Vector Space Access Structures Research Articles

A novel self‐healing key distribution scheme based on vector space access structure and MDS codes

SummarySelf‐healing group key distribution protocols are useful in applications that have a dynamic group structure. These include broadcast transmission systems and multicast networks such as pay‐per‐view television, embedded and sensor networks, and cellular and wireless networks. To cater to the requirements of these applications, several self‐healing group key distribution protocols are proposed in the literature. Many of these schemes are vulnerable to polynomial factorization or insider replay attacks. Some other schemes impose constraints on the users joining the group or revoked from the group. Motivated by these and other shortcomings of the existing schemes, we hereby propose a novel self‐healing group key distribution protocol. Some of the features of this scheme include that (a) the number and the set of revoked users is not constrained, (b) the communication group can consist of any set of users, and (c) a revoked user is allowed to rejoin the group in any of the later sessions. The scheme is analyzed for its security, and it is found to provide anywise forward and backward secrecy. It is also found to resist anywise collusion attack. Communication and computation complexity of the scheme is analyzed; while doing so, various possible realizations of the scheme is discussed. In addition to the theoretical analysis, the proposed scheme is experimentally verified for its correctness using OMNET++ network simulator.

Verifiable secret sharing scheme without dealer based on vector space access structures over bilinear groups

In general, most of the previous secret sharing schemes involve a secret dealer and a number of participants. The dealer generates a secret and then distributes the secret shares to the participants in a centralised manner, which may lead to the data tamper problem. A verifiable secret sharing scheme without dealer on vector space access structures over bilinear groups is presented. In particular, each participant can act as a ‘dealer’ and the secret in the proposed scheme is a random element generated by all the participants from a bilinear group. In addition, the performance of the proposed scheme in terms of security and computing complexity is analysed in detail and compared it with that of the scheme by Zhang and Zhang. The results show that the proposed scheme performs better in terms of computational cost.

Information-theoretical secure verifiable secret sharing with vector space access structures over bilinear groups and its applications

As a basic tool, Verifiable Secret Sharing (VSS) has wide applications in distributed cryptosystems as well as secure multi-party computations. A number of VSS schemes for sharing a secret from a finite field, both on threshold access structures and on general access structures, have been available. In this paper, we investigate the verifiably sharing of a secret that is a random element from a bilinear group on vector space access structures. For this purpose, we present an information-theoretical secure VSS scheme, and then convert it to a modified one with improved efficiency. The performance and the security of the proposed schemes are analyzed in detail. Two examples are given to illustrate the applications of our proposed VSS schemes. One is the secure sharing of an organization’s private key in Boneh and Franklin’s identity-based encryption system, and the other is the distributed key generation and distributed decryption for bilinear ElGamal encryption system, both with vector space access structures.

Extensions of access structures and their cryptographic applications

In secret sharing schemes a secret is distributed among a set of users $${\mathcal{P}}$$in such a way that only some sets, the authorized sets, can recover it. The family Γ of authorized sets is called the access structure. To design new cryptographic protocols, we introduce in this work the concept of extension of an access structure: given a monotone family $${{\it \Gamma} \subset 2^\mathcal{P}}$$and a larger set $${\mathcal{P}^{\prime} = \mathcal{P} \cup \tilde{\mathcal{P}}}$$, a monotone access structure $${{\it \Gamma}^{\prime}\subset 2^{\mathcal{P}^{\prime}}}$$is an extension of Γ if the following two conditions are satisfied: (1) The set $${\mathcal{P}}$$is a minimal subset of Γ′, i.e. $${\mathcal{P} \in {\it \Gamma}^{\prime}}$$and $${\mathcal{P} - \{R_i\}\notin {\it \Gamma}^{\prime}}$$for every $${R_i \in \mathcal{P}}$$, (2) A subset $${A \subset \mathcal{P}}$$is in Γ if and only if the subset $${A \cup \tilde{\mathcal{P}}}$$is in Γ′. As our first contribution, we give an explicit construction of an extension Γ′ of a vector space access structure Γ, and we prove that Γ′ is also a vector space access structure. Although the definition may seem a bit artificial at first, it is well motivated from a cryptographic point of view. Indeed, our second contribution is to show that the concept of extension of an access structure can be used to design encryption schemes with access structures that are chosen ad-hoc at the time of encryption. Specifically, we design and analyze a dynamic distributed encryption scheme and a ciphertext-policy attribute-based encryption scheme. In some cases, the new schemes enjoy better properties than existing ones.

