The exponentiations calculation of the form Cd mod n takes time O(logdM(n)), here M(n) is the cost of multiplying two n-bit integers and can be as O(log2 n). As soon as d = O(n), all these algorithms take time O(log3 n). CRT-RSA has dp = dq = O(√(n)) (so that log dp = log dq = O(log(n)/2)), for an overall cost of O(2(log(n)/2)3). This gives the Equation (15), with 4 times efficiency. Mprime RSA just extend the CRT-RSA for the decryption like it calculates like Equation (16) 〖 M〗_i= C^(d_i ) mod p_i (16) for 1 ≤ i ≤ k. Then by using CRT to the Equation (16) we can get the plain text M= Cd mod n. Mprime RSA has di = O(n1/k) (so that log