Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm
In this article we formalize in Mizar [1], [2] the maximum number of steps taken by some number theoretical algorithms, “right–to–left binary algorithm” for modular exponentiation and “Euclidean algorithm” [5]. For any natural numbers a, b, n, “right–to–left binary algorithm” can calculate the natur...
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| Language: | English |
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Sciendo
2019-04-01
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| Series: | Formalized Mathematics |
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| Online Access: | https://doi.org/10.2478/forma-2019-0009 |
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doaj-b8a2fd7a85f240f0855c2e9c16c19d672021-09-05T21:01:04ZengSciendoFormalized Mathematics1426-26301898-99342019-04-01271879110.2478/forma-2019-0009Maximum Number of Steps Taken by Modular Exponentiation and Euclidean AlgorithmOkazaki Hiroyuki0Nagao Koh-ichi1Futa Yuichi2Shinshu University, Nagano, JapanKanto Gakuin University, Kanagawa, JapanTokyo University of Technology, Tokyo, JapanIn this article we formalize in Mizar [1], [2] the maximum number of steps taken by some number theoretical algorithms, “right–to–left binary algorithm” for modular exponentiation and “Euclidean algorithm” [5]. For any natural numbers a, b, n, “right–to–left binary algorithm” can calculate the natural number, see (Def. 2), AlgoBPow(a, n, m) := ab mod n and for any integers a, b, “Euclidean algorithm” can calculate the non negative integer gcd(a, b). We have not formalized computational complexity of algorithms yet, though we had already formalize the “Euclidean algorithm” in [7].https://doi.org/10.2478/forma-2019-0009algorithmspower residueseuclidean algorithm68w4011a0511a1503b35 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Okazaki Hiroyuki Nagao Koh-ichi Futa Yuichi |
| spellingShingle |
Okazaki Hiroyuki Nagao Koh-ichi Futa Yuichi Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm Formalized Mathematics algorithms power residues euclidean algorithm 68w40 11a05 11a15 03b35 |
| author_facet |
Okazaki Hiroyuki Nagao Koh-ichi Futa Yuichi |
| author_sort |
Okazaki Hiroyuki |
| title |
Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm |
| title_short |
Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm |
| title_full |
Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm |
| title_fullStr |
Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm |
| title_full_unstemmed |
Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm |
| title_sort |
maximum number of steps taken by modular exponentiation and euclidean algorithm |
| publisher |
Sciendo |
| series |
Formalized Mathematics |
| issn |
1426-2630 1898-9934 |
| publishDate |
2019-04-01 |
| description |
In this article we formalize in Mizar [1], [2] the maximum number of steps taken by some number theoretical algorithms, “right–to–left binary algorithm” for modular exponentiation and “Euclidean algorithm” [5]. For any natural numbers a, b, n, “right–to–left binary algorithm” can calculate the natural number, see (Def. 2), AlgoBPow(a, n, m) := ab mod n and for any integers a, b, “Euclidean algorithm” can calculate the non negative integer gcd(a, b). We have not formalized computational complexity of algorithms yet, though we had already formalize the “Euclidean algorithm” in [7]. |
| topic |
algorithms power residues euclidean algorithm 68w40 11a05 11a15 03b35 |
| url |
https://doi.org/10.2478/forma-2019-0009 |
| work_keys_str_mv |
AT okazakihiroyuki maximumnumberofstepstakenbymodularexponentiationandeuclideanalgorithm AT nagaokohichi maximumnumberofstepstakenbymodularexponentiationandeuclideanalgorithm AT futayuichi maximumnumberofstepstakenbymodularexponentiationandeuclideanalgorithm |
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