Teichmüller Sets in Galois Rings

• The Teichmüller set in a Galois ring is a canonical collection of representatives that ensures unique p-adic expansions and preserves unit properties.
• It supports explicit Cauchy MDS matrix constructions via difference and sum forms, guaranteeing that all necessary elements remain invertible.
• Frobenius automorphisms and isomorphic mappings further extend these constructions, offering a vast family of cryptographic and coding matrices with proven MDS properties.

A Teichmüller set in the context of Galois rings plays a fundamental role in the construction and analysis of structured matrices, particularly Cauchy maximum distance separable (MDS) matrices over finite commutative local rings. The Teichmüller set provides canonical representatives that preserve crucial algebraic properties, enable efficient parameterizations for mathematical and cryptographic constructions, and facilitate the definition of automorphisms and isomorphisms. Specifically, in a Galois ring $GR(p^s,p^{s m})$, which generalizes finite fields to rings of characteristic $p^s$, the Teichmüller set underpins both the explicit algebraic structure of elements and the guarantees regarding invertibility and MDS properties necessary for matrix-based coding and cryptographic primitives (Ali et al., 22 Dec 2025).

1. Structure of the Galois Ring and Teichmüller Set

Let $p$ be a prime and $s, m \geq 1$. Given a monic basic-irreducible polynomial $f(x)\in\mathbb{Z}_{p^s}[x]$ of degree $m$, the Galois ring is

$GR(p^s, p^{sm}) \cong \mathbb{Z}_{p^s}[x]/(f(x)),$

a finite commutative local ring with characteristic $p^s$ and order $p^{sm}$.

Within $GR(p^s, p^{sm})$, a root $\xi$ of a basic-primitive polynomial of degree $m$ serves as a lift of a primitive element of $\mathbb{F}_{p^m}$, satisfying $\xi^{p^m-1} = 1$. The Teichmüller set is defined by

$T = \{ 0, 1, \xi, \xi^2, \ldots, \xi^{p^m-2} \},$

with $T^*$ denoting its nonzero part ($\{\xi^i\}_{i=0}^{p^m-2}$), a cyclic group of order $p^m - 1$. Any element $t \in GR(p^s, p^{sm})$ admits a unique $p$-adic expansion:

$$t = t_0 + p t_1 + p^2 t_2 + \ldots + p^{s-1} t_{s-1}, \quad t_i \in T.$$

The set of nilpotent elements coincides with the ideal $(p)$, explicitly $$\mathcal{N} = \{ a \mid a^N = 0 \text{ for some } N \} = (p)$$.

2. Explicit Cauchy MDS Matrix Constructions Using Teichmüller Representatives

Cauchy MDS matrices over $GR(p^s, p^{sm})$ are constructed such that each entry is the inverse of a unit. Utilizing Teichmüller representatives ensures that all required differences and sums remain units, guaranteeing MDS properties.

• Type I (difference form): For distinct $x_i, y_j \in T^* \setminus \{0\}$:

$A_{ij} = (x_i - y_j)^{-1}.$

All $x_i - x_j$, $y_j - y_i$, and $x_i - y_j$ are units. Every submatrix has the same form, maintaining the MDS property ((Ali et al., 22 Dec 2025), Theorem 3.1).

• Type II (sum form, $p \neq 2$): Let $$T' = \{0, 1, \xi, \ldots, \xi^{\lceil (p^m-2)/2 \rceil}\}$$, select distinct $x_i, y_j \in T' \setminus \{0\}$:

$A_{ij} = (x_i + y_j)^{-1}.$

Provided $x_i + y_j \neq 0$ in $\mathbb{F}_{p^m}$, all entries are invertible ((Ali et al., 22 Dec 2025), Theorem 3.2).

• Symmetric reduced-entry form with nilpotents: Fix $l \in \mathcal{N}$, choose distinct $x_1, \ldots, x_n \in T' \setminus \{0\}$, let $y_j = x_j + l$:

$A_{ij} = (x_i + x_j + l)^{-1}.$

This uses at most $n(n+1)/2$ unique ring elements, optimizing resource usage in implementations ((Ali et al., 22 Dec 2025), Theorem 3.2b).

3. MDS Conditions via Teichmüller Set Properties

A matrix is MDS over a ring if all $k \times k$ minors have unit determinant. For Cauchy matrices over $GR(p^s, p^{sm})$, this property holds if and only if denominators $x_i \pm y_j$ are units.

• Type I: No extra restriction on $p$; $x_i, y_j$ are selected as distinct nonzero Teichmüller representatives ($T^*\setminus \{0\}$).
• Type II: Requires $p \neq 2$ and restriction to "half" the Teichmüller set $T'$, ensuring $x_i + y_j \neq 0$ modulo $p$.

This reliance on the Teichmüller set underpins combinatorial uniqueness and prevents zero-divisors, as rigorously established in (Ali et al., 22 Dec 2025), Theorems 3.1–3.3.

4. Frobenius Automorphisms and MDS Family Generation

The Frobenius automorphism

$$\sigma(t) = t_0^p + p t_1^p + \dots + p^{s-1} t_{s-1}^p, \quad t_i \in T$$

has order $m$ on $GR(p^s,p^{sm})$, acting as $\sigma(\xi) = \xi^p$ and fixing $\mathbb{Z}_{p^s}$. Application of $\sigma$ or its iterates to a matrix $A$ yields:

$A^\sigma = [\sigma(A_{ij})]$

which remains MDS wherever $A$ is MDS, since $\det A^\sigma = \sigma(\det A)$. Scaling by units $c \in U(GR)$ extends this further. The total number of maps

$$f_{i,c}(t) = \sigma^i(t) c, \quad 0 \leq i < m, \quad c \in U(GR)$$

is at least $p^{(s-1)m}(p^m-1)$, ensuring a large variety of MDS matrices obtainable from a single seed ((Ali et al., 22 Dec 2025), Theorem 3.6).

5. Automorphisms, Isomorphisms, and Further Matrix Constructions

• Within-ring automorphisms: For $A = (a_{ij})$ MDS in $GR(p^s,p^{sm})$, matrices

$A' = (\sigma^i(a_{ij}) c)$

with $0 \leq i < m$ and $c \in U(GR)$ are also MDS, due to determinant preservation.

• Isomorphisms between presentations: For two Galois rings $GR(p^s, p^{sm})|_{h_1}$ and $GR(p^s, p^{sm})|_{h_2}$ with primitive roots $\eta_1, \eta_2$, mappings $\eta_1 \mapsto \eta_2^{s_u}$ (with $\gcd(s_u,p^m-1)=1$) induce isomorphisms $f$ such that, if $A$ is MDS over the first ring, $f(A)$ is MDS over the second ((Ali et al., 22 Dec 2025), Prop. 4.4). There are $p^{(s-1)m}(p^m-1)$ such compositions.

6. Practical Significance and Implementation Advantages

Teichmüller-set–based constructions ensure all matrix denominators are invertible, which is central to the MDS property. The usage of nilpotent elements for reduced-entry symmetric Cauchy matrices minimizes the inventory of required ring elements, leading to efficiencies in both hardware and software implementations. Frobenius automorphisms and isomorphism techniques allow the generation of extensive families of MDS matrices from a single seed, providing flexibility and cryptographic diversity in the design of diffusion layers for symmetric-key algorithms. Together, these approaches offer explicit formulae, provable MDS-ness, and a vast configuration space for design and optimization (Ali et al., 22 Dec 2025).