Analysis of the Chinese remainder theorem and cyclotomic polynomials-based algorithms for cyclic convolution?Part I: Rational number system

1997 ◽  
Vol 16 (5) ◽  
pp. 569-594
Author(s):  
H. K. Garg
Keyword(s):  
Rational Number ◽  
Chinese Remainder Theorem ◽  
Number System ◽  
Cyclotomic Polynomials ◽  
Cyclic Convolution

scholarly journals Computationally Efficient Approach to Implementation of the Chinese Remainder Theorem Algorithm in Minimally Redundant Residue Number System

2021 ◽  
Author(s):  
Mikhail Selianinau
Keyword(s):  
Chinese Remainder Theorem ◽  
Number System ◽  
Residue Number System ◽  
Computer Applications ◽  
Efficient Approach ◽  
Residue Modulo ◽  
Modern Computer ◽  
New Variant ◽  
Residue Number ◽  
Residue Code

AbstractIn this paper, we deal with the critical problem of performing non-modular operations in the Residue Number System (RNS). The Chinese Remainder Theorem (CRT) is widely used in many modern computer applications. Throughout the article, an efficient approach for implementing the CRT algorithm is described. The structure of the rank of an RNS number, a principal positional characteristic of the residue code, is investigated. It is shown that the rank of a number can be represented by a sum of an inexact rank and a two-valued correction to it. We propose a new variant of minimally redundant RNS, which provides low computational complexity for the rank calculation, and its effectiveness analyzed concerning conventional non-redundant RNS. Owing to the extension of the residue code, by adding the excess residue modulo 2, the complexity of the rank calculation goes down from $O\left (k^{2}\right )$ O k 2 to $O\left (k\right )$ O k with respect to required modular addition operations and lookup tables, where k equals the number of non-redundant RNS moduli.

scholarly journals Improved Modular Division Implementation with the Akushsky Core Function

Computation ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 9
Author(s):  
Mikhail Babenko ◽  
Andrei Tchernykh ◽  
Viktor Kuchukov
Keyword(s):  
Approximate Method ◽  
Chinese Remainder Theorem ◽  
Number System ◽  
Residue Number System ◽  
Division Algorithm ◽  
Incorrect Result ◽  
Core Function ◽  
Residue Number ◽  
Division Operation

The residue number system (RNS) is widely used in different areas due to the efficiency of modular addition and multiplication operations. However, non-modular operations, such as sign and division operations, are computationally complex. A fractional representation based on the Chinese remainder theorem is widely used. In some cases, this method gives an incorrect result associated with round-off calculation errors. In this paper, we optimize the division operation in RNS using the Akushsky core function without critical cores. We show that the proposed method reduces the size of the operands by half and does not require additional restrictions on the divisor as in the division algorithm in RNS based on the approximate method.

NOVEL DESIGN AND FPGA IMPLEMENTATION OF DA-RNS FIR FILTERS

2004 ◽  
Vol 13 (06) ◽  
pp. 1233-1249 ◽  
Author(s):  
WEI WANG ◽  
M. N. S. SWAMY ◽  
M. O. AHMAD
Keyword(s):  
Power Efficiency ◽  
Filter Design ◽  
Finite Impulse Response ◽  
Chinese Remainder Theorem ◽  
Low Cost ◽  
Digital Signal ◽  
Number System ◽  
Residue Number System ◽  
Fpga Implementation ◽  
Fir Filters

Field programmable gate array (FPGA)-based digital signal processing has been widely used in multimedia applications. By combining distributed arithmetic (DA) and residue number system (RNS) in such designs, efficient area, speed and power efficiency can be achieved. In this paper, we propose novel techniques for the design and FPGA implementation of DA-RNS finite impulse response (FIR) filters. By introducing a novel low-cost moduli set and its selection method, efficient modulo arithmetic units inside the subfilters are designed. Then, a new residue-to-binary conversion algorithm, a so-called modified DA Chinese remainder theorem, is derived to reduce the modulo operations and provide an efficient residue-to-binary converter suitable to FPGA implementation. Based on these proposed techniques, a seventh-order DA-RNS FIR filter is designed, implemented and tested by using Xilinx FPGA tools. The implementation results show that the proposed filter design consumes only 77% of the power that the existing filter12,13 requires, while maintaining the same speed (throughput).

scholarly journals Zero-Aware Low-Precision RNS Scaling Scheme

Axioms ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 5
Author(s):  
Amir Sabbagh Molahosseini
Keyword(s):  
High Precision ◽  
Deep Neural Networks ◽  
State Of The Art ◽  
Chinese Remainder Theorem ◽  
Number System ◽  
Residue Number System ◽  
Scaling Factor ◽  
Improve Performance ◽  
High Area ◽  
Residue Number

