Steganalysis of a PVD-based content adaptive image steganography

Signal Processing ] (]]]]) ]]]–]]]

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Contents lists available at SciVerse ScienceDirect

3

Signal Processing

5

journal homepage: www.elsevier.com/locate/sigpro

7 9 11 13

Steganalysis of a PVD-based content adaptive image steganography

15 Q1

Xiaolong Li a,b, Bin Li c, Xiangyang Luo d, Bin Yang a,n, Ruihui Zhu a

17

a

Institute of Computer Science and Technology, Peking University, Beijing 100871, China Beijing Key Laboratory of Internet Security Technology, Peking University, Beijing 100871, China College of Information Engineering, Shenzhen University, Shenzhen 518060, China d Zhengzhou Information Science and Technology Institute, Zhengzhou 450002, China b c

19 21 23 25 27 29 31

a r t i c l e i n f o

abstract

Article history: Received 15 October 2012 Received in revised form 25 January 2013 Accepted 21 March 2013

Pixel-value-differencing (PVD) is a well-known technique for content adaptive steganography. By this technique, secret data are embedded into the differences of adjacent pixels. Recently, a new PVD-based steganographic method is proposed by Luo et al. Besides realizing adaptive embedding using PVD, the new method also exploits a pairwise modification mechanism to reduce the distortion. In this work, a targeted detector is devised to detect the new PVD-based steganography. We show that although content adaptive approach may enhance the stego-security, Luo et al.'s PVD-based scheme is not a good choice for realizing adaptive embedding since it contains a serious design flaw in data embedding procedure and this flaw can lead to possible attacks. More specifically, by counting the differences of adjacent pixels in both vertical and horizontal directions, a folded difference-histogram is generated and we show that Luo et al.'s PVD-based method may arise significant artifact to this histogram which can be exploited for reliable detection. Experimental results verify that Luo et al.'s PVD-based method can be detected by the proposed detector even at a low embedding rate of 0.05 bits per pixel. & 2013 Published by Elsevier B.V.

Keywords: Content adaptive steganography Pixel-value-differencing Steganalysis

33 35 37 39 41 43 45 47 49 51 53 55 Q2 57 59 61

63 1. Introduction Steganalysis algorithms can be generally classified into two categories: targeted and universal [1–4]. Targeted algorithms aim to identify the existence of hidden data embedded by a specific steganographic method, whereas universal algorithms intend to detect a wide range of steganography. We consider digital image as cover data and study the technique of targeted steganalysis in this work. It is widely accepted that taking the characteristics of natural image into account may enhance stego-security. For example, it is obvious that embedding modifications operated in rough regions of a natural image are less perceptible than

n

Corresponding author. Tel.: +86 10 82529693; fax: +86 10 82529207. E-mail addresses: [email protected] (X. Li), [email protected] (B. Li), [email protected] (X. Luo), [email protected], [email protected] (B. Yang), [email protected] (R. Zhu).

that in flat regions. Besides, the slight modifications to rough regions cannot be easily perceived by analyzing usual image statistics since the embedding noise is covered by the inherent noise. Thus the content adaptive approach for steganography has the potential to provide a higher level of security. Based on this consideration, Wu et al. proposed the so-called pixelvalue-differencing (PVD) steganography [5], in which the difference value of a pixel pair is considered as a smoothness measurement and more data bits will be embedded into the pair if its difference is relatively large. Thereafter, numerous PVD-based methods are proposed [6–9] and their security are also discussed [10–13]. Recently, a new PVD-based method is proposed by Luo et al. [14]. By incorporating PVD with the pairwise embedding algorithm of Mielikainen [15], this method can realize content adaptive embedding and meanwhile provide a better PSNR compared with some previously proposed PVD-based methods. The experimental results reported in [14] show that this method is secure in resisting state-of-the-art steganalyzers.

0165-1684/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.sigpro.2013.03.029

Please cite this article as: X. Li, et al., Steganalysis of a PVD-based content adaptive image steganography, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.029i

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X. Li et al. / Signal Processing ] (]]]]) ]]]–]]]

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In this work, we propose a targeted detector to detect Luo et al.'s PVD-based method. We show that although content adaptive embedding is a way to enhance stegosecurity, Luo et al.'s PVD-based scheme is not a good choice for realizing adaptive embedding since it contains a serious design flaw in data embedding procedure and this flaw can lead to possible attacks. More specifically, by counting the differences of adjacent pixels in both vertical and horizontal directions, a folded difference-histogram is generated and we show that Luo et al.'s PVD-based method may arise significant artifact to this histogram which can be exploited for reliable detection. By our detector, Luo et al.'s PVD-based method can be detected even at a low embedding rate (secret data bits embedded per pixel, ER for short) of 0.05 bits per pixel (bpp). The rest of this paper is organized as follows. First, the embedding procedure of Luo et al.'s method is described in Section 2. Then, to better present our idea, we consider to detect a simple case of Luo et al.'s method in Section 3.1. A theoretical analysis of our method for this simple case is also provided. Next, the proposed detector for the general case of Luo et al.'s method is introduced in Section 3.2. Finally, experimental results are reported and conclusions are drawn in Sections 4 and 5, respectively.

