A process to split a secret (such as a private key into a number of shares, a subset of which can be later combined to recreate the original secret.
Based on the fact the $k$ points are required to define a polynomial of degree $k-1$
With our points being elements in a finite field $\mathbb{F}$ of size
$P$ where $0 < k le n < P; S<P$ and $P$ is a prime number.
Choose at random $k-1$ positive integers $a_1 .. a_{k-1}$
with $a_i <P$ and let $a_0 = S$
The person splitting the secret builds a polynomial where the secret is the constant term $a_0$
$f(x)=a_0 + a_1x + a_2x^2 + ... + a_{k-1}x^{k-1}$
Let us construct any $n$ points out of it, for instance set $i=1..n$ to retrieve $(i,f(i))$ .
Every participant is given a point
Given any subset of $k$ of these points, we can find the coefficients of the polynomial using interpolation.
The secret is the constant term $a_0$
For example
$x^4 + 3x^3 + 4x + 12$
