This article is another attempt of a comprehensive solution in the field of steganography data transmission. We also consider software model prototype that fully implements the process of hidden messages transmission in digital photographs. All stages of hidden message processing on the whole way from the sender to the recipient are taken into account. The programming model uses a discrete wavelet transform and new hiding algorithms based on the «Arnold cat map» decomposition. The efficiency of using noise-resistant coding methods and multithreshold decoding to ensure high probability of integrity and reliability of hidden messages when transmitting them through communication channels with a high level of noise is also shown.
The following features can characterize the current state of research in the field of steganography:
• Extensive use of discrete transformations of digital still images used as containers for steganography message, and the need to combine the well-known methods of discrete signal conversion with simple, well-tested old algorithms.
• Development of more and more new methods of steganography, which provide high secrecy, confidentiality and reliability of message delivery.
• Increasingly more complete account of the characteristics of data transmission channels and interference arising in it, especially in the conditions of their high noise level.1
Methods that allow messages to be embedded in the frequency domain first convert the container file and then only perform the embedding. These methods do not depend on image formats [Dav07], [Che08]. The information hiding the steganography methods is based on linear orthogonal transformations such as: • •
Mathematical and Natural-Scientific Training in Engineering Education", St.-Petersburg, Russia, 8–9 November, 2018, published at http://ceur-ws.org • Discrete cosine transform (DCT); • Discrete wavelet transform (DWT); • Singular value decomposition (SVD).
Among all the discrete transformations, the most popular are the discrete cosine transform (DCT) [Kus17] and the discrete wavelet transforms (DWT). The prevalence of these methods is explained by their wide use for image compression. Especially successfully, they are used in JPEG and JPEG2000 standards. The JPEG standard uses the DCT and the DWT is used the JPEG2000 image compression standard.
In [Kus17], the authors have successfully shown the use of combined stegoalgorithm on the basis of the method of the Least Significant Bit (LSB) in combination with DCT (LSB & DCT). In this paper, the authors tried to use LSB in combination with DWT (LSB & DWT). Let us dwell on this.
The General model of the steganography system is shown in figure 1. In this model, a secret graphic digitized message in 24-bit bmp format is used as an embedded message (figure 2).
The choice of a graphical object as an embedded message was made due to its higher resistance to noise in transformations. One of the main blocks in this model of the steganography system is the block implementing the function of embedding presented in figure 3. Consider its functionality.
First, it is designed to perform DWT and embed a pre-converted hidden message into it.
The main real indicators of concealing information methods in steganography are: also known that ACM is used to demonstrate the dynamics of chaotic processes (figure 6). • PSNR-peak signal to noise ratio; • Imbedded capacity; • Correlation.
PSNR is inversely proportional to capacity and directly proportional to correlation and vice versa. During the study, the correct ratio of PSNR to capacity and correlation was found, which suggests that the information can be sent over an unprotected channel of information transmission, without fear of unauthorized access by a third party. But also keep in mind that the larger the imbedded message, the greater its impact on the steganography container and for big data, you must choose a larger container.
In the proposed approach, DWT is used to decompose the image into high frequency and low frequency subbands.
In turn, imbedded message converted using the Arnold cat map (ACM). In mathematics, ACM is identified with the chaotic mapping on the torus first proposed by Vladimir Arnold [Div11].
ACM can be viewed as a two-dimensional map described by relations: " = (1) " = + 2 (1).
In these relations, the dash sign shows the dynamics of parameter values change at the next time step. As the phase space of ACM typically consider the surface of a torus. The parameter p on the torus specifies the coordinates of the Parallels, and the parameter q is the coordinate along the Meridian of the torus. The range of values of both parameters is limited by the interval from zero to one. Typically, a unit square with p and q coordinates is used as a graphical representation of ACM. The name ACM is because Vladimir Arnold illustrated it in a picture resembling a cat's head (figure 5) [Div11].
The record of these relations in the matrix representation has the form:
/11 21/ / / = /10 11/ /10 11/ //.
It should be noted that any picture subjected to ACM (for example, the cat's head) always retains its area. It is Also it should be noted that the ACM decomposition is iteratively reversible. Let us illustrate this fact by the example shown in figure 6. This figure shows that at a certain iteration step the image converted using ACM (in this case, the embedded message) necessarily takes the form of the original. In the example below, each picture has an iteration number corresponding to that picture. On iteration number 194 the image becomes equivalent to the original. Let's imagine that we used the original image as a hidden message, converting it to iteration number 50. Then when decoding this image to get the original message view, we need to perform 194-50 = 144 iterations! From here, we can conclude that the values 50 and 144 can be used as secret keys, respectively, at the stages of embedding and extracting the hidden message.
After the transformation is applied to the ACM algorithm, the hidden message is divided into RGB components, and embedded using LSB algorithm in the corresponding sub-band HL. After the implementation, the reverse DWT (RDWT) is applied, the components are assembled again and the filled stegocontainer is obtained in accordance with the embedding function (figure 3). ACM in this combination is used to increase safety. It allows you to extract a full secret message only to the recipient who has information about the method of embedding (key). The filled container is sent to the communication channel (figure 1) after the application of noiseresistant coding, where it is exposed to noise generated by the noise generator.
A noisy hidden message after the decoding procedure is passed to the input of the block that implements the extraction function (figure 1). The function of extracting a hidden message is performed in the following order. The DWT is first performed (similar to that shown in is assembled by applying a reverse DWT (RDWT) transformation to its RGB components and connecting them into a single unit.
Let us perform statistical analysis of the effectiveness of the developed stego-algorithm. Peak signal to noise ratio (PSNR) means the ratio between the maximum possible signal value and the noise power that distorts the signal values [Ami10]. This metric is used to show the difference between empty and filled containers:
Capacity is the relation of the hidden message size to container size. It is calculated by the formula: 2559 = 10 7
< .
