Cryptography Reference
In-Depth Information
Assume there exists a (
Qual
;
Forb
)-VCS with pixel expansion m. Let () be a one-bit
cyclical right shift function and let (h`) be odd.
Distributionphase.For each pixel of the secret image, the dealer:
runs the distribution phase of the underlying (
Qual
;
Forb
)-VCS; let s1i
1
i
be the share
distributed to participant i, for i = 1;:::;n;
for each ` = 2;:::;m, compute s
`
i
= (s
`
i
);
distributes the m-tuple (s1i
1
i
;:::;s
m
i
) to participant i.
Reconstructionphase.A qualified set fi
1
;:::;i
p
g of participants reconstructs the
secret pixel as follows:
superimpose their shares to get
`
= OR(s
`
i
1
;:::;s
`
i
p
), for ` = 1;:::;m;
compute = XOR(
1
;:::;
m
), wh
ic
h is the reconstructed pixel if (mh) is even,
otherwise the reconstructed pixel is .
FIGURE 9.6
Ideal contrast VCS with reversing starting from any VCS.
The security of the scheme directly follows from the security of the under-
lying VCS.
The reconstruction phase requires a qualified set of p participants to per-
form exactly 4(m 1) reversing operations and p 1 + 3(m 1) stacking
operations, since as seen in
Section 9.4.1
the XOR operation can be imple-
mented by means of 3 ORs and 4 NOTs operations. Finally, notice that in the
construction of
Figure 9.5
there is a loss of resolution, since each pixel in the
original image corresponds to m subpixels in the reconstructed image, where
m denotes the pixel expansion of the underlying VCS.
Example 8 Consider a (2; 3)-threshold VCS, which is not a perfect black,
where mh is odd. Assume the basis matrices S
0
and S
1
are:
2
3
2
3
1
0
0
1
0
0
S
0
=
4
5
S
1
=
4
5
:
1
0
0
0
1
0
1
0
0
0
0
1
The collections C
0
and C
1
are obtained by permuting the columns of the corre-
sponding basis matrix (S
0
for C
0
, and S
1
for C
1
) in all possible ways. Let S
0
be the matrix chosen by the dealer to share a white pixel. The shares generated
for participant 1 are s
1
= 100, s
1
= 010 and s
1
= 001, whereas, the shares
generated for the participant 2 are s
2
= 100, s
2
= 010, and s
2
= 001. During
the reconstruction phase, the two participants stack their shares and retrieve
1
= OR(s
1
;s
2
) = 100,
2
= OR(s
1
;s
2
) = 010, and
3
= O
R
(s
1
;s
2
) = 001.
By computing = XOR(
1
;
2
;
3
) = 111 and its reverse = 000 the two
participants reconstruct a white pixel.
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