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14 pages, 305 KB

Open AccessArticle

An Efficient Parallel Reverse Conversion of Residue Code to Mixed-Radix Representation Based on the Chinese Remainder Theorem

by Mikhail Selianinau and Yuriy Povstenko

Entropy 2022, 24(2), 242; https://doi.org/10.3390/e24020242 - 5 Feb 2022

Cited by 5 | Viewed by 3240

Abstract

In this paper, we deal with the critical problems in residue arithmetic. The reverse conversion from a Residue Number System (RNS) to positional notation is a main non-modular operation, and it constitutes a basis of other non-modular procedures used to implement various computational algorithms. We present a novel approach to the parallel reverse conversion from the residue code into a weighted number representation in the Mixed-Radix System (MRS). In our proposed method, the calculation of mixed-radix digits reduces to a parallel summation of the small word-length residues in the independent modular channels corresponding to the primary RNS moduli. The computational complexity of the developed method concerning both required modular addition operations and one-input lookup tables is estimated as

O (k^{2} / 2)

, where k equals the number of used moduli. The time complexity is

O (⌈\log_{2} k⌉)

modular clock cycles. In pipeline mode, the throughput rate of the proposed algorithm is one reverse conversion in one modular clock cycle. Full article

(This article belongs to the Special Issue Theory and Applications of Information Processing Algorithms)

17 pages, 909 KB

Open AccessArticle

A High-Speed Division Algorithm for Modular Numbers Based on the Chinese Remainder Theorem with Fractions and Its Hardware Implementation

by Nikolai Chervyakov, Pavel Lyakhov, Mikhail Babenko, Anton Nazarov, Maxim Deryabin, Irina Lavrinenko and Anton Lavrinenko

Electronics 2019, 8(3), 261; https://doi.org/10.3390/electronics8030261 - 27 Feb 2019

Cited by 12 | Viewed by 5129

Abstract

In this paper, a new simplified iterative division algorithm for modular numbers that is optimized on the basis of the Chinese remainder theorem (CRT) with fractions is developed. It requires less computational resources than the CRT with integers and mixed radix number systems (MRNS). The main idea of the algorithm is (a) to transform the residual representation of the dividend and divisor into a weighted fixed-point code and (b) to find the higher power of 2 in the divisor written in a residue number system (RNS). This information is acquired using the CRT with fractions: higher power is defined by the number of zeros standing before the first significant digit. All intermediate calculations of the algorithm involve the operations of right shift and subtraction, which explains its good performance. Due to the abovementioned techniques, the algorithm has higher speed and consumes less computational resources, thereby being more appropriate for the multidigit division of modular numbers than the algorithms described earlier. The new algorithm suggested in this paper has O (log₂ Q) iterations, where Q is the quotient. For multidigit numbers, its modular division complexity is Q(N), where N denotes the number of bits in a certain fraction required to restore the number by remainders. Since the number N is written in a weighed system, the subtraction-based comparison runs very fast. Hence, this algorithm might be the best currently available. Full article

(This article belongs to the Special Issue Approximate Computing: Design, Acceleration, Validation and Testing of Circuits, Architectures and Algorithms in Future Systems)

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