Abstract
Information hiding allows us to hide secret information into digital objects such as images without significantly distorting the objects. The object containing hidden information will be transmitted to a data receiver via a probably insecure channel. To securely transmit the object carrying hidden information, the distortion caused by data embedding should be as low as possible, which is referred to as the rate-distortion optimization problem. Many conventional methods optimize the data embedding procedure by a heuristic fashion, which may be not optimal in terms of the rate-distortion performance. In this chapter, we introduce novel approaches that use graph theory for information hiding. These graph models are general and can be used for improving the rate-distortion performance of information hiding systems. In addition to rate-distortion optimization, recent graph models used for system design of information hiding will be also reviewed. This chapter is intended as a tutorial introducing advanced graph models applied to information hiding.
Keywords
- Information hiding
- steganography
- steganalysis
- watermarking
- graph theory
- rate-distortion optimization
- security
- covert communication
1. Introduction
Information hiding [1] is referred to as the art of hiding secret data into a digital object such as image and video without significantly distorting the object content. The workflow of information hiding can be described as follows. A data hider uses a secret key to embed secret data into a digital object (typically also called
Information hiding is actually an emerging and interdisciplinary research area covering various applications, among which watermarking [2] and steganography [3] are two popular branches nowadays. In particular, the ease of use, reproduction, edit and distribution of digital commercial products has led increasing concern to copyright protection for digital media files, resulting in that, watermarking is still a major activity in digital media processing. Different from watermarking that aims to protect the copyright of digital product, steganography, as another important branch of information hiding, conceals the existence of the hidden information, and therefore can be used for secret communication. It works by hiding a secret message into an innocent cover such as text, image, video and audio, by modifying the noise-like component of the cover in such a way that the resulting stego has a low distortion to the cover and therefore will not arouse suspicion.
As mentioned above, information hiding requires us to select a digital object as the cover to be embedded. A straightforward idea to categorize information hiding algorithms is therefore based on the type of selected cover. For example, image is still the most popular cover type due to its wide distribution over social networks. A number of information hiding algorithms are originally designed for images, e.g., [4, 5, 6]. Other covers such as video sequences [7, 8, 9, 10], speech [11], natural language [12] are of increasing interest to researchers. Recently, studies have demonstrated that even social behaviors [13, 14] can be exploited for information hiding. In terms of data embedding strategies, information hiding can be roughly divided into four categories, i.e.,
From the information-theoretic view, information hiding can be modeled as a communication task, which aims to securely transmit a secret message to a receiver by embedding the secret message into a cover without significantly impairing the cover. On the one hand, we expect to embed as many secret bits as possible so that a sufficient payload can be carried by the cover. On the other hand, for a fixed size of secret message, we expect to keep the distortion caused by data embedding as low as possible so that the stego will be seemingly normal and thus will not arouse suspicion. Accordingly, a most commonly used method for evaluating information hiding algorithms is rate-distortion performance, where “rate” means the payload size and “distortion” measures the difference between the stego and the cover.
Many information hiding algorithms optimize the rate-distortion performance in a heuristic fashion. For example, reversible image watermarking [15, 16] allows both secret data and the original cover to be reconstructed at the data receiver side. A common approach to realize reversibility is shifting a so-called prediction-error histogram (PEH) in a reversible way, i.e., the original PEH can be fully recovered from the marked PEH. One has to select suitable shifting parameters such that the distortion caused by shifting can be kept low, thus resulting in good rate-distortion performance. Many works [17, 18, 19] heuristically select the shifting parameters for embedding, which may be not optimal in terms of rate-distortion optimization due to cover diversity. It motivates us to study deterministic optimization methods that are applicable to various covers and lead to optimal/near-optimal performance.
Based on the aforementioned analysis, in this chapter, we will introduce graph models that can be used for rate-distortion optimization of information hiding. All these models follow the identical framework. That is, an optimization problem to be addressed in information hiding is first modeled as a graph problem. Then, by using a graph algorithm, the optimal solution to the graph problem can be found, indicating that, one can use the optimal parameters or strategies derived from the optimal solution of the graph problem (under constraints) for information hiding.
The rest of this chapter is organized as follows. First, we provide basic concepts and algorithms in graph theory in Section 2. Then, in Section 3, we introduce graph models in information hiding. Finally, we conclude this chapter in Section 4.
