Z-OAXN: An Approach to Reducing the Pixel Changes in LSB_Based Image Steganography Abstract Image steganography is a way of hiding information in image files. LSB is one of the most commonly-used algorithms in this field which depends on embedding data in the least significant bits of bytes representing visual properties of pixels in the image file. This paper introduces a reversible logic function called the Zolfaghari function and exploits the properties of this function to propose an approach called Z-OAXN to modify the traditional LSB algorithm in order to reduce the average pixel change probability. Analytical modelings as well as experimental results are used to evaluate the approach and both demonstrate that it can reduce the average pixel change probability in contrast to the traditional LSB technique by 64 to 100 percent. 1 Introduction and Basic Concepts Steganography is a technique in which one type of data is hidden in another. For example, in image steganography, any type of data can be hidden in an image file. One of the most commonly used algorithms for image steganography is the Least Significant Bit (LSB) algorithm. LSB uses the least significant bits of the bytes presenting the pixels of the image in a bitmap file to embed the bits of the data to be hidden. For example, suppose that a short text message like “Hello” is to be hidden in a 24-bit bit map image file. The ASCII code for the character H is 01001000 , this code is 01100101 for e, 01101100 for l and 01101111 for 0. Thus, the message can be shown as 0100100001100101011011000110110001101111 in the form of a bit stream. The bit stream is thus accommodated in the image file as shown by figure 1. Figure 1: The message “Hello” embedded in a 24-bit bit map file by LSB Suppose that the traditional LSB algorithm is applied to a 24-bit bitmap file to hide a given stream of bits. If we designate the probability that a pixel is changed after embedding the data with Ppc ( LSB ) and the inverse probability (the probability that a pixel remains unchanged) with Ppnc (LSB) , we can interpret Ppnc (LSB) as the probability that neither of the LSBs of the 3 bytes representing a pixel are changed or each of the 3 LSBs is the same as the corresponding bit in the data to be hidden. Thus, Ppnc (LSB) can be given by equation 1. 1 1 1 1 . . 2 2 2 8 Accordingly, Ppnc can be calculated as follows: Ppnc ( LSB ) (Equation 1) 7 (Equation 2) 8 In this paper, we refer to the average probability that a pixel is changed as the Average Pixel Change Probability or APCP for short. Our proposed approach to reducing APCP is called Z-OAXN. We use 24-bit bitmap pictures to explain the approach but it is applicable to some other picture formats- especially those which use lossless compression algorithms such as GIF- with minor changes. The main idea behind the Z-OAXN approach is making use of a reversible function called the Zolfaghari function to replace the LSBs of bytes which present pixels of the image. In order to explain the Z-OAXN approach, we need to present some basic concepts and some definitions which follow. Definition: A gate -or a circuit at all- is reversible if it implements a bijective logic function . A bijective function is one that is surjective and injective. Reversible functions are all conservative functions, that is, they have identical numbers of input and output lines and also identical numbers of 1s in the input and the output. Thus, the output of a bijective function can be considered as a permutation of the input. In other words, a reversible function leaves its inputs unchanged except for the last input whose change depends on the values of the other inputs (and maybe the last input itself). Bijective functions (permutations) can be denoted with different notations one of which is the cycle notation . In this notation, disjoint cycles are used to show the permutation. For example, a function that swaps 000(0) and 001(1) and also swaps 110(6) and 111(7) in the input and the output is denoted by (0, 1) (6, 7). Definition: An m-CNOT gate is a gate with m+1 inputs (and the same number of outputs) which transfers its first inputs to the output without any changes and complements the last input if all the first inputs are equal to 1; otherwise the last output is left unchanged like the others. An m-CNOT gate can be implemented by AND and XOR function as figure2 shows. Ppc ( LSB ) 1 Ppnc ( LSB ) Figure 2: An implementation of a CNOT gate It can be observed from figure 2 that the last output of an m-CNOT gate whose inputs are I 1 , I 2 ,… I m 1 can be demonstrated as ( I1 .I 2 ...I m ) I m 1 . Definition: An m-input OAXN (OR-AND-XNOR) function is a function whose last output is 1 if all the inputs have identical values and is 0 otherwise. Other outputs are equal to their corresponding inputs. It is obvious that this function gives 1 as its last output in only two cases. The first case is when all the inputs are equal to 1. This is the only case in which the AND function of the m inputs will be equal to 1. The second case is when all the inputs have 0 values. The latter case is the only case in which the OR function of the inputs can be 0 or equally the NOR function is 1. Therefore the last output of an m-input OAXN function can be realized by an m-input AND function, an m-input NOR function and a two-input OR function as figure 3 shows. Figure 3: The last output of an m-input OAXN function But for simplicity in demonstrating the OAXN function in terms of well-known reversible gates, we propose another implementation