Ideal and Perfect Hierarchical Secret Sharing Schemes based on MDS codes

Secret sharing is a cryptographic primitive, which is used to distribute a secret among participants in such a way that an authorized subset of participants can uniquely reconstruct the secret and an unauthorized subset can get no information about the secret. It is a fundamental method used in secure multiparty computations, where various distrusted participants cooperate and conduct computation tasks based on the private data they provide. A secret sharing scheme is called ideal if the maximal length of shares and the length of the secret are identical. Secret sharing was first proposed independently by Blakley [5] and Shamir [23]. The scheme by Shamir relies on the standard Lagrange polynomial interpolation, whereas the scheme by Blakley is based on the geometric idea that uses the concept of intersecting hyper planes. The family of authorized subsets is known as the access structure. An access structure is said to be monotone if a set is qualified then its superset must also be qualified. Several access structures are proposed in the literature. They include the (t, n)-threshold access structure, the Generalized access structure and the Multipartite access structure. In the (t, n)-threshold access structure there are n shareholders, an authorized group consists of any t or more participants and any group of at most t− 1 participants is an unauthorized group. Let U be the set of n participants and let 2 be its power set. Then the ’Generalized access structure’ refers to situations where the collection of permissible subsets of U may be any collection Γ ⊆ 2 having the monotonicity property. In multipartite access structures, the set of players U is partitioned into m disjoint entities U1, U2, · · · , Um, called levels and all players in each level play exactly the same role inside the access structure.