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Volume 9
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Review

Review of Modular Multiplication Algorithms over Prime Fields for Public-Key Cryptosystems

by
Hai Huang
,
Jiwen Zheng
,
Zhengyu Chen
,
Shilei Zhao
*,
Hongwei Wu
,
Bin Yu
and
Zhiwei Liu
School of Computer Science and Technology, Harbin University of Science and Technology, Harbin 150080, China
*
Author to whom correspondence should be addressed.
Cryptography 2025, 9(2), 46; https://doi.org/10.3390/cryptography9020046
Submission received: 13 May 2025 / Revised: 9 June 2025 / Accepted: 12 June 2025 / Published: 17 June 2025
(This article belongs to the Section Cryptography Reviews)
Versions Notes

Abstract

Modular multiplication is a pivotal operation in public-key cryptosystems such as RSA, ElGamal, and ECC. Modular multiplication design is crucial for improving overall system performance due to the large-bit-width operation with high computational complexity. This paper provides a classification of integer multiplication algorithms based on their implementation principles. Furthermore, the core concepts, implementation challenges, and research advancements of multiplication algorithms are systematically summarized. This paper also gives a brief overview of modular reduction algorithms for various types of moduli and discusses the implementation principles, application scenarios, and current research results. Finally, the detailed research development of modular multiplication algorithms in four major classes over prime fields is deeply analyzed and summarized, making it essential as a guide for future research.
Keywords: public-key cryptosystems; prime fields; modular multiplication; integer multiplication; modular reduction public-key cryptosystems; prime fields; modular multiplication; integer multiplication; modular reduction

Share and Cite

MDPI and ACS Style

Huang, H.; Zheng, J.; Chen, Z.; Zhao, S.; Wu, H.; Yu, B.; Liu, Z. Review of Modular Multiplication Algorithms over Prime Fields for Public-Key Cryptosystems. Cryptography 2025, 9, 46. https://doi.org/10.3390/cryptography9020046

AMA Style

Huang H, Zheng J, Chen Z, Zhao S, Wu H, Yu B, Liu Z. Review of Modular Multiplication Algorithms over Prime Fields for Public-Key Cryptosystems. Cryptography. 2025; 9(2):46. https://doi.org/10.3390/cryptography9020046

Chicago/Turabian Style

Huang, Hai, Jiwen Zheng, Zhengyu Chen, Shilei Zhao, Hongwei Wu, Bin Yu, and Zhiwei Liu. 2025. "Review of Modular Multiplication Algorithms over Prime Fields for Public-Key Cryptosystems" Cryptography 9, no. 2: 46. https://doi.org/10.3390/cryptography9020046

APA Style

Huang, H., Zheng, J., Chen, Z., Zhao, S., Wu, H., Yu, B., & Liu, Z. (2025). Review of Modular Multiplication Algorithms over Prime Fields for Public-Key Cryptosystems. Cryptography, 9(2), 46. https://doi.org/10.3390/cryptography9020046

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