Shamir's secret sharing places a secret at the constant term of a random polynomial of degree at most and issues distinct nonzero evaluation pairs . Any shares recover by polynomial interpolation. Given fewer than shares, every candidate constant term has the same number of compatible coefficient tuples, which gives perfect secrecy.
Conditioned on any shares in Shamir's secret sharing, each candidate secret in remains equally likely. Appending to those shares determines exactly one degree-at-most- polynomial for every candidate , by the nonzero Vandermonde determinant.
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Shamir's Secret Sharing is a cryptographic algorithm conceived by Adi Shamir in 1979. It is designed to securely distribute a secret among a group of participants, in such a way that only a certain threshold of them can reconstruct the secret. The main idea behind the scheme is to split the secret into pieces, or "shares," using polynomial interpolation.