for it. This implementation is showed in figure 4. Figure 4: The XNOR Implementation of an m-input OAXN function In figure 4, the output of the XNOR gate is 1 only if the OR gate and the AND gate have identical outputs and this is possible only in two cases. The first case is when the AND and the OR both are 1, which requires all the inputs to be 1. The second case is when both AND and OR produce 0 outputs, which requires all the inputs to be 0. It is obvious that the implementation shown in figure 4 is equivalent to the one demonstrated in figure 3 from the output point of view. The name OAXN (OR-AND-XNOR) has been derived from the idea behind the implementation shown in figure 4. For consistency with previous works, we can show the OAXN function in terms of NOT and CNOT functions. To do this, we must note that if we show the last output of a CNOT gate by CNOTm and show the last output of an OAXN function by OAXN m , we will have: OAXNm ( I1 , I 2 ,..., I m ) ( I1.I 2 ...I m ) ( I1 I 2 ...I m ) I m 1 (Equation 3) Or: OAXN m ( I1 , I 2 ,..., I m ) ( I 1 .I 2 ...I m ) (( I1 .I 2 ...I m ) I m1 ) ( I 1 .I 2 ...I m ) CNOTm ( I1 , I 2 ,..., I m 1 ) CNOTm ( I 1 .I 2 ...I m ,CNOTm ( I1 , I 2 ,..., I m1 )) (Equation 4) Equation 4 shows that the OAXN function can be implemented by the NOT and CNOT gates as shown in figure 5. Figure 5: Implementing the OAXN function with NOT and CNOT gates The NOT gates on the output lines of the second CNOT gate in figure 5 are necessary because the OAXN function should not change its inputs except for the last one. Definition: An m-Zolfaghari function is a function with m+1 inputs and m+1 outputs which returns the complement of the last input as its last output if all its first m inputs have identical values and returns its last input unchanged in any other case. Such a function does not change its other inputs at all. The last output of an m-Zolfaghari function can be easily implemented by an OAXN between the first m inputs and an XOR function between the last input and the output of the OAXN function. Figure 6 shows such an implementation. Figure 6: The last output of an m-Zolfaghari function An m-Zolfaghari function is a permutation that can be designated in the cycle notation as (0, 1) ( 2 m 2,2 m 1 ) since swaps the 00…00 pattern with 00…01 and 11…10 with 11…11 in the output. According to figure 6, an m-Zolfaghari function can be implemented using an OAXN function and a CNOT gate as shown in figure 7. The reasoning is similar to that presented for the case of the OAXN function. Figure7: Implementing the Zolfaghari function using the OAXN and CNOT If we replace the OAXN function in figure 7 by the implementation given for it in figure 5, we will observe that the Zolfaghari function can be implemented using NOT and CNOT functions. Such an implementation has been shown in figure 8. Figure 8: The Zolfaghari function in terms of NOT and CNOT The Zolfaghari function has two important properties. The first is its reversibility. As figure 8 shows, The Zolfaghari function is reversible (because it can be implemented using the reversible NOT and CNOT functions). This is important because allows this function to be exploited in our proposed steganographic approach. In fact, this reversibility allows the hidden data to be revealed when required. The second important property of the Zolfaghari function is that it changes its last input with a relatively low probability. This property is important in steganography as well, because helps reduced the changes imposed to the cover image and makes it more difficult to discover the fact that some secret data has been hidden in it. The probability that the last input is changed by an m-Zolfaghari function is obviously equivalent to the probability that the output of the OAXN function of the first m inputs is 1. Since the OAXN function is 1 only in 2 cases among the 2 m possible cases, this probability can simply be calculated as follows. 2 1 (Equation 5) Plc (n) m m 1 2 2 The rest of this paper is organized as follows. Section 2 is dedicated to discussing the related works, Section 3 introduces the Z-OAXN approach, Section 4 evaluates the proposed approach and Section 5 is dedicated to conclusions and proposing further works. 2 Related Works There has been a lot of research regarding steganography and steganalysis in recent years. Among relevant topics on which large numbers of research works have been done, we can refer to topics such as developing steganography methods [19,32,33,38], developing techniques for steganography in video, audio, etc , applying statistical methods to steganography [5,35], benchmarking and performance evaluation of steganographic techniques [4,15,17,18,41], presenting techniques to choose among steganographic algorithms , presenting techniques for choosing proper host and cover files , developing steganalysis methods [20,22,23,39,47], applying statistical methods to steganalysis , applying artificial intelligence to steganalysis [6,16,24] and developing steganographic techniques that resist against steganalysis [3,7]. But the most relevant works to this paper are those that deal with analyzing, optimizing or introducing new variants of the LSB algorithm or finding new applications for it. In the following, we discuss some of these works. Chan and Cheng  proposed an optimization method which can augment the simple LSB algorithm and enhance the quality of the stego image with minor computational complexity and without any visually sensible changes in the cover image. They