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Software watermarking involves embedding a unique identifier, i.e., a watermark value, within a software to discourage software theft; to this end, several graph theoretic watermark methods encode the watermark values as graph structures and embed them in application programs using a wide range of algorithmic techniques. In this paper we propose an efficient method for encoding the same watermark value into several different graphs, we call it multiple encoding, answering thus the question we have recently left open. In particular, we propose an efficient algorithm which embed a cograph G[π*] into a reducible permutation graph F[π*] by first computing the cotree of G[π*] then computing a rooted binary tree having specific node-value and child-parent properties, and finally, based on these properties, producing a reducible permutation graph F[π*]. In light of our recent encoding algorithms which encode a watermark value w as a self-inverting permutation π* and the permutation π* into several cographs G1[π*], G2[π*],...,Gn[π*], we conclude that we can efficiently encode the same watermark value w into several reducible permutation graphs F1[π*], F2[π*],..., Fn[π*], n ≥ 2. This property causes a codec watermarking system resilient to attacks since we can embed multiple copies of the same watermark value w into an application program. We also propose decoding algorithms which efficiently extract the watermark value w from the reducible permutation graph F[π*].

Multiple encoding of a watermark number into reducible permutation graphs using cotrees