Smallest graphs achieving the Stinson bound
Mate Gyarmati Peter Ligeti

Reveal the non-local coherent nature from a dissipative Majorana teleportation
    Dong E. Liu
    December 30, 2023
==============================================================================




Perfect secret sharing scheme is a method of distribute a secret information s among participants such that only predefined coalitions, called qualified subsets of the participants can recover the secret, whereas any other coalitions, the unqualified subsets cannot determine anything about the secret. The most important property is the efficiency of the system, which is measured by the information ratio. It can be shown that for graphs the information ratio is at most (δ+1)/2 where δ is the maximal degree of the graph. Blundo et al. constructed  a family of δ-regular graphs  with information ratio (δ+1)/2 on at least c· 6^δ vertices. We improve this result by constructing a significantly smaller graph family on c· 2^δ vertices achieving the same upper bound both in the worst and the average case. 



Keywords: secret sharing, information ratio, entropy method, decomposition construction






§ INTRODUCTION

Secret sharing schemes were first introduced by Shamir <cit.> and Blakley <cit.> as a method of distribute a secret information s among participants such that only predefined coalitions, called qualified subsets of the participants can recover the secret. If the unqualified subsets have no information about s, then we call it a perfect secret sharing. The set of the qualified subsets is called access structure. If all minimal elements of the access structure have exactly  two elements, then the access structure can be represented with a graph: let the participants denote the vertices, and two vertices are supposed to be connected by an edge if the respective participants are qualified together. 

One of the most frequently examined problem on secret sharing is the efficiency of a particular system. This can be characterized by the amount of information a given participant must remember correlate to the size of the secret. This amount is called the information ratio of the share of this participant. From the whole system point of view, it is possible to investigate two, slightly different quantities: the worst case information ratio is the information that the most heavily loaded participant must remember; or the average case, which is the average of the information ratios of the participants. Determining the information ratio is a challenging but interesting problem for both  cases even for small access structures.
 
The exact value of the worst-case information ratio was determined for most of the graphs with at most six vertices <cit.>,<cit.>,<cit.>,<cit.>, <cit.> for trees <cit.>, for d-dimensional cubes <cit.>, for graphs with large girth <cit.>. On the other hand,  there are some sporadic results on the average case as well, see <cit.>, <cit.>, <cit.>, <cit.>. In  general however, there is a large gap between the  best known upper and lower bound. One notable universal upper bound was proved by Stinson in <cit.>. He showed that if the maximal degree of a graph is δ, then the information ratios are at most (δ+1)/2. Blundo et al.  in <cit.> showed that this bound is sharp for an infinite graph class with maximal degree δ. 

The graph class constructed by Blundo et al. with maximal degree δ has 2np^δ-3 vertices, where n,p≥6. We give a significant improvement  of that result by presenting graph classes with much less vertices, i.e. 2^δ for the worst case, and 3· 2^δ-1 for the average case. 


 















§ PRELIMINARIES

Let P be a finite set of participants. The access structure A on P is a  subset of 2^P which is monotone increasing in the sense, that if A∈𝒜 and A⊆ B, then B∈𝒜. 

 A perfect secret sharing scheme S realizing 𝒜 is a collection of random
variables ξ_i for every i∈𝒫 and ξ_s with a joint
distribution such that


  (i)  if A∈𝒜, then {ξ_i: i∈ A} determines ξ_s;
 
  (ii) if A∉A, then {ξ_i: i∈ A} is independent
of ξ_s.


Note that as a consequence of the monotonicity, the minimal elements of 𝒜 determine 𝒜. In this paper we consider only graph-based secret sharing schemes, where all the minimal elements of the access structure have two elements, hence from now on we will use a graph G=(V,E) instead of an 𝒜.

The size of a discrete random variable ξ is measured by it's Shannon-entropy, defined as H(ξ)=-∑_j p_jlog(p_j), where ξ has possible values x_j with probabilities p_j=P(ξ=x_j). The information ratio of a participant v in V is H(ξ_v)/H(ξ_s).


Let G=(V,E) be a graph. Then the information ratio of G is


  * σ(G)=inf_Smax_v∈ V H(ξ_v)/H(ξ_s) in the worst case;

  * σ(G)=inf_S ∑_v∈ VH(ξ_v)/H(ξ_s)|V| in the average case
 where every infimum is taken over all perfect secret sharing schemes S realizing G.


