Direct Integer Division in RNS and its Hardware Solutions
This work addresses a difficult and underdeveloped operation in RNS for hardware designers, offering incremental improvements in theory and practical implementation.
The paper tackles the challenge of direct integer division in Residue Number Systems (RNS) by reformulating an existing algorithm and introducing a power-based RNS with moduli as powers of primes, which increases dynamic range and improves bit efficiency, simplifying hardware implementation and enhancing scalability.
Residue Number Systems (RNS) offer efficient modular arithmetic and natural parallelism, but direct integer division in RNS remains a difficult and comparatively underdeveloped operation. This paper builds on the type-II division algorithm of Szabo and Tanaka and reformulates it for more efficient hardware implementation. A principal contribution is the introduction of a power-based RNS, in which moduli are selected as powers of natural primes, increasing dynamic range, improving bit efficiency, and providing greater flexibility for scaling during division. The paper further formalizes three decomposition methods required by the division process: multi-factor scaling for modulus-based division, mixed-radix conversion for base extension and comparison, and a new divisor decomposition method introduced in this work. Each method is supported by mathematical development, including analysis of modulus invalidation during computation. These results simplify the hardware structure of the algorithm and improve its scalability. Supported by hardware diagrams and performance tables, the work advances both the theory and practical implementation of direct RNS division.