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Secret sharing schemes on access structures with intersection number equal to one.

Jaume Martí-Farré, Carles Padró

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

The characterization of ideal access structures and the search for bounds on the optimal information rate are two important problems in secret sharing. These problems are studied in this paper for access structures with intersection number equal to one, that is, access structures such that there is at most one participant in the intersection of any two different minimal qualified subsets. The main result in this work is the complete characterization of the ideal access structures with intersection number equal to one. Besides, bounds on the optimal information rate are provided for the non-ideal case.

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One of the main open problems in secret sharing is the characterization of the ideal access structures. This problem has been studied for several families of access structures with similar results. Namely, in all these families, the ideal access structures coincide with the vector space ones and, besides, the optimal information rate of a non-ideal access structure is at most 2/3. An access structure is said to be r-homogeneous if there are exactly r participants in every minimal qualified subset. A first approach to the characterization of the ideal 3-homogeneous access structures is made in this paper. We show that the results in the previously studied families can not be directly generalized to this one. Nevertheless, we prove that the equivalences above apply to the family of the sparse 3-homogeneous access structures, that is, those in which any subset of four participants contains at most two minimal qualified subsets. Besides, we give a complete description of the ideal sparse 3-homogeneous access structures.

•Jan 1, 2003•0 citations

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A perfect secret-sharing scheme is a method to distribute a secret among a set of participants in such a way that only qualified subsets of participants can recover the secret and the joint share of participants in any unqualified subset is statistically independent of the secret. The collection of all qualified subsets is called the access structure of the perfect secret-sharing scheme. In a graph-based access structure, each vertex of a graph G represents a participant and each edge of G represents a minimal qualified subset. The average information ratio of a perfect secret-sharing scheme realizing the access structure based on G is defined as AR = (Pv2V (G) H(v))/(|V (G)|H(s)), where s is the secret and v is the share of v, both are random variables from and H is the Shannon entropy. The infimum of the average information ratio of all possible perfect secret-sharing schemes realizing a given access structure is called the optimal average information ratio of that access structure. Most known results about the optimal average information ratio give upper bounds or lower bounds on it. In this present structures based on bipartite graphs and determine the exact values of the optimal average information ratio of some infinite classes of them.

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Ideal and perfect multi-secret sharing schemes whose access structure are Q 3

Lv yingli, Xiaojun Xu, Zhang Xiao-feng +1 more

The characterization of access structure is one of the main open problems in secret sharing schemes. We are particularly interested in determining which Q 3 access structures can be realized and the requirement should be achieved even in the presence of an adversary who is able to corrupt some players in an ideal secret sharing scheme. The problems are studied in our paper. Above all, we present a concrete scheme that can meet all above requirements. In our scheme, it is not only a dynamic multi-secret and multi-use sharing protocol, but participants from access structure can recover the secret using their shares, while the same number of participants from adversary structure can't recover the secret using their shares, which is different from general access structure schemes. Moreover, in the whole dynamic process, the coefficients of polynomial and private secret shares always stay the same. Therefore, it enjoys high practical value.

•Aug 1, 2012•0 citations
The characterization of ideal access structures and the search for bounds on the optimal information rate are two important problems in secret sharing. These problems are studied in this paper for access structures with intersection number equal to one, that is, access structures such that there is at most one participant in the intersection of any two different minimal qualified subsets. The main result in this work is the complete characterization of the ideal access structures with intersection number equal to one. Besides, bounds on the optimal information rate are provided for the non-ideal case.
One of the main open problems in secret sharing is the characterization of the ideal access structures. This problem has been studied for several families of access structures with similar results. Namely, in all these families, the ideal access structures coincide with the vector space ones and, besides, the optimal information rate of a non-ideal access structure is at most 2/3. An access structure is said to be r-homogeneous if there are exactly r participants in every minimal qualified subset. A first approach to the characterization of the ideal 3-homogeneous access structures is made in this paper. We show that the results in the previously studied families can not be directly generalized to this one. Nevertheless, we prove that the equivalences above apply to the family of the sparse 3-homogeneous access structures, that is, those in which any subset of four participants contains at most two minimal qualified subsets. Besides, we give a complete description of the ideal sparse 3-homogeneous access structures.
‎The characterization of the ideal access structures is one of the main open problems in secret sharing and is important from both practical and theoretical points of views‎. ‎A graph-based 3−3-3−homogeneous access structure is an access structure in which the participants are the vertices of a connected graph and every subset of the vertices is a minimal qualified subset if it has three vertices and induces a connected graph‎. ‎In this paper‎, ‎we introduce the graph-based 3−3-3−homogeneous access structures and characterize the ideal graph-based 333-homogeneous access structures‎. ‎We prove that for every non-ideal graph-based 333-homogeneous access structure over the graph GGG with the maximum degree ddd there exists a secret sharing scheme with an information rate 1d+1\frac{1}{d+1}d+11​‎. ‎Furthermore‎, ‎we mention three forbidden configurations that are useful in characterizing other families of ideal access structures‎.
We study the information rate of secret sharing schemes whose access structure is bipartite. In a bipartite access structure there are two classes of participants and all participants in the same class play an equivalent role in the structure. We characterize completely the bipartite access structures that can be realized by an ideal secret sharing scheme. Both upper and lower bounds on the optimal information rate of bipartite access structures are given. These results are applied to the particular case of weighted threshold access structure with two weights.
In a bipartite secret sharing scheme, the set of participants is divided into two parts, and all participants in each part play an equivalent role. The ideal bipartite access structures were characterized by Padr´o and S´aez, but it is not known which is the optimal information rate of non ideal bipartite access structures. Determining the optimal information rate of general access structures is one of the major problems in secret sharing. We study this open problem restricted to the bipartite access structures, obtaining a new method to compute bounds on the optimal information rate. Namely, by using the connection between secret sharing schemes and polymatroids, we show a linear programming problem that determines, for each access structure, an upper bound on the information rate. Moreover, we show new optimal constructions for certain bipartite access structures.
A perfect secret-sharing scheme is a method to distribute a secret among a set of participants in such a way that only qualified subsets of participants can recover the secret and the joint share of participants in any unqualified subset is statistically independent of the secret. The collection of all qualified subsets is called the access structure of the perfect secret-sharing scheme. In a graph-based access structure, each vertex of a graph G represents a participant and each edge of G represents a minimal qualified subset. The average information ratio of a perfect secret-sharing scheme realizing the access structure based on G is defined as AR = (Pv2V (G) H(v))/(|V (G)|H(s)), where s is the secret and v is the share of v, both are random variables from and H is the Shannon entropy. The infimum of the average information ratio of all possible perfect secret-sharing schemes realizing a given access structure is called the optimal average information ratio of that access structure. Most known results about the optimal average information ratio give upper bounds or lower bounds on it. In this present structures based on bipartite graphs and determine the exact values of the optimal average information ratio of some infinite classes of them.
The characterization of access structure is one of the main open problems in secret sharing schemes. We are particularly interested in determining which Q 3 access structures can be realized and the requirement should be achieved even in the presence of an adversary who is able to corrupt some players in an ideal secret sharing scheme. The problems are studied in our paper. Above all, we present a concrete scheme that can meet all above requirements. In our scheme, it is not only a dynamic multi-secret and multi-use sharing protocol, but participants from access structure can recover the secret using their shares, while the same number of participants from adversary structure can't recover the secret using their shares, which is different from general access structure schemes. Moreover, in the whole dynamic process, the coefficients of polynomial and private secret shares always stay the same. Therefore, it enjoys high practical value.