And one more.
The following is quite strange:
sage: K=GF(2^8,'a',modulus=ZZ['x']("x^8 + x^7 + x^6 + x^4 + x^3 + x^2 + 1"))
sage: b=K.multiplicative_generator()
sage: c=K(0)
sage: c.log(b)
0
sage: c=K(1)
sage: c.log(b)
0
How to determine when c is 0 and when 1? Perhaps log should return
-1 for finite fields when c=0?
Kind regards,
Oleksandr
On Wednesday, June 13, 2012 3:02:46 PM UTC+2, Oleksandr Kazymyrov wrote:
>
> Hi all,
>
> Continuing the
> question<https://groups.google.com/forum/?fromgroups#!topic/sage-support/cvtz4Zh-WYQ>
> ...
>
> Input data:
> sage: K=GF(2^8,'a',modulus=ZZ['x']("x^8 + x^7 + x^6 + x^4 + x^3 + x^2 +
> 1"))
> sage: K.multiplicative_generator()
> a^4 + a^3 + a
> sage: P=PolynomialRing(K,'x')
> sage: pol=P.random_element(5)
> sage: pol
> (a^7 + a^5 + a^3 + 1)*x^5 + (a^5 + a^4 + a^3 + a^2 + a + 1)*x^4 + (a^7 +
> a^3 + a^2 + a)*x^3 + (a^5 + a^2)*x^2 + (a^6 + a^5 + a^4 + 1)*x + a^3
>
> 1. How to represent *pol *using multiplicative generator? The output must
> be b^(i1)*x^5 + b^(i2)*x^4 + b^(i3)*x^3 + b^(i4)*x^2 + b^(i5)*x + a^3,
> where b = K.multiplicative_generator(), i1-i5 logarithmic representation of
> (a^7
> + a^5 + a^3 + 1), (a^5 + a^4 + a^3 + a^2 + a + 1)...
>
> I have a correct function:
> sage: b=K.multiplicative_generator()
> sage: P(["({0})^{1}".format(b,i.log(b)) for i in pol.coeffs()]) == pol
> True
> sage: ["({0})^{1}".format(b,i.log(b)) for i in pol.coeffs()]
> ['(a^4 + a^3 + a)^27', '(a^4 + a^3 + a)^234', '(a^4 + a^3 + a)^195', '(a^4
> + a^3 + a)^134', '(a^4 + a^3 + a)^79', '(a^4 + a^3 + a)^101']
>
> If I have one program then the above method suits me. But if two, i.e. one
> program represents polynomial in the form b^(i1)*x^5 ... and returns b and
> another program tries to restore it, then I worry about the phrase
> 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."
>
> 2. So the question is how to uniquely reconstruct the original *pol*, if
> I have *modulus*, *multiplicative generator *and* representation of pol
> as *b^(i1)*x^5...?
>
> Regards,
> Oleksandr
>
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