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Ideal Homogeneous Access Structures Constructed from Graphs

Javier Herranz

Published Apr 7, 2020• 1 citation
Publisher

Abstract

Starting from a new relation between graphs and secret sharing schemes introduced by Xiao, Liu and Zhang, we show a method to construct more general ideal homogeneous access structures. The method has some advantages: it efficiently gives an ideal homogeneous access structure for the desired rank, and some conditions can be imposed (such as forbidden or necessary subsets of players), even if the exact composition of the resulting access structure cannot be fully controlled. The number of homogeneous access structures that can be constructed in this way is quite limited; for example, we show that (t, )-threshold access structures can be constructed from a graph only when t = 1, t = − 1 or t = .

DOI
10.3217/jucs-015-14-2881

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IEEE Transactions on Information Theory•Dec 6, 2021•6 citations
Starting from a new relation between graphs and secret sharing schemes introduced by Xiao, Liu and Zhang, we show a method to construct more general ideal homogeneous access structures. The method has some advantages: it efficiently gives an ideal homogeneous access structure for the desired rank, and some conditions can be imposed (such as forbidden or necessary subsets of players), even if the exact composition of the resulting access structure cannot be fully controlled. The number of homogeneous access structures that can be constructed in this way is quite limited; for example, we show that (t, )-threshold access structures can be constructed from a graph only when t = 1, t = − 1 or t = .
‎The characterization of the ideal access structures is one of the main open problems in secret sharing and is important from both practical and theoretical points of views‎. ‎A graph-based 3−3-3−homogeneous access structure is an access structure in which the participants are the vertices of a connected graph and every subset of the vertices is a minimal qualified subset if it has three vertices and induces a connected graph‎. ‎In this paper‎, ‎we introduce the graph-based 3−3-3−homogeneous access structures and characterize the ideal graph-based 333-homogeneous access structures‎. ‎We prove that for every non-ideal graph-based 333-homogeneous access structure over the graph GGG with the maximum degree ddd there exists a secret sharing scheme with an information rate 1d+1\frac{1}{d+1}d+11​‎. ‎Furthermore‎, ‎we mention three forbidden configurations that are useful in characterizing other families of ideal access structures‎.
One of the main open problems in secret sharing is the characterization of the ideal access structures. This problem has been studied for several families of access structures with similar results. Namely, in all these families, the ideal access structures coincide with the vector space ones and, besides, the optimal information rate of a non-ideal access structure is at most 2/3. An access structure is said to be r-homogeneous if there are exactly r participants in every minimal qualified subset. A first approach to the characterization of the ideal 3-homogeneous access structures is made in this paper. We show that the results in the previously studied families can not be directly generalized to this one. Nevertheless, we prove that the equivalences above apply to the family of the sparse 3-homogeneous access structures, that is, those in which any subset of four participants contains at most two minimal qualified subsets. Besides, we give a complete description of the ideal sparse 3-homogeneous access structures.
Given a set of parties {1, /spl middot//spl middot//spl middot/, n}, 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. The authors prove that this condition is also sufficient. They also show that being ideal over just one of the two domains does not suffice for universally ideal access structures. Finally, they give an exact characterization for each of these two conditions.>
In this paper we define a kind of decomposition for a quantum access structure. We propose a conception of minimal maximal quantum access structure and obtain a sufficient and necessary condition for minimal maximal quantum access structure, which shows the relationship between the number of minimal authorized sets and that of the players. Moreover, we investigate the construction of efficient quantum secret schemes by using these techniques, a decomposition and minimal maximal quantum access structure. A major advantage of these techniques is that it allows us to construct a method to realize a general quantum access structure. For these quantum access structures, we present two quantum secret schemes via the idea of concatenation or a decomposition of a quantum access structure. As a consequence, the application of these techniques allow us to save more quantum shares and reduce more cost than the existing scheme.
Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention since the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization is based on the well-known connection between ideal secret sharing schemes and matroids and, more specifically, on the connection between ideal multipartite secret sharing schemes and integer polymatroids. In particular, we prove that every hierarchical matroid port admits an ideal linear secret sharing scheme over every large enough finite field. Finally, we use our results to present a new proof for the existing characterization of the ideal weighted threshold access structures.
A secret-sharing scheme realizes the forbidden graph access structure determined by a graphG=(V,E)G=(V,E)G=(V,E)if the parties are the vertices of the graph and the subsets that can reconstruct the secret are the pairs of vertices inEEE(i.e., the edges) and the subsets of at least three vertices. Secret-sharing schemes for forbidden graph access structures defined by bipartite graphs are equivalent to conditional disclosure of secrets (CDS) protocols. We study the complexity of realizing a forbidden graph access structure by linear secret-sharing schemes, which are schemes in which the secret can be reconstructed from the shares by a linear mapping. We provide efficient constructions and lower bounds on the share size of linear secret-sharing schemes for sparse and very dense graphs, closing the gap between upper and lower bounds. Given a sparse (resp. very dense) graph withnnnvertices and at mostn1+βn^{1+\beta }n1+βedges (resp. at least(n2)−n1+β\binom {n}{2} - n^{1+\beta }(2n​)−n1+βedges), for some0≤β<10 \leq \beta < 10≤β<1, we construct a linear secret-sharing scheme realizing its forbidden graph access structure with total share sizeO~(n1+β/2)\tilde {O} (n^{1+\beta /2})O~(n1+β/2). Furthermore, we construct linear secret-sharing schemes realizing these access structures in which the size of each share isO~(n1/4+β/4)\tilde {O} (n^{1/4+\beta /4})O~(n1/4+β/4). We also provide constructions achieving different trade-offs between the size of each share and the total share size. We prove that almost all forbidden graph access structures require linear secret-sharing schemes with total share sizeΩ(n3/2)\Omega (n^{3/2})Ω(n3/2); this shows that the construction of Gay, Kerenidis, and Wee [CRYPTO 2015] is optimal. Furthermore, we show that for every0≤β<10 \leq \beta < 10≤β<1there exist a graph with at mostn1+βn^{1+\beta }n1+βedges and a graph with at least(n2)−n1+β\binom {n}{2}-n^{1+\beta }(2n​)−n1+βedges such that the total share size in any linear secret-sharing scheme realizing the associated forbidden graph access structures isΩ(n1+β/2)\Omega (n^{1+\beta /2})Ω(n1+β/2). Finally, we show that for every0≤β<10 \leq \beta < 10≤β<1there exist a graph with at mostn1+βn^{1+\beta }n1+βedges and a graph with at least(n2)−n1+β\binom {n}{2}-n^{1+\beta }(2n​)−n1+βedges such that the size of the share of at least one party in any linear secret-sharing scheme realizing these forbidden graph access structures isΩ(n1/4+β/4)\Omega (n^{1/4+\beta /4})Ω(n1/4+β/4). This shows that our constructions are optimal (up to poly-logarithmic factors).