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Universally Ideal Secret Sharing Schemes
 IEEE Trans. on Information Theory
, 1994
"... Given a set of parties f1; : : : ; ng, an access structure is a monotone collection of subsets of the parties. For a certain domain of secrets, a secret sharing scheme for an access structure is a method for a dealer to distribute shares to the parties. These shares enable subsets in the access stru ..."
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Given a set of parties f1; : : : ; ng, an access structure is a monotone collection of subsets of the parties. For a certain domain of secrets, a secret sharing scheme for an access structure is a method for a dealer to distribute shares to the parties. These shares enable subsets in the access structure to reconstruct the secret, while subsets not in the access structure get no information about the secret. A secret sharing scheme is ideal if the domains of the shares are the same as the domain of the secrets. An access structure is universally ideal if there exists an ideal secret sharing scheme for it over every finite domain of secrets. An obvious necessary condition for an access structure to be universally ideal is to be ideal over the binary and ternary domains of secrets. In this work, we prove that this condition is also sufficient. We also show that being ideal over just one of the two domains does not suffice for universally ideal access structures. Finally, we give an exac...
On secret sharing schemes, matroids and polymatroids
 Journal of Mathematical Cryptology
"... The complexity of a secret sharing scheme is defined as the ratio between the maximum length of the shares and the length of the secret. The optimization of this parameter for general access structures is an important and very difficult open problem in secret sharing. We explore in this paper the co ..."
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Cited by 18 (4 self)
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The complexity of a secret sharing scheme is defined as the ratio between the maximum length of the shares and the length of the secret. The optimization of this parameter for general access structures is an important and very difficult open problem in secret sharing. We explore in this paper the connections of this open problem with matroids and polymatroids. Matroid ports were introduced by Lehman in 1964. A forbidden minor characterization of matroid ports was given by Seymour in 1976. These results are previous to the invention of secret sharing by Shamir in 1979. Important connections between ideal secret sharing schemes and matroids were discovered by Brickell and Davenport in 1991. Their results can be restated as follows: every ideal secret sharing scheme defines a matroid, and its access structure is a port of that matroid. In spite of this, the results by Lehman and Seymour and other subsequent results on matroid ports have not been noticed until now by the researchers interested in secret sharing. Lower bounds on the optimal complexity of access structures can be found by taking into account that the joint Shannon entropies of a set of random variables define a polymatroid.
Ideal Multipartite Secret Sharing Schemes
 J. Cryptology
"... Multipartite secret sharing schemes are those having a multipartite access structure, in which the set of participants is divided into several parts and all participants in the same part play an equivalent role. In this work, the characterization of ideal multipartite access structures is studied wi ..."
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Cited by 12 (4 self)
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Multipartite secret sharing schemes are those having a multipartite access structure, in which the set of participants is divided into several parts and all participants in the same part play an equivalent role. In this work, the characterization of ideal multipartite access structures is studied with all generality. Our results are based on the wellknown connections between ideal secret sharing schemes and matroids and on the introduction of a new combinatorial tool in secret sharing, integer polymatroids. Our results can be summarized as follows. First, we present a characterization of multipartite matroid ports in terms of integer polymatroids. As a consequence of this characterization, a necessary condition for a multipartite access structure to be ideal is obtained. Second, we use representations of integer polymatroids by collections of vector subspaces to characterize the representable multipartite matroids. In this way we obtain a sufficient condition for a multipartite access structure to be ideal, and also a unified framework to study the open problems about the efficiency of the constructions of ideal multipartite secret sharing schemes. Finally, we apply our general results to obtain a complete characterization of ideal tripartite access structures, which was until now an open problem.
Secret Sharing Schemes With Three Or Four Minimal Qualified Subsets
, 2002
"... In this paper we study secret sharing schemes whose access structure has three or four minimal qualified subsets. The ideal case is completely characterized and for the nonideal case we provide bounds on the optimal information rate. ..."
