Homomorphic Matrix Transformations

• Homomorphic matrix transformations are algebraic techniques that preserve matrix structure, enabling operations on encrypted or structured data.
• They leverage conjugation and ring homomorphisms to convert and accelerate computations in structured linear algebra, reducing complexity.
• These transformations underpin privacy-preserving encryption schemes and high-throughput hardware implementations for secure matrix computations.

Homomorphic matrix transformations are a class of algebraic and algorithmic techniques that enable structural or function-preserving operations on matrices, frequently with the goal of facilitating computations in cryptographically secure, computationally efficient, or structurally compatible forms. These transformations underpin critical advances in privacy-preserving computation, structured linear algebra, algebraic combinatorics, and functional analysis. The term "homomorphic" here spans a broad territory: in cryptography, it refers to the ability to perform linear or polynomial operations on encrypted data; in algebra and operator theory, it often denotes structure-preserving (algebra homomorphism or module homomorphism) mappings between matrix algebras or modules.

1. Algebraic Classification of Homomorphic Matrix Maps

The notion of homomorphic transformations in the context of matrix algebras has a rigorous mathematical foundation in the classification of holomorphic (complex-analytic) maps that act linearly or polynomially on matrix spaces and preserve specific algebraic properties. For holomorphic transformations $H : M_m(\mathbb{C}) \to M_m(\mathbb{C})$ that are orthogonally additive and orthogonally multiplicative on self-adjoint inputs—that is, for all self-adjoint matrices $a, b$ with $ab=0$,

• $H(a+b) = H(a) + H(b)$ (additivity),
• $H(a)H(b) = 0$ (multiplicativity),

the classification theorem states that either:

• The range of $H$ consists of zero-trace matrices, or
• There exists an invertible $S \in M_m$ and scalars $\{\lambda_n\}$ such that, for all $x$,

$H(x) = \sum_{n\geq 1} \lambda_n S^{-1} x^n S$

or

$H(x) = \sum_{n\geq 1} \lambda_n S^{-1} (x^t)^n S$

Here, $x^t$ denotes the transpose of $x$. If $H$ also globally preserves zero products, the transpose form is excluded and only the conjugation by powers of $x$ remains. This result generalizes the classic result for linear homomorphisms of matrix algebras (inner automorphism or transpose-inner automorphism) to holomorphic, “power-series in $x$” functionals (Bu et al., 2014).

2. Homomorphic Matrix Transformations in Structured Linear Algebra

A central theme in fast computational linear algebra is the use of homomorphic transforms to map matrices between classical structured families: Toeplitz, Hankel, Vandermonde, and Cauchy. This transformation is executed via conjugation by carefully constructed multiplier matrices—typically Vandermonde, diagonal, or reflection matrices. The composite map $M \mapsto P M Q$ acts as a ring homomorphism between matrix classes, and:

• Preserves or increases displacement rank by at most a constant,
• Is invertible when $P, Q$ are invertible,
• Enables transferring nearly-linear algorithms (e.g., fast inversion, mat-vec, or polynomial evaluation/interpolation) across all four structured families.

The canonical transformation between Vandermonde and Cauchy matrices, for example, has the form

$$C_{s,t} = \mathrm{diag}(t(s_i)^{-1}) V_s V_t^{-1} \mathrm{diag}(t'(t_j))$$

where $t(x) = \prod_{j=1}^n (x - t_j)$. Applying these homomorphic transforms, efficient algorithms for inversion or mat-vec with Toeplitz, Vandermonde, or Cauchy structure reduce to $O(n \log n)$ time (Pan, 2013).

3. Homomorphic Matrix Transformations under Encryption

Homomorphic encryption (HE) allows algebraic operations to be performed directly on ciphertexts, enabling secure delegated computation on encrypted data. Homomorphic matrix transformations here denote:

• Matrix multiplication, linear transformation, or higher-degree polynomials applied to encrypted (integer or real-valued) matrices or vectors,
• Structural permutations or transpositions for ciphertext slot alignment,
• Circuit-level function evaluation (e.g., covariance, QR, SVD, eigen-decomposition).

State-of-the-art CKKS, BFV, and ElGamal-based schemes support such operations with varying efficiency and arithmetic expressivity. Efficient algorithms leverage batching/packing (SIMD), optimized rotation and hoisting for ciphertext slots, and key permutation decompositions. Recent advancements include:

• Highly optimized homomorphic matrix multiplication with Diagonal-Convergence Decomposition (DCD), BSGS, and hoisting for reduced rotation cost and key count (Ma, 2023),
• Ideal permutation decompositions for $O(\log n)$ rotation complexity and minimal rotation key count in matrix transposition and multiplication circuits (Ma et al., 2024),
• FPGA and AI-accelerator architectures for high-throughput, high-dimension matrix transformations exploiting the structural sparsity of linear maps and fusing rotation and key switching datapaths (Xu et al., 17 Dec 2025, Tong et al., 13 Jan 2025).

