Construction of Residue Number System Using Hardware Efficient Diagonal Function

Abstract

The residue number system (RNS) is a non-positional number system that allows one to perform addition and multiplication operations fast and in parallel. However, because the RNS is a non-positional number system, magnitude comparison of numbers in RNS form is impossible, so a division operation and an operation of reverse conversion into a positional form containing magnitude comparison operations are impossible too. Therefore, RNS has disadvantages in that some operations in RNS, such as reverse conversion into positional form, magnitude comparison, and division of numbers are problematic. One of the approaches to solve this problem is using the diagonal function (DF). In this paper, we propose a method of RNS construction with a convenient form of DF, which leads to the calculations modulo 2 n , 2 n − 1 or 2 n + 1 and allows us to design efficient hardware implementations. We constructed a hardware simulation of magnitude comparison and reverse conversion into a positional form using RNS with different moduli sets constructed by our proposed method, and used different approaches to perform magnitude comparison and reverse conversion: DF, Chinese remainder theorem (CRT) and CRT with fractional values (CRTf). Hardware modeling was performed on Xilinx Artix 7 xc7a200tfbg484-2 in Vivado 2016.3 and the strategy of synthesis was highly area optimized. The hardware simulation of magnitude comparison shows that, for three moduli, the proposed method allows us to reduce hardware resources by 5.98-49.72% in comparison with known methods. For the four moduli, the proposed method reduces delay by 4.92-21.95% and hardware costs by twice as much by comparison to known methods. A comparison of simulation results from the proposed moduli sets and balanced moduli sets shows that the use of these proposed moduli sets allows up to twice the reduction in circuit delay, although, in several cases, it requires more hardware resources than balanced moduli sets. such as field-programmable gate array (FPGA) and application-specific integrated circuit (ASIC). All these attractive features increase interest to RNS in the areas where large amounts of computation are needed. The applications of RNS are digital signal processing [2][3][4], cryptography [5][6][7], digital image processing [8], cloud computing [9], Internet of Things [10] and others. In [11], the authors propose a technique to estimate real-valued numbers by means of the Chinese remainder theorem (CRT), employing for this goal a Kroenecker based M-Estimation, to improve robustness. A new method based on the Chinese remainder theorem (CRT) is proposed for absolute position computation in [12]. This has advantages in terms of hardware and flexibility because it does not use memory. The authors of [13] offer to use RNS to improve the performance of the convolutional neural network developed for pattern recognition tasks. Reference [14] describes the method of construction for finite impulse response filers using RNS.However, the limitations of RNS include some operations such a...

Classification of Moduli Sets for Residue Number System With Special Diagonal Functions

“…It was observed that diagonal modulus of special binary representations (very low or very high Hamming weight) are useful in the construction of RNS with good performance. This point was described in [7].…”

“…Example. Consider quadruples (43, 51, 79, 91), (23, 43, 87, 143), (5,29,93,313) and (3,7,43,2323) however, in practice preference should be given to the quadruple (43, 51, 79, 91), since it is very well balanced, which is confirmed by the smallest value .…”

“…We can't see any perfectly balanced quadruples but have one quadruple with = 0.25. In the previous work [7], only triples {3, 5, 14} and {7, 9, 124} and quadruple {5, 29, 93, 313} were analyzed in the experimental part. These moduli sets have = 0.67, = 2.89 and = 5.00, respectively.…”

“…One way to implement non-modular operations in RNS is to use a diagonal function from [7][8][9][10] requiring division with the remainder by the diagonal module SQ. Division with a remainder in the general case is the most complex arithmetic operation in computer technology.…”

“…Under the condition = 2 − 1 the remainder of the division by SQ is calculated as the sum of the k-bit parts of the dividend modulo SQ [11]. Algorithms for RNS constructing with special SQ forms based on number-theoretical properties are presented in [7] and [8]. The main drawbacks of those algorithms is the impossibility of RNS construction with predetermined dynamic ranges as well as obtaining unbalanced RNSs (with too much difference in modules bit-width) in most cases.…”