Ieee a secure algorithm for image based information hiding with one-dimensional chaotic systems

Ieee a secure algorithm for image based information hiding with one-dimensional chaotic systems

• 1.
  A Secure Algorithmfor Image Based Information Hiding with One-dimensional Chaotic Systems Yinglei Song School of Electronics and Information Science Jiangsu University of Science and Technology Zhenjiang, China e-mail: syinglei2013@163.com Jia Song Department of Electronic and Information Technology Suzhou Vocational University Suzhou, 215104, China e-mail: yigibird@yahoo.com.cn Junfeng Qu Department of Computer Science and Information Technology Clayton State University Morrow, GA 30260, USA e-mail: junfengqu@clayton.edu Abstract—The goal of information hiding is to securely hide important information within digital documents such as files, images and videos. In this paper, we develop a new method that can efficiently and securely embed a sequence of binary bits into a gray-scale image in two stages. In the first stage, the sequence is divided into a few subsequences of equal length and a set of one-dimensional logistic mappings are used to shuffle the binary bits in the sequence. In the second stage, the shuffled sequence is divided into regions of equal length and the regions are sequentially embedded into the corresponding pixels in the given gray-scale image. The embedding of a region in the shuffled sequence is performed by replacing the least significant bits of the gray value of the corresponding pixel with the bits in the region. Analysis and testing results show that this information hiding algorithm can securely embed information into a gray-scale image without altering the content of image significantly and thus can potentially be used in practice for secure image-based information hiding. Keywords-information hiding; gray-scale image; one- dimensional logistic systems; key space I. INTRODUCTION The past two decades have witnessed rapid development in information technologies. As a result, a large amount of digital information and data has been generated. In order to protect the intellectual property rights associated with crucial data and information, researchers have developed numerous methods and techniques for the encryption of important data and information. Information hiding is one of such techniques [1]. Specifically, the goal of information hiding is to embed important data and information into digital documents such as files, images and videos without changing the content of the documents significantly. A document where the embedding is performed is often referred to as a cover media and the document that results from an embedding is a stego media. In general, a stego media is highly similar to the corresponding cover media and the difference between them often cannot be recognized by human eyes. Since images are important for the storage and transmission of multimedia data, the embedding of information into gray scale images is an important technology in the area of information hiding. A large number of methods for image based information hiding have been developed, including the information hiding technology based on pixel value difference and LSB replacement [2], the side match technology [3], the hiding technology based on pixel difference expansion and modulus function [4]. Most of these methods compute the difference in gray values of certain neighboring pixels. The gray values of both pixels are then changed to encode a part of the data that needs to be embedded. In [5], a technique based on similar sample blocks is developed for image based information hiding. The technique searches in a standard image for a sample block and uses the sample block as the reference for the embedding of information. Most of these methods can successfully embed the information into a given image without altering the content of the image significantly. However, most of them either need the cover media or a standard image to recover the embedded information from the stego media, which may not be convenient in certain applications. In addition, the information embedded into the image can be obtained with a straightforward approach when both the stego media and cover media are available, the security of the embedded information is thus weak and not guaranteed. Recently, a large number of algorithms have been developed to encrypt images with chaotic systems. In [6], an algorithm that can encrypt an image with three sets of pseudorandom sequences generated with a high-dimensional chaotic system is developed. In [7], an image encryption algorithm that combines permutation and diffusion is developed. In [8], an image encryption algorithm for color images by shuffling pixels is developed. The shuffling of pixels is performed with chaotic systems. Our previous work [9] develops a new method that can robustly encrypt an image with two sets of one-dimensional logistic systems. It is thus interesting to consider whether chaotic systems can be 1824 2017 3rd IEEE International Conference on Computer and Communications 978-1-5090-6352-9/17/$31.00 ©2017 IEEE
• 2.