analytically modeled the worst case for their proposed method from the point of view mean square error between the cover image and the stego image. Dabeer, et al [48, 54, 55], proposed an approach to exploit hypothesis theory in steganalysis of images containing data hidden by LSB algorithm. Their approach reduces the problem of composite hypothesis testing to a simple hypothesis testing problem for an image with a known Probability Mass Function (PMF). They used images with known PMFs to obtain tests based on estimation of the host PMF and showed that their tests have superior self calibration and receiver operating characteristics compared with previously known tests. Goljan  presented a modular method for estimating the length of a bit stream embedded in randomly distributed of an image using the LSB algorithm. They also refined their approach to produce more accurate results for different types of natural images. They showed that their proposed approach has the advantage that fits well to statistical image models. Celik, et al , modified the traditional LSB method to obtain a reversible technique for embedding data. Their approach makes it possible to fully recover the host data when recovering the embedded data. The main idea behind their technique is transmitting parts of the host data that are suspectible to embedding distortion along with the hidden data after compression. Cvejic and Seppänen [45, 52, 53] proposed an LSB audio watermarking approach which decreases the distortion imposed to the host audio. In their proposed approach higher robustness against noise through embedding the watermark bits in higher LSB layers. They showed true listening tests that their approach also improves the perceptual quality of the watermarked audio. Ker [46, 49, 54, 37] argued that LSB matching is harder to discover than the LSB replacement method and therefore proposed modified variants of this method for embedding data in gray scale and color bitmaps. He argued that the Histogram Characteristic Function introduced by Harmsen  is effective for color images but ineffective to use on gray scale images. He used two techniques to make HCF more applicable. The first technique is calibrating the output using a down sampled image and the second is exploiting the adjacency histogram instead of the usual histogram. He demonstrated that his approach outperforms previous the traditional LSB matching technique especially in cases that the cover image is stored in JPEG files. Ker  combined the structural and combinatorial descriptors exploited by previous frameworks to introduce a general framework for steganalysis of simple LSB replacement in image files which can result in more powerful detection algorithms. He also introduced and studied a novel descriptor and showed through experiments that their suggested descriptor can perform that previously known ones. Wu, et al , introduced a new steganographic method which combines the ideas of LSB replacement and Pixel Value Differencing (PVD) in order to obtain high embedding capacity and minimize the difference between the cover image and the stego image. Their method obtains a difference between adjacent pixels to distinguish smooth areas from edge areas. Then hides the data in smooth areas using the PVD technique and hides in the edge areas through the use of LSB replacement. They showed that the security level of the stego image produced by their method is similar to that of the stego images produced by pure PVD method, but its hiding capacity shows notable improvement over PVD. Draper, et al , proposed a statistical approach based on Probability Mass Functions (PMFs) and frequency counts of pixel intensities for revealing messages hidden through LSB replacement. They generalized the tests proposed by westfeld and Pfitzmann  and Dabeer, et al[48,55], by ignoring the assumption that pixel intensities are independent random values identically distributed and considering PMFs of neighboring pixel intensities. They evaluated their test method and compared it to the RS test suggested by Fridrich, et al , which does not exploit PMFs. They showed that their test outperforms its PMF-based predecessors but the RS test performs better than their test and used these results to state that making use of statistical methods best on PMFs to reveal messages hidden by LSB replacement is inherently inefficient. =Brisbane, et al , argued that the steganographic method proposed by Seppänen, Makela and Keskinarkaus causes a high level noise and unreasonably decreases the quality of the image although it exhibits high embedding capacity. They presented a novel coding structure that causes low amounts of noise in addition to high embedding capacity. The maximum size of the coding structure was limited and this enhanced the capacity to distortion ratio. They also proposed an algorithm that helps identify pixels having high capacity to distortion ratios. Yu, et al , proposed a steganalysis method for LSB in cases that the data has been embedded in L>0 least significant bits instead of being hidden only in the LSB. They proved that the method had a high accuracy in both detecting the hidden message and estimating its length. They showed the efficiency of their method through the use of experimental results as well as analytical evaluation. Lee, et al , proposed a steganalysis methodology based on analyzing chains of horizontally and vertically adjacent pixels. They used experimental results to show that their methodology outperforms its predecessors and can detect