2. Basic concepts and algorithms in graph theory
A graph
2.1 Graph traversal and coloring
Graph traversal is referred to as the process of visiting each node in a graph. It can be done by using depth-first search (DFS) or breadth-first search (BFS) [20]. Taking DFS for explanation, a node
During the DFS or BFS, each node can be assigned with a color, which refers to graph (node) coloring [20]. We can use two different colors (say “red” and “black”) for graph coloring so that each node is colored with either red or black. Meanwhile, for each node (that is not isolated), there is always another adjacent node that has the different color to the present node. It can be done during the visiting process. For example, assuming that we have reached a node
2.2 Weighted bipartite graph matching
3. Graph models in information hiding
Figure 1 shows the general framework of information hiding, where an image is used as the cover. In this chapter, unless mentioned, we will use a grayscale image as the cover object. Referring to Figure 1, let
For data extraction, a data receiver should be able to extract
In some applications, e.g., reversible watermarking [16], it is further required that
3.1 Multi-bit mapping using graph coloring
A simplest information hiding method is least significant bit (LSB) replacement, which enables us to modify
where 1 ≤
For example, a secret bit 1 (i.e.,
Let
Mathematically, given a
In the first layer, by using DFS or BFS, each node in the original graph
Assuming that, the
The multi-layer graph coloring algorithm is finished when there has no new bit produced in a certain layer. For example, if there are 64 nodes in the original graph, the number of layers will be at most 6 since 64 = 26. For better understanding the multi-layer coloring procedure, we provide an example in Figure 3, in which there are six nodes and nine edges in the graph. Our goal is to map each node in the graph to a bit-stream with an indefinite length. First of all, we can use DFS to assign either “0” or “1” to each node. In detail, assuming that the visiting order by DFS is
It is inferred from Figure 3 that, for each node, we can always select such a node from the adjacent set that the selected node must match a prefix of the secret bit-stream to be embedded, namely, the graph in Figure 3 can be used for information hiding. For example, assuming that we have a cover sequence
3.2 Reversible embedding using graph matching
Reversible (data) embedding, also called reversible watermarking or reversible data hiding, not only allows a data receiver to extract secret data from the stego, but also enables the cover to be perfectly reconstructed from the stego. It is quite useful for sensitive applications that require no permanent distortion of the cover such as military imaging. A straightforward idea to realize reversible embedding is lossless compression, which losslessly compresses the noise-like component of the cover to reserve extra room for data embedding, i.e., the extra room will be used to carry the secret data and the compressed code. As shown in Figure 4, by extracting the losslessly compressed code from the stego, one can reconstruct the original cover without any error. However, since it is not easy to significantly compress the noise-like component in a lossless way, the size of the secret data will be low.
Unlike lossless compression, histogram shifting (HS) [21] embeds secret data into a histogram determined from the cover by reversibly modifying the histogram. Figure 5 shows an example of HS based reversible embedding in a gray-scale image. In Figure 5, a histogram determined from the original image is used to embed data. We first select two bins (peak bin/zero bin) from the histogram. The peak bin has the maximum occurrence, and the zero bin has no occurrence. Obviously, in Figure 5, the peak bin is “1” and the zero bin is “3”. Then, we shift all bins between the two bins towards the zero bin by one step, i.e., the bin “2” will be shifted to the right. It is equivalent to modifying all pixels having a value of 2 as that have a new value of 3. Thereafter, we use the peak bin “1” to carry secret data. Assuming that, the secret data is “0100”, for each pixel having a value of 1, it will be unchanged if the present secret bit to be embedded is “0”; otherwise, it will be replaced with a new value of 2 if the secret bit is “1”. In this way, the secret data can be embedded, resulting in a new image containing secret data (also called marked image). It is easy for a data receiver to extract secret data and reconstruct the original image. First of all, secret data can be retrieved by processing all pixels having a value of 1 or 2 in the marked image. Then, all pixels that have a value of 2 in the marked image will be modified with a value of 1. Finally, for those pixels having a value of 3, they will be modified with a value of 2. In this way, the resulting image will be equal to the original image. We refer a reader to [21] for more details about HS based reversible embedding.