By this means, the stego pixel pair ðx′; y′Þ can be given in a concise way

ðx′; y′Þ ¼

8 ðx; yÞ > > > > < ðx þ 1; yÞ ðx; y 7 1Þ > > > > : ðx−1; yÞ

65 if m ¼ 0 if m ¼ 1

ð4Þ

if m ¼ 2

29 31 33 35 37 39

71

where m ¼ b1 þ 2b2 −x−2y ðmod 4Þ:

ð5Þ

Particularly, when m ¼2, there are two solutions ðx; y þ 1Þ and ðx; y−1Þ, and one of them will be selected randomly. Step 4 (adjustment): After Step 3, x′ or y′ may be out of the range of ½0; 255 (i.e., overflow/underflow may occur) or the new difference jx′−y′j may be less than T. In such cases, to avoid overflow/underflow and remain the set EU (T) unchanged, ðx′; y′Þ is adjusted to ðx″; y″Þ ðx″; y″Þ ¼ arg minju−xj þ jv−yj

ð6Þ

2. Embedding procedure of Luo et al.'s method The data embedding procedure of Luo et al.'s method is described step by step as follows. Some remarks are also included in the description. Step 1 (pixel pair partition): First, for a pre-selected integer Bz∈f1; 4; 8; 12g, divide the cover image into nonoverlapped blocks of Bz  Bz pixels. Then, for each pixel block, rotate it by a pseudo-random degree chosen from {01,901,1801,2701}. Next, rearrange the resulting image as a row vector V by raster scanning. Finally, divide V into nonoverlapped pairs consisting of every two consecutive pixels. Step 2 (threshold selection): For every integer t≥0, define a set EUðtÞ D V as

Then take a threshold T as the largest t∈f0; …; 31g satisfying 2jEUðtÞj≥M, where j  j means the cardinal number of a set and M is the message length. Step 3 (PVD-based embedding): Permute pixel pairs of EU(T) in a pseudo-random order, and sequentially, embed 2 bits into each pair using Mielikainen's algorithm [15] until the message is embedded. We remark that Mielikainen's algorithm is a special case of the method proposed in [16]. Then, for a pixel pair ðx; yÞ∈EUðTÞ and 2 bits ðb1 ; b2 Þ to be embedded, the stego pixel pair ðx′; y′Þ is in fact determined as follows:

51 53 55

ðx′; y′Þ ¼ arg minju−xj þ jv−yj

ð1Þ

ð2Þ

ðu;vÞ∈S1

where

61

79 81 83 85

where

 Neglecting the rare cases of overflow/underflow, the 

S1 ¼ fðu; vÞ : −1≤u−x; v−y≤1; u þ 2v≡b1 þ 2b2 ðmod 4Þg: ð3Þ

89 91

Here we give some remarks:

solution to (6) for T≥1 can also be given in a concise way (see Table 1). When T ¼0, the adjustment step is unnecessary and the stego pixel pair is just ðx′; y′Þ determined by Step 3. It is essentially degraded to Mielikainen's algorithm [15] in this case. So the adaptive embedding can only be realized when T≥1.

Step 5 (rotation restoration): Finally, rotate back the blocks and the stego image is obtained. Take the Lena image for an example (see Fig. 1(a)), the positions of modified pixels due to the above embedding procedure are shown in Fig. 1(b), for an ER of 0.1 bpp with block size Bz¼1. One can see that the modifications are located in rough regions, and thus content adaptive embedding is realized by this PVD-based algorithm.

93 95 97 99 101 103 105 107 109 111 113

Table 1 Solution ðx″; y″Þ to (6) for T≥1. jx−yj4 T

x−y ¼ T

x−y ¼ −T

m¼ 0 m¼ 1

(x,y) ðx þ 1; yÞ

(x,y) ðx þ 1; yÞ

(x,y) ( ðx þ 1; y 7 2Þ ðx þ 1; y þ 2Þ

m¼ 2 m¼ 3

ðx; y7 1Þ ðx−1; yÞ

ðx; y−1Þ ( ðx−1; y7 2Þ

if T ¼ 1

ðx; y þ 1Þ ðx−1; yÞ

ðx−1; y−2Þ

if T 4 1

57 59

77

ðu;vÞ∈S2

ð7Þ

43

49

75

87

EUðtÞ ¼ fðx; yÞ∈V : jx−yj≥tg:

47

73

S2 ¼ fðu; vÞ : u≡x′ ðmod 4Þ; v≡y′ ðmod 2Þ; ju−vj≥T; 0≤u; v≤255g:

41

45

67 69

if m ¼ 3

25 27

63

115 117 if T ¼ 1 if T 4 1

Please cite this article as: X. Li, et al., Steganalysis of a PVD-based content adaptive image steganography, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.029i

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1

63

3

65

5

67

7

69

9

71

11

73

13

75

15

77

17

79 81

19 Fig. 1. (a) The 512  512 sized gray-scale cover image Lena. (b) Positions of modified pixels by Luo et al.'s embedding algorithm, for ER ¼ 0.1 bpp and Bz¼ 1.

