Multipartite Access Structures Research Articles

Ideal uniform multipartite secret sharing schemes

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. Secret sharing schemes for these access structures have gained attention, as they generalize threshold secret sharing. This work focuses on the construction of linear secret sharing schemes for ideal uniform multipartite access structures, while avoiding the inefficiencies and randomness of known constructions. To achieve this, we have developed two approaches. The first approach combines polymatroid-based techniques with Gabidulin codes, which efficiently realize a particular family of ideal uniform multipartite access structures. The efficiency of this method is derived from the unique properties of Gabidulin codes, which allow for polynomial share and secret sizes relative to the number of participants. Our second approach applies linear algebraic techniques and polymatroid-based techniques to explicitly design linear secret sharing schemes for arbitrary ideal uniform multipartite access structures. The efficiency of this method stems from the properties of polynomials over finite fields. The schemes from this method are efficient for smaller numbers of parts relative to the number of participants. In summary, we present explicit constructions of linear secret sharing schemes for ideal uniform multipartite access structures, which have important applications in cryptography.

Access structures determined by uniform polymatroids

Abstract In this article, all multipartite access structures obtained from uniform integer polymatroids were investigated using the method developed by Farràs, Martí-Farré, and Padró. They are matroid ports, i.e., they satisfy the necessary condition to be ideal. Moreover, each uniform integer polymatroid defines some ideal access structures. Some objects in this family can be useful for the applications of secret sharing. The method presented in this article is universal and can be continued with other classes of polymatroids in further similar studies. Here, we are especially interested in hierarchy of participants determined by the access structure, and we distinguish two main classes: they are compartmented and hierarchical access structures. The main results obtained for access structures determined by uniform integer polymatroids and a monotone increasing family Δ \Delta can be summarized as follows. If the increment sequence of the polymatroid is non-constant, then the access structure is connected. If Δ \Delta does not contain any singletons or the height of the polymatroid is maximal and its increment sequence is not constant starting from the second element, then the access structure is compartmented. If Δ \Delta is generated by a singleton or the increment sequence of the polymatroid is constant starting from the second element, then the obtained access structures are hierarchical. They are proven to be ideal, and their hierarchical orders are completely determined. Moreover, if the increment sequence of the polymatroid is constant and ∣ Δ ∣ > 1 | \Delta | \gt 1 , then the hierarchical order is not antisymmetric, i.e., some different blocks are equivalent. The hierarchical order of access structures obtained from uniform integer polymatroids is always flat, that is, every hierarchy chain has at most two elements.

Efficient Explicit Constructions of Multipartite Secret Sharing Schemes

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. Secret sharing schemes for multipartite access structures have received considerable attention due to the fact that multipartite secret sharing can be seen as a natural and useful generalization of threshold secret sharing. This work deals with efficient and explicit constructions of ideal multipartite secret sharing schemes. Most ideal multipartite secret sharing schemes in the literature can be classified as either hierarchical or compartmented. The main results are the constructions for ideal hierarchical access structures, a family that contains every ideal hierarchical access structure as a particular case such as the disjunctive hierarchical threshold access structure and the conjunctive hierarchical threshold access structure, the constructions for three families of compartmented access structures, and the constructions for two families compartmented access structures with compartments. We present an efficient method to construct ideal linear schemes realizing these access structures by linear algebraic techniques on the basis of the relationship between multipartite secret sharing schemes, polymatroids and matroids.

An Efficient Compartmented Secret Sharing Scheme Based on Linear Homogeneous Recurrence Relations

Multipartite secret sharing schemes are those that have multipartite access structures. The set of the participants in those schemes is divided into several parts, and all the participants in the same part play the equivalent role. One type of such access structure is the compartmented access structure, and the other is the hierarchical access structure. We propose an efficient compartmented multisecret sharing scheme based on the linear homogeneous recurrence (LHR) relations. In the construction phase, the shared secrets are hidden in some terms of the linear homogeneous recurrence sequence. In the recovery phase, the shared secrets are obtained by solving those terms in which the shared secrets are hidden. When the global threshold is t , our scheme can reduce the computational complexity of the compartmented secret sharing schemes from the exponential time to polynomial time. The security of the proposed scheme is based on Shamir’s threshold scheme, i.e., our scheme is perfect and ideal. Moreover, it is efficient to share the multisecret and to change the shared secrets in the proposed scheme.

