A Steganographic Protocol Based on Linear Error-Block Codes
Rabiî Dariti and El Mamoun Souidi
Laboratory of Mathematics, Computer Science and Applications, Faculty of Science,
Mohammed V University-Agdal , BP 1014 Rabat, Morocco
Keywords: Steganography, Linear Error-Block Codes, Heterogeneity, Bit Planes.
Abstract: We present a steganographic protocol based on linear error-block codes. Recent works have showed that
these codes allow to increase the number of information carrier bits within a given cover by exploiting
multiple bit planes (not only LSB plane) from pixels which would not have a perceptible influence on the
cover. We employ a parameter, called heterogeneity, to assess the ability of pixels to be modified without
perturbing the cover. The quality of the modified cover is handled by tuning a vector of heterogeneity
thresholds which determines the number of bit planes that we are allowed to use for each pixel in the cover.
1 INTRODUCTION
The objective of code based steganography is to
create a cover , with minimum modifications
regarding the original cover , such that the
syndrome of is the secret message . In other
words, the vector must be the closest vector to
whose syndrome is . Concretely, let be a linear
error correcting code of parity-check matrix . The
retrieval map requires simply computing the
syndrome of the modified cover :
. The embedding algorithm consists of three
steps. First compute :, then find
, a
word of smallest weight among all words of
syndrome . In other words,
must be the leader of
the coset of syndrome . This is exactly the
syndrome decoding problem. Finally, modify the
cover by computing ,:
. It is easy to
verify that we have
,
1
.
The idea of using error correcting codes in
steganography was firstly brought up by (Crandall,
1998). His main objective was to use an error
correcting code to randomize the bits used to hide
the secret message. His method was called matrix
embedding. In 2001, Westfeld implemented an
algorithm called the F5 algorithm, which was the
first steganographic scheme based on matrix
embedding (Westfeld, 2001). He used the Hamming
code of length 7, which allows embedding 3
message bits within each 7 cover bits by altering at
most one of these bits.
To ensure the imperceptibility of embedding, the
majority of steganographic protocols modify only
least significant bits. The main condition to be able
to use further bits remains the imperceptibility of
their alteration. Consider an 8-bit gray-scale image.
The number 8 is called the bit-depth; it determines
the number of bits within which a pixel is encoded.
A set of bits corresponding to a given bit position
between 1 and 8 in all pixels is called a bit
signification plane or, simply, a bit plane. The first
bit plane, corresponding to the set of bits in position
1, consists of the least significant bits (LSB) of all
the pixels. Obviously, the first bit plane is the most
suitable to be modified. There are many embedding
methods that use bit signification planes in different
ways (Zhang, 2007; Zhang, 2007; Liao, 2008; Dariti,
2011). In our method, we use multiple bit planes if
their modification does not cause a noticeable
change to the cover. We assess the ability of pixels
to support alteration by calculating a parameter
called heterogeneity, which measures the
resemblance between a pixel and the pixels adjacent
to it. The more a pixel is different, the higher is its
heterogeneity value and the more it is suitable to
carry information by modifying its value. We benefit
from pixels which have significantly high
heterogeneity values to exploit further bit planes.
178
Dariti R. and Souidi E..
A Steganographic Protocol Based on Linear Error-Block Codes.
DOI: 10.5220/0005014501780183
In Proceedings of the 11th International Conference on Security and Cryptography (SECRYPT-2014), pages 178-183
ISBN: 978-989-758-045-1
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
We define a list of heterogeneity thresholds in order
to determine the highest bit plane we can use and the
number of pixels for which a given bit plane is to be
used. The added value of linear error-block codes
comes from their block structure. When blocks are
of different sizes, a random single bit error is more
probably to occur in a block of big size than in a
block of small size. Therefore, we construct block
vectors by putting sensitive bits in small size blocks
and less sensitive bits in big size blocks. Bits
belonging to similar bit planes are put in blocks of
the same size. The decoding procedure of linear
error-block codes ensures that the probability of
modifying small size blocks is less than the
probability of modifying bigger ones. Thus the
modified cover is expected to have good quality
parameters. Moreover, these parameters can be
handled by adjusting the heterogeneity thresholds.
