O(n)-depth circuit algorithm for modular exponentiation

An O(n)-depth polynomial-size combinational circuit algorithm is proposed for n-bit modular exponentiation, i.e., for the computation of “x^y mod m” for arbitrary integers x, y and m. Represented as n-bit binary integers, within bounds 2^n-1⩽m<2ⁿ and 0⩽x,y<m. The algorithm is a generalization of the square-and-multiply method. An obvious implementation of the square-and-multiply method yields a circuit of depth O(nlogn) and size O(n³). In the proposed algorithm, the terms x² mod m´s for all i´s ε{O,…,(n-1)} are computed in [n-1/[αlogn]] parallel rounds, each of which computes [αlogn] consecutive terms, where 1/logn⩽α. The circuit implementing a round has depth O((1+α)log n) and size O(n^2(1+α)) yielding a circuit for modular exponentiation of depth O((1+α/α)n) and size O(n^3+2α/αlogn)