Detecting LSB Matching by Applying Calibration Technique for Difference Image

Transcription

1 Detecting LSB atching by Applying Calibration Technique for Difference Image iaolong Li Institute of Computer Science & Technology Peking University 87, Beijing, P. R. China Tieyong Zeng CLA, ENS Cachan, CNRS, UniverSud 6 Avenue du President Wilson, F-93 Cachan, France zeng@cmla.enscachan.fr Bin Yang Institute of Computer Science & Technology Peking University 87, Beijing, P. R. China yangbin@icst.pku.edu.cn ABSTRACT In this paper, we investigate the calibration technique used in steganalysis of LSB matching. Instead of working on the original image, we propose to calculate the calibration-based detectors (e.g. Calibrated HCF CO) on the difference image, which is defined as the difference of the adjacent pixels of an image. The theoretical reliability of the new detectors is carefully studied. oreover, several practical observations for enhancing the detectability are also given. The extensive experimental results clearly illustrate that the new detectors outperform the previous. Indeed, the new ones perform well even when the embedding rate is low. Categories and Subject Descriptors D.. [Software Engineering]: Software Architectures information hiding General Terms Security Keywords Steganalysis, LSB atching, Calibration, Difference Image. INTRODUCTION The goal of steganography is to embed a message within an innocuous looking cover data so that casual inspection of the resulting medium will not reveal the presence of the message. On the contrary, the purpose of steganalysis is to detect whether the observed data contains secret message. Correspondance author. This work was supported by National Key Technology R&D Program of China under contract No.6BAHA. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. &Sec 8, September 3, 8, Oxford, United Kingdom. Copyright 8 AC /8/9...$5.. The scope of more advanced steganalysis also includes the feature (e.g. length) estimation and even recovering the secret message. Generally, a cover data might appear as digital image, audio, video and text, etc. In this paper, we consider digital image as cover data and concentrate our efforts on the steganalytic techniques. Least significant bit (LSB) replacement and LSB matching are two widely-used steganographic schemes [, ]. The embedding process of LSB replacement is rather simple: first, converting the secret data into a stream of bits; then, choosing cover pixels in a pseudo-random order generated by a shared secret key; and finally, replacing the LSB of each selected cover pixel by the correspondent secret data bit. For LSB matching, it is a minor modification of LSB replacement: if the secret data bit does not match the LSB of the cover image, then is randomly either added to or subtracted from the cover pixel value. Some recent works [, 5, 8] have shown that LSB replacement can be easily detected even when the embedding rate (secret data bits embedded per pixel) is very low. However, the study on steganalysis of LSB matching is just in the early stage. Note that, in LSB replacement, cover pixels with even value are either unchanged or increased by, while the inverse is true for odd-valued pixels. This embedding asymmetry has been considered as the starting point for almost all the steganalytic methods of LSB replacement. Nevertheless, the situation of LSB matching is different. For any fixed pixel value, the probability of increasing or decreasing is equal and thus symmetric. Therefore, the usual steganalytic methods of LSB replacement can not directly used for LSB matching and the later is proved much harder to detect. In [3], Harmsen et al. proposed a steganalytic method based on the so-called HCF CO for the detection of additive noise based steganography. In [7], Ker pointed out that this method performs rather well for detecting LSB matching in RGB color images, but it is not reliable for gray-scale images. Indeed, for gray-scale images, although the HCF CO decreases after LSB matching embedding, its ranges for cover and stego image are heavily overlapped. Hence, the HCF CO can not distinguish well between the cover and the stego for gray-scale images. In order to overcome this shortcoming, [7] proposed an approach based on the calibration (downsample) technique, which is proved much more effective. The main idea of [7] is that the procedure of downsample can reduce the embedding noise. Furthermore,

