OPTIMAL INFORMATION RATIO OF SECRET SHARING SCHEMES ON DUTCH WINDMILL GRAPHS

. One of the basic problems in secret sharing is to determine the exact values of the information ratio of the access structures. This task is important from the practical point of view, since the security of any system degrades as the amount of secret information increases. A Dutch windmill graph consists of the edge-disjoint cycles such that all of them meet in one vertex. In this paper, we determine the exact information ratio of secret sharing schemes on the Dutch windmill graphs. Furthermore, we determine the exact ratio of some related graph families.


Introduction
Let P = {p 1 , p 2 , . . . , p n } be the set of participants among which the dealer wants to share some secret s in such a way that only the qualified subsets of P can reconstruct the secret s. A secret sharing scheme is called perfect if the non-qualified subsets of P can not obtain any information about the secret s. 2 P denotes the set of all subsets of the set P , and Γ is a collection of subsets of P . We say that Γ is monotone over P if A ∈ Γ and A ⊆ A ′ , then A ′ ∈ Γ. In the secret sharing schemes, the access structure Γ over P is a collection of all qualified subsets of P that is monotone and ∅ / ∈ Γ. A qualified subset is minimal if it is not a proper subset of any qualified subset. The collection of all minimal qualified subsets is called the basis. Since Γ is monotone, it is fully determined by its basis. The information ratio of a secret sharing scheme is the ratio between the maximum length (in bit) of the shares given to the participants and the length of the secret.
Secret sharing scheme was introduced by Blakley and Shamir [2,20]. Ito et al. showed that there exists a secret sharing scheme to realize any access structure Γ [15,16]. Also, for arbitrary monotone access structures, Benaloh and Leichter proposed another construction to realize secret sharing schemes [1]. Perfect secret sharing schemes for graphical access structures have attracted the interest of the scientific community [7,10,17,22,24]. Stinson introduced the decomposition construction method to obtain an upper bound for the information ratio of graphical access structure [21]. Jackson and Martin in [17] have studied the information ratio of perfect secret sharing schemes on five participants. The information ratio of secret sharing schemes on graphical access structures with six vertices has been studied by Van Dijk in [24]. Sun and Chen proposed the weighted decomposition construction for secret sharing schemes. Csirmaz and Tardos in [9] determined the exact information ratio for all trees. Csirmaz and Liegti have investigated the information ratio of an infinite family of graphs in [12].
In this paper, we firstly give some preliminaries and introduce some important concepts. After that, our main results are introduced in details. In section 3, we determine the exact information ratio of secret sharing schemes on the Dutch windmill graphs. Furthermore, the information ratio of secret sharing schemes on the cone graphs is studied and the information ratio of secret sharing schemes on some variations of the studied graphs is determined. Finally, we propose some research problems.