83

21 3. Proposed steganalysis method 23 3.1. A case study with the block size Bz ¼1 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

We consider in this subsection a simple case of Luo et al.'s method where the block size Bz is fixed at 1. Such a case equivalently skips the processing before image raster scanning in Step 1 of Section 2. Define the difference-histogram of cover image as jfðx; yÞ∈V : x−y ¼ tgj ð8Þ jVj where V is the set defined in Step 1 of Section 2. The corresponding difference-histogram of stego image noted as hs can be defined in a similar way. From Table 1, the change of difference-histogram due to data embedding can be quantified. To this end, we first define a scalar parameter β as hc ðtÞ ¼

M ð9Þ 2jEUðTÞj where M is the message length and EU is the set defined in (1). This parameter represents the ratio of pixel pairs in EU (T) used to carry hidden data. That is to say, in Step 3 of Section 2, βjEUðTÞj pixel pairs in EU(T) are randomly selected for embedding while other pairs remain unchanged. We now analyze the change of difference value due to data embedding. For a pixel pair (x,y) used in data embedding, when jx−yj 4 T, referring to Table 1, we see that β¼

 if m ¼0, the difference value t ¼ x−y is unchanged after data embedding, where m is defined in (5),

 if m ¼1, t is changed to ðx þ 1Þ−y ¼ t þ 1,  if m¼2, t is changed to x−ðy þ 1Þ ¼ t−1 or x−ðy−1Þ ¼ t þ 1 

equiprobably, if m ¼3, t is changed to ðx−1Þ−y ¼ t−1.

Noticing that m is uniformly distributed in {0,1,2,3}, we may conclude that after data embedding, t will change to t, t þ 1, and t−1 with probability 14, 38, and 38, respectively.

Similarly, one can derive that, when T 41

 if t ¼ x−y ¼ T, t will change to t and t þ 1 with prob-

85

ability

1 4

and 34, respectively,

87

ability

1 4

and 34, respectively.

89

 if t ¼ x−y ¼ −T, t will change to t and t−1 with prob-

In addition, since βjEUðTÞj pairs in EU(T) are randomly selected for embedding others remain unchanged. We may assume that, for every t satisfying jtj≥T, each pair in the set fðx; yÞ∈V : x−y ¼ tg is selected for embedding data with probability β. That is to say, in the set fðx; yÞ∈V : x−y ¼ tg, a proportion of β pairs is used for data embedding while other pairs are not modified. Based on the above discussions, we may conclude that for a pixel pair (x,y) with difference t ¼ x−y, when T 4 1

 if jx−yj 4 T, t will change to t, t þ 1, and t−1 with probability ð1−βÞ þ

1 4β

¼ 1− 34 β, 38 β,

and

3 8 β,

respectively,

 if x−y ¼ T, t will change to t and t þ 1 with probability 1− 34 β

and

3 4 β,

respectively,

and

3 4 β,

respectively.

 if x−y ¼ −T, t will change to t and t−1 with probability 1− 34 β

91 93 95 97 99 101 103 105 107

Then, for T 41, the relation between hc and hs can be established as follows. As the difference-histogram is symmetric to 0 (i.e., we assume that hc ðtÞ ¼ hc ð−tÞ), only the case of t≥0 is shown 8 h ðtÞ; 0≤t o T > > > c > < ð1−2γÞhc ðtÞ þ γhc ðt þ 1Þ; t¼T hs ðtÞ ¼ 2γhc ðt−1Þ þ ð1−2γÞhc ðtÞ þ γhc ðt þ 1Þ; t ¼ T þ 1 > > > > : γhc ðt−1Þ þ ð1−2γÞhc ðtÞ þ γhc ðt þ 1Þ; t 4T þ 1 ð10Þ where γ ¼ 38 β. It should be mentioned that the relation (10) also holds for T ¼1 due to the symmetry assumption. The detailed demonstration is omitted here.

Please cite this article as: X. Li, et al., Steganalysis of a PVD-based content adaptive image steganography, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.029i

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X. Li et al. / Signal Processing ] (]]]]) ]]]–]]]

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1 3 5 7 9

hs ðTÞ≤ð1−2γÞhc ðTÞ þ γhc ðTÞ ¼ ð1−γÞhc ðTÞ≤hc ðTÞ hs ðT þ 1Þ≥2γhc ðTÞ þ ð1−2γÞhc ðT þ 1Þ≥hc ðT þ 1Þ:

13

where ra(t) is defined by, for t≥0 and a∈fc; sg

15

ha ðt þ 1Þ : r a ðtÞ ¼ ha ðtÞ

19

In addition, by (10) and (11), we can also prove that ð15Þ

35

qa ðtÞ ¼ 2r a ðtÞ−r a ðt þ 1Þ−r a ðt−1Þ:

37

Referring to Fig. 2(d), as expected, jqs j has a much higher peak at T compared with jqc j. Based on the above discussion, we take

25 27 29 31

39

da ¼ max jqa ðtÞj 2≤t≤31

41 43 45 47

p

ð14Þ

33

23

ð18Þ

where θ ¼ e−1=λ , λ 4 0 is the scale parameter, p 40 is the shape parameter, and Z is a normalization constant. This model provides a good fit to the derivative statistics of natural images [17,18] and the typical value of p is not larger than 1. For example, we show in Fig. 3 the true difference-histogram (blue) and its estimation (red) for Lena image. One can see that the GGD distribution can well model the difference-histogram. Moreover, we show in Fig. 4 the distribution of p for the database BOSSBase v1.00 [19] which contains 10,000 512  512 sized grayscale images. For those shape parameters, the maximum is 1.7586, the mean value is 0.4504, and only 1.41% of them are larger than 1. The shape parameters are estimated using Mallat's method [17].

