國立暨南國際大學國立暨南國際大學National Chi Nan UniversityNational 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)
3
encryption
decryption
share
Introduction – (k, n)-threshold Secret Sharing
(k, n) = (2, 3)
4
decryption
encryption
Introduction – Progressive Visual Secret Sharing
5
Progressive visual secret sharing (PVSS)
Introduction – Naor and Shamir (1995)
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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.
Introduction – Naor and Shamir (1995)
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C0 : white pixel; C1 : black pixel (2, 4)
C0 = C1 =
1000
0100
0010
0001
0001
0001
0001
0001
OR
R1 and R2 0001 0011
0001 0111
0001 1111
R1, R2 and R3
R1, R2, R3 and R4
R1
R2
R3
R4
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
VC scheme :
n × n n × 4 n
Introduction – Fang et al. (2008)
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They construct a (k, n)-threshold secret sharing scheme in VC without expansion.
They use the “Hilbert-curve” method.
Preliminary
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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 , where the weight of a column vector means the sum of each entry in this column vector.
M(4, 2) =
110100
101010
011001
000111
m = C2 = 6 4
n
Preliminary
10
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.
A = , B = , [2A||B] =
0
0
0
0
0001
0010
0100
1000
000100
001000
010000
100000B2A
Preliminary
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Definition 3.Light transmission rate = #white pixel #all
pixel = 1 (#black pixel #all pixel).
The (k, n)-threshold Secret Sharing Scheme
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It must follow the two conditions : (C0, t) = (C1, t) for 1 t k.
(C0, t) (C1, t) for t k.
Algorithm
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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)
C0 =
C1 =
else
C0 =
C1 =
( 3) / 22 2 22 2 21
( , 2) || ( ,0) || ( , 2 1)kn k n t
k ttM n C M n C M n n t
( 3) / 23 2 3
3 2 31( 1) ( ,1) || ( , ) || ( , 2 )
kn n tk k tt
n k M n C M n n C M n n t
/ 2 22 3 2 32 3 2 31
( , 2) || ( ,0) || ( , ) || ( , 2 )kn k n n t
k k ttM n C M n C M n n C M n n t
/ 2 1 2 2
2 21( 1) ( ,1) || ( , 2 1)
k n tk tt
n k M n C M n n t
Algorithm
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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) ;
0001
0100
1000
C0 =
m
R1
R2
Rn…
Proof
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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.
Experimental Results
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Example: (4, 5) C0 : [M(5, 2) || 3 M(5, 0) || 2 M(5, 5)]
C1 : [2 M(5, 1) || M(5, 4)]
110000000001111
110000001110001
110000110010010
110001010100100
110001101001000
111100000100001
111010001000010
110110010000100
101110100001000
011111000010000
Experimental Results
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(4, 5)
Experimental Results
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(4, 6)
Experimental Results (5, 6)
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Conclusion
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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
Hilbert-curve approach,” 2008.
NS scheme[1] FLL scheme[2] Our scheme
contrast in (4, 5) 1/4261 1/4261 1/15
contrast in (4, 6) 1/4261 1/4261 1/24
contrast in (5, 6) 1/12820 1/12820 1/30
contrast in (6, 8) 1/152200 1/152200 1/128
Thanks for your listening
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