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 , , , “right–to–left binary algorithm” can calculate the natural...
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| Vydáno v: | Formalized mathematics Ročník 27; číslo 1; s. 87 - 91 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Bialystok
Sciendo
01.04.2019
De Gruyter Brill Sp. z o.o., Paradigm Publishing Services |
| Témata: | |
| ISSN: | 1426-2630, 1898-9934 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | 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
,
,
, “right–to–left binary algorithm” can calculate the natural number, see (Def. 2), Algo
) :=
mod
and for any integers
,
, “Euclidean algorithm” can calculate the non negative integer gcd(
). We have not formalized computational complexity of algorithms yet, though we had already formalize the “Euclidean algorithm” in [7].
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+log
⌋ and for “Euclidean algorithm”, we formalize the Lamé’s theorem [6], which says the required number of the divisions in this algorithm is at most 5 log
min(
). Our aim is to support the implementation of number theoretic tools and evaluating computational complexities of algorithms to prove the security of cryptographic systems. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1426-2630 1898-9934 |
| DOI: | 10.2478/forma-2019-0009 |