The breadth of Shamir's secret-sharing scheme

In 1979 Shamir and Blakley introduced the concept of secret sharing through threshold schemes. Their models were based on polynomials and finite geometries. Since 1979 many researchers have taken the basic concept of a threshold scheme and used other mathematical structures to adapt threshold schemes to meet the needs of many practical situations. In this paper the authors take Shamir's construction and show how it can be used to realize the adapted models. In particular, the authors give new constructions for multipart, multilevel, democratic and prepositioned schemes. It will also be demonstrated how known methods for detecting cheaters and disenrolling participants can be incorporated into Shamir's scheme.