What is an isogeny? (2019) | Hacker News Reader

What is an isogeny? (2019)

johndcook.com

22 pointshasheddan10 comments

I found this talk by Luca De Fao is a very good introduction to isogneys and SIDH (caveat emptor: I can grok some algebra, but am not cryptographer) https://m.youtube.com/watch?v=EAe5dqWcxh4

This is one of those things where I think there are 2 or 3 layers of math concepts above the concept of isogeny that I have to understand before I can understand isogenies.

I tried following a trail upwards from isogeny and most of the related terms I didn't understand either. Nice.

>" Two curves E/K and E‘/K are isogenous over K if there exists a morphism φ : E → E‘ with coefficients in K mapping the neutral element of E to the neutral element of E‘."

What does it mean for a morphism to "have elements in K" ????

Not sure I understand this so-called definition.

A map between algebraic curves is defined by polynomials. That the map is defined over K means you can find a coordinate system such that the equations of the curves and the equations of the morphisms have coefficients in K and not some larger ring, eg the complex numbers or a large extension field of F_p(field of p elements)

> A map between algebraic curves is defined by polynomials.

That's a statement that requires IMO justification.

Is that really always the case?

Does it somehow stem from the fact that the source and destination sets are - albeit large - finite?

Again: not obvious.

A map of algebraic curves is not a special case of a map between two sets. There aren't really source and destination sets to speak of. The fact that it is given by polynomials really is a definition.

The moral "justification" of the definition is something like "algebraic geometry is precisely the study of such objects" or "we want definitions that are stable under ring base change, and this implies polynomials", etc., but we don't formally need to justify definitions.

[One could take a different route to defining those things, in which this becomes a theorem instead of a definition. For example one can define algebraic curves over a field k as contravariant functors from k-algebras to sets satisfying certain additional properties, and then maps of algebraic curves are natural transformations between those functors. The fact that they are given by polynomial equations is then a theorem. Just stating the "additional properties" for a curve is a rather daunting task, though, unfortunately.]

Hmm.. What's the difference between this and a homomorphism? Are there homomorphisms which are not isogenies? I'm still not very clear on this.

Great question. There's a subtle point the article glosses over "... with coefficients in K." If you have two elliptic curves that just happen to be cyclic groups over finite fields, you can construct homomorphisms between them with finite kernels. But, you won't have any hope of finding morphisms unless those finite fields are the same. (same characteristic? It's been a while)

In less fancy language: isogenies are homomorphisms you can describe with simple formulas involving (x, y) coordinates.

Good question, no there is no difference for elliptic curves, which you can think of as 1-dimensional geometric objects (curves) which posses group structure. A good map in this category should respect the geometry (be a so called rational map, ie defined by polynomials in a suitable coordinate system) and the group structure. Interestingly all these maps are either constant (map everything to 0) or surjective.

For higher dimensional geometric groups (abelian surfaces etc) one usually wants to make a distinction and calls the surjective homomorphisms with finite preimage isogenies.

Very glad to read the link and find out this isn't a highly potent form of intersectional bigotry that was created in a lab by critical studies theorists.

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