|_p \le \max\{|x|_p, |y|_p\}$, which underpins the notion of orthogonality in $p$-adic vector spaces.

A $p$-adic lattice is a free $\Z_p$-submodule $\Lat(\alpha_1, \ldots, \alpha_m) = \{ \sum_{i=1}^m a_i \alpha_i : a_i \in \Z_p \}$ in a $\Q_p$-vector space $V$, with $m$ linearly independent vectors. Orthogonality is defined norm-theoretically: a basis $\{\alpha_i\}$ is orthogonal if $|\sum b_i \alpha_i| = \max_{i} |b_i \alpha_i|$ for any scalars $b_i$ (Zhang et al., 30 Dec 2025, Zhang et al., 2024, Deng, 2023).

2. Existence and Characterization of Orthogonal Bases

Weil's theorem guarantees any finite-dimensional normed $\Q_p$-vector space $V$ admits a direct sum decomposition $V = V_1 \oplus \cdots \oplus V_n$ with $\dim V_i = 1$ such that $|\sum v_i| = \max_i |v_i|$ for $v_i \in V_i$ (Zhang et al., 30 Dec 2025, Deng, 2023). Bases respecting this decomposition are inherently orthogonal. The uniqueness is up to permutation and scaling by units.

The explicit construction leverages reduction modulo uniformizers and residue field theory. For $K/\Q_p$ of degree $n=ef$, with uniformizer $\pi$ and elements $s_1, \ldots, s_f$ in $\O_K$ whose images form a basis of the residue field $\O_K/(\pi)$ over $\F_p$, the family $$\{ s_i\,\pi^j : 1 \le i \le f,\, 0 \le j \le e-1 \}$$ is an orthogonal $\Q_p$-basis of $K$ (Zhang et al., 30 Dec 2025, Zhang et al., 2024). Totals for totally ramified cases ($f=1$) and unramified cases ($e=1$) admit simpler canonical forms.

3. Constructive and Algorithmic Orthogonalization

Algorithmic orthogonal basis extraction for $p$-adic lattices centers on polynomial-time procedures:

• Maximal order computation: Starting with $\Z_p[\theta]$ (for $K = \Q_p(\theta)$), reduced ring computations and Frobenius action identify the $p$-radical (Zhang et al., 30 Dec 2025).
• Uniformizer and residue field base construction: The maximal $|\cdot|_p$ element in the radical yields $\pi$, and random search in $\F_p[X]$ finds a monic irreducible polynomial for the residue field basis.
• Assembly: Theorem 4.3 in (Zhang et al., 30 Dec 2025) ensures $\{s_i\,\pi^j\}$ forms an orthogonal basis.

P-adic Gram–Schmidt orthogonalization adapts Weil's proof: recursive closest vector problem (CVP) resolution and subtraction realize the orthogonal basis, eschewing classical inner product computation (Deng, 2023).

Step	Description	Complexity
Maximal order and $p$-radical	Linear algebra over $\F_p$, $\Z_p$	$O(n^3)$/round, $O(n)$ rounds
Uniformizer extraction	Maximum valuation selection	Polynomial in $n, \log p$
Residue basis (field search)	Find irreducible in $\F_p[X]$	$O(f^2 \log^2 p)$ expected
Gram–Schmidt (CVP recursive)	Recursive CVP solving	$O(n \cdot \text{poly}(n,N))$

4. Structure, Properties, and Duality of Orthogonal Bases

The orthogonal basis decomposes $p$-adic lattices into 1-dimensional components, with successive minima $\lambda_i$ coinciding with basis vector norms: $\lambda_i(\Lambda) = \|\alpha_i\|_{\max}$ for an $M$-orthogonal basis $\{\alpha_i\}$ (Zhang, 2024). Elementary operations (scaling by units, swaps, limited linear combinations with bounded $p$-adic norm) preserve orthogonality.

A fundamental duality holds: the dual basis to an orthogonal basis (via $(B^{-1})^{\top}$ for $B$ the basis matrix) is again $M$-orthogonal. Furthermore, exact transference relations arise: $$\lambda_i(\Lambda) \cdot \lambda_{n+1-i}(\Lambda^*) = 1$$ and the volume (product of successive minima) matches $|\det B|_p$, paralleling Minkowski's second theorem (Zhang, 2024). This mutual orthogonality of the basis and its dual underpins applications in lattice theory and harmonic analysis.

5. Applications: Lattice Algorithms, Cryptography, and Harmonic Analysis

Given an orthogonal basis, the Closest Vector Problem (CVP) and Longest Vector Problem (LVP) admit efficient, explicit solutions: decomposition in the basis and $p$-adic rounding solves CVP in polynomial time (Zhang et al., 30 Dec 2025, Zhang et al., 2024). The strong correspondence between residue field independence and ultrametric norm independence allows trivial computation of local invariants such as determinants and zeta-factors.

Recent cryptographic schemes exploiting $p$-adic lattices depend on the hardness of recovering an orthogonal basis. Explicit constructions with large residue degree $f$ thwart known attacks relying on totally ramified extensions ($f=1$) (Zhang et al., 2024). The ability to compute a basis efficiently implies schemes must hide norm access or restrict basis publication.

In harmonic analysis, orthogonal bases such as the real eigenbases for $L^2(\Q_p)$ diagonalize $p$-adic pseudo-differential operators, relate to $p$-adic wavelet theory, and exploit underlying $p$-adic lattice structures for multiresolution analysis (Bikulov et al., 2015, Lukomskii, 2012, Albeverio et al., 2010).

6. Quadratic and Bilinear Structures: Hyperbolic Orthogonal Bases

In the context of quadratic forms and alternating bilinear forms, chains of $p$-elementary lattices over $p$-adic integers admit common hyperbolic bases, as shown via explicit bilinear algorithms (Schulze-Pillot, 2018). Here, bases $\{e_i,f_i\}$ realize simultaneous isotropic orthogonality in multiple chains, with clear exponents and modular splittings indexed by Gram valuations. Each such space orthogonally decomposes into hyperbolic planes plus any anisotropic complement, with structure directly informing Bruhat–Tits building theory and arithmetic of quadratic forms.

7. Classification, Isometry, and Future Directions

Orthogonal bases segregate $p$-adic lattices into isometric types determined by norm and residue-field data. The interplay between these components yields new classification results, facilitating local-global analyses of quadratic forms and catalyzing advances in quantum-resistant cryptographic primitives. Extensions to higher dimensions, more exotic lattice types, and multiresolution settings on Vilenkin groups reinforce the centrality of orthogonal bases for $p$-adic analysis (Lukomskii, 2012, Albeverio et al., 2010).

Research continues into explicit computation of orthogonal bases in general ring settings, optimization of lattice algorithms, the full bounds of transference theorems, and structural characterization of $p$-adic MRA decompositions. The richness of these bases lies in their interlocking ultrametric, arithmetic, and geometric properties, which together unlock deep solutions to both theoretical and applied problems in $p$-adic mathematics.