Assume that I have a finite field $\mathbb{F}$ of prime order $p$, that is $\mathbb{F} = \mathbb{F}_p = \mathbb{Z} / p\mathbb{Z}$. Let's also assume that I fix a certain integer $e$ that will act as the degree of the extension I am looking to work on.
It is of common knowledge that to obtain the extension field of $\mathbb{F}_p$ of order $e$, denoted $\mathbb{F}_{p^e}$, I have to first find an irreducible polynomial $f$ of degree $e$ over $\mathbb{F}_p[X]$ and from this polynomial one obtains that $\mathbb{F}_{p^e} = \mathbb{F}_p[X] / (f)$.
My question concerns the election of the irreducible polynomial: Assuming it exists, how can I obtain the polynomial that produces the best arithmetic in the extension? I.e., I am looking for the polynomial that gives me the most efficient arithmetic in terms of time, meaning that I use as few field elements as possible in the computation of the multiplication of two elements in the extension, for example.
I have seen that the Conway polynomials are the standard way of choosing the irreducible polynomial and that they have some special properties, but I think that they are not the best polynomials for what I am looking for.