This paper is organized as follows. Section 2
recalls some preliminaries about linear error-block
codes and describes their convenience in
steganography. Section 3 presents the method used
to assess the quality of the modified cover. Section 4
shows how can linear error-block codes ensure a big
quantity of payload at the same time as keeping a
good quality of the cover. Experimental results are
given in Section 5. Finally, Section 6 gives
concluding remarks.
2 LINEAR ERROR-BLOCK
CODES IN STEGANOGRAPHY
Linear error-block codes are a generalization of
linear error correcting codes, introduced by (Feng,
2006). The Hamming metric is replaced by a more
general distance called the -metric, which is
defined as follows. For a given non null integer ,
…
is said to be a partition of if
⋯
where
⋯
1 and is a non null integer. In the case
…
…
…
…
we write
…
where
⋯
1 and and
for 1,2,…, are
non null integers. The integers
, 1,2,…, and
, 1,2,…,, are called summands of . Let
be a prime power and
be the finite field with
elements. Given a partition
…
of
, the vector space
over
can be viewed as the
product
…
noted
.
Consequently, each vector ∈
can be uniquely
written as
…
, where
∈
. For each
pair and in
, the -weight
of and the
-distance
, between and are defined by
#|1,
0∈
and
,
–#
|1,
.
An
-linear subspace of
is called an
,
linear error-block code of type over
,
where dim
. Notice that a classical linear
error correcting code is a linear error-block code of
type
1
.
The idea of using linear error-block codes in
steganography was introduced by (Dariti, 2011). As
vectors in
are concatenations of blocks, a one-
block error vector is a vector in
which has a
single non-zero block. This does not necessarily
mean that only a single one of its bits is non-zero,
but that all of its non-zero bits are located within the
same block. In steganographic terms, assume that to
embed a -bit message we have to modify bits of
the -bit cover. These bits are located in ′ blocks
where ′. Thus we have to provoke ′ errors,
whilst with a classical code we provoke errors.
Therefore, the correction capacity of linear error-
block codes does not need to be as big as in the
classical case. A drawback of this method is the
following. Recall that a coset leader is a vector of
minimum -weight within its coset. Let us consider
a code where the two vectors
11000 and
00010 belong to the same coset. If the code is
of type
3
1
the vector
can be considered
as coset leader. If we use it to embed some message
then we need to modify 2 bits in the cover, whereas
with a classical code (type is
1
) we need to
modify only one bit since the vector
will be the
coset leader. Actually, in this method the cover bits
are reorganized into blocks of specific sizes in order
to give preferential treatment to each block of bits.
We explain in the following two facts justifying that
it might be more suitable to flip a whole block of
relatively big size rather than a single bit from
another block of smaller size.
In general, pictures feature areas that can better
hide distortion than other areas. The human vision
system is unable to detect changes in heterogeneous
areas of a digital media due to the complexity of
such areas. For example, if we modify the gray
values of pixels in smooth areas of a gray-scale
image, they will be more easily noticed by the
human eye. Furthermore, the pixels in edged areas
may tolerate larger changes of pixel values without
causing noticeable modifications. So we can keep
the changes in the modified image unnoticeable by
embedding more data in edged, or heterogeneous,
ASteganographicProtocolBasedonLinearError-BlockCodes
179
areas than in smooth, or homogeneous, areas. Thus
we can use large blocks for the most heterogeneous
pixels, and smaller blocks for less heterogeneous
pixels. The second fact consists of using further bit
planes to the least significant bit plane. It is clear
that modifying some bits from a given bit plane is
more suitable than modifying fewer bits from a
higher bit plane. This gives another solution by
putting least significant plane bits in large blocks
and higher plane bits in smaller and smaller blocks.
The method we present in this paper combines these
two solutions. Firstly, we select heterogeneous
pixels and use their first bit plane, then we tighten
our heterogeneity selection to get only the most
heterogeneous ones among them and, if we get
enough pixels, we use their second bit plane. We
continue this process to get further bits from higher
bit planes. Thus, we get more information carrier
bits. Meanwhile, our selection criterion ensures a
minimum perceptibility of modifications, and the
decoding algorithm of linear error-block codes
ensures that modifications are most likely to occur
over suitable bits, as we are going to see thoroughly
within the next sections.
3 COVER QUALITY
An important step in any steganographic scheme is
selecting the bits from the cover which are going to
carry the message. These bits are called the selection
channel. Basically, a steganographic scheme aims to
minimize the modifications made on the selection
channel. Furthermore, a well-chosen selection
channel reduces the possibility of detecting the
modifications. Hence, our aim is to hide as many
bits as possible by making the minimum
modifications in the cover.
