supersingular elliptic curve group structure I am trying to understand the basic idea behind supersingular isogeny cryptography. With very little knowledge about the group theory and elliptic curve, I find it very hard to thoroughly understand the underlying math of this protocol. Then I read several friendly introductions, and am still confused about the following questions. To be honest, I am not sure if I am asking the right question, or maybe I have misunderstood some concepts. Any answer will be helpful.
According to the pp8 of introductions, supersingular curves $E/\mathbb{F}_{p^2}$ always have their full rational $(p-1)$ or $(p+1)$-torsion defined over $\mathbb{F}_{p^2}$. Taking $E/\mathbb{F}_{431^2}$ for example, $E/\mathbb{F}_{431^2}$ is precisely the $(p+1)$-torsion. Also, according to $$ker([p+1]) \cong \mathbb{Z}_{p+1} \times \mathbb{Z}_{p+1}$$
We have $$E/\mathbb{F}_{p^2}\cong \mathbb{Z}_{p+1} \times \mathbb{Z}_{p+1}$$
Then all $E/\mathbb{F}_{p^2}$ are isomorphic to each other, which I think should be wrong. Could anyone tell me where is wrong?
 A: There are a few possible explanations to your dilemma. Without a clearer statement of what exactly your problem is, it is impossible to be more precise.

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*Not all supersingular elliptic curves over $\mathbb{F}_{p^2}$ are isomorphic as abelian groups over $\mathbb{F}_{p^2}$. For example, taking $p=431$, the curve $E : y^2 = x^3 + x$ has group structure $E(\mathbb{F}_{p^2}) \cong (\mathbb{Z}/(p+1)\mathbb{Z})^2$ whereas its quadratic twist $E' : y^2 = x^3 + (2+i)x$ has group structure $E'(\mathbb{F}_{p^2}) \cong (\mathbb{Z}/(p-1)\mathbb{Z})^2$.


*Even if two elliptic curves are isomorphic over $\mathbb{F}_{p^2}$ as abelian groups over $\mathbb{F}_{p^2}$, this does not mean they are isomorphic as elliptic curves. An elliptic curve has more structure than just merely that of an abelian group.


*Even if you are only looking at abelian groups, isomorphic abelian groups may not be equivalent for the purposes of cryptography. For example the abelian groups $G_1 = (\mathbb{Z}/p\mathbb{Z})^*$ and $G_2 = \mathbb{Z}/(p-1)\mathbb{Z}$ are isomorphic as abelian groups, but the discrete logarithm problem is believed to be hard on $G_1$ whereas the same problem is known to be easy on $G_2$. In this case the issue is that the isomorphism, though it exists, is not efficiently computable in both directions, which makes a difference in cryptography.