Every construction yields an upper bound for the information ratio, one notable example is the decomposition theorem of Stinson <cit.> yielding an upper bound derived from any covering of the graph. Let us mention, that this fundamental result has  more general possible interpretations though, here we present a simple consequence  of covering with stars only:


Let G be a graph with maximal degree δ. Then 

    σ(G)≤ (δ+1)/2 
    σ(G)≤ (δ+1)/2



Note that in the average case this bound can be sharp only for regular graphs.

Let S be a perfect secret sharing scheme based on graph G=(V,E) with shares ξ_v for v∈ V and secret ξ_s, and define the values

    f(A)=H(ξ_v:v∈ A)/H(ξ_s)

for each A⊆ V. Using the standard properties of the entropy function and the definition of the perfect secret sharing we get the following, so called Shannon-inequalities:


  (a) f(∅)=0, and in general f(A)≥ 0 (positivity);

  (b) if A⊆ B ⊆ V then f(A)≤ f(B) (monotonicity);

  (c) f(A) + f(B) ≥ f(A∩ B) + f(A∪ B) (submodularity).

  (d) if A⊆ B, A is an independent set and B is not, then
   f(A) + 1 ≤ f(B) (strong monotonicity);

  (e) if neither A nor B is independent but A∩ B is so, then
  f(A)+f(B) ≥ 1+f(A∩ B)+f(A∪ B) (strong submodularity).

Let f be any function satisfying these five type of inequalities. Then max_v∈ V f(v) is a lower bound for the information ratio in the worst case, and ∑_v∈ Vf(v)/|V| is a lower bound for the information ratio in the average case. This is the so-called entropy method. In some cases we shall use the rearrangement of the inequalities. 

For  simplicity we usually write AB instead of A∪ B for subsets of vertices, and a instead of {a} for vertices. 







§ RESULTS



As we noted above, the information ratio of any graph is at most (d+1)/2, where d is the maximal degree of the graph as a simple consequence of Stinson's decomposition theorem <cit.>. Within this section we construct infinite classes of graphs with this best achievable information ratio both in the worst and the average case. Additionally, these graphs are significantly smaller than the recently known examples satisfying this property. 




 §.§ Construction for the worst case


Let C_d^* be a graph with 2^d+1 vertices, built from a d-dimensional cube, 2^d independent vertices and a 1-factor between the vertices of the cube and the independent vertex set.






Let f be any function satisfying the Shannon-inequalities for C_d^*, and A,B,X⊂ V such that A⊂ X. We shall use the following notation.















    [[X,B,A]]=∑_b∈ Bf(bX)-∑_a∈ Af(X-a)



Split the vertices of C_d^* into two equal independent sets the chessboard like fashion: C_d^*=X_d∪ Y_d, |X_d|=|Y_d|=2^d, X_d∩ Y_d=∅, and X_d and Y_d are independent. 
Let C_d be the d-dimensional subcube of C_d^* and A_d=C_d∩ X_d and B_d=C_d∩ Y_d. Clearly A_d and B_d are equal, independent and disjoint sets and A_d∪ B_d=C_d.


       

    ∑_v∈ C_d f(v)≥ [[X_d,B_d,A_d]]+d· 2^d-1



First we check for d=1. C_d^* is a path of length 3, with vertices x,b,a,y respectively. The inequality in <ref> becomes 

    f(a)+f(b)≥ [[ax,b,a]]+1=f(abx)-f(x)+1.

Using the strong submodularity 

    f(ab)+f(ax)≥ f(a)+f(abx)+1
 and the submodularities 

    f(a)+f(b)≥ f(ab) 
    
    f(a)+f(x)≥ f(ax),

 we get:

    f(abx)-f(x)≤ f(ab)+f(ax)-f(a)-1-f(x)≤ f(a)+f(b)-1,

which is the statement of the lemma.

Now suppose that <ref> holds for the d dimensional case. C_d+1^* consist of two disjoint copy of C_d^* and the vertices of the d-dimensional cubes are connected with a perfect matching M. Split this two copies of C_d^* the chess-board like fashion to X_d∪ Y_d, and X_d'∪ Y_d' respectively such that edges of M are between X_d and Y_d', and X_d' and Y_d. A_d, B_d, A_d', B_d', A_d+1, and B_d+1 are define as above. Clearly A_d∪ A_d'=A_d+1, B_d∪ B_d'=B_d+1, X_d∪ X_d'=X_d+1 and Y_d∪ Y_d'=Y_d+1, and M is between A_d+1 and B_d+1. 



