Cryptography Reference
In-Depth Information
choices are possible. It is easy to see that r
0
= m
0
!
m
0
. An explanation of
this expression is that the binomial coecient gives the number of possible
ways of choosing m
0
vectors among the r of the collections, and the factorial
coecient accounts for all the possible permutations for each choice.
Fix a qualified set Q of participants. To compute p
0
bjb
(Q), the number of
matrices of C
B
(S) that yield a black reconstructed pixel is needed. Notice
that, for a qualified set, the number of matrices (vectors) C
B
(S) that would
give the correct reconstruction of a black subpixel (i.e., a black pixel) is rp
bjb
,
while the remaining rrp
bjb
, would give a wrong reconstruction of a black
secret pixel (i.e., a white pixel). Recall that since S is canonical, p
bjb
does not
depend on Q. Hence, the number of matrices in C
B
(S
0
) that will have a certain
number z of black subpixels, is given by
rp
bjb
z
rrp
bjb
m
0
z
m
0
!
because z vectors can be selected from the rp
bjb
vectors that yield a recon-
structed black subpixel, and m
0
z vectors from the rrp
bjb
ones that yield
a reconstructed white subpixel. Hence, z is constrained by 0 z rp
bjb
and
0 m
0
z rrp
bjb
, which implies that binomial coecients are defined.
(By definition
0
= 0 when b > a.) Hence, the values for
the probabilities of the resulting scheme can be expressed as follows:
= 1 and that
b
P
m
0
z=h
0
rp
bjb
rrp
bjb
m
0
z
z
p
0
bjb
(Q) =
m
0
(5.2)
and similarly
P
m
0
z=h
0
rp
bjw
rrp
bjw
m
0
z
z
p
0
bjw
(Q) =
m
0
(5.3)
rrp
wjw
z
rp
wjw
m
0
z
P
`
0
z=0
p
0
wjw
(Q) =
m
0
(5.4)
rrp
wjb
z
rp
wjb
m
0
z
P
`
0
z=0
p
0
wjb
(Q) =
m
0
:
(5.5)
Since from Equations 5.2{5.5, probabilities of S
0
do not depend on the
particular qualified set Q, it is possible to conclude that S
0
is canonical.
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