Preliminaries
Let S be the set of all secrets and p ∈ P be an arbitrary participant. The set of all possible shares given to the participant p is denoted by K(p). A secret sharing scheme can be seen as a distribution rule by which the dealer distributes a secret s ∈ S, according to some probability distribution, among the participants in P by giving a share to each participant of P . Thus, each secret sharing scheme induces random variables on the sets S and K(p), where p ∈ P . The Shannon entropy of the random variable taking values in S is denoted by H(S). Also, for each A = {p i1 , . . . , p ir } ⊆ P , the Shannon entropy of the random variable taking values in K(A) = K(p i1 ) × · · · × K(p ir ) is denoted by H(A) (for more details see [5]). Let Γ be an access structure on the set of participants P , and let S be the set of secrets. In terms of the entropy, a secret sharing scheme Σ for access structure Γ and the set of secrets S is called perfect secret sharing scheme if the two following conditions hold: Suppose Γ is an access structure with the participants set P . Suppose Σ is a secret sharing scheme to realize Γ. The information ratio of a participant p ∈ P in Σ, denoted by σ p (Σ, Γ), is defined by σ p (Σ, Γ) = H(p)/H(S). Also, the information ratio of the scheme Σ is defined by σ(Σ, Γ) = max p∈P σ p (Σ, Γ). The optimal information ratio of the access structure Γ is defined as where the infimum is taken over all secret sharing schemes Σ for the access structure Γ. A secret sharing scheme for an access structure Γ is ideal if σ(Γ) = 1. The intuition and the basics of the entropy method can be found in here [14,24]. Lemma 2.1 which was introduced in [9], is a good tool to obtain some lower bounds for the information ratio of an arbitrary access structure Γ. Lemma 2.2. Suppose that Σ is a secret sharing scheme with access structure Γ on a set of participants P . Let S be the set of secrets and X, Y, Z ⊆ P . If X ∪ Z ∈ Γ, Y ∪ Z ∈ Γ and Z / ∈ Γ, then H(XZ) ≥ H(XZ|Y ) + H(S).
M. Van Dijk proved the following property in [24].
, Corollary 2.2, Corollary 2.6). Suppose that Σ is a secret sharing scheme with access structure Γ on a set of participants P . Let S be the set of secrets.
In the above lemma, if we set Z = ∅, then the fact H(S) ≤ H(X) is concluded (for further details see [7]).
A complete graph on n vertices is denoted by K n , and the n-vertex cycle is C n . A n-vertex complete multipartite graph with partition sets X 1 , . . . , X k is denoted by K n1,...,n k , where n = k i=1 n i and |X i | = n i for i = 1, . . . , k. When necessary, the vertices and edges of a graph G are denoted by V (G) and E(G), respectively. Also, v ′ ∼ v if and only if vv ′ ∈ E(G). Let G and H be two graphs, with vertices x and y, respectively. If we identify the vertices x and y, the resulting graph is called the coalescence of G and H at x and y (we denote it by C({G, H} : x, y)). Let F k = {G 1 , . . . , G k } be a family of k connected graphs. Let x 1 , . . . , x k be the vertices of the graphs G 1 , . . . , G k , respectively. By the coalescence over the family F k we mean that we identify the vertices x 1 , . . . , x k of the graphs G 1 , . . . , G k , respectively. The resulting coalescence graph over the family F k is denoted by C(F k : x 1 , . . . , x k ). If every graph of the family F k is isomorphic to the complete graph K 2 , then n ) if every graph of the family F k is isomorphic to the cycle graph C n . We say that C(F k : n ) if every graph of the family F k is isomorphic to the complete graph K n . In the case n = 3, the graph D is quite famous and is called the friendship graph (denoted by F k ) [13]. In the graph G, the length of the shortest induced cycle is called the girth of G, and is denoted by g(G). Suppose that the complete graph K 1 is denoted by a single vertex v. The union of disjoint copies of the graphs G and H is denoted by G ∪ H. The join G∇H of (disjoint) graphs G and H is a graph that is obtained from G ∪ H by joining each vertex of G to each vertex of H. The graph v∇H is called the cone over H with cone vertex v. If H is a complete multipartite graph, then v∇H is a complete multipartite graph.
The independent sequence method (lemma 2.4) was introduced by Blundo, et all in [4] and was generalized by Padró and Sáez in [18]. Suppose Γ is an access structure with the participants set P . A sequence of subsets ∈ Γ be a sequence of subsets that is made independent by A ⊂ P . Then The following theorem, which was proved in [6], characterizes the information ratio of the realised secret sharing schemes on the complete multipartite graphs.
To find lower bound on the information rate of the access structures several methods have been introduced. The λ−decomposition method is one of them, which was introduced by Stinson in [21]. A λ−decomposition of an access structure Γ is the family Γ 0,1 , . . . , Γ 0,r ⊂ Γ 0 such that Γ 0,1 ∪. . .∪Γ 0,r = Γ 0 , and every element of Γ 0 is covered at least λ times by the family Γ 0,1 , . . . , Γ 0,r . The following lemma is a direct consequence of theorem 2.1 of [21].
Lemma 2.7. Let Γ be an access structure on the set of participants P . Assume that Γ 0 is the basis of Γ and Γ 0,1 , . . . , Γ 0,r ⊂ Γ 0 is the λ−decomposition of Γ. Suppose that Γ i is the access structure with basis Γ 0,i and P i = A∈Γ0,i A. Also, suppose that for every i ∈ {1, . . . , r} there exists a secret sharing scheme Σ i to realize the access structure Γ i . Then

