Bug in GAP conversion of finite field elements

[vdelecroix]

As reported in ​this question on ask we have a problem with our finite fields and GAP

sage: K = GF(16,'a')
sage: a = K.gen()
sage: b = a**2 + a
sage: K(b._gap_())
Traceback (most recent call last):
...
TypeError: unable to coerce from a finite field other than the prime subfield

Strangely, it works with GF(8).

[pbruin]

This is because the function sage.interfaces.gap.gfq_gap_to_sage, which converts a GAP element into an element of a Sage finite field F, constructs an intermediate field which does not necessarily have a canonical embedding into F. I am now testing a patch which avoids the intermediate field.

[pbruin]

Notes:

• Before this patch, it was implicitly assumed that non-prime finite fields are defined by Conway polynomials, because this is how GAP represents finite fields. This is now an explicit requirement.
• Elements of a GAP prime field Z(p) are now converted into a Sage FiniteField(p), as opposed to IntegerModRing(p).

[pbruin]

In addition to the doctest, I tested the example given in the link in the ticket description 200 times and did not get any errors.

[vdelecroix]

Hello,

That's wonderful that you did have a look! I put this to positive review as it fixes an important bug (and might hopefully be in sage-6.6).

On a related note, the code uses a lot of string parsing. This is ugly and not very safe. I think that we can get rid of most of it with

• Characteristic(z): the characteristic of the field
• DegreeFFE(z): degree of the smallest base field containing the element z
• LogFFE(z,base): logarithm in finite field

I do not like the fact that we need to call LogFFE to have access to the good power of the generator. If you think about a good solution for that I will open an other ticket.

Vincent