# Characterizing Watermark Numbers encoded as Reducible Permutation Graphs against Malicious Attacks

## TL;DR

This paper analyzes the resilience of a watermark encoding method using reducible permutation graphs against attacks, characterizing watermark strength based on permutation structures and minimal modifications needed for validity.

## Contribution

It provides a theoretical characterization of watermark strength in permutation-based graph encodings and quantifies the minimal modifications to alter watermark validity.

## Key findings

- Characterizes strong and weak watermarks via permutation structure
- Calculates minimal binary modifications to produce valid watermarks
- Proves resilience properties of the W-RPG watermarking scheme

## Abstract

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 $F[\pi^*]$ through the use of self-inverting permutations $\pi^*$. 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 $F[\pi^*]$. In particular, we characterize the integer $w\equiv\pi^*$ as strong or weak watermark through the structure of self-inverting permutations $\pi^*$ which encodes 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 $w$, 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 $\pi^* \equiv w$.