2 in [6], Ker explained theoretically why the calibration technique is useful for detecting LSB matching. In [9], Li et al. further investigated the calibration technique and proposed to downsample only for non-oscillating pixels. They considered the ratio of the histogram s discrete Fourier transform (DFT) coefficients of the image to the corresponding coefficients of the downsampled image, and utilized a linear combination of these ratios as a detector. Experimental results have shown that, as compared to the method in [7], the detectors in [9] are more reliable, especially for uncompressed images. We also remark that, besides the above mentioned target detectors for LSB matching, there exists also the universal detectors which are intended to detect a wide range of steganographic algorithms, including previously unknown methods. For instance, the WA (wavelet absolute moment) detector proposed in [] is reported to outperform Ker s detectors in [7]. Usually, the universal detectors extract the features in certain domain and then apply the SV (support vector machine) or FLD (Fisher linear discriminant) to built -class classifier. In the literature, the calibration-based detectors are few steganalytic detectors which are able to give theoretical proof (as opposed to empirical demonstrations) of their correctness. In this paper, we investigate the calibration technique used in steganalysis of LSB matching. We propose to calculate the calibration-based detectors (e.g. Calibrated HCF CO) on the difference image, which is defined as the difference of adjacent pixels of an image. In Section, we give a brief introduction to the previous works [3, 7, 6, 9] for detecting LSB matching. In Section 3, we theoretically discuss the calibration-based detectors which are calculated on the difference image. Then in Section, extensive experimental results are reported. As compared to the previous works, the results illustrate the excellent performance of the new detectors. Finally, we conclude our discussion in Section 5.. PREVIOUS CALIBRATION-BASED DE- TECTORS Let I be a gray-scale image, h be the histogram of I and b h be the DFT of h. In [3], Harmsen et al. called b h the histogram characteristic function (HCF) of I and defined the center of mass of the HCF (HCF CO) as: k= C(I) = k b h(k) k= b h(k) (N = 56). () Note that the symbols used here are slightly different from [3]. Let I c be a gray-scale image, I s be its stego image by LSB matching with embedding rate α, h c and h s are their histograms. As a consequence of LSB matching embedding, we know that h s is a regularization of h c: h s = f α h c, () where the convolutional kernel f α is the distribution of embedding noise: f α() = α, fα() = fα( ) = α. (3) It follows that, in the DFT domain, ch s(k) = c f α(k) b h c(k) = ` α sin (kπ/n) b hc(k). () By Eq.() and the discrete Čebyšev inequality (Chap., []), we can get C(I s) C(I c). This illustrates that after LSB matching embedding, the HCF CO will decrease. Based on this observation, [3] proposed to use HCF CO as a detector to distinguish the cover and the stego image. ore precisely, for an image I, we first calculate C(I), and then classify the cover and the stego image according to C(I) T or C(I) < T, where T is a predefined threshold. [7] modified this detector. Let e I be the downsampled image of I, where its pixel value is given by ei i,j = (I i,j + I i+,j + I i,j+ + I i+,j+)/. (5) Experimental results have shown that, for cover image I, we have C(I) C( I). e However, C(I) < C( I) e holds for most stego images subjected to LSB matching. Therefore, [7] proposed to use C(I)/C( I) e as a dimensionless detector and showed experimentally that this detector is much more reliable than C(I). Here, the downsampled image serves as a calibration of the full-sized image for CO. In [6], Ker explained why the downsample technique is useful for detecting LSB matching. We summarize his idea in the following theorem (see [9] for details of the proof). Theorem. Suppose that, Y, Y,..., Y is a sequence of independent discrete random variables, is uniformly distributed on {,,..., } and Y i satisfies: P (Y i = ) = α/ and P (Y i = ) = P (Y i = ) = α/, where α ], ] is a constant. Then, we have j P + k j i= P Yi g (α), t = ; = t = g (α), t = ±, where g (α) = mλ m, λ m = P Y i = m. (6) i= oreover, when >, we have g (α) < α. Let I e c ( I e s, resp.) be the downsampled image of I c (I s, resp.). Now, we assume that the sum of four cover pixels {(i + i, j + j ) I c : i, j {, }} is uniformly distributed for mod. Then, by taking = in Thm., we can conclude that: Is e can be regarded as the stego image of I e c by LSB matching with embedding rate g (α) < α, the procedure of downsample reduces the difference between the cover and the stego image. oreover, as a consequence, the histograms of I e c and I e s: hc e and h f s, satisfy fh s = f eα e h c, (7) where eα = g (α) and the function f eα is defined in Eq.(3) (α is replaced by eα). [7] called the detector C (I)/C ( I) e as Calibrated HCF CO and proposed another two detectors based on two-dimensional histogram: Adjacency HCF CO and Calibrated Adjacency HCF CO. In [9], Li et al. proposed some improvements for Calibrated HCF CO and Calibrated Adjacency HCF CO. Let s introduce some notations first. For an image I, let d(k, I) = b h(k) / b e h(k), (8) where e h is the histogram of the downsampled image e I, b e h is the DFT of e h. Then, as a consequence of Eq.() and Eq.(7), we can get: d(k, I s) = α sin (kπ/n) eα sin d(k, Ic), (kπ/n)

3 which yields that: d(k, I s) d(k, I c). (9) Therefore, same as the HCF CO, the ratio d(k, I) decreases after LSB matching embedding. Then, to avoid the instability of d(k, I) in high-frequency (i.e. when k is large), [9] considered the cut-off of d(k, I): d (k, I) = min{d(k, I), } and use the following quantity as a detector: D (I) = k= s kd (k, I) k= s, () k where s k are weighted parameters. Note that, as a immediate consequence of Eq.(9), we have D (I s) D (I c), which provides a theoretical reliability for the detector defined in Eq.(). oreover, [9] suggested to downsampling only for smooth pixels. Precisely, they defined a subset ps(i) of the image I: ps(i) = {S i,j : D i,j < T }, where S i,j = {(i + i, j + j ) I : i, j {, }} is the set of four connected pixels, T is a predefined threshold and D i,j = I i,j I i,j+ + I i,j+ I i+,j+ + I i+,j+ I i+,j + I i+,j I i,j describes the oscillation for four connected pixels S i,j. Their idea is that the pixels with smaller oscillation will less change the histogram (for cover image) under the downsample procedure, and thus the better performance of detection is expected. Then, the final detector proposed in [9] is that D ps (I) = D(ps(I)). ore precisely, we first choose ps(i) for a certain threshold T ; then we downsampling ps(i) and calculate d(k, ps(i)) according to Eq.(8) (note that the image I is replaced by ps(i)); finally, we get the detector D (ps(i)) by Eq.(). In [9], the authors also considered the twodimensional histogram based detector D ps (I), we omit the detailed presentation due to the limitation of the space. 3. THE NEW ETHOD Considering the difference image I d, which is defined as the difference of adjacent pixels of a gray-scale image I: I d i,j = I i,j I i+,j + 55, () where the pixel value of I d varies form to 5. Our idea is that the difference image I d will well present the embedding noise as compared to the original image I when the image is wrapped by LSB matching, since the distribution of pixel value of I d is rather concentrated and the maximal modification changes from (for original image) to (for difference image) after embedding. Note that the key point of the calibration-based detector is that the procedure of downsample will reduce the embedding noise. Hence, we will investigate the change of histogram (from cover to stego) for the downsampled image of I d. First, we point out that the cover pixel value Ii,j d changes to (Ii,j d + s + t) after LSB matching embedding (with embedding rate α), where s and t are two independent random variables with distribution function f α (which is defined in Eq.(3)). Then we have: h d s = f α f α h d c, () where h d c (h d s, resp.) is the histogram of I d c (I d s, resp.), I d c (I d s, resp.) is the difference image of the cover (stego, resp.) image I c (I s, resp.). Now, we define the downsampled image (noted by e I d ) of I d as: ei d i,j = (I d i,j + I d i,j+)/. (3) Then, we give the following theorem before a further discussion. Theorem. Suppose that, Y, Y,..., Y is a sequence of independent discrete random variables, is uniformly distributed on {,,..., } and Y i satisfies: P (Y i = ) = α/ and P (Y i = ) = P (Y i = ) = α/, where α ], ] is a constant. Let j P + µ t = P i= Yi Then, we have g (α), µ = µ µ = µ k = t. () g (α) where the function g (α) is defined in Eq.