Size of Parameters in Polynomial Key-Splitting Algorithm
All of this arithmetic must be done modulo $p$ (for some prime $p$ that is large enough so that it'll be larger than any conceivable $K$ you might ever want to use).  You need to pick $p$ in advance.

Once you've picked $p$, then you choose $a$ and $b$ uniformly at random from the set of integers modulo $p$, i.e., uniformly at random from the set $\{0,1,2,\dots,p-1\}$.

The values $x_i$ can be, so long as they are distinct.  A simple choice is to set $x_i=i$, i.e., $x_1=1$, $x_2=2$, $x_3=3$, $x_4=4$.  That suffices.

(Working modulo $2^n$ is not secure.  You need to work modulo a prime $p$.)

(Technically, you can work in any finite field; working modulo $p$ is just one way to get a finite field.  However, if you don't know what a finite field is, you can ignore this remark.  I'm adding this remark only to keep my fellow cryptographers happy.)