secret data embedded with low rates. Raja, et al , combined the LSB algorithm with Discrete Cosine Transform (DCT) and compression techniques to present a novel image steganography method that improves the security of the embedded payload. Their proposed method uses the LSB algorithm to embed the payload in the cover image; DCT to transform the resulting stego image from spatial domain to frequency domain and in the last step exploits quantization and run length coding algorithms to compress the transformed image. They showed that their proposed method has the advantage to its predecessors that allows the transfer of images with low Bit Error Rates (BERs) without any need for passwords. Luo, et al , introduced an LSB steganography method that can not be revealed by steganalysis methods such as RS, SPA and DIH which are based on sample pair analysis. The idea behind their method is adopting chaotic and dynamic compensation techniques. They showed that their technique can make sample pair analysis based methods to estimate very small embedding rates even if the actual embedding rate is almost 100%. Rocha and Goldenstein  introduced an approach called Progressive Randomization (PR), to reveal data embedded in LSBs. Their approach is based on generating different images from the input image that differ only in the LSBs. Each of the produced images indicates a different possible stream of bits hidden in the image. The PR approach increases size and entropy in each step. They demonstrated that their approach can perform equally well or even better than the previously known techniques. Liu, et al , proposed a method based on feature extraction and pattern recognition to reveal data hidden using LSB matching. Their proposed method measures picture complexity using Generalized Gaussian Distribution (GGD) in the wavelet domain. They trained and classified their proposed feature sets by the use of several statistical pattern recognition algorithms. They demonstrated that their proposed method will be more convenient for color images and low complexity gray scale images. Ker  stated that highly-sensitive bit replacement methods do not perform very well on images in which the payload is concentrated at the start of the cover image and proposed an approach to attack this problem. Their proposed approach is in fact a modified variant of the Weighted Stego (WS) image steganalysis technique introduced by Fridrich and Goljan. They demonstrated that the modified WS method can exhibit about 10 times more accuracy that the traditional WS technique. Ker  modified standard LSB steganalysis methods to give them the ability to reveal data hidden in two Least Significant Bits instead of one LSB in an image. He demonstrated that his approach was much more sensitive and accurate than the only previously proposed method to reveal data hidden in multiple LSBs. He also showed that embedding data in two LSBs can be preferable to embedding in only one LSB. Charles, et al , suggested a new method to reveal data embedded by both LSB replacement and Â±1 LSB embedding techniques. Their method is based on a support vector machine classifier that uses statistics produced by a lossless compression method. They compared their method with one of the most renowned methods called pairs method. And showed that both performed equally well but their proposed approach can detect Â±1 LSB embedding while the Paris method could not detect data hidden by this technique. Luo, et al , referred to Sample Pairs Steganalysis (SPA)  as one of the most powerful methods and proposed a variant of the LSB steganography method that resists against the mentioned steganalysis method. Their proposed technique applies a dynamic compensation on the stego image that contains a secret data in randomly chosen pixels selected by a chaotic system. The dynamic compensation causes the SPA steganalysis to obtain very small estimate values and consequently fail to make correct judgments about the stego image, even in cases were the embedding rate is close to 100%. They demonstrated that their steganographic technique can also resist against Different Image Histogram (DIH) [60, 61] and Regular and Singular groups (RS)  steganalysis methods as well as different variants of SPA and RS methods. Zhang et al.  argued that with previous steganalysis methods, high-frequency noises are often mistaken as messages hidden in the picture using LSB matching techniques. They proposed a novel approach to overcome this shortcoming. The main idea behind their proposed approach is the fact that the intensity histograms of stego images show smeller absolute differences between local minima/maxima and heir neighbors in contrast to those of cover images. They demonstrated that their proposed approach is more convenient for cases such as never-compressed scans of photographs or video. Liu, et al [1,12], proposed combining a dynamic evolving neural fuzzy inference system (DENFIS) with a feature selection of support vector machine recursive feature elimination (SVMRFE) to introduce an approach based on feature mining and pattern classification to discover LSB matching steganography in gray-scale images. They also proposed a special feature set and argued that this set could outperform previously known sets of features. Liu, et al , introduced image complexity as a new metric for performance evaluation of steganalysis methods. They also used feature mining and pattern recognition ideas to build a steganalysis scheme that can be applied to images that contain secret data embedded by LSB matching. They demonstrated that in addition to the embedding rate, the image complexity influences the significance of features and the performance of the detection method and this dependencies cause their approach to outperform other previously propose techniques. 