=Paper= {{Paper |id=Vol-3200/paper25 |storemode=property |title=Use of the Normalized Gap of Maximum Singular Value of the Image Block to Evaluate the Capacity of the Steganographic Channel |pdfUrl=https://ceur-ws.org/Vol-3200/paper25.pdf |volume=Vol-3200 |authors=Ivan Bobok,Alla Kobozieva,Nataliya Kushnirenko }} ==Use of the Normalized Gap of Maximum Singular Value of the Image Block to Evaluate the Capacity of the Steganographic Channel == https://ceur-ws.org/Vol-3200/paper25.pdf

Use of the Normalized Gap of Maximum Singular Value of the
Image Block to Evaluate the Capacity of the Steganographic
Channel
Ivan Bobok1, Alla Kobozieva2 and Nataliya Kushnirenko3
1,2, 3
Odessа Polytechnic State University, Shevchenko av., 1, Odesa, 65044, Ukraine

Abstract
The Least Significant Bit (LSB) method is one of the most widespread and demanded
steganographic methods nowadays. Detection and decoding the hidden information, embedded
in a container using the LSB, is a challenging task, in particular, in conditions of low capacity
of the hidden communication channel. The existing steganalysis algorithms developed to detect
the LSB, as a rule, solve the main problem of steganalysis - the detection of a hidden
communication channel. However, the problem of the additional information recovery remains
unfulfilled. The important step in solving this problem is the evaluation of the hidden
(steganographic) channel capacity. In the current work, a digital image is used as container. All
the results obtained can also be applied to digital video, which is considered as a sequence of
frames. The aim of the work is to get estimates for the value of the capacity of the hidden
communication channel, formed by the LSB method. To achieve the aim of the work the
following studies carried out: performed additional in-depth investigation of properties of the
normalized gap of the maximum singular value of non-intersecting image blocks, obtained by
standard splitting; studied properties of a discrete function y(QF), that determines the number of
image blocks in which the normalized gap of maximum singular value increases when the image
is re-saved to lossy format with quality factor QF. As a result of the research, the estimates of
the value of the capacity of the hidden communication channel, created using the LSB method
and based on a container in a lossy format, were obtained.

Keywords
Steganalysis method, digital image, the capacity of the hidden communication channel, the LSB
method, the normalized gap of singular value

1. Introduction steganalysis [3]. Powerful efforts of scientists
around the world today are aimed at solving the
main task of steganalysis - to identify the presence
Steganography today is one of the most
of hidden (additional) information in information
powerful and widely used areas of information
content [4]. However, in the condition of the
security. One of the main questions here is who
information confrontation, that takes place in the
holds such a powerful means of protection, since
modern world [2], these actions are not sufficient.
the use of steganography, unfortunately, can lead
Only the decoding of hidden information, its
to the setting up of hidden communication with
recovery will allow achieving the goal of
anti-state, illegal, inhuman goals [1,2]. In such
steganalysis to the fullest. The extracting of
cases, early detection of hidden communication is
hidden information and its decoding are the most
critical. The main "weapon" for here is

III International Scientific And Practical Conference “Information
Security And Information Technologies”, September 13–19, 2021,
Odesa, Ukraine
EMAIL: onu_metal@ukr.net (A. 1); alla_kobozeva@ukr.net (A. 2);
infsec2011@gmail.com (A. 3)
ORCID: 0000-0003-4548-0709 (A. 1); 0000-0001-7888-0499
(A. 2); 0000-0003-3722-0229 (A. 3)
©️ 2021 Copyright for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
complicated tasks. It can be facilitated by be considered promising for evaluating the
determining/evaluating the capacity of the steganographic channel capacity. To ensure the
organized steganographic channel [3,5], which is possibility of determining/evaluating the capacity
what this work is aimed at. of the hidden channel, additional studies of the
Today, one of the most widespread and properties of the function y (QF ) are required.
demanded steganographic methods is the least The aim of the work is to obtain estimates for
significant bit modification method - LSB [3]. the value of the capacity of the hidden
However, modern steganalysis methods, as a rule, communication channel, formed using the LSB
do not evaluate the capacity of the hidden method, by identifying the corresponding
communication channel [6,7,8]. additional properties of y (QF ) .
In [9], a steganalysis method was proposed,
which aimed at detecting a hidden communication
channel with low capacity. The method was 2. Main Body
based on properties of the normalized gap of the
maximum singular value of image matrix block. Let the image were initially saved in a lossy
In particular, it took into account the number of format; F1 is the matrix of the image, which is
image blocks, obtained by standard matrix subject to examination. Formally, it is saved in a
splitting, in which the normalized gap of the lossless format. If F1 is a steganographic
maximum singular value increased due to re- message, then we will assume that it is obtained
saving of image to a lossy format with different
on the basis of a Jpeg container with a matrix F
quality factors QF. This number was reflected by
. Let us apply to F1 the steganographic
the discrete function y (QF ) that was built for the
transformation with the low capacity of the
image under examination. Let us introduce the
hidden communication channel (for example,
appropriate notation.
1%), which formally represented as [11]:
Let F be the matrix of the digital image,
F1,1 = F1 + F , (1)
which is split in a standard way into non-
intersecting l×l-blocks with singular values [10] where F is the matrix representation of the
1   2  ...  l  0 , which form vector of additional information, F1,1 is the matrix of the

