Cryptography Reference
In-Depth Information
that embedding by J-Steg is detectable using the chi-square attack [19] since
it is based on flipping the least significant bits (LSBs). F5 cannot be detected
by the chi-square attack. However it can be detected by a specific technique
[20] which exploits a significant change on the histogram of quantized DCT
coe
cients caused by embedding.
This section presents two histogram preserving JPEG steganographic
methods. The first one is a histogram quasi-preserving method which uses
quantization index modulation (QIM) [21] at quantization step of DCT co-
e
cients. Since a straightforward application of QIM causes a significant
histogram change, a device is introduced in order not to change the after-
embedding histogram excessively. The second one is a histogram preserving
method which uses a histogram matching technique. Here we call the first
method QIM-JPEG steganography and the second one histogram matching
JPEG (HM-JPEG) steganography.
8.6.1 QIM in DCT Domain
Consider applying QIM [21] with two different quantizers to embed binary
data at the quantization step of DCT coe
cients in JPEG compression. Each
bit (zero or one) of binary data is embedded in such a way that one of
two quantizers is used for quantization of a DCT coe
cient, which corre-
sponds to embed zero, and the other quantizer is used to embed one. Given
a quantization table and a quality factor for JPEG compression, quanti-
zation step size ∆
k
, 1≤k≤64 for each frequency component is deter-
mined. Then, two codebooks, C
0
and C
1
, for two quantizers are chosen as
C
0
=2j∆
k
; j∈Z,C
1
=(2j +1)∆
k
; j∈Zfor k-th frequency. Given a
k-th frequency DCT coe
cient x,2q with q = arg min
j
becomes
the quantized coe
cient in case of embedding zero, for example, and 2q +1
with q = arg min
j
x−2j∆
k
in case of embedding one.
Assuming that the probabilities of zero and one are the same in binary
data to be embedded, and considering how histograms of quantized DCT
coe
cients change after embedding. In the following, we assume that DCT
coe
cients belonging to k-th frequency are divided by its quantization step
size ∆
k
in advance and then two codebooks, C
0
and C
1
can be defined as
C
0
=2j; j∈Z,C
1
=2j +1;j∈Zfor all frequency components. Let
h
i
,i∈Z denote the number of DCT coe
cients whose values x areinthe
interval i−0.5 <x<i+0.5, which is described as h
i
= N (i−0.5 <x<
i +0.5). Let h
x−(2j +1)∆
k
i
and h
i
denote the number of DCT coe
cients in the interval
i−0.5 <x<iand that in i<x<i+0.5, respectively and therefore
h
−
−
i
+ h
i
= h
i
. After embedding by QIM, the histogram h
i
is changed to h
′
i
as
1
2
h
i
+
1
2
(h
i−1
+ h
h
′
i
−
=
i+1
).
(8.19)
The change in Eq. (8.19) can be understood as follows. If i is an even number,
i.e., i∈C
0
and C
0
is used for embedding zero, half of DCT coe
cients
