Speed up PARI finite field operations

[johanbosman]

Let us perform a simple addition of finite field elements that are represented using PARI.

sage: F.<a> = GF(3^11)
sage: x = F.random_element()
sage: y = F.random_element()
sage: %timeit x + y
625 loops, best of 3: 13.2 µs per loop

Let us now measure how much time the *actual* addition takes:

sage: vx = x._FiniteField_ext_pariElement__value
sage: vy = y._FiniteField_ext_pariElement__value
sage: %timeit vx + vy
625 loops, best of 3: 4.6 µs per loop

This means two thirds of the execution time is used to wrap a ribbon around the result and only one third for the actual addition!

But in fact Pari has a faster implementation for finite fields than this! This was already mentioned at ​http://groups.google.com/group/sage-nt/browse_thread/thread/e2dbbc72caeb589a

sage: def pari_ffelt(x): parix=pari(x); return parix.lift().subst(parix.variable(), "ffgen((%s).mod)"%parix) 

sage: px = pari_ffelt(x)
sage: py = pari_ffelt(y)
sage: %timeit px + py
625 loops, best of 3: 2.43 µs per loop

For multiplication, we have the following timings:

sage: %timeit x * y
625 loops, best of 3: 18.2 µs per loop
sage: %timeit vx * vy
625 loops, best of 3: 8.4 µs per loop
sage: %timeit px * py
625 loops, best of 3: 3.33 µs per loop

This ticket implements an interface to PARI's FFELT type for non-prime finite fields. It can be tested by constructing finite fields as follows, for p prime and n >= 2:

sage: F.<a> = FiniteField(p^n, impl='pari_ffelt')

This implementation should probably become the default for non-prime finite fields of characteristic > 2 and cardinality > 2¹⁶, superseding the existing PARI polmod implementation. This switch is the goal of ticket #14888.

Apply:

• trac_12142-FiniteField_pari_ffelt.improved.patch​
• trac_12142-singular_conversion.patch​
• trac_12142-reviewer.patch​
• trac_12142-doctest.patch​
• trac_12142-remove_cinit.patch​

[jpflori]

Just a reminder for myself, it would also be nice to have a common cython interface to deal with finite fields elements, surely using templates as for polynomials.

[pbruin]

I have been working on an interface to PARI's implementation of finite fields (t_FFELT). Like for the other interfaces (except the existing PARI polmod implementation), the field class is written in Python and the element class in Cython. I hope to upload a patch soon after cleaning up the documentation and fixing a minor problem.

The speed improvement for addition and multiplication relative to the current implementation appears to vary roughly from a factor 3 to a factor 12, depending on the size of the field.

For small fields, this PARI interface is (of course) much slower than Givaro (a factor 4-6 for F_{3¹⁰}). For fields of characteristic 2, it is slower than NTL+gf2x, but the difference is not that dramatic.

[pbruin]

This ticket led to several other patches, which I put into separate tickets; see the list of dependencies. The dependence on #14817 is optional.

A patch for this ticket is now practically ready.

[pbruin]

The attached patch implements a new class FiniteField_pari_ffelt and associated element class FiniteFieldElement_pari_ffelt. There is currently one doctest failure, related to conversion from and to Singular.

[pbruin]

OK, converting between FiniteFieldElement and Singular was an easy generalisation of existing code. I will try to make a small table comparing the performance of the various implementations. In the meantime this is ready to be reviewed.

[pbruin]

Here are some timings for addition and multiplication in the various implementations:

x, y random elements of a field of p^n elements
all times in microseconds

Addition:  %timeit -c -r 1 x + y

(p, n)     ntl/gf2e  givaro  pari_ffelt  pari_mod

(2, 15)    0.85      0.45    0.95        15.1
(2, 5000)  1.06      ----    1.84        3300

(3, 10)    ----      0.47    1.14        12.5
(3, 80)    ----      ----    1.75        44

(251, 2)   ----      0.51    1.30        9.0

Multiplication:  %timeit -c -r 1 x * y

(p, n)     ntl/gf2e  givaro  pari_ffelt  pari_mod

(2, 15)    1.11      0.61    1.51        22
(2, 1000)  5.0       ----    61          1060

(3, 10)    ----      0.57    4.0         19
(3, 80)    ----      ----    36          99

(251, 2)   ----      0.56    1.69        12.1

[pbruin]

Initialisation of finite field elements is too slow, and computing a/b raises a ZeroDivisionError if a = 0... A fix will be ready soon.

[pbruin]

based on 5.11.beta3

[pbruin]

more speedups in element construction; initialisation from None allowed

[jpflori]

Reviewer patch, trivial typo.

[jpflori]

Patches look fine, doctests pass and there is indeed a very nice speedup.

[pbruin]

For patchbot:

Apply trac_12142-FiniteField_pari_ffelt.improved.patch, trac_12142-singular_conversion.patch, trac_12142-reviewer.patch

[pbruin]

Apply trac_12142-FiniteField_pari_ffelt.improved.patch, trac_12142-singular_conversion.patch, trac_12142-reviewer.patch 

[pbruin]

add doctest

[pbruin]

The next patch removes the Cython constructor __cinit__(), which is unnecessary since it just sets an attribute to NULL which is already guaranteed to be NULL. Actually, it just comments it out with an explanation for future reference. This should fix the remaining doctest failure and may give another slight speed improvement.

(I'm leaving this as "positive_review" since I can't set it back to "needs_review" immediately. The change is not entirely trivial, though.)

[jpflori]

This change looks ok to me, and is very well documented.

I did not really pay attention to the doctests missing as you were copying from a large previous Sage file. Maybe we could take this opportunity to fix it as well and so increase coverage.

[jdemeyer]

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.

[pbruin]

Yes, it works and there are doctests. Finite field elements are not implemented as type gen (as in libs/pari/gen.pyx), so the fact that pickling does not work for a gen that happens to be a t_FFELT does not affect this ticket.

[pbruin]

Replying to jpflori:

I did not really pay attention to the doctests missing as you were copying from a large previous Sage file. Maybe we could take this opportunity to fix it as well and so increase coverage.

Actually the doctest coverage of all files in sage/rings/finite_rings is 100%, except integer_mod.pyx, which has a coverage of only 99/150 functions. Apart from that, I think I saw one or two colons that should be double colons. I don't think it is really worth the trouble to spend time on that in this ticket.