The Construction Of Two Kinds Of Ideal Access Structures In Secret Sharing

The concept of secret sharing is produced in practical application, which is a method of sharing a secret among some participants, so that only participants in a predetermined authorized subset can recover the secret, and those in a unauthorized subset can not reconstruct it. In 1979, Shamir and Blakley independently proposed the concept of secret sharing. Since then, many scholars at home and abroad have begun to study the secret sharing theory and model, and have achieved fruitful results. Constructing ideal secret sharing schemes and finding more ideal access structures are core issues in the field of secret sharing, which requires us to further study. This paper mainly aims to study the construction of the ideal access struc-tures. On the one hand, this paper constructs a kind of ideal access structures based on privileged coalitions in Shamir’s type scheme, where finding privileges coalitions is an important problem, so that this paper gives an algorithm for calculating the privileges coalitions; on the other hand, this paper constructs a kind of ideal access structures based on a type of special hypergraph, and gives the number of this kind of ideal access structures.Main contributions of this paper are summarized as follows:1. This paper analyzes that using privileges coalitions in Shamir’s type scheme can construct a kind of ideal access structures, as well as the limitations of the algorithm of computing privileges coalitions designed by S.Spiez et al. Therefore, this paper firstly proves that Shamir’s type scheme must be a kind of secret shar-ing schemes based on vector space, then, using this result, designs an algorithm on Mathemathica software platform for the privileged coalitions of arbitrary length on any finite field. Compared with the previous one designed by S.Spiez et al, the algo-rithm of this paper includes it, which extends the privilege coalitions of maximum length to that of arbitrary length, and thus it solves the open problem proposed by S.Spiez et al. Thus, we can find all the privilege coalitions in Shamir’s type scheme, resulting in all ideal access structures of this class.2. This paper firstly defines a type of special hypergraphs-r-uniform hyper-graphs with complete k partition, and constructs ideal access structures based on this kind of hypergraphs. According to the conclusion that access structures based on complete multipartition graphs are ideal, we try to find a kind of ideal hyper-graph access structures in the hypergraph access structures. This kind of new ideal access structures is very rich in structure. When the number of participants is n, the number of ideal access structures based on r-uniform hypergraphs with complete k partition is ∑kn=2(k-1)B(n,k).Moreover, this kind of ideal access structures have the (+,+) homomorphism property of secret sharing scheme in vector space, which can be applied to electronic voting protocol and so on.