ON THE TRANSFORMATION OF REPRESENTATION OF NUMBERS IN THE RESIDUO FROM ONE MODULE SYSTEM TO ANOTHER
DOI:
https://doi.org/10.15802/stp2016/74735Keywords:
residual classes, number, complex operations, positional characteristic, system of modules, iterationAbstract
Purpose. The purpose of this work is the theoretical foundation of one of the approaches to improve the effectiveness of the number system in nonpositional residual classes non-modular, so-called complex operation, the realization of which requires knowledge of all the digits of operands charges. The operation consists in transformation of the representation of one of the modules of its representation in the other system of modules. Methodology. The tools of research methodology are the system analysis, theory of numbers, the Chinese remainder theorem. The technique uses a representation of number as its residues, and in the polyadic code and is based on the determination of the balance of the module based on the obtained residues on the remaining modules of the original system. Such a determination is performed by sequentially subtracting of constants from the obtained residues of the original number and summing of these constants to the results, which are formed by the required modules. Thus, constant at each iteration are selected from pre-calculated tables depending on the value of the residue in the analyzed discharge. The proposed method is algorithmically simple and at circuit implementation can create the computational structures of high performance and reliability. Findings. The theoretical justification for this approach to obtain effective solutions of non-modular transformation operation in the system of residual classes for transition from representation of the number by the one system of units to its representation by the other system of modules. Originality. A theoretical justification of the proposed approach to the solution of a non-modular conversion operations in the residue number system for the transition from representation of number in one system of units to its representation in the other system of modules was proposed. This approach is appropriate to consider as one of the areas of research ways to improve the computational efficiency. Practical value. It follows from the importance of the theoretical conclusions and results of the study. It consists in the fact that it is justified a simple and effective approach to the problem of implementation of non-modular conversion operations in the residue number system for the transition from representation of the number in one system of units to its representation in the other system of modules. The above mentioned solutions have a high speed and may be effective in the development of modular computing structures for advanced information technologies.
References
Akushskiy I.Ya., Yuditskiy D.I. Mashinnaya arifmetika v ostatochnykh klassakh [Machine arithmetic in the residual classes]. Moscow, Sovetskoye radio Publ., 1968. 440 p.
Vdovin V.M., Surkova L.Ye., Valentinov V.A. Teoriya sistem i sistemnyy analiz [Systems theory and systems analysis]. Moscow, Dashkov i K° Publ., 2010. 640 p.
Gabidulin E.M., Kshevetskiy A.S., Kolybelnikov A.I. Zashchita informatsii [Data protection]. Moscow, MFTI Publ., 2011. 262 p.
Irkhin V.P. Tablichnaya realizatsiya operatsiy modulyarnoy arifmetiki [Tabular implementation of modular arithmetic operations]. Trudy yubileynoy Mezhdunarodnoy nauchno-tekhnicheskoy konferentsii «50 let modulyarnoy arifmetiki» [Proc. of Anniversary Intern. Sci. and Techn. Conf. «50 years of modular arithmetic»]. Moscow, 2015, pp. 268-273.
Knut D. Iskusstvo programmirovaniya. Tom 2. Poluchislennyye algoritmy (Programming art. Vol. 2. Semi-numerical algorithms). Moscow, Dialektika-Vilyams Publ., 2013. 832 p.
Kolesnykova T.O. Integratsiya ukrainskoy otraslevoy nauchnoy periodiki v mirovoye nauchno-informatsionnoye prostranstvo: problemy i resheniya [Integration of Ukrainian industry scientific periodicals into world scientific information space: problems and solutions]. Nauka ta prohres transportu – Science and Transport Progress, 2013, no. 6 (48), pp. 7-22. doi: 10.15802/stp2013/19835.
Chervyakov N.I. Lavrinenko I.N., Lavrinenko S.V., Mezentseva O.S. Metody i algoritmy okrugleniya, masshtabirovaniya i deleniya chisel v modulyarnoy arifmetike [Methods and algorithms for rounding, scaling, and dividing numbers in modular arithmetic]. Trudy yubileynoy Mezhdunarodnoy nauchno-tekhnicheskoy konferentsii «50 let modulyarnoy arifmetiki» [Proc. of Anniversary Intern. Sci. and Technical Conf. «50 years of modular arithmetic»]. Moscow, 2015, pp. 291-310.
Chervyakov N.I. Modulyarnyye parallelnyye vychislitelnyye struktury neyroprotsessornykh sistem [Modular parallel computing structure of neuroprocessor systems]. Moscow, Fizmatlit Publ., 2003. 288 p.
Nesterenko Yu.V. Teoriya chisel [Number theory]. Moscow, Akademiya Publ., 2008. 272 p.
