Implement from_integer and to_integer for all finite fields, extending and replacing fetch_int and integer_representation

[yyyyx4]

Finite fields have .fetch_int() to decode integers into finite-field elements by reinterpreting the base-p representation as a polynomial:

sage: F.<t> = GF(7^10)
sage: el = F.fetch_int(2424); el
t^4 + 3*t + 2

However, the inverse operation is a bit cumbersome:

sage: el.polynomial().change_ring(ZZ)(el.parent().characteristic())
2424

In this patch, we add an inverse to .fetch_int(), named .to_integer() in resemblance with the Python method int.to_bytes().

For symmetry, we then rename .fetch_int() to .from_integer(): I, for one, could never remember if .fetch_int() refers to "fetching an element from an int" or "fetching this element into an int", so I think the new name makes a lot more sense.

Also, some code cleanup and optimization while we're at it.

CC: @DaveWitteMorris @slel

Component: finite rings

Author: Lorenz Panny

Branch/Commit: 0eabc46

Reviewer: Kwankyu Lee

Issue created by migration from https://trac.sagemath.org/ticket/33941

[sagetrac-git]

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

4c82bfe

rename .fetch_int() to .from_integer()

[sagetrac-git]

Changed commit from e354108 to 4c82bfe

[yyyyx4]

comment:3

I just found out that #31605 already tracks this issue and that binary finite fields have .integer_representation().

This leaves us with three options for naming these methods:

1. .fetch_int() / .integer_representation() [backwards compatible]
2. .from_integer() / .integer_representation() [one deprecation]
3. .from_integer() / .to_integer() [two deprecations]

Personally I'm still in favor of (3) because the others lack symmetry. Opinions?

[videlec]

comment:5

Replying to @yyyyx4:

I just found out that #31605 already tracks this issue and that binary finite fields have .integer_representation().

This leaves us with three options for naming these methods:

1. .fetch_int() / .integer_representation() [backwards compatible]
2. .from_integer() / .integer_representation() [one deprecation]
3. .from_integer() / .to_integer() [two deprecations]

Personally I'm still in favor of (3) because the others lack symmetry. Opinions?

+1 to option 3

[slel]

comment:6

Also +1 for (3) for the sake of symmetry and consistency.

[sagetrac-git]

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

c658823	Merge tag '9.7.beta6' into public/implement_inverse_of_finite_field_fetch_int
b340a29	references for .integer_representation(), and use it if available as it's faster
6230ed7	correct file title for NTL binary-field elements

[sagetrac-git]

Changed commit from 4c82bfe to 6230ed7

[yyyyx4]

comment:8

Thanks for the responses! It seems the only remaining question is whether we want to deprecate .integer_representation() or not.

[slel]

comment:9

Replying to @yyyyx4:

[...] whether [...] to deprecate .integer_representation() or not.

The only reason I would see not to would be consistency
with other software.

Do Magma, Maple, Mathematica have similar functions,
and if so how they are named?

[yyyyx4]

comment:10

I don't know. Some quick online searches didn't find anything.

[sagetrac-git]

Changed commit from 6230ed7 to 42d0e5f

[sagetrac-git]

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

dd12ebc	Merge tag '9.7.rc0' into public/implement_inverse_of_finite_field_fetch_int
42d0e5f	Merge tag '9.8.beta2' into public/implement_inverse_of_finite_field_fetch_int

[sagetrac-git]

Changed commit from 42d0e5f to aeb34ec

[sagetrac-git]

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

aeb34ec

Merge tag '9.8.beta3' into public/implement_inverse_of_finite_field_fetch_int

[kwankyu]

comment:14

Replying to Samuel Lelièvre:

Do Magma, Maple, Mathematica have similar functions,
and if so how they are named?

I could not find one for Magma and Maple. Mathematica has ToElementCode and FromElementCode

I think Sage is the leading software in this regard. No worry for consistency.

+1 to option 3.

[sagetrac-git]

Changed commit from aeb34ec to 12411a0

[sagetrac-git]

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

d544fe3	Merge tag '9.8.beta4' into public/implement_inverse_of_finite_field_fetch_int
12411a0	deprecate .fetch_int() and .integer_representation()

[kwankyu]

comment:16

Thanks.

One more thing. I think the hyphen in finite-field element is legitimate but not really necessary. The hyphen is there to mean "(finite field) element" not "finite (field element)". But sage is mathematics software and there is zero risk that one reads it as "finite (field element)". "finite field", "elliptic curve", "cyclic group", etc are all mathematical terms, and rarely written in two forms (hyphened and not hyphened). For consistency, let us stick to one form. Please reconsider.

[sagetrac-git]

Changed commit from 12411a0 to 9e2e6ee

[sagetrac-git]

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

9e2e6ee

remove hyphens per reviewer request

[yyyyx4]

comment:18

Done. (IMHO the hyphens improve readability even when they are not strictly needed to disambiguate, but it's obviously a matter of personal preference.)

[sagetrac-git]

Changed commit from 9e2e6ee to 5e92893

[sagetrac-git]

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

5e92893

Minor edits on spaces

[sagetrac-git]

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

0eabc46

One more edit on spaces

[sagetrac-git]

Changed commit from 5e92893 to 0eabc46

[kwankyu]

comment:21

Replying to Lorenz Panny:

Done. (IMHO the hyphens improve readability even when they are not strictly needed to disambiguate, but it's obviously a matter of personal preference.)

Thanks! The boundary of consistency and personal preference is blurred and unfortunately is a source of perpetual debates.

I made minor edits on spaces. Otherwise, the branch looks good to me. Set it positive review once tests pass.

[kwankyu]

Reviewer: Kwankyu Lee

[yyyyx4]

comment:23

Bot looks green. Thank you!

[vbraun]

Changed branch from public/implement_inverse_of_finite_field_fetch_int to 0eabc46