New visual secret sharing schemes using probabilistic method

1
New visual secret sharing schemes using
probabilistic method

• Author Ching-Nung Yang
• Source Pattern Recognition Letters, Vol.25,
  March 2004, pp. 481- 494
• Speaker Yu-Cheng Wang
• Date 2004/02/19

2
Outline

• Introduction
• The basic VSS scheme (Moni Naor and Adi Shamirs
  method)
• The proposed VSS scheme
• Conclusions

3
Introduction
Shared secret
(k,n) visual secret sharing scheme
n3,k2
recovered image
shadow image
Bob
shadow1
Mary
shadow2
Tom
shadow3
4
The basic VSS scheme(Naor and Shamir)

• Example (3,3) visual secret sharing

Original image
shadow 1
shadow 2
shadow 3
5
The proposed VSS scheme

• Non-expansible shadow size
• Probabilistic method
• p0 the appearance probability of white pixel in
  the white area of the recovered image.
• p1 the appearance probability of white pixel in
  the black area of the recovered image.

6
The proposed VSS scheme
Shared secret
Recovered image
p0 8/16 1/2
p1 0/16 0
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The proposed VSS scheme (Define)

• A ProbVSS Scheme is considered valid if the
  following conditions are met
• For these n?(resp. n?) matrices in the set
  C0(resp. C1), L(V) is the OR-ed value of any
  k-tuple column vector V. These values of all
  matrices form a set ? (resp. ?).
• The two sets ? and ? satisfy that p0 gt pTH and
  p1lt pTH a, where p0 and p1 are the appearance
  probabilities of the 0(white color) in the set
  ? and ?, respectively.
• For any subset i1,i2,,iq of 1,2,,n with qlt
  k, the p0 and p1 are the same.

• C0 white set consisting of n? n1 matrices
• C1 black set consisting of n? n1 matrices

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The proposed VSS scheme (Example n3, k3)
white set
black set
p0 1/4
p1 0
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A (2,2) ProbVSS scheme (Theorem)

• Construction
• Let , and .
  Then, C0 and C1 are the white and black sets
  consisting of 2x1 matrices for a (2,2) ProbVSS
  scheme.
• Theorem
• The scheme from Construction is a (2,2) ProbVSS
  scheme with non-expansible shadow size and the
  parameters threshold probability
  and the contrast .

• µ i,j the set of all nx1 column matrices.
• i the Hamming weight of every column vector
• j denotes the matrices belonging to Cj .
• Example n2, µ 1,1 are two 2x1 column matrices
• and belongs to C1.

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A (2,2) ProbVSS scheme (Proof.)

• Since the two sets
• So
• when stacking two shadows. The appearance
  probabilities of white
• color in ? and ? are p00.5 and p10, the
  threshold probability
• pTH0.5 and the contrast a 0.5.
• Security.

a p0-p1 p0 gt pTH and p1lt pTH a
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A (2,2) ProbVSS scheme(Example)
recovered image shadow1 shadow2
secret image
shadow1
shadow2
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A (2,n) ProbVSS scheme

• Method 1
• the parameters threshold probability pTH 0.5 and
  the contrast

,if n is even
,if n is odd
,if n is even
,if n is odd
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A (2,n) ProbVSS scheme Method 1 (Example a
(2,3) ProbVSS scheme)
pTH0.5 , a 1/3
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A (2,n) ProbVSS scheme

• Method 2
• the parameters threshold probability pTH 1/n
  and the contrast

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A (2,n) ProbVSS scheme Method 2 (Example a
(2,3) ProbVSS scheme)
pTH1/3 , a 1/3
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A (k,k) ProbVSS scheme

• C0 µi,0, where i is even and 0 lt ilt k
• C1 µi,1, where i is odd and 0 lt ilt k
• gt
• the parameters threshold probability pTH
  1/2k-1 and the contrast a 1/2k-1.

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A (k,k) ProbVSS scheme(Example a (3,3) ProbVSS
scheme)
pTH1/4 , a 1/4
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A general (k,n) ProbVSS scheme

• Let B0 and B1 be the two n x m white and black
  matrices, respectively, as defined in the
  conventional (k,n) VSS scheme.
• m shadow size
• The Hamming weight of OR-ed V of any k of the n
  rows in white(resp. black) matrix is m-h (resp.
  m-l ) and h gtl .
• C0 T(B0) and C1 T(B1)
• gt
• The parameters threshold probability pTHh /m and
  contrast a (h -l )/ m

Transfer operation T() Let Bbij be an n x m
Boolean matrix, where 1ltiltn and 1ltjltm. Then
T(B) is transferred to a set of m n x 1 column
matrices
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A general (k,n) ProbVSS scheme (Example a (3,4)
ProbVSS scheme)
A (3,4) ProbVSS scheme
A shamirs (3,4) VSS scheme
transfer
pTH1/3 , a 1/6
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Recognition of small areas in the secret image
for the ProbVSS scheme
EX.
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Recognition of small areas in the secret image
for the ProbVSS scheme
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Conclusions

• 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 by using Transfer operation T().