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

Lv yingli, Xiaojun Xu, Zhang Xiao-feng, Lian-Lian Zhang

Published Aug 1, 2012• 0 citations
OpenAlex

Abstract

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.

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Secret sharing schemes on sparse homogeneous access structures with rank three.

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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.

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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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IEEE Transactions on Information Theory•Sep 1, 2011•8 citations
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.
‎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.
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 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.
For more than two decades, proving or refuting the following statement has remained a challenging open problem in the theory of secret sharing schemes (SSSs): every ideal access structure admits an ideal perfect multi-linear SSS. The class of group-characterizable (GC) SSSs include the multi-linear ones. Hence, if the above statement is true, then so is the following weaker statement: every ideal access structure admits an ideal perfect GC SSS. One contribution of this paper is to show that ideal SSSs are not necessarily GC. Our second contribution is to study the above two statements with respect to several variations of weakly-ideal access structures. Recently, Mejia and Montoya studied ideal access structures that admit ideal multi-linear schemes and provided a classification-like theorem for them. We additionally present some tools that are useful to extend their result.
Abstract. An n -player -secure robust secret sharing scheme is a ( t , n )-threshold secret sharing scheme with the additional property that the secret can be recovered, with probability at least , from the set of all shares even if up to t players provide incorrect shares. The existing constructions of robust secret sharing schemes for the range have the share size larger than the secret size. An important goal in this area is to minimize the share size. In the paper, we propose a new unconditionally-secure robust secret sharing scheme for the case with share size equal to the secret size. This is the minimum possible size as dictated by the perfect secrecy of the scheme. We further extend our scheme to realize a class of multilevel access structures that satisfy a special condition. The property that the share size is equal to secret size is preserved in the extended scheme. The proposed scheme is the first known robust secret sharing scheme realizing multilevel access structure.
Consider a model of secret sharing schemes with cheaters. We say that a secret sharing scheme is error decodable if we can still recover the secretscorrectly from a noisy share vector (share1', ..., sharen'). In this paper, we first prove that a perfect secret sharing scheme is error decodable if and only if the adversary structure Γ satisfies a certain condition calledQ3. Next, for such Γ , we show a scheme such that the decoding algorithm runs in polynomial-time in |S| and the size of a linear secret sharing scheme which realizes Γ. We finally show an application to 1-round perfectly secure message transmission schemes (PSMT).