<abstract xmlns="http://www.w3.org/1999/xhtml">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 <italic>a</italic>, <italic>b</italic>, <italic>n</italic>, “right–to–left binary algorithm” can calculate the natural number, see (Def. 2), AlgoBPow(<italic>a, n, m</italic>) := <italic>ab</italic> mod <italic>n</italic> and for any integers <italic>a</italic>, <italic>b</italic>, “Euclidean algorithm” can calculate the non negative integer gcd(<italic>a, b</italic>). 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+log2<italic>n</italic>⌋ 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 log10 min(<italic>|a|, |b|</italic>). Our aim is to support the implementation of number theoretic tools and evaluating computational complexities of algorithms to prove the security of cryptographic systems.</abstract>

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].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+log2n⌋ 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 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.

Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 Public License.

{"article-title":"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...

Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm

Published Online: May 16, 2019

Page range: 87 - 91

Accepted: Mar 11, 2019

DOI: https://doi.org/10.2478/forma-2019-0009

Keywords
algorithms, power residues, Euclidean algorithm

© 2019 Hiroyuki Okazaki et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 Public License.

Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm

Published Online: May 16, 2019

Page range: 87 - 91

Accepted: Mar 11, 2019

DOI: https://doi.org/10.2478/forma-2019-0009

Keywordsalgorithms, power residues, Euclidean algorithm

© 2019 Hiroyuki Okazaki et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 Public License.

Keywords
algorithms, power residues, Euclidean algorithm