Montgomery model for elliptic curves

[yyyyx4]

Sage implements all its elliptic-curve functionality on long Weierstrass curves. This makes sense, but sometimes we'd like to interface with (or debug) other software, most commonly in cryptography, where other curve models are much more popular.

This patch adds the method EllipticCurve_generic.montgomery_model(), which computes a Montgomery model

`BY^2 = X^3 + AX^2 + X`

for a given elliptic curve. Both "untwisted" Montgomery curves with B=1 and "twisted" Montgomery curves with arbitrary B are supported. The latter cannot be represented as an EllipticCurve_generic, so we return a ProjectivePlaneCurve instead. Arithmetic on that curve can be emulated by transfering back and forth to the Weierstrass form; a suitable morphism to do so can optionally be returned by the new method.

(In the future, we should also add Edwards curves, but this raises more questions with non-obvious answers: Edwards curves are singular and the correspondence with Weierstrass curves is only a birational equivalence, not an isomorphism.)

CC: @defeo @JohnCremona @categorie

Component: elliptic curves

Author: Lorenz Panny

Branch/Commit: 4e380e2

Reviewer: Kwankyu Lee

Issue created by migration from https://trac.sagemath.org/ticket/33707

[sagetrac-git]

Changed commit from 3347fb1 to c093464

[sagetrac-git]

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

240cf80	Merge branch 'develop' into t/33707/public/montgomerize_elliptic_curves-test
c093464	Some edits mostly improving readibility of code

[kwankyu]

Reviewer: Kwankyu Lee

[kwankyu]

comment:4

I did some edits mostly improving readability of the code. I also wanted to change the added line in AUTHORS section to ...: added montgomery_model(), but did not, to respect your taste. But the added line still looks difficult to read in the file. My taste is such that the AUTHORS section should contain plain text. But you may object.

Otherwise, all looks good to me. You can set positive review if you agree with my edits.

[sagetrac-git]

Changed commit from c093464 to ff391f9

[sagetrac-git]

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

ff391f9

One more edit

[sagetrac-git]

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

af5081a	Merge tag '9.6.rc3' into public/montgomerize_elliptic_curves
4e380e2	some more style tweaks

[sagetrac-git]

Changed commit from ff391f9 to 4e380e2

[yyyyx4]

comment:7

Thank you! I just applied a few more small stylistic tweaks (including the one you suggested above).

[kwankyu]

comment:8

Thanks.