Efficient modular exponentation of powers Is there any way to efficiently compute $(((\mathrm{base}^{M_1})^{M_2})^{M_3} \dots )^{M_n}$ modulo $P$, where $P$ is prime? One way is to repeatedly do modular exponentiation for each of the powers. Another way, we could compute the Multiplicative order and powers thereafter would just be 1. Are there any alternative ways to do this efficiently?

IF $P$ divides $b,$ the remainder will be $0$

Else $(b,P)=1$

$$((b^{M_1})^{M_2})^{M_3}\cdots)^{M_n}=b^{M_1\cdot M_2\cdot M_3\cdots M_n}$$

As $P$ is prime, $a^{P-1}\equiv\pmod P$ if $(a,P)=1$ using Fermat's Little Theorem.

So, we need to find $M_1\cdot M_2\cdot M_3\cdots M_n\pmod {(P-1)}$

Then use repeated exponentiation.