I'd like to get just the mechanics of multivariate-quadratic cryptography, so I'm trying to understand the definitions involved. From Wolfram's website, I get that a quadratic map is for example a recurrence of the form $x_{n+1} = 2 + (x_n)^2$. I will need an initial value too, right? I choose $x_0 = 3$. What's the inverse of this map and how do I get it?
I'm guessing an inverse of a quadratic map would be the transformation that takes $x_n$ back to $x_{n-1}$ so the using the example given in the first paragraph, $x_0 = 3, x_1 = 11, x_2 = 123$. So given $x_2 = 123$, I can return to 11 by subtracting 2 and taking square root: that is, $x_{n-1} = \sqrt{x_n - 2}$ is the inverse map of my example.
It's just a guess. I've never seen these things before. Can't find any reference that explains this in baby-steps.
Also, I'm wondering what would be an example of a quadratic map whose input is say a vector. My example is a 1x1 vector. What would it look if my input would be a 2x1 vector?