		 [scale=2]
		[line width= 1.2] (0:0) node (A) – (90:1) node (B);
		[line width= 1.2] (45:1) node (D) – (67.5:1.85) node (C);
		[line width= 1.2] (A) – (D);
		[line width= 1.2] (B) – (C);
		[line width= 1.2] (0:1) node (E) – (45:1.41) node (F);
		[line width= 1.2] (45:2.41) node (G) – (22.5:1.85) node (H);
		[line width= 1.2] (E) – (H);
		[line width= 1.2] (F) – (G);
		(A) – (180:1) node (A1);
		(B) – +(180:1) node (B1);
		(C) – +(180:1) node (C1);
		(D) – +(180:1) node (D1);
		(E) – +(0:1) node (E1);
		(F) – +(0:1) node (F1);
		(G) – +(0:1) node (G1);
		(H) – +(0:1) node (H1);
		
		[line width= 1.2, color=red, dash pattern=on 2pt off 2pt] (A) – (E);
		[line width= 1.2, color=red, dash pattern=on 2pt off 2pt] (B) – (F);
		[line width= 1.2, color=red, dash pattern=on 2pt off 2pt] (C) – (G);
		[line width= 1.2, color=red, dash pattern=on 2pt off 2pt] (D) – (H);
		[fill=white] (0:0) circle (1.5 pt);
		[fill=black] (90:1) circle (1.5 pt);
		[fill=white] (67.5:1.85) circle (1.5 pt);
		[fill=black] (45:1) circle (1.5 pt);
		[fill=black] (0:1) circle (1.5 pt);
		[fill=white] (45:1.41) circle (1.5 pt);
		[fill=black] (45:2.41) circle (1.5 pt);
		[fill=white] (22.5:1.85) circle (1.5 pt);
		
		[fill=black] (0:0)+(180:1) circle (1.5 pt);
		[fill=white] (90:1)+(180:1) circle (1.5 pt);
		[fill=black] (67.5:1.85)+(180:1) circle (1.5 pt);
		[fill=white] (45:1)+(180:1) circle (1.5 pt);
		[fill=white] (0:1)+(0:1) circle (1.5 pt);
		[fill=black] (45:1.41)+(0:1) circle (1.5 pt);
		[fill=white] (45:2.41)+(0:1) circle (1.5 pt);
		[fill=black] (22.5:1.85)+(0:1) circle (1.5 pt);
		
		(90:1.2) node (B2)b;
		(135:0.2) node (A2)a;
		(50:1.5) node (C2)a';
		




Let b∈ B_d arbitrary, and its pair in M is a'∈ A_d'. By submodularity 

    f(bX_d)-f(X_d) ≥ f(bX_dX_d'-a')-f(X_dX_d'-a')


Let a∈ A_d be arbitrary neighbor of b. Then bX_d' and abX_d'-a' are both qualified, but their intersection, bX_d'-a' is independent, hence by the strong submodularity. 

    f(bX_d')-f(bX_d'-a')≥ f(abX_d')-f(abX_d'-a')+1


Using the submodularity, we can write two additional inequalities:

    f(X_d)-f(X_d'-a')   ≥ f(bX_d')-f(bX_d'-a') 
    
    f(abX_d')-f(abX_d'-a')    ≥ f(bX_dX_d')-f(bX_dX_d'-a')


Adding the inequalities <ref>, <ref>, <ref>, <ref>, we get

    f(bX_d)-f(X_d)+f(X_d')-f(X_d'-a') ≥ f(bX_dX_d')-f(X_dX_d'-a')+1


Similarly if we choose arbitrary b'∈ B_d' and a∈ A_d is the pair of b' in M, then

    f(b'X_d')-f(X_d')+f(X_d)-f(X_d-a) ≥ f(b'X_dX_d')-f(X_dX_d'-a)+1


All edges between B_d and A_d', and B_d' and A_d yield an inequality as <ref> and <ref> respectively. Adding up all these 2^d inequalities, on the left hand side all f(X_d) and f(X_d') cancel out the remaining is

    ∑_b∈ B_d f(bX_d)-∑_a'∈ A'_d f(X_d'-a')+   ∑_b'∈ B'_d f(b'X_d')-∑_a∈ A_d f(X_d-a)
    ≥
    ∑_b∈ B_d f(bX_dX_d')-∑_a'∈ A'_d f(X_dX_d'-a')+   ∑_b'∈ B'_d f(b'X_dX_d')-∑_a∈ A_d f(X_dX_d'-a)+2^d.