Design of ideal secret sharing based on new results on representable quadripartite matroids

With the rapid development of the next generation of mobile networks and communications (5G networks), group-oriented applications show the great potential ability in 5G and multipartite group communications occur frequently in a single network, where a multipartite access structure is denied to be a collection of the subsets of users who may come from different parts of the network such that only users in an authorized subset of users can use their shares to build up a group key for a secure group communication. Most existing group key establishment schemes based on a secret sharing target on building up a group key for a threshold access structure and need to compute a t-degree interpolating polynomial in order to encrypt and decrypt the secret group key. This approach is not suitable and inefficient in terms of computational complexity for multipartite group environments which need to realize the multipartite access structures. In this paper, we study the characterization of ideal access structures related to representable multipartite matroids. By employing some existing results, such as the well-known connection between ideal secret sharing and matroids, the recent discovery of ideal multipartite access structures, and the connection between multipartite matroids and discrete polymatroids, we obtain the characterization of a family of representable multipartite matroids, including a sufficient condition for an access structure to be ideal. Based on this result, we provide a complete characterization of representable quadripartite matroids, showing that these access structures related to representable quadripartite matroids are the ideal ones. From these ideal access structures, we can then construct ideal secret sharing schemes for multipartite group communications. Our results are attractive for efficient and secure multi-group-oriented applications.

Compartmented Secret Sharing Schemes and Locally Repairable Codes

Multipartite secret sharing is an important research object in the area of secret sharing schemes. Compartmented access structures are an interesting class of multipartite access structures. The constructions of ideal linear schemes realizing compartmented access structures are studied by codes in this paper. We find that compartmented secret sharing is related to a class of codes called locally repairable codes which are being used widely in distributed and cloud storage systems. We study secret sharing schemes for compartmented access structures with upper bounds, compartmented access structures with lowers bounds and compartmented access structures with upper and lower bounds. We establish the relationships between secret sharing schemes for these compartmented access structures and locally repairable codes. Based on the relationships and some locally repairable codes, we obtain ideal linear schemes realizing these compartmented access structures by efficient methods.

Efficient explicit constructions of compartmented secret sharing schemes

Multipartite secret sharing schemes have been an important object of study in the area of secret sharing schemes. Two interesting families of multipartite access structures are hierarchical access structures and compartmented access structures. This work deals with efficient and explicit constructions of ideal compartmented secret sharing schemes, while most of the known constructions are either inefficient or randomized. We construct ideal linear secret sharing schemes for three types of compartmented access structures, such as compartmented access structures with upper bounds, compartmented access structures with lower bounds, and compartmented access structures with upper and lower bounds. There exist some methods to construct ideal linear schemes realizing these compartmented access structures in the literature, but those methods are inefficient in general because non-singularity of many matrices has to be determined to check the correctness of the scheme. Our constructions do not need to do these computations. Our methods to construct ideal linear schemes realizing these access structures combine polymatroid-based techniques with Gabidulin codes. Gabidulin codes play a fundamental role in the constructions, and their properties imply that our methods are efficient.

Practical (fully) distributed signatures provably secure in the standard model

A distributed signature scheme allows participants in a qualified set to jointly generate a signature which cannot be forged even when any unqualified set of participants collude together. In this paper, we propose an efficient scheme that supports any monotone access structures and show its unforgeability and robustness under the computational Diffie–Hellman (CDH) assumption in the standard model. For 192-bit security, its secret key shares and signature fragments are as short as 511 bits and 1022 bits, which are shorter than existing schemes assuming random oracle. We then propose two extensions. The first one allows new participants to dynamically join the system without any help from the dealer. The second one supports a type of multipartite access structures, where the participant set is divided into multiple disjoint groups, and each group is bounded so that a distributed signature cannot be generated unless a pre-defined number of participants from multiple groups work together. Finally, we present a fully distributed signature scheme such that the centralized trusted dealer can be removed from the system, and the secret keys (shares) can be jointly computed by the involved participants.

Further ideal multipartite access structures from integer polymatroids

Ideal access structures admit ideal secret sharing schemes where the shares have the minimal size.As multipartite access structures can well mirror the real social organizations, of which the participants arepartitioned into disjoint groups according to their properties, it is desirable to find expressive ideal multipartiteaccess structures. Integer polymatroids, due to their close relationship with ideal multipartite access structures,have been shown as a powerful tool to study the ideality of some multipartite access structures. In this paper, tocater for flexible applications, we consider several ideal multipartite access structures that further extend someknown results. We first explore a type of compartmented access structures with strictly lower bounds, whichprovide fairness among all the participant groups when recovering the secret. Then, we investigate ideal benchaccess structures where the participant set is divided into two parts, that is, line-up section and bench section.The participants in line-up section can delegate their capabilities to the participants in bench section in such away that the participants in bench section can take over the role of their delegators in line-up section, which isapplicable to emergency situations when there are no enough participants in line-up section for recovering thesecret. Finally, we propose two types of ideal partially hierarchical access structures which are suitable to morerealistic hierarchical social organizations than existing results.

