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2022-12-14T04:15:09Z
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Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm
Okazaki, Hiroyuki
Nagao, Koh-ichi
Futa, Yuichi
© 2019 Hiroyuki Okazaki et al., published by SciendoThis work is licensed under the Creative Commons Attribution-ShareAlike 4.0 Public License.
algorithms
power residues
Euclidean algorithm
In this article we formalize in Mizar the maximum number of steps taken by some number theoretical algorithms, “right–to–left binary algorithm” for modular exponentiation and “Euclidean algorithm” 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). W have not formalized computational complexity of algorithms yet, though we had already formalize the “Euclidean algorithm” in. For “right-to-left binary algorithm”, we formalize the theorem, which says that the required number of the modular squares and modular products in this algorithms are 1+blog2 nc and for “Euclidean algorithm”, we formalize the Lam´e's theorem, which says the required number of the divisions in this algorithm is at most 5 log10 min(|a|, |b|). Our aim is to support the implementation of number theoretic tools and evaluating computational complexities of algorithms to prove the security of cryptographic systems.
Article
FORMALIZED MATHEMATICS 27(1) : 87-91(2019)
SCIENDO
2019-05-16
eng
journal article
VoR
http://hdl.handle.net/10091/0002001207
https://soar-ir.repo.nii.ac.jp/records/2001207
https://doi.org/10.2478/forma-2019-0009
10.2478/forma-2019-0009
1898-9934
FORMALIZED MATHEMATICS
27
1
87
91
https://soar-ir.repo.nii.ac.jp/record/2001207/files/17K00182_1.pdf
application/pdf
2022-10-25