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and
p wjw (Q) p wjb (Q)
return then a measure of the goodness of the scheme for the given qualified set
Q. The bigger the above differences the better is the scheme. If both quantities
are equal to 1 for all qualified sets, then the scheme is again a deterministic
one, with no possible error in the reconstruction.
If there exists a positive constant such that for any qualied set Q it
holds
p bjb (Q) p bjw (Q)
and
p wjw (Q) p wjb (Q) :
then the scheme is called -probabilistic, meaning that denotes the possible
error in the reconstruction. Notice that when every reconstructed pixel is
either white or black, that is, the reconstructed pixel has at most ` black
subpixels or at least h black subpixels, we have that for any Q,
p wjw (Q) = 1 p bjw (Q)
and that
p bjb (Q) = 1 p wjb (Q)
and thus,
p wjw (Q) p wjb (Q) = p bjb (Q) p bjw (Q):
The value ` and h are the thresholds that define the number of subpix-
els needed to correctly distinguish between a white and black pixel in the
reconstructed image. For some scheme, for some qualified set Q, it could be
possible to obtain a secret pixel with a number of black subpixels strictly
greater than ` and strictly less than h, where the reconstructed pixel is
neither white nor black. In this case the value p bjb (Q) p bjw (Q) might be
different from p wjw (Q) p wjb (Q). To avoid such kind of ambiguous situ-
ations, it is possible to x the value ` = h 1, so that the equation
p wjw (Q) p wjb (Q) = p bjb (Q) p bjw (Q) always holds.
Probabilistic schemes are then characterized by a further parameter: the
probabilistic factor . In the following a probabilistic scheme will be described
by all these parameters (that is, ;k;n;`;h, and m) which are referred to as
the characteristic parameters of the scheme.
It is now possible to provide the formal denition of a -probabilistic
threshold visual cryptography scheme with characteristic parameters
(k;n;`;h;m), for short -probabilistic (k;n;`;h;m)-VCS. The denition nat-
urally extends also to general access structures.
Denition 2 A -probabilistic (k;n;`;h;m)-VCS consists of two collections
of nm binary matrices, C W and C B , satisfying the following properties:
 
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