Hi,
yes, you are right. It is modulo 2 but not modulo 3.
Can you tell me if x^4+x+2 is primitive irreducible?
LG,
Sara
-------- Original-Nachricht --------
> Datum: Tue, 19 Feb 2013 15:41:24 +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)
> Sara, x^4+x+1 is not irreducible over F3. So that will make a difference.
>
> (%i1) display2d : false$
> (%i2) gf_irreducible_p(x^4+x+1, 2);
> (%o2) true
>
> OK modulo 2, but
>
> (%i3) gf_irreducible_p(x^4+x+1, 3);
> (%o3) false
>
> The factors over F3:
>
> (%i4) gf_factor(x^4+x+1, 3);
> (%o4) (x+2)*(x^3+x^2+x+2)
>
> (%i5) gf_set(3, x^4+x+1);
> (%o5) [false,x^4+x+1]
>
> If I use x^4+x+1 for reduction this is the answer of gf_set in Maxima
> 5.29.1.
>
> The answer in 5.30 will be of more information. I load the recent git
> version of src/numth.lisp:
>
> (%i6) load("/home/volker/Maxima/git_130219/numth.lisp")$
>
> (%i7) gf_set(3, x^4+x+1);
> (%o7) gf_data(3,4,x^4+x+1,false,81,52,[[2,2],[13,1]])
>
> If display2d will be true, gf_set will answer what gf_info does:
> (%i8) gf_info();
> (%o8) "characteristic = 3, exponent = 4, reduction = x^4+x+1,
> primitive = false, cardinality = 81, order = 52, factors_of_order =
> [[2,2],[13,1]]"
>
> That means the group (F3/(x^4+x+1))* has no generator, so it is not
> cyclic. F3/(x^4+x+1) has 81 elements, the group (F3/(x^4+x+1))* only
> 52. F3/(x^4+x+1) is not a field. E.g. x+2 is no unit.
>
> (%i9) gf_inv(x+2);
> (%o9) false
>
> Volker van Nek
>
>
> 2013/2/19 Sara Mussie <saramussie at gmx.net>:
> > 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
> >> > _______________________________________________
> >> > Maxima mailing list
> >> > Maxima at math.utexas.edu
> >> > http://www.math.utexas.edu/mailman/listinfo/maxima