Hi Volker,
x^4+x+2 is irreducible over F3 and you tell me this in your first answer.
My question was if it is also 'primitive'. There are irreducible polynomials and primitive irreducible polynomials. For my construction I need the 'primitive irreducible polynomials' to find the subgroup. The primitive irreducible polynomials gives me that subgroup.
There is only one real subgroup of GF(3)^4 with the order 4 that meets the condition
a*a = b and b*b = a and a*b = 1 and generates all the side classes of a quotient space, which I need to generate all 2-dimensional subspaces.
Maybe you can give me the necessary command to generate all irreducible polynomials like x^4+x+2 over F3 and then I can find the primitive irreducible polynomial under these irreducible polynomials.
Thanks.
Regards,
Sara
-------- Original-Nachricht --------
> Datum: Tue, 19 Feb 2013 22:21:34 +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)
> Yes, x^4+x+2 is irreducible over F3. Please read again my first answer.
>
> Volker van Nek
>
> 2013/2/19 Sara Mussie <saramussie at gmx.net>:
> > 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
> >> >> > _______________________________________________
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> >> >> > Maxima at math.utexas.edu
> >> >> > http://www.math.utexas.edu/mailman/listinfo/maxima