ð13Þ

r s ðT−1Þ≤r c ðT−1Þ:

69 71 73 75 77 79 81 83 85

0.15

87

difference−histogram estimated GGD

89 91 0.1

93 95

ð16Þ

ða∈fc; sgÞ

65

p

ð12Þ

The facts (13) and (15) motivate us to consider the ratio of two adjacent difference-histogram bins as a trace of data embedding. Let us now present some experimental results. Fig. 2(a) and (b) shows the difference-histograms hc and hs respectively for Lena image. One can observe that the change occurs mainly at bins T and T þ 1. The ration functions rc and rs are plotted in Fig. 2(c). One can see that r c ðT−1Þ is lowered from 0.92 to 0.54, r c ðT þ 1Þ is also lowered from 0.87 to 0.64, while rc(T) becomes much larger and it changes from 0.88 to 2.10. The change is significant since rc takes its value in a relatively small range ½0:7; 1:1. In summary, we conclude that rs has a peak at T and two valleys at T 7 1, whereas rc changes smoothly. We then consider the following function, for t≥1 and a∈fc; sg

21

θjtj hc ðtÞ ¼ Z

Thus the ration hc ðT þ 1Þ=hc ðTÞ becomes larger, i.e., r s ðTÞ≥r c ðTÞ

63

67

ð11Þ

while hc ðT þ 1Þ becomes larger since

11

17

The theoretical reliability of our detector can be established by the following theorem with the cover differencehistogram modeled as a generalized Gaussian distribution (GGD) centered at 0

By (10), we observe that the data embedding may alter hc(t) for t≥T. Furthermore, by a reasonable assumption that hc(t) is decreased for t≥0, we can prove that after data embedding, hc(T) becomes smaller since

97 0.05

99 101

ð17Þ

103

as a detector for the simple case of Luo et al.'s method. Notice that the value qa(t) for t¼1 is excluded in (17) since a flat cover image may have a very high peak hc ð0Þ and thus generate a small r c ð0Þ. In this situation, jqc ð1Þj is large enough and it may be larger than jqs ðTÞj. This may lead to false detection if taking t¼1 into account in (17).

0 −50

0

50

105

Fig. 3. The true difference-histogram (blue) and its estimation (red) using Mallat's method [17], for Lena image. (For interpretation of the 107 references to color in this figure caption, the reader is referred to the web version of this article.) Q3

109

−3

x 10

49 51 53

0.08

hs

hs

0.07 0.06

0.04

113

|qs |

115

2

1.5

1.5 4

1

117

1

0.02 0.5

0 T

31

119

0.5

2 0

61

3

rs

2

c

2.5

6

0.01

59

|q |

8

0.03

57

rc

hc

10

0.05

55

111

3.5 hc

T−5

T

T+10

121

0 0

T

31

1

T

31

Fig. 2. Change of difference-histogram for Lena image (ER ¼0.1 bpp, Bz ¼ 1). (a) Right sides of hc and hs. (b) Zoom in of (a). (c) rc and rs. (d) jqc j and jqs j.

Please cite this article as: X. Li, et al., Steganalysis of a PVD-based content adaptive image steganography, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.029i

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Theorem 1. Suppose the cover difference-histogram hc is a zero-mean GGD with a shape parameter p∈ð0; 1. Then we have dc ≤0:25. Furthermore, if the following two conditions hold: (a) 1≥hc ð0Þ þ 2hc ð1Þ þ 3hc ð2Þ, (b) 2≤T≤30, we have ds ≥0:25 and thus the reliability of our detector is guaranteed. See Appendix for the proof of this theorem. The condition (a) in Theorem 1 is satisfied for most natural images. For example, in the database BOSSBase v1.00, 9,931 out of 10,000 images satisfy this condition and only some very flat images are exceptions. We now discuss the condition (b). Fig. 5 shows the proportions of images whose embedding thresholds satisfy T¼31, T∈f0; 1g, and T∈f2; …; 30g. One can see that the proportion for T¼ 31 decreases dramatically when ER increases. Besides, the case T∈f0; 1g may occur with a large probability only for high ER. Thus (b) can be satisfied for most images when ER is neither very low nor high. Moreover, it should be mentioned that the two conditions in Theorem 1 are just sufficient conditions and one can get much better detection result in practice than expected by exploiting this theorem. Fig. 6(a) shows the receiver operating characteristic (ROC) curves of our method, for BOSSBase v1.00 with different ER. One can see that a rather good performance is achieved when ER¼0.05 bpp. Moreover, it is worth mentioning that our detector still works for a very low ER of 0.01 bpp. In addition,

