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 An efficient implementation of the Chinese Remainder Theorem in minimally redundant Residue Number System

2020 ◽  
Vol 21 (2) ◽  
Author(s):  
Mikhail Selianinau
Keyword(s):  
Chinese Remainder Theorem ◽  
Number System ◽  
Residue Number System ◽  
Computer Applications ◽  
The Novel ◽  
Residue Modulo ◽  
Modern Computer ◽  
Residue Number ◽  
Novel Method ◽  
Redundant Residue Number System

The Chinese Remainder Theorem (CRT) widely used in many modern computer applications. This paper presents an efficient approach to the calculation of the rank of a number, a principal positional characteristic used in the Residue Number System (RNS). The proposed method does not use large modulo addition operations compared to a straightforward implementation of the CRT algorithm. The rank of a number is equal to a sum of an inexact rank and a two-valued correction factor that only takes on the values 0 or 1. We propose a minimally redundant RNS, which provides low computational complexity of the rank calculation. The effectiveness of the novel method is analyzed concerning conventional non-redundant RNS. Owing to the extension of the residue code, by adding the extra residue modulo 2, the complexity of rank calculation goes down from \(O(k^2)\) to \(O(k)\), where \(k\) equals the number of residues in non-redundant RNS.

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.

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 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.

Application of Chinese remainder theorem with fractions to error correction process optimization in the residue number system

2018 ◽  
Vol 16 (2) ◽  
pp. 157-168
Author(s):  
N.I. Chervyakov ◽  
◽  
P.A. Ljahov ◽  
M.G. Babenko ◽  
I.N. Lavrinenko ◽  
...  
Keyword(s):  
Error Correction ◽  
Process Optimization ◽  
Chinese Remainder Theorem ◽  
Number System ◽  
Residue Number System ◽  
Correction Process ◽  
Residue Number

scholarly journals Neural network method for base extension in residue number system

2020 ◽  
Author(s):  
M. Babenko ◽  
E. Shiriaev ◽  
A. Tchernykh ◽  
E. Golimblevskaia
Keyword(s):  
Neural Network ◽  
Elliptic Curve ◽  
Chinese Remainder Theorem ◽  
Homomorphic Encryption ◽  
Computational Cost ◽  
Number System ◽  
Residue Number System ◽  
Pseudorandom Sequence ◽  
Base Extension ◽  
Residue Number

Confidential data security is associated with the cryptographic primitives, asymmetric encryption, elliptic curve cryptography, homomorphic encryption, cryptographic pseudorandom sequence generators based on an elliptic curve, etc. For their efficient implementation is often used Residue Number System that allows executing additions and multiplications on parallel computing channels without bit carrying between channels. A critical operation in Residue Number System implementations of asymmetric cryptosystems is base extension. It refers to the computing a residue in the extended moduli without the application of the traditional Chinese Remainder Theorem algorithm. In this work, we propose a new way to perform base extensions using a Neural Network of a final ring. We show that it reduces 11.7% of the computational cost, compared with state-of-the-art approaches.

An Efficient Implementation of the CRT Algorithm Based on an Interval-Index Characteristic and Minimum-Redundancy Residue Code

2019 ◽  
Vol 17 (10) ◽  
pp. 2050004
Author(s):  
Mikhail Selianinau
Keyword(s):  
Computational Complexity ◽  
Chinese Remainder Theorem ◽  
Computer Applications ◽  
Ancient China ◽  
New Approach ◽  
Low Computational Complexity ◽  
Modern Computer ◽  
Residue Code ◽  
Number Formula ◽  
Efficiency Factors

The Chinese remainder theorem (CRT), which appeared in ancient China, is widely used in many modern computer applications. This paper presents the CRT implementation by using the interval-index characteristic and minimum redundancy residue code. The proposed algorithm does not use large modulo addition operations and provides low computational complexity compared to conventional non-redundant RNS. The efficiency factors of using the minimally redundant RNS increase with the number [Formula: see text] of non-redundant moduli, asymptotically approaching the threshold [Formula: see text]. The new approach presented here will have a significant impact on many non-modular operations in RNS arithmetic, which currently use the CRT.

scholarly journals A Division Algorithm in a Redundant Residue Number System Using Fractions

2020 ◽  
Vol 10 (2) ◽  
pp. 695
Author(s):  
Nikolay Chervyakov ◽  
Pavel Lyakhov ◽  
Mikhail Babenko ◽  
Irina Lavrinenko ◽  
Maxim Deryabin ◽  
...  
Keyword(s):  
Chinese Remainder Theorem ◽  
Arithmetic Operation ◽  
Number System ◽  
Residue Number System ◽  
Experimental Simulation ◽  
Division Algorithm ◽  
Reverse Conversion ◽  
Base Extension ◽  
Mixed Radix Conversion ◽  
Residue Number

The residue number system (RNS) is widely used for data processing. However, division in the RNS is a rather complicated arithmetic operation, since it requires expensive and complex operators at each iteration, which requires a lot of hardware and time. In this paper, we propose a new modular division algorithm based on the Chinese remainder theorem (CRT) with fractional numbers, which allows using only one shift operation by one digit and subtraction in each iteration of the RNS division. The proposed approach makes it possible to replace such expensive operations as reverse conversion based on CRT, mixed radix conversion, and base extension by subtraction. Besides, we optimized the operation of determining the most significant bit of divider with a single shift operation of the modular divider. The proposed enhancements make the algorithm simpler and faster in comparison with currently known algorithms. The experimental simulation using Kintex-7 showed that the proposed method is up to 7.6 times faster than the CRT-based approach and is up to 10.1 times faster than the mixed radix conversion approach.

scholarly journals Construction of Residue Number System Using Hardware Efficient Diagonal Function

Electronics ◽  
2019 ◽  
Vol 8 (6) ◽  
pp. 694 ◽  
Author(s):  
Maria Valueva ◽  
Georgii Valuev ◽  
Nataliya Semyonova ◽  
Pavel Lyakhov ◽  
Nikolay Chervyakov ◽  
...  
Keyword(s):  
Chinese Remainder Theorem ◽  
Number System ◽  
Residue Number System ◽  
Magnitude Comparison ◽  
Reverse Conversion ◽  
Circuit Delay ◽  
Residue Number ◽  
Hardware Costs ◽  
Positional Number System ◽  
Diagonal Function

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.

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