1. Cheating and Prevention in Visual Secret Sharing
陳昱圻 博士
中央研究院
資訊科學研究所

2. Outline Secret Sharing Visual Secret Sharing (VSS)
Cheating Problem in VSS
How to Prevent Cheating Attacks
Cheating Immune VSS Schemes
Conclusions
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3. Treasure Map  Secret Sharing
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4. Treasure Map  Secret Sharing
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5. Secret Sharing Secret is the treasure or the map.
How to share the secret?
Three pieces of the treasure map.
How to get the secret?
Only three men together. Any one cannot.
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6. How to Share a Secret
Secret sharing: distributing a secret among a group of participants.
A dealer (a trusted party) allocates each of participants a share of the secret.
The secret can only be recovered when the shares are combined together.
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7. Threshold Secret Sharing: 2-out-of-3 SS
Dealer
secret
(2, 3)-secret sharing
Captain America
Iron Man
Hulk
shares
secret retrieval
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8. Share Generation (2-out-of-3)
Secret: K=3
[Shamir’s method]
It is super easy to know
(1, 5), (2, 7)  f(x)=2x+3
(1, 5), (3, 9)  f(x)=2x+3
(2, 7), (3, 9)  f(x)=2x+3
K=3
Pick a=2
Generate f(x)=ax+K
f(x)=2x+3
(1, 5)
(2, 7)
(3, 9)
Get three points (1, 5), (2, 7), (3, 9)
a=2, K=3
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9. Share Generation (2-out-of-3)y
S3=(3,9)
(0,K=3)
S2=(2,7)
S1=(1,5) x
Any two points can decide the line.
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10. How about k-out-of-n 1. Pick random ak-1, ak-2, …, a1
2. f(x) = ak-1xk-1 + ak-2xk-2 + … a1x + K
3. Generate (x1,y1), (x2,y2), …, (xn, yn)
4. Distribute (xi,yi) to User i
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11. Secret Recovery 1. Pick random ak-1, ak-2, …, a1
2. f(x) = ak-1xk-1 + ak-2xk-2 + … a1x + K
3. Generate (x1,y1), (x2,y2), …, (xn, yn)
k users have (x1,y1), … (xk,yk)
4. Distribute (xi,yi) to User i
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12. Security Solve ak-1, …, a1, K We must have k following equations
(x1, y1) y1 = ak-1(x1)k-1 + … a1(x1) + K …
(xk, yk) yk = ak-1(xk)k-1 + … a1(xk) + K
Any k-1 users cannot compute ak-1, …, a1, K
Only k-1 equations.
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13. Outline Secret Sharing Visual Secret Sharing (VSS)
Cheating Problem in VSS
How to Prevent Cheating Attacks
Cheating Immune VSS Schemes
Conclusions
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14. Secret Sharing without …
Visual Secret Sharing is secret sharing.
No computation in share recovery
We do not use any computer.
Human visual system
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15. Treasure Map  VSS
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16. Visual Secret Sharing k-out-of-n VSS Secret: a secret image
Total n participants
Any k can get the secret
Secret: a secret image
Share generation, run by the dealer
Share recovery, run by k participants
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17. Result of 2-out-of-3 VSS A secret retrieval: stacking secret image
(2, 3)-visual secret sharing
Transparencies
(shares/shadows)
secret retrieval: stacking
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18. Share Generation A Input: a secret image
Deal with black and white pixels separately
A pixel will be extended to some subpixels
secret image
Ex:
2-out-of-3
1 pixel
3 subpixels
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19. Share Generation Use base matrices Encode white pixels by using CW
Encode black pixels by using CB
Column permutation P1 P2 P3
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20. Share Generation (2,3)-VSS
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21. A
secret image T1 T2 T3
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22. Share Recovery Stack the shares (OR operation) 1 T1 T1 T2 T1+T2 T31 T1 T1 T2
T1+T2 T3
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23. Share Recovery 1 1 1 1 1 1 T1 T1+T2 1 1 T2 T2+T3 1 1 1 1 T3 T1+T3 OR
T1+T2 1 T2 1 1
T2+T3 T3 1
T1+T3 1 OR T1 1
T1+T2 1 T2 1 1
T2+T3 T3 1
T1+T3 1
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24. Review A secret retrieval: stacking secret image
(2, 3)-visual secret sharing
Transparencies
(shares/shadows)
secret retrieval: stacking
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25. (2, n) visual secret sharing
Generally, we can use the following base matrices to construct a (2,n)-VSS. n n
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26. Outline Secret Sharing Visual Secret Sharing (VSS)
Cheating Problem in VSS
How to Prevent Cheating Attacks
Cheating Immune VSS Schemes
Conclusions
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27. Cheating Dishonest collusive parties want to fool an honest one
Assume n-1 cheaters and one victim.
Probability = 1
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28. Cheating
We want to fool Hulk.
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29. How to do for white pixel
Hulk’s is [1 0 0].
