I'm trying to understand how to calculate the Frobenius endomorphism for an elliptic curve.
Specifically, If $E$ is defined over $\mathbb{F}_q$, then the Frobenius endomorphism $\pi$ is defined as
$\pi : E \rightarrow E,\,\,\,\,\,\,\,\,\,$ $(x,y) \rightarrow (x^q,y^q)$
This definition doesn't make sense to me b/c the Frobenius endomorphism is defined as carrying an element, $a$ to $a^p$ where $p$ is the characteristic of the ring (and is prime). So it seems like the definition should be $(x, y) \rightarrow (x, y)^q$ = $(x, y) + (x, y) + ... (x, y)$, $q$ times since elliptic curve addition is between 2 points. Why are the $x$ and $y$ values calculated separately?
Here is a concrete example I don't understand.
- Take the curve $E/\mathbb{F}_q : y^2 = x^3 + 4x + 3$ where $q = 67$
- Let $P = (15, 50)$. We know $[q](P) = P + P + ... + P$ (q times) = $[67]P = (24, 56)$
- The Frobenius endomorphism, $\pi$ maps $P$ to itself. However, $(15, 50) \rightarrow (15, 50)$ is different from $(15, 50) \rightarrow (24, 56)$.
- Basically, I believe my assumption is that the Frobenius endomorphism applies the elliptic curve addition law to a point, $P$ in $\mathbb{F}_q$, $q$ times, however, it seems like this assumption is wrong.