(6). g (α), Proof. We decompose P i= Yi as a sum of two terms ( P i= Yi) + (P i=+ Yi), then by the definition of λm in Eq.(6) for m, it is easy to get: µ = µ = = m s+t + s+t + s+t=+m λ sλ t (s + t )λ sλ t st λsλt P, stλ sλ t = mλm s= t= which yields that µ = µ (g (α)/). Similarly, we can get: where µ = µ = F (s, t)λ sλ t + λ F (s, t) = s= t= j max(s, t), s + t ; min(s, t), s + t >. mλ m, Observing that λ + P λm =, by Eq.(6), we have: g (α) µ = = λ + s= s= t= t= g (α) λ m F (s, t)λ sλ t + λ a s,tλ sλ t, mλ m mλ m mλ m where a s,t = (s + t F (s, t)) st. Note that a s,t = a s, t = a s, t = a s,t, then when is odd (the

4 even case is similar), the above equation can be rewritten as g (α) g (α) µ = aa N a T, N = ( )/, where a = (λ λ, λ λ,..., λ N λ N ), A N = (a s,t) s,t N is symmetric. oreover, we can verify that: ««BN A N = + B A N, B N = + C N N, where B N = (b s,t) s,t N, b s,t = max{s, t}, C N = (c s,t) s,t N, c s,t = when s, t < N and c s,t = when s = N or t = N. Then, by the semi-positivity of C N, we can prove inductively that the matrix A N and B N are semipositive. This completes the proof. Let f h d c ( f h d s, resp.) be the histogram of e I d c ( e I d s, resp.), where e I d c ( e I d s, resp.) is the downsampled image of I d c (I d s, resp.) as defined in Eq.(3). Then, under the assumption that the sum of two pixels (I d c ) i,j + (I d c ) i,j+ is uniformly distributed for mod, and by taking = in Thm., we can get fh d s = e f α f h d c, (5) where e f α(t) = µ t, and µ t is defined in Eq.(). Now, by taking = in Thm. and Thm., we have µ = µ < It follows that α and µ = µ < α α. ef α(t) < (f α f α)(t), t {±, ±}. (6) Then reviewing Eq.(), Eq.(5) and Eq.(6), we can conclude that, as compared to the original (non downsampled) image, the probability of the pixel value p changing to (p+t) is reduced in the downsampled case, where t {±, ±}. In other words, the procedure of downsample reduces the embedding noise, for difference image. Furthermore, in the DFT domain, we have and ch d s(k) = (α α ) sin θ α sin θ ch d c (k), cfh d s(k) = ` µ sin θ µ sin θ c fh d c (k), where θ = kπ/n. Here, the value of N is 5 (instead of 56) since the histogram of difference image is defined on [, 5]. Then, same as Eq.(9), we have d d (k, I s) d d (k, I c), (7) where d d (k, I) = h cd (k) / c hd f(k). Eq.(7) guarantees the theoretical reliability of the detector defined in Eq.() for difference image, i.e. I is replaced by I d when we calculate the detector by Eq.(). After the above theoretical analysis, we now present some examples to show that the application of calibration-based detectors on difference image can significantly improve the detection performance. We also remark that the above theoretical analysis also holds for the sum image I s defined as: I s i,j = I i,j + I i+,j... 3 uncompressed images embedding rate is compressed images embedding rate is.. Figure : The ROC curves for the Calibrated HCF CO which is calculated on the difference image (solid), the sum image (dotted) and the original image (dashed), respectively. Nevertheless, the detection performance of the detectors carried on sum image is much worse than the one using difference image. Let s see Fig., which shows the comparison of Calibrated HCF CO calculated on three different types (difference, sum and original) of image: ) Detector C(I d )/C( I ed ) (solid), which is calculated on difference image, i.e. we first obtain I d by Eq.(), then the resampled image ei d by Eq.(3) and the HCF COs C(I d ), C( I ed ) by Eq.(). Note that here the value of Nin Eq.() is 5 instead of 56. This detector will be denoted by F d in the upcoming section. ) Detector C(I s )/C( I es ) (dotted), which is calculated on sum image and will be denoted by F s in the upcoming section. 3) Detector C(I)/C( I) e (dashed), which is the original Calibrated HCF CO proposed in [7]. From Fig., we can clearly see that the detector based on difference image is much better than its original version and the one applying for the sum image.. EPERIENTAL RESULTS First, we would like to point out two important observations. ) As the natural images are somewhat contin-

