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In the domain of software watermarking, we have proposed several graphtheoretic watermarking codec systems for encoding watermark numbers $w$ asreducible permutation flow-graphs $F[\pi^*]$ through the use of self-invertingpermutations $\pi^*$. Following up on our proposed methods, we theoreticallystudy the oldest one, which we call W-RPG, in order to investigate and proveits resilience to edge-modification attacks on the flow-graphs $F[\pi^*]$. Inparticular, we characterize the integer $w\equiv\pi^*$ as strong or weakwatermark through the structure of self-inverting permutations $\pi^*$ whichencodes it. To this end, for any integer watermark $w \in R_n=[2^{n-1},2^n-1]$, where $n$ is the length of the binary representation $b(w)$ of $w$, wecompute the minimum number of 01-modifications needed to be applied on $b(w)$so that the resulting $b(w')$ represents the valid watermark number $w'$; notethat a number $w'$ is called valid (or, true-incorrect watermark number) if$w'$ can be produced by the W-RPG codec system and, thus, it incorporates allthe structural properties of $\pi^* \equiv w$.

Characterizing Watermark Numbers encoded as Reducible Permutation Graphs against Malicious Attacks