Residue Number System Comparison revisited, a software perspective

TL;DR

This paper introduces a new method for comparing two numbers in Residue Number System using an additional modulus, enabling full-range comparison without special moduli constraints, and offers computational efficiency improvements.

Contribution

The paper presents a novel comparison method in RNS that works universally across bases and reduces complexity, enhancing RNS operation capabilities.

Findings

Allows full-range comparison in RNS without special moduli

Single conversion to mixed radix with O(n^2) complexity

Parallelization reduces complexity to O(log n)

Abstract

This paper presents a novel method to compare two numbers in Residue Number System (RNS) using an additional modulus, which is often already available because it is required in modular computations and digital signal processing scaling.Our method provides the comparison of two integers in the full range of the RNS base. It does not require moduli of a special form, unlike other state-of-the-art methods that are restricted to specific RNS bases or require bounds on input numbers. Our approach only requires one single conversion to a mixed radix representation with a complexity of O(n2), which can be reduced to O(log(n)) in time with parallelization. This provides a significant advantage over classical methods and more recent competitive methods which work under restrictions. This opens perspectives for advancements in challenging RNS operations such as division, scaling, and cryptographic applications.