High-Performance Computation in Residue Number System Using Floating-Point Arithmetic

Abstract

Residue number system (RNS) is known for its parallel arithmetic and has been used in recent decades in various important applications, from digital signal processing and deep neural networks to cryptography and high-precision computation. However, comparison, sign identification, overflow detection, and division are still hard to implement in RNS. For such operations, most of the methods proposed in the literature only support small dynamic ranges (up to several tens of bits), so they are only suitable for low-precision applications. We recently proposed a method that supports arbitrary moduli sets with cryptographically sized dynamic ranges, up to several thousands of bits. The practical interest of our method compared to existing methods is that it relies only on very fast standard floating-point operations, so it is suitable for multiple-precision applications and can be efficiently implemented on many general-purpose platforms that support IEEE 754 arithmetic. In this paper, we make further improvements to this method and demonstrate that it can successfully be applied to implement efficient data-parallel primitives operating in the RNS domain, namely finding the maximum element of an array of RNS numbers on graphics processing units. Our experimental results on an NVIDIA RTX 2080 GPU show that for random residues and a 128-moduli set with 2048-bit dynamic range, the proposed implementation reduces the running time by a factor of 39 and the memory consumption by a factor of 13 compared to an implementation based on mixed-radix conversion.

Improved Modular Division Implementation with the Akushsky Core Function

“…In addition, the value Y = (1, 10, 6, 4) is fed to the inverter. The result is −Y = (10, 3,11,15). After calculating the core function by the constant multiplication and addition blocks, C(−Y) = 130, 980 is obtained.…”

“…We get |X| = (6, 9,14,14). Representations of powers "2" in RNS can be calculated in advance, depending on the range of RNS (the highest occurrence power of 2 n is log 2 K ) and stored in memory: 2 14 = (5, 4, 13, 6), 2 13 = (8, 2, 15, 3), 2 12 = (4, 1,16,11), 2 11 = (2, 7,8,15), 2 10 = (1, 10, 4, 17), 2 9 = (6, 5, 2, 18), 2 8 = (3, 9, 1, 9), 2 7 = (7, 11,9,14), 2 6 = (9, 12, 13, 7), 2 5 = (10, 6,15,13), 2 4 = (5, 3,16,16), 2 3 = (8, 8,8,8),…”

“…Using the formula (2), we calculate ∆ 1 = X − 2 7 • Y: 9,14,14) − (7, 11,9,14)• (1,10,6,4) = (10, 3,11,15), ∆ 1 < 0.…”

“…It means that 2 5 is included in the representation of the quotient Q. Let us check 2 4 , by calculating ∆ 4 = ∆ 3 − 2 4 • Y: 11,16,10) − (5, 3,16,16) • (1, 10,6,4) = (4, 7, 5, 3), ∆ 4 > 0.…”

“…An important issue is to choose an effective method for encoding numerical information, i.e., selecting a number representation for fast processing. The residue number system (RNS) is used to improve the efficiency of data encryption algorithms [2][3][4], cloud computing [5][6][7], digital signal processing [8,9], wireless networks [10], matrix computing [11,12], and artificial neural networks [13,14]. One of the computationally complex operations in RNS is the Euclidean division or remainder division.…”