Hi Volker,
that helps to get started. I use the irreducible primitive polynomial x^4+x+1. It shouldn't make any different right? but will look at the package now.
Thank you.
Regards,
Sara
-------- Original-Nachricht --------
> Datum: Tue, 19 Feb 2013 13:25:00 +0100
> Von: Volker van Nek <volkervannek at gmail.com>
> An: Sara Mussie <saramussie at gmx.net>
> CC: Maxima at math.utexas.edu
> Betreff: Re: [Maxima] GF(3^4)
> 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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