3 The Z-OAXN Approach The Z-OAXN approach exploits the properties of the Zolfaghari function to improve the LSB steganography algorithm from the pixel change probability point of view. Before explaining the way Z-OAXN works, we will need some definitions which are presented in the following. Definition: The cover image is the image in which some data is going to be hidden. Definition: The secret data is the data to be hidden in the cover image. Definition: The change domain is the set of pixels in the cover image which are selected to accommodate the secret data. These pixels may be adjacent, randomly distributed over the cover image or selected due to some predetermined strategy. Definition: The Storage Capacity indicates the number of bits of the secret data that can be embedded in each byte of the change domain in the cover image file. For example, the storage capacity of the traditional LSB technique is equal to 1, since it can store one bit of the secret data in each byte of the change domain in the cover file. Now we can explain the Z-OAXN technique. This technique works as follows. n-1 triples of bytes in the cover image file are selected which are representatives for n-1 pixels of the image. We designate these triples by p1 , p 2 ,…, pn 1 . Then the Least Significant Bits (LSBs) of corresponding bytes in these triples are given to 3 n-Zolfaghari functions as the first inputs. Each of the n-Zolfaghari functions has n+1 inputs. The n-th input of each n-Zolfaghari function is the LSB of the corresponding byte of the n-th pixel and the last input of each n-Zolfaghari function is one bit of the secret data. The last output of each n-zolfaghari function is also stored in the corresponding LSB of the n-th pixel. This way, 3 bits of the secret data are stored in the (n+1)th pixel. Figure 9 demonstrates the details of the mentioned mechanism for n=3. This figure shows the case where the 3 bits to be hidden are 000. Figure 9: Storing 3 bits of the secret data in the fourth pixel In figure 9, the values in the squares designate the bits of the secret data. The technique can also be explained in terms of the Zolfaghari function as figure 10 shows. Figure 10: The Z-OAXN technique explained in terms of the Zolfaghari function Figure 11 shows the LSBs of the (n+1)th pixel after embedding the 000 sequence by the Z-OAXN approach when n=3. Figure 11: The first four pixels after embedding the first three bits of the secret data As figure 11 shows, the first n-1 pixels remain unchanged. Thus, we can feed the LSBs of n-1 of them into the Zolfaghari function which helps store the next 3 bits of the secret data. The next 3 bits of the secret data are stored in p n 1 . This is done through giving corresponding LSBs of p 2 ,…, p n along with the bits of the secret data to nZolfaghari functions and storing the outputs of the functions in the corresponding LSBs of p n 1 . This is demonstrated in figure 12. Figure 12: Storing the next 3 secret bits in the next pixel of the cover image Embedding the next triples of the bits of the secret data continues this way until the triple n-1 is embedded in p2 n 2 . To do this, LSBs of the pixels pn 1 , … , p 2 n 3 are used. When this is done, 3 * ( n 1) bits of the secret data have been embedded in 2 * ( n 1) pixels of the cover image while the first n 1 pixels are absolutely unchanged and the next n 1 pixels may have or have not been changed depended on the outputs of the n-Zolfaghari functions. This way, 3 * ( n 1) bits of the secret 2 * ( n 1) pixels or equally data can be embedded in each 3 * 2 * ( n 1) bytes of the change domain in the cover file. Thus, the storage capacity of the Z-OAXN approach can be obtained from equation 6. 3 * ( n 1) 1 bit SC Z OAXN (Equation 6) byte 3 * 2 * ( n 1) 2 Equation 6 shows that the storage capacity of the Z-OAXN is one half that of the traditional LSB technique regardless of the value chosen for n. This means that the ZOAXN approach is useful for applications in which pixel change probability is more important than storage capacity. We can further clarify the way Z-OAXN works by giving a numerical example. To do this we have selected a 16*16 window of a real 24-bit bitmap image and hidden the message “Z-OAXN” in it by the use of the Z-OAXN approach with n=3. The changes made to the original image by the Z-OAXN approach are visually difficult to distinguish but we have not brought the original image and the modified image. The reason is that they do not have high visual qualities because of their small sizes. The string “Z-OAXN” will be converted to the bit stream“010 110 100 100 010 101 001 111 010 000 010 101 100 001 001 110” by the ASCII code. The length of this bit stream is equal to 48. We know that each pixel includes 3 LSBs in 24-bit bitmap format and the 1 storage capacity of the Z-OAXN approach is equal to . Therefore the number of the 2 pixels required to embed the mentioned message will be equal to 48 3 32 1 2 . The 32 pixels have been selected from the beginning of the image that is just after the header. g Table 1 shows the contents of the pixel before embedding the message. Pixel 1 2 3 4 5 6 7 8 Table 1: The contents of the pixels before embedding the message Content Pixel Content Pixel Content Pixel 424E58 5C685C 