Some Ideal Secret Sharing Schemes

In a secret sharing scheme, a dealer has a secret. The dealer gives each participant in the scheme a share of the secret. There is a set \(\Gamma\) of subsets of the participants with the property that any subset of participants that is in \(\Gamma\) can determine the secret. In a perfect secret sharing scheme, any subset of participants that is not in \(\Gamma\) cannot obtain any information about the secret. We will say that a perfect secret sharing scheme is ideal if all of the shares are from the same domain as the secret. Shamir and Blakley constructed ideal threshold schemes, and Benaloh has constructed other ideal secret sharing schemes. In this paper, we construct ideal secret sharing schemes for more general access structures which include the multilevel and compartmented access structures proposed by Simmons.