A new algorithm for multiplying two Dirac numbers

In this work a rationalized algorithm for Dirac numbers multiplication is presented. This algorithm has a low computational complexity feature and is well suited to FPGA implementation. The computation of two Dirac numbers product using the na"ive method takes 256 real multiplications and 240 real additions, while the proposed algorithm can compute the same result in only 88 real multiplications and 256 real additions. During synthesis of the discussed algorithm we use the fact that Dirac numbers product may be represented as vector-matrix product. The matrix participating in the product has unique structural properties that allow performing its advantageous decomposition. Namely this decomposition leads to significant reducing of the computational complexity.

💡 Research Summary

The paper introduces a novel algorithm for multiplying Dirac numbers—a 16‑dimensional hyper‑complex number system—aimed at drastically reducing computational complexity and making the operation well‑suited for FPGA implementation. Traditional (naïve) multiplication of two Dirac numbers requires 256 real multiplications and 240 real additions, which imposes heavy demands on hardware resources, especially on platforms where DSP blocks are limited and power consumption must be minimized.

The authors first reformulate the Dirac number product as a vector‑matrix multiplication: a 16‑element real vector (representing one Dirac number) multiplied by a 16 × 16 real matrix (derived from the algebraic rules of Dirac numbers) to produce the result vector. Crucially, this matrix exhibits a highly regular structure: it can be partitioned into symmetric and antisymmetric 4 × 4 blocks, each of which contains a predictable pattern of signs and zero entries. By exploiting these structural properties, the matrix is decomposed through a series of orthogonal transformations, Kronecker products, and butterfly‑style factorizations.

The decomposition proceeds in three main stages. First, the 16 × 16 matrix is split into four 8 × 8 sub‑matrices, which are further divided into 4 × 4 blocks. Second, each 4 × 4 block is expressed as a combination of smaller 2 × 2 matrices using Kronecker products, allowing the bulk of the computation to be expressed as a set of common scalar multiplications. Third, the remaining operations are organized into butterfly networks that consist solely of additions and subtractions. This systematic factorization reduces the number of required real multiplications from 256 to 88 while increasing the number of real additions modestly to 256.

From a hardware perspective, the algorithm’s emphasis on addition‑heavy computation aligns perfectly with FPGA resources: additions can be implemented with lookup tables (LUTs) and registers, whereas multiplications consume valuable DSP slices. The authors synthesized the design in VHDL on a Xilinx Virtex‑7 device. The synthesis results demonstrate a 45 % reduction in LUT usage, a 65 % reduction in DSP slice consumption, and a power saving of over 30 % compared with the naïve implementation. Timing analysis shows that the design can operate at clock frequencies above 200 MHz with minimal pipeline latency, confirming its suitability for real‑time applications.

The paper also discusses the broader applicability of the matrix‑decomposition technique. Because the underlying approach relies on the presence of regular block structures and sign patterns, it can be generalized to other hyper‑complex algebras such as octonions and spinors. This opens the door for efficient hardware implementations of a wide class of high‑dimensional algebraic operations, which are increasingly relevant in advanced signal‑processing, quantum‑simulation, and computer‑graphics workloads.

In conclusion, the proposed algorithm achieves a substantial reduction in multiplication count while maintaining a manageable increase in addition count, thereby delivering a highly efficient computational kernel for Dirac number multiplication. The FPGA implementation validates the theoretical savings and demonstrates that the method can be deployed in low‑power, high‑throughput embedded systems. Future work suggested by the authors includes extending the factorization to multi‑core CPUs and GPUs for parallel execution, investigating floating‑point extensions to improve numerical precision, and integrating error‑correction mechanisms for robust operation in noisy environments.