## Non-supersingular curve

As a part of brain exercise, I’m spending times on weekend doing calculations for a non-supersingular curve over $F_2^4$ with $f(z)=z^4+z+1$ and $E : y^2+xy=x^3+z^3x^2+(z^3+1)$.

To determine points in the curve, I have to solve quadratic equations for that elliptic curve. Rosing said that the variable has to be changed to eliminate $x$.

Let $y = xz$

and that gives $(xz)^2+x^2z+f(x)=0$

By multiplying the entire equation by $x^{-2}$, then the equation will be:

$z^2+z+c$

where

$c= f(x).x^{-2}$.

$z^2+z+c$ will only has solution when the Trace of $c$ is 0.

Hmmm, I have done calculation for simpler E equation, never for this kind of equation. Well, most of the times, calculating math in numbers is not that simple! I mean, there are definitions and equations, and looking at them make you think that you already understand them. But when you try to apply them using numbers, then it is another hard work to do 😀