國立暨南國際大學
National Chi Nan University
A Study of (k, n)-threshold Secret Image
Sharing Schemes in Visual Cryptography
without Expansion
Presenter： Ying-Yu Chen
Authors: Ying-Yu Chen, Justie Su-Tzu Juan
Department of Computer Science and Information Engineering
National Chi Nan University
Puli, Nantou Hsien, Taiwan
Outline

Introduction

Preliminary

The (k, n)-threshold Secret Sharing Scheme

Experimental Results

Conclusion
2
Introduction –
Visual Cryptography

Visual cryptography (VC)
encryption
share
decryption
3
Introduction –
(k, n)-threshold Secret Sharing

(k, n) = (2, 3)
decryption
encryption
4
Introduction –
Progressive Visual Secret Sharing

Progressive visual secret sharing (PVSS)
5
Introduction –
Naor and Shamir (1995)
They construct a (k, n)-threshold secret
sharing scheme in VC with expansion.

 : The relative difference in weight between white pixel
and black pixel of stacking k shares. If contrast  is larger, it
represents the image is clearer to visible.
6
n×n
n×4n
Introduction –
VC scheme :
Naor and Shamir (1995)

1000
C0 : white pixel; C1 : black pixel

(2, 4)
C0 =
1

1

1

1
0
0
0
0
0
0
0
0
0

0

0

0
C1 =
1

0

0

0
R1
0
0
1
0
0
1
0
0
0

0

0

1
OR
0100
R2
0010
R3
0001
R4
R1 and R2 1
0
0
0
1
1
0
0
R1, R2 and R3 1
0
0
0
1
1
1
0
R1, R2, R3 and R4 1
0
0
0
1
1
1
1
7
Introduction –
Fang et al. (2008)
They construct a (k, n)-threshold secret
sharing scheme in VC without expansion.

They use the “Hilbert-curve” method.
8
Preliminary

Definition 1.
An n  m 0-1 matrix M(n, j) is called totally symmetric if
each column has the same weight, say j, and m equals to Cj ,
n
where the weight of a column vector means the sum of each
entry in this column vector.
M(4, 2) =
1

1

0

0
1
1
0
0
0
0
1
1
1
0
1
0
0
1
0
1
m = C24 = 6
0

0

1

1
9
Preliminary

Definition 2.
Given an n  m1 matrix A and an n  m2 matrix B, we define
1. [A||B] be an n  (m1 + m2) matrix that obtained by
concatenating A and B;
2. [a  A||b  B] be an n  (a  m1+ b  m2) matrix that be
obtained by concatenating A for a times and B for b times.
B
2A
A＝
0 
 
0
 
, B
0 
 
0 
＝
0

0

0

1
0
0
1
0
1

1 0

,
[2A||B]
0 0

0 0
0
＝
0

0

0

0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
0
0
1

0

0

0
10
Preliminary

Definition 3.
Light transmission rate  = #white pixel  #all pixel
= 1  (#black pixel  #all pixel).
11
The (k, n)-threshold Secret
Sharing Scheme

It must follow the two conditions :

(C0, t) = (C1, t) for 1  t  k.

(C0, t)  (C1, t) for t  k.
12
Algorithm


Input : A binary secret S with size w  h and the value of n and k.
Output : n shares R1, R2, …, Rn, each with size w  h.
1. if (k mod 2 == 1)
C
M ( n , n  2 t  1) 
C0 =  M ( n , 2) || C M ( n , 0) || 

C
M (n, n  2t ) 
C1 =  ( n  k  1) M ( n ,1) || C M ( n , n ) || 

else
M ( n, n  2t ) 
C0 =  M ( n , 2) || C M ( n , 0) || C M ( n , n ) ||  C

M ( n , n  2 t  1) 
C1 =  ( n  k  1) M ( n ,1) ||  C

( k 3) / 2
nk 2
2
t 1
( k 3) / 2
n3
k 3
nk 2
2
t 1
n3
k 3
k / 2 1
t 1
n2t 2
k 2t2
n2t3
k 2t 3
k / 22
t 1
n2t3
k  2t 3
n2t 2
k 2t2
13
Algorithm
2. for (1  i  h; 1  j  w)
x = random(1…m)
for (1  t  n)
if ( S(i, j) == 0 )
Rt(i, j) = C0(t, x) ;
else
Rt(i, j) = C1(t, x) ;
0
0
0
1

0
0
m
1

0



0
R1
R2
…
C0 =
0

0



1
Rn
14
Proof
Theorem 1.
In the proposed scheme, if we stack at least k
shares, the secret can be revealed; and if we stack
the number of share less than k, the secret cannot
be revealed.
Proof
(C0, t) = (C1, t) for 1  t  k.
(C0, t)  (C1, t) for t  k.
15
Experimental Results

Example: (4, 5)

C0 : [M(5, 2) || 3  M(5, 0) || 2  M(5, 5)]
0

0

0

1
1


0
0
1
0
0
1
0
1
1
0
0
0
1
0
1
0
0
1
0
1
0
1
0
0
0
1
1
0
0
1
0
0
1
1
0
0
0
0
1
0
0
0
1
1
1
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
1
1

1

1

1
1
C1 : [2  M(5, 1) || M(5, 4)]
0

0

0

0
1

0
0
0
1
0
0
0
0
1
1
1
1
1
0
0
1
0
0
0
0
1
0
1
1
1
0
0
1
0
0
0
0
1
0
0
1
1
0
1
1
0
0
0
0
1
0
0
0
1
0
1
1
0
0
0
0
1
0
0
0
0
0
1
1
1
0

1

1

1
1 
16
Experimental Results

(4, 5)
17
Experimental Results

(4, 6)
18
Experimental Results

(5, 6)
19
Conclusion
contrast  in (4, 5)
contrast  in (4, 6)
contrast  in (5, 6)
contrast  in (6, 8)


NS scheme[1] FLL scheme[2] Our scheme
1/15
 1/4261
 1/4261
1/24
 1/4261
 1/4261
1/30
 1/12820
 1/12820
1/128
 1/152200
 1/152200
There is no expansion in our scheme.
With larger contrast  we proposed, the stacked image is
clearer.
[1] M. Naor and A. Shamir, “Visual cryptography,” 1995.
[2] W.-P. Fang, S.-J. Lin, and J.-C. Li, “Visual cryptography (VC) with non-expanded shadow images: a
20
Hilbert-curve approach,” 2008.
Thanks for your listening
21