A Method for Implementing the Operation of Modulo Addition of the Residues of Two Numbers in the Residue Number System

Krasnobayev, Victor; Koshman, Sergеy; Kuznetsov, Alexandr

doi:10.1007/s10559-020-00323-9

Cited by 14 publications

(9 citation statements)

References 5 publications

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“…It is possible to increase (ensure) reliability if the computer system will have a certain property, the use of which will allow it to be done. Such a property is defined and called fault tolerance [11][12][13]. With respect to the computer system, the concept of fault tolerance can be understood as the property of the computer system to ensure its operational state in case of failures of the elements included in their composition.…”

Section: Problem Statementmentioning

confidence: 99%

The Structure of the Computer System in the Residual Classes

Koshman

Krasnobayev

Nikolsky

et al. 2023

A.I.S.

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The subject of the article is the formulation and solution of the inverse problem of optimal redundancy in the system of residual classes (RNS) based on the use of the dynamic programming method. The solution of this problem makes it possible to improve the reliability of the operation of computer systems and components (CSC) in the RNS. The purpose of the article is to increase the reliability of the functioning of CSC, which are built on the basis of the use of RNS, without reducing the speed of calculations, as well as to calculate and compare the reliability, in terms of the probability of failure-free operation, of CSC in RNS and a tripled computing system that operates in a positional binary number system (PNS). Tasks: to analyze the influence of the number system used on the reliability of the CSC, taking into account the primary and secondary redundancy; to synthesize a computing system in RNS for a l-byte bit grid based on the use of the passive fault tolerance method (constant structural redundancy); formulate and solve the inverse problem of optimal redundancy in RNS based on the use of the dynamic programming method; to check the correctness of the results obtained, calculate the conditional amount of computer system equipment in the residual classes; evaluate the efficiency of using RNS to improve reliability when building a redundant CSC in relation to a redundant CSC in the PNS. Research methods: methods of analysis and synthesis of computer systems, number theory, coding theory in RNS, reliability theory. The following results are obtained. The paper shows that the use of PNS as a number system does not allow a radical increase in the performance and reliability of CSC. In this regard, the article developed the concept of using RNS as a number system for constructing a CSC. Based on this, the inverse problem of optimal redundancy in RNS is formulated and solved. Conclusions. As shown by the results of calculations and comparative analysis, the use of RNS provides a higher reliability of the CSC than the majority three-channel computing system in the PNS. The obtained research results can be used for the synthesis of fault-tolerant computer structures in RNS.

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Section: Problem Statementmentioning

confidence: 99%

The Structure of the Computer System in the Residual Classes

Koshman

Krasnobayev

Nikolsky

et al. 2023

A.I.S.

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“…The Chinese remainder theorem states that if we know the remainder of the Euclidean division of an integer n by several integers, we can uniquely determine the remainder of the division of n by the product of these integers, provided that the divisors are prime to each other [32][33][34][35]. CRT algorithm is used to convert from RNS to the binary system [36][37][38].…”

Section: Chinese Remainder Theorem (Crt)mentioning

confidence: 99%

An efficient image cryptosystem based on the residue number system and hybrid chaotic maps

Kari¹,

Ghomian

2023

Preprint

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This paper presents a new method for encrypting gray-scale digital images. The proposed method uses a combination of the residue number system and chaotic maps to avoid the complexities of high-dimensional chaotic maps and improve the security of encryption and processing speed. In this method, the image pixel coordinates confusion operation is performed by Arnold's cat map, and the pixel content diffusion step is performed in two phases. In the first phase, the contents of the image pixels are converted to the introduced {(r-1)a, rb, (r + 1)c} residue number system and stored in a matrix. At the same time, a chaotic system produces a combination of Sinusoidal and Logistic maps of chaotic sequences, and after quantization, they are converted into the introduced residue number system and stored in another matrix of the same size. In the second phase of diffusion, XOR operation is performed between these two matrices. The implementation results show that the use of the residue number system, in addition to improving the evaluation parameters, improves the processing time, and the average processing time for encryption is 0.15 seconds. Also, the uniform histogram, the entropy of about 8, and the correlation coefficient close to 0 of the encrypted images all demonstrate the high security of the proposed method.

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“…At the stage of creating a CS, in designing, to determine the degree of compliance with the reliability of the CS, the specified requirements or when performing a comparative analysis of the reliability of various data processing tools, various methods of calculating reliability indicators (RI) are used [13]. The results of the calculation and comparative analysis of RI CS make it possible to solve a number of scientific and practical problems.…”

Section: Problem Statementmentioning

confidence: 99%

Mathematical Model of Computer System Reliability in Residual Classes

Krasnobayev

Koshman²,

Nikolsky³

et al. 2022

A.I.S.

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The subject of the article is the construction of a mathematical model of the reliability of a computer system (CS) that operates in a system of residual classes (RNS). This mathematical model is based on the use of structural sliding reservation. The purpose of the article is to increase the reliability of the CS, which operates in the system of residual classes, as well as to calculate and compare the reliability, in terms of the probability of failure-free operation, of the CS in the RNS and the ternary computing system, which operates in the positional binary number system (PBNS). Tasks: to analyze the influence of the number system used on the reliability of the CS; to investigate the properties of the RNS and determine their influence on the structure of the CS in the RNS; build a mathematical model of reliability and, on its basis, perform the calculation and comparative analysis of the reliability of the CS in the RNS and the tripled CS in the PBNS. Research methods: methods of analysis and synthesis of computer systems, number theory, coding theory in RNS, reliability theory. The following results are obtained. The paper shows that the provision of a given level of reliability in the design of the CS using the RNS is due to the presence of various types of redundancy at the same time: structural, informational, temporal, functional and load. In RNS, these types of redundancy can be effectively used to improve the reliability of the CS. The CS in the RNS represents a computational structure that is identical in structure and principle of operation to the sliding structural redundancy model in the PBNS in the case of a loaded (hot) mode of operation of the redundant elements. Proceeding from this, the paper presents a mathematical model of the reliability of the CS in the RNS. In this paper, the calculation and comparative analysis of the reliability of a tripled computing structure in a PBNS with an ideal majority element and a CS in an RNS with an ideal reliability automaton are carried out. Conclusions. As shown by the results of calculations and comparative analysis in some time intervals of operation, the probability of non-failure operation of the CS in the RNS is higher than the probability of non-failure operation of the aircraft in the PBNS with a tripled majority structure. This implies the effective use of the RNS to improve the reliability of the CS.

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A Method for Implementing the Operation of Modulo Addition of the Residues of Two Numbers in the Residue Number System

Cited by 14 publications

References 5 publications

The Structure of the Computer System in the Residual Classes

The Structure of the Computer System in the Residual Classes

An efficient image cryptosystem based on the residue number system and hybrid chaotic maps

Mathematical Model of Computer System Reliability in Residual Classes

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