Dot Product via Modular Addition (DoMA)

• DoMA is a modular arithmetic framework that replaces traditional multiplication–accumulation workflows with parallel, compressed operations across classical, quantum, and photonic systems.
• It employs Q-adic packing, Kronecker substitution, and coding techniques to perform efficient dot products and matrix multiplications with robust error correction.
• DoMA underpins applications in privacy-preserving computations, secure multi-party protocols, and advanced quantum-photonic processing, ensuring scalability and high performance.

Dot Product via Modular Addition (DoMA) encompasses a spectrum of arithmetic and hardware-accelerated strategies for accelerating dot product and matrix multiplication tasks over modular domains. These techniques are foundational in both classical and quantum computing, as well as photonic and privacy-preserving linear algebra. DoMA’s unifying principle is the replacement of traditional multiplication–accumulation workflows with schemes that exploit modular arithmetic, polynomial evaluation, combinatorial coding, or interference-based physical processes. By restructuring dot product evaluation in this fashion, DoMA achieves computational parallelism, efficient packing in hardware, robust error correction, and enhanced privacy guarantees.

1. Modular Packing and Compression

The primary instantiation of DoMA is in compressed modular matrix multiplication (0803.1975, 0809.0063), where residues modulo a prime $p$ are packed into a single machine word via a $Q$-adic (base-$Q$) representation. For a vector $a = [a_0, \ldots, a_d]$, the packed word is $\bar{a} = \sum_{i=0}^d a_i Q^i$. The packing is designed such that addition, subtraction, and certain types of dot product operations over $\mathbb{Z}/p\mathbb{Z}$ can be executed through word-level arithmetic, leveraging the parallelism of hardware arithmetic units.

In practical terms, dot products in the compressed representation are realized through polynomial multiplication. Specifically, storing one operand in reverse $Q$-adic order enables extraction of the dot product via the $Q^d$ coefficient following a single integer (or floating point) multiplication:

$$\sum_{i=0}^d a_i b_{d-i} \equiv \text{Coeff}_{Q^d} \Big( \bar{a}_{\text{rev}} \cdot \bar{b} \Big) \pmod{p}$$

Matrix multiplication is performed by compressing matrix rows or columns, multiplying in the compressed domain, and then decompressing—often in bulk using the REDQ algorithm for simultaneous modular reduction.

2. Kronecker Substitution and Efficient Polynomial Dot Product

When extending modular dot product computation to small finite fields and polynomial rings (0809.0063), Kronecker substitution is deployed. A polynomial $a(X) = \sum_{i=0}^{k-1} a_i X^i$ is packed using the Discrete $Q$-adic Transform (DQT):

$\text{DQT}(a) = \sum_{i=0}^{k-1} a_i q^i$

This substitution translates the problem into integer arithmetic, allowing the dot product (or matrix multiplication) to be performed rapidly via classical hardware. The key algorithmic advancement is REDQ, which enables simultaneous unpacking and modular reduction of all coefficients after a single bulk multiplication:

• Compute $r = \sum a_i b_{j-i} q^j$
• Extract and reduce coefficients via

$$u_i = \lfloor r / q^i \rfloor - p \lfloor (r/p) / q^i \rfloor$$

and corrections

$\mu_i = (u_i - q u_{i+1}) \mod p$

This approach reduces the number of expensive modular reductions and facilitates high-throughput modular polynomial algebra rivaling numerical BLAS performance.

3. Fault-Tolerant Dot Product Engines and Coding Integration

In hardware dot product engines (DPEs), particularly those implemented in crossbar arrays or resistive memories, DoMA is fundamental to enabling self-protecting computation (Roth, 2017). Here, the matrix storing the operands is systematically encoded to embed redundancy directly in the modular arithmetic of the DPE’s operation. Error correction and detection is possible in both the $L_1$–metric (Lee/Manhattan metric) and Hamming metric.

Coding schemes operate by appending check columns computed via modular congruences. For example, single-error correcting schemes use code locators $\alpha_j$ so that each row $a$ of the matrix satisfies:

$$\sum_{j=0}^{k-1} a_j \alpha_j + \sum_{j=0}^{m-1} a_{k+j} \alpha_{k+j} \equiv 0 \pmod{2n+1}$$

Decoding and error location are accomplished by syndrome computation. This construction leverages exclusively modular additions, aligning precisely with DoMA’s arithmetic. Extensions to double-error correction and recursive multilayer coding further enhance reliability and distance properties for neuromorphic and analog-digital architectures.