4C5961 9 17 25 535D64 596957 525D61 10 18 26 4E575B 5B6C57 525B5E 11 19 27 515959 5D6D5B 515B5B 12 20 28 5C6360 5D6D5C 555E5B 13 21 29 556663 5B695D 59625F 14 22 30 59605B 5A675F 5B655F 15 23 31 586059 5A6562 5A655D 16 24 32 Content 586359 576756 586A59 5C6D5F 5C6D60 596860 586660 5A6765 In table 1, the columns entitled “Pixel” show the numbers of pixels and the columns entitled “Content” show the contents of the pixels. Table 2 gives the contents of the first 32 pixels after hiding the message in the image. Pixel 1 2 3 4 5 6 7 8 Table 2: The contents of the pixels after embedding the message Content Pixels Content Pixels Content Pixels 424E58 5C685C 4C5961 9 17 25 535D64 596957 525D61 10 18 26 4E575B 11 5A6D56 19 535A5E 27 515858 12 5D6D5B 20 505A5B 28 5C6360 5D6D5C 555E5B 13 21 29 556663 5B695D 59625F 14 22 30 59605A 15 5B675F 23 5B645F 31 586158 16 5A6462 24 5A655C 32 Content 586359 576756 596B58 5C6D5E 5C6D60 596860 596660 5A6665 In table 2, the new contents of the changed pixels have been highlighted. This table 15 47% of the 32 pixels have been changed. As we will see in shows that 32 section 4, the expected average pixel change probability for n=3 is almost 22%. This difference is due to the similarity between corresponding LSBs in adjacent pixels. In fact, this example shows a bad case for the Z-OAX approach. However if we consider an adequate number of images and messages, we will reach more close results like those given in table 4. As shown in figure 8, the zolfaghari function can be explained in terms of the reversible NOT and CNOT functions. Thus, this function is reversible and we can use this reversibility in order to restore the hidden data using the original image and the cover image in which the data has been embedded. The reverse of a CNOT function is a CNOT function and the case is similar for a NOT function. Therefore, If we consider the implementation shown in figure 8, the reverse of the Zolfaghari function can be implemented as demonstrated in figure 13. Figure 13: The reverse of the Zolfaghari function in terms of NOT and CNOT According to figures 9, 10 and 12, the values of the LSBs of the pixels in the cover image are changed during the embedding of the secret data as equation 7 shows. P i / 3 (( [i %3] new i / 3 1 i / 3 1 j i / 3 n 1 j j i / 3 n 1 Pj [i%3]new .Pi / 3 [i%3]old ) ( P [i%3] new Pi / 3 [i %3]old )) I i (Equation 7) In equation 7, P j [k ]old is the LSB of the k-th byte of the j-th pixel before embedding the secret data, P j [k ]old is the same LSB after hiding the secret data and I i is the i-th bit number of the secret data which has been hidden in the cover image. Equation 8 which has been derived from equation 7, gives us the way to reveal each individual bit of the hidden data. I i Pi / 3 [i %3]new (( i / 3 1 i / 3 1 j i / 3 n 1 j j i / 3 n 1 Pj [i%3]new .Pi / 3 [i%3]old ) ( P [i%3] new Pi / 3 [i %3]old )) (Equation 8) As equation 8 shows, each bit of the hidden data can be directly revealed without a need for retrieving previous or next bits. But to do this, the original image and the cover image in which the data has been embedded should both be available. Applying equation 8 to the old and new values of the LSBs of the pixels in tables 1 and 2 will reveal the bit sequence “010 110 100 100 010 101 001 111 010 000 010 101 100 001 001 110” which is the ASCII coded equivalent of the message “Z-OAXN”. 4 Performance Evaluations In this section, we evaluate the impact of the Z-OAXN approach on the average pixel change probability (or APCP for short) through analytical modeling and experimental methods. Section 4-1 is dedicated to deriving an analytical model in order to predict the impact of the approach on the pixel change probability and section 4-2 uses experimental results to verify the correctness of the derived model. 4-1 The Analytical Model Suppose that the first 3 bits of the secret data are going to be embedded in p n .The probability that an individual LSB in p n is changed is equal to the probability that the OAXN function of the previous n-1 corresponding LSBs and I 1 (the first bit of the secret data) is equal to 1. This probability can obviously be obtained from equation 7. 1 1 (Equation 9) Z OAXN bc (n) 2 * n n 1 2 2 Thus, the probability that an individual LSB in the last pixel does not change can be calculated as follows: n 1 1 1 2 (Equation 10) Z OAXN bnc (n) 1 n 1 1 2 The probability that the last pixel is not changed is equal to the probability that all 3 LSBs in the pixel remain unchanged and is obtained as follows: Z OAXN n 1 pnc ( n) 2 2 1 n 1 n 1 .2 1 n 1 n 1 .2 2 1 n 1 2 n 1 2 (Equation 11) Now we can obtain the probability that the n-th pixel is changed as equation 12 shows. Z OAXN ( n) 1 2 n 1 pc 1 3 3n 3 3n 3 2 2 2 n 1 1 3 3n 3 2 (Equation 12) Or: 2n2 Z OAXN pc ( n) 3*2 n 1 3*2 3n 3 2 1 3n 3 2 (Equation 13) Equation 13 can be rewritten in the following form. 3 3 1 Z OAXN pc (n) 2 n 1 2 2 n 2 2 3n 3 (Equation 14) Now let us analyze the case where the second 3 bits of the secret data are going to be embedded in p n 1 . In this case the probability that an individual LSB in p n 1 is changed should be calculated as the sum of two distinct probabilities. The first is the probability that the LSB is changed while the corresponding LSB in p n has been changed too. We designate this probability by Z OAXN pc 1 ( n 1) . The second probability is related to the case where the LSB in p n 1 is changed while the corresponding LSB in p n has not been changed. The latter probability is demonstrated by Z OAXN pc 2 ( n 1) . We know that the change in any LSB depends on the related OAXN function. The first probability demonstrates the case that the OAXN function related to the corresponding LSB in p n and its counterpart in p n 1 both generate 1s in their out puts. Therefore, we can calculate the first probability as follows. 