Generalized Secret Sharing and Monotone Functions

Secret Sharing from the perspective of threshold schemes has been wellstudied over the past decade. Threshold schemes, however, can only handle a small fraction of the secret sharing functions which we may wish to form. For example, if it is desirable to divide a secret among four participants A , B: C, and D in such a way that either A together with B can reconstruct the secret or C together with D can reconstruct the secret, then threshold schemes (even with weighting) are provably insufficient. This paper will present general methods for constructing secret sharing schemes for any given secret sharing function. There is a natural correspondence between the set of “generitlized” secret sharing functions and the set of monotone functions, and tools developed for simplifying the latter set can be applied equally well t o the former set.