Modular Exponentiation

Modular exponentiation (or powmod, or modpow) is a calculation on integers composed of a power followed by a modulo. This type of calculation is widely used in modern cryptography.

A direct method is to calculate the value of the power then to extract the modulo from it (the remainder in division by n).

In practice, the numbers generated by the powers are gigantic, and mathematicians and computer scientists use simplifications, especially fast exponentiation.

Calculating the last $ x $ digits of $ a^b $ is equivalent to calculating $ a^b \mod n $ with $ n = 10^x $ (the number $ 1 $ followed by $ x $ zeros)

There are several algorithms, but the most efficient one, called (modular) fast exponentiation, uses a property on the binary writing of $ e $.

Writing $ e=\sum_{i=0}^{m-1}a_{i}2^{i} $ over $ m $ bits with $ a_i $ the binary values (0 or 1) in writing in base 2 of $ e $ (with $ a_{m-1} = 1 $)

Then $ b^e $ can be written $$ b^e = b^{\left( \sum_{i=0}^{n-1} a_i \cdot 2^i \right)} = \prod_{i=0}^{n-1} \left( b^{2^i} \right)^{a_i} $$

And so $$ b^e \mod n \equiv \prod_{i=0}^{n-1} \left( b^{2^i} \right)^{a_i} \mod n $$

Here is the implementation of fast modular exponentiation in pseudocode:// pseudocode
function powmod(base b, exponent e, modulus m) {
r = 1
b = b % m
if (b == 0) return 0
while (e > 0) {
if (e % 2) r = (r * b) % m
e = e >> 1
b = (b ** 2) % m
}
return r
}

In theory, the fast powmod algorithm (above) is also the one with the fewest steps. It needs $ m $ steps, with $ m $ the size in bits of the number $ b $ in binary.

In practice, for small values of $ a $, $ b $ and $ n $ calculating the power then the modulo (with Euclidean division) is more instinctive.

This calculation is known as the discrete logarithm problem. Some solutions can be found by brute force but there is no trivial general solution.

Calculus uses exponent and modulus that are generally defined over the natural number domain set N. It is possible to use rational numbers but it is not handled here.