Meta-ticket: Further work on isogenies and endomorphisms of elliptic curves #7368

chriswuthrich · 2009-11-01T14:26:11Z

Working on ticket #7096, I realised that there are many functionalities concerning morphisms of elliptic curves that are not implemented or implemented badly.

See also #6887, and #7262.

Here is a wish-list:

General isogenies, not only cyclic ones should be implemented.
Possibly this could be done in a clever way, by having it
internally factored into cyclic isogenies. → composite elliptic-curve isogenies #32744
Inseparable isogenies (e.g. Frobenii) should be possible too. → inseparable elliptic-curve isogenies #33915 (See also Retrieving the multiplication-by-p isogeny on elliptic curves over fields of characteristic p fails. #6413.)
One should be able to compose isogenies. → Compose isogenies #16245, composite elliptic-curve isogenies #32744
There should be an addition for isogenies with the same domain
and codomain.
There should be an endomorphism_ring for elliptic curves whose
elements are isogenies.
Similarly, automorphisms should give a group of isogenies.
Isogenies of large prime degree ℓ can be computed in time Õ(√ℓ) using https://velusqrt.isogeny.org/ → √élu algorithm: asymptotically faster elliptic-curve isogenies #34303
Is there a way of constructing a (not necessarily normalized)
isogeny knowing the degree and the domain and codomain?

CC: @sagetrac-weigandt @pjbruin @sagetrac-sbesnier @yyyyx4

Component: elliptic curves

Keywords: isogeny, isogenies, endomorphism ring

Issue created by migration from https://trac.sagemath.org/ticket/7368

sagetrac-adhalanay · 2015-10-20T18:28:39Z

comment:5

added

…curves This ticket adds `EllipticCurveHom_scalar`, a new class encapsulating scalar multiplications on elliptic curves. This serves two main purposes: 1. It solves one of the motivations behind sagemath#8014 (faster `multiplication_by_m_isogeny`). 2. Wrapping scalar multiplications as an `EllipticCurveHom` is an important step towards implementing endomorphism rings (see sagemath#7368). We also deprecate `.multiplication_by_m_isogeny()`: It should be replaceable by `.scalar_multiplication()` in all cases. URL: https://trac.sagemath.org/32826 Reported by: lorenz Ticket author(s): Lorenz Panny Reviewer(s): John Cremona

We implement an `EllipticCurveHom` child class `EllipticCurveHom_frobenius` encapsulating purely inseparable ''Frobenius isogenies''. As every isogeny decomposes into a separable and a purely inseparable part, we can (together with `EllipticCurveHom_composite`) now express any isogeny between two elliptic curves in Sage. One immediate application (also implemented in the patch) is that separable isogenies of degree divisible by the characteristic now have a working `.dual()` method. Other than that, changes to the existing codebase are kept minimal. This is one of the items on the "isogeny wish-list" sagemath#7368. It is also an important step towards implementing endomorphism rings later; cf. comment:3:ticket:32826. Diff without the dependency: https://git.sagemath.org/sage.git/diff?id2= 79ae468&id=e953939d23995c0c26964dc969fa 69cea52ee1c4 URL: https://trac.sagemath.org/33915 Reported by: lorenz Ticket author(s): Lorenz Panny, Mickaël Montessinos Reviewer(s): John Cremona

chriswuthrich added c: elliptic curves labels Nov 1, 2009

chriswuthrich assigned JohnCremona Nov 1, 2009

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yyyyx4 changed the title ~~Further work on isogenies and endomorphisms of elliptic curves~~ Meta-ticket: Further work on isogenies and endomorphisms of elliptic curves Aug 8, 2022

defeo mentioned this issue Apr 26, 2014

Inconsistent domain, codomain and parent in EllipticCurveIsogeny #12880

Closed

sagetrac-sbesnier mannequin mentioned this issue Sep 8, 2014

Compose isogenies #16245

Closed

This was referenced Sep 19, 2021

elliptic-curve isogenies probably shouldn't be mutable #32388

Closed

scalar-multiplication endomorphisms of elliptic curves #32826

Closed

inseparable elliptic-curve isogenies #33915

Closed

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Meta-ticket: Further work on isogenies and endomorphisms of elliptic curves #7368

Meta-ticket: Further work on isogenies and endomorphisms of elliptic curves #7368

chriswuthrich commented Nov 1, 2009

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Meta-ticket: Further work on isogenies and endomorphisms of elliptic curves #7368

Meta-ticket: Further work on isogenies and endomorphisms of elliptic curves #7368

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chriswuthrich commented Nov 1, 2009

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