(
singular values  = 1 ,2 ,...,l )T ; the normalized image-steganographic message. Let us define
( )
functions y QF for F1 and F1,1 re-saving them
(
vector of singular values  = 1 ,2 ,...,l ) is
T
with losses with all possible values of the quality
determined by factor QF. For a particular QF, as a rule, the value
=  , y (QF ) for F1,1 will be greater than that for F1 .
where  is a norm of vector  . Then the Geometrically it means that the y (QF ) graph for
normalized gap of the singular value i , i = 1,l is F1,1 will be higher along the ordinate than the
determined as follows [9]: y (QF ) graph for F1 whether the message or the
svdgapn (i ) = min  j − i ,
i j container matches the matrix F1 . However, the
whence it follows that the normalized gap of the difference between the values of the function
maximum singular value is y (QF ) (between the corresponding graphs) will
svdgapn (1) = 1 − 2 be different depending on whether the matrix F1
and corresponds to the original image or
0  svdgapn (1)  1 steganographic message.
The efficiency of algorithmic implementation Let F1 be the matrix of the container, then
of the method proposed in [9] exceeds the modern the steganographic message (1) for it will be the
analogues in terms of the detection of the hidden first and only one. If F1 corresponds to the
communication channel in conditions of a low
steganographic message, then for it (1) is a
capacity. It means, that the mathematical basis of
repeated steganographic transformation. Let us
the method provides sensitivity to small
show that the primary transformation (1) with the
disturbances of the container in the process of
steganographic transformation, and therefore can help of the matrix F will "lift" the y QF ( )
graph higher along the ordinate compared to the will decrease as a result of the steganographic
graph constructed for F1 , than repeated transformation, the less the normalized gap of the
transformation using the same matrix F . maximum singular value in the blocks F1
The steganographic transformation of the involved in the steganographic transformation,
Jpeg container almost always leads to an increase the “higher” will be the graph of the function
in the smallest singular values and decrease in the y (QF ) , obtained when re-saving F1 with losses.
normalized gap of the maximum singular value in
When re-embedding additional information into a
the blocks involved in the steganographic
steganographic message formed with a relatively
transformation, thereby increasing the likelihood
significant primary capacity of the hidden
of the growth of the normalized gap of the
channel, there will be a significant number of
maximum singular value when the image is re-
blocks, where, after repeated steganographic
saved into a lossy format. If the additional
transformation, the normalized gap of the
information is embedded in the image-
maximum singular value will increase, rather than
steganographic message, then the normalized gap
decrease, in comparison with the normalized gap
of the maximum singular value in blocks,
of the maximum singular value in the block of the
involved in the primary steganographic
input steganographic message. This will lead to
transformation, is less than in the corresponding
the fact that when re-saving a steganographic
blocks of the original container. After the
message obtained as a result of a double
additional information is embedded in the
steganographic transformation, although the
( )
steganographic message, the smallest singular
values of the corresponding blocks involved in graph of the function y QF will be higher than
repeated steganographic transformation, which the graph of a similar function for the input
are no longer comparable to zero in those blocks steganographic message (obtained as a result of a
that were involved in the primary transformation, single steganographic transformation), this
can both decrease and increase. This fact can lead difference will be the smaller, the larger was the
to both an increase and decrease in the normalized capacity of the hidden channel of primary
gap of the maximum singular value. Re- steganographic transformation. It was confirmed
embedding the additional information in the in practice by the results of a computational
steganographic message will generally increase experiment, in which the following sets of digital
the resulting capacity of the hidden images were involved:
communication channel, additionally disturbing
the singular values, but the relative change in the • M Tif – 500 images in lossless format (Tiff)
smallest singular values of the container blocks be (150 images from 4cam_auth base [12], 275
greater than in the smallest singular values of images from img_Nikon_D70s base [13], 75
steganographic message with the same images taken by non-professional camera);
disturbance. Thus, the number of blocks in which • M Jpeg ,70 , M Jpeg ,75 , M Jpeg ,80 – each contained
the normalized gap of the maximum singular
value will increase at re-saving with losses of the 500 images, obtained by re-saving of images
steganographic message, obtained as a result of from the set M Tif to the Jpeg format with
consecutive double steganographic
QF=70, 75, 80 respectively (the most
transformation will be greater, than when re-
frequently used quality factors in practice).
saving the primary steganographic message.
At the first stage, additional information was
However, the degree of this increase will be less
embedded into the original image (with or without
than the degree of increase using the same (which
loss) with the capacity of the hidden
is characterized by matrix F ), but the primary communication channel of 1, 5, 10%. The original
steganographic message on an empty container. image-container and the obtained steganographic
Moreover, the degree of increase will be smaller messages were re-saved into lossy format (Jpeg)
the more the capacity of the hidden with all quality factors QF  1,2,...100  . As a
communication channel of the primary
result, discrete functions y0 (QF ) (for the
steganographic transformation. Indeed, the more
the capacity of the hidden communication channel container), y1 (QF ) , y5 (QF ) , y10 (QF ) ,
of the primary steganographic message, the more QF  1,2 ,...100  for the steganographic message
the number of container blocks, in which the were determined, respectively. A value
normalized gap of the maximum singular value characterizing the change in the function
y0 (QF ) was considered as a quantitative format (Jpeg) and the quality factor used
characteristic of the image change as a result of (QF=75). Using a different lossy format (for
the primary steganographic transformation: example, Jpeg2000) or a different quality factor
1 will only change the quantitative indicators of the
 100 2 2 histograms.

