### Supersingular Elliptic Curves

#### Definition

Let $$p$$ is a prime and let $$q$$ be a power of $$p$$. Let $$E$$ be an elliptic curve over $$\mathbb{F}_q$$. If $$E[p]=O$$, then $$E$$ is a Supersingular elliptic curve, if $$E[p]=\mathbb{Z}/p\mathbb{Z}$$ then $$E$$ is an Ordinary elliptic curve.

#### Example

For $$p=3$$, the curve $$E: y^2=x^3-x$$ is supersingular over the field $$\bar{F}_3$$. Here we see that $$[3]*(x,y)=O$$ for $$(x,y) \neq O$$ if and only if $$3x^4-6x^2-1=0$$, but such $$x$$ does not exist since $$\bar{F}_3$$ has characteristic $$3$$. Thus $$E[3]=O$$

#### Properties

Theorem [[Sil09], Chapter V.3, Theorem 3.1] These following conditions are equivalent:

1. $$E[p^r]=0$$ for all $$r \geq 1$$.

2. $$End(E)$$ is an order in a quaternion algebra.

3. The map $$[p]: E \rightarrow E$$ is purely inseperable and $$j(E) \in \mathbb{F}_{p^2}$$.

As we see, all Supersingular elliptic curves are isomorphic to a curve in $$F_{p^2}$$, up to isomorphism, therefore the number of these curves are finite. It is natural that we want to count the number of these curves. Fortunately, we have a formula for the number of supersingular elliptic curves, as stated below:

Theorem. [[Sil09], Chapter V.4, Theorem 4.1] The number of supersingular elliptic curves up to isomorphism is $$\left\lfloor \dfrac{p}{12} \right\rfloor+z$$, where

• $$z=0$$ if $$p \equiv 1 \pmod{ 12}$$

• $$z=1$$ if $$p \equiv 5,7 \pmod{ 12}$$

• $$z=2$$ if $$p \equiv 11 \pmod{ 12}$$

In the next chapter, we introduce the graph where the vertices are the Supersingular elliptic curves (up to isomorphism). This graph has several interesting properties that make it a candidate for constructing post quantum cryptosystems.