Cryptography Reference
In-Depth Information
N
d
−
d
and that of negative coe
cients N
in the dead zone are described as
N
d
= N (0 <x<t
d
)andN
−
d
−
d
<x<0), respectively. t
d
−
d
= N (t
and t
are
determined by optimum N
d
−
d
and N
values which minimize the histogram
changes for 0 and1bins.
After-embedding histogram h
′
i
, shown in Eq. (8.19), it can be rewritten as
1
2
(h
i
i+1
)+
1
h
′
i
=
−
−
i
+ h
i−1
),
+ h
2
(h
(8.20)
where (h
i
i
+ h
i−1
)/2 correspond to half of DCT coe
cients
in the interval i<x<i+ 1 and half in i−1 <x<i, respectively, and they
are quantized to i. By introducing the aforementioned dead zone, part of h
0
and h
−
i+1
)/2 and (h
−
+ h
−
0
, i.e., h
0
−N
d
−
0
−
d
and h
−N
are used for embedding, and therefore
h
′
0
, h
′
1
and h
′
−1
become as follows,
1
2
1
2
h
′
0
= N
d
(h
0
−N
d
−
1
−
d
−
0
−
d
)+h
+
−1
+
)+h
+ N
+
(h
−N
, (8.21)
1
2
(h
1
+ h
2
)+
1
h
′
1
=
−
−
1
+(h
0
−N
d
h
),
(8.22)
2
1
2
1
2
(h
h
′
−1
=
h
+
−1
+(h
−
0
−
d
−
−1
+ h
+
−2
).
−N
)+
(8.23)
The optimum values for N
d
−
d
can be derived by minimizing the sum
of squared histogram changes over 0 and1bins,
and N
1
i=−1
(h
i
−h
′
i
)
2
.
Note that in the proposed QIM-JPEG steganography, quantized coe
-
cients 0s cannot be treated as zeroes embedded in them, because they cannot
be discriminated from 0s in the dead zone. Therefore, if 0 coe
cient is se-
lected to embed a zero at quantization step, the embedding process to embed
the zero should be continued until other even coe
cient except 0 is chosen.
This causes a serious problem on the histogram that a surplus of even coef-
ficients is produced resulting in a significant change of the after-embedding
histogram. This problem can be resolved by applying the exclusive OR (XOR)
operation to a message bit with a random bit whenever embedding a message
bit. That is, an embedding bit is the XOR-applied message bit. When 0 coef-
ficient is selected to embed a zero of embedding bit, which means no message
embedded, an embedding bit at subsequent embedding is not necessarily kept
same though the message bit is same until non-0 coe
cient is selected. The
message bits can be recovered from the embedded XOR-applied message bits
using the same random binary sequence as one used in embedding.
8.6.3 Histogram Preserving HM-JPEG Steganography
Consider histogram matching at quantization step of DCT coe
cients, as-
suming that the probabilities of zero and one are same in binary data to be
embedded. Histogram matching is here considered separately for positive coef-
ficient part and negative one, since there sometimes exists asymmetry between
