Secret Sharing

Secret Sharing is used to split a secret (usually a key) into several pieces which are then given to distinct persons so that some of these persons must cooperate to reconstruct the secret.

A Simple Approach

One simple approach to split a secret number $D$ into $n$ pieces $D_1,D_2,\dots ,D_n$ such that any $k$ pieces are sufficient (and necessary) to reconstruct $D$ is using a $k-1$ polynomial.

When splitting the secret a random polynomial $f(x)=a_0+a_1\cdot x+a_2\cdot x+\dots a_{k-1}{x}^{k-1}$ with $a_0=D$ is generated. The $D_i$ are calculated as $D_i=f(i)\text{ for }i=1,\dots ,n$.

Given any $k$ $D_i$ it is possible to interpolate the polynomial and calculate $f(0)$ which gives the original secret $D$.