The complexity of the connected graph access structure on seven participants

Massoud Hadian Dehkordi; Ali Safi

doi:10.1515/jmc-2016-0017

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The complexity of the connected graph access structure on seven participants

Massoud Hadian Dehkordi and Ali Safi

Published/Copyright: February 23, 2017

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From the journal Journal of Mathematical Cryptology Volume 11 Issue 1

Abstract

In this paper, we study an important problem in secret sharing that determines the exact value or bound for the complexity. First, we use the induced subgraph complexity of the graph G with access structure Γ to obtain a lower bound on the complexity of the graph G. Then, applying decomposition techniques, we obtain an upper bound on the complexity of the graph G. We determine the exact values of the complexity for each of the ten graph access structures on seven participants. Also, we improve the value bound of the complexity for the six graph access structures with seven participants.

Keywords: Access structure; complexity; secret sharing scheme; connected graph; induced subgraph

MSC 2010: 68R10; 94A62

1 Introduction

A secret sharing scheme is a method which allows a secret K to be shared among a set of participants P in such a way that only qualified subsets of participants can recover the secret. The first secret sharing schemes were considered by Shamir [9] and Blakley [1]; in their schemes any proper subset A of the participants set P such that |A|≥t can recover the secret K, and for every subset A of P such that |A|<t no one can obtain any information about the secret. Such schemes are called (t,n)-threshold schemes, where t is the threshold scheme and n is the size of the participants set P.

The access structure Γ of a secret sharing scheme defined on P with secret K is the collection Γ of subsets of P that are desired to be able to reconstruct the value of K from their pooled shares. In general, an access structure Γ on the set P is considered to be monotone. This means that if A∈Γ and A⊆A′⊂P, then A′∈Γ. A subset of P that belongs to Γ is called a qualified subset and those that do not belong to Γ are known as unqualified subsets. A minimal qualified subset A∈Γ is a subset of participants such that A′∉Γ for all A′⊂A. The basis of Γ is the collection of all minimal qualified subsets of P, which is denoted by [Γ]-. We will say that a scheme is a perfect secret sharing scheme realizing the access structure Γ when the following two properties are satisfied [12]:

If the participants in a qualified subset A⊆P pool their shares, then they can determine the value of K.
If the participants in an unqualified subset A⊆P pool their shares, then they can determine nothing about the value of K.

The efficiency of a perfect secret sharing scheme can be assessed in terms of its complexity, which is the ratio between the maximum size of the shares given to any participant in P and the size of the secret. The complexity of an access structure Γ, denoted by σ⁢(Γ), is defined as the infimum of the complexities of all secret sharing schemes with access structure Γ. A perfect secret sharing scheme is called ideal if its complexity is equal to one. The graph access structure is an access structure that contains only minimal qualified subsets of cardinality two. The complexity of the graph access structures has been studied in the literature by several authors since the 90s [16, 14, 3, 2, 4, 12, 11, 15, 10]. There have been particular endeavors to determine the most efficient secret sharing schemes for all graphs with a small number of vertices. In line with this, lower bounds have been obtained based on entropy considerations, as well as upper bounds by using a decomposition or weighted decomposition technique for constructing good schemes.

In [16], Jackson and Martin studied the complexity of connected graph access structures on five participants. Van Dijk studied the complexity of the 112 graph access structures on six participants, and in 94 cases he determined the exact values of the complexity [14]. With results obtained in [13, 7, 5], the complexity in several cases out of eighteen cases of secret sharing schemes which remained unsolved for connecting graphs with six participants were determined. In [11, 15], Wang, Li and Song studied the complexity of connected graph access structures on seven participants with six, seven, eight, nine, and ten edges. They determined the exact values of the complexity for 189 cases. Also, Song et al. [10] studied the complexity of the 272 graph access structures on nine participants with eight and nine edges, and in 231 cases they determined the exact values of the complexity.