  utilized to enhancethe security of an information hiding algorithm. In this paper, we develop a novel image based information hiding algorithm that can efficiently and securely embed a sequence of binary bits into a gray scale image with a set of one-dimensional logistic mappings. In the first stage of the algorithm, the sequence is divided into subsequences of equal length and every bit in the sequence can then be mapped to a pair of two integers, which correspond to a data point in a two-dimensional space. The whole sequence thus corresponds to a grid in the two- dimensional space and the bits in the sequence can be shuffled based on a set of one-dimensional logistic systems. In the second stage of the algorithm, the shuffled sequence is divided into regions of equal length and the regions are sequentially embedded into the gray scale image to complete the embedding. The embedding of a region is performed by replacing the least significant bits of the gray value of the corresponding pixel with the bits in the region. The keys for recovering the embedded sequence are the parameters of the one-dimensional logistic mappings used in the first stage of the algorithm. The algorithm does not need the cover image or a standard image to recover the information. Our analysis shows that the key space of this algorithm is large and robust against attacks based on exhaustive search. In addition, our testing results show that this algorithm can efficiently generate secure embedded results and the stego images generated by this algorithm is closer to the cover images than those generated by two other methods for image based information hiding. II. THE ALGORITHM A. The Suffling of the Sequence Wherever we use B to denote a sequence of binary bits that need to be embedded into a gray scale image. A sequence of real numbers ,...,,...,, 110 kk yyyy can be defined based on a one dimensional logistic mapping R as follows. )1(1 kkk yyy (1) where satisfies 40 and is the parameter of the mapping, 0k is the integer index. The initial value 0y of the sequence satisfies 10 0y . It is straightforward to see that 10 ky holds for each 0k . All numbers in the sequence are thus positive and less than 1. A well known fact on logistic mappings is that the sequence period becomes infinite and the initial value 0y significantly affects the numbers in the sequence when 457.3 [4], A significant amount of difference in ky can be created by a tiny perturbation in 0y for large enough k . A logistic mapping thus has the property of a one- dimensional chaotic system. A large number of image encryption algorithms have been developed based on the chaotic property of one-dimensional logistic mappings [10, 11]. Given a positive integer p and sequence B , we use BL to denote the length of B and )(iB to denote the i th bit in B . We assume BLp and 1/ pLq B . It is not difficult to see that B can be divided into q subsequences and each subsequence contains p bits. The last subsequence may contain less than p bits and we can pad enough bits of 0 to the end of B such that the last subsequence also contains p bits. We therefore assume that B contains q subsequences of length p in the rest of the paper. We thus can map )(iB to a pair of integers ),( ts such that pis / and pit mod . s is the first coordinate of the )(iB and t is its second coordinate. Each bit in B can thus be uniquely mapped to a point in a two-dimensional space. All bits that have the same value of the first coordinate together form a row. Similarly, all bits that have the same value of the second coordinate together form a column. Two bits are in the same row if they have the same first coordinate and in the same column if their second coordinates are equal. We use integers 2,1 …, q to number the rows formed by bits in B and p,...,2,1 for columns. For row s in B , a logistic mapping suR is needed to shuffle the pixels in the row. Similarly, for column t in the image, a logistic mapping tvR is needed to shuffle the pixels in the column. Parameters pq vvvuuu ,...,,,,...,, 2121 are set to be numbers between 3.6 and 4.0 and are the keys of the information hiding algorithm. The relocation of bits in B starts by shuffling the bits row by row. For row s ( qs1 ) in B , a sequence of real numbers pyyy ,...,, 21 can be computed with a given initial value 0y from suR . From pyyy ,...,, 21 , the algorithm computes a sequence of integers plll ,...,, 21 as follows. pMyl kk mod)( (2) where pk1 , M is a large integer and p is not a factor of M . The bits in row s are then shuffled based on plll ,...,, 21 . Specifically, the bit whose first and second coordinates are s and k respectively is moved to the location where the first and second coordinates are s and kl in B . We denote the resulting sequence by 1B . The bits in 1B are then shuffled column by column. For column t ( pt1 ) in 1B , a sequence of real numbers 1825
• 3.
