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.
 
Search WWH ::




Custom Search