New Results on Ideal Multipartite Secret Sharing and its Applications to Group Communications

With the rapid development of various group-oriented services, multipartite group communications occur frequently in a single network, where a multipartite access structure is defined to be a collection of the subsets of users who may come from different parts of the network such that only users in an authorized subset of users can use their shares to build up a group key for a secure group communication. Most existing group key establishment schemes based on a secret sharing target on building up a group key for a threshold access structure, and need to compute a $$t$$t-degree interpolating polynomial in order to encrypt and decrypt the secret group key. This approach is not suitable and inefficient in terms of computational complexity for multipartite group environments which need to realize the multipartite access structures. In 1991, Brickell et al. proved that an ideal access structure is induced by a matroid and furthermore, an access structure is ideal if it is induced by a representable matroid. In this paper, we study the characterization of representable matroids. By using the connection between ideal secret sharing and matroids and, in particular, the recent results on ideal multipartite access structures and the connection between multipartite matroids and discrete polymatroids, we introduce a new concept on $$R$$R-tuple, which is determined by the rank function of the associated discrete polymatroid. Using this new concept, we come up a new and simple sufficient condition for a multipartite matroid to be representable (in fact, every matroid and every access structure are multipartite). In other words, we have developed a sufficient condition for an access structure to be ideal. These new results can be applied to establish multipartite group keys efficiently in secure group communications.

Ideal Multipartite Secret Sharing Schemes

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 well-known 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.

New Results on Multipartite Secret Sharing Matroids

In a secret-sharing scheme, a secret value is distributed among a set of participants by giving each participant a share. The requirement is that only predefined subsets of participants can recover the secret from their shares. The family of the predefined authorized subsets is called the access structure. An access structure is ideal if there exists a secret-sharing scheme realizing it in which the shares have optimal length, that is, in which the shares are taken from the same domain as the secrets. Brickell and Davenport (J. of Cryptology, 1991) proved that ideal access structures are induced by matroids. Subsequently, access structures induced by matroids and properties of matroids have received a lot of attention. Since every matroid is multipartite and has the corresponding discrete polymatroid, in this paper, by dealing with the bases of discrete polymatroids, we obtain a necessary condition for a set of vectors to be the bases of a discrete polymatroid, which implies a necessary condition for a multipartite access structure to be ideal. Concretely, we describe the properties of the bases of bipartite, tripartite and quadripartite matroids respectively, which show the necessary conditions for multipartite access structures induced by bipartite, tripartite and quadripartite matroids to be ideal accordingly. Our results give new contributions to the characterization of ideal access structures, which is one of the main open problems in secret sharing.

Multipartite matroids and secret sharing

In a secret-sharing scheme, a secret value is distributed among a set of participants by giving each participant a share. The requirement is that only predefined subsets of participants can recover the secret from their shares. The family of the predefined authorized subsets is called the access structure. An access structure is ideal if there exists a secret-sharing scheme realizing it in which the shares have optimal length, that is, in which the shares are taken from the same domain as the secrets. Brickell and Davenport proved that ideal access structures are induced by matroids. Subsequently, ideal access structures and access structures induced by matroids have received a lot of attention. Seymour gave the first example of an access structure induced by a matroid namely the Vamos matroid, that is non-ideal. Since every matroid is multipartite and has the associated discrete polymatroid, in this paper, by dealing with the rank functions of discrete polymatroids, we obtain a sufficient condition for a multipartite access structure to be ideal. Furthermore, we give a new proof that all access structures related to bipartite and tripartite matroids coincide with the ideal ones. Our results give new contributions to the open problem, that is, which matroids induce ideal access structures.

Multipartite Secret Sharing by Bivariate Interpolation

Given a set of participants that is partitioned into distinct compartments, a multipartite access structure is an access structure that does not distinguish between participants belonging to the same compartment. We examine here three types of such access structures: two that were studied before, compartmented access structures and hierarchical threshold access structures, and a new type of compartmented access structures that we present herein. We design ideal perfect secret sharing schemes for these types of access structures that are based on bivariate interpolation. The secret sharing schemes for the two types of compartmented access structures are based on bivariate Lagrange interpolation with data on parallel lines. The secret sharing scheme for the hierarchical threshold access structures is based on bivariate Lagrange interpolation with data on lines in general position. The main novelty of this paper is the introduction of bivariate Lagrange interpolation and its potential power in designing schemes for multipartite settings, as different compartments may be associated with different lines or curves in the plane. In particular, we show that the introduction of a second dimension may create the same hierarchical effect as polynomial derivatives and Birkhoff interpolation were shown to do in Tassa (J. Cryptol. 20:237–264, 2007).

New results on multipartite access structures

In a multipartite access structure, the set of players is divided into K different classes in such a way that all players of the same class play the same role in the structure. Not many results are known about these structures, when K ≥ 3.Although the total characterisation of ideal multipartite access structures seems a very ambitious goal, we nevertheless take a first step in this direction. On the one hand, we detect some conditions that directly imply that a multipartite structure cannot be ideal. On the other hand, we introduce a new strategy which helps to prove that a multipartite access structure is ideal, and we apply this strategy to three wide families of multipartite access structures.

Ideal secret sharing schemes with multipartite access structures

The concept of quasi-threshold multipartite access structures for secret sharing schemes is introduced. These access structures allow one distinguished class of participants to retain control over the reconstruction of the secret, while allowing flexibility over the quorum of participants from other classes. While ideal bipartite access structures have been classified, the case for multipartite access structures is still open. Here it is shown that quasi-threshold multipartite access structures are ideal by constructing a projective space representation of the associated matroids.