On estimation of secret message length in JSteg-like steganography

Abstract

Recenty, image steganalysis has attracted more attention. In this paper, a new method for detecting the secret message and estimating the secret message length of bit-streams embedded using J-Steg like steganography is proposed. First, based on the model of statistical distribution of quantized DCT coefficients, the histogram of the cover image is estimated from the stego image. Then, the secret message is detected as well as the estimation of the secret message length using the estimated cover histogram. Interestingly, our study on steganalysis of J-Steg like embedding offers a proof of Zhang et al. 's method (2003). The methodology described in this paper is a framework which can also be applied to many other stegaographic methods such as JPEG2000 file format embedding.

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On Estimation of Secret Message Length in JSteg-like Steganography

Xiaoyi Yu, Yunhong Wang, Tieniu Tan
National Laboratory of Pattern Recognition, Institute of Automation,
Chinese Academy of Sciences, P.O. Box 2728, Beijing, 100080, P.R. China
E-mails: {xyyu,wangyh,tnt }@nlpr.ia.ac.cn

Abstract

Image steganalysis has attracted more attention
recently. In this paper, a new method for detecting secret
message and estimating the secret message length of bit-
streams embedded using J-Steg like steganography is
proposed. First, based on the model of statistical
distribution of quantized DCT coefficients, the histogram
of cover image is estimated from stego image. Then the
secret message is detected and the secret message length
is estimated with the estimated cover histogram.
Interestingly, our study on steganalysis of J-Steg like
embedding offers a proof of Zhang et al.’s method [6].
The methodology described in this paper is a framework
which can also be applied to many other stegaographic
methods such as JPEG2000 file format embedding.

1. Introduction

Steganography, a science
communication, aims to hide the very presence of
communication. That is to say, the essential goal of
steganography is to conceal the facts of a hidden message.
The frequently used steganographic method in JPEG
format is the J-Steg like algorithm, which proposed by D.
Upham [1]. It works by embedding message bits as the
LSBs of the quantized DCT (Discrete Cosine Transform)
coefficients. The embedding mechanism skips all
coefficients with the values of ‘0’ or ‘1’. There are two
embedding ways according to the selection of coefficients.
One is sequential embedding; the other is random
embedding whose coefficients selection is usually
determined by a secret stego key shared by the
communicating parties.
During the last few years, many powerful steganalytic
methods [2,5] capable of detecting J-Steg like embedding
were proposed. J-Steg with sequential message embedding
is detectable using the chi-square attack [2]. J-Steg with
random straddling is detectable using the generalized chi-
square attack [5]. Provos point out that the method he
proposed can work well [5]. However he hasn’t given any
further details for this generalized approach. Both
methods can’t estimate the number of changes due to
embedding and thus can’t estimate the secret message
length. Zhang proposed a method which can reliable
and art of secret
detect the existence of hidden message and even could
estimate the length of hidden message exactly [6].
Although existing methods can detect hidden messages,
these methods have their limitations. They either can’t
estimate the length of hidden message or lack theoretical
foundation. In this paper, we present a systematic
approach for estimating the number of embedding changes
due to J-Steg like algorithm steganography in JPEG file
formats. Our method can also be applied to many other
stegaographic methods such as JPEG2000 file format
embedding. It is a framework to detect the hidden
message and estimate the length of hidden message if one
can find a optimal statistical image models for the
stegaographic method he want to detect. The detection
and estimation algorithm is introduced in Section 2. In
Section 3, we apply the method described in Section 2 to
steganalysis J-Steg like steganography. The verification
of Zhang et al.’s Method is outlined in Section 4. The
experimental results and the conclusion are in the last 2
sections of the paper.

