It is no big secret that exponentiation is just multiplication in disguise. It is a short hand way to write an integer times itself multiple times and is especially space saving the larger the exponent becomes. In the same vein, a serious problem with calculating numbers raised to exponents is that they very quickly become extremely large as the exponent increases in value. The following rule provides a great computational advantage when doing modular exponentiation.

The rule for doing exponentiation in modular arithmetic is:

This states that if we take an integer , raise it to an integer power and calculate the result modulo we will get the same result as if we had taken modulo first, raise it to , and calculate that product modulo .

### Proof

I’m winging it on this one, but the quotient remainder theorem hasn’t let me down yet, so I am going to start by defining and using it:

where this means that

where this means that

This time we will just substitute for in the left hand side of our exponentiation rule:

The binomial theorem gives us an easy way to expand the binomial we just introduced:

Fortunately, because we are taking the result modulo c, we can eliminate any terms in the expanded binomial that contain c, leaving us with:

b choose b is equal to 1, so:

From our definition of using quotient remainder theorem, :

And that, as they say, is all she wrote.