The Math Behind

Imagine having to draw a line passing through a single point; there are infinitely many possibilities. However, if two different points are required, only one unique line is possible. This simple idea is at the heart of Shamir's Secret Sharing and Lagrange's interpolation, upon which Qasmat is built.

The line is replaced by a polynomial of degree ( k-1 ), whose value at zero is the secret to be protected, and other coefficients are chosen at random. Then, at least ( k ) distinct points of this polynomial are distributed across distant servers.

To recover the secret, at least ( k ) points of the polynomial are retrieved from different servers, and the polynomial is reconstructed using Lagrange's interpolation (visualize here). ( P(0) ), corresponding to the secret, is then computed.

The number of points distributed (( n )) must be greater than ( k ). These two values constitute a threshold (( k, n )), whose value is determined according to the required security level.

Since the coefficients are integers, all operations are performed modularly. The modulus is a prime number, which helps avoid basic attacks.