2. Method Description

As described in [7], if we have a model which captures
some main statistical properties of the carrier media and
which will be changed by steganographic process, we can
reliably detect the existence of hidden message. We use
the terms as [7]. Let x denote an instance of a class of
potential carrier media, such as pixel values or quantized
DCT coefficients of an image. If we treat x as an instance
of a random variable X, we can model X using the
probability distribution PX(x) .
To detect whether there is a hidden message embedded
in a suspected media is to perform a hypothesis test to find
out whether the instance x obey the probability
distribution PX(x) . But how to estimate the length of
hidden message? In [3], Fridrich summarized the
principles to estimate the length of secret message. For
most steganographic techniques, it is usually not too
difficult to identify a macroscopic quantity S(m) that
predictably changes (for
increases) with the length of the embedded secret message
m. We can calculate an estimate of the unknown message
length m by solving the equation S(m) = Sstego for m, where
Sstego is the value of S for the stego image under
investigation. Fridrich call S(m) the distinguishing
example, monotonically
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statistics. In general, the function S has several
undetermined parameters which can be determined by
estimating some extreme values of S, such as S(0) (i.e. the
cover image). To obtain the value S(0), Fridrich cropping
the stego image by four pixels to obtain the macroscopic
properties that well approximate the properties of the
cover image. In this paper, we apply the same principle to
estimate the length of hidden message. Since we can
model the carrier media, it is not necessary to crop the
stego image to approximate the macroscopic properties of
the cover image. If we can model x precisely, our method
can obtain more precise results than Fridrich’s method.
We will show how our proposed method works on
estimation of the secret message length in JSteg-like
steganography.

3. Detection of Secret Message & Estimation
of Secret Message Length in JSteg-like
Steganography

In this section, we will show the process of detection
the secret message and estimation the length in JSteg-like
steganography. We first model the statistical distribution
of quantized DCT coefficients and then use estimated
parameters to detect the secret message and estimate the
capacity of secret message.

3.1. Model the Statistical Distributions of
Quantized DCT Coefficients

In the JPEG compression standard, images are divided
into 8×8 blocks. Each block is passed through a DCT
(Discrete Cosine Transform) to produce 64 DCT
coefficients and then the coefficients are quantized
according to a quantization table and encoded using an
entropy encoder. It is generally believed that the
distribution of DCT coefficients is Laplacian except for
the [0,0] coefficients [3]. These coefficients are called AC
coefficients. The Laplacian distribution
( )
2
If the coefficients are quantized, a generalized
Laplacian can still be used to fit the resulting histogram
with integer width bins. In [7], Sallee used a specialized
form of a generalized Cauchy distribution instead of the
generalized Laplacian:
( )
s
2
For this distribution, there is a closed form solution for
the cumulative density function which makes it easy to
integrate the model density for individual histogram
bins. As Sallee described, when taking into account a
||x
exp
λ−
λ
=
(1)
p
s
xp
xp
−
−
−
=
) 1|(|
1
(2)
more accurate estimation of the quantization effects, one
would find this distribution appears to fit DCT
coefficients better than
Laplacian/Gaussian. In this paper, we’ll use this
distribution to model quantized DCT coefficient.
Now, we come to the core of the problem. We only
know the stego image, how can we model the cover image?
As described in Section 1, J-Steg like algorithm works by
embedding message bits as the LSBs of the quantized
DCT coefficients except ‘0’ and ‘1’. Let h(d), d=…-2,-
1,0,1,2,…, be the histogram of the quantized DCT
coefficients from the cover image. Let H(d) be the
histogram of cover image after embedding m pseudo-
random bits in the LSBs, the histograms H(d) and h(d)
will have relations as follow equations.
) 1 () 1 (),0 () 0 (hHhH
==
(3)
[
2 ()2 ()2 ()2 (
−−=
hihihiH
α
[
2 () 12 () 12 (
++=+
hihiH
α
where
,... 2 , 1±±=
i
,
m
α

the generalized
] ) 1
+
−
h
i
)
(4)
] ) 12 (
+
ii
(5)

∑
≠
, 0
≠
=
1
)(2
ii
iH
From (4) and (5), we obtain
H(2i)+H(2i+1)=h(2i)+h(2i+1). The sum of histogram
values keeps unchanged after message embedding. This
condition must be satisfied
pairs( h(2i),h(2i+1)).
Let
) 2 ()(
+=
iHiHα
)(),1 (),0 ( iHHH
α
to fit the model (2). In fact,
)(iHα
is low precision histogram of AC coefficients
from the cover image. The model parameters s and p can
be determined from
)(iHα
. The fit process is the same
as [7]. Once the model is fit to the histograms for a stego
image, it is used to estimate the histogram of the cover
)(ˆdh
be the estimated histogram of cover
image.
) 1 (ˆ), 0 () 0 (ˆ hHh
=
)2 (
)()2 (ˆ
iP
α
2 (
)() 12 (ˆ
P
5 . 12
2 ∫
−
i
∫
−
5 . 02i
Figure 1 shows the original coefficient histogram of an
image and the estimated histogram after message
for all histogram
) 1
+
2 (iH
. We use
image. Let
) 1 (H
=
(6)
)(
P
iP
iHih
α
=
(7)
)(
) 1
+
i
i
iHih
α
)dx
α
=+
(8)
where
,()(
5 . 0
+
i
=
xpiPα

