Applications of Visual Cryptography - Visual Cryptography and Secret Image Sharing - page 335

probability than other pairs. But the partially random patterns are made in
a way such that this probability decreases in the overall image.
Let us now turn to the slides on the previous page: If the slide was used
more than n times, then Eve can try the following attack: She chooses an area
of q = xy pixels somewhere in the image, then she tries each combination of
positions of the 0 in each of the corresponding q clusters on the slide ((2 n ) q
possibilities) and checks if it is consistent with each encrypted image in the
sense that there are 4 symbols in F overlapping the area. This takes 4xyf
steps. And thus 2 nq 4 xyf steps in total.
The number of possible subimages of a possible original image, where four
symbols overlap, is approximately (xy f) 4 . This means each observed en-
crypted image can help Eve to exclude a sucient number of possible choices
if 2 q >> (x y f) 4 . We therefore estimate the number of steps for Eve as
>> (xyf) 4n 4xyf = 4(xyf) 4n+1 . Now assume we use x = y = 10
and a huge font F with many possibilities to depict a symbol that leads to
f = 1000 and roughly 10 20n steps for Eve. Considering many obvious and
also less obvious improvements of the algorithm for Eve, we believe that the
attack is still too expensive for Eve for n = 3.

12.5 Using Refraction
In [4] we generalize the slide from a 2-dimensional array over = f0; 1g to a
2-dimensional array over f1g[RRwith the idea that each pixel on the
slide can either be black or contains a prism (x;y) that refracts the light from
a region on the encrypted image or, from the perspective of the user as shown
in Figure 12.12, refracts the view to a region that is shifted by (x;y) from the
pixel, which is directly behind the pixel on the slide. For example (0; 0) would
correspond to the 0 in the case of usual Visual Cryptography just showing
the pixel directly behind. One possible application would be to use clusters of
22 = 4 pixels for each pixel of t he original and r andomly choose one of the 4

(1; 0) (0; 0)

(1;1) (0;1)

pixels to be visible. For example would direct the view to the

upper right pixel on the encrypted image. This corresponds to construction 2
in Section 12.4. Using the slide two times is information theoretically secure
and the contrast is 1.
If we use lenses or fragments of lenses instead of prisms, the view can be
focused on a point inside a pixel. This has the advantage that the positioning
of the slide allows an error of up to half of a pixel. An example is shown in