Jan 122016

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:

a^b \bmod{c} = ((a \bmod{c})^b) \bmod{c}

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

Continue reading »