+
=
5 . 02
)()2 (P
i
dx xpi,
∫
+
+
=+
5 . 12
5 . 02
)() 12 (P
i
i
dxxpi
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embedding. The coefficients are all AC coefficients. It can
be seen from the figure that we can almost exactly
estimate the histogram of cover image from a stego image.
Our estimation method is better than Fridrich’s cropping
method, because it isn’t necessary for us to consider
double compression effect.
3.2. Detection of secret message and estimation of
secret message length

Since we can accurately estimate the histogram of the
cover image, the detection of hidden message and
estimation of hidden message length becomes easy. The
detection is determined using the Chi-square test. The
2
χ statistic is given as
∑
±=
i
±
−
h (ˆ
=
k
i
iH
)
ih (ˆ
1
2
2
))()(
χ
with
2k-1 degree of freedom. We can perform Chi-square test
at significance level α and 2k-1 degree of freedom to
decide whether a suspect images contains secret message
or not. Furthermore, we can reliably estimate the length of
the message bits. For random embedding method,
equation (4) and (5) express the distinguishing statistics as
a function of secret message m and the original histogram.
The original histogram can be estimated using the method
described in Section 3.1. We calculate α using the
following equation which is derived from (4).
∑
±=
i
1
The least square approximation in the above equation
leads to the following formula for α
∑
±
±=
i
ih
1
)(
±
+−−−=
k
ih2 (ˆih2 (ˆiH
2
))1)) 1 () 2 ((min
α
arg
ααα
∑
±
±=
+−
+−−
=
k
k
i
ih2 (ˆ
ih2 (ˆih 2 (ˆih2 (ˆ iH
2
1
))12 (ˆ
))1)))()2 ((
α
(9)
where k is the maximum quantized DCT AC
coefficient. Thus the length of unknown message can be
calculated as
∑
≠≠
1, 0ii
Let C denotes the whole coefficients, S the subset of C,
SCS
−=
the complement of S in C,
)(
ˆ
dhA
the estimate histogram of set
A, where A could be
S,
embedding method, we construct S the same way as
sequential embedding method, and ensure
ˆ
dhS
are approximately equal using Chi-Square test.
Then the embedding ratio is
∑∑
≠≠
1 , 0 1 , 0ii
and the length of unknown message is
∑
≠
1 , 0i
3.3. Experimental Results
=
)(2iHM
α
(10)
)(dHA
the
actual histogram and
S
or C. For sequential
)(dHS
and
)(
=
)()(
CS
iHiH
β
(11)
=
)(
SiHM
(12)
For testing purpose, we have used the CBIR image
database from Washington University (824 JPEG images
totally) [8]. Since we can estimate the histogram of the
cover image, message embedding in sequential and
random ways can be detected and estimated in the way
described in Section 3.2. In our experiments, we have
generated 8 stego images (4 for sequential embedding and
4 for random) for each image and the length of hidden
messages are 20, 50, 80 and 100 percentage of hiding
capacity, corresponding to α
2
convenience of display, only parts of the results of the
experiments are show in Figure 2. ‘△’, ‘*’,‘o’, ‘+’ and
‘ ☆ ’ represent the estimated percentages of message
=0.2, 0.5, 0.8, 1.0. For the
Figure
histograms between original histogram of cover
image and estimated histogram of stego image
1. A comparison of coefficient
Figure 2. Estimates of embedding ratio on
part images from ICBR image database
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Page 4

capacity, corresponding to α
Table 1 shows the mean
deviation
) 2 ( ασ
Table 1. Statistical Results
For Sequential
2
βα /

)(βµ
0 0.0748
0.2 0.2562
0.5 0.5365
0.8 0.8160
1.0 1.0009
2
=1.0, 0.8, 0.5, 0.2, 0.
/
)(βµ
and the standard
for each embedding ratio over
)2 ( αµ
)
/
2 ( ασ
the whole 824 stego images. From Figure 2 and Table 1,
we can see the experimental results are satisfied. And we
also can see the standard deviation increases as
embedding ratio decreases. This is because less message
embedded into the cover image cause little change of
DCT coefficients. We should model DCT coefficients
more precisely.

