A high-speed division algorithm for modular numbers based on the chinese remainder theorem with fractions and its hardware implementation
In this paper, a new simplified iterative division algorithm for modular numbers that is optimized on the basis of the Chinese remainder theorem (CRT) with fractions is developed. It requires less computational resources than the CRT with integers and mixed radix number systems (MRNS). The main idea...
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ir-20.500.12258-50792020-02-18T13:19:26Z A high-speed division algorithm for modular numbers based on the chinese remainder theorem with fractions and its hardware implementation Chervyakov, N. I. Червяков, Н. И. Lyakhov, P. A. Ляхов, П. А. Babenko, M. G. Бабенко, М. Г. Nazarov, A. S. Назаров, А. С. Deryabin, M. A. Дерябин, М. А. Lavrinenko, I. N. Лавриненко, И. Н. Lavrinenko, A. V. Лавриненко, А. В. Division algorithm Modular arithmetic Residue number system (RNS) In this paper, a new simplified iterative division algorithm for modular numbers that is optimized on the basis of the Chinese remainder theorem (CRT) with fractions is developed. It requires less computational resources than the CRT with integers and mixed radix number systems (MRNS). The main idea of the algorithm is (a) to transform the residual representation of the dividend and divisor into a weighted fixed-point code and (b) to find the higher power of 2 in the divisor written in a residue number system (RNS). This information is acquired using the CRT with fractions: higher power is defined by the number of zeros standing before the first significant digit. All intermediate calculations of the algorithm involve the operations of right shift and subtraction, which explains its good performance. Due to the abovementioned techniques, the algorithm has higher speed and consumes less computational resources, thereby being more appropriate for the multidigit division of modular numbers than the algorithms described earlier. The new algorithm suggested in this paper has O (log 2 Q) iterations, where Q is the quotient. For multidigit numbers, its modular division complexity is Q(N), where N denotes the number of bits in a certain fraction required to restore the number by remainders. Since the number N is written in a weighed system, the subtraction-based comparison runs very fast. Hence, this algorithm might be the best currently available 2019-04-08T10:05:31Z 2019-04-08T10:05:31Z 2019 Статья Chervyakov, N., Lyakhov, P.E., Babenko, M., Nazarov, A., Deryabin, M., Lavrinenko, I., Lavrinenko, A. A high-speed division algorithm for modular numbers based on the chinese remainder theorem with fractions and its hardware implementation // Electronics (Switzerland). - 2019. - Volume 8. - Issue 3. - Номер статьи 261 https://www.scopus.com/record/display.uri?eid=2-s2.0-85063445845&origin=resultslist&sort=plf-f&src=s&st1=%09A+high-speed+division+algorithm+for+modular+numbers+based+on+the+chinese+remainder+theorem+with+fractions+and+its+hardware+implementation&st2=&sid=892029d19f7c728d4bff772c02120f7f&sot=b&sdt=b&sl=153&s=TITLE-ABS-KEY%28%09A+high-speed+division+algorithm+for+modular+numbers+based+on+the+chinese+remainder+theorem+with+fractions+and+its+hardware+implementation%29&relpos=0&citeCnt=0&searchTerm= http://hdl.handle.net/20.500.12258/5079 en Electronics (Switzerland) application/pdf application/pdf MDPI AG |
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Division algorithm Modular arithmetic Residue number system (RNS) |
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Division algorithm Modular arithmetic Residue number system (RNS) Chervyakov, N. I. Червяков, Н. И. Lyakhov, P. A. Ляхов, П. А. Babenko, M. G. Бабенко, М. Г. Nazarov, A. S. Назаров, А. С. Deryabin, M. A. Дерябин, М. А. Lavrinenko, I. N. Лавриненко, И. Н. Lavrinenko, A. V. Лавриненко, А. В. A high-speed division algorithm for modular numbers based on the chinese remainder theorem with fractions and its hardware implementation |
| description |
In this paper, a new simplified iterative division algorithm for modular numbers that is optimized on the basis of the Chinese remainder theorem (CRT) with fractions is developed. It requires less computational resources than the CRT with integers and mixed radix number systems (MRNS). The main idea of the algorithm is (a) to transform the residual representation of the dividend and divisor into a weighted fixed-point code and (b) to find the higher power of 2 in the divisor written in a residue number system (RNS). This information is acquired using the CRT with fractions: higher power is defined by the number of zeros standing before the first significant digit. All intermediate calculations of the algorithm involve the operations of right shift and subtraction, which explains its good performance. Due to the abovementioned techniques, the algorithm has higher speed and consumes less computational resources, thereby being more appropriate for the multidigit division of modular numbers than the algorithms described earlier. The new algorithm suggested in this paper has O (log 2 Q) iterations, where Q is the quotient. For multidigit numbers, its modular division complexity is Q(N), where N denotes the number of bits in a certain fraction required to restore the number by remainders. Since the number N is written in a weighed system, the subtraction-based comparison runs very fast. Hence, this algorithm might be the best currently available |
| format |
Статья |
| author |
Chervyakov, N. I. Червяков, Н. И. Lyakhov, P. A. Ляхов, П. А. Babenko, M. G. Бабенко, М. Г. Nazarov, A. S. Назаров, А. С. Deryabin, M. A. Дерябин, М. А. Lavrinenko, I. N. Лавриненко, И. Н. Lavrinenko, A. V. Лавриненко, А. В. |
| author_facet |
Chervyakov, N. I. Червяков, Н. И. Lyakhov, P. A. Ляхов, П. А. Babenko, M. G. Бабенко, М. Г. Nazarov, A. S. Назаров, А. С. Deryabin, M. A. Дерябин, М. А. Lavrinenko, I. N. Лавриненко, И. Н. Lavrinenko, A. V. Лавриненко, А. В. |
| author_sort |
Chervyakov, N. I. |
| title |
A high-speed division algorithm for modular numbers based on the chinese remainder theorem with fractions and its hardware implementation |
| title_short |
A high-speed division algorithm for modular numbers based on the chinese remainder theorem with fractions and its hardware implementation |
| title_full |
A high-speed division algorithm for modular numbers based on the chinese remainder theorem with fractions and its hardware implementation |
| title_fullStr |
A high-speed division algorithm for modular numbers based on the chinese remainder theorem with fractions and its hardware implementation |
| title_full_unstemmed |
A high-speed division algorithm for modular numbers based on the chinese remainder theorem with fractions and its hardware implementation |
| title_sort |
high-speed division algorithm for modular numbers based on the chinese remainder theorem with fractions and its hardware implementation |
| publisher |
MDPI AG |
| publishDate |
2019 |
| url |
https://www.scopus.com/record/display.uri?eid=2-s2.0-85063445845&origin=resultslist&sort=plf-f&src=s&st1=%09A+high-speed+division+algorithm+for+modular+numbers+based+on+the+chinese+remainder+theorem+with+fractions+and+its+hardware+implementation&st2=&sid=892029d19f7c728d4bff772c02120f7f&sot=b&sdt=b&sl=153&s=TITLE-ABS-KEY%28%09A+high-speed+division+algorithm+for+modular+numbers+based+on+the+chinese+remainder+theorem+with+fractions+and+its+hardware+implementation%29&relpos=0&citeCnt=0&searchTerm= https://dspace.ncfu.ru/handle/20.500.12258/5079 |
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