5 uous and the difference image describes the variation between adjacent pixels, hence the histogram of the difference image should be concentrated in an interval around 55. Experiments suggest that we can take this interval as [55 8, ]. Then, a natural idea is that we can calculate the histogram as below, for k {,,..., 56}: h d (k) = {(i, j) : I d (i, j) = 7 + k}, (8) and then normalize h d. Now, note that h d is defined on [, 56], then when we calculate the HCF CO by Eq.(), the value of N is 57. ) When calculating the calibration based detectors for diffrence image, the application of lowfrequency DFT coefficients usually leads to a better performance, e.g. we might use the first 6 DFT coefficients ch d (k) for HCF CO in Eq.(), instead of 8 (or, in other words, we might replace the upper index N/ by N/ for the sum). The comparison experiments illustrating this phenomena will be reported below. Next, we describe the image sets used in our experiments. ) Image Set (IS-): same as [7], we downloaded 3 images from the USDA NRCS Photo Gallery. For testing, we resampled each of them to the /3 of the original size (the size of the result images are about 7 5) and converted each image to gray-scale. ) Image Set (IS-): this set contains 5 images with good quality. These images were collected from several types of digital cameras and then resampled to make all the images with the size from to 8 8 and changed to gray-scale. 3) Image Set 3 (IS- 3): JPEG version of IS- with quality factor 9. ) Image Set (IS-): JPEG version of IS- with quality factor 75. For experiments, we consider the following detectors: Detector A: Calibrated HCF CO [7]. Detector B: Adjacency HCF CO [7]. Detector C: Calibrated Adjacency HCF CO [7]. Detector D: the detector D ps (I) defined in [9] with s k = k and the threshold T =. Detector E: the detector D ps (I) defined in [9] with s k,k = and the threshold T =. Detector F d : the detector C(I d )/C( I ed ). Detector F s : the detector C(I s )/C( I es ). Detectors G 6, G 8, G ps 6, Gps 8 : the detectors C(Id )/C( I ed ) (for G 6 and G 8) or C(ps(I d ))/C( ps(i d )) (for G ps 6 and G ps 8 ). Here we use Eq.(8) to calculate the histogram, the index 6 or 8 of the detectors means to calculate the HCF CO in Eq.() by taking the first 6 or 8 DFT coefficients, respectively, the index ps means to get the detector by using the technique of pixel selection proposed in [9], i.e. we choose a subset ps(i d ) of image I d as follows: ps(i d ) = n I d i,j, I d i,j+ : I d i,j I d i,j+ < T where T is a predefined threshold (we choose T = 3 in our tests), then we consider ps(i d ) as an image and calculate the detectors G ps 6 or Gps 8. Detectors H 6, H 8, H ps 6, Hps 8 : the detectors D(Id ) (for H 6 and H 8) or D (ps(i d )) (for H ps 6 and Hps 8 ) defined in Eq.() with s k = k. Here we use Eq.(8) to calculate the histogram, and the meaning of index 6, 8 or ps is identical as the detector G. Now, let s see Table. and Table., they show the false positive rate when the false negative rate is.5, which can o, Table : False-positive rate at which false-negative rate is.5, for uncompressed images. E-R means embedding rate. Image Set IS- IS- IS- IS- IS- IS- E-R Det. A [7] Det. B [7] Det. C [7] Det. D [9] Det. E [9] Det. F d Det. F s Det. G Det. G Det. H Det. H be regarded as a measure of performance for steganalytic detectors []. For each column (indicating a fixed image set), we underline three smallest values which stand for three best detectors for this image set. From the two tables, we can see that: ) F d is better than F s. This emphasizes the advantage of difference image, as compared to sum image. ) The difference image based detectors F d, G 6, G 8, G ps 6 and H 6, H 8, H ps 6 are better than the detectors A E, especially when the embedding rate is low. 3) The pixel selection technique can improve the performance of the detectors which use low-frequency DFT coefficients (for instance, we can compare G 6 with G ps 6, H6 with Hps 6 ), but it is useless for the detectors using all the DFT coefficients (note that the DFT coefficients are symmetric). ) When the embedding rate is low, the utilization of low-frequency DFT coefficients is useful. 