I have come up with the following rudimentary example of how ECC relates to asymmetric keys.
Is this a valid explanation of ECC and its relationship to asymmetry?
To only be deciphered by the person with the private key, it is necessary to use methods nontrivial in cryptography. To randomize the key, a set would be required to contain multiple solutions of which only one is the key. Primes are also used in ECC, so a useful approximation for implementation would be Gauss' approximation to primes.
$ x^2+xy+x−y^3$ will be used as the basis for this example.
A random point may be selected from within the area of the curve, generated by a set of functions and a rotating exchange of variables.
The shake in the cipher would be:
$k(n)+(−y^3)+x+xy+x^2=kG$ for:
$k(n)+(−y^3)+x+xy+x^2=k$
There are four solutions for $y$ which allows one to be a random key in this example. The asymmetry comes from the rotation of solutions to $y$, coupled with having to know that $y$ is the variable to be solved for.
Is this a good basic example of ECC?
For proof let Alice create the message $k$, and then create $kG$. Bob receives $kG$ and has been told by Alice in a separate message that the key is negative and a solution for $y$. Thus Bob knows to decipher $kG$ he must use:
$$y = -\frac{2^{1/3} x}{(\sqrt{(-27 k(n)+27 k-27 x^2-27 x)^2-108 x^3}-27 k(n)+27 k-27 x^2-27 x)^{1/3}} - \frac{(\sqrt{(-27 k(n)+27 k-27 x^2-27 x)^2-108 x^3}-27 k(n)+27 k-27 x^2-27 x)^{1/3}}{32^{1/3}}$$
To summarize and make my question as explicit as possible, is this a workable example for how asymmetry relates to ECC?