A class to manage embedding between non-prime fields

[dlucas]

In Sage, there is no general mechanism to manage the embedding of elements of a finite extension field in one of its subfields.

We propose here a class which takes care of that.

Considering a big, non-prime field Fqm and one of its subfields Fq, this class is able to give either a polynomial or a vectorial representation of an element of Fqm in Fq.

[dlucas]

I pushed my code, and I open this ticket for review.

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New commits:

​49cebee

New class to manage embeddings between non-prime fields

[dlucas]

By the way, as I'm not sure where to put my code, I left it (for now) in sage/coding...

[jsrn]

Just to be precise: Sage already supports embedding a small field GF(p^s) into a larger one GF(p^(sm)) (by using Fsmall.extension(m, map=True)). It also supports the inverse map (returned by the section() method on the embedding map).

The new functionality of this ticket is to support representing any GF(p^(sm)) element in a basis over GF(p^s). AFAIK, the current implementation supports this with only one particular choice of bases: if 1, beta, ..., beta^(sm-1) is the basis of GF(p^(sm)) that Sage currently uses natively, then it is always the case that 1, beta, ..., beta^(m-1) is a basis of GF(p^(sm)) over GF(p^s). This is the basis employed by the current ticket.

More precisely, it gives a function that takes an element e in GF(p^(sm)) and returns a vector (v[0],...,v[m-1]) in GF(p^s)^m, such that

    e = sum_{i=0}^{m-1} phi(v[i]) * beta^i

where phi is a given embedding from GF(p^s) into GF(p^(sm)).

Best, Johan

[git]

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

​7ffd2c0	Update to 7.3beta2
​74a8c23	representation_matrix is now a private method
​4f4c0a6	Changed names: Fqm over Fq is now designated as a relative finite field extension
​87deeba	Changed return value of big_field_representation
​0eb1d5a	Changed naming: small field -> relative field, big field -> absolute field
​38b77f9	Added method to check if an element of the absolute field is in the relative field under the embedding

[dlucas]

Hello,

I made some changes to the class:

• I completely changed its nomenclature, which is now based on the one defined for ​relative extensions.
• I changed the behaviour of some methods.
• I added a method to check if an element of the "big field" is in the relative field under the embedding.

This is still open for review.

Best,

David

[vdelecroix]

Two naive questions:

• Why is this in sage/coding?
• Why not extending the preceding existing embedding class?

Vincent

[jsrn]

Replying to vdelecroix:

Two naive questions:

• Why is this in sage/coding?
• Why not extending the preceding existing embedding class?

Because we did not know where to put it, and we didn't get input from anyone. We need the functionality for several things in sage/coding, so we just wanted to add it. And it's on the agenda for Sage Days 75 to merge it much more sensibly with the rest of Sage.

Incidentally, will you be coming for SD75?

[vdelecroix]

Replying to jsrn:

Replying to vdelecroix:

Two naive questions:

• Why is this in sage/coding?
• Why not extending the preceding existing embedding class?

Because we did not know where to put it, and we didn't get input from anyone. We need the functionality for several things in sage/coding, so we just wanted to add it. And it's on the agenda for Sage Days 75 to merge it much more sensibly with the rest of Sage.

Would make more sense with everything about finite field, no? That is to say sage/rings/finite_rings/.

Incidentally, will you be coming for SD75?

Sadly not. I will be in Chile at that time.

[jpflori]

I'll be there at sd75. And I do agree it would nice to use such functionalities in a more general setting. We could still put it in coding first and move it later.

[dlucas]

Hello,

And I do agree it would nice to use such functionalities in a more general setting. We could still put it in coding first and move it later.

I would indeed prefer to put it in coding first, as it locks several very useful tickets for us (#20039, #20100, #20335) while it's not reviewed. Furthermore, we need it for our GSoC project on rank-metric codes.

I can put an experimental warning in it: this way, we will be able to move it later on without being worried by deprecation warnings.

Best,

David

[jsrn]

I can put an experimental warning in it: this way, we will be able to move it later on without being worried by deprecation warnings.

+1

[git]

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

​311b960	Added experimental warning for RelativeFiniteFieldExtension
​2492c31	Added relative_finite_field_extension to sage.coding's index.rst

[dlucas]

I added this experimental warning to the class.

I also added this module in sage.coding's index.

[arpitdm]

I went through the reviewer's checklist and all the tests pass. The naming conventions are quite suitable and make it easier to interpret the functions. Giving it a positive review.