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 $D_{1},D_{2},\ldots ,D_{n}$ 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 $f(x)=a_{0}+a_{1}\cdot x+a_{2}\cdot x+\ldots a_{k-1}x^{k-1}$ with $a_{0}=D$ is generated. The $D_{i}$ are calculated as $D_{i}=f(i){\mbox{ for }}i=1,\ldots ,n$.

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