Encoding watermark numbers as reducible permutation graphs using self-inverting permutations

Abstract

Several graph theoretic watermark methods have been proposed to encode numbers as graph structures in software watermarking environments. In this paper we propose an efficient and easily implementable codec system for encoding watermark numbers as reducible permutation flow-graphs and, thus, we extend the class of graphs used in such a watermarking environment. More precisely, we present an algorithm for encoding a watermark number w as a self-inverting permutation π * , an algorithm for encoding the self-inverting permutation π * into a reducible permutation graph F [π * ] whose structure resembles the structure of real program graphs, as well as decoding algorithms which extract the permutation π * from the reducible permutation graph F [π * ] and the number w from π * . Both the encoding and the decoding process takes time and space linear in the length of the binary representation of w. The two main components of our proposed codec system, i.e., the self-inverting permutation π * and the reducible permutation graph F [π * ], incorporate the binary representation of the watermark w in their structure and possess important structural properties, which make our system resilient to attacks; to this end, we experimentally evaluated our system under edge modification attacks on the graph F [π * ] and the results show that we can detect such attacks with high probability.

Characterizing Watermark Numbers encoded as Reducible Permutation Graphs against Malicious Attacks

“…Indeed, they extended the class of software watermarking codec algorithms and graph structures by proposing efficient and easily implemented algorithms for encoding numbers as reducible permutation flow-graphs (RPG) through the use of self-inverting permutations (or, for short, SiP). More precisely, they have presented an efficient method for encoding first an integer w as a self-inverting permutation π * and then encoding π * as a reducible permutation flow-graph F [π * ] [2]; see, also [4]. The watermark graph F [π * ] incorporates properties capable to mimic real code, that is, it does not differ from the graph data structures built by real programs.…”

“…In this section we briefly present the codec system, which we shall call W-RPG, proposed by Chroni and Nikolopoulos [2][3][4]. We firstly discuss the proposed structural components of their model, namely self-inverting permutation (or, for short, SiP) π * and reducible permutation graph (or, for short, RPG) F [π * ], and their properties of methods components, which help prevent edge and/or node modifications attacks.…”

“…Let F [π * ] be the reducible permutation graph which encodes the watermark number w or, equivalently, the self-inverting permutation π * . The graph F [π * ] is constructed by computing the function [4]. Now we perform valid modifications on the elements of the SiP π * .…”