Running Time of Modular Exponentiation

Question

I am trying to understand why the modular exponentiation algorithm has a running time of about $\log(n)$ where $n$ is the exponent. Here is the algorithm:

function modular_pow(base, exponent, modulus)
    result := 1
    while exponent > 0
        if (exponent mod 2 == 1):
           result := (result * base) mod modulus
        exponent := exponent >> 1
        base = (base * base) mod modulus
    return result

I think it has something to do with the number of bits for the binary representation of the exponent. What is also an intuitive numerical example?

Answer 1

The main idea is that $a^x \cdot a^x = a^{2x}$.

Now, in order to get, say, $a^{10}$ you can do $res = res * a$ for 10 times. On the other hand you can do compute $a^2=a\cdot a$, then $a^4=a^2\cdot a^2$, then $a^8 = a^4 \cdot a^4$, and get the final result by multiplying $a^8$ with $a^2$. That is, we "break" the computation in a similar way to the binary representation of the exponent.

Let me give do the example in more details, and then give the general statement. note that $10 = 1010_{(2)}$, and can think of each '1' bit as an element we need to multiply in order to get the final answer (here $a^8$ for the 4th bit, and $a^2$ for the 2nd lsb bit. The power is the same as the "value" of that bit in binary..).

Generaly, in order to compute $a^{exp}$, consider $exp_{(2)}$: the MSB is always 1, and you will have to compute the element $a^{2^{\lfloor \log exp \rfloor}}$, which will take you $\log exp$ (modular) multiplications, and generate $a^2, a^4, a^8, a^{16}, ..., a^{2^{\lfloor \log exp \rfloor}}$. Finally, with another up to $\log exp$ multiplication you will multiply all the elements that correspond to a '1' bit in the binary representation of $exp$.