Cryptography Reference
In-Depth Information
high-end applications. Two TiOISSSs [6, 9] have been proposed. Both schemes
can stack shadows to decode a halftone secret image by the HVS in the first
stage, and then can perfectly reconstruct the gray-level secret image in the
second stage. A possible application scenario of TiOISSS is described below.
In a distributed multimedia system, n shadows of PISSS can be delivered in
a distributed system where each shadow is stored in any distributed storage
node. The failure of (nk) shadows during transmission does not aect the
reconstruction phase, as the secret image can be perfectly restored using k
shadows. Suppose a fake shadow is received. The receiver spends consider-
able computation and finally finds the received shadow is wrong. Because the
reconstruction phase of PISSS is very computationally intensive, we can ap-
ply the TiOISSS to save the computational time for verifying the validity of
shadows. The receiver can first verify the shadows by visually previewing the
secret without computation. After the successful verification, the receiver then
recovers the original gray-level secret image by computation. In this chapter,
we will briefly describe two TiOISSSs [6, 9], and also introduce our recent
research result on TiOISSS published in [20].
17.2 Preliminaries
17.2.1 PISSS
A polynomial-based (k, n) secret sharing scheme was first proposed by Shamir
[12], in which the secret data is encrypted into n shadows. Any k shadows can
be used to reconstruct the secret, but any k 1 or fewer shadows learn no
information. By taking the secret data as g
0
(constant term) in the following
(k1)-degree polynomial g(x) where p is a prime number, we could construct
n shadows (x
i
;g(x
i
)) by choosing n different xi,
i
, i 2 [1;n].
g(x) = (g
0
+ g
1
x + ::: + g
k1
x
k1
) mod p:
(17.1)
Any k shadows (without loss of generality, we use k shadows (xi,
i
;g(x
i
));i 2
[1;k]) can be used to reconstruct the (k 1)-degree polynomial g(x) by La-
grange interpolation as follows. Afterwards, the secret data g
0
can be deter-
mined from g
0
= g(0).
g(x) = g(x
1
)
(xx
2
)(xx
3
):::(xx
k
)
(x
1
x
2
)(x
1
x
3
):::(x
1
x
k
)
+ g(x
2
)
(xx
1
)(xx
3
):::(xx
k
)
(x
2
x
1
)(x
2
x
3
):::(x
2
x
k
)
+ + g(x
k
)
(17.2)
(xx
1
)(xx
2
):::(xx
k1
)
(x
k
x
1
)(x
k
x
2
):::(x
k
x
k1
)
mod p:
Through Shamir's secret sharing scheme, we could take every secret pixel
as g
0
in a (k 1)-degree polynomial g(x) to construct n random grayscale
values in shadows to construct a (k;n)-PISSS. At this time, the prime number
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