Construction (I)

Before we can literally describe our first construction, there are some more no-tations needed to be introduced. For any l disjoint sets of vertices V1, V2, . . . , Vl,

we use K(V1, V2, . . . , Vl) to denote the complete multipartite graph with partite sets V1, V2, . . . and Vl. Let Gl = W (|A1|, . . . , |Al|, |C1|, . . . , |Cl|) be the l-weighted graph with vertex set (Sl

i=1Ai) ∪ (Sl

Lemma 2.3.1. Π^B_l is a complete multipartite covering of Bl where

Π^B_l = of Bl are then all used up. For odd l, the argument is similar.

With these notations in mind, we are able to give our complete multipar-tite covering Πk of Gk. Let Πk be obtained recursively by letting Π1 = {G1}, subgraphs virtually make up the k-weighted graph G_k. We have the following lemma.

Lemma 2.3.2. The collection Π_k stated above is a complete multipartite covering of G_k with mimimum edge occurence one.

Our next goal is to evaluate the vertex-number sum mkof Πk. Due to the complexity of the enumeration, we consider the reduced forms first. We call G⁰_k = W (1, . . . , 1, 1, . . . , 1) the reduced form of a general k-weighted graph W (a1, . . . , ak, c1, . . . , ck). We also let B_l⁰, M_l⁰_1,l2 and H_j⁰ be the graphs defined in the same ways as Bl, Ml1,l2 and Hj respectively, except that ai’s and cj’s involved are all set to be one. Then G⁰_kand B_k⁰have the complete multipartite covering Π⁰_k and Π^B_k⁰ reduced from Πk and Π^B_k respectively. Note here that G⁰_k has 2k vertices. By applying suitable splitting and expanding operations mentioned in Section 2.2 accordingly to the reduced form G⁰_k, one can recover the general k-weighted graph W (a1, . . . , ak, c1, . . . , ck). For the description of the evaluation of the vertex-number sum m⁰_k of Π⁰_k, we introduce a specially designed binary tree.

Figure 2.1: The binary tree for Construction (I)

Note that we have decomposed G⁰_k into B_⌊⁰k+1

G⁰_2j+3. By recursively repeating this process, we observe that all G⁰_k’s can be built up from some B_l⁰’s, M_l⁰_1,k’s and just G1, G2 and G3. We illustrate this relation by means of a binary tree in Figure 2.1. In this tree, each path from the root represents the conformation of a k-weighted graph of the reduced form in our covering. For example, the leftmost path from the root Gj to G4j+6 represents that G⁰_2j+2 is composed of G⁰_j, B_j+1⁰ and M_j+2,2j+2⁰ and then G⁰_4j+6 is composed of G⁰_2j+2, B_2j+3⁰ and M_2j+4,4j+6⁰ . Hence the path shows how G⁰_4j+6 is built up. The 2^x paths of length x from the root give the conformations of the 2^x k-weighted graphs where k ranges from (j + 2)2^x− 2 to (j + 3)2^x− 3, j = 1, 2, 3.

Theorem 2.3.3. Let Γ = {A ⊆ P|w(A) ≥ t} be an access structure rep-resented by a k-weighted graph G⁰_k of reduced form, k1 = (j + 2)2^x− 2 and k₂ = (j + 3)2^x − 3, x ≥ 1, j = 1, 2, 3. If k₁ ≤ k ≤ k₂, then there exists a secret-sharing scheme Σ for the access structure Γ whose average information ratio AR_Σ satisfies

= (1

4(l²+ 8l), if l is even;

1

4(l²+ 8l − 1), if l is odd;

(1) First, we consider G⁰_k1 whose composition process is shown by the leftmost path of length x from the root. Adding up the orders of all subgraphs involved, we have are able to construct a secret-sharing scheme with average information ratio ARΣ1 = ^m

0 k1

2k1.

(2) We consider G⁰_k2 whose composition process is shown by the rightmost path of length x from the root. Similar to (1), we have

m⁰_k2 = m⁰_j +

With this covering of G⁰_k2, we have constructed a secret-sharing scheme with average information ratio ARΣ2 = ^m

0 k2

2k0. The result then follows.