Choosing the selection channel in image covers
is an approximate process. Although there are many
parameters that evaluate the quality of a modified
image, no parameter can precisely compare the
similarity between two images. The Mean-Squared
Error (MSE) and the Peak Signal to Noise Ratio
(PSNR) are widely used in the literature. The mean-
squared error between two images, presented by
their pixel value matrices
, and
,, is
defined by
∑
∈,∈
,
,
(2)
where and are the numbers of rows and
columns respectively in the input images. The mean-
squared error depends strongly on the image
intensity scaling. An MSE of 100.0 for an 8-bit
image (with pixel values in the range 0255) looks
dreadful, but for a 10-bit image (pixel values in
01023) it is barely noticeable.
The Peak Signal-to-Noise Ratio avoids this
problem by scaling the MSE according to the image
range:
10log
(3)
where is the maximal value possible for a pixel of
the image, i.e. 2
1 where is the bit-depth
of the image. PSNR is measured in decibels (). It
is a good measure for comparing restoration results
for the same image, but comparing two completely
different images by PSNR is meaningless. Values
over 36 in PSNR are acceptable in terms of
degradation, which means no significant degradation
is observed by human eye (Tan, 2003).
Nevertheless, both MSE and PSNR rely on pixel
intensities instead of image structure.
Figure 1: Adjacent pixels to a pixel p.
In order to predict the influence of modifying a
given pixel on the cover image, the method we
propose consists first of assessing the heterogeneity
of all cover pixels. Heterogeneity is a statistical
value that presents the resemblance between a pixel
and the pixels adjacent to it (see Figure 1).
Modifying pixels of large heterogeneity values
should produce images with unnoticeable changes
(small MSE and big PSNR values, relatively). We
set heterogeneity of a pixel to be the sum of
the absolute values of the differences between the
value of the pixel and the values of pixels adjacent
to it, divided by the number of the adjacent pixels.
For pixels which are not located within the edge of
the image we have the following formula.
1
8
|
|
(4)
Heterogeneity of pixels within the edges is
assessed by an adapted formula using only the
existing adjacent pixels. For smooth pixels we
SECRYPT2014-InternationalConferenceonSecurityandCryptography
180
expect to be a small value and it will have a
larger value for more complex pixels.
By calculating the heterogeneity values, we
construct a matrix, corresponding to the cover
image, the entries of which indicate the location of
the selection channel. Moreover we assign to each
pixel the number of bit planes we are allowed to
exploit. This is done by determining a list of
thresholds that present the heterogeneity value
required for modifying a given bit plane. A simple
formula to calculate the thresholds is given by
max
∈
min
∈
#
,for1,2,…,
(5)
where is the list of all cover pixels and is the
greatest bit plane we decided to use. The values
can be adjusted to allow us to have more or less
pixels in a given level. For example, we consider the
8-bit 256256 gray-scale image shown in Figure 2
and we set the list of heterogeneity thresholds to
30,50,80. This produces three sets of pixels that
we call pixel levels, which we illustrate in Figure 3.
First pixel level contains the pixels whose
heterogeneity value is greater than 30. Only the first
bit plane of these pixels is exploited. In the second
pixel level we exploit the first and the second bit
planes. The heterogeneity values of these pixels
must be greater than 50. The third pixel level is the
most resistant to alteration; the heterogeneity value
of its pixels must be greater than 80, so we can
exploit all of their first 3 bit planes. The pixels of
heterogeneity value less than 30 are not used.
Figure 2: The cover image.
1
st
level pixels 2
n
d
level pixels 3
r
d
level pixels
Figure 3: Heterogeneity levels.
In this way we construct an overall matrix, called
pixel levels matrix, whose entries are in
0,1,2,...,. They are represented in Figure 4 by
gray values starting from 0 (white) for pixels which
are not used, to 3 (black) for 3
rd
level pixels. The
sender and the receiver should agree on this matrix
in addition to the code’s parity-check matrix before
communicating.
Figure 4: Pixel levels matrix for thresholds 30,50,80.
