Shamir's Secret Sharing Algorithm

Shamir's secret sharing is an algorithm for splitting a private piece of information into multiple shares distributed among a group of people, designed so the secret can only be reconstructed once a minimum number of those shares, the threshold, are brought back together, while any smaller number of shares reveals nothing about it at all. It works through polynomial interpolation: the secret becomes the constant term of a specially constructed polynomial whose other coefficients are chosen at random, and each participant receives one point from that polynomial's curve, exploiting the mathematical fact that a set number of points uniquely pins down a polynomial of the matching degree. Cryptographer Adi Shamir devised the scheme in 1979, and because its protection rests on this mathematical structure rather than on the difficulty of any computational problem, it offers information theoretic security, meaning even an attacker who gathers shares below the threshold learns nothing about the secret no matter how much computing power is available. The technique is used today to protect cryptographic keys, cryptocurrency wallet recovery phrases and access to password managers and physical safes.