Hi,
there is a package for Galois fields. See
share/contrib/gf/gf_manual.pdf
for documentation. It is work in progress, so it will be slightly
different in the next version. In 5.29.1 a session looks like this.
(%i1) display2d : false$
(%i2) gf_set(3, 4);
(%o2) [x,x^4+x+2]
This defines F3^4 with x as a generator of the group (F3^4)* of order 80
(%i3) [gf_primitive(), gf_order()];
(%o3) [x,80]
and with x^4+x+2 as the irreducible reduction polynomial.
(%i4) [gf_reduction(), gf_irreducible_p(x^4+x+2)];
(%o4) [x^4+x+2,true]
Now I print 0 thru 80 literally, in base 3 and viewed as a polynomial in F3^4.
(%i5) for n:0 thru 80 do printf(true, "~d : ~3R : ~a ~%", n, n, gf_n2p(n))$
0 : 0 : 0
1 : 1 : 1
2 : 2 : 2
3 : 10 : x
4 : 11 : x+1
5 : 12 : x+2
6 : 20 : 2*x
7 : 21 : 2*x+1
8 : 22 : 2*x+2
9 : 100 : x^2
10 : 101 : x^2+1
11 : 102 : x^2+2
12 : 110 : x^2+x
13 : 111 : x^2+x+1
14 : 112 : x^2+x+2
15 : 120 : x^2+2*x
16 : 121 : x^2+2*x+1
17 : 122 : x^2+2*x+2
18 : 200 : 2*x^2
etc.
Hope that helps to get started.
Volker van Nek
2013/2/19 Sara Mussie <saramussie at gmx.net>:
> Hi everybody,
>
> hope all is well with you.
>
> I am working on a (2,n)-Threshold Secret Sharing Scheme based on the vector space construction. My example works over the elements of GF(2)^4. I am not sure about the right setting of my parameters for any random number of participants.
>
> Can anybody tell me the elements of GF(3)^4 expressed as polynomials?
>
> V = GF(2)^4 ; 16 elements
>
> expressed as polynomials:
>
> V = {0, 1, X, X+1, X^2, X^2+1, X^2+X, X^2+X+1, X^3, X^3+1, X^3+X, X^3+X+1,
> X^3 + X^2, X^3 + X^2 + 1, X^3 + X^2 + X, X^3 + X^2 + X + 1}
>
> T = GF(3)^4 ; 81 elements
> expressed as polynomials:
> ?
>
> Thanks you.
>
> Regards,
> Sara
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