In [11], the complexity of the 111 connected graph access structures on seven participants with six, seven and eight edges were studied among which the exact value of 91 cases were determined and the other cases provided the upper and lower bound. With results obtained in [8], the complexity in four out of twenty cases of secret sharing schemes which remained unsolved for connecting graphs with seven participants were determined. In this paper, we obtain the exact value of the complexity of ten connected graph access structures from the remaining sixteen cases by using the following method: First, a six-vertex graph access structure is reached by removing the vertex appropriate to its edges of the seven-vertex graph access structure. Because of the complexity of the most six-vertex graph access structures which was determined in [14, 13, 7, 5], we obtain the lower bound for complexity of the seven-vertex graph access structures. On the other hand, by using decomposition techniques, we get an upper bound equal to the lower bound. In addition to the above method we improve the value bound of the complexity for the six graph access structures with seven participants.

2 Definitions and basic properties

In this section, we introduce definitions and theorems which are used to determine the exact values of the complexity of the connected graph access structures with seven vertices.

Theorem 2.1

Theorem 2.1 ([14])

Let G=(V,E) be a graph with vertices a,b∈V. Let d∈V. Then ad is an edge if and only if bd is an edge. Define G′ by deleting vertex a and edges ad for all vertices d. Then σ⁢(ΓG)=σ⁢(ΓG′).

Definition 2.2

Let G be a graph with vertex set V and edge set E. If V1⊆V, then the induced subgraphG⁢[V1] is defined to have the vertex set V1 and edge set {v⁢w∈E⁢(G):v,w∈V1}.

Definition 2.3

Let G be a connected graph with vertex set V. Suppose that V can be partitioned into subsets V1,…,Vn such that the edges in G are defined by all pairs of vertices from different subsets. Then G is called a complete multipartite graph.

Theorem 2.4

Theorem 2.4 ([3])

The access structure Γ based on a connected graph G is ideal if and only if G is a complete multipartite graph.

Theorem 2.5

Theorem 2.5 ([2])

If G is not a multipartite graph, then σ⁢(G)≥32.

Theorem 2.6

Theorem 2.6 ([2])

Suppose G is a graph and G′ is an induced subgraph of G. Then σ⁢(ΓG′)≤σ⁢(ΓG).

Definition 2.7

Definition 2.7 ([12])

Suppose Γ is an access structure having basis [Γ]-. Let further λ≥1 be an integer. A λ-decomposition of [Γ]- consists of a collection (i.e. a multiset) {Γ1,…,Γt} such that the following properties are satisfied:

Γh⊆[Γ]- for 1≤h≤t.
For each A∈[Γ]- there exist λ indices 1≤i1<⋯<iλ≤t such that A∈Γij for 1≤j≤λ.

Theorem 2.8

Theorem 2.8 ([12])

Let P={Pi:1≤i≤n} be the set of n participants and let Γ be an access structure having basis [Γ]-. If the collection {Γ1,…,Γt} is a λ-decomposition of [Γ]- such that Γh is ideal and

R=max⁡{|{h:pi∈Ph}|:1≤i≤n}

for 1≤h≤t, then σ⁢(Γ)≤Rλ.

Example 2.9

The 2-decomposition for the graph Γ12 is shown in Figure 1. According to Theorem 2.8, we have σ⁢(Γ)≤32.

Definition 2.10

Definition 2.10 ([13])

A weighted access structure for a secret sharing scheme, denoted by Γw, is a set

{(X,W⁢(X)):W⁢(X)≥0,W⁢(X)∈ℤ,X∈2P}.

The weight of Γw is defined by W⁢(Γw)=max⁡{W⁢(X):X∈2P}.

A secret sharing scheme for a weighted access structure Γw is a method which allows W⁢(Γw) secrets of the same size to be shared among a set of participants P in such a way that each subset of participants X can exactly recover W⁢(X) secrets out of the W⁢(Γw) secrets. It is obvious that if X⊆X′⊆P, then W⁢(X)≤W⁢(X′).

