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

Abstract

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.