27 0.16

29

0.14

33 35

proportion

0.12

31

0.1

63 65 67 69 71 73

3.2. Proposed detector 75 In Section 3.1, we only consider the simple case of Luo et al.'s method where randomization on block size and block rotation is disabled. When random block size Bz∈f1; 4; 8; 12g and block rotation mechanism are used, the pixel pair locations (i.e., the set V) cannot be determined by steganalyzer. Thus the difference-histograms hc and hs defined in (8) cannot be obtained in this situation. We should provide a way to compute the detector when randomization exists. Since Luo et al.'s method is based on PVD, for a pixel pair (x,y) used in data embedding, y must be one of the four nearest neighbors of x. One can expect that the histogram abnormality also exists if all pixel pairs in both vertical and horizontal directions are counted. The detailed computation procedure of our detector is summarized below. For a given n1  n2 sized image I, consider first the set Vn composed of all pixel pairs (in overlapped manner) in both vertical and horizontal directions, i.e., we take V ¼ fðI i;j ; I i;jþ1 Þj1≤i≤n1 ; 1≤j≤n2 −1g ∪fðI i;j ; I iþ1;j Þj1≤i≤n1 −1; 1≤j≤n2 g:

0.08 0.06 0.04

0 0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

shape parameter

39 41

the values of dc and ds with ER¼0.05 bpp, for the first 100 images of BOSSBase v1.00, are plotted in Fig. 6(b), where obvious discrimination can be observed. Finally, we remark that in practice, the value of dc computed from cover image is a little larger than its theoretical upper bound 0.25. This is mainly due to the oscillation of histogram tails. We will see later that when computing the difference-histogram using both vertical and horizontal adjacent pixels, the resulting difference-histogram is more regular and the value of dc fits well to its theoretical bound.

77 79 81 83 85 87 89 91 93

n

0.02

37

5

Fig. 4. Distribution of the shape parameter of cover difference-histogram, for the database BOSSBase v1.00 [19].

ð19Þ

The size of Vn is clearly n1 ðn2 −1Þ þ n2 ðn1 −1Þ ¼ 2n1 n2 −n1 −n2 . Then we define the folded difference-histogram H as 8 jfðx; yÞ∈V n : x−y ¼ tgj > > if t ¼ 0 > < jV n j ð20Þ HðtÞ ¼ > jfðx; yÞ∈V n : jx−yj ¼ tgj > > if t 4 0: : n 2jV j

95 97 99 101 103 105

43 1

45

0.9

47

0.8

T = 31 0 ≤ T≤ 1 2 ≤ T ≤ 30

107 109

49 51

proportion

0.7

111

0.6 0.5

113

0.4

115

53 0.3

55 57 59 61

117

0.2 0.1

119

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ER (bpp) Fig. 5. Proportions of images whose embedding thresholds satisfy T ¼31, T∈f0; 1g, and T∈f2; …; 30g, for BOSSBase v1.00 with Bz ¼1 and different ER.

Please cite this article as: X. Li, et al., Steganalysis of a PVD-based content adaptive image steganography, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.029i

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6

1

1

63

3.5 d (cover) 3

probability of detection

3 5 7 9 11

2.5 0.6

67

2

69

1.5

0.4

71

1

73

0.05 bpp 0.03 bpp 0.01 bpp

0.2

0.5

75

0

0 0

15

65

ds (stego)

0.8

13

17

c

0.2

0.4

0.6

0.8

1

0

20

40

probability of false positive

60

80

100

77

image index

Fig. 6. (a) ROC curves of our method for the simple case of Luo et al.'s method where Bz ¼1. (b) Values of dc (cover) and ds (stego, ER ¼ 0.05 bpp, Bz¼ 1). The figure is plotted according to the sorted values of ds.

81

19 21 23 25 27 29 31 33

0.1

1.1

0.016 Hc Hs

0.08

Hc Hs

0.014

0.06 0.04 0.02 0

0.01

0.95

0.008

0.9

0.006

0.85

0.004

0.8

0.002

0.75

T

31

T−5

T

T+10

85

0.5

87

0.4 0.3

89

0.2

91

0.1

0.7

0 0

|Qc| |Qs|

0.6

1

0.012

83

0.7 Rc Rs

1.05

0 0

T

31

1

T

31

Fig. 7. Change of difference-histogram for Lena image (ER¼0.1 bpp, Bz¼ 4). (a) Folded difference-histogram H, for cover (noted as Hc) and stego (noted as Hs) images. (b) Zoom-in of (a). (c) Function R. (d) Function jQ j.

Next, we take RðtÞ ¼

39

Hðt þ 1Þ HðtÞ þ C

ð21Þ

41

where C is a small constant to avoid dividing by 0 and it is empirically chosen as e−8 in our implementation. Finally, we take

43

D ¼ max jQ ðtÞj 2≤t≤31

45 47 49 51 53 55 57 59 61

93 95 97

35 37

79

ð22Þ

where Q ðtÞ ¼ 2RðtÞ−Rðt þ 1Þ−Rðt−1Þ:

ð23Þ

The quantity D is taken as a detector to carry out detection, i.e., the image I will be classified as stego if D is larger than a predefined threshold or classified as cover otherwise. The ROC curve demonstrating the detector's performance can be obtained by varying the threshold. Referring to Fig. 7(a) and (b), one can see that H changes also at bins T and T þ 1 where the block size Bz is taken as 4 in data embedding. Though the change is not significant compared with that of Bz¼1 (see Fig. 2), the histogram abnormality also exists according to Fig. 7(c) and (d). The fact is a general phenomenon and it occurs for other images. Figs. 8 and 9 illustrate the cases of the first and last image for BOSSBase v1.00. The same abnormality can also be observed from those figures.

