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 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]