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 $F_1[π^*]$ and $F_2[π^*]$ 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 $F_1[π^*]$ or $F_2[π^*]$ and also efficiently decoded from theses graph structures. The proposed flow-graphs $F_1[π^*]$ and $F_2[π^*]$ 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.