Abstract
The main purpose of this paper is to give a new and general way to obtain steganographic schemes from perfect codes on Cayley graphs, motivated by the F5 algorithm based on binary Hamming codes. We obtain the steganography based on perfect Hamming codes as a special case and also give various equivalent conditions for the existence of a perfect code on a k-regular Abelian Cayley graph. Then we show that a perfect code on an Abelian Cayley graph produces a proper steganographic scheme. We further compute the various parameters for the steganographic scheme of type [n, k] over a finite field \(\mathbb {F}_q\) arising from a linear [n, n − k, d] code over \(\mathbb {F}_q\) and find also parameters for some steganographic schemes from perfect codes in k-regular Abelian Cayley graphs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Crandall R.: Some notes on steganography (1998).
Dorbec P., Mollard M.: Perfect codes in Cartesian products of 2-paths and infinite paths. J. Comb. 12, 65 (2005).
Davidoff G., Sarnak P., Valette A.: Elementary Number Theory, Group Theory, and Ramanujan Graphs. Cambridge University Press, Cambridge (2003).
Fridrich J.: Steganography in Digital Media. Cambridge University Press, New York (2010).
Fridrich J., Linsonĕk P.: Grid Colorings Steganogr. IEEE Trans. Inf. Thoery 53, 1547–1549 (2007).
Golomb S.W., Welch L.R.: Perfect codes in the Lee metric and the packing of polyominoes. SIAM J. Appl. Math. 18, 302–317 (1970).
Kim J.-L., Kim S.-J.: The 2-distance coloring of the Cartesian product of cycles using optimal Lee codes. Discret. Appl. Math. 159, 2222–2228 (2011).
Krotov D.S.: On calculation of the interweight distribution of an equitable partition. arXiv:1303.0002v2 [math.CO] 1 Jul (2013).
Martínez C., Beivide R., Gabidulin E.M.: Perfect codes for metrics induced by circulant graphs. IEEE Trans. Inf. Thoery 53, 3042–3052 (2007).
Martínez C., Beivide R., Gabidulin E.M.: Perfect codes from Cayley graphs over Lipschitz integers. IEEE Trans. Inf. Thoery 55, 3552–3562 (2009).
Mollard M.: On perfect codes in Cartesian product of graphs. Eur. J. Comb. 32, 398–403 (2011).
Munuera C.: Steganography from a coding theory point of view. In: Algebraic Geometry Modelling in Information Theory. Edited by Martinez-Moro E., World Scientific, pp. 83–128, (2012).
Schönfeld D, Winkler A.: Embedding with syndrome coding based on BCH codes. In: Proceedings of the 8th Workshop on Multimedia and Security. ACM (2006).
Spacapan S.: Optimal Lee-type local structures in Cartesian products of cycles and paths. SIAM J. Discret. Math. 21, 750–762 (2007).
Tietäväinen A.: On the nonexistence of perfect codes over finite fields. SIAM J. Appl. Math. 24, 88–96 (1973).
Van Dijk M., Willems F.: Embedding information in grayscale images. In: Proceedings of the 22nd Symposium on Information and Communication Theory in the Benelux. Enschede, The Netherlands (2001).
Zhang W., Li S.: A coding problem in steganography. Des. Codes Cryptogr. 46(1), 67–81 (2008).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. H. Koolen.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
J.-L. Kim was supported by Basic Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2005172).
Rights and permissions
About this article
Cite this article
Kim, JL., Park, J. & Choi, S. Steganographic schemes from perfect codes on Cayley graphs. Des. Codes Cryptogr. 87, 2361–2374 (2019). https://doi.org/10.1007/s10623-019-00624-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-019-00624-x