Hi,
K.gen() returns the thing we represent field elements in, but not necessarily
the field generator.
1) However, the documentation is wrong:
"""
Return a generator of self. All elements x of self are expressed as
log_{self.gen()}(p) internally
"""
This should be fixed and a reference to multiplicative_generator() added.
2) K.multiplicative_generator() currently returns a random generator, but for
the GivaroGFq implementation we should simply return the generator which has
internal representation '1', i.e., the guy we actually use to represent the
elements. Once this done log_repr() will also start to make sense again.
Olexandr, would those fixes solves your confusion/problem?
On Sunday 03 Jun 2012, Oleksandr Kazymyrov wrote:
> I have used log_repr() and expect that it return y of equation x^y = z. I
> also believed hat k.gen() return generator of the field.
>
> Now I must use following construction for solving my problem
> sage: R.<x>=ZZ[]
> sage: k=GF(2^8,name='a',modulus=x^8+x^4+x^3+x+1)
> sage: a = k.multiplicative_generator()
> sage: a^ZZ(k(ZZ(3).digits(2)).log(a)) == k(ZZ(3).digits(2))
>
> By the way, any of next functions don't return the value y of equation
> x^y=z sage: b=K(ZZ(3).digits(2))
> sage: b
> a + 1
> sage: b.log_repr()
> '1'
> sage: b.log_to_int()
> 3
>
> I think that log_repr() should has the same logic as int_repr() or
> integer_representation(), i.e.
> sage: a=K.multiplicative_generator()
> sage: ZZ(K(ZZ(3).digits(2)).log(a))
> 164
>
> What do you think?
>
> Kind regards,
> Oleksandr
>
> On Tuesday, May 29, 2012 12:52:16 PM UTC+3, AlexGhitza wrote:
> > Hi,
> >
> > On Mon, May 21, 2012 at 9:29 AM, Oleksandr Kazymyrov
> >
> > <vrona.aka.ham...@gmail.com> wrote:
> > > I have encountered the following problem In Sage 5.0:
> > > sage: R.<x>=ZZ[]
> > > sage: k=GF(2^8,name='a',modulus=x^8+x^4+x^3+x+1)
> > > sage: k(ZZ(3).digits(2))
> > > a + 1
> > > sage: k.gen()^ZZ(k(ZZ(3).digits(2)).log_repr())
> > > a
> > > sage: k.gen()^ZZ(k(ZZ(3).digits(2)).log_repr()) == k(ZZ(3).digits(2))
> > > False
> > > sage: k("a+1")^ZZ(k(ZZ(3).digits(2)).log_repr()) == k(ZZ(3).digits(2))
> > > True
> > >
> > > It easy see that k.gen() or k.multiplicative_generator() is not a
> >
> > generator
> >
> > > of the finite field:
> > > sage: k.multiplicative_generator()
> > > a^4 + a + 1
> >
> > Why is it clear that a^4+a+1 is not a multiplicative generator? I think
> > it is:
> >
> > sage: k.<a> = GF(2^8, names='a', name='a', modulus=x^8+x^4+x^3+x+1)
> > sage: (a^4+a+1).multiplicative_order()
> > 255
> >
> > Indeed, so is a+1:
> > sage: (a+1).multiplicative_order()
> > 255
> >
> > The docs for multiplicative_generator() say: "return a generator of
> > the multiplicative group", then add "Warning: This generator might
> > change from one version of Sage to another."
Cheers,
Martin
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