Shamir Secret Sharing
Shamir’s Secret Sharing is the (t,n) threshold scheme proposed in 1979, based on polynomial interpolation over GF(p), information-theoretically secure.
Purpose
It splits a secret into n shares, t of which reconstruct it, while t-1 colluding shares leak nothing — the de facto standard of threshold cryptography.
Mechanism
Choose a prime p and build a degree t-1 polynomial f(x) = S + a_1 x + ... + a_{t-1} x^{t-1} (mod p); the shares are (i, f(i)). Reconstruction uses Lagrange interpolation: f(0) = Σ f(i)·λ_i (mod p).
Use cases
- Cryptocurrency wallet backups
- Threshold protection of CA private keys
- A foundation of MPC protocols