  qzzz ,...,, 21can be computed based on a given initial value 0z and tvR . The algorithm computes a sequence of integers qrrr ,...,, 21 from qzzz ,...,, 21 as follows. qNyr uu mod)( (3) where qu1 and N is a large integer and q is not a factor of N . The bits in column t are then shuffled based on qrrr ,...,, 21 . The bit whose first and second coordinates are u and t respectively is relocated to the location where the first and second coordinates are ur and t . We denote the resulting sequence by 2B . B. The Embedding of the Sequence Let I be the gray scale image where B needs to be embedded into. We assume I contains m rows and n columns and ),( hgI is the gray value for the pixel in row g and column h in I . Since the bits in B have been relocated and the resulting sequence is 2B , the algorithm sequentially embeds the bits in 2B into I in the second stage. The bits in 2B are embedded into I in groups. Each group contains equal number of bits. We use w to denote the number of bits in each group. Since the gray value of a pixel contains 8 bits, wmust satisfy 81 w . The bits in 2B are thus sequentially divided into 1/ wpqb subsequences, each subsequence contains w bits. We use integers b,...,2,1 to sequentially number the subsequences. Let bmne / , the bits in subsequence l is embedded into the gray value of pixel ),( ll hg in I , where lg and lh are computed as follows. 1/ nlegl (4) nlehl mod (5) The gray value of pixel ),( ll hg is updated to include the w bits in subsequence l as follows. l w llll ChgIhgI 2/),(),(2 (6) where ),(2 ll hgI is the revised gray value for pixel ),( ll hg and lC is the binary value of the w bits in subsequence l . It is clear from equation (6) that the least significant bits are replaced by the bits in subsequence after the embedding is complete. C. The Recovery of the Embedded Sequence Let 2I be the resulting stego image after B has been embedded into I . Given the values of wqp ,, and the keys used in the shuffling of B , the recovery of B from 2I can be performed in two stages. In the first stage, the bits in sequence 2B can be efficiently recovered by accessing certain pixels in 2I and obtaining the w least significant bits from the gray values of these pixels. Specifically, the binary value encoded by the w bits in the subsequence l in 2B can be computed with equation (7). w lll hgIC 2mod),(2 (7) where lC is the binary value encoded by the w bits in the subsequence l in 2B ; the values of lg and lh are computed as shown equations (4) and (5). The subsequence l in 2B can then be obtained from lC . We assume the initial values used for the shuffling of rows and columns in are 0y and 0z respectively while the keys for the shuffling of the mapped rows and columns in B are quuu ,...,, 21 and pvvv ,...,, 21 . In the second stage of the recovery process, the algorithm shuffles the bits in 2B column by column. For column t ( pt1 ) in 2B , a sequence of real numbers qzzz ,...,, 21 are computed with the initial value 0z and tvR . The algorithm computes a sequence of integers qrrr ,...,, 21 from qzzz ,...,, 21 as follows. qNzr uu mod)( (8) where qu1 . The bits in column t are then relocated with qrrr ,...,, 21 . Specifically, the bit whose first and second coordinates are ur and t respectively is moved to the location where the first and second coordinates are u and t . We denote the resulting sequence by 1B . Finally, The bits in 1B are shuffled row by row. For row s ( qs1 ) in 1B , a sequence of real numbers pyyy ,...,, 21 are computed with the initial value 0y and suR . A sequence of integers plll ,...,, 21 can be computed from pyyy ,...,, 21 as follows. pMxl kk mod)( (9) where pk1 . The bits in row s are then relocated with plll ,...,, 21 . The bit whose first and second coordinates are 1826
• 4.