5) The best two detectors are G ps 6 and Hps 6, which combine several techniques such as the application of difference image, the procedure of pixel selection and the utilization of low-frequency DFT coefficients. 6) The HCF CO based detector G ps 6 is slightly better than H ps 6, which uses the linear combination of the ratios for the corresponding DFT coefficients. Finally, to better illustrate the excellent performance of the new detectors, we present the comparisons of the ROC curves in Fig. for the following detectors: Detector G ps 6 (solid), Detector H ps 6 (dotted), Detector A: Calibrated HCF CO (dashed), Detector D: D ps (I) (dashdot). Here, the detectors A and D are ineffective while both the detectors G ps 6 and Hps 6 perform rather well. 5. CONCLUSIONS This paper investigated the calibration-based detectors calculated on the difference image to detect LSB matching. Combining techniques of pixel selection and utilizing lowfrequency DFT coefficients, the new detectors outperform the previous ones and are capable of detecting LSB matching in gray-scale image even when the embedding rate is low,

6 Table : False-positive rate at which false-negative rate is.5, for compressed images. E-R means embedding rate. Image Set IS-3 IS-3 IS-3 IS- IS- IS- E-R Det. A [7] Det. B [7] Det. C [7] Det. D [9] Det. E [9] Det. F d Det. F s Det. G Det. G Det. H Det. H uncompressed images (IS ) embedding rate is compressed images (IS 3) embedding rate is... Figure : The ROC curves for different detectors: G ps 6 (solid), Hps 6 (dotted), Calibrated HCF CO (dashed) and D ps (I) (dashdot). especially for compressed images. Our opinion is that the calibration (or, in other words, self-reference) technique is very useful for steganalysis, though this study is just in the initial stage. The more philosophical calibration techniques are expected in the future works. For instance, noting that the difference image is closely related to the Haar wavelet coefficients, a natural extension of our work is to considering the calibration technique in the transform domain. oreover, the combination of the calibration based detectors with SV or FLD also might be a valuable experimental task. 6. ACKNOWLEDGENTS We would like to thank the two anonymous reviewers for their helpful comments on this paper. 7. REFERENCES [] J. Fridrich,. Goljan, and R. Du. Detecting LSB steganography in color and gray-scale images. IEEE ultimedia, 8(): 8, October December. []. Goljan, J. Fridrich, and T. Holotyak. New blind steganalysis and its implications. In Security, Steganography, and Watermarking of ultimedia Contents VIII, volume 67 of Proc. SPIE, pages 3, 6. [3] J. J. Harmsen and W. A. Pearlman. Steganalysis of additive noise modelable information hiding. In Security and Watermarking of ultimedia Contents V, volume 5 of Proc. SPIE, pages 3, 3. [] A. D. Ker. Quantitative evaluation of Pairs and RS steganalysis. In Security, Steganography, and Watermarking of ultimedia Contents VI, volume 536 of Proc. SPIE, pages 83 97,. [5] A. D. Ker. A general framework for structural steganalysis of LSB replacement. In Proc. of the 7th International Workshop on Information Hiding, volume 377 of Springer LNCS, pages 96 3, 5. [6] A. D. Ker. Resampling and the detection of LSB matching in color bitmaps. In Security, Steganography, and Watermarking of ultimedia Contents VII, volume 568 of Proc. SPIE, pages 5, 5. [7] A. D. Ker. Steganalysis of LSB matching in grayscale images. IEEE Signal Process. Lett., (6):, June 5. [8] A. D. Ker. A fusion of maximum likelihood and structural steganalysis. In Proc. of the 9th International Workshop on Information Hiding, volume 567 of Springer LNCS, pages 9, 7. [9]. Li, T. Zeng, and B. Yang. A further study on steganalysis of LSB matching by calibration. Accepted by IEEE ICIP 8, to appear. [] D. itrinovic, J. Pecaric, and A. Fink. Classical and New Inequalities in Analysis. Kluwer. Academic Publishers, Dordrecht, The Netherlands, 993. [] F. A. P. Petitcolas, R. J. Anderson, and. G. Kuhn. Information hiding - A survey. Proc. of the IEEE, 87(7):6 78, July 999. [] T. Sharp. An implementation of key-based digital signal steganography. In Proc. of the th International Workshop on Information Hiding, volume 37 of Springer LNCS, pages 3 6,.