Fix factory keys for finite fields to avoid repeated construction

[jdemeyer]

The following sequence of commands somehow breaks the coercion framework. This is on vanilla Sage 6.4.beta1. The problem only occurs if impl="pari_ffelt" is explicitly given, even though that is the default:

sage: k1.<a> = GF(17^14, impl="pari_ffelt")
sage: _ = PolynomialRing(k1, 'x')(a/2)
sage: k2.<a> = GF(17^14, impl="pari_ffelt")
sage: _ = PolynomialRing(k2, 'x')(a/2)
Traceback (most recent call last):
...
TypeError: no coercion defined

The underlying reason is:

sage: k1 is k2
False

This double construction of the same finite field is caused by slightly different UniqueFactory keys arising during the construction of a common parent (for the computation of a/2) by the coercion system.

Related: #16855

[pbruin]

I don't claim to know what is going on, but it may be caused by equality and identity not being the same for finite fields; #16855 seems to fix this bug.

[pbruin]

This guess is corroborated by the fact that the following change also appears to fix the bug:

src/sage/rings/finite_rings/element_pari_ffelt.pyx

@@ -202 +202 @@
         cdef Integer xi
 
         if isinstance(x, FiniteFieldElement_pari_ffelt):
-            if self._parent is (<FiniteFieldElement_pari_ffelt>x)._parent:
+            if self._parent == (<FiniteFieldElement_pari_ffelt>x)._parent:
                 pari_catch_sig_on()
                 self.construct((<FiniteFieldElement_pari_ffelt>x).val)
             else:

After #16855, this change effectively does nothing.

[jdemeyer]

Replying to pbruin:

src/sage/rings/finite_rings/element_pari_ffelt.pyx

@@ -202 +202 @@
         cdef Integer xi
 
         if isinstance(x, FiniteFieldElement_pari_ffelt):
-            if self._parent is (<FiniteFieldElement_pari_ffelt>x)._parent:
+            if self._parent == (<FiniteFieldElement_pari_ffelt>x)._parent:
                 pari_catch_sig_on()
                 self.construct((<FiniteFieldElement_pari_ffelt>x).val)
             else:

Would you propose to make the above change, or simply to close as ticket as dupe of #16855?

[pbruin]

Replying to jdemeyer:

Would you propose to make the above change, or simply to close as ticket as dupe of #16855?

I would say either we close this as a duplicate of #16855, or we try to fix the other bug (comment:11:ticket:16855) here:

sage: k1.<a> = GF(17^14, impl="pari_ffelt")
sage: _ = a/2
sage: k2.<a> = GF(17^14, impl="pari_ffelt")
sage: k1 is k2
False

[jdemeyer]

OK, let's wait and see what happens with #16855 then...

[pbruin]

The construction of the non-identical k1 and k2 appears to be due to a subtlety involving the FiniteFieldFactory.other_keys() method and an intermediate construction of another finite field (defined by the same data but with impl=None) during the construction of a common parent for the computation of a/2. The keys returned by other_keys() only differ in the impl keyword. I think the best solution is to always set the impl keyword when constructing the key (as opposed to allowing it to be None). Then the use of this other_keys() method, which feels to me somewhat like a hack, can be avoided.

[pbruin]

This branch depends on #16930 to avoid merge conflicts. It merges with, and works independently of, #16855.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​8923777	Trac 16934: remove method FiniteFieldFactory.other_keys()
​5e2681e	Trac 16934: more cleaning up
​ffb74fa	Trac 16934: add doctest

[pbruin]

Diff for easier reviewing: ​http://git.sagemath.org/sage.git/diff/?id2=8cdbd7920c73713a3a8dbea697451649f0434f0a&id=ffb74fa8116d1a60e255300370ca4467e3b36732

[jdemeyer]

Please restore the warning

warn("the 'modulus' argument is ignored when constructing prime finite fields")

for all prime finite field implementations because the modulus really is ignored:

sage: x = polygen(GF(7)); GF(7, modulus=x-2, impl="givaro").modulus()
x + 6

Of course we could change that behaviour (perhaps we should), but let's do that on a new ticket.

