Opened 12 years ago

Closed 11 years ago

# linbox minpoly over small finite fields is TOTALLY BROKEN

Reported by: Owned by: was was critical sage-4.3.3 linear algebra sage-4.3.3.alpha0 William Stein Yann Laigle-Chapuy N/A

```
On Wed, Jun 10, 2009 at 6:03 PM, Yann<yannlaiglechapuy@gmail.com> wrote:
>
> ----------------------------------------------------------------------
> | Sage Version 4.0.1, Release Date: 2009-06-06                       |
> | Type notebook() for the GUI, and license() for information.        |
> ----------------------------------------------------------------------
sage: A=matrix(GF(3),2,[0,0,1,2])
sage: R.<x>=GF(3)[]
sage: D={ x:0 , x+1:0 , x^2+x:0 }
sage: for i in range(10000): D[A._minpoly_linbox()]+=1

sage: D
{x: 38266, x + 1: 29397, x^2 + x: 32337}
>

You're absolutely right!  This *sucks* -- it seems like nothing we have ever wrapped in Linbox is right at first.  Hopefully the issue is that somehow the algorithm is only supposed to be probabilistic, and we're just misusing it in sage (quite possible).
```

### comment:1 Changed 12 years ago by was

from a linbox devel:

```Well, I think this was corrected in linbox-1.1.6:

The minpoly algorithm used depends on which method you are using from
LinBox of course but,
If you use the solution "minpoly" you will get the blackbox algorithm
(just like if you specify "minpoly(pol, mat, Method::Blackbox())")
then (since sept 2008 and 1.1.6) we will end up using an extension field
to compute the minpoly (on my machine it will be GF(3^10)) and then
I e.g. got the following result for one try (the algorithm is still
probabilistic, but has a much larger success rate, roughly around 1/3^10):

> 99993 minimal Polynomials are x^2 +x, 3 minimal polynomial are x+1, 4
minimal polynomials are x

Now for a so small matrix it could be better to use a dense version,
which can be called by "minpoly(pol,mat,Method::Elimination())".
If i am correct this dense version is also probabilistic (choice of the
Krylov non-zero vector) and therefore should also pick vectors from an
extension.
This is not the case in 1.1.6.
Clément can you confirm this ? If so it should be easy to fix, the same
way we fixed Wiedemann.

For your example matrix in some of the cases, when vectors [1,1], and
[2,2] are chosen the Krylov space has rank 1, whereas for other non zero
vectors  it has rank 2 and
thus the dense minbpoly will be x^2+x or x+1 ...

btw, the returned polynomial is always a factor of the true polynomial,
therefore to get a 1/3^{10k} probability  of success it will be
sufficient to perform the lcm of k runs.

Best,

--
Jean-Guillaume Dumas.
```

My remarks

```Hi Yann (and sage-support),

This is from a linbox developer (see below).   This will be fixed by:

(1) upgrading -- actually, we *already* use linbox-1.1.6 in sage, so ...

(2) making it so minpoly by default just raises a NotImplementedError, however
minpoly(proof=False) will call minpoly a bunch of times and return
the lcm of the
results.

It turns out that maybe linbox doesn't seem to have a proof=True
minpoly algorithm yet (they are hard to write), so our wrapping of
linbox is wrong, given that in Sage the default is proof=True
everywhere.

Yann -- if you want to work on improving the situation wrt any of the

William
```

### comment:2 Changed 11 years ago by was

• Description modified (diff)
• Report Upstream set to N/A

### comment:3 follow-up: ↓ 4 Changed 11 years ago by was

• Description modified (diff)
• Status changed from new to needs_review

### comment:4 in reply to: ↑ 3 Changed 11 years ago by ylchapuy

We should at least take the lcm of the results so far:

line 974: g = g.lcm(self._minpoly_linbox(var)

otherwise, it seems ok.

Yann

this is part 2

### comment:5 Changed 11 years ago by ylchapuy

• Status changed from needs_review to positive_review

Ok positive review.

As an aside, is their any reason the result is cached but never fetched?

Yann

### comment:6 Changed 11 years ago by mpatel

• Authors set to William Stein
• Merged in set to sage-4.3.3.alpha0
• Resolution set to fixed
• Reviewers set to Yann Laigle-Chapuy
• Status changed from positive_review to closed

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