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Cited by 10 (4 self)
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In this paper we study secret sharing schemes whose access structure has three or four minimal qualified subsets. The ideal case is completely characterized and for the nonideal case we provide bounds on the optimal information rate.
Secret Sharing Schemes on Access Structures With Intersection Number Equal To One
, 2002
"... The characterization of ideal access structures and the search for bounds on the optimal information rate are two important problems in secret sharing. ..."
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Cited by 9 (3 self)
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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.
Finding Lower Bounds on the Complexity of Secret Sharing Schemes by Linear Programming
 LATIN 2010: THEORETICAL INFORMATICS. LECTURE NOTES IN COMPUT. SCI. 6034
, 2010
"... Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and longstanding open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that ..."
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Cited by 5 (1 self)
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Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and longstanding open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants. By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program nonShannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of nonrepresentable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing scheme.
Secret Sharing Schemes for Very Dense Graphs
, 2012
"... A secretsharing scheme realizes a graph if every two vertices connected by an edge can reconstruct the secret while every independent set in the graph does not get any information on the secret. Similar to secretsharing schemes for general access structures, there are gaps between the known lower ..."
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A secretsharing scheme realizes a graph if every two vertices connected by an edge can reconstruct the secret while every independent set in the graph does not get any information on the secret. Similar to secretsharing schemes for general access structures, there are gaps between the known lower bounds and upper bounds on the share size for graphs. Motivated by the question of what makes a graph “hard” for secretsharing schemes (that is, require large shares), we study very dense graphs, that is, graphs whose complement contains few edges. We show that if a graph with n vertices contains () n 1+β 2 − n edges for some constant 0 ≤ β < 1, then there is a scheme realizing the graph with total share size of Õ(n5/4+3β/4). This should be compared to O(n2 /logn) – the best upper bound known for the share size in general graphs. Thus, if a graph is “hard”, then the graph and its complement should have many edges. We generalize these results to nearly complete khomogeneous access structures for a constant k. To complement our results, we prove lower bounds for secretsharing schemes realizing very dense graphs, e.g., for linear secretsharing schemes we prove a lower bound of Ω(n1+β/2) for a graph with n
Secret sharing schemes on sparse homogeneous access structures with rank three
, 2004
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On linear secret sharing for connectivity in directed graphs
 Proc. of the Sixth Conference on Security and Cryptography for Networks
"... Abstract. In this work we study linear secret sharing schemes for st connectivity in directed graphs. In such schemes the parties are edges of a complete directed graph, and a set of parties (i.e., edges) can reconstruct the secret if it contains a path from node s to node t. We prove that in every ..."
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Abstract. In this work we study linear secret sharing schemes for st connectivity in directed graphs. In such schemes the parties are edges of a complete directed graph, and a set of parties (i.e., edges) can reconstruct the secret if it contains a path from node s to node t. We prove that in every linear secret sharing scheme realizing the stcon function on a directed graph with n edges the total size of the shares is Ω(n 1.5). This should be contrasted with st connectivity in undirected graphs, where there is a scheme with total share size n. Our result is actually a lower bound on the size monotone span programs for stcon, where a monotone span program is a linearalgebraic model of computation equivalent to linear secret sharing schemes. Our results imply the best known separation between the power of monotone and nonmonotone span programs. Finally, our results imply the same lower bounds for matching. 1
unknown title
, 2013
"... among a set of n participants in such a way that only qualified subsets of participants can r [22] and Blakley the set P emes forsharing sc with additional capacities were proposed [8,11,13,14,16,17,21,23,27]. The reader is referred to [1] for a comprehensi vey. Secret sharing has been an interesti ..."
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among a set of n participants in such a way that only qualified subsets of participants can r [22] and Blakley the set P emes forsharing sc with additional capacities were proposed [8,11,13,14,16,17,21,23,27]. The reader is referred to [1] for a comprehensi vey. Secret sharing has been an interesting branch of modern cryptography. 00200255/ $ see front matter 2013 Elsevier Inc. All rights reserved.