no bug!
generator() does not necessarily return a primitive element, which
generates the cyclic group (F-{0},*)! It returns an element which generates
the algebra, so the result is correct.
See comment:
generator : () -> %
++ generator() returns a root of the defining polynomial.
++ This element generates the field as an algebra over the ground field.
Perhaps you want to use
primitiveElement()$F16
(84) -> w := primitiveElement()$FFP(PF 2, p)
(84) %E + 1
Type:
FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?^3+?^2+?+1)
(85) -> order w
(85) 15
Type: PositiveInteger
Am 23.08.18 um 23:22 schrieb Paul Onions:
> Hi All,
>
> It seems that calling generator() on a field created with
> FiniteFieldExtensionByPolynomial may not return a primitive element of
> the field. Specifically, when the field is created using an
> irreducible (but not primitive) polynomial.
>
> Example:-
>
> (1) -> p : UP(x,PrimeField(2)) := x^4 + x^3 + x^2 + x + 1
>
> 4 3 2
> (1) x + x + x + x + 1
> Type: UnivariatePolynomial(x,PrimeField(2))
> (2) -> F16 := FiniteFieldExtensionByPolynomial(PrimeField(2), p)
>
> (2) FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?^3+?^2+?+1)
> Type: Type
> (3) -> g := generator()$F16
>
> (3) %A
> Type: FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?^3+?^2+?+1)
> (4) -> g^5
>
> (4) 1
> Type: FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?^3+?^2+?+1)
>
> Looking at the source (ffp.spad) it looks like the implementation of
> generator() assumes that the defining polynomial is primitive, but the
> comments at the head of the file clearly state that the only
> requirement on the defining polynomial is that it be irreducible.
>
> Am I misunderstanding something here?
> Paul
>
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