Secret Sharing
Secret sharing splits a secret S into n shares so that any t shares reconstruct S while t-1 shares reveal nothing about it (information-theoretically secure).
Purpose
It stores secrets distributively to eliminate single points of failure and serves as a building block of threshold cryptography and MPC.
Mechanism
Shamir’s scheme: pick a random polynomial f(x) of degree t-1 over GF(p) with f(0)=S; the shares are f(i). Any t shares reconstruct f(0)=S via Lagrange interpolation, while t-1 shares are uniformly distributed over GF(p) and leak nothing.
Use cases
- Distributed key storage
- Threshold signatures
- A foundation of MPC