Finite field homomorphisms and Frobenius endomorphisms

[xcaruso]

Here is a patch implementing:

1. Frobenius endomorphisms over finite fields (generic implementation, plus special implementations for prime finite fields and givaro finite fields)
2. embeddings between finite fields (with again a special implementation for givaro finite fields)
3. hashing and rich comparisons for general morphisms

---

Apply: attachment: trac_13214-folded.patch

Depends on #8335

CC: @sagetrac-tfeulner @xcaruso @darijgr @jpflori @pjbruin @defeo @sagetrac-JCooley @sagetrac-dfesti

Component: finite rings

Keywords: frobenius finite fields sd51

Author: Xavier Caruso, Peter Bruin

Reviewer: Paul Zimmermann, Peter Bruin, Volker Braun

Merged: sage-5.13.beta2

Issue created by migration from https://trac.sagemath.org/ticket/13214

[sagetrac-tfeulner]

comment:2

I guess your patch should also contain some *.pxd files since I receive the following error message.

Building modified file sage/rings/finite_rings/hom_finite_field.pyx.
Traceback (most recent call last):
  File "setup.py", line 829, in <module>
    queue = compile_command_list(ext_modules, deps)
  File "setup.py", line 789, in compile_command_list
    dep_file, dep_time = deps.newest_dep(f,m)
  File "setup.py", line 696, in newest_dep
    for f in self.all_deps(filename, ext_module):
  File "setup.py", line 677, in all_deps
    for f in self.immediate_deps(filename, ext_module):
  File "setup.py", line 659, in immediate_deps
    self._deps[filename] = self.parse_deps(filename, ext_module)
  File "setup.py", line 647, in parse_deps
    raise IOError, msg
IOError: could not find dependency sage/rings/finite_rings/hom_finite_field.pxd included in sage/rings/finite_rings/hom_prime_finite_field.pyx.
Error installing modified sage library code.

[zimmermann6]

comment:4

Xavier, if I define an element of GF(q²) which is in fact an element of GF(q), will this patch enable to convert it efficiently to the smaller field? If so, please can you tell me how?

Paul

[xcaruso]

comment:5

Oh sorry, I forgot *.pxd files. Thanks!

Paul, the answer to your question is probably yes... at least if q is small enough so that you can work with givaro fields (otherwise, it won't be so efficient).

Actually, I have not implemented automatic coercions between finite fields (essentially because embeddings between finite fields are not canonical). As a consequence it's not so easy to create embeddings. Nevertheless, here is what you can do:

sage: K.<x> = GF(5^3)
sage: L.<y> = GF(5^6)
sage: from sage.rings.finite_rings.hom_finite_field_givaro import FiniteFieldEmbedding_givaro
sage: f = FiniteFieldEmbedding_givaro(K,L); f
Embedding of Finite Field in x of size 5^3 into Finite Field in y of size 5^6
sage: g = f.section(); g
Section of Embedding of Finite Field in x of size 5^3 into Finite Field in y of size 5^6
sage: g(y^126)
3*x^2 + 1
sage: g(y)
Traceback (most recent call last):
...
ValueError: y is not in the image of Embedding of Finite Field in x of size 5^3 into Finite Field in y of size 5^6

If you are sure that you won't never use another embedding of K into L, you can even define f as a coerce map as follows:

sage: K._unset_coercions_used()
sage: K._populate_coercion_lists_(embedding=f)

Then the following will work:

sage: L(x)
2*y^5 + 3*y^4 + 3*y^2 + y + 2
sage: z = x*y; z
3*y^5 + 3*y^4 + 4*y^2 + 2*y + 1
sage: K(z/y)
x
sage: K(y)
Traceback (most recent call last):
...
ValueError: y is not in the image of Embedding of Finite Field in x of size 5^3 into Finite Field in y of size 5^6

[jdemeyer]

comment:7

Please fill in your real name as Author.

[xcaruso]

Author: Xavier Caruso

[zimmermann6]

comment:9

Xavier, I tried to apply the patch on top of Sage 5.1 but got:

sage: hg_sage.import_patch("trac_13214_hom_finite_field.patch")
cd "/usr/local/sage-5.1-linux-64bit-ubuntu_12.04_lts-x86_64-Linux/devel/sage" && sage --hg import   "/tmp/trac_13214_hom_finite_field.patch"
applying /tmp/trac_13214_hom_finite_field.patch
** unknown exception encountered, please report by visiting
**  http://mercurial.selenic.com/wiki/BugTracker
** Python 2.7.3 (default, Jul  9 2012, 15:05:17) [GCC 4.6.3]
** Mercurial Distributed SCM (version 1.8.4)
** Extensions loaded: color, mq, pager, rebase, relink
Traceback (most recent call last):
  File "/usr/local/sage-5.1-linux-64bit-ubuntu_12.04_lts-x86_64-Linux/local/bin/hg", line 38, in <module>
    mercurial.dispatch.run()
  File "/usr/local/sage-5.1-linux-64bit-ubuntu_12.04_lts-x86_64-Linux/local/lib/python/mercurial/dispatch.py", line 16, in run
    sys.exit(dispatch(sys.argv[1:]))
  File "/usr/local/sage-5.1-linux-64bit-ubuntu_12.04_lts-x86_64-Linux/local/lib/python/mercurial/dispatch.py", line 36, in dispatch
    return _runcatch(u, args)
...
  File "/usr/local/sage-5.1-linux-64bit-ubuntu_12.04_lts-x86_64-Linux/local/lib/python/mercurial/patch.py", line 339, in readgitpatch
    gp.setmode(int(line[-6:], 8))
ValueError: invalid literal for int() with base 8: '</pre>'

Should I apply it on top of Sage 5.2 or later?

Paul

[zimmermann6]

comment:10

please discard my last comment, I downloaded the patch in html form...

Paul

[zimmermann6]

Work Issues: failing doctests

[zimmermann6]

Reviewer: Paul Zimmermann