4 HANDLING COVER BITS
WITH LINEAR ERROR-BLOCK
CODES
The pixel levels matrix defines, for each pixel level
1,2,…,, a sequence of
bits. Generally, in
any significant image most areas are homogeneous,
thus the number of pixels within high heterogeneity
thresholds goes fewer and fewer. This is suitable for
our method since they will be put in small size
blocks. However, the number of bit planes we are
going to use should provide enough pixels within
high thresholds. If this was not the case, we should
simply decrease heterogeneity thresholds, although
this would also decrease the quality parameters.
Our embedding process starts by splitting the
secret message into parts of bits. To embed a
given part, we construct a carrier vector of bits
using different pixel levels in the following way.
First we consider a partition
…
of the integer where
⋯
1 and is the number of bit
planes to be used. From each pixel level 1,...,
we take an
-bit vector
and construct a carrier
vector in the form
,
,…,
where
∈
. The maximum number of carrier vectors
we can construct is min
,1,2,...,. This
number is handled (increased or decreased) by our
choices of partition and/or heterogeneity thresholds.
Let us index the elements of and write it in the
form
...
where
...
1. For one bit modification in the -bit carrier
vector, the probability that this bit is located in the
ASteganographicProtocolBasedonLinearError-BlockCodes
181
block is
. Consequently, the modification of the
first plane bits is more probable than the
modification of the second plane bits, and the
modification of the second plane bits is more
probable than the modification of the third plane
bits, and so on. This ensures a good quality of the
modified image beforehand.
Let us see now what happens in the classical case
where
1
. In this case only one heterogeneity
threshold is used and it determines the lowest
heterogeneity value a modifiable pixel must have.
Thus, the whole message is to be embedded within
least significant bits (first bit plane). This ensures a
very good quality of the modified cover on one
hand, but on the other hand the number of
information carrier bits will be small. We can
provide more bits by decreasing the heterogeneity
threshold, but their number remains limited in
comparison with the general case. Therefore, the
classical code based steganography would work only
for small size messages.
The choice of partition can be improved by the
following. On one hand, in order to have a minimum
influence on the cover quality, we take
1 so
that a minimum number of
plane bits will be
used. On the other hand, changing an
plane bit of
a pixel gives the same MSE and PSNR values as
changing four 1
plane bits. This leads us to
set the summands of the partition in such a way that
each summand is four times its successor.
Consequently, the partition should have the form
4
4
…
4
1
. This requires
the code length to be related to the number of used
bit planes by
∑
4
. However,
the selection channel can provide better choices of
partition by analyzing the image structural
properties. The number of bit planes to use depends
essentially on the density of pixels with high
heterogeneity value in the cover. If the cover bits
were supposed to have equal influence on image
quality, for example all the pixels have close
heterogeneity values and/or only the LSB plane can
be used, then turns to the classical type
1
.
Now we recapitulate the steps of our protocol.
Let be an -bit message.
Secret sharing:
(a) Compute pixels heterogeneity, choose the
number of bit planes to use , set
heterogeneity thresholds and compute the
pixel levels matrix,
(b) Choose an parity-check matrix
and a partition
...
with
distinct summands.
Embedding:
(a) Using the pixel levels matrix construct an -
bit carrier vector of the form
…
,
…
,…,
,…,
,
(b) Compute
,
(c) Find the leader of the coset with syndrome
,
(d) Compute ,
(e) Update the image by replacing each bit
by
the bit
.
Retrieval:
(a) Using the pixel levels matrix construct
…
,
…
,…,
,…,
,
(b) Compute
.
5 RESULTS
We tested our protocol by embedding the 8-bit
6464 gray-scale image shown in Figure 5, which
contains 32768 information bits, within the 8-bit
256256 gray-scale image shown in Figure 2,
whose each bit plane contains 65536 bits.
Figure 5: The message image.
In the following we present the results of
applying several partitions with the [7,3] linear
error-block code given by the parity-check matrix
1110001
0111100
1001010
1010010
This code embeds a 4-bit message in a carrier of
7 bits. Thus, our message requires 8192 carrier
vectors. The maximum number and the average
number of altered bits cannot be described by the
code parameters as can be done in the classical case
by the covering radius value and the ratio
respectively. These parameters do not give a precise
assessment in our case since they do not depend on
the partition summands. Even if we consider the
block-covering radius
(Dariti, 2013) it would
give misleading values, since the alteration of a
SECRYPT2014-InternationalConferenceonSecurityandCryptography
182
Table 1: Image quality using a 7,3 linear error-block code.
