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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