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Full Characterization of a Class of Graphs Tailored for Software Watermarking

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Abstract

Digital watermarks have been regarded as a promising way to fight copyright violations in the software industry. In some graph-based watermarking schemes, identification data is disguised as control-flow graphs of dummy code. Recently, Chroni and Nikolopoulos proposed an ingenious such scheme whereby an integer is encoded into a particular kind of permutation graph. We give a formal characterization of the class of graphs generated by their encoding function, which we call canonical reducible permutation graphs. A linear-time recognition algorithm is also given, setting the basis for a polynomial-time algorithm to restore watermarks that underwent the malicious removal of some edges. Finally, we give a simpler decoding algorithm for Chroni and Nikolopoulos’ watermarks.

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Acknowledgements

Funding was provided by Conselho Nacional de Desenvolvimento Cientìfico e Tecnológico (Grant Nos. 400609/2015-0, 487061/2013-6, 309234/2015-8, 421007/2016-8), Pronametro (Grant No. 52600.017257/2013), Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (BR) (Grant Nos. JCNE-203889, TECNOVA-143360) and Financiadora de Estudos e Projetos (Grant No. TI-MAIOR-SBV936).

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Authors and Affiliations

  1. Univ. Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Lucila M. S. Bento, Vinícius G. Pereira de Sá & Jayme L. Szwarcfiter

  2. Inst. Nac. de Metrologia, Qualidade e Tecnologia, Rio de Janeiro, Brazil

    Davidson R. Boccardo & Raphael C. S. Machado

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  1. Lucila M. S. Bento
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  2. Davidson R. Boccardo
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  3. Raphael C. S. Machado
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  4. Vinícius G. Pereira de Sá
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  5. Jayme L. Szwarcfiter
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Correspondence to Vinícius G. Pereira de Sá.

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Part of this paper was presented as an extended abstract entitled “Towards a provably resilient scheme for graph-based watermarking” at the 39th International Workshop on Graph Theoretic Concepts in Computer Science, WG 2013, and appeared in Lecture Notes in Computer Science8165 (2013), 50–63.

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Bento, L.M.S., Boccardo, D.R., Machado, R.C.S. et al. Full Characterization of a Class of Graphs Tailored for Software Watermarking. Algorithmica 81, 2899–2916 (2019). https://doi.org/10.1007/s00453-019-00557-w

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