Visual Cryptography Schemes with Reversing - Visual Cryptography and Secret Image Sharing - page 257

contrast in their VCS with reversing is on the average q c . The price to pay
to get an almost ideal contrast is the number of transparencies stored by each
participant, which corresponds to the number c of runs. Moreover, there is a
loss of resolution in their scheme, since each pixel in the original image corre-
sponds to m subpixels in the reconstructed image, where m denotes the pixel
expansion of the underlying VCS.
In the distribution phase of the VCS with reversing in [13], the encoding
of the secret image is handled pixel by pixel, where each pixel is considered
independently of the others. For each pixel of the original image, the dealer
runs c times independently a ( Qual ; Forb )-VCS with a perfect reconstruction
of black pixels and pixel expansion m; we denote by s`i i the share for partici-
pant i in run `, for i = 1;:::;n and ` = 1;:::;c (notice that the transparency
corresponding to such a share contains m subpixels for each pixel of the orig-
inal image). In the reconstruction phase, any qualified set of participants can
recover the original secret image by performing a sequence of stacking and
reversing operations on their transparencies. The construction is described in
Figure 9.1.

Assume there exists a ( Qual ; Forb )-VCS with a perfect reconstruction
of black pixels.

Distributionphase.For each pixel of the secret image, the dealer:

runs c times independently of the distribution phase of the underlying
( Qual ; Forb )-VCS; let s ` i be the share distributed to participant i in
the `-th run, for i = 1;:::;n and ` = 1;:::;c;
distributes the c-tuple (s1i 1 i ;:::;s c i ) to participant i.
Reconstructionphase. A qualified set fi 1 ;:::;i p g of participants
reconstructs the secret pixel as follows:

superimpose their shares to get ` = OR(s ` i 1 ;:::;s ` i p ), for ` =
1;:::;c;
reverse the results of the previous step to obtain ` , for ` = 1;:::;c;
supe ri mpose the results of the previous step to get =

OR( 1 ;:::; c );
nally, reverse the result of the previous step to obtain , which is
the reconstructed pixel.

FIGURE 9.1
A construction for almost ideal contrast ( Qual ; Forb )-VCS with reversing.

Notice that in the VCS with rev er sing described in Figure 9.1, the com-
putation of the r eco nst ructed pixel corresponds to performing c 1 AND
operations, since = OR( 1 ;:::; c ) = AND( 1 ;:::; c ). From the truth