4. Algebraic and Combinatorial Structure over Finite Rings

DoMA principles are closely linked with the algebraic theory of dot products over finite rings (The et al., 2020). For a subset $E \subset (\mathbb{Z}/n\mathbb{Z})^d$, the dot-product set

$\Pi(E) = \{ x \cdot y : x, y \in E \}$

behaves surjectively under sufficient density conditions on $E$, as established by Fourier analytic techniques. This property ensures that all residue classes in the ring are realizable via dot products from $E$, underpinning randomness and distribution properties central to applications in coding theory, cryptography, and spectral graph theory. The framework enables robust incidence geometry and combinatorics in modular contexts.

5. Quantum and Photonic Implementations

DoMA generalizes elegantly into quantum and photonic computation domains. In quantum protocols for inner product, matrix addition, and matrix multiplication (Zenchuk et al., 2023), amplitudes are manipulated via unitary transformations such as multiqubit Toffoli-type gates and modular additions implemented through Hadamard transforms. Ancilla qubits facilitate garbage removal, and the depth of quantum circuits for addition is $O(1)$, with logarithmic scaling for products. The modular-additive approach ensures that the computational flow is efficiently mapped onto quantum gates and measurement probabilities.

In photonic engines, inverse-designed optical cavities realize the DoMA framework physically (Mathur, 18 Jul 2025). Inputs $x$ and $y$ are encoded optically; the cavity executes

$(x+y)^2 - (x-y)^2 = 4xy$

by means of engineered interference and phase shifts—constructive interference corresponds to addition, destructive to subtraction. The measured photocurrents at the outputs directly yield the dot product, eliminating the need for explicit modular addition hardware. Inverse design techniques optimize the entire cavity morphology to achieve this mapping, enabling dramatic reductions in area (88%), power (23.43%), and energy in practical transformer training. The resultant analog computation is robust and scalable.

6. Privacy-Preserving Binary DoMA in Multi-Party Computation

DoMA is also central to secure multi-party computation protocols for binary dot products (Dehkordi et al., 18 Oct 2025). Here, logical AND operations (i.e., bitwise multiplication) are reformulated algebraically:

$$a_i \wedge b_i = \frac{(a_i + b_i) - [(a_i + b_i) \bmod 2]}{2}$$

The aggregate dot product over $n$ coordinates is the sum of these. For $l$-party computations, the analogous formula generalizes to modular reduction over $l$:

$$d = \frac{(a_1 + \ldots + a_l) - [(a_1 + \ldots + a_l) \bmod l]}{l}$$

Private computation proceeds with random masking in an expanded field and a three-party oblivious transfer protocol to enforce modular reduction privacy. The approach significantly reduces overhead compared with classical secret sharing and scales efficiently in distributed user environments.

7. Comparison, Limitations, and Implications

DoMA techniques supplant multiplication–accumulation pipelines with compressed modular arithmetic, polynomial evaluation, combinatorial error control, quantum circuit design, or engineered photonic interference. Classical implementations are bounded in packing capacity by $Q$, $p$, and machine word size: $(d+1)(p-1)^2 < Q$ and $Q^{d+1} < 2^\beta$ for mantissa size $\beta$. Quantum and photonic schemes leverage hardware parallelism and interference to optimize for depth and latency, with bounded resource requirements determined by the number of qubits or cavity design.

In privacy-preserving settings, DoMA replaces heavyweight cryptographic primitives with modular arithmetic, achieving information-theoretic guarantees with minimal computational and communication complexity.

The spectrum of DoMA applications encompasses classical linear algebra over finite fields, polynomial arithmetic, fault-tolerant dot product engines, quantum and photonic processing, combinatorial geometry over rings, and privacy-preserving machine learning.

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Summary Table: DoMA Implementations Across Modalities

Modality	Key Construction	Efficiency/Implication
Classical Comp.	Q-adic/packed polynomial dot product	Hardware parallelism, reduced reductions
Coding/Hardware	Redundant modular encoding	Error correction in DPEs, in situ reliability
Algebraic	Fourier/Combinatorial ring analysis	Surjectivity results, code design
Quantum	Toffoli/Hadamard/amplitude protocols	$O(1)$/logarithmic depth, garbage removal
Photonic	Inverse-designed cavity, interference	Area/power/energy reduction, analog speed
Privacy-Preserving	Modular addition, masking, OT	Efficient, scalable MPC for binary vectors

DoMA comprises a versatile framework for implementing efficient, robust, and scalable dot product and matrix multiplication operations, leveraging modular arithmetic across computation, hardware, quantum, photonic, and secure distributed systems.