Consider a weighted graph G⁢(V,E) with weights We⁢(G) (We⁢(G)≥1). Then W⁢(G)=max⁡{We⁢(G):e∈E}, for e∈E, is a perfect secret sharing scheme which satisfies the following requirements:

Any pair of participants corresponding to an edge e of G can obtain We⁢(G) secrets out of the W⁢(G) secrets.
Any subset of participants containing no edge of G has no information on the W⁢(G) secrets.

Example 2.11

Consider the weighted graph G that is shown in Figure 2; later on the perfect secret sharing scheme for it will be given. Assume the secret s=(s1,s2) is selected randomly from (G⁢F⁢(q))2, where q is a prime number.

The shares are given as follows:

a:(r2),

b:(r2+s2,r1),

c:(r1+s1,r2),

d:(r1),

e:(r1+r2),

where r1 and r2 are selected randomly from (G⁢F⁢(q)) and all operations are calculated at (G⁢F⁢(q)). The sets {a,b}, {c,d}, {b,e}, and {c,e} obtain s2, s1, s2, and s1 respectively. Also, the participants set {b,c} can obtain the secrets s1 and s2.

$Figure 1 The 2-decomposition for the graph Γ12${{\Gamma_{12}}}$.$

Figure 1

The 2-decomposition for the graph Γ12.

Figure 2

Graph G.

Definition 2.12

Definition 2.12 ([13])

Suppose Γ is an access structure. A λ-weighted decomposition of Γ consists of a collection {Γ1,…,Γt} such that the following requirements are satisfied:

Each Γh is a weighted access structure for 1≤h≤t.
For each X∈Γ there exist some indices, say i1<⋯<ik, such that ∑ijW⁢(X;Γij)≥λ, where W⁢(X;Γij) is the weight of X in Γij for 1≤j≤k.
For each X∉Γ there holds ∑h=1tW⁢(X;Γh)=0.

Theorem 2.13

Theorem 2.13 (Weighted decomposition construction for graphs, [13])

Let G be a graph of access structure on n participants and suppose that {G1,…,Gt} is a λ-weighted decomposition of G. Assume that for each weighted graph Gh, 1≤h≤t, there exists a perfect secret sharing scheme with complexity σi⁢h for each pi∈Ph. Then there exists a perfect secret sharing scheme for G with complexity

σ⁢(Σ)=max⁡{∑{h:pi∈Ph}W⁢(Gh)⁢σi⁢hλ:1≤i≤n}.

Example 2.14

The weighted decomposition construction for the graph Γ106 with seven participants is shown in Figure 3.

So the secret sharing scheme for Γ106 is set up as follows: Let the secret K=(K1,K2,K3) be taken randomly from (G⁢F⁢(q))3, where q is a prime power. Let f⁢(x)=K1+K2⁢x+K3⁢x2(modq). The values yi computed from f⁢(x) are yi=f⁢(i)(modq) for i=1,…,9. It is clear, given four yi’s, that f⁢(x) can be determined uniquely, and hence the secret K can be recovered. On the contrary, one who does not have any knowledge of these yi’s obtains no information on the secret K. The shares for a, b, c, d, e, f, g are assigned as follows:

Sa:(r2,r3,r4+y4,r5+y5,r6),

Sb:(r2,r3,r4-y4,r5+,r6+y6),

Sc:(r2,r3,r9),

Sd:(r1,r7,r8),

Se:(r1+y1,r2,r3,r7+y7,r8+y8),

Sf:(r1+r2,r3,r7,r8,r9),

Sg:(r1,r2+y2,r3+y3,r4,r9+y9).

Thus,

{a,b} can recover y4,y5,y6,

{a,g} can recover y2,y3,y4,

{b,g} can recover y2,y3,y4,

{c,g} can recover y2,y3,y9,

{d,e} can recover y1,y7,y8,

{e,f} can recover y1,y7,y8,

{e,g} can recover y1,y2,y3,

{f,g} can recover y2,y3,y9.

$Figure 3 The 3-weighted decomposition for the graph Γ106${{\Gamma_{106}}}$.$

Figure 3

The 3-weighted decomposition for the graph Γ106.

