New visual secret sharing schemes
using probabilistic method
Ching-Nung Yang
Pattern Recognition Letters 25 , 2004
指導老師：李南逸
Speaker：黃資真
1
Outline


Introduction
ProbVSS scheme






A k-out-of-n ProbVSS scheme
A 2-out-of-2 ProbVSS scheme
A 2-out-of-n ProbVSS scheme
A k-out-of-k ProbVSS scheme
A general k-out-of-n ProbVSS scheme
Conclusion
2
Introduction

White pixels show the contrast of recovered
image.

The new scheme is non-expansible shadow.
3
ProbVSS scheme
n
k
C0
shadows
Get the shared secret by stacking their shadows.
n*1 Boolean matrices of sharing a white pixel.
C1 n*1 Boolean matrices of sharing a black pixel.
L(V) ‘OR-ed’ operation of this k-tuple column vector V.
λ
L( C0 ) values
γ
L( C1 ) values
α
contrast α>0
pTH The threshold probability 0≦ pTH≦1
p0
The appearance probability of white pixel in the write area.
p1
The appearance probability of white pixel in the black area.
Probabilistic scheme use the abbreviation ProbVSS scheme.
4
A k-out-of-n ProbVSS scheme

A (k,n) ProbVSS Scheme is conditions ：

n*1 matrices in the set C0 and
C1 , L(V) operation
values of all matrices form two sets λ and γ.


The two sets λ and γ satisfy that
p1 ≦ pTH -α.
p≧0
pand
TH
For any subset { i1 , i2 ,..., iq } of {1,2,…,n} with q<k, the p0
and p1 are the same.
5
A 2-out-of-2 ProbVSS scheme
i , j denotes the set of all n*1 column matrices.
ex：3*1
 2,0

 1   0  1  

 , 1  ,  0  
1
 
     
  0  1  1  
      
Construction：
C0  {0,0 ,2,0 }
Proof.
C0  {0,0 ,2,0 }={  00 ， 11  }
C1 ={1,1}
C1 ={1,1}={  01 ， 10  }
C0 and C1 is sets consisting
so
of 2*1 matrices.
  {L(  00  ) ，L(  11  )}={0,1} , p 0  0.5

Theorem：
pTH = 0.5

  {L(  01  ) ，L(  10  )}={1,1} , p1  0
 = 0.5
6
A 2-out-of-2 ProbVSS scheme

Proof of third condition “security”：


Shadow 1 λ={L([0]) , L([1])} = {0,1} ,
γ={L([0]) , L([1])} = {0,1} ,
Shadow 2 λ={L([0]) , L([1])} = {0,1},
γ={L([1]) , L([0])} = {1,0},
p0=0.5
p1 =0.5
p0 =0.5
p1 =0.5
7
A 2-out-of-n ProbVSS scheme

Construction 1：
C0  {0,0， n ,0 }
C1  {[ n / 2],1}　(even n)
C1  {[ n / 2],1，[ n / 2]1,1}
(odd n)
C0 and C1 is sets consisting of n*1 matrices

Theorem 1：
p TH  0.5
n

(even n)
4n  4
n+1

(odd n)
4n
8
A 2-out-of-n ProbVSS scheme

Proof： A (2,3) ProbVSS scheme
C0
C1
  0  1 


1 
 { 0,0
，3,0 }=  
0
，
   
  0  1 
    
 {1,1
， 2,1}
 0  0  1   1  0   1 


1 
0 
1 
1 
0 
= 
0
，
，
，
，
，
           
 1   0   0   0  1  1  
            



 0  


 L     , L 

 0  



1  
1   
  

0,1

 0  
 0  
 1   
 L     , L     , L     ,

 0  
 1  
 0  
 


1


1


0









L
,
L
,
L














 1 
 0  
 1   

 {0,1,1,1,1,1}
p 0 =1/2
p TH =0.5
p1 =1/6
 =1/3
9
A 2-out-of-n ProbVSS scheme

Proof： A (2,4) ProbVSS scheme
C0
C2
  0  1 
    
 0
1 
 { 0,0
， 4,0 }=   
，

  0  1 
   

  0  1 

 { 2,1}
 1   0  1   0   0  1  
            
 1
0
0
1 
1 
0 
=  
，
，
，
，
，

 0  1  1   0  1   0  
           

 0  1   0  1   0  1  


 0  
 1  



   


  
 L     , L       0,1
0
1



 1 
 0  
 1   
 L     , L     , L     ,

 1 
 0  
 0  
 

 0  
 1  
 0  

 L  1   , L   0   , L  1   
 
 
  