[1 0 0]
[1 0 0]
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30. How to do for black pixel
Hulk’s is [1 0 0].
[0 0 1]
[0 1 0]
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31. The original pixel is black
We want to fool Hulk.
Ex:
The original pixel is black
[0 1 0]
[0 0 1]
[1 0 0]
[1 0 0]
[1 0 0] + [1 0 0] = [1 0 0] White
Generate a fake transparency collusively.
Give FT to Hulk
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32. Haha.. The real secret is A.
Oh! The secret is F.
Haha.. The real secret is A.
FT+T3
T1+T2
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33. Result of Cheating in (2,3)-VSS
T1+T2
T1+T3
T2+T3 FT
FT+T3
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34. Outline Secret Sharing Visual Secret Sharing (VSS)
Cheating Problem in VSS
How to Prevent Cheating Attacks
Cheating Immune VSS Schemes
Conclusions
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35. Two directions to prevent
The base matrices can inherently prevent cheating.
The victim can verify that the share is valid or not.
Share and blind authentication
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36. Outline Secret Sharing Visual Secret Sharing (VSS)
Cheating Problem in VSS
How to Prevent Cheating Attacks
Cheating Immune VSS Schemes
Conclusions
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37. Horng et al.’s scheme 1 Verification logo LC Stacking VC and SA, LC is
shown on the left-top corner
Stacking VC and SB, LC is
shown on the right-top corner
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38. Logo Verification
Insecure region
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39. Horng et al.’s scheme 2
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40. 2-out-of-(n+1)-VSS T1 T1 T2 T2 T3 T3
WTH, T3 could be [ ] or [ ] T1 T2
Pr = ½
Black pixels are secure
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41. 2-out-of-(n+1)-VSS T1 T1 T2 T2 T3 T3 Haha…, we know T3 is [1 0 0 0] T1
Pr = 1
White pixels are insecure
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42. Set a verification image
Hu-Tzeng’s scheme B
Set a verification image
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43. Hu-Tzeng’s scheme Shares generating (base matrices): CW=
CB=
Extra subpixels (verifying).
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44. Hu-Tzeng’s scheme Verification shares generating:
If the pixel in VI is black, the sibpixels are [ ]. (which corresponds to the permutation of basic matrices)
If the pixel in VI is white, the sibpixels are [ ].
Extra subpixels.
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45. Result Example: T1[1 0 1 0 0] (white) T1, T2, T3 T2[1 0 0 1 0] (white)
T1+T2, T1+T3, T2+T3
T1+T2[ ] (black)
V1, V2, V3
V2[ ] (white +)
V1+T2, V1+T3, V2+T1
T1+V2[ ] (black)
V2+T3, V3+T1, V3+T2
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46. Problem of Hu-Tzeng’s scheme
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47. Problem of Hu-Tzeng’s scheme
Black or white pixel is insecure in Hu-Tzeng’s scheme.
V1 = [ ]
T1 = [ ]
Added columns
Original columns
T3 = [ ]
V2 = [ ]
T2 = [ ]
T1+T2 = [ ] is Black
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48. Improvement (Chen et al.)
Shares generating (base matrices):
CW=
CB=
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49. Share Authentication
Horng et al. 2006: Verification logo and verification shares  Insecure
Hu-Tzeng 2007: Added columns and verification shares  Insecure
Chen et al. 2012: Added columns and verification shares  Secure
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50. Share Authentication Horng et al. 2006: Insecure
Hu-Tzeng 2007: Insecure
Chen et al. 2012: Secure
Verification patterns with verification shares
Verification logos without verification shares
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51. Verification patterns with verification shares
T1 and V1
T2 and V2
T3 and V3
Number of patterns: n1
Number of patterns: n2
n1, n2 & n3
Number of patterns: n3
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52. This figure is from J. Vis. Commun. Image R. 23 (2012) page 1229.
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53. Verification logos without verification shares
Logo: L1
Logo: L2
L1, L2 & L3
Logo: L3
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54. Ex: if T1+T? and no two D, we will find T? is FT.
L1 = D, L2 = A, L3 = B, Secret = CC
Ex: if T1+T? and no two D, we will find T? is FT.
This figure is from Digital Signal Processing 23 (2013)page 1501.
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55. Comparisons Horng et al. (Verification logos): Insecure
Hu-Tzeng (Added columns): Insecure
Chen et al. (Improvement from HT): Secure
Chen et al. (Verification patterns): Secure
Chen et al. (Verification logos): Secure
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56. Outline Secret Sharing Visual Secret Sharing (VSS)
Cheating Problem in VSS
How to Prevent Cheating Attacks
Cheating Immune VSS Schemes
Conclusions
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57. Conclusions Cheating is a security issue in visual secret sharing.
Two notions are presented to prevent cheating.
Some cheating immune schemes are introduced.
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58. Thanks for your attention
Yu-Chi Chen
Institute of Information Science
Academia Sinica
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