Therefore, all pairs above can obtain three yi’s, and they can recover the secret K. On the contrary, no other pair can obtain any information on the yi’s. It is clear that the complexity for these shares are σa=53, σb=53, σc=1, σd=1, σe=53, σf=53, and σg=53. Hence the complexity for the constructed secret sharing scheme is 53. Thus, according to the definition of the complexity of the access structure, we have σ⁢(Γ106)≤53.

3 The results

The purpose of this section is to determine the exact value of the complexity of the following graph access structures with seven participants P={a,b,c,d,e,f,g}:

[Γ12]-={a⁢b,a⁢g,b⁢c,c⁢d,d⁢e,e⁢f,f⁢g}, ⁢[Γ43]-={a⁢b,a⁢f,a⁢g,b⁢c,b⁢f,d⁢e,e⁢f},

=-{ab,af,ag,bf,bc,cd,de,ef}, ⁢[Γ60]-={a⁢b,a⁢f,a⁢g,b⁢e,c⁢d,d⁢e,e⁢f,f⁢g},

=-{ab,ag,bc,bd,bf,df,ef,fg}, ⁢[Γ70]-={a⁢b,a⁢c,b⁢c,c⁢d,c⁢g,d⁢e,e⁢f,f⁢g},

=-{ab,ag,cd,de,df,dg,ef,fg}, ⁢[Γ85]-={a⁢b,b⁢c,c⁢d,c⁢f,c⁢g,d⁢e,d⁢f,f⁢g},

=-{ab,ag,bg,cg,de,ef,eg,fg}, ⁢[Γ108]-={a⁢b,a⁢d,a⁢g,b⁢c,b⁢d,d⁢e,d⁢f,e⁢f},

Further, we improve the value bound of the complexity of the following graph access structures with seven participants P={a,b,c,d,e,f,g}:

[Γ61]-={a⁢b,a⁢e,a⁢g,b⁢c,b⁢e,c⁢d,d⁢e,e⁢f}, ⁢[Γ62]-={a⁢b,a⁢f,a⁢g,b⁢c,c⁢d,c⁢f,e⁢f,f⁢g},

=-{ab,ag,bc,bg,cd,cf,ef,fg}, ⁢[Γ64]-={a⁢b,a⁢c,a⁢g,c⁢e,c⁢f,d⁢e,e⁢f,f⁢g},

=-{ab,af,ag,bc,be,cd,df,ef}, ⁢[Γ94]-={a⁢b,b⁢c,b⁢g,c⁢d,c⁢e,c⁢g,e⁢f,e⁢g}.

The lower bound and upper bound obtained for the complexity of the above graph access structures are listed in Table 1; see [11].

For the seven-vertex graph access structure Γ12, from Theorem 2.5, we can get the lower bound of σ⁢(Γ12) to be 32, and by using the 2-decomposition in the provided Example 2.9, we get that the upper bound for σ⁢(Γ12) is equal to 32, so σ⁢(Γ12)=32.

For the seven-vertex graph access structures Γ43 and Γ47, removing the vertex d and edges connected to it, we get the six-vertex graph access structure Γ9 with the exact value of complexity being 74 (see [13]). Theorem 2.6 implies that the lower bounds of σ⁢(Γ43) and σ⁢(Γ47) are at least 74. Theorem 2.13 together with the 4-weighted decomposition provided at the end of the paper (see Figures 4 and 5) implies that the upper bound for σ⁢(Γ43) and σ⁢(Γ47) is equal to 74; thus σ⁢(Γ43)=σ⁢(Γ47)=74.

We remark that the result for Γ43 can be obtained by using the complexity graph access structures with nine-vertex Γ257 or Γ268 in [10] and Theorem 2.1. We can see that σ⁢(Γ43)=74.