Using the facts that X_d ∪ X_d'=X_d+1, A_d∪ A_d'=A_d+1, and B_d∪ B_d'=B_d+1 this can be written as

    [[X_d,B_d,A_d]]+[[X_d',B_d',A_d']]≥[[X_d+1,B_d+1,A_d+1]]+2^d.


Applying the inductive hypothesis we get 


    [[X_d,B_d,A_d]]+[[X_d',B_d',A_d']]    ≤
    ∑_v∈ C_d f(v)-d· 2^d-1  +∑_v∈ C_d' f(v) -d· 2^d-1   =∑_v∈ C_d+1 f(v) -d· 2^d



This inequality and  <ref> together yield the statement of the lemma. 

 

    [[X_d,B_d.A_d]]≥ 2^d


C_d is a regular bipartite graph with parts A_d and B_d hence there is a perfect matching between A_d and B_d. Let (a,b) be a pair in the perfect matching, a∈ A_d, b∈ B_d, and let y be the leaf neighbor of a. X_d and X_d-a are independent, but bX_d and yX_d are not, thus by the strong monotonicity and the submodularity:

    f(bX_d)-f(X_d)≥ 1 
    
    f(yX_d)-f(yX_d-a)≥ 1 
    
    f(yX_d-a)+f(X_d)-f(X_d-a)-f(yX_d) ≥ 0


The sum of this three inequality gives

    f(bX_d)-f(X_d-a)≥ 2


 Adding up <ref> for all the 2^d-1 edges in the perfect matching between A_d and B_d gives the statement of the lemma 

 

    ∑_v∈ C_d f(v)≥ (d+2)2^d-1



This is just an easy corollary of lemma <ref> and lemma <ref>.


 
 The information ratio of C_δ-1^* in the worst case is 

    σ(C_δ-1^*)=δ+1/2




The maximal degree of C_δ-1^* is δ, hence we get σ(C_δ-1^*)≤δ+1/2 from the Stinson-bound <ref>.


On the other hand, C_δ-1 has 2^δ-1 vertices, hence by Lemma <ref> for at least one vertex 
    f(v)≥(δ+1)2^δ-2/2^δ-1=δ+1/2
 
holds, which completes the proof. 



 §.§ Construction for the average case


We construct the graph class Δ_d as follows. Let C_d^(0), C_d^(1) and C_d^(2) be three disjoint d-dimensional cubes. The vertices of each cube can divide into two independent sets of vertices the chessboard-like fashion, C_d^(i)=A_d^(i)∪ B_d^(i). Now we get D_d by adding arbitrary 1-factors between A_d^(i) and B_d^(i+1) for i=1,2,3. The family Δ_d consists of all graphs D_d constructed by this method, see the following example of a D_3∈Δ_3:



[scale=1.2]
		at (0,0) (A);
		at (1,0) (B);
		at (1,1) (C);
		at (0,1) (D);
		[line width= 1.2] (A) – (B);
		[line width= 1.2] (C) – (D);
		[line width= 1.2] (A) – (D);
		[line width= 1.2] (B) – (C);
		at (0.7,0.7) (E);
		at (1.7,0.7) (F);
		at (1.7,1.7) (G);
		at (0.7,1.7) (H);		
		[line width= 1.2] (E) – (F);
		[line width= 1.2] (H) – (G);
		[line width= 1.2] (E) – (H);
		[line width= 1.2] (F) – (G);
		[line width= 1.2] (A) – (E);
		[line width= 1.2] (B) – (F);
		[line width= 1.2] (C) – (G);
		[line width= 1.2] (D) – (H);

		
		at (3,2) (A2);
		at (4,2) (B2);
		at (4,3) (C2);
		at (3,3) (D2);
		[line width= 1.2] (A2) – (B2);
		[line width= 1.2] (D2) – (C2);
		[line width= 1.2] (A2) – (D2);
		[line width= 1.2] (B2) – (C2);
		at (3.7,2.7) (E2);
		at (4.7,2.7) (F2);
		at (4.7,3.7) (G2);
		at (3.7,3.7) (H2); 
		[line width= 1.2] (E2) – (F2);
		[line width= 1.2] (H2) – (G2);
		[line width= 1.2] (E2) – (H2);
		[line width= 1.2] (F2) – (G2);
		
