Improved Modular Multiplication Algorithms Using Solely IEEE 754 Binary Floating-Point Operations

In this paper, we propose three modular multiplication algorithms that use only the IEEE 754 binary floating-point operations. Several previous studies have used floating-point operations to perform modular multiplication. However, they considered only positive integers and did not utilize the dedic...

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Veröffentlicht in:IEEE transactions on emerging topics in computing Jg. 13; H. 3; S. 1259 - 1271
Hauptverfasser: Sugizaki, Yukimasa, Takahashi, Daisuke
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.07.2025
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
Approximation algorithms
Arithmetic
Codes
Computer arithmetic
Floating point arithmetic
Instruction sets
Integers
Libraries
multiple precision arithmetic
numerical algorithms
Polynomials
Processors
Training
Transforms
Vectors
ISSN:2168-6750, 2168-6750
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Zusammenfassung:In this paper, we propose three modular multiplication algorithms that use only the IEEE 754 binary floating-point operations. Several previous studies have used floating-point operations to perform modular multiplication. However, they considered only positive integers and did not utilize the dedicated sign bit in the floating-point representation. Our first algorithm is an extension of these studies, which are based on Shoup multiplication. By allowing operands to be negative, we increased the maximum supported modulus size by approximately 1.21 times. Our remaining two algorithms are based on Montgomery multiplication for positive and signed integers, respectively. Although these algorithms require more round-to-integral operations, they support a modulus size of up to twice as large as that for Shoup multiplication for positive integers. For processors with relatively low round-to-integral performance, we propose versions of the three algorithms without the round-to-integral operation. Evaluations on four CPUs with different levels of instruction performance show that floating-point-based algorithms, including the proposed algorithms, can be regarded as alternatives to integer-based algorithms for mid-sized moduli, especially when floating-point operations are faster on the processors.
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ISSN:2168-6750
2168-6750
DOI:10.1109/TETC.2025.3582551