We remark that for the flat image “1.pgm” shown in Fig. 8(a), it has a very high peak H c ð0Þ (see Fig. 8(b)). This results in a large jQ c ð1Þj such that jQ c j and jQ s j have the same peak at bin 1 (see Fig. 8(e)). In this situation, the histogram abnormality cannot be captured if taking the maximum for 1≤t≤31. This example illustrates why we exclude the bin 1 when considering the maximum of jQ j. Let us see Fig. 10(a), it shows the values of D for cover (noted as Dc) and its corresponding stego image (noted as Ds) with ER ¼0.05 bpp and a random Bz. From this figure, obvious discrimination of cover and stego can also be observed. In addition, we show the distribution of Dc in Fig. 10(b). We observe that more than 90% of images in BOSSBase v1.00 satisfy Dc ≤0:25, which fits well to the theoretical bound. The performance of the proposed detector D will be reported in the next section.

99 101 103 105 107 109 111 113 115

4. Experimental results 117 The experiments are performed on the BOSSBase v1.00 database [19] which contains 10,000 512  512 gray-scale images. The stego images are obtained by Luo et al.'s method with random Bz∈f1; 4; 8; 12g and different ER. Then the detector D defined in (22) is computed for both cover and stego images.

Please cite this article as: X. Li, et al., Steganalysis of a PVD-based content adaptive image steganography, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.029i

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71 Fig. 8. Change of difference-histogram for the image “1.pgm” in BOSSBase v1.00 (ER¼ 0.1 bpp, Bz ¼8).

11 13

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0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

15 17 19 21

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1.1

c

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3 2 0

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x 10−3

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T

1 31 T−5

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0 T

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1

T

31

Fig. 9. Change of difference-histogram for the image “10000.pgm” in BOSSBase v1.00 (ER ¼0.1 bpp, Bz¼ 12).

83 85 87

25 0.7

0.08

Dc (cover)

27 0.6

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proportion

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91

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Ds (stego)

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40 60 image index

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Fig. 10. (a) Values of Dc (cover) and Ds (stego, ER ¼0.05 bpp, Bz is randomly selected from f1; 4; 8; 12g), for the first 100 images of BOSSBase v1.00. The figure is plotted according to the sorted values of Ds. (b) Proportion of Dc, for BOSSBase v1.00 database. Here, 9007 out of 10,000 images satisfy the condition Dc ≤0:25.

49 51 53 55 57

The proposed detector is evaluated by comparing it with following prior arts: the second order SPAM proposed by Pevny et al. [20], and the recently proposed detector [12] of Tan and Li which is specifically designed for Luo et al.'s method. SPAM utilizes 686 features extracted from the empirical probability transition matrix of difference image. For SPAM, we use the parameter independent classifier linear SVM1 to train and test. In each experiment, we randomly choose 80% of cover and 80% of stego images for training, and the remaining 20% for testing. The procedure is repeated 10 times for 5-fold cross-validation and ROC curves are vertically averaged. Tan and Li's method uses B-spline

59 61

105 107

45 47

103

1

http://www.linearsvm.com/

fitting to approximate the difference-histogram and the approximation residue is employed as features for detection. The comparisons results are shown in Fig. 11. One can see that the proposed detector yields a superior performance over the other two methods. For even an ER of 0.01 bpp, the proposed detector still shows detection power to some extent whereas both SPAM and the method [12] are failure to detect. When ER increases to 0.05 bpp or larger, better detection results can be obtained by the proposed detector. The proposed detector is also evaluated using the detection error under equal probability of cover and stego images: P E ¼ minP FP ðP FP þ P FN Þ=2, where PFP and PFN stand for the probabilities of false positive (detecting cover as stego) and false negative (detecting stego as cover), respectively. Let us see Fig. 12. It shows the comparisons of PE for low ER. One can see that our improvement over

Please cite this article as: X. Li, et al., Steganalysis of a PVD-based content adaptive image steganography, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.029i

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11 Fig. 11. Comparisons of ROC curves for the proposed detector, second-order SPAM [20], and the targeted detector [12].

75

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proposed method Pevny et al. Tan and Li

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0

25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 ER (bpp) Fig. 12. Comparisons of detection errors for the proposed detector, second-order SPAM [20], and the targeted detector [12].

20

40

60

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image index

Fig. 13. Estimated ER for Luo et al.'s method, for the first 100 images of BOSSBase v1.00. The true ER is 0.1 bpp.

