Where All Roads Meet: Modular Exponentiation

Abstract

In addition to the calculational rules for addition, subtraction, and multiplication in residue classes we can also define an operation of exponentiation, where the exponent specifies how many times the base is to be multiplied by itself. Exponentiation is carried out, as usual, by means of recursive calls to multiplication: For a in ℤ _m we have a⁰:= ī and a^e+1 := a · a^e.

For a long time on that spot I stood,

Where two roads converged in the wood and I thought:

“Someone going the other way

Might someday stop here for the sake

Of deciding which path to take.”

But my direction lay where it lay.

And walking on, I felt a sense

Of wonder at that difference.

—Ilya Bernstein, Attention and Man