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Applied Sciences
Volume 15
Issue 10
10.3390/app15105332
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

An Efficient Implementation of Montgomery Modular Multiplication Using a Minimally Redundant Residue Number System

by
Mikhail Selianinau
and
Bożena Woźna-Szcześniak
*
Department of Mathematics and Computer Science, Faculty of Science and Technology, Jan Dlugosz University in Czestochowa, al. Armii Krajowej 13/15, 42-200 Czestochowa, Poland
*
Author to whom correspondence should be addressed.
Appl. Sci. 2025, 15(10), 5332; https://doi.org/10.3390/app15105332 (registering DOI)
Submission received: 14 March 2025 / Revised: 24 April 2025 / Accepted: 8 May 2025 / Published: 10 May 2025
(This article belongs to the Special Issue Novel Insights into Cryptography and Network Security)
Versions Notes

Abstract

This paper presents an implementation of modular multiplication based on Montgomery’s scheme within the Residue Number System (RNS). The key innovation of the proposed approach lies in utilizing minimally redundant residue arithmetic, where the rank of a number serves as the primary positional characteristic of the residue code. Additionally, integer numbers are represented in rank form during base extension operations. Due to the low computational complexity of rank calculation in minimally redundant RNS and the specific constraints imposed on the RNS moduli sets, the proposed modular multiplication method achieves up to a 1.5 times performance improvement over non-redundant RNS counterparts. This approach is particularly suited for applications in public key cryptosystems.
Keywords: residue number system; Montgomery multiplication; residue arithmetic; rank of a number; base extension operation; modular exponentiation residue number system; Montgomery multiplication; residue arithmetic; rank of a number; base extension operation; modular exponentiation

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MDPI and ACS Style

Selianinau, M.; Woźna-Szcześniak, B. An Efficient Implementation of Montgomery Modular Multiplication Using a Minimally Redundant Residue Number System. Appl. Sci. 2025, 15, 5332. https://doi.org/10.3390/app15105332

AMA Style

Selianinau M, Woźna-Szcześniak B. An Efficient Implementation of Montgomery Modular Multiplication Using a Minimally Redundant Residue Number System. Applied Sciences. 2025; 15(10):5332. https://doi.org/10.3390/app15105332

Chicago/Turabian Style

Selianinau, Mikhail, and Bożena Woźna-Szcześniak. 2025. "An Efficient Implementation of Montgomery Modular Multiplication Using a Minimally Redundant Residue Number System" Applied Sciences 15, no. 10: 5332. https://doi.org/10.3390/app15105332

APA Style

Selianinau, M., & Woźna-Szcześniak, B. (2025). An Efficient Implementation of Montgomery Modular Multiplication Using a Minimally Redundant Residue Number System. Applied Sciences, 15(10), 5332. https://doi.org/10.3390/app15105332

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