Supersingular Elliptic Curves


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.


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 \(E[3]=O\)


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.