A common paradigm is to express homomorphic matrix transformation via linearized circuits that reduce, after slot-level permutation and multiplication, to a sum over diagonal factors: $$\mathrm{HLT}_U([m]) = \sum_{t=0}^{d-1} \mathrm{CMult}(\mathrm{Rot}([m]; z_t), u^{(z_t)})$$ where $U$ has $d$ nonzero diagonals at shifts $z_t$ (Xu et al., 17 Dec 2025).

4. Homomorphic Module Homomorphisms and Tensors

In abstract algebra and multilinear analysis, higher-order tensors associated to circulant-based products give rise to module homomorphisms acting on spaces of matrices with vector-valued scalars. Taking group rings $R[G]$ (for a finite abelian group $G$ and commutative ring $R$), the set of third-order tensors $T \in R^{n \times n \times n}$ is isomorphic to the ring of $R[G]$-linear endomorphisms on $R^{n \times n}$ under convolutive multiplication. This lifts the classical correspondence—matrices as endomorphisms of $R^n$—to higher order tensor–module pairs: $$R[G]^{n \times n \times |G|} \cong \mathrm{End}_{R[G]}(M_n(R))$$ resulting in a closed algebra for higher-order operators, tools for spectral theory, and the possibility of extending eigendecompositions and SVD to tensors as module endomorphisms (Navasca et al., 2010).

5. Constructive Homomorphic Transformations in Combinatorics and Algebra

Algebra homomorphisms underpin a variety of combinatorial matrix constructions, such as Butson–Hadamard matrix expansion. Embedding homomorphisms (field embeddings via companion matrices and their entry-wise or block-wise extensions) enable construction of larger $BH(mn, t)$ matrices from smaller $BH(n, k)$ instances, with explicit preservation of matrix product and involution. The block-Kronecker approach with algebra homomorphisms ensures that orthogonality and root-of-unity structure transfer to the lifted matrix (Cathain et al., 2019).

6. Implementation, Acceleration, and Practical Impact

Algorithmic and hardware implementation of homomorphic matrix transformations have become central to privacy-preserving machine learning and scientific computing:

• CKKS- and BFV-based protocols can offload secure matrix multiplications, transposition, and general linear algebraic workflows (PCA, QR, SVD, eigen-decomposition) to the cloud or edge-device, with end-to-end FHE protection (Ma, 2023, Bae et al., 20 Mar 2025).
• FPGA and ASIC AI-chip integration achieves orders-of-magnitude practical speedup for matrix transformations by fusing sub-operations at the arithmetic and memory datapath level, exploiting rotation/diagonal structure, and mapping the high-precision arithmetic of HE to dense GEMMs via compiler lifts (Xu et al., 17 Dec 2025, Tong et al., 13 Jan 2025).
• In additively homomorphic encryption (AHE), compression–reconstruction algorithms minimize expensive scalar–ciphertext multiplications by trading for cheap point additions, yielding an order-of-magnitude acceleration for large matrix dimensions on resource-constrained devices (Ramapragada et al., 20 Apr 2025).

The table below summarizes representative transformation paradigms across domains:

Domain	Transformation Form	Structural/Computational Role
Algebraic matrix maps	$S^{-1}x^n S$ conj. powers	Functional calculus, operator theory (Bu et al., 2014)
Structured matrices	$P M Q$ with structured $P, Q$	Structure reduction, fast solvers (Pan, 2013)
Encrypted linear algebra	Rotation-diagonal sum in SIMD slots	Matrix-matrix/vector on ciphertexts (Ma, 2023)
Tensor algebra	Convolution in $R[G]$-module	Generalized operator theory (Navasca et al., 2010)
Combinatorics (BH matrices)	Field embedding/block lift	Recursive expansion (Cathain et al., 2019)

7. Generalizations, Open Questions, and Outlook

• The correspondence between module homomorphisms and tensor convolutional structure suggests extensibility to arbitrary abelian groups and commutative base rings, opening ways to define and compute spectral theory for higher-order arrays (Navasca et al., 2010).
• Homomorphic transformations in the FHE context are subject to bandwidth, key, and arithmetic depth constraints; ongoing improvements in permutation decomposition (Ma et al., 2024), circuit fusion, and hardware co-design are closing the performance gap relative to plaintext linear algebra (Bae et al., 20 Mar 2025).
• The algebraic structure of holomorphic, orthogonally multiplicative maps potentially generalizes to infinite-dimensional settings, but classification results are more complex and non-uniqueness can arise (Bu et al., 2014).
• Trade-offs between invertibility, low displacement rank, and arithmetic overhead continue to play a central role in the search for optimal transforms—both classically and in cryptographically secure computation.

Homomorphic matrix transformations, thus, constitute a unifying theme that bridges classical algebraic analysis, fast and structure-exploiting linear algebra, and privacy-preserving computation at scale.