Optimal Information Ratio of Secret Sharing Schemes on Dutch Windmill Graphs

Advances in Mathematics of Communications doi:10.3934/amc.2019005 Volume 13, No. 1, 2019, 89–99 OPTIMAL INFORMATION RATIO OF SECRET SHARING SCHEMES ON DUTCH WINDMILL GRAPHS Bagher Bagherpour∗, Shahrooz Janbaz and Ali Zaghian Applied Mathematics and Cryptography Department Malek Ashtar university of technology, Isfahan, Iran (Communicated by Jens Zumbraegel) Abstract. 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. 1. Introduction Let P = {p1,p2,...,pn} 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. 2P 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 2010 Mathematics Subject Classification: Primary: 94A60, 94A62. Key words and phrases: Secret sharing, information ratio, access structure, Dutch windmill graphs. ∗ Corresponding author: Bagher Bagherpour. 89 c 2019 AIMS 90 BagherBagherpour,ShahroozJanbazandAliZaghian 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. 2. 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 = {pi1 ,...,pir } ⊆ P , the Shannon entropy of the random variable taking values in K(A) = K(pi1 ) ×···× K(pir ) 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: 1. ∀A ∈ Γ, H(S|A)=0 2. ∀A∈ / Γ, H(S|A)= H(S). 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 σp(Σ, Γ), is defined by σp(Σ, Γ) = H(p)/H(S). Also, the information ratio of the scheme Σ is defined by σ(Σ, Γ) = maxp∈P σp(Σ, Γ). The optimal information ratio of the access structure Γ is defined as σ(Γ) = inf(σ(Σ, Γ)), 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 2.1 ([9]). 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 A, B ⊆ P . Then we have 1) if A ⊆ B, B ∈ Γ, but A∈ / Γ, then H(B) ≥ H(A)+ H(S), 2) if A, B ∈ Γ but A ∩ B∈ / Γ, then H(A)+ H(B) ≥ H(A ∪ B)+ H(A ∩ B)+ H(S) An equivalent form of the second condition of lemma 2.1 is that if X,Y,Z ⊂ P,X ∪ Z ∈ Γ, Y ∪ Z ∈ Γ and Z∈ / Γ, then we have I(X; Y |Z) ≥ H(S). The following lemma is immediate from H(XZ)= H(XZ|Y )+ I(X; Y |Z)+ I(Y ; Z). Advances in Mathematics of Communications Volume 13, No. 1 (2019), 89–99 Dutch windmill graph access structures 91 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]. Lemma 2.3 ([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. Then for all X,Y,Z ⊆ P , if Y ∪ Z∈ / Γ but X ∪ Y ∪ Z ∈ Γ, then H(S) ≤ H(X|Y ). 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 Kn, and the n-vertex cycle is Cn. A n-vertex complete multipartite graph with partition sets X1,...,Xk is denoted k by Kn1,...,nk , where n = i=1 ni and |Xi| = ni 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 ifPvv′ ∈ 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 Fk = {G1,...,Gk} be a family of k connected graphs. Let x1,...,xk be the vertices of the graphs G1,...,Gk, respectively. By the coalescence over the family Fk we mean that we identify the vertices x1,...,xk of the graphs G1,...,Gk, respectively. The resulting coalescence graph over the family Fk is denoted by C(Fk : x1,...,xk). If every graph of the family Fk is isomorphic to the complete graph K2, then C(Fk : x1,...,xk) is isomorphic to the star graph K1,n. Also, C(Fk : x1,...,xk) (k) is called the Dutch Windmill graph (denoted by Dn ) if every graph of the family Fk is isomorphic to the cycle graph Cn. We say that C(Fk : x1,...,xk) is the (k) Windmill graph (denoted by Wn ) if every graph of the family Fk is isomorphic to (k) the complete graph Kn. In the case n = 3, the graph D3 is quite famous and is called the friendship graph (denoted by Fk)[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 K1 is denoted by a single vertex v. The union of disjoint copies of the graphs G and H is denoted by G ∪ H.

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