Hi Volker,
thank you very much. Am developing a 'threshold secret sharing scheme based on the generalized vector space construction and your work helps me so much cause now I can easily determine the primitive irreducible polynomial with which I can construct the required subspaces which I give to my users as their matrices ;-)
I can't wait to go through your work in details tomorrow Volker.
Thanks also to Adam.
With best regards,
Sara
-------- Original-Nachricht --------
> Datum: Thu, 25 Oct 2012 17:14:10 +0200
> Von: Volker van Nek <volkervannek at gmail.com>
> An: Sara Mussie <saramussie at gmx.net>
> CC: Maxima at math.utexas.edu
> Betreff: Re: [Maxima] Irreducible Polynomial
> Hi,
> I am working on a package for Galois Fields. I nearly finished coding the
> functions but unfortunately at the moment most of them are not documented.
>
> Having Maxima 5.28 available you can do the following
>
> (%i1) display2d : false$
>
> (%i2) gf_irr_p(x^4+x+1, 2);
> (%o2) true
>
> which means that x^4+x+1 is irredicible over F2,
> while it is not over F3:
>
> (%i3) gf_irr_p(x^4+x+1, 3);
> (%o3) false
>
> (%i4) gf_factor(x^4+x+1, 3);
> (%o4) (x+2)*(x^3+x^2+x+2)
>
> How to compute all irredicible polynomials over e.g. F2 up to a certain
> degree:
>
> First set an arbitrary field over F2:
>
> (%i5) gf_set(2,4);
> (%o5) [x,x^4+x+1]
>
> This sets F2^4 (primitive element:x, poly for reduction: x^4+x+1) and
> tells
> gf_n2p which converts a number to the corresponding polynomial to use the
> modulus 2. (It might be a good idea to allow the modulus to be an optional
> parameter to gf_n2p in the future. Then setting a field wouldn't be
> necessary.)
>
> (%i6) for n:1 thru 2^6-1 do (p : gf_n2p(n), if gf_irr_p(p) then print(p));
> x
> x+1
> x^2+x+1
> x^3+x+1
> x^3+x^2+1
> x^4+x+1
> x^4+x^3+1
> x^4+x^3+x^2+x+1
> x^5+x^2+1
> x^5+x^3+1
> x^5+x^3+x^2+x+1
> x^5+x^4+x^2+x+1
> x^5+x^4+x^3+x+1
> x^5+x^4+x^3+x^2+1
> (%o6) done
>
> These are all irredicible polynomials over F2 of degree 1 through 5.
>
> Hope that helps
> Volker van Nek
>
> 2012/10/25 Sara Mussie <saramussie at gmx.net>
>
> > Dear members,
> >
> > hope all is well with you.
> >
> > Can I get all irreducible polynomials of any degree with Maxima? If yes,
> > can you please tell me how?
> >
> > Thank you.
> >
> > Regards,
> > Sara
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