O(n)-depth modular exponentiation circuit algorithm

An O(n)-depth polynomial-size combinational circuit algorithm is proposed for n-bit modular exponentiation, i.e., for the computation of x^v mod m for arbitrary integers x, y, and m represented as n-bit binary integers, within bounds 2^n-1⩽m<2ⁿ and 0⩽Ix, y<m. The algorithm is a generalization of the square-and-multiply method. The terms (x(2^l)mod m)s for all is ε{0, n-1} are computed in [n-1/[αlogn]] parallel rounds, each of which computes [αlogn] consecutive terms, where α⩾1/logn. The circuit implementing a round has depth O((1+α)logn) and size O(n^2(1+α)) yielding a circuit for modular exponentiation of depth O(1+α/αn) and size O(n^3+2α/αlogn)