I am working on a small computer program to generate multiplication table of general finite field in form of GF(p^m) where p is a prime number and m is an integer greater than one. While the special case of p=2 has been extensively investigated, I can find little on addressing the general form of GF(M) where M=p^m and p!=2.
First I need to mention that for my specific application I work with relatively small values of M, let's say M<1000. I already have the addition and multiplication table of GF(p^1) and I used irreducible polynomials for tackling GF(2^m). On the general case if GF(p^m) I have found few algorithms including the work of Adleman(1) and also a paper by Couveignes(2) . Again special cases of 3^3 has been discussed here(3). Moreover there is a discussion on the subject here(4) that has been clearly mentioned that construction of irreducible polynomials for a general case is not a trivial problem and also the work of Adlmen has been mentioned and discussed. I have found all those works too complicated for my specific application that the processing time is not very prohibitive and probably even a brute force solution will not be too expensive for me (considering p^m<1000). I came across this paper(5) and the author described a simple algorithm on pages 5 and 6 under the title "II. THE MASSEY-OMURA ALGORITHM". I found the original work but it was specialized for GF(2^m) so I use this paper to ask my question: On top of page 6 there is the following equation:
A=A^(m-1)=||a_{hk}||
My understanding is that this matrix refers to the multiplication table of GF(p^1). Am I right? If not can someone point on how I can construct this matrix or any brute force algorithm to generate multiplication table of GF(p^m) for small values of p^m, let's say p^m<1000? By the way I am an engineer and not a mathematician so I appreciate your elaborate responses in advance.
(1) "dl.acm.org/citation.cfm?id=12166" (2) "arxiv.org/pdf/0905.1642.pdf" (3) "math.stackexchange.com/questions/1050966/find-all-irreducible-monic-polynomials-of-degree-3-in-mathbb-z-3-mathbb-zx?rq=1" (4) "math.stackexchange.com/questions/150646/construction-of-irreducible-polynomials-over-finite-fields" (5) "dspace.mit.edu/bitstream/handle/1721.1/3025/P-1716-19450906.pdf?sequence=1"