Fast exponentiation using the truncation operation

AbstractGiven an integerk, and anarbitrary integer greater than $$2^{2^k } $$ , we prove a tight bound of $$\Theta (\sqrt k )$$ on the time required to compute $$2^{2^k } $$ with operations {+, −, *, /, ⌊·⌋, ≤}, and constants {0, 1}. In contrast, when the floor operation is not available this computation requires Ω(k) time. Using the upper bound, we give an $$O(\sqrt {\log n} )$$ time algorithm for computing ⌊log loga⌋, for alln-bit integersa. This upper bound matches the lower bound for computing this function given by Mansour, Schieber, and Tiwari. To the best of our knowledge these are the first non-constant tight bounds for computations involving the floor operation.