1 1 1 Z OAXN bc 1 ( n 1) * (Equation 15) n 1 2 2 2n 1 In equation 15, is the probability that the corresponding LSB in p n has been 2 n 1 changed (the immediate result is that all the first n-1 LSBs have had identical values) and 1 is the probability that the corresponding bit of the secret data has the same value. 2 The second probability is the representative of a case where the OAXN function related to p n has generated a 0 while its corresponding OAXN function in p n 1 is generating a 1. This is possible only in the case that the corresponding LSBs 1 3 in p 2 ,…, p n have identical values but either the LSB in p1 or the related bit of the secret data differs from them in value. Therefore, the second probability can be obtained from equation 16. 2 1 Z OAXN bc 2 ( n 1) * (1 ) (Equation 16) n 1 4 2 2 In the left side of the equation 16, is the probability that the corresponding LSBs 2 n 1 1 in p 2 ,…, p n have identical values and (1 ) shows the probability that the 4 input or the LSB in the first pixel has a different value. This equation can be rewritten as follows. 3 Z OAXN bc 2 ( n 1) (Equation 17) 2n Now we can obtain the probability that an individual LSB in p n 1 is changed as shown by equation 18. Z OAXN bc (n 1) Z OAXN bc 1 (n 1) Z OAXN bc 2 (n 1) (Equation 18) Or: 1 3 4 1 Z OAXN bc ( n 1) n (Equation 19) n n n2 2 2 2 2 Accordingly, equation 19 shows the probability that an individual LSB in p n 1 remains unchanged. 2n2 1 (Equation 20) 2n2 2n2 It is obvious that the probability that the whole pixel p n 1 remains unchanged can be calculated as follows. ( 2 n 2 1) 3 Z OAXN pnc ( n 1) (Equation 21) 2 3n 6 Now we can calculate the probability that the pixel p n 1 is changed as demonstrated by equation 22. ( 2 n 2 1) 3 3 * 2n4 3 * 2n2 1 Z OAXN pc ( n 1) 1 2 3n 6 2 3n 6 (Equation 22) Another form of equation 22 can be as follows. 3 3 1 Z OAXN pc ( n 1) 2 n 4 3n 6 (Equation 23) n2 2 2 2 Through similar reasoning, the change probability for the next pixels is obtained as follows Z OAXN bnc ( n 1) 1 1 Z OAXN pc ( n i ) 3 2 n 1 i 3 2 2 ( n 1 i ) 1 2 3 ( n 1 i ) i [0, n 2] (Equation 24) Among each group of 2n-2 pixels, the first n-1 pixels remain unchanged. Therefore, the average pixel change probability of the Z-OAXN approach can be calculated as follows. n2 n2 n2 3 3 1 n 1 i 2 ( n 1 i ) 3 ( n 1 i ) 2 i 0 2 i 0 2 Z OAXN pc i 0 2n 2 (Equation 25) Each of the sums in the numerator in equation 25 is the sum of a geometric progression with n-1 terms and we know that the sum of n-1 terms in each geometric progression can be obtained from equation 26. * (1 d n 1 ) S n 1 (Equation 26) 1 d In equation 26, is the first term and d is the common ratio of the progression. On 3 1 the other hand, in the first progression, we have 1 and d 1 .Thus n 1 2 2 the first sum can be written as follows. 3 1 * (1 n 1 ) n2 n 1 3 3 * ( 2 n 1 1) 2 2 (Equation 27) n 1 i 2 n 3 1 2 i 0 2 1 2 3 1 1 In the second progression, we have 2 and d 2 . 2n2 2 4 2 2 Therefore, equation 28 can be used to calculate the second sum. 3 1 * (1 2 n 2 ) n2 2 n 2 3 ( 2 2 n 2 1) 2 2 2 ( n 1 i ) 1 2 4n 6 i 0 2 1 2 2 (Equation 28) As for the third progression, we observe that 1 1 1 3 3 n 3 and d 3 3 . Thus, we can obtain the third sum from 8 2 2 equation 29. 1 1 1 * ( 2 3 n 3 1) * (1 3 n 3 ) n2 3 n 3 1 7 2 2 3 ( n 1 i ) 1 2 2 6 n 9 i 0 1 3 2 (Equation 29) Thus, the average pixel change probability of the Z-OAXN approach can be obtained from equation 30. 1 ( 2 3n 3 1) 3 * (2 1) 2 1 7 2 n 3 4n 6 2 2 2 6 n 9 2n 2 n 1 Z OAX pc 2n2 21* 2 4 n 6 * ( 2 n 1 1) 7 * 2 2 n 3 * ( 2 2 n 2 1) ( 2 3n 3 1) 7 * 2 6 n 9 ( 2n 2) 21* 2 5 n 7 7 * 2 4 n 5 21* 2 4 n 6 2 3n 3 7 * 2 2 n 3 1 7 * 2 6 n 9 ( 2n 2) (Equation 30) The derivative of the Z OAXN pc function shows that it gets smaller and smaller as n grows. This trend can be analyzed better through the calculation of the following limit. 21*2 5 n 7 Lim n Z OAXN pc Lim n 7 * 2 6 n 9 ( 2n 2) 3 Lim n n 2 0 (Equation 31) 2 ( 2 n 2) Equation 31 shows that when n grows large enough the average change probability of an individual pixel in the Z-OAXN approach declines until it approaches zero. Now let us quantitatively examine the improvement in the average pixel change probability gained by the Z-OAXN approach. Since the APCP of the traditional LSB 7 technique is equal to , the improvement ratio of this probability can be calculated as 8 follows. 7 Z OAXN pc 7 8 * Z OAXN pc 8 I pc 7 7 8 (Equation 32) Or: I pc 7 2 3 * Z OAXN pc 7 1 2 3 * Z OAXN pc 7 (Equation 33) According to equation 30, equation 33 can be rewritten as follows after some simple algebraic operations. 21* 2 5 n 4 7 * 2 4 n 2 21* 2 4 n 3 2 3 n 7 * 2 2 n 8 I pc 1 49 * 2 6 n 9 * ( 2n 2) (Equation 34) The importance of equation 34 becomes more obvious when we realize that we will have I pc 0.64 for n 2 (It is obvious that in our approach n must be grater than or equal to 2) and I pc quickly approaches unity with the growth of n. This trend can be more clearly shown by the following limit. 