		[line width= 1.2] (A2) – (E2);
		[line width= 1.2] (B2) – (F2);
		[line width= 1.2] (C2) – (G2);
		[line width= 1.2] (D2) – (H2);
		
		at (-3,3) (A3);
		at (-2,3) (B3);
		at (-2,4) (C3);
		at (-3,4) (D3);
		[line width= 1.2] (A3) – (B3);
		[line width= 1.2] (D3) – (C3);
		[line width= 1.2] (A3) – (D3);
		[line width= 1.2] (B3) – (C3);
		
		at (-2.3,3.7) (E3);
		at (-1.3,3.7) (F3);
		at (-1.3,4.7) (G3);
		at (-2.3,4.7) (H3);
		[line width= 1.2] (E3) – (F3);
		[line width= 1.2] (H3) – (G3);
		[line width= 1.2] (E3) – (H3);
		[line width= 1.2] (F3) – (G3);
		
		[line width= 1.2] (A3) – (E3);
		[line width= 1.2] (B3) – (F3);
		[line width= 1.2] (C3) – (G3);
		[line width= 1.2] (D3) – (H3);


		[color=red](H) – (D2);
		[color=red](F) – (B2);
		[color=red](A) – (E2);
		[color=red](C) – (G2);
		
		[color=blue](A3) – (D);
		[color=blue](C3) – (B);
		[color=blue](F3) – (G);
		[color=blue](H3) – (E);
		
		[color=green](A2) – (B3);
		[color=green](C2) – (E3);
		[color=green](F2) – (D3);
		[color=green](H2) – (G3);
		
		[fill=white] (0,0) circle (2 pt);
		[fill=black] (1,0) circle (2 pt);
		[fill=white] (1,1) circle (2 pt);
		[fill=black] (0,1) circle (2 pt);
		[fill=black] (0.7,0.7) circle (2 pt);
		[fill=white] (1.7,0.7) circle (2 pt);
		[fill=black] (1.7,1.7) circle (2 pt);
		[fill=white] (0.7,1.7) circle (2 pt);
		
		[fill=white] (3,2) circle (2 pt);
		[fill=black] (4,2) circle (2 pt);
		[fill=white] (4,3) circle (2 pt);
		[fill=black] (3,3) circle (2 pt);
		[fill=black] (3.7,2.7) circle (2 pt);
		[fill=white] (4.7,2.7) circle (2 pt);
		[fill=black] (4.7,3.7) circle (2 pt);
		[fill=white] (3.7,3.7) circle (2 pt);		

		[fill=white] (-3,3) circle (2 pt);
		[fill=black] (-2,3) circle (2 pt);
		[fill=white] (-2,4) circle (2 pt);
		[fill=black] (-3,4) circle (2 pt);
		[fill=black] (-2.3,3.7) circle (2 pt);
		[fill=white] (-1.3,3.7) circle (2 pt);
		[fill=black] (-1.3,4.7) circle (2 pt);
		[fill=white] (-2.3,4.7) circle (2 pt);		






 
The average information ratio of every  graph  D_δ-1∈Δ_δ-1 is

    σ(D_δ-1)=δ+1/2



All vertices of any graphs D_δ-1 have a degree of δ, hence by Stinson-bound <ref> the information ratio is at most (δ+1)/2.

Restrict an arbitrary fixed graph D_δ-1∈Δ_δ-1 to H=C_δ-1^(1)∪ B_δ-1^(2)∪ A_δ-1^(3). It is easy to see, that H is isomorphic to C_δ-1^*. C_δ-1^(1) is a d-1 dimensional cube, and B_δ-1^(2)∪ A_δ-1^(3) is an independent set. Furthermore there is a perfect matching between A_δ-1^(1)and B_δ-1^(2), and between B_δ-1^(1) and A_δ-1^(3), hence there is a perfect matching between C_δ-1^(1) and B_δ-1^(2)∪ A_δ-1^(3). This shows that H is an induced subgraph of D_δ-1 isomorphic to C_δ-1^* hence we can apply lemma <ref> for H.