  s and klrespectively is moved to the location where the first and second coordinates are s and k . The resulting sequence is the original sequence B . It is straightforward to see that the recovery of the embedded information does not require the availability of the cover image or a standard image. III. EXPERIMENTAL RESULTS AND DISCUSSIONS We have implemented this information hiding algorithm into a computer program in MATLAB and tested its performance on embedding information into four benchmark gray-scale images. One of the images is the well known benchmark image Lena, the other three images are downloaded from the Berkeley Segmentation Data Set and Benchmarks 500 (BSD500), which can be accessed at site https://www2.eecs.berkeley.edu/Research/Projects/CS/vision /grouping/resources.html. To compare the overall performance of this algorithm with existing methods, we use our algorithm, two other existing methods to embed randomly generated binary sequences into the benchmark images and compare the performance of three methods. The keys for the shuffling of the embedded sequences are real numbers randomly generated between 3.7 and 4.0. The values of 00 , zy are set to be 0.3 and NM , are both set to be 7 10 . All images are scaled to the same size, which is 512×512. Figure 1 (a) (b) (c) and (d) show the four tested images. (a) (b) (c) (d) Figure 1. (a) image Lena; (b) image 3063 from BSD500; (c) image 5096 from BSD500; (d) image 8068 from BSD500. We randomly generate 100 sequences of binary bits for embedding tests. Each sequence contains 100 randomly generated binary bits. qp, are both set to be 10. The effects of won stego images are also evaluated by embedding the same sequence into the same benchmark image with different values of w. The quality of a stego image can be evaluated based on Peak Signal of Noise Ratio (PSNR). Table I, II and III compare the means and standard deviations of the PSNRs of the stego images obtained with our algorithm and the method developed in [12] on the 100 testing sequences when the value of w is 1, 3 and 5 respectively. It is clear from Tables I, II and III that our approach can achieve higher mean values of PSNRs on all benchmark images, which suggests that our approach outperforms the method developed in [12]. In addition, the standard deviations of PSNRs confirm that the performance of our approach is reliable and robust. To compare the performance of our approach with the method developed in [13], we embed the testing sequences into the four benchmark images with the method developed in [13] and compute the values of PSNRs and their standard deviations for the stego images obtained with the method. The results are shown in Tables IV, V, and VI. It can be seen from Tables IV, V, and VI that the mean values of PSNRs obtained with the method in [13] are lower than those obtained with our approach on all benchmark images. This shows that our approach also outperforms the method developed in [13]. TABLE I. A COMPARISON OF OUR ALGORITHM WITH THE METHOD DEVELOPED IN [12] WHEN wIS 1. Stego Image Our Approach The Method in [12] PSNR STD PSNR STD Lena 85.42 0.40 80.33 0.37 3063 85.27 0.42 81.42 0.39 5096 85.32 0.41 79.61 0.35 8068 85.46 0.43 80.58 0.34 TABLE II. A COMPARISON OF OUR ALGORITHM WITH THE METHOD DEVELOPED IN [12] WHEN wIS 3. Stego Image Our Approach The Method in [12] PSNR STD PSNR STD Lena 84.58 0.41 79.52 0.40 3063 84.44 0.40 80.60 0.38 5096 84.47 0.42 78.79 0.36 8068 84.63 0.41 79.74 0.33 1827
• 5.
  TABLE III. ACOMPARISON OF OUR ALGORITHM WITH THE METHOD DEVELOPED IN [12] WHEN wIS 5. Stego Image Our Approach The Method in [12] PSNR STD PSNR STD Lena 83.97 0.42 78.92 0.38 3063 83.81 0.43 80.03 0.39 5096 83.89 0.41 78.13 0.35 8068 84.03 0.40 79.11 0.36 TABLE IV. A COMPARISON OF OUR ALGORITHM WITH THE METHOD DEVELOPED IN [13] WHEN wIS 1. Stego Image Our Approach The Method in [13] PSNR STD PSNR STD Lena 85.42 0.40 81.69 0.43 3063 85.27 0.42 80.31 0.45 5096 85.32 0.41 81.53 0.46 8068 85.46 0.43 80.62 0.39 TABLE V. A COMPARISON OF OUR ALGORITHM WITH THE METHOD DEVELOPED IN [13] WHEN wIS 3. Stego Image Our Approach The Method in [13] PSNR STD PSNR STD Lena 84.58 0.41 80.83 0.41 3063 84.44 0.40 79.48 0.44 5096 84.47 0.42 80.70 0.38 8068 84.63 0.41 79.75 0.42 TABLE VI. A COMPARISON OF OUR ALGORITHM WITH THE METHOD DEVELOPED IN [13] WHEN wIS 5. Stego Image Our Approach The