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August  2012, 6(3): 273-285. doi: 10.3934/amc.2012.6.273

## Wet paper codes and the dual distance in steganography

 1 Department of Applied Mathematics, University of Valladolid, Avda Salamanca SN, 47014 Valladolid, Castilla 2 Computer Science Laboratory, École Polytechnique, 91 128 Palaiseau CEDEX, INRIA Saclay, ÎIle de France

Received  March 2011 Revised  January 2012 Published  August 2012

In 1998 Crandall introduced a method based on coding theory to secretly embed a message in a digital support such as an image. Later, in 2005, Fridrich et al. improved this method to minimize the distortion introduced by the embedding; a process called wet paper. However, as previously emphasized in the literature, this method can fail during the embedding step. Here we find sufficient and necessary conditions to guarantee a successful embedding, by studying the dual distance of a linear code. Since these results are essentially of combinatorial nature, they can be generalized to systematic codes, a large family containing all linear codes. We also compute the exact number of embedding solutions and point out the relationship between wet paper codes and orthogonal arrays.
Citation: Carlos Munuera, Morgan Barbier. Wet paper codes and the dual distance in steganography. Advances in Mathematics of Communications, 2012, 6 (3) : 273-285. doi: 10.3934/amc.2012.6.273
##### References:
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show all references

##### References:
 [1] D. Augot, M. Barbier and C. Fontaine, Ensuring message embedding in wet paper steganography,, in, (2011), 244. Google Scholar [2] J. Brierbrauer and J. Fridrich, Constructing good covering codes for applications in steganography,, in, (2008), 1. doi: 10.1007/978-3-540-69019-1_1. Google Scholar [3] F. Chai, X. S. Gao and C. Yuan, A characteristic set method for solving Boolean equations and applications in cryptanalysis of stream ciphers,, J. Sys. Sci. Compl., 21 (2008), 191. doi: 10.1007/s11424-008-9103-0. Google Scholar [4] C. Cooper, On the rank of random matrices,, Random Struc. Algor., 16 (2000), 209. doi: 10.1002/(SICI)1098-2418(200003)16:2<209::AID-RSA6>3.0.CO;2-1. Google Scholar [5] R. Crandall, Some notes on steganography,, \url{http://os.inf.tu-dresden.de/ westfeld}, (). Google Scholar [6] C. Fontaine and F. Galand, How Reed-Solomon codes can improve steganographic schemes,, in, (2007), 130. Google Scholar [7] J. Fridrich, M. Goljan, P. Lisonek and D. Soukal, Writing on wet paper,, IEEE Trans. Signal Proc., 53 (2005), 3923. doi: 10.1109/TSP.2005.855393. Google Scholar [8] J. Fridrich, M. Goljan and D. Soukal, Efficient wet paper codes,, in, (2005). doi: 10.1007/b104759. Google Scholar [9] J. Fridrich, M. Goljan and D. Soukal, Steganography via codes for memory with defective cells,, in, (2005), 1521. Google Scholar [10] J. Fridrich, M. Goljan and D. Soukal, Wet paper codes with improved embedding efficiency,, IEEE Trans. Inform. Forens. Secur., 1 (2006), 102. doi: 10.1109/TIFS.2005.863487. Google Scholar [11] B. J. Hamilton, SINCGARS system improvement program (SIP) specific radio improvement,, in, (1996), 397. Google Scholar [12] R. W. Hamming, "The Art of Probability for Scientists and Engineers,'', Westview Press, (1994). Google Scholar [13] A. S. Hedayat, N. J. A. Sloane and J. Stufken, "Orthogonal Arrays: Theory and Applications,'', Springer-Verlag, (1999). Google Scholar [14] M. Keinänen, "Techniques for Solving Boolean Equation Systems,'', Ph.D thesis, (2006). Google Scholar [15] V. F. Kolchin, "Random Graphs,'', Cambridge University Press, (1999). Google Scholar [16] F. J. MacWilliams and N. J. Sloane, "The Theory of Error-Correcting Codes,'', North-Holland Publishing Co., (1977). Google Scholar [17] MinT, Online database for optimal parameters of $(t, m, s)$-nets, $(t, s)$-sequences, orthogonal arrays, linear codes, and OOAs,, available at \url{http://mint.sbg.ac.at/}, (). Google Scholar [18] C. Munuera, On the generalized Hamming weights of geometric Goppa codes,, IEEE Trans. Inform. Theory, 40 (1994), 2092. doi: 10.1109/18.340488. Google Scholar [19] M. Nadler, A 32-point $n=12$, $d=5$ code,, IRE Trans. Inform. Theory, 8 (1962). doi: 10.1109/TIT.1962.1057670. Google Scholar [20] D. Schönfeld and A. Winkler, Embedding with syndrome coding based on BCH codes,, in, (2006), 214. Google Scholar [21] D. Schönfeld and A. Winkler, Reducing the complexity of syndrome coding for embedding,, in, (2007), 145. Google Scholar [22] B.-Z. Shen, A Justesen construction of binary concatenated codes that asymptotically meet the Zyablov bound for low rate,, IEEE Trans. Inform. Theory, 39 (1993), 239. doi: 10.1109/18.179365. Google Scholar [23] D. Stinson, Resilient functions and large sets of orthogonal arrays,, Congressus Numerantium, 92 (1993), 105. Google Scholar [24] C. Studholme and I. F. Blake, Random matrices and codes for the erasure channel,, Algoritmica, 56 (2010), 605. doi: 10.1007/s00453-008-9192-0. Google Scholar [25] J. H. van Lint, A new description of the Nadler code,, IEEE Trans. Inform. Theory, 18 (1972), 825. doi: 10.1109/TIT.1972.1054904. Google Scholar [26] J. H. van Lint, "Introduction to Coding Theory,'', Springer-Verlag, (1982). Google Scholar [27] V. K. Wei, Generalized Hamming weights for linear codes,, IEEE Trans. Inform. Theory, 37 (1991), 1412. doi: 10.1109/18.133259. Google Scholar [28] W. Zhang, X. Zhang and S. Wang, Maximizing embedding efficiency by combining Hamming codes and wet paper codes,, in, (2008), 60. Google Scholar
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