## Trace function

Target : to embed plaintext to points in elliptic curve

What to do first : solving quadratic equation of the elliptic curve.

Problems : a quadratic equations only has a solution when the Trace of $c$ is 0

Example:
$E=y^{2}+xy=x^{3}+x^{2}+1$

by converting the right-hand side to a simple form, say $f\left (x \right )$, then bring that over to the left-hand side to become :
$y^{2}+xy+f\left (x \right )=0$

Now let
$y = xz$

then substitute it to the previous equation so it becomes:
$\left (xz \right )^{2}+x^{2}z+f\left (x \right )=0$

Multiply the entire equation by $x^{-2}$, so we get:
$z^{2}+z+c=0$

where:
$c=f\left (x \right ).x^{-2}$

Once we know the $Trace\left (c \right )=0$, we can solve for $z$. It turns out $z+1$ is also a solution. So let $z$ be one solution and $z{}'$ be another solution. After we find one solution, the other one is trivial. After the two solutions are recovered, then our data can be embedded on the curve.

[rewrite from this book]