The impact of the theoretical properties of the Logistic Function on the generation of optically detectable watermarks

A digital watermark is a visible, or preferably invisible, identification code that is permanently embedded in digital media, to prove owner authentication and provide protection for security or defence documents. In this paper, we present an approach for the generation of watermarks using a 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. We have previously presented a study where an optical correlator was suitable for the detection of chaotically generated watermarks. We have also studied the impact of shot noise present in an optical detector for watermarks generated using the logistic function. 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 on watermark generation and their highpass and lowpass properties, which when embedded in digital media, are suitable for optical detection.