The previous post said that isogenies between elliptic curves are the basis for a quantum-resistant encryption method, but we didn’t say what an isogeny is.

It’s difficult to look up what an isogeny is. You’ll find several definitions, and they seem incomplete or incompatible.

If you go to Wikipedia, you’ll read “an isogeny is a morphism of algebraic groups that is surjective and has a finite kernel.” The possibly misleading term here is “algebraic group.” It may sound like they’re throwing in the term “algebraic” for clarification as if to say “a group like the kind you’d talk about in math, as opposed to the kind of group you might talk about somewhere else like in sociology.” An algebraic group is indeed a group, but one with additional structure. A “morphism” is a structure-preserving map, and so here “morphism” means a function that preserves the algebraic and topological structure of an algebraic group.

The algebraic groups we care about are elliptic curves, so let’s specialize to elliptic curves. For the rest of this post all definitions and theorems will come from Handbook of Elliptic and Hyperelliptic Curve Cryptography by Cohen and Frey. This book defines *isogeny* implicitly by defining what it means for two curves to be *isogenous*.

Two curves

E/KandE‘/Kare isogenous overKif there exists a morphism φ :E→E‘ with coefficients inKmapping the neutral element ofEto the neutral element ofE‘.

Here *K* is the field the curves are defined over. The neutral element is the identity element of the curve as a group, the point at infinity.

Something strange is going on here. The definition talks about two curves being isogenous. That sounds like a symmetric relationship, but the definition is not symmetric. It specifies the existence of a morphism in one direction but not in the other. But the book goes on to explain that if an isogeny exists in one direction, there necessarily exists a unique isogeny in the opposite direction, the dual isogeny, though it’s not obvious why this should be.

Another puzzling thing is that it doesn’t seem to take much for a function to be an isogeny, just map the group identity to the group identity. But there are other requirements implicit in the statement that φ is a morphism in the context of algebraic groups. Cohen and Frey do not require φ to be a homomorphism, as some authors do, but they point out that in fact φ will turn out to be a group homomorphism. They say “for more background on isogenies, we refer to section 4.3.4.” OK, let’s go there.

In that section the authors say that a morphism between Abelian varieties (a special case of algebraic groups which includes elliptic curves) is an isogeny if and only if it is surjective and has a finite kernel. That seems like a much stronger requirement than the definition above. However, in this context simply requiring a morphism to map the group identity to the group identity implies that the morphism will be surjective and have a finite kernel, and vice versa.

## An isogeny is not an isomorphism

An isogeny is a group homomorphism, but not an isomorphism, not in the modern usage of the term. Historically, however, the term “isomorphism” was used for what we now call an isogeny.

We said above that the existence of an isogeny in one direction implied the existence of a dual isogeny in the opposite direction. These functions are not inverses of each other: their composition is not the identity. However, their composition does have a simple form: it is multiplication by an integer *n*, called the degree of the isogeny. (Multiplication here means repeated group addition, i.e. the composition takes an element *x* to the sum of *n* copies of *x* using the group law on the elliptic curve.)

## Example

Cohen and Frey give the example

of an isogeny of degree 2 between the elliptic curves

*y*² = *x*³ + 1132*x* + 278

and

*y*² = *x*³ + 500*x* + 1005

over the field with 2003 elements. The curves are not isomorphic because the group structure of the former is a cyclic group and the group structure of the latter is not.

Incidentally, there is a theorem that says two elliptic curves over a finite field are isogenous if and only if they have the same number of points. So a brute-force approach to showing that the curves above are isogenous would be to show that they both have the same number of points. And indeed, both have 1956 points.

I think you have a small typo: “morphism between affine varieties” should be “morphism between abelian varieties”. Morphisms between abelian varieties are especially nice precisely because they are projective and not affine.

Thanks. Fixed.

I think there’s a missing condition: for the curves to be isogenous, φ should be non-constant. Otherwise you could take φ to be the constant map to the neutral element of E’. Earlier in Section 4.3.4, Definition 4.62, isogenies are required to be surjective by definition. (On the other hand, Silverman’s “The Arithmetic of Elliptic Curves” allows isogenies to be constant, then uses non-zero isogenies in the definition of isogenous.)

In the definition of isogeny of curves, merely asking that the neutral element is preserved allows for the constant map, which doesn’t fit the ‘surjective with finite kernel’ definition. Apparently the literature is not 100% consistent whether the constant map is allowed to be an isogeny or not…

But if a map of curves preserves the neutral element and is not constant, then it *is* surjective.

There’s a small typo in the 2nd curve equation. It should be

y² = x³ + 500x + 1005

instead of

y² = x³ + 500x + 105

Thanks, Nicolas.

Hi! I feel like I’m a bit late to the party but I was reading this post while trying to understand isogenies and your post has indeed helped me but I do have 2 remarks:

1.- You mention that “there necessarily exists a unique isogeny in the opposite direction, the dual isogeny, though it’s not obvious why this should be.”. In case anyone was as curious as me, I have found a proof of this in theorem 6.1 of the third chapter of Silverman’s “The arithmetic of elliptic curves”.

2.- I don think the statement “An isogeny is a group homomorphism, but not an isomorphism” is correct. As you mention before that, isogenies are morphisms but they are not group homomorphisms because that would imply that they preserve group structure. But for example in the case of elliptic curves, the group structure provided by the addition law is not preserved by isogenies.

@Marta PS…. The key focus here is that, an isogeny preserves the group structure of the elliptic curve, which means that it maps the identity element to the identity element, preserves the group operation, and maps each point to a unique point on the target curve. This ensures that the isogeny has a unique inverse, which is another isogeny that “undoes” the effect of the original isogeny, but may not necessarily preserve the curve structure.

On the other hand, an isomorphism is a bijective map between two elliptic curves that preserves not just the group structure, but also the curve structure. This means that an isomorphism preserves the underlying geometry of the curve, including its topology and algebraic properties. Because an isomorphism preserves the curve structure, it necessarily has an inverse that is also an isomorphism.

So, an isogeny can have a unique inverse without being an isomorphism because the uniqueness of the inverse is a consequence of the group structure that is preserved by the isogeny, while the preservation of the curve structure required for an isomorphism is a stricter condition that may not be satisfied by an isogeny in general.