Size of Parameters in Polynomial Key-Splitting Algorithm I've seen the elegant way of splitting a key among different people so that only a certain number need to be present to re-compute the key, yet nobody has enough information to re-compute the key on their own. For example, if I want to split $K$ between 4 people, such that any 3 can re-compute the key, I first form a polynomial $y(x)=ax^2+bx+K$.
Then I compute and distribute 4 pairs of the form $(x_i, f(x_i))$, one to each of the people. 

It seems like $a, b, x$ should certainly be chosen with a cryptographically secure RNG, but is there any advice on how big they should be? Should they be as big as $K$? Big enough so that the term is about the same size as $K$ (which would make $a\approx K/x_i^2$, where $\approx$ means of the same order of magnitude)? As big as I can make them to the point that time to re-compute the key becomes an issue?
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.)