ABOUT COMPLEX OPERATIONS IN NON-POSITIONAL RESIDUE NUMBER SYSTEM

Yu. D. Polissky

Abstract


Purpose. The purpose of this work is the theoretical substantiation of methods for increased efficiency of execution of difficult, so-called not modular, operations in non-positional residue number system for which it is necessary to know operand digits for all grade levels. Methodology. To achieve the target the numbers are presented in odd module system, while the result of the operation is determined on the basis of establishing the operand parity. The parity is determined by finding the sum modulo for the values of the number positional characteristics for all of its modules. Algorithm of position characteristics includes two types of iteration. The first iteration is to move from this number to a smaller number, in which the remains of one or more modules are equal to zero. This is achieved by subtracting out of all the residues the value of one of them. The second iteration is to move from this number to a smaller number due to exclusion of modules, which residues are zero, by dividing this number by the product of these modules. Iterations are performed until the residues of one, some or all of the modules equal to zero and other modules are excluded. The proposed method is distinguished by its simplicity and allows you to obtain the result of the operation quickly. Findings. There are obtained rather simple solutions of not modular operations for definition of outputs beyond the range of the result of adding or subtracting pairs of numbers, comparing pairs of numbers, determining the number belonging to the specific half of the range, defining parity of numbers presented in non-positional residue number system. Originality. The work offered the new effective approaches to solve the non-modular operations of the non-positional residue number system. It seems appropriate to consider these approaches as research areas to enhance the effectiveness of the modular calculation. Practical value. The above solutions have high performance and can be effective in developing modular computing structures.


Keywords


residue classes; number; complex operations; positional characteristic; parity number; iteration

References


Akushskiy I.Ya., Yuditskiy D.I. Arifmetika v ostatochnykh klassakh [Machine arithmetic in the residual classes]. Moscow, Sovetskoye radio Publ., 1968. 440 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 (23.11.–25.11.2005)» [Proc. of Anniversary Intern. Sci. and Techn. Conf. «50 years of modular arithmetic»]. Moscow, 2015, pp. 268-273.

Knut D. Iskusstvo programmirovaniya [Programming art]. 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 progres transportu – Science and Transport Progress, 2013, no. 6 (48), pp. 7-22. doi: 10.15802/stp2013/19835.

Magomedov Sh.G. Preobrazovaniye predstavleniy chisel v modulyarnoy arifmetike v sistemakh ostatochnykh klassov s raznymi osnovaniyami [Transformation of numeration in a modular arithmetic in systems of remaining classes with different bases]. Vestnik Astrakhanskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya: «Upravleniye, vychislitelnaya tekhnika, informatika» [Bulletin of Astrakhan state and technical University. Series: «Management, computer technology, informatics»]. Astrakhan, 2014, no. 4, pp. 32-39.

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 rounding algorithms, scaling and dividing numbers in modular arithmetic]. Trudy yubileynoy Mezhdunarodnoy nauchno-tekhnicheskoy konferentsii «50 let modulyarnoy arifmetiki (23.11.–25.11.2005)» [Proc. of Anniversary Intern. Sci. and Techn. Conf. «50 years of modular arithmetic»]. Moscow, 2005, pp. 291-310.

Chervyakov N.I., Sakhnyuk P.A., Shaposhnikov A.V., Ryadnov S.A. Modulyarnyye parallelnyye vychislitelnyye struktury neyroprotsessornykh system [Modular parallel computing structure of neuroprocessor systems]. Moscow, Fizmatlit Publ., 2003. 288 p.

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 University of Railway Transport named after Academician V. Lazaryan], 2007, issue 16, pp. 68-72.

Polisskiy Yu.D. Algoritm vypolneniya slozhnykh operatsiy v sisteme ostatochnykh klassov s pomoshchyu predstavleniya chisel v obratnykh kodakh [Algorithm to perform complex operations in the residual classes system using representation of numbers in reverse codes]. Elektronnoye modelirovaniye – Electronic modeling, 2014, vol. 36, no. 4, pp. 117-122.

Polisskiy Yu.D. O vypolnenii slozhnykh operatsiy v sisteme ostatochnykh klassov [About the implementation of complex transactions in the system of residual classes]. Elektronnoye modelirovaniyeElectronic modeling, 2006, vol. 28, no. 3, pp. 117-123.

