Graph Access Structures Research Articles

The lower bound and exact value of the information rate of some developed graph access structures

Various studies have focused on secret sharing schemes and in all of them, each shareholder is interested in a shorter share. The information rate of a secret sharing scheme shows the ratio between the size of the secret to the maximum number of shares given to each shareholder. In this regard, the researchers investigated the optimal information rate of the graph access structure. This paper aims to discover the exact values for the optimal information rates of the two graph access structures, which remained as open problems in Van Dijk’s paper. Furthermore, we introduced the developed multipartite graph and the developed cycle graph and calculated the exact value of their optimal information rate. Moreover, we presented a lower bound on the information rate for other developed graph access structures.

Optimal linear secret sharing schemes for graph access structures on six participants

The efficiency of a secret sharing scheme is measured via its information ratio. The optimal information ratio (OIR) of an access structure is the infimum of the information ratios of all secret sharing schemes realizing it. When the infimum is taken over all linear secret sharing schemes, it is called the optimal linear information ratio (OLIR). We review the problem of finding the OIRs and OLIRs of graph access structures on six participants. Study of such access structures was initiated by van Dijk et al. (1998) [43]. In a sequence of works, the OIRs of nine access structures, out of the 18 initially unsolved non-isomorphic ones, were determined. Very recently, the known lower bounds on the OLIRs of the other nine cases have been improved (Farràs et al., 2018 [20]), determining the OLIRs for another two cases. In this paper, for each of the remaining seven cases, we provide a new upper bound on the OIR which matches that of the recently found lower bound on the OLIR. Improved upper bounds are achieved by constructing a linear secret sharing scheme, using a suitable decomposition technique for each access structure. A new decomposition technique, called (λ,ω)-weighted decomposition, is also introduced which is a generalization of all known decomposition methods.

On the Pixel Expansion of Visual Cryptography Scheme

In this paper, we first follow Ateniese et al.'s work that provides upper bounds of the pixel expansion of visual cryptography schemes(VCSs) for more kinds of graph access structures, in which we require that a subset of parties can determine the secret if they contain an edge of the graph G. The constructive upper bounds are derived by the graph decomposition technique. Then we generalize Ateniese et al.'s method of comparing the optimal pixel expansion of VCSs with two different access structures.

The complexity of the connected graph access structure on seven participants

Abstract In this paper, we study an important problem in secret sharing that determines the exact value or bound for the complexity. First, we use the induced subgraph complexity of the graph G with access structure Γ to obtain a lower bound on the complexity of the graph G. Then, applying decomposition techniques, we obtain an upper bound on the complexity of the graph G. We determine the exact values of the complexity for each of the ten graph access structures on seven participants. Also, we improve the value bound of the complexity for the six graph access structures with seven participants.

The optimal information rate for graph access structures of nine participants

The information rate is an important metric of the performance of a secret-sharing scheme. In this paper we consider 272 non-isomorphic connected graph access structures with nine vertices and eight or nine edges, and either determine or bound the optimal information rate in each case. We obtain exact values for the optimal information rate for 231 cases and present a method that is able to derive information-theoretical upper bounds on the optimal information rate. Moreover, we apply some of the constructions to determine lower bounds on the information rate. Regarding information rate, we conclude with a full listing of the known optimal information rate (or bounds on the optimal information rate) for all 272 graphs access structures of nine participants.

The Optimal Information Rates of the Graph Access Structures on Seven Participants

The information rate is an important metric of the performance of a secret-sharing scheme. In this paper, we deal with determining the exact values for the optimal information rates of the six graph access structures and improving the information rate of a graph access structure on seven participants, which remained as open problems in Song's and Wang's paper([1,2]). We prove that the optimal information rate for each of the six graph access structures is equal to 4/7

Graph access structures with optimal pixel expansion three

Let G=(V,E) be a graph without isolated vertices. Let Γ(G) denote the strong access structure whose basis is the edge set E and let m⁎(G) denote the optimal pixel expansion of a visual cryptography scheme for Γ(G). In this paper we obtain a characterization of all connected graphs G with m⁎(G)=3.

Perfect secret sharing schemes for graph access structures on six participants

Perfect secret sharing schemes for graph access structures on six participants

Finding lower bounds on the complexity of secret sharing schemes by linear programming

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 long-standing 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 non-Shannon 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 non-representable 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.

The complexity of the graph access structures on six participants

In this paper, we deal with determining the exact values for the complexity of the five graph access structures on six participants, which remained as open problems in van Dijk's paper (Des Codes Cryptogr 6:143---169, 1995). We prove that the complexity for each of the five graph access structures is equal to 7/4.

On the Pixel Expansion of Hypergraph Access Structures in Visual Cryptography Schemes

In a visual cryptography scheme, a secret image is encoded into n shares, in the form of transparencies. The shares are then dis- tributed to n participants. Qualified subsets of participants can recover the secret image by superimposing their transparencies, but non-qualified subsets of participants have no information about the secret image. Pixel expansion, which represents the number of subpixels in the encoding of the secret image, should be as small as possible. Optimal schemes are those that have the minimum pixel expansion. In this paper we study the pixel expansion of hypergraph access structures and introduce a number of upper bounds on the pixel expansion of special kinds of access struc- tures. Also we demonstrate the minimum pixel expansion of induced matching hypergraph is sharp when every qualified subset is exactly one edge with odd size. Furthermore we explain that the minimum pixel ex- pansion of every graph access structure Pn is exactly n+1 2 . It indicates the lower bound mentioned in (4) is sharp.

Decomposition Construction for Secret Sharing Schemes with Graph Access Structures in Polynomial Time

The purpose of this paper is to describe a new decomposition construction for perfect secret sharing schemes with graph access structures. The previous decomposition construction proposed by Stinson is a recursive method that uses small secret sharing schemes as building blocks in the construction of larger schemes. When the Stinson method is applied to the graph access structures, the number of such “small” schemes is typically exponential in the number of the participants, resulting in an exponential algorithm. Our method has the same flavor as the Stinson decomposition construction; however, the linear programming problem involved in the construction is formulated in such a way that the number of “small” schemes is polynomial in the size of the participants, which in turn gives rise to a polynomial time construction. We also show that if we apply the Stinson construction to the “small” schemes arising from our new construction, both have the same information rate.

Visual cryptography and threshold schemes

Visual cryptography is a secret sharing scheme that uses the human visual system to perform computations. In this article, we present some background to traditional secret-sharing schemes, then explain visual schemes, describing some of the basic construction techniques used. topics discussed include: two out of two scheme, two out of n schemes, and graph access structures.