Make FiniteField_pari_ffelt the default for generic finite fields

[pjbruin]

Ticket #12142 implements an interface to PARI's t_FFELT type for finite fields, which is much faster than the one currently used by Sage, in which finite field elements are PARI objects like Mod(Mod(1, 3)*a, Mod(1, 3)*a^17 + Mod(2, 3)*a + Mod(1, 3)). The attached patch makes the new implementation the default for those finite fields that currently use the slow PARI implementation, namely those of cardinality pⁿ with p > 2 prime, n > 1 and pⁿ > 2¹⁶.

The actual switch is just a trivial change in the FiniteField constructor. The only other real addition is construction of Integer from t_FFELT; the rest of the patch fixes doctests. Almost all fixes are for doctests that assumed FiniteField_elt_pari to be the default. One somewhat notable point is that certain finite elements now compare differently (see the changed doctest in polynomial_zz_pex.py). This is because gcmp_sage (in sage/libs/pari/misc.h) compares all non-real PARI types via their string representations. With the new FiniteFieldElement_pari_ffelt, string comparison gives the same result as lexicographic comparison of polynomials expressing finite field elements in terms of the chosen generator, which is nice.

Apply:

• attachment: trac_14888-switch_to_pari_ffelt.patch
• attachment: 14888_fix_32bit.patch​
• attachment: 14888_review.patch

Depends on #12142
Depends on #15124
Depends on #15125

Component: finite rings

Keywords: FiniteField performance

Author: Peter Bruin, Jeroen Demeyer

Reviewer: Jean-Pierre Flori, Jeroen Demeyer, Peter Bruin

Merged: sage-5.13.beta0

Issue created by migration from https://trac.sagemath.org/ticket/14888

[pjbruin]

Attachment: trac_14888-switch_to_pari_ffelt.patch.gz

switch to FiniteField_pari_ffelt; based on 5.11.beta3 + #12142

[pjbruin]

Description changed:

--- 
+++ 
@@ -1,3 +1,6 @@
 Ticket #12142 implements an interface to PARI's t_FFELT type for finite fields, which is much faster than the one currently used by Sage, in which finite field elements are PARI objects like `Mod(Mod(1, 3)*a, Mod(1, 3)*a^17 + Mod(2, 3)*a + Mod(1, 3))`.  The attached patch makes the new implementation the default for those finite fields that currently use the slow PARI implementation, namely those of cardinality *p<sup>n</sup>* with *p* > 2 prime, *n* > 1 and *p<sup>n</sup>* > 2<sup>16</sup>.
 
 The actual switch is just a trivial change in the `FiniteField` constructor.  The only other real addition is construction of `Integer` from `t_FFELT`; the rest of the patch fixes doctests.  Almost all fixes are for doctests that assumed `FiniteField_elt_pari` to be the default.  One somewhat notable point is that certain finite elements now compare differently (see the changed doctest in `polynomial_zz_pex.py`).  This is because `gcmp_sage` (in `sage/libs/pari/misc.h`) compares all non-real PARI types via their string representations.  With the new `FiniteFieldElement_pari_ffelt`, string comparison gives the same result as lexicographic comparison of polynomials expressing finite field elements in terms of the chosen generator, which is nice.
+
+Apply: [attachment: trac_14888-switch_to_pari_ffelt.patch](https://github.com/sagemath/sage-prod/files/10658130/trac_14888-switch_to_pari_ffelt.patch.gz)
+

[jpflori]

Reviewer: Jean-Pierre Flori

[jpflori]

comment:2

Looks good and works fine for what I tried, good work Peter!

[jdemeyer]

comment:3

Haven't read the code of #12142, but does pickling work properly and has it been doctested? If not, then #14888 should be set back to needs_work.

[pjbruin]

comment:4

There shouldn't be any problems with pickling, see the comment I just made at #12142.