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 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.