Chervyakov N.I. Metody, algoritmy i tekhnicheskaya realizatsiya osnovnykh problemnykh operatsiy, vypolnyaemykh v sisteme ostatochnykh klassov [Methods, algorithms and technical implementation of the basic problem of operations performed in the system of residual 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 (23.11.–25.11.2005)» [Proc. of Anniversary Intern. Sci. and Techn. 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. Intern. 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. Intern. 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. Intern. Journal of Security and Its Applications, 2012, vol. 6, no. 4, pp. 1-12.


GOST Style Citations


  1. Акушский, И. Я. Машинная арифметика в остаточных классах / И. Я. Акушский, Д. И. Юдицкий. – Москва : Сов. радио, 1968. – 440 с.
  2. Ирхин, В. П. Табличная реализация операций модулярной арифметики / В. П. Ирхин // 50 лет модулярной арифметики : тр. юбил. Междунар. науч.-техн. конф. (23.11.–25.11.2005) / Моск. ин-т электрон. техники. – Москва, 2015. – С. 268–273.
  3. Кнут, Д. Искусство программирования / Д. Кнут. – Москва : Диалектика-Вильямс, 2013. – 832 с.
  4. Колесникова, Т. А. Интеграция украинской отраслевой научной периодики в мировое научно-информационное пространство: проблемы и решения / Т. А. Колесникова // Наука та прогрес транспорту. – 2013. – № 6 (48). – С. 7–22. doi: 10.15802/stp2013/19835.
  5. Магомедов, Ш. Г. Преобразование представлений чисел в модулярной арифметике в системах остаточных классов с разными основаниями / Ш. Г. Магомедов // Вестн. Астрах. гос. техн. ун-та. Серия : «Управление, вычислител. техника, информатика». – Астрахань, 2014. – № 4. – С. 32–39.
  6. Методы и алгоритмы округления, масштабирования и деления чисел в модулярной арифметике / Н. И. Червяков [и др.] // 50 лет модуляр. арифметики : тр. юбил. Междунар. науч.-техн. конф. (23.11.–25.11.2005) / Моск. ин-т электрон. техники. – Москва, 2005. – С. 291–310.
  7. Модулярные параллельные вычислительные структуры нейропроцессорных систем : монография / под ред. Н. И. Червякова. – Москва : Физматлит, 2003. – 288 с.
  8. Полисский, Ю. Д. Алгоритм выполнения операции деления чисел на два в системе остаточных классов / Ю. Д. Полисский // Вісн. Дніпропетр. нац. ун–ту залізн. трансп. ім. акад. В.Лазаряна. – Дніпропетровськ, 2007. – Вип. 16. – С. 68–72.
  9. Полисский, Ю. Д. Алгоритм выполнения сложных операций в системе остаточных классов с помощью представления чисел в обратных кодах / Ю. Д. Полисский // Электронное моделирование. – 2014. – Т. 36, № 4. – С. 117–122.
  10. Полисский, Ю. Д. О выполнении сложных операций в системе остаточных классов / Ю. Д. Полисский // Электронное моделирование. – 2006. – Т. 28, № 3. – С. 117–123.
  11. Червяков, Н. И. Методы, алгоритмы и техническая реализация основных проблемных операций, выполняемых в системе остаточных классов / Н. И. Червяков // Инфокоммуник. технологии / Поволж. гос. ун-т телеком. и информ. – Самара, 2011. – № 4. – С. 4–12.
  12. Червяков, Н. И. Методы и принципы построения модулярных нейрокомпьютеров / Н. И. Червяков // 50 лет модулярной арифметики : тр. юбил. Междунар. науч.-техн. конф. (23.11.–25.11.2005) / Моск. ин-т электрон. техники. – Москва, 2005. – С. 232–242.
  13. Boateng, K. O. A Smith-Waterman Algorithm Accelerator Based on Residue Number System / K. O. Boateng, E. Y. Baagyere // Intern. J. of Electronics and Communication Engineering. – 2012. – Vol. 5, № 1. – P. 99–112.
  14. Tomczak, T. Hierarchical residue number systems with small moduli and simple converters / T. Tomczak // Intern. J. of Applied Mathematics and Computer Science. – 2011. – Vol. 21. – Iss. 1. – P. 173–192. doi: 10.2478/v10006-011-0013-2.
  15. Youssef, M. I. Multi-Layer Data Encryption Using Residue Number System in DNA Sequence / M. I. Youssef, A. E. Emam, M. Abd Elghanym // Intern. J. of Security and Its Applications. – 2012. – Vol. 6, № 4. – P. 1–12.


DOI: https://doi.org/10.15802/stp2016/67297

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