4. Verification of Zhang et al.’s Method

As we introduced in Section 1, Zhang et al. proposed a
steganlytic technique to estimate the hidden message
length using J-Steg like algorithms. Our analysis presented
in the preceding section can prove their estimation
formula.
From equation (4) and (5), we can derive four
equations as follows.
∑∑
>>
00
ii
∑∑
>>
00
ii
∑∑
<<
00ii
∑∑
<<
00
ii
using the symmetric property of the histogram , we
combine (13) and (16) to obtain equation (17), and
combine (14) and (15 ) to obtain equation (18)
∑∑
><
00
ii

∑∑
><
00
ii
∑
>
i
+
) 1
++−=
0
) 12 (h) 2 (h) 1 () 2 (iiiH
αα
(13)
∑
>
i
) 1
∑
i
+−=+
0
(15)
)2 (h2 (h)1 () 12 (iiiH
αα
(14)
∑
<
i
i
++−=
0
+
2 (h)2 (h)1 ()2 (iiiH
αα
<
+−=+
0
)2 (h) 12 (h)1 () 12 (iiH
αα
(16)
∑
>
i
− h
α
∑
>
0
i
∑
<
i
++=++
00
) 1 2 (h) 2 (h ) 12 ()2 (iiiHiH
) 1
i
(−
2 (h
(17)
∑
<
0
i
) 1(−+ h
α
(18)
) 1
,
++=++
)2 (h) 1) 2 () 12 (iiHiH

Florida, USA, March 2003.
[7] P. Sallee, “Model Based Steganography”, In
Proceedings of International Workshop on Digital
Watermarking 03, Seoul, Korea, September, 2003.
[8] CBIR Image Database, University of Washington,
http://ww.cs.washington.edu/research/magedatabase/
roundtruth/.
Let
∑
>
i
2 (H
∑
<
i
∑
<
++=
00
0
2 ()2 (iHiHf
) 1 ()2 () 1
+
00
1
HiHif
ii
++=∑
>
,
we have
) 1 (2) 1
−
)
0
(2
01
Hhf
=
f
αα=
) 1 ( H
=
(
−
β
, let
α
2
β
=
,
we obtain
estimation formula.

5. Conclusion

We have presented a new approach to steganalysis.
The approach, based on statistical model, provides a
framework for steganalysis. We have shown that the
method can not only detect the existence of secret
message, but also estimate the length of secret message.
We also offered a proof of Zhang et al.’s method. The
methodology described in this paper can also be a
framework to be applied to many other stegaographic
methods such as JPEG2000 file format embedding.

Acknowledgments

The work is sponsored by the Natural Sciences Foundation of
China under grants No. 60172054, No. 60121302 and No.
60335010.

6. References

[1] JPEG-JSteg-V4, http://www.funet.fi/pub/crypt/
steganography/jpeg-jsteg-v4.diff.gz.
[2] A.Westfeld and A. Pfitzmann. “Attacks on
Steganographic Systems”. IHW 99, Dresden, Germany,
1999.
[3] S. R. Smoot and L. A. Rowe “Study of DCT
coefficient distributions”. In Proceedings of the SPIE
Symposium on Electronic Imaging, volume 2657, San
Jose, CA, January 1996.
[4] J. Fridrich1, M. Goljan, D. Hogea, and D. Soukal,
“Quantitative Steganalysis of Digital Images: Estimating
the Secret Message Length”, ACM Multimedia Systems
Journal, Special issue on Multimedia Security, Vol. 9(3),
2003, pp. 288-302.
[5] N. Provos and P. Honeyman, “Detecting
Steganographic Content on the Internet”, CITI Technical
Report,2001 pp. 01-11.
[6] T. Zhang and X. Ping, "A Fast and Effective
Steganalytic Technique against Jsteg-Like Algorithms", In
Proceedings of ACM Symposium on Applied Computing,
1
ff −
. This is Zhang’s
For Random
)2 ( αµ

0.0638
0.2352
0.5003
0.7948
1.0004

)(βσ
0.0159
0.0105
0.0042
0.00073
0.00013

)2 ( ασ
0.0237
0.0240
0.0212
0.0279
0.0131

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