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."
>
>
> --
> Best,
> Alex
>
> --
> Alex Ghitza -- Lecturer in Mathematics -- The University of Melbourne
> http://aghitza.org
>
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