87 89 91

Then, we take

93

prior arts is significant. Comparing our detector with SPAM, the decrease of PE ranges from 0.12 (ER¼0.01 bpp) to 0.41 (ER¼ 0.08 bpp). On the other hand, comparing our detector with Tan and Li's method, the decrease of PE ranges from 0.05 (ER¼0.1 bpp) to 0.2 (ER¼0.03 bpp). The reason why SPAM cannot work at low ER may be due to the fact that only sharp edges in cover image are involved in Luo et al.'s data embedding. But SPAM features that extracted from the empirical probability transition matrix which is truncated by a threshold only capture the dependency between small pixel differences. Thus it fails to capture the changes in large pixel differences. Compared with Tan and Li's targeted detector which uses the approximation difference-histogram residue, our detector exploits the relation among adjacent bins which can more directly reveal the change of difference-histogram. Moreover, we remark that, our method may capture the threshold T (see Figs. 7(d), 8(e), and 9(e)). In other words, for the stego image, the peak of the function jQ j defined in (23) is probably T. As a byproduct, we may estimate ER for Luo et al.'s method. Reviewing the set EU defined in (1), we know that the message length M satisfies the condition 2jEUðTÞj≥ M 42jEUðT þ 1Þj. Then the average value of 2jEUðTÞj and 2jEUðT þ 1Þj, jEUðTÞj þ jEUðT þ 1Þj, can be used as an estimation of M. Practically, since the sets EU(T) and EUðT þ 1Þ cannot be obtained by steganalyzer, we then propose to estimate ER by using the folded difference-histogram. For a stego image, we first determine the peak of jQ j as

as the estimated ER, where H is the folded differencehistogram defined in (20). Referring to Fig. 13, one can observe that (25) may give a good estimation with only few exceptions. Finally, it should be mentioned that our detector is invalid when T¼0. For this case, the embedding pixel pair is randomly selected from the entire image regardless of the difference value. It is just a kind of LSB matching with reduced distortion and the adaptive embedding has been disabled. The relation (10) no longer holds and we have hs ¼ f β nhc instead, where f β is a convolutional kernel: f β ð0Þ ¼ 1− 34 β and f β ð1Þ ¼ f β ð−1Þ ¼ 38 β. Unlike the case of T 40, it leads to low pass filtering the differencehistogram and will not arise significant abnormality. However, since T¼ 0 corresponds to the conventional LSB matching, we may consider using merged features. For example, we show in Fig. 14 the ROC curves by using both our detector and SPAM (1+686 ¼687 features in total). One can see that by using merged features, it will not lose detection power for low ER, while for high ER such as 1.0 bpp, it can significantly enhance the detection ability such that the steganography is detected. 5. Conclusion

119

In this work, we devised a targeted detector for detecting the recently proposed PVD-based embedding method [14]. We have shown that this PVD-based method

121

T n ¼ arg maxjQ ðtÞj:

59 61

0

2≤t≤31

ð24Þ

HðT n Þ þ 2 ∑ HðtÞ t4T

ð25Þ

n

Please cite this article as: X. Li, et al., Steganalysis of a PVD-based content adaptive image steganography, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.029i

95 97 99 101 103 105 107 109 111 113 115 117

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ER = 0.05 bpp

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probability of false positive

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1

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may arise significant artifact to the difference-histogram, and it can be well detected even for an ER as low as 0.05 bpp. In this light, we conclude that the PVD-based method [14] is not a good choice to realize content adaptive embedding.

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r s ðTÞ≥

The authors would like to thank Dr. Shunquan Tan of Shenzhen University, Shenzhen, China, for providing us the source code in [12].

77

2γ þ r c ðTÞ; 1−γ

ð32Þ

79

We first prove (31). Notice that hc ðt−1Þ þ hc ðt þ 1Þ ≥2hc ðtÞ is equivalent to

81

r s ðT þ 1Þ≤r c ðT þ 1Þ:

83

ð33Þ

where a ¼ ðt−1Þp −t p and b ¼ ðt þ 1Þp −t p . By the definition and estimation of f, we have −a≥b≥0. Thus, (33) can be proved by θa þ θb ¼

Appendix A

73 75

we have

θa þ θb ≥2

Acknowledgments

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probability of false positive

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ER = 1.0 bpp 1

Fig. 14. ROC curves for the proposed detector and merged features by using both our detector and SPAM [20].

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probability of detection

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probability of detection

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 −a  b 1 1 þ θb ≥ þ θb ≥2: θ θ

85 87 89

ð34Þ 91 2

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Only key points for the proof of Theorem 1 are presented here. For the shape parameter p∈ð0; 1 and an integer t≥2, let t p −ðt−1Þp f ðtÞ ¼ : ðt þ 1Þp −t p

ð26Þ

By the fundamental theorem of calculus, f(t) can be rewritten as R1 ðt−1 þ xÞp−1 dx f ðtÞ ¼ 0R 1 : ð27Þ p−1 dx 0 ðt þ xÞ Notice that ðt−1 þ xÞp−1

t ≤2 ≤ t−1

ð28Þ

45

1≤

47

holds for every x∈½0; 1. Thus f satisfies 1≤f ðtÞ≤2. By (14) and (18), we see r c ðt−1Þ ¼ ðr c ðtÞÞf ðtÞ . Then, since 0≤r c ðtÞ≤1 and 1≤f ðtÞ≤2, we have ðr c ðtÞÞf ðtÞ ≤r c ðtÞ and ðr c ðtÞÞf ðtÞ ≥r c ðtÞ2 . So, r c ðtÞ2 ≤r c ðt−1Þ≤r c ðtÞ. Hence,

49 51

ðt þ xÞp−1

0≤r c ðtÞ−r c ðt−1Þ≤r c ðtÞ−r c ðtÞ2 ¼ 0:25−ðr c ðtÞ−0:5Þ2 ≤0:25: ð29Þ

53 55 57 59 61

This yields an estimation of qc(t) for t≥2, −0:25≤r c ðtÞ−r c ðt þ 1Þ≤qc ðtÞ≤r c ðtÞ−r c ðt−1Þ≤0:25:

