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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