As a matter of fact, the vertex-number sum m⁰_k of each G⁰_k can be evalu-ated in a similar way. The resulting expression only slightly differs from the ones for m⁰_k1 and m⁰_k2 at some nonleading coefficients.

After dealing with the reduced forms we shall turn back to the general forms. Let us introduce some more notations to simplify our description. Let

~z_l = (1 1 2 1 2 1 2 1 · · · 2 1), ~yl = (₂^l + 1) ₂^l ₂^l (₂^l − 1) (₂^l − 1) · · · 2 2 1
and ~1l = (1 1 · · · 1) be three l-dimensional vectors. For l1 ≤ l₂, let ~a(l1, l2) = (al1 al1+1 al1+2 · · · a_l2) and ~c(l1, l2) = (cl1 cl1+1 cl1+2 · · · c_l2) where ai = |Ai| and ci = |Ci|, i = l₁, l1+ 1, . . . , l2.

Lemma 2.3.4. For k = 3 · 2^x− 2 and x ≥ 1, the vertex-number sum m_k of the covering Πk is given as follows.

m_k = of whose coefficients represents the occurrence of the vertices of that part in the covering Πk.

(1) First, let us examine the occurrence of vertices of Bl, whose partite sets are Sl

are exactly the first l coordinates in ~zl+1. Similarly, the vertices in C1 have occurrence ^l+1₂ (in K(A1, C1) and H2i+1’s, i = 1, . . . ,^l−1₂ ), the vertices in C2j, j = 1, . . . ,^l−1₂ , have occurrence ^l−1₂ − j + 1 (in H2i+1’s, i ≥ j) and the vertices in C2j+1, j = 1, . . . ,^l−1₂ , have occurrence ^l−1₂ − j + 1 (in H2i+1’s, i ≥ j + 1 and K(A2j+1, C2j+1)). Hence, the occurrences of the vertices in C1, C2, . . . , Cl are exactly the first l coordinates in ~yl+1− ~1l+1.

(2) Let us consider the value of mk now. We prove the result by induction on x. When x = 1, m4 = a1+ 2a2+ a3+ 2a4+ 2c1+ 2c2+ c3+ 2c4 by direct counting the occurrences of vertices in Π4. So, the result holds when x = 1.

Next, for k = 3 · 2^x+1 − 2, Gk = W (a1, . . . , ak, c1, . . . , ck) is composed of B3·2^x−1, M_3·2^x_,3·2^x+1₋₂ and G3·2^x−2. For convenience, denote M_3·2^x_,3·2^x+1₋₂ by M for now. Observe that the vertices in Ai, 1 ≤ i ≤ 3 · 2^x − 1, have the same occurrences in Πk as they do in the covering Π^B_3·2^x₋₁ because they do not lie in M and G3·2^x−2, while the vertices in Ci, 1 ≤ i ≤ 3 · 2^x − 1, gain

one more occurrences in Πk than they do in Π^B_3·2^x₋₁ because they also occur in M. Notice that the vertices in A3·2^x and C3·2^x only occur once in Πk. Besides, the vertices in Ai’s and Ci’s, i = 3 · 2^x+ 1, . . . , k, also gain one more occurrence in Πk than they do in the covering Π3·2^x−2 of G3·2^x−2. Therefore, by (1) and the induction hypothesis, we have

m3·2^x+1−2

This lemma presents a sophisticated expression for mk in terms of ai’s and ci’s. In what follows, we give the conditions on the values of ai’s and ci’s under which mk attains its minimum value when n =Pk

i=1(ai+ ci) is fixed.

Thereby, the lowest possible average information ratio of the secret-sharing scheme constructed via this covering is obtained.

Theorem 2.3.5. Let Γ be a weighted threshold access structure represented by a k-weighted graph G = W (a1, . . . , ak, c1, . . . , ck) of order n and k = 3·2^x− similar to the reduced form. So, we make an adjustment in the expression for m⁰_k1 (with j = 1) in the proof of Theorem 2.3.3 to derive what we need here. secret-sharing scheme constructed with this covering attains its minimum value ^m_n^k and the proof is completed.

Our result appears to be quite good if k is relatively small compared with n. In fact, as k fixed, the ratio given in Theorem 2.3.5 asymptotically approaches “1” which is the optimal value for this ratio.