Problem
Research questions and friction points this paper is trying to address.
Residue Number System
number comparison
modular arithmetic
full dynamic range
RNS base
Residue Number System Comparison revisited, a software perspective
🤖 AI Summary
This work addresses the long-standing challenge of general integer comparison in residue number systems (RNS) by proposing an efficient method based on the introduction of an auxiliary modulus and a single mixed-radix conversion. The approach is applicable to arbitrary RNS moduli sets without imposing restrictions on the input range, thereby overcoming limitations inherent in existing techniques that require specific modulus forms or bounded dynamic ranges. The algorithm achieves a time complexity of O(n²), which can be parallelized to O(log n), significantly outperforming both classical and recent state-of-the-art methods constrained by such assumptions. This advancement provides a novel and practical solution to a critical bottleneck in RNS-based applications, including division, scaling, and cryptographic operations.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Residue Number System
number comparison
mixed radix conversion
parallel complexity
modular arithmetic
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