Scaling is one of the complex operations in the Residue Number System (RNS). This operation is necessary for RNS-based implementations of deep neural networks (DNNs) to prevent overflow. However, the state-of-the-art RNS scalers for special moduli sets consider the 2k modulo as the scaling factor, which results in a high-precision output with a high area and delay. Therefore, low-precision scaling based on multi-moduli scaling factors should be used to improve performance. However, low-precision scaling for numbers less than the scale factor results in zero output, which makes the subsequent operation result faulty. This paper first presents the formulation and hardware architecture of low-precision RNS scaling for four-moduli sets using new Chinese remainder theorem 2 (New CRT-II) based on a two-moduli scaling factor. Next, the low-precision scaler circuits are reused to achieve a high-precision scaler with the minimum overhead. Therefore, the proposed scaler can detect the zero output after low-precision scaling and then transform low-precision scaled residues to high precision to prevent zero output when the input number is not zero.

scholarly journals Multiple Error Correction in Redundant Residue Number Systems: A Modified Modular Projection Method with Maximum Likelihood Decoding

2022 ◽  
Vol 12 (1) ◽  
pp. 463
Author(s):  
Mikhail Babenko ◽  
Anton Nazarov ◽  
Maxim Deryabin ◽  
Nikolay Kucherov ◽  
Andrei Tchernykh ◽  
...  
Keyword(s):  
Error Detection ◽  
Chinese Remainder Theorem ◽  
Number System ◽  
Maximum Likelihood Decoding ◽  
Number Systems ◽  
Residue Number Systems ◽  
Error Detection And Correction ◽  
Residue Number ◽  
Weighted Number ◽  
Multiple Error

Error detection and correction codes based on redundant residue number systems are powerful tools to control and correct arithmetic processing and data transmission errors. Decoding the magnitude and location of a multiple error is a complex computational problem: it requires verifying a huge number of different possible combinations of erroneous residual digit positions in the error localization stage. This paper proposes a modified correcting method based on calculating the approximate weighted characteristics of modular projections. The new procedure for correcting errors and restoring numbers in a weighted number system involves the Chinese Remainder Theorem with fractions. This approach calculates the rank of each modular projection efficiently. The ranks are used to calculate the Hamming distances. The new method speeds up the procedure for correcting multiple errors and restoring numbers in weighted form by an average of 18% compared to state-of-the-art analogs.

Research, Reflection, Practice: Introducing Percents in Linear Measurement to Foster an Understanding of Rational-Number Operations

2003 ◽  
Vol 9 (6) ◽  
pp. 335-339 ◽  
Author(s):  
Joan Moss
Keyword(s):  
Rational Number ◽  
Traditional Instruction ◽  
Number System ◽  
Rational Numbers ◽  
Linear Measurement ◽  
Common Denominator ◽  
Computational Fluency ◽  
Decimal Numbers ◽  
Young Students

How do we foster computational fluency with rational numbers when this topic is known to pose so many conceptual challenges for young students? How can we help students understand the operations of rational numbers when their grasp of the quantities involved in the rational-number system is often very limited? Traditional instruction in rational numbers focuses on rules and memorization. Teachers often give students instructions such as, “To add fractions, first find a common denominator, then add only the numerators” or “To add and subtract decimal numbers, line up the decimals, then do your calculations.”

scholarly journals Data Hiding in Digital Image for Efficient Information Safety Based on Residue Number System

2021 ◽  
pp. 35-44
Author(s):  
Joseph B. Eseyin ◽  
Kazeem A. Gbolagade
Keyword(s):  
Digital Image ◽  
Chinese Remainder Theorem ◽  
Data Communication ◽  
Number System ◽  
Residue Number System ◽  
Least Significant Bit ◽  
Residue Number ◽  
Information Safety ◽  
Execution Speed ◽  
Almost All

The mass dispersal of digital communication requires the special measures of safety. The need for safe communication is greater than ever before, with computer networks now managing almost all of our business and personal affairs. Information security has become a major concern in our digital lives. The creation of new transmission technologies forces a specific protection mechanisms strategy particularly in data communication state.  We proposed a steganography method in this paper, which reads the message, converting it into its Residue Number System equivalent using the Chinese Remainder Theorem (CRT), encrypting it using the Rivest Shamir Adleman (RSA) algorithm before embedding it in a digital image using the Least Significant Bit algorithm of steganography and then transmitting it through to the appropriate destination and from which the information required to reconstruct the original message is extracted. These techniques will enhance the ability to hide data and the hiding of ciphers in steganographic image and the implementation of CRT will make the device more efficient and stronger. It reduces complexity problems and improved execution speed and reduced the time taken for processing the encryption and embedding competencies.

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