Computationally secure self-healing key distribution with revocation in wireless ad hoc networks

This paper introduces a novel self-healing technique for key distribution with revocation and demonstrates how it improves the efficiency over the previous approaches while using the existing idea of secret sharing or revocation polynomial in designing self-healing key distribution schemes with revocation. Unlike the existing approaches, our self-healing mechanism does not need to send the history of revoked users and consequently enables better performance gain over the previous approaches in terms of storage, communication and computation complexity. We propose and analyze a generalized self-healing key distribution using a vector space access structure in order to reach more flexible performance of the scheme. We describe three efficient constructions for scalable self-healing key distribution with t-revocation capability. We provide a rigorous treatment of the security of our constructions in an appropriate security framework and show that they are computationally secure and achieve both forward and backward secrecy.

On Codes, Matroids, and Secure Multiparty Computation From Linear Secret-Sharing Schemes

Error-correcting codes and matroids have been widely used in the study of ordinary secret sharing schemes. In this paper, the connections between codes, matroids, and a special class of secret sharing schemes, namely, multiplicative linear secret sharing schemes (LSSSs), are studied. Such schemes are known to enable multiparty computation protocols secure against general (nonthreshold) adversaries. Two open problems related to the complexity of multiplicative LSSSs are considered in this paper. The first one deals with strongly multiplicative LSSSs. As opposed to the case of multiplicative LSSSs, it is not known whether there is an efficient method to transform an LSSS into a strongly multiplicative LSSS for the same access structure with a polynomial increase of the complexity. A property of strongly multiplicative LSSSs that could be useful in solving this problem is proved. Namely, using a suitable generalization of the well-known Berlekamp-Welch decoder, it is shown that all strongly multiplicative LSSSs enable efficient reconstruction of a shared secret in the presence of malicious faults. The second one is to characterize the access structures of ideal multiplicative LSSSs. Specifically, the considered open problem is to determine whether all self-dual vector space access structures are in this situation. By the aforementioned connection, this in fact constitutes an open problem about matroid theory, since it can be restated in terms of representability of identically self-dual matroids by self-dual codes. A new concept is introduced, the flat-partition, that provides a useful classification of identically self-dual matroids. Uniform identically self-dual matroids, which are known to be representable by self-dual codes, form one of the classes. It is proved that this property also holds for the family of matroids that, in a natural way, is the next class in the above classification: the identically self-dual bipartite matroids.

Generation of key predistribution schemes using secret sharing schemes

In a key predistribution scheme a trusted authority distributes pieces of information among a set of users in such a way that users belonging to a specified privileged subset can compute individually a secret key common to this subset. In such a scheme, a family of forbidden subsets of users cannot obtain any information about the value of the secret. In this paper, we present a new construction of a key predistribution scheme using a family of vector space secret sharing schemes. The set of privileged users and the family of forbidden subsets is described in terms of the family of vector space access structures. A generalization using linear secret sharing schemes is given. We show that a particular case of this construction is any key predistribution scheme in which pieces of information and secrets are linear combinations of random numbers. Using this result, we show explicitly that the most important key predistribution schemes can be seen as a particular case of this construction. For this construction, the question of when given secrets can be predistributed is discussed.

Generation of Key Predistribution Schemes using Secret Sharing Schemes

Abstract In a key predistribution scheme a trusted authority distributes pieces of information among a set of users of such a way that specified privileged subsets of users can compute a secret key individually. In such a scheme, a family of forbidden subsets of users cannot obtain any information about the value of the secret. In this paper we present a new construction of a key predistribution scheme using a family of vector space secret sharing schemes. The set of privileged users and the family of forbidden subsets is described in terms of the family of vector space access structures. A generalization using linear secret sharing schemes is given. We show that a particular case of this construction is any key predistribution scheme in which pieces of information and secrets are linear combination of random numbers. Using this result we show explicitly that the most important key predistribution schemes can be seen as a particular case of this construction. For this construction, the question of when given secrets can be predistributed is discussed.