 {1,0,1,1,1,1}
p 0 =1/2
p TH =0.5
p1 =1/6
 =1/3
10
A 2-out-of-n ProbVSS scheme

Construction 2：
n-1
C0  {0,0，n ,0 ... n ,0 }
C1  {n -1,1}
C0 and C1 is sets consisting of n*1 matrices

Theorem 2：
pTH  1/ n
  1/ n
11
A 2-out-of-n ProbVSS scheme

Proof： A (2,3) ProbVSS scheme
  0  1 1 

，1，1 
C0  { 0,0
，3,0
，3,0 }=  
0
     
  0  1 1 
      
C1  { 2,1}
 1 0 1 

， 1 ， 0  
= 
1
     
  0  1  1  
      

 1 
 1  
  0 

   L     , L     , L       0,1,1

 1 
 1  
  0 



 1 
 1  
 0  


  
 {1,1,1}
 
   

  L     , L     , L    
1
0
1
p 0 =1/3
p TH =1/3
p1 =0
 =1/3
12
A k-out-of-k ProbVSS scheme

Construction 2：
C0  i ,0 , where i is even and 0  i  k
C1  i ,1 , where i is odd and 0  i  k
C0 and C1 is sets consisting of n*1 matrices

Theorem 2：
pTH =1/2k-1
 =1/2k-1
13
A k-out-of-k ProbVSS scheme

Proof： A (3,3) ProbVSS scheme
 0  1  1  0 


C0  {0,0， 2,0 }=   0 ， 1 ， 0 ， 1  
 0 0 1 1 
        
 1   0   0  1 


C1  {1,1
，3,1} =   0 ，1 ， 0 ，1 
  0   0   1  1 
        
  0 
 1 
 1 
 0  
 







   L   0  ，L  1  ，L   0  ，L  1     0,1,1,1
  0 
 0  
 1  
 1   











   
 
  1  
 0 
 0 
 1  
 







   L   0  ，L  1  ，L   0  ，L  1    {1,1,1,1}
  0 
 0 
 1  
 1  
 
 
   
  
p 0 =1/4
p TH =1/4
p1 =0
 =1/4
14
A general k-out-of-n ProbVSS scheme
h
l
m
B0
B1
T(.)
The ‘whiteness’ of white pixel
The ‘whiteness’ of black pixel
shadow size
n*m Boolean matrices of sharing a white pixel.
n*m Boolean matrices of sharing a black pixel.
T(.) is transferred to a set of ‘m’ n*1 column
matrices.
15
A general k-out-of-n ProbVSS scheme

Construction :
C0  T ( B0 )
C1  T ( B1 )

Theorem：
h
p TH 
m
hl

m
16
A general k-out-of-n ProbVSS scheme

Proof： A Shamir’s (3,4) VSS scheme with white and black matrices
0
0
(1) B0  
0

0
0 1 1 1 0
0 1 1 0 1 
0 1 0 1 1

0 0 1 1 1
1
1
B1  
1

1
1 0 0 0 1
1 0 0 1 0 
1 0 1 0 0

1 1 0 0 0
  0 
 0 
 1  
 0   0  1  1  1   0  





 , L  0  , L  1  , 
            
L
0

0
0
1
1
0
1
 
  
 





 1  
(2) C0  T ( B0 )     ,   ,   ,   ,   ,   
 

0

  
  0














0
0
1
0
1
1

(3)   
  1,1, 0,1,1,1













1


1


0










 0  0   0  1  1  1  

 L  1   , L   0   , L  1   
 
  
  
 1 1  0  0  0  1  




 1   




0
1

 
 


  

            
0
1
0 
 1 1 0
  1 
 1 
 0  
C1  T ( B1 )     ,   ,   ,   ,   ,   

 
 
  
L
1
,
L
1
,
L
 1 1  0  1   0   0  
  
 
 0  ,













 1 
 0  
 


 
  
  1


 1 1 1   0   0   0  

  1,1, 0,1,1,1

0


0


1









 L   0   , L  1   , L  0   
 
   
  




 0   





 0  
   
  1  
 
(4) p0  1/ 3
p1  1/ 6
pTH  1/ 3  =1/6
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Conclusion

New (k,n) ProbVSS schemes with non-expansible
shadow size based on the probabilistic method.

The conventional VSS scheme can be transferred
to ProbVSS scheme.

The ProbVSS scheme is a different view of the
conventional VSS scheme.
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