Two-Decoding-Option Image Sharing Method - Visual Cryptography and Secret Image Sharing - page 471

have

m
w

m
m=2

m (L)

LIN = (n=3) m 8=log 2

= ((8 mn)=3) =log 2

m(m1)(m=2+1)
(m=2)(m=21)1

= ((8 mn)=3) =log 2

> ((8 mn)=3) =log 2 m m=2 = (16n=3) =log 2 (m) = (16n=3) =
(log 2 (2n 2))
> ((4n 4)=3) = (log 2 (2n 2)) = (2=3) (2n 2) = (log 2 (2n 2))
> 2=3 = m=(kg) = m (L)

PRO (since g = m=2 and k = 3):

(17.11)
When using Naor{Shamir (2, n)-VCS with m = n;w = m 1;g = 1, we
have

m
w

m
m 1

m (L)

LIN = (n=2) m 8=log 2

= (4 mm) =log 2

= 4m 2 =log 2 (m) = (4m) (m=log 2 (m)) > 4m
> m=2 = m=(kg) = m (L)

PRO (since g = 1 and k = 2):

(17.12)

From Equations (17.10), (17.11), and (17.12), we obtain m (L)

LIN > m (L) PRO .

The proof is completed.

2
Table 17.1 lists the pixel expansions of some (k;n)-TiOISSSs. Our scheme
(respectively, Lin and Lin's scheme) could choose dierent g (respectively, w )
in the same m to reduce the shadow size. For example, both (2, 2)-TiOISSSs

110
101

110
110

10
01

10
10

can use B 1 = , B 0 = instead of B 1 = , B 0 = to

reduce the pixel expansions from m (L)

PRO = 1 and m (L) LIN = 16 to m (L)

PRO = 3=4

and m (L)

LIN = 15:14, respectively. Our (2, 3)-TiOISSS could reduce the pixel

2 3 2 3

100
010
001

100
100
100

4 5 , B 0 = 4 5 with

expansion from 3/2 to 3/4 by replacing B 1 =

2 3 2 3

110
011
101

110
110
110

4 5 , B 0 = 4 5 , but the pixel expansion m (L)

B 1 =

PRO = 22:71 of Lin

m
w

3
2

3
1

and Lin's scheme cannot be further reduced since = = .

Our pixel expansions m (L)

PRO = m=(k g): when using Naor{Shamir (2, n)-
VCS, (3, n)-VCS and (n;n)-VCS, are n/2, 2/3 and 2/n, respectively. From
Equations (17.4), (17.7), and (17.8), it is observed that m (L)

LIN is n-intensive
and the value will increase when n increases whereas the other two schemes
are n-invariant. On the other hand, Lin and Lin's scheme and our scheme
obtain more eective performance for large k than Jin et al.'s scheme because
their pixel expansions are inversely proportional to the value of k.
File sizes of shadows in Table 17.1 are represented in bits when using a
512 512 gray-level secret image. The values in parentheses imply that we