Cryptography Reference
In-Depth Information
Then we construct the base matrices for the 3-color scheme as follows:
2
3
11 2 2 3 3
11 22 33
1 1 2 2 3 3
4
5
B
1
= [B
!1
jB
!2
jB
!3
] =
2
3
1 1 22 3 3
11 22 33
1 1 2 2 3 3
B
2
= [B
!1
jB
!2
jB
!3
] =
4
5
2
3
1 1 2 2 33
11 22 33
1 1 2 2 3 3
B
3
= [B
!1
jB
!2
jB
!3
] =
4
5
Using as a building block the original (k;n)-threshold scheme provided in
the paper by Naor and Shamir [8], whose pixel expansion is 2
k1
, the c-color
schemes so obtained have pixel expansion m = c2
k1
. This greatly improves
on the pixel expansion of [2, 10].
Finally, as observed, also in [11], we can delete from the base matrices the
columns that have all pixels with color . Using the base matrices provided in
the paper by Naor and Shamir [8], for n even we always have one such column
in each base matrix, while for n odd we always have c 1 such columns
in each base matrix. Hence, the pixel expansion can be further improved to
m = c 2
k1
1 for n even and to m = c 2
k1
c + 1 for n odd. This is
important as we will see that for k = n this improved pixel expansion matches
a lower bound proved in [3].
The contrast property considered is the weak one. The scheme have pa-
rameters h = 1 and ` = 0 (recall that for the special case of ` = 0 the weak
contrast property is equivalent to the strong one). The annihilator presence is
= (m 1)=m.
The same idea used for the construction of c-color (k;n)-threshold schemes
starting from black and white (k;n)-threshold schemes, can be used also for
general access structure schemes. The pixel expansion of the c-color scheme is
c times the pixel expansion of the black and white scheme that we start with.
2.4.4 The CDD Schemes and a Lower Bound
Paper [3] denes the contrast as = (h`)=m and considers the weak contrast
property. The following theorems are proved in [3]:
Theorem 1 In
the
SC
model,
the
optimal
contrast
of
a c-color (n;n)-
threshold scheme is
(
1
c2
n1
1
; if n is even
1
opt
=
c2
n1
c+1
; if n is odd.
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