Generation and Detection of Watermarks Derived from Chaotic Functions

A digital watermark is a visible, or preferably invisible, identification code that is permanently embedded in digital media, to prove owner authentication thereby providing a level of document protection. In this paper, we review several approaches for the generation of watermarks using chaotic functions, and in particular, the logistic chaotic function. Using this function, in conjunction with seed management, it is possible to generate chaotic sequences that may be used to create highpass or lowpass digital watermarks. A slight change in the initial conditions will quickly lead to a significant change in the subsequent states of the system, and thus will generate substantially different watermarks. This technique has been shown to offer an added security advantage over the more traditionally generated watermarks created from pseudorandom sequences, in that only the function seed needs to be stored. It also has the advantage that, through examination of the theoretical properties of the function, it is possible to choose seeds that lead to robust, lowpass watermarks. We review various detection techniques including correlation and statistical methods, and present an analysis of the impact of noise present in a model optical detector. The logistic function presented in this paper is ill defined for certain seed values and has not been fully investigated for the purpose of watermark generation. We consider the impact of the theoretical properties of the logistic function for several chaos-based watermark generation techniques, in particular, their highpass and lowpass properties, which when embedded in digital media, are suitable for correlation and statistical based detection methods.