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78
Anonymous Secret Sharing Schemes
 Designs, Codes and Cryptography
, 1996
"... In this paper we study anonymous secret sharing schemes. Informally, in an anonymous secret sharing scheme the secret can be reconstructed without knowledge of which participants hold which shares. In such schemes the computation of the secret can be carried out by giving the shares to a black box t ..."
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Cited by 85 (7 self)
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In this paper we study anonymous secret sharing schemes. Informally, in an anonymous secret sharing scheme the secret can be reconstructed without knowledge of which participants hold which shares. In such schemes the computation of the secret can be carried out by giving the shares to a black box that does not know the identities of the participants holding those shares. Phillips and Phillips gave necessary and sufficient conditions for there to exist an anonymous secret sharing scheme where the size of the shares given to each participant is equal to the size of the secret. In this paper, we provide lower bounds on the size of the share sets in any (t; w) threshold scheme, and for an infinite class of nonthreshold access structures. We also discuss constructions for anonymous secret sharing schemes, and apply them to access structures obtained from complete multipartite graphs. 1 Introduction Informally, a secret sharing scheme is a method of distributing a secret key among a set ...
Secret Sharing Schemes with Bipartite Access Structure
, 1998
"... 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 struct ..."
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Cited by 32 (8 self)
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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.
On the Information Rate of Secret Sharing Schemes
 Theoretical Computer Science
, 1992
"... We derive new limitations on the information rate and the average information rate of secret sharing schemes for access structure represented by graphs. We give the first proof of the existence of access structures with optimal information rate and optimal average information rate less that 1=2 + ff ..."
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Cited by 30 (5 self)
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We derive new limitations on the information rate and the average information rate of secret sharing schemes for access structure represented by graphs. We give the first proof of the existence of access structures with optimal information rate and optimal average information rate less that 1=2 + ffl, where ffl is an arbitrary positive constant. We also consider the problem of testing if one of these access structures is a substructure of an arbitrary access structure and we show that this problem is NPcomplete. We provide several general lower bounds on information rate and average information rate of graphs. In particular, we show that any graph with n vertices admits a secret sharing scheme with information rate\Omega\Gammate/3 n)=n). 1 Introduction A secret sharing scheme is a method to distribute a secret s among a set of participants P in such a way that only qualified subsets of P can reconstruct the value of s whereas any other subset of P ; nonqualified to know s; cannot ...
Tight Bounds on the Information Rate of Secret Sharing Schemes
 Designs, Codes and Cryptography
, 1997
"... A secret sharing scheme is a protocol by means of which a dealer distributes a secret s among a set of participants P in such a way that only qualified subsets of P can reconstruct the value of s whereas any other subset of P; nonqualified to know s; cannot determine anything about the value of ..."
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Cited by 30 (0 self)
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A secret sharing scheme is a protocol by means of which a dealer distributes a secret s among a set of participants P in such a way that only qualified subsets of P can reconstruct the value of s whereas any other subset of P; nonqualified to know s; cannot determine anything about the value of the secret. In this paper we provide a general technique to prove upper bounds on the information rate of secret sharing schemes. The information rate is the ratio between the size of the secret and the size of the largest share given to any participant. Most of the recent upper bounds on the information rate obtained in the literature can be seen as corollaries of our result. Moreover, we prove that for any integer d there exists a dregular graph for which any secret sharing scheme has information rate upper bounded by 2=(d + 1). This improves on van Dijk's result [14] and matches the corresponding lower bound proved by Stinson in [22]. Index terms : Secret Sharing, Data Security,...
Characterizing Ideal Weighted Threshold Secret Sharing
 Second Theory of Cryptography Conference, TCC 2005. Lecture Notes in Comput. Sci. 3378
, 2005
"... Abstract. Weighted threshold secret sharing was introduced by Shamir in his seminal work on secret sharing. In such settings, there is a set of users where each user is assigned a positive weight. A dealer wishes to distribute a secret among those users so that a subset of users may reconstruct the ..."