T0 ,i = 
 1
y0 (QF ) − yi (QF )  ,

(2) Analysis of the numerical values of (2), (3)
using the obtained histograms (Fig. 1) allows us
i  1,5,10. to make conclusions, that the following points are
At the second stage, additional information important for evaluation the value of the capacity
was re-embedded with the channel capacity of 1% of the hidden channel:
into the steganographic messages generated at the • If for image, which is under examination
first stage (the matrix of additional information the value Ti ,1  125 , then the
F was randomly generated, the same matrix steganographic transformation were not
was used for steganographic messages with the applied to it;
channel capacity of 1, 5, 10%, formed on the basis • If 61  Ti ,1 , then for analyzed image the
of one container ). Steganographic messages
obtained after the repeated steganographic capacity of the hidden channel is <5%,
transformation were re-saved with losses (Jpeg here the image can be a "clean" container;
format) with QF  1,2,...100  . As a result, discrete • If 26  Ti ,1 , then for analyzed image the
functions y1,1 (QF ) , y5,1 (QF ) , y10 ,1 (QF ) , capacity of the hidden channel is <10%.
The results obtained at this stage of the
QF  1,2 ,...100  were obtained for
research are not final, the quantitative estimates
steganographic messages with the channel obtained for the capacity of the hidden channel
capacity of the primary steganographic are one-sided (upper estimates), such that they
transformation of 1, 5, 10%, respectively. By depend on the value of the capacity of the hidden
analogy with (2), the following value was channel of the primary stegano-transformation
considered as a quantitative characteristic of the of the image in the Jpeg format (QF=75). By
change in the image-steganographic message after expanding the computational experiment, by
repeated transformation: increasing the variety of values of the capacity
1
of the hidden channel for the primary stegano-
 100 2 2

Ti ,1 = 
 1
yi (QF ) − yi ,1 (QF )  ,

(3) transformation (for example, from 1 to К% with a
step h%), the results obtained can be made more
i  1,5,10. precise, what will be done in the development of
The experimental results for the original the direct method for evaluating the capacity of
images in the lossy format for the case of the Jpeg the hidden communication channel. Using a
format with the quality factor QF=75 are shown different lossy format (for example, Jpeg2000) or
in Fig. 1 and in Table 1, where can be observed a different quality factor QF will change the
the general tendency of qualitative changes in the quantitative indicators of histograms, therefore,
values of estimates (2), (3) with increasing the the development of a method requires quantitative
capacity of the hidden channel of the primary characteristics for all possible (most used) values
steganographic transformation: decreasing the of the quality factor. Taking into account their
mode of the histogram of values Ti ,1 with a possible variety, the preliminary step of
determining QF for a container in a lossy format
simultaneous increase in the value in the mode; is required before using the method for estimating
decreasing the length of the interval of possible the capacity of the hidden communication
values Ti ,1 by decreasing the maximum value. Ti ,1 channel. It can be done using, for example, the
. The quality results obtained are typical for lossy method proposed in [14,15].
images, regardless of the specifics of the
a b

c d
Figure 1: Histograms of values Ti ,1 , i  0 ,1,5,10 , for the original image-container, saved in Jpeg with
QF=75: а – T0,1 (the mode equals 13, the value in mode is 19); b – T1,1 (the mode is 10, the value in
mode is 24); c – T5,1 (the mode is 6, the value in mode is 30); d – T10,1 (the mode is 5, the value in
mode is 41)

Table 1
Maximum and minimum values for the experiment Ti ,1 , i  0 ,1,5,10 for image-containers, initially
saved in Jpeg with QF=75
T0,1 T1,1 T5,1 T10,1
Max Min Max Min Max Min Max Min
146 2.1 124 2.4 61 1.7 26 2

3. Conclusions the maximum singular value increases when
re-saving with losses, is greater, than in mage-
container regardless of the container format
The paper studied the properties of the
(with/without losses);
normalized gap of the maximum singular value of
2. The primary steganographic transformation
the image matrix blocks, a discrete function
y (QF ) , that corresponds to the image in the of a digital image using a matrix F changes
(«lifts» along the ordinate) the graph of the
function y (QF ) higher, than a repeated
conditions of its re-saving with losses with
different quality factors and represents the
number of blocks in which the normalized gap of steganographic transformation using the same
the maximum singular value increases as a result matrix F ;
of re-saving. 3. The higher the capacity of the hidden
It is found that: communication channel of the primary
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