Implement FiniteFieldExtension::getPrimitiveElement()

[rheitjoh]

(This issue has been imported from the Gitlab repository because it seems to not have been addressed yet)

Note: FiniteFieldExtension has been deleted from the project. The comments may still be relevant, though.

Original Text (Issue 201)

In "Mathematics of Public Key Cryptography" (Galbrath 2012), there is an algorithm for this on page 51

Comment by Denis Diemert

Also related Zp::getPrimitiveElement()

Comment by Denis Diemert

Maybe ExtensionField::getPrimitiveElement() might be also affected, currently here null is returned. If that is intentionally, we need to add javadoc.

[rheitjoh]

We use the notion "primitive element" to mean generator of the unit group. Note that there seem to be some competing definitions here, because "primitive element" is also defined to be the element that generates the field extension from the base field (e.g. https://en.wikipedia.org/wiki/Simple_extension#Definition). Seems a bit confusing for ExtensionField.

I would suggest maybe renaming the method to getUnitGroupGenerator?

[rheitjoh]

So implementing this via algorithm 5 from section 2.15.3 from Galbraith (Online version 2, Oct 2018) would require first factorizing the order of the unit group, and then applying algorithm 5.