Modular Exponentiation

It consists in an exponentiation followed by a modulus, but it exists optimized algorithms with big numbers to return a fast result without having to actually perform the calculation (called fast, thanks to mathematical simplifications).

Example: $$ 12^{34} \equiv 16 \mod 56 $$

The word power indicates the name of the operation, and exponent to indicate the operand.

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)

Example: $ 3 ^ 9 = 19683 $ and $ 3 ^ 9 \mod 100 = 83 $ (the last 2 digits)

There are several algorithms, here is the shortest one in pseudocode:// pseudocode
function powmod(base b, exponent e, modulus m)
if m = 1 then return 0
var c := 1
for var a from 1 to e
c := (c * b) mod m
end for
return c

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