I am not sure if I am talking to the correct community (perhaps stack overflow is best). I want to be able to compute $g^x = \underbrace{g \cdot g \cdot \dots \cdot g}_\text{x amount of times}$ really fast, where $g \in GF\left(2^m \right)$.
Here m is rather large, $m = 256$, $384$, $512$, etc. so lookup tables are not the solution. I know that there are really fast algorithms for a similar idea, modpow for $\mathbb{Z}/n\mathbb{Z}$ (see page 619-620 of HAC).
- What is a fast, non-table based way to compute cycles (i.e. $g^x$)?
- This is definitely a wishful question but here it comes: Can the idea of montgomery multiplication/exponention be 'recycled' to galois fields?