Threshold Signature Schemes: Concepts

Traditional digital signatures use a single private key held by one party. If that key is compromised, stolen, or lost, all signatures made with it are potentially forged and the key cannot be recovered. High-value signing operations — code signing for operating systems, certificate authority key ceremonies, cryptocurrency wallet control — require stronger protection than any single party can provide.

Shamir's Secret Sharing (SSS), invented in 1979, splits a secret s into n shares such that any t shares can reconstruct s, but fewer than t shares reveal nothing. The scheme uses polynomial interpolation over a finite field: s is the constant term of a random degree-(t-1) polynomial, and the shares are evaluations of this polynomial at distinct points. SSS is information-theoretically secure.