    ∑_v∈ C_δ-1^(1) f(v)≥ (δ+1)2^δ-2


Similarly, if we consider C_δ-1^(2)∪ B_δ-1^(3)∪ A_δ-1^(1) and C_δ-1^(3)∪ B_δ-1^(1)∪ A_δ-1^(2) we obtain

    ∑_v∈ C_δ-1^(2) f(v) ≥ (δ+1)2^δ-2
    ∑_v∈ C_δ-1^(3) f(v) ≥ (δ+1)2^δ-2

Combining <ref>, <ref>, <ref>, we get 

    ∑_v∈ D_δ-1 f(v)≥ 3(δ+1)2^δ-2.

D_δ-1 has 3· 2^δ-1 vertices, hence the information ratio in the average case is at least (δ+1)/2, which completes the proof. 





§ CONCLUSION


In this paper we present new families of graphs of maximal degree δ achieving the best possible information ratio value given by the Stinson bound (δ+1)/2. These graphs are asymptotically the smallest ones recently: the first graph class achieves the bound in the worst case on 2^δ vertices and the other graph class constructed for the average case has 3/2·2^δ vertices in contrast to the best known constructions on c·6^δ vertices.



§ ACKNOWLEDGEMENT


This research has been partially supported by  the European Union, co-financed by the European Social Fund (EFOP-3.6.2-16-2017-00013, Thematic Fundamental Research Collaborations Grounding Innovation in Informatics and Infocommunications). The authors thank the members of the Crypto Group of the Rényi Institute, and especially László Csirmaz and Gábor Tardos, the friutful comments and discussions.


9


blakley
G. R. Blakley, Safeguarding cryptographic keys, in: Proc. of the Nat.
  Comp. Conf., 48, (1979) pp. 313–317.

blundo_tightbounds C. Blundo, A. De Santis, R. De Simone, U. Vaccaro, Tight bounds on the information rate of
secret sharing schemes, Des., Codes and Crypt. 11 (2) (1997) pp. 107–110.

blundo_avr C. Blundo, A. De Santis, L. Gargano, U. Vaccaro, On the information rate of
secret sharing schemes, Theor. Comp. Sci. 154 (2) (1996) pp. 283–306.

blundo1
C. Blundo, A. De Santis, D. R. Stinson and U. Vaccaro, Graph decomposition and
  secret sharing schemes, J. of Crypt. 8 (1995), 39–64.
  
stinson6
E. F. Brickell and D. R. Stinson, Some improved bounds on the information rate
  of perfect secret sharing schemes, J. of Crypt. 5 (1992),
  153–166.
   
csirmaz_log
L. Csirmaz, Secret sharing schemes on graphs, Studia Math. Hung.
  44 (2007), 297–306. 

csirmaz_cube
Csirmaz, L.: Secret sharing on the d-dimensional cube. Des. Codes Cryptogr. 74, 719–729
(2015)

csl_girth
L. Csirmaz and P. Ligeti, Secret sharing on large girth graphs, submitted to  Crypt. and Comm. (2018)

csirmaztardos
L. Csirmaz and G. Tardos, Optimal information rate of secret sharing schemes on
  trees, IEEE Tr. on Inf. Theory 59 (2013), 2527–2630.


dijk6
M. van Dijk, On the information rate of perfect secret sharing schemes,
  Des., Codes and Crypt. 6 (1995), 143–160.  

finished_six
O. Farràs, T. Kaced, S. Martin, C. Padró, Improving the Linear Programming Technique in the Search for Lower Bounds in Secret Sharing. Cryptology ePrint Archive, Report 2017/919,2017; available at https://eprint.iacr.org/2017/919

fulu_tree
H.-L. Fu, H.-C. Lu, The exact values of the average information ratio for tree-based access structures of perfect secret sharing schemes, Des., Codes and Crypt. 73 (1) (2014), 37–46. 

fulu_unicycle
H.-L. Fu, H.-C. Lu, The optimal average information ratio of secret-sharing schemes for the access structures based on unicycle graphs and bipartite graphs, Disc. Appl. Math. 233  (2017), 131–142. 


jacksonfive
W. Jackson and K. M. Martin, Perfect secret sharing schemes on five
  participants, Des., Codes and Crypt. 9 (1996), 233–250.

shamir
A. Shamir, How to share a secret, Comm. of the ACM 22 (1979),
  612–613.

stinston_decconstr D. R. Stinson, Decomposition construction for secret sharing schemes, IEEE Trans. on Inf.
Theory, 40 (1) (1994) pp. 118–125.

sun_six
H.L. Sun and B.L. Chen, Weighted decomposition construction for perfect secret
  sharing schemes, Comp. and Math. with Appl. 43 (2002),
  877–887.