The information ratio of the Dutch Windmill graphs
In this section we compute the exact information ratio of the secret sharing schemes for Dutch Windmill graphs.
Let F k = {C n1 , . . . , C n k } be a family of k cycles C n1 , . . . , C n k of length n 1 , . . . , n k , respectively. For each cycle graph C ni , we denote its vertex set by V (C ni ) = {v i 1 , . . . , v i ni }. For simplicity, we denote the coalescence graph C(F k : v 1 1 , . . . , v k 1 ) briefly by C(F k ). Also, if we add a pendant (vertex of degree one is called pendant) to the central vertex (vertex with maximum degree) of the graph C(F k ), then we denote the resulted graph by C ′ (F k ). In the following, we determine the exact information ratio of the coalescence graph over the family F k whose girth is at least five. We notate that the results of the paper [12] does not yield our results.
Theorem 3.1. If the girth of the graphs C(F k ) and C ′ (F k ) is at least five, then Proof. We prove that σ(C(F k )) = (4k − 1)/2k. Let us denote the identified vertices v 1 1 , . . . , v k 1 in C(F k ) by v c (for the sake of simplicity, we use V (C ni ) = {v c , v i 2 , . . . , v i ni } to denote the vertex set of the induced subgraph C ni of the graph C(F k )). To determine the lower bound for the information ratio of the graph C(F k ), it suffices to show According to lemma 2.5 we have It can be seen that

Substituting (2) into the inequality (1) yields
To prove the upper bound for the information ratio of the graph C(F k ), we give a covering set for C(F k ) which is a trivial decomposition. Consider the subgraphs of the graph C(F k ) which are introduced in the table 1. We know that in the path Table 1. Subgraphs of the graph C(F k ) . . , k}} i ∈ {1, . . . , k} graph of length at least 4, the initial and last vertices have information ratio 1, and the mid vertices have information ratio 3/2. Therefore, Π = {Π 1 , Π 2 , Π 3 , Π 4 , Π 5 }, is a 2k-decomposition of C(F k ) (we note that the path P i 2 is maybe star or only one edge). Thus, by lemma 2.7 we have σ(C(F k )) ≤ (4k − 1)/2k. Now, we prove that σ(C ′ (F k )) = (4k + 1)(2k + 1). Suppose v ′ is a pendant which has been added to the vertex v c . Firstly, we show that the information ratio of the graph C ′ (F k ) is lower bounded by (4k + 1)/(2k + 1). By similar argument, which was used in the proof of the information ratio of the graph C(F k ), we have Therefore, σ(C ′ (F k )) ≥ (4k + 1)/(2k + 1). To prove the upper bound, consider the subgraphs of the graph C ′ (F k ) which are introduced in the table 2. Let Π = Table 2. Subgraphs of the graph C ′ (F k ) It can be seen that Π is a (2k+1)-decomposition of C ′ (F k ) (we note that the path P i 2 is maybe star or only one edge). Thus, by lemma 2.7 we have σ(C ′ (F k )) ≤ (4k + 1)/(2k + 1).
Let F k = {C n1 , . . . , C n k } be a family of cycle graphs. If all cycles in the set F k are isomorphic to the cycle graph C n , then the coalescence graph C(F k ) is isomorphic to the Dutch windmill graph D (k) n . Therefore, by using the theorem 3.1, the exact information ratio of the graph D n , then the exact information ratio of the resulted graph is (4k + 1)/(2k + 1). Hitherto, we have determined the exact information ratio of secret sharing schemes on Dutch windmill graphs with girth at least 5. In the sequel of this section, we determine the exact information ratio of secret sharing schemes on Dutch windmill graphs with girth three and four.
where G ′ is the graph G when a pendant has been added to the vertex v c .  Therefore, σ(G ′ ) ≥ (2k + 1)/(k + 1).
The friendship graph F k is a cone graph over the k complete graphs K 2 . We calculated the exact information ratio of this graph. The below question can be studied for future work: Problem. Let G be an arbitrary graph and v∇G denotes the cone graph over G. Determine the exact information ratio of the graph v∇G.