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?