For the seven-vertex graph access structure, removing the vertex c in Γ60, we can obtain the six-vertex graph access structure Γ31 with σ⁢(Γ31)=53 (see [14]). For the seven-vertex graph access structure, removing the vertex a in Γ70, we can obtain the six-vertex graph access structure Γ18 with σ⁢(Γ18)=53 (see [14]). We conclude that the complexity of Γ60 and Γ70 are at least 53. Using the 3-decomposition provided at the end of the paper (see Figures 6 and 13) and from Theorem 2.8, we can get that the upper bound for σ⁢(Γ60) and σ⁢(Γ70) is equal to 53; thus σ⁢(Γ60)=σ⁢(Γ70)=53.

Removing the vertex c from the seven-vertex graph access structure Γ65 and removing the vertex d and its connected edges from Γ106, we get the six-vertex graph access structures Γ33 and Γ25, respectively, with exact complexity value of 53 (see [14]). Theorem 2.6 implies that the lower bounds for Γ65 and Γ106 are 53. Using Theorem 2.13 and the 3-weighted decomposition provided at the end of the paper (see Figure 11) and in Example 2.14, we get that the upper bound for σ⁢(Γ65) and σ⁢(Γ106) is equal to 53; thus σ⁢(Γ65)=σ⁢(Γ106)=53.

For the seven-vertex graph access structure Γ83, removing vertex b and for Γ85 removing vertex a and its connected edges, we can get the six-vertex graph access structure Γ22 with σ⁢(Γ22)=74. For the seven-vertex graph access structure Γ108, removing vertex e and its connected edges, we can get the six-vertex graph access structure Γ9 with σ⁢(Γ9)=74 (see [13, 7]). From Theorem 2.6 we conclude that the lower bound for σ⁢(Γ83), σ⁢(Γ85) and σ⁢(Γ108) is 74. With use of Theorem 2.13 and the 4-weighted decomposition provided at the end of the paper (see Figures 14, 15 and 17) we obtain that the upper bound for σ⁢(Γ83), σ⁢(Γ85) and σ⁢(Γ108) is 74; thus σ⁢(Γ83)=σ⁢(Γ85)=σ⁢(Γ108)=74.

For the seven-vertex graph access structures, removing the vertex d in Γ61 and vertex f in Γ94, we obtain the six-vertex graph access structures Γ9 and Γ22 with σ⁢(Γ9)=σ⁢(Γ22)=74 (see [6, 7]). Theorem 2.6 implies that the complexity of Γ61 and Γ94 is at least 74. Using the 8-weighted decomposition provided at the end of the paper (see Figures 7 and 16) and by Theorem 2.13, we can get that the upper bound for σ⁢(Γ61) and σ⁢(Γ94) is equal to 158.

Removing the vertex e from the seven-vertex graph access structures Γ62 and removing the vertex f and its connected edges from Γ68, we get the six-vertex graph access structures Γ31 and Γ4, respectively, with exact complexity value of 53 (see [14]). Theorem 2.6 implies that the lower bound for Γ62 and Γ68 is 53. Using Theorem 2.13 and the 4-weighted decomposition provided at the end of the paper (see Figure 8) for Γ62, respectively using Theorem 2.8 and the 4-decomposition provided at the end of the paper (see Figure 12) for Γ68, we get that the upper bound for σ⁢(Γ62) and σ⁢(Γ68) is equal to 74.

For the seven-vertex graph access structure, removing the vertex d in Γ63, we can obtain the six-vertex graph access structure Γ31 with σ⁢(Γ31)=53 (see [14]). Theorem 2.6 implies that the complexity of Γ63 is at least 53. Using the 8-decomposition provided at the end of the paper (see Figure 9) and from Theorem 2.8 we can get that the upper bound for σ⁢(Γ63) is equal to 158.

Removing the vertex d from the seven-vertex graph access structure Γ64, we get the six-vertex graph access structure Γ31 with exact complexity value of 53 (see [14]). Theorem 2.6 implies that the lower bound for Γ64 is equal to 53. Using Theorem 2.13 and the 4-weighted decomposition provided at the end of the paper (see Figure 10) for Γ64, we get that the upper bound for σ⁢(Γ64) is equal to 74.

The results are given in Table 2.

