Re: [sage-support] Generator of Finite Field
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 -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
Re: Re: [sage-support] Generator of Finite Field
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
--
name: Martin Albrecht
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_otr: 47F43D1A 5D68C36F 468BAEBA 640E8856 D7951CCF
_www: http://martinralbrecht.wordpress.com/
_jab: martinralbre...@jabber.ccc.de
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Re: Re: [sage-support] Generator of Finite Field
PARI doesn't have a log_repr() to begin with, so I don't see how that's relevant. On Sunday 03 Jun 2012, Jeroen Demeyer wrote: On 2012-06-03 12:14, Martin Albrecht wrote: 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. On the other hand, doing this for the Givaro interface but not for the PARI interface (where logs are never used) is also confusing. I would say: leave things as they are. Note that the generator is not random, it is deterministic. Cheers, Martin -- name: Martin Albrecht _pgp: http://pgp.mit.edu:11371/pks/lookup?op=getsearch=0x8EF0DC99 _otr: 47F43D1A 5D68C36F 468BAEBA 640E8856 D7951CCF _www: http://martinralbrecht.wordpress.com/ _jab: martinralbre...@jabber.ccc.de -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
Re: [sage-support] Generator of Finite Field
On 2012-06-03 22:37, Martin Albrecht wrote: PARI doesn't have a log_repr() to begin with, so I don't see how that's relevant. Yes, never mind my comment. -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
Re: [sage-support] Generator of Finite Field
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 -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
[sage-support] Generator of Finite Field
Hello all. 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 sage: k.gen() a sage: k.list() [0, a + 1, a^2 + 1, a^3 + a^2 + a + 1, a^4 + 1, a^5 + a^4 + a + 1, a^6 + a^4 + a^2 + 1, ... ] Generator is a+1! How to get generator of Finite Field? It was fine in Sage 4.8. Ubuntu 12.04 Linux hamsin 3.2.0-24-generic #37-Ubuntu SMP Wed Apr 25 08:43:52 UTC 2012 i686 i686 i386 GNU/Linux Best regards, Oleksandr -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
