# Why we use specifically Jacobian Groups for HECC?

The following is stated in this answer on "What is so special about elliptic curves?":

But for these curves, an excellent geometric rule does not exist to add points, like in conics and elliptic curves. So, we are forced to work in the Jacobian group of these curves, which is not the group of points anymore but of divisors (which are like polynomials of points, if that makes any sense).

I've got the following questions about those statements:

1. What makes us work specifically in the Jacobian Group and not some other group?
2. We care in Hyper Elliptic Curves to have points adding to "zero" (for genus = 2). So, for instance $$R_1 + (-R_1) = O$$ etc. Why is that?
• Hint: Jacobian is a variety. Commented May 26 at 2:12
• Can you think of any other algebraic varieties that we can use in cryptography? Commented May 28 at 7:12

1. Diffie-Hellman Key Exchange (DH/ECDH) requires the underlying group to be cyclic so that DLP be hard.
2. We prefer storing the data in cryptography which can be represented efficiently in space because space is issue in cryptography.

What makes us to work specifically in Jacobian Group and not some other group?

No we are not working in Jacobian group. We are working in the group of divisors.

Before proceeding further let us understand what a variety is? A variety is, superficially speaking, a collection of algebraic objects satisfying certain conditions.

For example Let $$L$$ be a degree $$1$$ polynomial over any field then the set of solutions to such a polynomial in the form $$(x,y)$$ would constitute a projective variety.

Now can you define group structure over this variety? Clearly, no. Hence, even though it is a variety it doesn’t constitute a group (leave cyclic, this is not even a group) and thus we would need to find another hack to use this.

This hack in terms of hyper-elliptic curve is called Jacobian variety.

(+ I don’t think Jacobian group terminology in the context is correct. The correct term would be Jacobian variety.)

HECC constitute:

1. A curve $$C$$ whose solution constitute a projective variety.
2. Jacobian variety, an abelian variety, of the curve, $$J(C)$$, which have a natural map to the projective variety.
3. The group of divisors on $$C$$, which is isomorphic to the Jacobian variety.

Thus, it can be said that we are actually working in group of divisors, which is an actual group that satisfies DH Key exchange conditions because it is cyclic.

The Jacobian is there to map the divisors to point so that they can be stored. See the definition of Jacobian Variety to know more about it.

We care in Hyper Elliptic Curves to have points adding to "zero" (for genus = 2). So, for instance $$R1+(−R1)=O$$ etc. Why is that?

We care in Hyper Elliptic Curve group theory (DH uses groups) about things adding to zero (identity element) because existence of inverse is one of the group axioms.