The Average Information Ratio of Secret-Sharing Schemes for Access Structures Based on Coalescence of Graphs

A perfect secret-sharing scheme is a method of distributing a secret among a set of participants in such a way that only qualified subsets of participants can recover the secret and the participants in any unqualified subset cannot obtain any information about the secret. The collection of all qualified subsets is called the access structure of the scheme. In a graph-based access structure, each vertex of a graph G represents a participant and each edge of G represents a minimal qualified subset. The average information ratio of a perfect secret-sharing scheme realizing a given access structure is the ratio of the average length of the shares given to the participants to the length of the secret. The infimum of the average information ratio of all possible perfect secret-sharing schemes realizing an access structure is called the optimal average information ratio of that access structure. In this paper, we study the optimal average information ratio of access structures based on coalescence graphs. We investigate how the optimal average information ratio changes under graph coalescence.


I. INTRODUCTION
In a secret-sharing scheme, there is a dealer who has a secret, a finite set P of participants and a collection Γ of subsets of P called the access structure. Each subset in Γ is a qualified subset. An access structure must be monotone, that is, any subset of P containing a qualified subset must also be qualified. A secret-sharing scheme is a method by which the dealer distributes a secret among the participants in P such that only the participants in a qualified subset can recover the secret from the shares they received. If, in addition, the participants in any unqualified subset cannot get any information about the secret, then the secret-sharing scheme is called perfect. Since all secret-sharing schemes considered in this paper are perfect, we will simply use ``secret-sharing scheme" for ``perfect secret-sharing scheme". Therefore, an access structure Γ is completely determined by the family of all its minimal subsets which is called the basis of Γ.
In 1979, the first kind of secret-sharing schemes called the (t,n)-threshold schemes was introduced independently by Shamir [1] and Blakley [2]. In such a scheme, the basis of the access structure consists of all t-subsets of the participant set of size n. Their work has raised a great deal of interest in the research of many aspects of secretsharing problems. The reader is referred to [3] for a survey on recent progress and applications on the topic. The information ratio and the average information ratio of secret-sharing schemes have been the main subjects of discussion. The information ratio of a secret-sharing scheme is the ratio of the maximum length (in bits) of the share given to a participant to the length of the secret, while the average information ratio of a secret-sharing scheme stands for the ratio of the average length of the shares given to the participants to the length of the secret. Since these ratios respectively represent the maximum and the average number of bits a participant has to remember for each bit of the secret, they are expected to be as low as possible. Constructing secret-sharing schemes with the lowest ratios becomes an important task to achieve. Given an access structure Γ, the infimum of the (average) information ratio of all possible secretsharing schemes realizing this access structure Γ is referred to as the optimal (average) information ratio of Γ.
In this paper, we consider graph-based access structures. Given a simple graph G, let the vertex set V(G) of G be the set of participants. The access structure based on G has the edge set E(G) of G as its basis. A secretsharing scheme  for the access structure based on G is a collection of random variables S Given a discrete random variable X with possible values {x 1 , x 2 , …, x n } and a probability distribution 1 { ( )} n ii px  , the Shannon entropy of X is defined as which is a measure of the average uncertainty associated with X. This value reflects the average number of bits needed to represent the element in X faithfully [4]. Using this well-known Shannon entropy, the information ratio of the scheme  can be defined as () For simplicity, with the same symbol G, we will denote both the graph as well as the access structure based on it. For instance, ``a secret-sharing scheme on G" refers to ``a secret-sharing scheme for the access structure based on G". As mentioned above, the optimal information ratio R(G) of G and the optimal average information ratio AR(G) of G are the infimum of the information ratio R  and the average information ratio AR  over all possible secret-sharing schemes  on G respectively. It is well known that ( ) ( ) 1 R G AR G  [5] and that R(G)=1 if and only if AR(G)=1. A secret-sharing scheme  with the optimal ratio 1 R   or 1 AR   is called ideal. An access structure G is ideal if there exists an ideal secret-sharing scheme on it.
In this paper we investigate the average information ratio of coalescence of graphs. Let G and H be two graphs. The coalescence of G and H, denoted as , is the graph obtained by making coalescence of these two graphs through identifying the vertices u and v as a new vertex () uv w in the resulting graph. Therefore, the coalescence graph . Determining the exact values of R(G) and AR(G) is quite challenging. Most known results give bounds on them [6][7][8][9][10][11][12][13][14][15][16][17][18]. The exact values of the optimal average information ratio of most graphs of order no more than five and the optimal information ratio of most graphs of order no more than six have been determined [9,14,15]. Before 2007, apart from a specially defined class of graphs [6], the paths and cycles are the only infinite classes of graphs which have known exact values of the optimal information ratio and the optimal average information ratio. Csirmaz and Tardos's [19] determined the exact values of the optimal information ratio of all trees in 2007. In 2009, Csirmaz and Ligeti [20] made an even greater achievement by showing that R(G)=2-1/k, where k is the maximum degree of G, for any graph G satisfying the following properties: (i) every vertex has at most one neighbor of degree one; (ii) vertices of degree at least three are not connected by an edge, and (iii) the girth of G is at least six. In 2012, Lu and Fu [21] went on settling the exact values of the optimal average information ratio of all trees. In this paper, we consider the optimal average information ratio of coalescence graphs.
This paper is organized as follows. In Section II, we recall basic definitions and restate some known results to be used in our discussion. In Section III, we investigate how the optimal average information ratio changes under graph coalescence. Subsequently, the exact values or bounds on the optimal average information ratio of some classes of graphs are obtained. A final remark is given in the conclusion.

