Characterizing Watermark Numbers encoded as Reducible Permutation Graphs against Malicious Attacks
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In the domain of software watermarking, we have proposed several graph theoretic watermarking codec systems for encoding watermark numbers w as reducible permutation flow-graphs F[π^*] through the use of self-inverting permutations π^*. Following up on our proposed methods, we theoretically study the oldest one, which we call W-RPG, in order to investigate and prove its resilience to edge-modification attacks on the flow-graphs F[π^*]. In particular, we characterize the integer w≡π^* as strong or weak watermark through the structure of self-inverting permutations π^* which encodes it. To this end, for any integer watermark w ∈ R_n=[2^n-1, 2^n-1], where n is the length of the binary representation b(w) of w, we compute 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'; note that 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 all the structural properties of π^* ≡ w.
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