Some more remarks and considerations:
The last "::gf16" in Bill's statement is not necessary!
(19) -> eval(p::POLY gf16, x,a)
2
(19) %A + 1
Type: Polynomial(FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?+1))
However, what is really strange, is that the following does not wok:
(20) -> eval(p::POLY gf16, x=a)
There are 3 exposed and 0 unexposed library operations named
equation having 2 argument(s) but none was determined to be
applicable. Use HyperDoc Browse, or issue
)display op equation
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named
equation with argument type(s)
Variable(x)
FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?+1)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
which should be corrected as the following works without problems:
(20) -> q := 3*x^2+x-4
2
(20) 3x + x - 4
Type: Polynomial(Integer)
(21) -> eval(q, x=3)
(21) 26
Type: Polynomial(Integer)
(22) -> eval(q, x, 3)
(22) 26
Type: Polynomial(Integer)
While
(23) -> eval(p, x, a)
>> Error detected within library code:
coerce: element doesn't belong to smaller field
directly tells you what the problem is, however, this is not consequent, look
at the following:
(23) -> eval(q, x, 4/3)
8
(23) -
3
Type: Polynomial(Fraction(Integer))
hence the interpreter extends to a larger coefficient domain.
Who can and wants to improve the coercion mechanism to have a predictable
behaviour in all cases?
Am 11.08.2011 um 00:39 schrieb Bill Page:
> Paul,
>
> p is a polynomial over the base field. eval expects to return a
> polynomial of the same type.
>
> Perhaps this is what you intended?
>
> (6) -> eval(p::POLY gf16, x,a)::gf16
>
> 2
> (6) %A + 1
> Type: FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?+1)
>
> Regards,
> Bill Page.
>
> On Wed, Aug 10, 2011 at 5:13 PM, Paul Onions <[email protected]> wrote:
>> I have the following problem when I try to evaluate a polynomial in a
>> finite-field extension:
>>
>> -- START OF TRANSCRIPT --
>> gf2 := PrimeField 2 -- the base field
>>
>> (1) PrimeField(2)
>>
>> Type: Type
>> gf16 := FiniteFieldExtensionByPolynomial(gf2, x**4 + x + 1) -- the
>> extension field
>>
>> (2) FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?+1)
>>
>> Type: Type
>> a := index(2)$gf16 -- a primitive element of the extension field
>>
>> (3) %A
>> Type: FiniteFieldExtensionByPolynomial(PrimeField(2),?
>> ^4+?+1)
>> order a -- check primitiveness
>>
>> (4) 15
>> Type:
>> PositiveInteger
>> p := x**2 + 1 :: POLY gf2 -- a polynomial over the base field
>>
>> 2
>> (5) x + 1
>> Type:
>> Polynomial(PrimeField(2))
>> eval(p, x=a)
>>
>> There are 3 exposed and 0 unexposed library operations named
>> equation having 2 argument(s) but none was determined to be
>> applicable. Use HyperDoc Browse, or issue
>> )display op equation
>> to learn more about the available operations. Perhaps
>> package-calling the operation or using coercions on the
>> arguments
>> will allow you to apply the operation.
>>
>> Cannot find a definition or applicable library operation named
>> equation with argument type(s)
>> Variable(x)
>> FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?+1)
>>
>> Perhaps you should use "@" to indicate the required return
>> type,
>> or "$" to specify which version of the function you need.
>> -- END OF TRANSCRIPT --
>>
>> where I expect
>> eval(p,x=a)
>> to give the same result as
>> a**2 + 1
>> which does indeed work, as follows:
>>
>> -- START OF TRANSCRIPT --
>> (6) -> a**2 + 1
>>
>> 2
>> (6) %A + 1
>> Type: FiniteFieldExtensionByPolynomial(PrimeField(2),?
>> ^4+?+1)
>> -- END OF TRANSCRIPT --
>>
>> Any idea what I'm doing wrong?
>>
>> Thanks,
>> Paul
>>
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>>
>
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Mit freundlichen Grüßen
Johannes Grabmeier
Prof. Dr. Johannes Grabmeier
Köckstraße 1, D-94469 Deggendorf
Tel. +49-(0)-991-2979584, Tel. +49-(0)-171-5503789
Tel. +49-(0)-991-3615-100 (d), Fax: +49-(0)-1803-5518-17745
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