A Remark on Hierarchical Threshold Secret Sharing

The main results of this paper are theorems which provide a solution to the open problem posed by Tassa [1]. He considers a specific family Γν of hierarchical threshold access structures and shows that two extreme members Γ and Γv of Γν are realized by secret sharing schemes which are ideal and perfect. The question posed by Tassa is whether the other members of Γν can be realized by ideal and perfect schemes as well. We show that the answer in general is negative. A precise definition of secret sharing scheme introduced by Brickell and Davenport in [2] combined with a connection between schemes and matroids are crucial tools used in this paper. Brickell and Davenport describe secret sharing scheme as a matrix M with n+1 columns, where n denotes the number of participants, and define ideality and perfectness as properties of the matrix M. The auxiliary theorems presented in this paper are interesting not only because of providing the solution of the problem. For example, they provide an upper bound on the number of rows of M if the scheme is perfect and ideal