ð30Þ

Thus dc ≤0:25 is proved. We now consider to estimate qs(T). We claim that based on (10) and the following properties of hc hc ðt−1Þ þ hc ðt þ 1Þ≥2hc ðtÞ;

hc ðt−1Þhc ðt þ 1Þ≥hc ðtÞ2

ð31Þ

For the inequality hc ðt−1Þhc ðt þ 1Þ≥hc ðtÞ , it is equivalent to θaþb ≥1. This is direct since a þ b≤0. We now prove (32). In fact, one can verify that

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2hc ðTÞ2 þ hc ðT þ 2Þhc ðTÞ−hc ðT þ 1Þ2 r s ðTÞ−r c ðTÞ ¼ γ hs ðTÞhc ðTÞ hc ðTÞ 2γ : ≥ hs ðTÞ 1−γ

97 ð35Þ

99

Thus r s ðTÞ≥2γ=ð1−γÞ þ r c ðTÞ is proved. On the other hand, notice that

101

≥2γ

r s ðT þ 1Þ−r c ðT þ 1Þ ≤γ

hc ðT þ 1Þ2 þ hc ðT þ 1Þhc ðT þ 3Þ−2hc ðTÞhc ðT þ 2Þ ≤0: hs ðT þ 1Þhc ðT þ 1Þ ð36Þ

ð37Þ

2hc ð2Þ : 1−hc ð0Þ−2hc ð1Þ

113 115 117 119

jEUðT þ 1Þj ∑t≥Tþ1 hc ðtÞ ∑t≥3 hc ðtÞ ¼ ≥ jEUðTÞj ∑t≥T hc ðtÞ ∑t≥2 hc ðtÞ

¼ 1−

107

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Furthermore, based on the facts: M 4 2jEUðT þ 1Þj (see the definition of T in Section 2), hc ðtÞ=ð∑s≥t hc ðsÞÞ increases when t decreases (this is a direct result of (31)), and hc ð0Þ þ 2∑t≥1 hc ðtÞ ¼ 1, we have β4

105

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Thus r s ðT þ 1Þ≤r c ðT þ 1Þ is proved. By (15), (32) and (30), we can get, when T≥2 4γ þ qc ðTÞ≥4γ−0:25 ¼ 1:5β−0:25: qs ðTÞ≥ 1−γ

103

121 ð38Þ

Please cite this article as: X. Li, et al., Steganalysis of a PVD-based content adaptive image steganography, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.029i

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1

Finally, by (37) and (38), we conclude that

3

qs ðTÞ≥1:25−

5

Thus ds ≥0:25 holds since hc ð0Þ þ 2hc ð1Þ þ 3hc ð2Þ≤1.

7

References

9 11 13 15 17 19 21 23 25

3hc ð2Þ : 1−hc ð0Þ−2hc ð1Þ

ð39Þ

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[9] C.H. Yang, C.Y. Weng, S.J. Wang, H.M. Sun, Adaptive data hiding in edge areas of images with spatial LSB domain systems, IEEE Transactions on Information Forensics and Security 3 (3) (2008) 488–497. [10] C.H. Yang, S.J. Wang, C.Y. Weng, Analyses of pixel-value-differencing schemes with LSB replacement in stegonagraphy, in: Proceedings of the IIH-MSP, 2007, pp. 445–448. [11] L. Ji, X. Li, B. Yang, Z. Liu, A further study on a PVD-based steganography, in: Proceedings of the MINES, 2010, pp. 686–690. [12] S. Tan, B. Li, Targeted steganalysis of edge adaptive image steganography based on LSB matching revisited using B-spline fitting, IEEE Signal Processing Letters 19 (6) (2012) 336–339. [13] J. Fridrich, J. Kodovsky, Rich models for steganalysis of digital images, IEEE Transactions on Information Forensics and Security 7 (3) (2012) 868–882. [14] W. Luo, F. Huang, J. Huang, Edge adaptive image steganography based on LSB matching revisited, IEEE Transactions on Information Forensics and Security 5 (2) (2010) 201–214. [15] J. Mielikainen, LSB matching revisited, IEEE Signal Processing Letters 13 (5) (2006) 285–287. [16] X. Li, B. Yang, D. Cheng, T. Zeng, A generalization of LSB matching, IEEE Signal Processing Letters 16 (2) (2009) 69–72. [17] S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Transactions on Pattern Analysis and Machine Intelligence 11 (7) (1989) 674–693. [18] J. Huang, D. Mumford, Statistics of natural images and models, in: Proceedings of the IEEE CVPR, 1999, pp. 541–547. [19] P. Bas, T. Filler, T. Pevny, Break our steganographic system—the ins and outs of organizing boss, in: Proceedings of the 13th International Workshop on Information Hiding, Lecture Notes in Computer Science, vol. 6958, Springer, 2011, pp. 59–70. [20] T. Pevny, P. Bas, J. Fridrich, Steganalysis by subtractive pixel adjacency matrix, IEEE Transactions on Information Forensics and Security 5 (2) (2010) 215–224.

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