II. PRELIMINARIES
In this section, we introduce some basic notions and important results for our discussion in the next section. All graphs considered in this paper are simple graphs without loops and isolated vertices, not necessarily connected. We start with the result from Birckell and Davenport [10] which gives a complete characterization of ideal graph-based access structures.
Theorem 2.1 [10] Suppose that G is a connected graph. Then R(G) = AR(G) =1 if and only if G is a complete multipartite graph.
For a non-ideal access structure G, the average information ratio of any secret-sharing scheme on G makes a natural upper bound on AR(G). Stinson's decomposition construction [18] enables us to build up a secret-sharing scheme on a graph through its complete multipartite covering. A complete multipartite covering of a graph is a collection of complete multipartite subgraphs is a complete multipartite covering of a graph G of order n. Then there exists a secret-sharing scheme  on G with . This theorem suggests that a complete multipartite covering with the least vertex-number sum is what we need to construct a secret-sharing scheme on a graph with lower average information ratio. If every subgraph in a complete multipartite covering is a star, this covering is also called a star covering. Using star covering is in general most suitable for graphs of larger girth. This is our main tool used in this paper for deriving upper bounds on AR(G).
For any () A star covering with the least vertex-number sum gives the largest deduction. We also denote the maximum deduction of a star covering of G as d*(G) called the deduction of G. A star covering  with * ( ) d d G   is referred to as an optimal star covering of G. Stinson's result then provides a useful upper bound on AR(G) in terms of deduction as follows.
Theorem 2.3 [18] If  is a star covering of a graph G with deduction d  , then For the derivation of lower bounds on AR(G), we use the method in [21] based on information theoretic approach [4,[7][8][9]12,13,15,19,20]. Csirmaz et al. [20] defined a core of G as a set of vertices Theorem 2.4 [21] If  is a core cluster of a graph G, then Theorem 2.5 [21] The inequality cd    holds for any star covering  and core cluster g of a graph G. In Theorem 2.6 [21] If there exists a star covering  and a core cluster g of a graph G such that cd    , then

III. MAIN RESULT
Now, we discuss the average information ratio of the coalescence graph of two graphs.
If H  is also a core clusters of Hv  , then there is a core In this proof, we denote the graph uv GH  as K.
Let us assume that u is a vertex of a star u S in the star covering G  and v is a vertex of the star v S in the star On the other hand, let us consider the core cluster of K. Suppose that u belongs to the core 1 G C in the core cluster (2) In this case, the star covering Next, we define the core cluster as the collection  In the case where H  is a core cluster of Hv  , we suppose that u belongs to the core 1 G C in the core cluster (3) In this case, the star covering The core cluster In addition, if u and v are centers of the stars u S and v S in G  and H  respectively, then the star covering In the case where H  is a core cluster of Hv  , let . This result naturally leads to a bound on the optimal average information ratio of coalescence graphs as follows.

Corollary 3.6
Suppose that G is a realizable graph with 2 ( ( )) u V G   and T is a tree. If there is a vertex of T which has two leaf neighbors v and v', then uv GT  is realizable.
Proof. Let 0 v be the vertex which has two leaf neighbors v and v'. Since 0 v has two leaf neighbors, the vertex v' can be the designated outside neighbor for 0 v in any core cluster. Any optimal core cluster of T is also a core cluster of Tv  . We then have the result by Corollary 3.3.

IV. CONCLUSION
In this paper, we discuss the optimal average information ratio of coalescence graphs and subsequently obtain the exact value or bounds on the optimal average information ratio of some classes of graphs. Our result also suggests that while considering the optimal average information ratio of large graphs, it is possible to view the large graph as the coalescence of some small graphs. Therefore, the exact value or bounds on the optimal average information ratio of the large graph can be obtained through examining small graphs.