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Cited by 28 (6 self)
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Abstract. Weighted threshold secret sharing was introduced by Shamir in his seminal work on secret sharing. In such settings, there is a set of users where each user is assigned a positive weight. A dealer wishes to distribute a secret among those users so that a subset of users may reconstruct the secret if and only if the sum of weights of its users exceeds a certain threshold. On one hand, there are nontrivial weighted threshold access structures that have an ideal scheme – a scheme in which the size of the domain of shares of each user is the same as the size of the domain of possible secrets (this is the smallest possible size for the domain of shares). On the other hand, other weighted threshold access structures are not ideal. In this work we characterize all weighted threshold access structures that are ideal. We show that a weighted threshold access structure is ideal if and only if it is a hierarchical threshold access structure (as introduced by Simmons), or a tripartite access structure (these structures generalize the concept of bipartite access structures due to Padró and Sáez), or a composition of two ideal weighted threshold access structures that are defined on smaller sets of users. We further show that in all those cases the weighted threshold access structure may be realized by a linear ideal secret sharing scheme. The proof of our characterization relies heavily on the strong connection between ideal secret sharing schemes and matroids, as proved by Brickell and Davenport.
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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Cited by 21 (8 self)
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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...
Online Secret Sharing
 In Proc. of the 5th IMA Conf. on Cryptography and Coding
, 1995
"... . We propose a new construction for computationally secure secret sharing schemes with general access structures where all shares are as short as the secret. Our scheme provides the capability to share multiple secrets and to dynamically add participants online, without having to redistribute new ..."
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Cited by 19 (0 self)
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. We propose a new construction for computationally secure secret sharing schemes with general access structures where all shares are as short as the secret. Our scheme provides the capability to share multiple secrets and to dynamically add participants online, without having to redistribute new shares secretly to the current participants. These capabilities are gained by storing additional authentic (but not secret) information at a publicly accessible location. 1 Introduction Secret sharing is an important and widely studied tool in cryptography and distributed computation. Informally, a secret sharing scheme is a protocol in which a dealer distributes a secret among a set of participants such that only specific subsets of them, defined by the access structure, can recover the secret at a later time. Secret sharing has largely been investigated in the informationtheoretic security model, requiring that the participants' shares give no information on the secret, i.e. that the res...
Fully Dynamic Secret Sharing Schemes
 Theoretical Computer Science
, 1994
"... We consider secret sharing schemes in which the dealer is able (after a preprocessing stage) to activate a particular access structure out of a given set and/or to allow the participants to reconstruct different secrets (in different time instants) by sending them the same broadcast message. In this ..."
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Cited by 19 (1 self)
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We consider secret sharing schemes in which the dealer is able (after a preprocessing stage) to activate a particular access structure out of a given set and/or to allow the participants to reconstruct different secrets (in different time instants) by sending them the same broadcast message. In this paper we establish a formal setting to study secret sharing schemes of this kind. The security of the schemes presented is unconditional, since they are not based on any computational assumption. We give bounds on the size of the shares held by participants, on the size of the broadcast message, and on the randomness needed in such schemes. 1 Introduction A secret sharing scheme is a method of dividing a secret s among a set P of participants in such a way that: if the participants in A ` P are qualified to know the secret then by pooling together their information they can reconstruct the secret s; but any set A of participants not qualified to know s has absolutely no information on the...
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 Hierarchical Secret Sharing Schemes
"... The search of efficient constructions of ideal secret sharing schemes for families of nonthreshold access structures that may have useful applications has attracted a lot of attention. Several proposals have been made for access structures with hierarchical properties, in which the participants are ..."
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Cited by 17 (3 self)
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The search of efficient constructions of ideal secret sharing schemes for families of nonthreshold access structures that may have useful applications has attracted a lot of attention. Several proposals have been made for access structures with hierarchical properties, in which the participants are distributed into levels that are hierarchically ordered. Here, we study hierarchical secret sharing in all generality by providing a natural definition for the family of the hierarchical access structures. Specifically, an access structure is said to be hierarchical if every two participants can be compared according to the following natural hierarchical order: whenever a participant in a qualified subset is substituted by a hierarchically superior participant, the new subset is still qualified. We present a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. We use the well known connection between ideal secret sharing and matroids and, in particular, the fact the every ideal access structure is a matroid port. In addition, we use recent results on ideal multipartite access structures and the connection between multipartite matroids and discrete polymatroids. We prove that every ideal hierarchical access structure is the port of a representable matroid and, more specifically, we prove that every ideal structure in this family admits ideal linear secret sharing schemes over fields of all characteristics. This generalizes previous results on weighted threshold access structures. Finally, we use our results to find a new characterization of the ideal weighted threshold access structures that is more precise than the existing one.