Table 1

Upper and lower bounds provided for the complexityof the 16 remaining access structures in [11].

Γi	σ⁢(Γi)	Γi	σ⁢(Γi)	Γi	σ⁢(Γi)	Γi	σ⁢(Γi)
Γ12	32∼117	Γ43	53∼2	Γ47	53∼2	Γ60	53∼2
Γ61	53∼2	Γ62	32∼2	Γ63	53∼2	Γ64	53∼2
Γ65	53∼2	Γ68	32∼2	Γ70	53∼2	Γ83	53∼2
Γ85	53∼2	Γ94	53∼2	Γ106	32∼2	Γ108	53∼2

Table 2

The obtained results.

Γi	σ⁢(Γi)	Γi	σ⁢(Γi)	Γi	σ⁢(Γi)	Γi	σ⁢(Γi)
Γ12	32	Γ43	74	Γ47	74	Γ60	53
Γ61	74∼158	Γ62	53∼74	Γ63	53∼158	Γ64	53∼74
Γ65	53	Γ68	53∼74	Γ70	53	Γ83	74
Γ85	74	Γ94	74∼158	Γ106	53	Γ108	74

$Figure 4 The 4-weighted decomposition for the graph Γ43${\Gamma_{43}}$.$

Figure 4

The 4-weighted decomposition for the graph Γ43.

$Figure 5 The 4-weighted decomposition for the graph Γ47${\Gamma_{47}}$.$

Figure 5

The 4-weighted decomposition for the graph Γ47.

$Figure 6 The 3-weighted decomposition for the graph Γ60${\Gamma_{60}}$.$

Figure 6

The 3-weighted decomposition for the graph Γ60.

$Figure 7 The 8-weighted decomposition for the graph Γ61${\Gamma_{61}}$.$

Figure 7

The 8-weighted decomposition for the graph Γ61.

$Figure 8 The 4-weighted decomposition for the graph Γ62${\Gamma_{62}}$.$

Figure 8

The 4-weighted decomposition for the graph Γ62.

$Figure 9 The 8-weighted decomposition for the graph Γ63${\Gamma_{63}}$.$

Figure 9

The 8-weighted decomposition for the graph Γ63.

$Figure 10 The 4-weighted decomposition for the graph Γ64${\Gamma_{64}}$.$

Figure 10

The 4-weighted decomposition for the graph Γ64.

$Figure 11 The 3-weighted decomposition for the graph Γ65${\Gamma_{65}}$.$

Figure 11

The 3-weighted decomposition for the graph Γ65.

$Figure 12 The 4-weighted decomposition for the graph Γ68${\Gamma_{68}}$.$

Figure 12

The 4-weighted decomposition for the graph Γ68.

$Figure 13 The 3-weighted decomposition for the graph Γ70${\Gamma_{70}}$.$

Figure 13

The 3-weighted decomposition for the graph Γ70.

$Figure 14 The 4-weighted decomposition for the graph Γ83${\Gamma_{83}}$.$

Figure 14

The 4-weighted decomposition for the graph Γ83.

$Figure 15 The 4-weighted decomposition for the graph Γ85${\Gamma_{85}}$.$

Figure 15

The 4-weighted decomposition for the graph Γ85.

$Figure 16 The 8-weighted decomposition for the graph Γ94${\Gamma_{94}}$.$

Figure 16

The 8-weighted decomposition for the graph Γ94.

$Figure 17 The 4-weighted decomposition for the graph Γ108${\Gamma_{108}}$.$

Figure 17

The 4-weighted decomposition for the graph Γ108.

Remark

We point out that our results can be applied for some graph access structures of nine participants with ten or more edges by using Theorem 2.1.

Communicated by Carlo Blundo

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Received: 2015-11-25

Accepted: 2017-1-15

Published Online: 2017-2-23

Published in Print: 2017-3-1

Articles in the same Issue

https://doi.org/10.1515/jmc-2016-0017

Keywords for this article

Access structure; complexity; secret sharing scheme; connected graph; induced subgraph