Nesterova L.Yu., Feklistov S.V. Kitayskaya teorema ob ostatkakh v oblasti glavnykh idealov [Chinese remainder theorem in principal ideal]. Molodoy uchenyy – Young Scientist, 2014, no. 19, pp. 1-4.
Pereslavtseva O.N. Rasparallelivanie algoritmov s primeneniyem kitayskoy teoremy ob ostatkakh [Parallelization of the algorithms using the Chinese remainder theorem]. Vestnik Tambovskogo universiteta. Seriya: Estestvennyye i tekhnicheskiye nauki [Bulletin of Tambov University. Series: Natural and Technical Sciences], 2009, no. 4, pp. 779-781.
Polisskiy Yu.D. Algoritm vypolneniya operatsii deleniya chisel na dva v sisteme ostatochnykh klassov [The algorithm of operation performing of dividing the number by two in the system of residual classes]. Visnyk Dnipropetrovskoho natsionalnoho universytetu zaliznychnoho transportu imeni akademika V. Lazariana [Bulletin of Dnipropetrovsk National Univesity of Railway Transport named after Academician V. Lazaryan], 2007, issue 16, pp. 68-72.
Polisskiy Yu.D. Vypolneniye slozhnykh operatsiy v modulyarnykh vychislitelnykh strukturakh [Performing of complex operations in modular computing structures]. Dopovidi XV mіzhnarodnoi konferentsii z avtomatichnoho upravlіnnia «Avtomatika-2008» [Proc. of the XV Intern. Conference on Automatic Control «Automation-2008»]. National University «Odessa Maritime Academy», Odessa, 2008, Part І, pp. 444-446.
Polisskiy Yu.D. O vypolnenii slozhnykh operatsiy v sisteme ostatochnykh klassov [On the implementation of complex operations in the residue number system]. Elektronnoye modelirovaniye – Electronic Modeling, 2006, vol. 28, no. 3, pp.117-123.
Polisskiy Yu.D. Pro odyn metod rozshyrennia diapazonu zobrazhennia chysel u systemi zalyshkovykh klasiv [A method for expanding the range of image numbers in the system of residual classes]. Matematychne modeliuvannia – Mathematical Modelling, 2007, no. 2, pp. 16-17.
Chervyakov N.I., Averbukh V.M., Babenko M.G. Priblizhennyy metod vypolneniya nemodulnykh operatsiy v sisteme ostatochnykh klassov [An approximate method for performing non-modular operations in the residue number system]. Fundamentalnyye issledovaniya – Fundamental Research, 2012, no. 6 (1), pp. 189-193.
Chervyakov N.I. Metody, algoritmy i tekhnicheskaya realizatsiya osnovnykh problemnykh operatsiy, vypolnyayemykh v sisteme ostatochnykh klassov [Methods, algorithms and technical implementation of the basic problem operations performed in the system of remaining classes]. Infokommunikatsionnyye tekhnologii – Information and Communication Technologies, 2011, no. 4, pp. 4-12.
Chervyakov N.I. Metody i printsipy postroyeniya modulyarnykh neyrokompyuterov [Methods and principles of construction of modular neural computers]. Trudy yubileynoy Mezhdunarodnoy nauchno-tekhnicheskoy konferentsii «50 let modulyarnoy arifmetiki» [Proc. of Anniversary International Sci. and Technical Conf. «50 years of modular arithmetic»]. Moscow, 2005, pp. 232-242.
Boateng K.O., Baagyere E.Y. A Smith-Waterman Algorithm Accelerator Based on Residue Number System. International Journal of Electronics and Communication Engineering, 2012, vol. 5, no. 1, pp. 99-112.
Tomczak T. Hierarchical residue number systems with small moduli and simple converters. International Journal of Applied Mathematics and Computer Science, 2011, vol. 21, issue 1, pp. 173-192. doi: 10.2478/v10006-011-0013-2
Youssef M.I., Emam A.E., Abd Elghanym M. Multi-Layer Data Encryption Using Residue Number System in DNA Sequence. International Journal of Security and Its Applications, 2012, vol. 6, no. 4, pp. 1-12.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 Yu. D. Polissky
This work is licensed under a Creative Commons Attribution 4.0 International License.
Copyright and Licensing
This journal provides open access to all of its content.
As such, copyright for articles published in this journal is retained by the authors, under the terms of the Creative Commons Attribution 4.0 International License (CC BY 4.0). The CC BY license permits commercial and non-commercial reuse. Such access is associated with increased readership and increased citation of an author's work. For more information on this approach, see the Public Knowledge Project, the Directory of Open Access Journals, or the Budapest Open Access Initiative.
The CC BY 4.0 license allows users to copy, distribute and adapt the work in any way, provided that they properly point to the author. Therefore, the editorial board of the journal does not prevent from placing published materials in third-party repositories. In order to protect manuscripts from misappropriation by unscrupulous authors, reference should be made to the original version of the work.