The root mean square error (MSE) – determines the difference between the intensities of the filled and empty containers:
H
D 1 × = > >((, ) − ("(, )))9.
IFG EFG
Where f(i,j) is an empty container and f'(i,j) is a filled container. A large MSE value indicates that the original image is of poor quality, and Vice versa.
Capacity = hidden message pixels number container pixels number .
Correlation used to display a linear relationship between empty and filled containers [Mut11]: _` = ∑IdFG(I − )(I − )
.
( − 1)_`
Table 1 presents the main performance indicators of this algorithm. As can be seen from the table, the proposed method copes with hiding data in the image.
Recently, in the field of digital signal transmission in channels with a high level of noise, methods of noisecorrecting coding based on the use of multi-threshold decoders (MTD) of self-orthogonal codes (SOC) are intensively used. The prototype of MTD is a simple decoder of Massey [Mas69]. New technical solutions used in the MTD represent the implementation of an effective algorithm of noise-correcting coding. Distinctive features of MTD are: • Linear computational complexity; • High efficiency of error correction; • Iterative error correction process that constantly brings the decoding process closer to the optimal decoder; • Easy technical implementation; • Ability to work efficiently with different code speeds in high-interference channels; • High performance and significant energy gain.
Let us consider in more detail the device of MTD [Zol12]. An example of a multi-threshold character block coder (MTBC) scheme for a self-orthogonal code with one information branch is shown in figure 7. It can be seen that the encoder consists only of a shift register and a group of adders modulo q, where q=256.
In this example, the group of adders determined in accordance with the image of the polynomial P=x0+x1+x4+x6.
uj
2 The scheme of the multi-threshold character block decoder (MTCBD) for such code has the form shown in figure 8. The information register performs the role of the information branch here, and the role of the verification branch is the syndrome register.
The MTCBD scheme consists only of shift registers, adders, subtracts modulo q and a threshold element (TE).
TE task is to count the most common characters in the corresponding positions of the syndrome and difference registers. For example, the symbol q1 occurs a1 times, and the symbol q2 occurs b1 times. Then, the value of |a1 – b1| compare with some set threshold value in the majority element with further correction of the associated elements in case of exceeding the threshold value.
An example of the MTBC scheme with two information branches presented in figure 9. The parameters of the communication channel: q- synchronous channel
Number of information branches: nk=4 Number of test branches: nr=4 Number of possible symbol values: q= 256 Code rate: R=0.50
Code length: 9704 Number of decoding iterations: 7 The probability of error in the channel: P0=0.16000 Number of blocks transmitted = 21 Number of information characters transmitted = 101892
0 1 2 3 4 5 6 7 Number of errors at the output of different decoding iterations:
16347 14338 7384 259 0 0 0 0 Probability of error at the output of different decoding iterations: 1.60e-001 1.41e-001 7.25e-002 2.54e-003 0.00e+000 0.00e+000 0.00e+000 0.00e+000 The probability of an error on the symbol at the output of a non-binary decoder was 9.81e-006 The probability of an error on the block at the output of a non-binary decoder was 0.00e+000
The simulation was carried out with the help of MTCBD, which has 4 information and 4 verification branches. The output file of the simulation results contained information on the simulation parameters, the encoder and decoder used in the simulation, as well as information on the estimated error probability at the output of the MTCBD and the number of errors remaining after decoding iteration for each error probability in the communication channel. An example of a typical entry in the results file is shown in table 2. As can be seen in table 2, the MTCBD is very effective. Given a sufficiently high probability of error in the channel 0.16 (16 decibels) and the size of the source character file consisting of 101892 characters (bytes), combined into 21 blocks, all errors, the total number of which in the source file was 16347, were eliminated at the fourth iteration.
A binary synchronous communication channel (BSC) with an independent error stream (channel without memory) subject to Gauss distribution was chosen as a communication channel model.
Let us consider the results of the stenographic process modeling as a whole using the model shown in figure 1.
Figure 10 shows a 24-bit BMP graphic file used as a steganography container.
After the function, embedding secret message, filled stegocontainer are supplied to MTCBC, and after the encoding process is transmitted in noisy BSC. The output of the BSC file has the form shown in figure 11.
From the output of the BSC stegocontainer are supplied to the MTCBD where all the noise is removed, and the file stegocontainer are taking on the appearance it had at the entrance to the BSC.
Output MTCBD the steganography container are supplied to the removal unit, the output of which then issued a secret message.
The following parameters were set for the final simulation: • The code rate is 0.5; • The probability of error in channel P0 = 0.25; • Container size - 93640 bytes; • Number of generated errors - 23543; • Number of decoding iterations until all errors are fully retrieved - 18;
In the opinion of the authors, this paper describes a successful attempt of a comprehensive solution in the field of steganography container data transmission. The authors present a prototype of a software model based on four main components:
• Discrete wavelet transformation the steganography container are;
• Pre-coding the hidden message using Arnold's cat decomposition;
• Embedding encoded message on LSB algorithm in wavelet transformations the stegocontainer are; • Application of noise-resistant coding in the communication channel using advanced technologies of multi-threshold decoding using MTD (multi-threaded decoder).
Conventionally, this combination of four components can be designated as DWT & ACM & LSB & MTD.
The authors believe that this model fully implements the process of transmission of hidden messages in digital still images. The model takes into account and agrees on the features of all processing hidden message stages.
Studies carried out on this topic were carried out with partial financial support from RFBR grants (No. 16-2909482-ofi-m), under the budget theme No. 0073–2019– 0004.