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Adi Shamir introduced a polynomial-interpolation construction for threshold secret sharing in 1979. In Shamir's secret sharing, a secret is the constant term of a random polynomial of degree less than $k$ over a finite field; shares are its values at distinct nonzero field elements. Any $k$ shares recover the polynomial and hence the secret.^[ref]

Fewer than $k$ shares leave every possible secret equally compatible when the other coefficients are uniformly random. This is an information-theoretic threshold guarantee, rather than a claim that recovering the secret merely takes a long time. The construction makes distributed authorization and resilience to missing shares compatible. It requires enough field elements for all distinct share positions.

By Adi Shamir. Published in Communications of the ACM 22(11), pages 612–613.

Publisher: doi.org/10.1145/359168.359176

Primary paper: people.cs.pitt.edu/~litman/courses/cs2001/lectures/lecture-02-p612-shamir.pdf

Earlier MIT technical-report record: hdl.handle.net/1721.1/148961

Pages 612–613 construct Shamir's secret sharing by polynomial interpolation and prove the threshold property.