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 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 Transparent Memory Tests Based on the Double Address Sequences

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 894
Author(s):  
Ireneusz Mrozek ◽  
Vyacheslav N. Yarmolik
Keyword(s):  
Control Systems ◽  
Time Complexity ◽  
Computer Applications ◽  
Testing Methods ◽  
Functional Diagnostics ◽  
Automated Control ◽  
New Approach ◽  
Modern Computer ◽  
Network Servers ◽  
Periodic Testing

An important achievement in the functional diagnostics of memory devices is the development and application of so-called transparent testing methods. This is especially important for modern computer systems, such as embedded systems, systems and networks on chips, on-board computer applications, network servers, and automated control systems that require periodic testing of their components. This article analyzes the effectiveness of existing transparent tests based on the use of the properties of data stored in the memory, such as changing data and their symmetry. As a new approach for constructing transparent tests, we propose to use modified address sequences with duplicate addresses to reduce the time complexity of tests and increase their diagnostic abilities.

Closed Sets of Boolean Terms in Relational Databases

1991 ◽  
Vol 14 (3) ◽  
pp. 367-385
Author(s):  
Andrzej Jankowski ◽  
Zbigniew Michalewicz
Keyword(s):  
Computational Complexity ◽  
Incomplete Information ◽  
Relational Databases ◽  
The Other ◽  
New Approach ◽  
Closed Sets ◽  
Null Value ◽  
Algebraic Operations

A number of approaches have been taken to represent compound, structured values in relational databases. We review a few such approaches and discuss a new approach, in which every set is represented as a Boolean term. We show that this approach generalizes the other approaches, leading to more flexible representation. Boolean term representation seems to be appropriate in handling incomplete information: this approach generalizes some other approaches (e.g. null value mark, null variables, etc). We consider definitions of algebraic operations on such sets, like join, union, selection, etc. Moreover, we introduce a measure of computational complexity of these operations.

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