21* 2 5 n 5 Lim n I pc Lim n (1 ) 49 * 2 6 n 9 * ( 2n 2) 3 Lim n (1 ) 1 (Equation 35) n4 49 * 2 * ( 2n 2) Equation 35 demonstrates that can improve the APCP from 64% to nearly 100% when compared to the traditional LSB algorithm. Table 3 shows the average pixel change probability obtained from the model for n 2,20. Table 3: The average pixel change probabilities obtained from the model N APCP Improvement 0.3125000000 0.2182617188 0.0987497965 0.0419649482 0.0177703314 0.0076087060 0.0033045799 0.0014553028 0.0006489219 0.6428571429 0.7505580357 0.8871430897 0.9520400592 0.9796910499 0.9913043360 0.9962233373 0.9983367968 0.9992583749 n APCP Improvement 0.9996657236 11 0.0002924919 0.9998479323 12 0.0001330592 0.9999302739 13 0.0000610103 0.9999678122 14 0.0000281643 0.9999850541 15 0.0000130776 0.9999930249 16 0.0000061032 0.9999967303 17 0.0000028610 0.9999984613 18 0.0000013463 0.9999992734 19 0.0000006358 0.9999996558 20 0.0000003012 In table 3, the first column shows the value of n, the second column shows the average pixel change probability for each value of n and the third column shows the improvement over the traditional LSB technique for each value of n from the average pixel change probability point of view. As table 3 shows, the greatest possible value for APCP in the Z-OAXN approach is 0.3125 which is obtained for n=2. A comparison to the APCP of 7 0.875 ) shows that the minimum improvement over LSB the traditional LSB ( 8 will be grater that 64% and this improvement quickly approaches 100% with the growth of n. 2 3 4 5 6 7 8 9 10 Figures 14 and 15 show the curves which demonstrate how the APCP varies with the value of n and how much improvement over LSB can be gained through the use of the ZOAXN approach. Figure 14: The average pixel change probability obtained from the model In figure 14, the horizontal axis shows the values of n and the vertical axis shows APCP. Figure 15: The improvement over the traditional LSB In figure 15, the horizontal axis shows the values of n and the vertical axis shows the improvement of the APCP over traditional LSB. 4-2 Experimental Results In this section, we present the APCPs obtained from experiments in order to compare with those obtained from the analytical model. Section 4-2-1 explains the experimental methodology and section 4-2-2 gives the obtained results. 4-2-1 Experimental Methodology To verify the analytical model, 1000 experiments have been done to calculate the APCP for each n 2,20. For each value of n, 100 secret messages with lengths between 500 to 1000 bytes have each been stored in 10 different cover images with sizes between 500 to 800 Kbytes and the average percents of changed pixels have been measured. The cover images vary in visual properties in addition to size. They have been selectively chosen from natural scenes, personal photos, paintings, etc. 4-2-2 Results Table 4 shows the experimentally obtained APCPs along with their differences from their analytically obtained counterparts (in percents). Table 4: The results obtained from experiments n APCP Difference n 0.3215244414 0.2182993178 0.0985954124 0.0422001758 0.0181132798 0.0076202679 0.0033123614 0.0012815252 0.0006490419 0.0288782124 0.0001722657 0.0015633864 0.0056053355 0.0192989338 0.0015195609 0.0023547726 0.1194098931 0.0001849362 11 12 13 14 15 16 17 18 19 20 APCP Difference 0.0002913697 0.0038366736 0.0001330546 0.0000342919 0.0000610234 0.0002152021 0.0000281384 0.0009211001 0.0000158550 0.2123796732 0.0000060114 0.0150447919 0.0000028529 0.0028150073 0.0000013481 0.0013639322 0.0000006407 0.0077048549 0.0000003013 0.0004866451 Average 0.0223047089 In table 4, the second column shows the APCPs and the third column gives their differences from their corresponding values obtained from the model (given in table 3) in percents. The single cell labeled “Average” shows the average difference ratio between the values obtained from experiments from those obtained from the model. As shown in table 4, the average difference ratio is a little more than 0.02 and this verifies the correctness of the model. To observe the similarity between the APCPs presented in table 4 and those shown by table 3 in a more visual form, we have shown the experimentally obtained APCPs in the form of a curve in figure 16. 2 3 4 5 6 7 8 9 10 Figure 16: The pixel change probabilities given by experiments The similarity between the above diagram and the one in figure 14 means that experimental results verify the correctness of the model with a high precision. 4-3 Disadvantages and Shortcomings The Z-OAXN approach has an important disadvantage from the storage capacity point of view despite its improved pixel change probability. In fact, this approach can store the bits of the secret data in LSBs of n-1 bytes among each 2n-2 bytes and therefore its n 1 1 while the traditional LSB can store in 2n 2 2 each LSB and has a storage capacity equal to 1. In other words, Z-OAXN reduces the storage capacity by 50% in contrast to the traditional LSB. In other words, the Z-OAXN approach improves the APCP at the cost of the reduction in the storage capacity and since the improvement in the APCP grows with n, this approach will obviously work more and more efficiently as we chose larger values for n. storage capacity is equal to 5 Conclusions and Further Works This paper proposed and evaluated a novel approach called Z-OAXN that uses the properties of a reversible logic function called the Zolfaghari function to improve the traditional LSB from the average pixel change probability point of view. The improvement gained by this approach varies from 64 to 100 percent. On the other hand, this approach reduces the storage capacity by 50% compared with the traditional LSB. 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