We consider secret sharing schemes in which the dealer has the feature of being 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 to all participants the same broadcast message. In this paper we establish a formal setting to study such secret sharing schemes. 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 and on the site of the broadcast message in such schemes.

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Fully dynamic secret sharing schemes

Access control ensures that only the authorized users of a system are allowed to access certain resources or tasks. Usually, according to their roles and responsibilities, users are organized in hierarchies formed by a certain number of disjoint classes. Such hierarchies are implemented by assigning a key to each class, so that the keys for descendant classes can be efficiently derived from classes higher in the hierarchy. However, pure hierarchical access may represent a limitation in many real-world cases. In fact, sometimes it is necessary to ensure access to a resource or task by considering both its directly responsible user and a group of users possessing certain credentials. In this paper, we first propose a novel model that generalizes the conventional hierarchical access control paradigm, by extending it to certain additional sets of qualified users. Afterward, we propose two constructions for hierarchical key assignment schemes in this new model, which are provably secure with respect to key indistinguishability. In particular, the former construction relies on both symmetric encryption and perfect secret sharing, whereas, the latter is based on public-key threshold broadcast encryption.

Hierarchical and Shared Access Control

A (t,k,n,S) ramp scheme is a protocol to distribute a secret s chosen in S among a set P of n participants in such a way that: (1) sets of participants of cardinality greater than or equal to k can reconstruct the secret s; (2) sets of participants of cardinality less than or equal to t have no information on s, whereas (3) sets of participants of cardinality greater than t and less than k might have "some" information on s. In this correspondence we analyze multiple ramp schemes, which are protocols to share many secrets among a set P of participants, using different ramp schemes. In particular, we prove a tight lower bound on the size of the shares held by each participant and on the dealer's randomness in multiple ramp schemes.

Multiple ramp schemes

Randomness is a useful computation resource due to its ability to enhance the capabilities of other resources. Its interaction with resources such as time, space, interaction with provers and its role in several areas of computer science has been extensively studied. In this paper we give a systematic analysis of the amount of randomness needed by secret sharing schemes and secure key distribution schemes. We give both upper and lower bounds on the number of random bits needed by secret sharing schemes. The bounds are tight for several classes of secret sharing schemes. For secure key distribution schemes we provide a lower bound on the amount of randomness needed, thus showing the optimality of a recently proposed key distribution protocol.

Randomness in Distribution Protocols

In this paper we provide upper and lower bounds on the randomness required by the dealer to set up a secret sharing scheme for infinite classes of access structures. Lower bounds are obtained using entropy arguments. Upper bounds derive from a decomposition construction based on combinatorial designs (in particular, t-(v,k,λ) designs). We prove a general result on the randomness needed to construct a scheme for the cycle Cn; when n is odd our bound is tight. We study the access structures on at most four participants and the connected graphs on five vertices, obtaining exact values for the randomness for all them. Also, we analyze the number of random bits required to construct anonymous threshold schemes, giving upper bounds. (Informally, anonymous threshold schemes are schemes in which the secret can be reconstructed without knowledge of which participants hold which shares.)

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On the Dealer‘s Randomness Required in Secret Sharing Schemes

The problem we deal with in this paper is the research of upper and lower bounds on the randomness required by the dealer to set up a secret sharing scheme. We give both lower and upper bounds for infinite classes of access structures. Lower bounds are obtained using entropy arguments. Upper bounds derive from a decomposition construction based on combinatorial designs (in particular, t-(v, k, λ) designs). We prove a general result on the randomness needed to construct a scheme for the cycle C n; when n is odd our bound is tight. We study the access structures on at most four participants and the connected graphs on five vertices, obtaining exact values for the randomness for all them. Also, we analyze the number of random bits required to construct anonymous threshold schemes, giving upper bounds. (Informally, anonymous threshold schemes are schemes in which the secret can be reconstructed without knowledge of which participants hold which shares.)

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On the dealer's randomness required in secret sharing schemes

Probability of shares in secret sharing schemes

We study the amount of private information that must be given to participants and the amount of randomness needed to set up secret sharing schemes. These quantities are important from the practical point of view since the security of any systems tends to degrade as the amount of private information increases; moreover, since random bits are a natural (and hard to find) computational resource, the amount of randomness used in the computation is an important issue in the field of randomized algorithms. >

A. Giorgio Gaggia

Papers

On the Dealer‘s Randomness Required in Secret Sharing Schemes

On the dealer's randomness required in secret sharing schemes

Probability of shares in secret sharing schemes

Lower bounds in secret sharing schemes