Security measure is of great importance in both steganography and steganalysis. Considering that statistical feature perturbations caused by steganography in an image are always nondeterministic and that an image is considered nonstationary, in this paper, the steganography is regarded as a fuzzy process. Here a steganographic security measure is proposed. This security measure evaluates the similarity between two vague sets of cover images and stego images in terms of
Security of the steganographic system is the fundamental issue in the field of the information hiding. Image steganography is the technique of hiding information in digital image and trying to conceal the existence of the secret information. The image with and the image without hidden information are called stego image and cover image, respectively [
Now, the study of the security measure becomes one of the hotspots in the steganography research field. Researchers have put forward their views from different viewing angels. From the point of view of information theory, Cachin [
The security measures mentioned above all assume that accurate statistical estimations can be obtained from the finite data samples. However, an image is a nonstationary process; its local statistical correlation will change when image is changed slightly. So the statistical features change is nondeterministic after steganography processing. Meanwhile, for a steganographic system, the warden is lack of the knowledge of the cover distribution. Thus, the distribution estimates of the cover and stego image are not stable. So the security measures defined under the deterministic statistical model are hard to apply due to the lack of the accurate distribution.
To address this problem, we regard the steganography as a fuzzy and indeterministic process. The goal of this paper is to provide a practical security measure in terms of the vague sets similarity measure between cover images and the stego images. Particularly, the sequence of image pixels is modeled as an We derive a security measure for a steganographic system which is different from the deterministic ones. The existing security measures are defined by evaluating the difference between cover images and stego images. In contrast, the new security measure is defined by evaluating the similarity between cover images and their stego version. The Simulation results verify the effectiveness of the new security measure by benchmarking several popular steganographic schemes. When embedding rate is low, the new security measure is more sensitive to reveal the statistical features change than other security measures. Thus, the proposed security measure can provide a better guidance for the design of steganography and steganalysis.
The rest of the paper is organized as follows. Section
Suppose
In fact, we have little information about the PMF involved due to the large dimensionality of the set
To account for the dependence of the pixels, Sullivan et al. [
The two security measures mentioned above are defined based on the Shannon information theory under the assumption that the image data statistical distribution is deterministic. Most of the security measures proposed later are also defined under the same assumption. However, the image data shows the sceneries in the aspects of gray, texture, shape, and so forth. There are many a kind of indeterministic factors (such as noise) in a steganography process. Therefore, the security measures with the deterministic statistical distribution model cannot measure the security accurately.
The weakness of the above two security measure lies in the fact that the image model such as i.i.d and first-order Markov are too simple to capture interpixel dependency. Therefore, here we model the sequence of image pixels as an
There are at least two reasons for us to select
Let
The generating process of the empirical matrixes of first-order and second-order Markov chain.
In Figure
Since the cover sources are strongly correlated, the probabilities of two adjacency samples are equal or nearly equal. As a result, in the empirical matrixes, the masses are more concentrated near the main diagonal in a correlated source. In [
Empirical matrixes of a cover image and its stego image.
The original empirical matrixes
The zoomed empirical matrixes
The vague sets similarity measure [
Roughly speaking, a fuzzy set is a class with fuzzy boundaries. The fuzzy set
Let
Let
Let
As discussed in Section
Suppose
Let
Moreover, a steganographic system is called perfectly secure if
Let
(1) Boundedness is
(2) Commutativity is
(3) Unity is
In this section, we report experimental results that demonstrate the capability of the new security measure. First of all in Section
For the experimental validation we used two image databases. The first one is BOWS2 [
Some images of image database.
Some images of BOWS2
Some images of NRCS
To evaluate the performance of the proposed method for measuring the security of the steganographic algorithms, the new security measure with different orders is used to measure the security of different steganographic algorithms with different embedding rates. First, we select some spatial-domain steganographic algorithms, including LSBM (least significant bit matching) [
The same order security measure for different steganographic methods with different embedding rate.
Zero-order security measure
First-order security measure
Second-order security measure
In Figure
Furthermore, in order to evaluate the measuring ability of different order security measures, we compare the security for the same steganographic algorithm using different order security measures. Figure
The different order security measures for the same steganography with different embedding rate.
For HUGO
For LSBM
For LSB2
To further verify the effectiveness of the proposed security measure. We used it to benchmark JPEG steganographic algorithms schemes on different database. And we focus on low payloads to see if any of the test steganographic schemes becomes distinguishable by using the vague sets security measure with finite image sample.
We selected 1500 images from NRCS Photo Gallery. All images were converted into grayscale and central cropped to a size of 512 × 512 for experimental purposes. The images were embedded with pseudorandom payloads with 5%, 10%, 15%, and 20% bpac (bits per nonzero AC coefficient). The tested stego schemes include F3, F5 without shrinkage (nsF5) [
Different order vague sets security measure for different steganography methods.
Steganography method | Embedding rate (bpac) | Zero-order | First-order | Second-order |
---|---|---|---|---|
F3 | 5% | 0.9768 | 0.9662 | 0.9569 |
10% | 0.9755 | 0.9647 | 0.9569 | |
15% | 0.9714 | 0.9608 | 0.9498 | |
20% | 0.9683 | 0.9584 | 0.9477 | |
|
||||
nsF5 | 5% | 0.9865 | 0.9736 | 0.9593 |
10% | 0.9847 | 0.9711 | 0.9542 | |
15% | 0.9840 | 0.9687 | 0.9531 | |
20% | 0.9818 | 0.9656 | 0.9515 | |
|
||||
MB1 | 5% | 0.9994 | 0.9879 | 0.9785 |
10% | 0.9991 | 0.9866 | 0.9699 | |
15% | 0.9965 | 0.9849 | 0.9673 | |
20% | 0.9959 | 0.9837 | 0.9656 | |
|
||||
MB2 | 5% | 0.9999 | 0.9868 | 0.9687 |
10% | 0.9996 | 0.9842 | 0.9624 | |
15% | 0.9987 | 0.9922 | 0.9617 | |
20% | 0.9982 | 0.9818 | 0.9609 |
The data in Table
To show the superiority of the proposed security measure
Looking at Figures
Vague sets similarity measure is a simple yet effective tool for measuring the similarity between two vague sets. In this work, a novel security measure for a steganographic system in terms of the vague sets similarity measure is proposed to measure the similarity between cover images and stego images. Particularly, in the new security measure, the sequence of image pixels is modeled as an
(1)
Since
Hence
(2) According to the definition of the
(3) From the proving procedure of property (1), we have
Hence
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Foundation of China (nos. 61462046, 61363014), the Science and Technology Research Projects of Jiangxi Province Education Department (nos. GJJ16079, GJJ160750), the Natural Science Foundation of Jiangxi Province (nos. 20151BAB207026, 20161BAB202050, and 20161BAB202049), Jinggangshan University Doctoral Scientific Research Foundation (nos. JZB1311, JZB15016), and Key Laboratory of Watershed Ecology and Geographical Environment Monitoring of NASG (nos. WE2015012, WE2016013).