# American Institute of Mathematical Sciences

August  2011, 5(3): 425-433. doi: 10.3934/amc.2011.5.425

## $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes in Steganography

 1 Department of Computer Science and Multimedia, Universitat Oberta de Catalunya, 08018-Barcelona, Spain 2 Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain

Received  May 2010 Revised  March 2011 Published  August 2011

Steganography is an information hiding application which aims to hide secret data imperceptibly into a cover object. In this paper, we describe a novel coding method based on $\mathbb{Z}_2\mathbb{Z}_4$-additive codes in which data is embedded by distorting each cover symbol by one unit at most ($\pm 1$-steganography). This method is optimal and solves the problem encountered by the most efficient methods known today, concerning the treatment of boundary values. The performance of this new technique is compared with that of the mentioned methods and with the well-known rate-distortion upper bound to conclude that a higher payload can be obtained for a given distortion by using the proposed method.
Citation: Helena Rifà-Pous, Josep Rifà, Lorena Ronquillo. $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes in Steganography. Advances in Mathematics of Communications, 2011, 5 (3) : 425-433. doi: 10.3934/amc.2011.5.425
##### References:
 [1] J. Bierbrauer and J. Fridrich, Constructing good covering codes for applications in steganography, in "Trans. on Data Hiding and Multimedia Security III,'' (2008), 1-22. [2] J. Borges and J. Rifà, A characterization of 1-perfect additive codes, IEEE Trans. Inform. Theory, 45 (1999), 1688-1697. doi: 10.1109/18.771247. [3] R. Crandall, Some notes on steganography, available from http://www.dia.unisa.it/~ads/corso-security/www/CORSO-0203/steganografia/LINKS%20LOCALI/matrix-encoding.pdf, 1998. [4] J. Fridrich and P. Lisoněk, Grid colorings in steganography, IEEE Trans. Inform. Theory, 53 (2007), 1547-1549. doi: 10.1109/TIT.2007.892768. [5] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland Publishing Company, 1977. [6] P. Moulin and R. Koetter, Data-hiding codes, Proc. IEEE, 93 (2005), 2083-2126. doi: 10.1109/JPROC.2005.859599. [7] H. Rifà-Pous and J. Rifà, Product perfect codes and steganography, Digit. Signal Process., 19 (2009), 764-769. doi: 10.1016/j.dsp.2008.11.005. [8] B. Ryabko and D. Ryabko, Asymptotically optimal perfect steganographic systems, Probl. Inform. Transm., 45 (2009), 184-190. doi: 10.1134/S0032946009020094. [9] A. Westfeld, High capacity despite better steganalysis (F5 - A steganographic algorithm), Lecture Notes in Comput. Sci., 2137 (2001), 289-302. doi: 10.1007/3-540-45496-9_21. [10] F. M. J. Willems and M. van Dijk, Capacity and codes for embedding information in grayscale signals, IEEE Trans. Inform. Theory, 51 (2005), 1209-1214. doi: 10.1109/TIT.2004.842707.

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##### References:
 [1] J. Bierbrauer and J. Fridrich, Constructing good covering codes for applications in steganography, in "Trans. on Data Hiding and Multimedia Security III,'' (2008), 1-22. [2] J. Borges and J. Rifà, A characterization of 1-perfect additive codes, IEEE Trans. Inform. Theory, 45 (1999), 1688-1697. doi: 10.1109/18.771247. [3] R. Crandall, Some notes on steganography, available from http://www.dia.unisa.it/~ads/corso-security/www/CORSO-0203/steganografia/LINKS%20LOCALI/matrix-encoding.pdf, 1998. [4] J. Fridrich and P. Lisoněk, Grid colorings in steganography, IEEE Trans. Inform. Theory, 53 (2007), 1547-1549. doi: 10.1109/TIT.2007.892768. [5] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland Publishing Company, 1977. [6] P. Moulin and R. Koetter, Data-hiding codes, Proc. IEEE, 93 (2005), 2083-2126. doi: 10.1109/JPROC.2005.859599. [7] H. Rifà-Pous and J. Rifà, Product perfect codes and steganography, Digit. Signal Process., 19 (2009), 764-769. doi: 10.1016/j.dsp.2008.11.005. [8] B. Ryabko and D. Ryabko, Asymptotically optimal perfect steganographic systems, Probl. Inform. Transm., 45 (2009), 184-190. doi: 10.1134/S0032946009020094. [9] A. Westfeld, High capacity despite better steganalysis (F5 - A steganographic algorithm), Lecture Notes in Comput. Sci., 2137 (2001), 289-302. doi: 10.1007/3-540-45496-9_21. [10] F. M. J. Willems and M. van Dijk, Capacity and codes for embedding information in grayscale signals, IEEE Trans. Inform. Theory, 51 (2005), 1209-1214. doi: 10.1109/TIT.2004.842707.
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