Representing finite fields

Question

I was reading Field Theory. Few basic things I know are-

1. For every prime $p$ and natural number $n$, there exist a finite field of order $p^n$.
2. Multiplicative group of finite field is cyclic. So, if I know the generator then I can find the field. Is there way to find the generator in general?
3. We take irreducible polynomial over $F_p$ of degree $n$ or the splitting field of the polynomial $x^{p^n}-x$ to construct the finite field.

I was curious to know, if there are some other ways to represent finite fields. Because, even if I want to write all the elements of $F_{16}$, it becomes very tedious to calculate all the elements in quotient ring $F_p[x]/\langle f(x)\rangle$, where $f(x)$ is irreducible polynomial over $F_p$ of degree $n$.

To sumup, I want to know is there any method by which I can represent elements of finite field in some compact form.

Answer 1

Zech Logarithms are useful for performing addition when you represent elements as powers of a multiplicative generator.

They depend on a choice of the minimal polynomial of the generator. Although all finite fields of order $p^n$ are isomorphic, the minimal polynomial is not unique. Conway polynomials see here or here provide a useful method of choosing the minimal polynomial.

Computer algebra systems use Conway polynomials and Zech logarithms for computing in finite fields of small or moderate size. A problem is that they have to be computed in advance and, although it only has to be done once for each prime power, computing the Conway polynomial gets increasingly difficult and time-consuming as the field grows larger. Some promising alternatives have been proposed.

Answer 2

Is there way to find the generator in general?

One method is to check that the order of a possible generator = q-1. Create a list of all combinations of the prime factors of q-1 (normal math), and test all product combinations except for q-1. The main issue with this method is factoring q-1 if q-1 is large.

For example, for GF(2^8), q-1 = 255, the factors are 3, 5, 17, and the combinations to check are (3 × 5) = 15, (3 × 17) = 51, (5 × 17) = 85. So to test for a possible generator α:

α^15 != 1
α^51 != 1
α^85 != 1

These exponentiations can be sped up using exponentiation by squaring:

https://simple.wikipedia.org/wiki/Exponentiation_by_squaring