Two RPG Flow-graphs for Software Watermarking using Bitonic Sequences of Self-inverting Permutations

Software watermarking has received considerable attention and was adopted by the software development community as a technique to prevent or discourage software piracy and copyright infringement. A wide range of software watermarking techniques has been proposed among which the graph-based methods that encode watermarks as graph structures. Following up on our recently proposed methods for encoding watermark numbers w as reducible permutation flow-graphs F [π∗] through the use of self-inverting permutations π∗, in this paper, we extend the types of flow-graphs available for software watermarking by proposing two different reducible permutation flow-graphs F1[π ∗] and F2[π ∗] incorporating important properties which are derived from the bitonic subsequences composing the self-inverting permutation π∗. We show that a self-inverting permutation π∗ can be efficiently encoded into either F1[π ∗] or F2[π ∗] and also efficiently decoded from theses graph structures. The proposed flow-graphs F1[π ∗] and F2[π ∗] enrich the repository of graphs which can encode the same watermark number w and, thus, enable us to embed multiple copies of the same watermark w into an application program P . Moreover, the enrichment of that repository with new flow-graphs increases our ability to select a graph structure more similar to the structure of a given application program P thereby enhancing the resilience of our codec system to attacks.