Direct Integer Division in RNS and its Hardware Solutions

TL;DR

This paper introduces a power-based RNS and new decomposition methods to enable efficient hardware implementation of direct integer division in residue number systems.

Contribution

It reformulates the type-II division algorithm for RNS, introduces a power-based RNS, and formalizes three decomposition methods to enhance hardware efficiency and scalability.

Findings

Increased dynamic range and bit efficiency with power-based RNS.

Simplified hardware structure for RNS division.

Improved scalability demonstrated through hardware diagrams and performance tables.

Abstract

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.