---

New commits:

​33c86ef

Merge remote-tracking branch 'origin/develop' into ticket/16934

[pbruin]

This commit adds back the warning. Alternatively, it could have been put outside the if impl == 'modn', but in this way it is easier to remove again for non-modn implementations.

[jdemeyer]

sage: GF(7, impl="givaro", modulus="primitive")
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-2-c6a1d2461111> in <module>()
----> 1 GF(Integer(7), impl="givaro", modulus="primitive")

/usr/local/src/sage-git/local/lib/python2.7/site-packages/sage/structure/factory.so in sage.structure.factory.UniqueFactory.__call__ (build/cythonized/sage/structure/factory.c:1160)()
                                                                                                                                                                        
/usr/local/src/sage-git/local/lib/python2.7/site-packages/sage/rings/finite_rings/constructor.pyc in create_key_and_extra_args(self, order, name, modulus, names, impl, proof, check_irreducible, **kwds)                                                                                                                                       
    521                 if not modulus.is_irreducible():                                                                                                                
    522                     raise ValueError("finite field modulus must be irreducible but it is not.")
--> 523             if modulus is not None and modulus.degree() != n:
    524                 raise ValueError("the degree of the modulus does not equal the degree of the field.")
    525 

AttributeError: 'str' object has no attribute 'degree'

[jdemeyer]

Replying to pbruin:

This commit adds back the warning. Alternatively, it could have been put outside the if impl == 'modn', but in this way it is easier to remove again for non-modn implementations.

I would prefer to have the warning in one place. If we ever allow a custom modulus, we should allow it for all implementations.

My vision for this "modulus" argument is that the following should assign to a a primitive root:

sage: K.<a> = GF(7, modulus='primitive')

so perhaps we should expand the warning to say

warn("currently, the 'modulus' argument is ignored when constructing prime finite fields. This might change in the future and this will probably also change the output of the .gen() and .modulus() methods on the finite field.\nSee http://trac.sagemath.org/16934 for details.")

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​9cbdcde

Trac 16934: fix warning about ignored 'modulus' argument

[nbruin]

Replying to jdemeyer:

Replying to pbruin:

This commit adds back the warning. Alternatively, it could have been put outside the if impl == 'modn', but in this way it is easier to remove again for non-modn implementations.

I would prefer to have the warning in one place. If we ever allow a custom modulus, we should allow it for all implementations.

My vision for this "modulus" argument is that the following should assign to a a primitive root:

sage: K.<a> = GF(7, modulus='primitive')

The word "modulus" here is definitely wrong. If anything is a modulus for GF(7), it is 7, because the natural way to construct GF(7) as quotient of ZZ by 7ZZ.

In general, even for field *extensions*, the word modulus is confusing. That only applies if you're making the field as a quotient of a polynomial ring. For the purpose proposed, generator would cover the notion much better. I see from the doc that we have legacy against us on this one.

[jdemeyer]

Replying to nbruin:

Replying to jdemeyer:

My vision for this "modulus" argument is that the following should assign to a a primitive root:

sage: K.<a> = GF(7, modulus='primitive')

The word "modulus" here is definitely wrong. If anything is a modulus for GF(7), it is 7, because the natural way to construct GF(7) as quotient of ZZ by 7ZZ.

Depends on how you look at it. For me, there is a big philosophical difference between IntegerModRing(p) and GF(p). I see IntegerModRing(m) as ZZ/(m), while GF(p^n) is IntegerModRing(p)[x]/(f(x)) for an irreducible polynomial f(x) of degree n.

For fields GF(p^n) (n > 1), the word "modulus" for f(x) is certainly more standard than "generator". For me, the "generator" is the image of x in GF(p^n) = IntegerModRing(p)[x]/(f(x)), which is returned by the gen() method.

[jdemeyer]

Replying to jdemeyer:

My vision for this "modulus" argument is that the following should assign to a a primitive root:

sage: K.<a> = GF(7, modulus='primitive')

By the way, this is now implemented at #16983.