Heterogeneity
thresholds
Number of blocks provided
by each bit plane
MSE
PSNR ()
1
1.8 8286 0.1798 55.5816
4
1
4,7
9399,8387)
0.3711 52.4349
43
4,7 9399,8387 0.6487 50.0104
52
4,7
7519,12580)
Not enough 1
s
t
plane blocks
52
3.5,11.3
8207,8327)
0.2951 53.4308
421
4.7,11,20 8521,8521,8514 0.7438 49.4159
given block can be caused by the alteration of any
number of its bits, which will give the same value
for different number of altered bits within the same
block. Consequently, the reliability of our protocol is
only assessed by the influence of alteration on the
image, presented by the MSE and PSNR values. In
Table 1 we also give the number of blocks provided
by each bit plane to construct carrier vectors. The
classical protocol corresponds to the classical type
1
given in the first row of the table. Note that it
uses only least significant bits. Therefore, to get
enough information carrier bits, we had to set the
heterogeneity threshold to a very low value.
By comparing MSE and PSNR values, it is clear
that partition
4
1
is better than 43. Both of
them use 2 bit planes and require the same number
of bits, but the first partition has a better structure as
was described in Paragraph 5, Section 4. Partition
52 with heterogeneity thresholds 4,7 does not
provide enough bits to embed the message. We
adjusted these to 3.5,11.3 which provided higher
embedding capacity, though at the cost of some
cover quality degradation. Notice that partition
421 as it uses 3 bit planes, provides a large
embedding capacity, which allowed us to set high
heterogeneity thresholds, and obtain close MSE and
PSNR values to the other partitions.
6 CONCLUSIONS
Comparing to classical code based steganography
protocols, our scheme based on linear error-block
codes increases the number of exploitable bits in a
given cover. Specifically, multiple bit planes in an
image are exploited whilst maintaining good MSE
and PSNR values. A major factor to get the
maximum benefit from this scheme is the choice of
the cover, as heterogeneous pixels within an image
promote using multiple bit planes. Heterogeneity
thresholds determine the number of bit planes to use
in each pixel in order to keep good image quality
parameters.
In this paper we compared the results of
embedding using different types of a given code,
including the classical LSB embedding which
corresponds to the classical type. Forthcoming
works involve finding optimal codes to use with this
method. Also, by statistical studies, the list of
heterogeneity thresholds is likely to be optimized.
REFERENCES
Crandall, R., 1998. Some notes on steganography. Posted
on the steganography mailing list.
Westfeld, A., 2001. F5 - a steganographic algorithm. In
Ira S. Moskowitz, editor, Information Hiding, Lecture
Notes in Computer Science, vol. 2137, pp. 289-302.
Springer Berlin Heidelberg.
Zhang W., Zhang, X., Wang, S., 2007. A double layered
"plus-minus one" data embedding scheme. IEEE
Signal Processing Letters, vol. 14, no. 11, pp. 848-
851.
Zhang, X., Zhang, W., Wang, S., 2007. Efficient double-
layered steganographic embedding. Electronics
Letters, vol. 43, no. 8, pp. 482-483.
Liao, X., Wen, Q. Y., 2008. Embedding in two least
significant bits with wet paper coding. In Computer
Science and Software Engineering, vol. 3, pp. 555-
558. IEEE Computer Society.
Dariti, R., Souidi, E. M., 2011. Improving Code-Based
Steganography with Linear Error-Block Codes. In
Digital Information Processing and Communications,
CCIS, vol. 189, pp. 310-321, Springer Berlin
Heidelberg.
Feng, K., Xu, L., Hickernell, F. J., 2006. Linear error-
block codes. Finite Fields and Their Applications, vol.
12, no. 4, pp. 638-652.
Tan, J., Luo, X., Cheng, Q., 2003. A lossless data
embedding scheme for medical in application of e-
diagnosis. In Proceedings of the 25th Annual
International Conference of the IEEE EMBS Cancun.
Dariti, R., Souidi, E. M., 2013. Packing and Covering
Radii of Linear Error-Block Codes. International
Journal of Mathematical, Computational Science and
Engineering, vol. 7, no. 4, pp. 14-18. World Academy
of Science, Engineering and Technology.
ASteganographicProtocolBasedonLinearError-BlockCodes
183
