XOR-Based Visual Cryptography - Visual Cryptography and Secret Image Sharing - page 166

contrast VC scheme even when the underlying VC scheme with reversing is
nonperfect black. At this time the decoding needs an XOR operation.
Yang et al.'s scheme [25] is described in a pseudo-code style below in terms
of its construction procedure and the revealing procedure.

Yang et al.'s algorithm for a(k,n)-VC scheme

Distribution phase.
Step 1: Given a secret image, the dealer performs a (k;n) nonperfect
black VC scheme(NPBV CS) to generate n shadows, s 1 ; ;s n for the
rst run.
Step 2: The dealer generates the shadows s j = (s r j ) for the rth run,
r = 2; ;m. Note that the shadow should be labeled as to which run it
is, for easy management by the participant.
Step 3: The dealer distributes m shadows s j ; ;s j to Participant j,
j = 1; ;n.
Step 4: Finally, every participant holds m shadows.
Reconstruction phase.
Step 1: To recover the secret within m runs, at least k participants,
Participants j 1 ; ;j k , oer their (kR) shadows s j1 ; ;s jk , i = 1;R,
for reconstruction.
Step 2: Stack the shadows s j1 ; ;s jk to reconstruct the image T r in the
rth run.

T r = s j1 + s j2 + + s jk ;r = 1; ;m
Step 3: Finish m runs by using an XOR operation to reconstruct U 0 =
T 1 T m .
Step 4: If "mh" is even (i.e., "ml" is odd) th en the reconstructed
image is U 0 ; otherwise, the reconstructed image is U 0 .

Theorem 9 [25] The whiteness percentages of the white and black secret pix-
els for Yang et al.'s algorithm are P W = 100% and P W = 0%, respectively,
after finishing m runs.

Proof There are "mh" B "h" W (respectively "ml" B "l" W) subpixels
for the white (respectively black) secret pixel. When shifting right one bit
m times, there are mh (respectively ml) black subpixels for the white
(respectively black) secret pixels in U 0 . Suppose "mh" is even (respectively
odd); an XOR ope rat ion will result in all white subpixels for the white pixels
in U 0 (respectively U 0 ). Thus, the P W of the white secret pixel is 100%. On the
other hand, even (respectively odd) "m h" means odd (respectively even)
"ml" since 'hl' is odd. It is evident that an XOR ope rat ion will result in
all black subpixels for the black pixels in U 0 (respectively U 0 ), i.e., the P W of
the black secret pixel is 0%.
An example (2; 4)NPBV CS is given below as a demonstration for Yang
et al.'s algorithm.