Figure 5 corresponds to the classical HS operation, which can be improved by using two techniques. The first one is using a prediction-error histogram (PEH) for reversible data embedding, rather than the histogram directly determined from the original cover. The second one is using more histogram bins for data embedding, rather than only one peak bin in Figure 5. For better understanding, Figure 6 shows an example for constructing a PEH. In Figure 6, one can determine a histogram directly from the cover image, i.e., “Histogram I”. It can be seen that, the maximum occurrence is less than 3000, meaning that, we can embed no more than 3,000 bits if we use only one peak bin. However, it can be significantly improved if we can produce a PEH, e.g., by dividing the pixels into two sets (i.e., cross set and dot set), the pixels in the dot set (which are unchanged during data embedding) can be used to predict the pixels in the cross set (e.g., using the average value of the four adjacent pixels to predict the present pixel), allowing us to determine the prediction errors (PEs) of all pixels in the cross set. The PEs can be then used to build a PE histogram (PEH), i.e., “Histogram II”, from which we can find that the maximum occurrence is more than 150,000, indicating that, we can embed more than 150,000 secret bits by using a single PE bin. Obviously, we can use more PE bins for data embedding, which will result in the better rate-distortion behavior.
Mathematically, given a PEH
Shifting PEH bins requires us to accordingly modify the image pixels so that the marked image is corresponding to the shifted PEH. Regardless of the detailed steps of modifying the image pixels, it can be easily inferred that, in Figure 7, for a fixed
Each edge in
Therefore, we can conclude that, once
For better understanding, Figure 9 shows an example for searching the optimal (
3.3 Graph steganography in social networks
Many conventional algorithms use media objects as the cover. Recently, social behaviors are exploited for steganography. Comparing with media objects, social behaviors can be more easily concealed by the very large number of normal social activities. Moreover, both the data sender and the data receiver can easily integrate into the social networks. It is possible for the data receiver that he observes social behaviors for data extraction, without taking any other actions, which can well protect the real identify of the data receiver. From the point of algorithm design, one can extend existing methods originally designed for media objects to social behaviors, which may not well exploit the characteristics of social networks. On the other hand, by modeling social networks as graphs, we can use graph theory for conveying secret data, which is referred to as
Figure 10 shows the general framework for graph steganography. In Figure 10, the data sender (or say data hider) first converts secret data to a (directed) graph. Then, the graph will be embedded into social network by producing a sequence of seemingly-normal social interactions. For data receiver, he can retrieve the secret graph by observing the interactions without taking any other actions. In this way, he can finally recover secret data from the graph by inverting the graph generation procedure. In the following, we show the core steps for the general framework.
Given secret data
and
In this way,
In order to construct
The data receiver has the ability to perfectly reconstruct
After constructing
3.4 Other graph strategies in information hiding
In addition to the aforementioned graph models, graph theory can be also used for other purposes in information hiding. The reason is that graph theory enables us to model various types of relations and processes.
4. Conclusion and discussion
This chapter has introduced advanced graph models applicable to information hiding. In detail, we first model the data embedding operation in a graph, in which all cover elements are mapped to nodes and the modification relationship between cover elements are mapped to edges between the corresponding nodes. In this way, by applying the graph coloring technique to nodes, each node can be mapped to a bit-stream with an indefinite length, which allows a data hider to modify the cover elements according to the mapped bit-streams. Since the mapped bit-streams have fully exploited the modification relationship, a high data-embedding capacity can be achieved. We also model the parameter optimization task of HS based reversible watermarking as a weighted graph matching problem. In the built bipartite graph, the nodes are corresponding to the PEH bins to be shifted, and the edges show all the possible shifting operations. The weight of an edge equals the distortion caused by shifting one node in the edge to the other. By finding the MWMM in the built weighted bipartite graph, we can identify the optimal shifting solution for a fixed peak-set. In addition, we have also introduced graph steganography, which uses a graph to represent secret data and sends the graph via a social network platform by producing a sequence of ordered interactions such as “liking” and “forwarding”. In summary, all the above graph models first use nodes to denote cover elements and use edges to represent somehow relationship between nodes. Then, the target problem to be addressed is modeled as a graph optimization problem. By solving the graph optimization problem, the optimal or near-optimal information hiding strategies can be determined, accordingly providing superior information hiding performance. We have also discussed other potential graph strategies that can be used in information hiding. Since graph theory enables us to model various types of relations and processes in information hiding, it is believed that, graph models and methods will play an increasingly important role in information hiding.
Acknowledgments
It was supported by the National Natural Science Foundation of China under Grant No. 61902235, the Shanghai “Chen Guang” Project under Grant No. 19CG46.
Conflict of interest
No conflict of interest exists in this chapter, and it is approved by all authors for publication.
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