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

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

Helena Rifà-Pous 1, , Josep Rifà 2, and  Lorena Ronquillo 2,

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.
Keywords: $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes., Steganography.
Mathematics Subject Classification: Primary: 68P30; Secondary: 94A60, 94B6.
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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

J. Fridrich and P. Lisoněk, Grid colorings in steganography, IEEE Trans. Inform. Theory, 53 (2007), 1547-1549. doi: 10.1109/TIT.2007.892768.  Google Scholar

[5]

F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland Publishing Company, 1977.  Google Scholar

[6]

P. Moulin and R. Koetter, Data-hiding codes, Proc. IEEE, 93 (2005), 2083-2126. doi: 10.1109/JPROC.2005.859599.  Google Scholar

[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.  Google Scholar

[8]

B. Ryabko and D. Ryabko, Asymptotically optimal perfect steganographic systems, Probl. Inform. Transm., 45 (2009), 184-190. doi: 10.1134/S0032946009020094.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

J. Fridrich and P. Lisoněk, Grid colorings in steganography, IEEE Trans. Inform. Theory, 53 (2007), 1547-1549. doi: 10.1109/TIT.2007.892768.  Google Scholar

[5]

F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland Publishing Company, 1977.  Google Scholar

[6]

P. Moulin and R. Koetter, Data-hiding codes, Proc. IEEE, 93 (2005), 2083-2126. doi: 10.1109/JPROC.2005.859599.  Google Scholar

[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.  Google Scholar

[8]

B. Ryabko and D. Ryabko, Asymptotically optimal perfect steganographic systems, Probl. Inform. Transm., 45 (2009), 184-190. doi: 10.1134/S0032946009020094.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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