Prof. Amos Beimel

Given a set of parties {1,….n}, an access structure is a monotone collection of subsets of the parties. For a certain domain of secrets, a secret sharing scheme for an access structure is a method for a dealer to distribute shares to the parties, such that only subsets in the access structure can reconstruct the secret. A secret sharing scheme is ideal if the domains of the shares are the same as the domain of the secrets. An access structure is universally ideal if there is an ideal secret sharing scheme for it over every finite domain of secrets. An obvious necessary condition for an access structure to be universally ideal is to be ideal over the binary and ternary domains of secrets. In this work, we prove that this condition is also sufficient. In addition, we give an exact characterization for each of these two conditions, and show that each condition by itself is not sufficient for universally ideal access structures.