Method in [13] PSNR STD PSNR STD Lena 83.97 0.42 80.10 0.42 3063 83.81 0.43 78.92 0.41 5096 83.89 0.41 80.04 0.45 8068 84.03 0.40 79.03 0.43 It is also clear that the algorithm needs qp keys in total to embed a sequence that contains up to pq bits. The key space is thus of size qp c , where c is the number of real numbers that can be selected by a computer from the interval between 3.6 and 4.0. It is straightforward to see that 10 2c and the size of the key space is thus at least )10 2 qp . For a sequence that contains more than 100 binary bits, this number is at least 200 2 , which suggests that our approach is secure against attacks based on exhaustive search when the embedded sequence is of moderate length. IV. CONCLUSIONS In this paper, we develop a new algorithm for image based information hiding. The algorithm embeds a sequence of binary bits into a given gray scale image in two stages. In the first stage, the binary bits in the sequence are shuffled based on a set of one-dimensional logistic mappings. In the second stage, the shuffled sequence is divided into subsequences of equal lengths and the subsequences are sequentially embedded into the gray values of the corresponding pixels in the given gray scale image. Our analysis and testing results show that this approach does not need the cover image or a standard image to recover the embedded information and outperforms two other methods for image based information hiding. In addition, our analysis shows that the embedded information is secure against attacks based on exhaustive search. It can be seen clearly from the description of the algorithm that its time complexity is linear, which suggests that the algorithm is computationally efficient and is thus potentially useful for image based information hiding in internet applications. ACKNOWLEDGMENT Y. Song’s work is fully supported by The Fund of Specially Appointed Professors in Jiangsu Province with the grant number: 1034901501. REFERENCES [1] A.P. Fabien, R J. Anderson, and M.G. Kuhn, “Information hiding-a survey,” Proceeding of the IEEE Special Issue on Protection of Multimedia Content, vol. 87, no. 7, pp. 1062-1078, 1999. [2] K.H. Jung, “High-capacity steganographic method based on pixel- value differencing and LSB replacement methods”, The Imaging Science Journal, vol.58, no. 4, pp. 213-221, 2013. [3] S.L. Li, K.C. Leung, L.M. Chen and C.K. Chan, “Performance evaluation of steganographic method for digital images using side match”, Proceedings of 2006 International Conference on Innovative Computing, pp. 54-57, 2006 [4] C.M. Wang, N.I. Wu, C.S. Tsai, and M.S. Hwang, “A high quality steganographic method with pixel-value differencing and modulus function,” The Journal of Systems and Software, vol. 81, no. 1, pp. 150-158, 2008. [5] T. Lu, S. Liao, and C. Chang, “The information hiding technology based on the similar sample blocks of grayscale image”, Proceedings of the Sixth International Conference on Intelligent Information Hiding and Multimedia Signal Processing, pp. 17-20, 2010. [6] X.Y. Wang, L. Yang, R. Liu, and A. Kadir, “A chaotic image encryption algorithm based on perceptron model,” Nonlinear Dynamics, vol. 62, no. 3, pp. 615–621, 2010. 1828
• 6.
  [7] Y. Wang,K.W. Wong, X. Liao, and G. Chen, “A new chaos-based fast image encryption algorithm,” Applied Soft Computing Journal, vol. 11, no. 1, pp. 514–522, 2011. [8] C.K. Huang and H.H. Nien, “Multi chaotic systems based pixel shuffle for image encryption,” Optics Communications, vol. 282, no. 11, pp. 2123–2127, 2009. [9] Y. Song, J. Song and J. Qu, “A secure image encryption algorithm based on multiple one-dimensional chaotic systems”, Proceedings of the Second IEEE International Conference on Computer and Communications, pp. 584-587, 2016. [10] M.S. Baptista, “Cryptography with chaos,” Physics Letters, Section A, vol. 240, no. 1-2, pp. 50–54, 1998. [11] O. Edward, Chaos in Dynamical Systems, Cambridge University Press, Cambridge, UK, 2nd edition, 2003. [12] C.C. Chang and H.W. Tseng, “A steganographic method for digital images using side match,” Pattern Recognition Letters, vol. 25, no.12, pp. 1431-1437, 2004. [13] H.C. Wu, N.I. Wu, C.S. Tsai, and M.S. Hwang, “Image steganographic scheme based on pixel-value differencing and LSB replacement methods”, IEE Proceedings Vision, Image &Image Signal Process, vol. 152, no. 5, pp. 611-615, 2005. 1829