Estimating the Secret Key of Spread Spectrum Watermarking Based on Equivalent Keys

The security of spread spectrum (SS) watermarking largely depends on the difficulty of estimating its secret key. Some estimators have been proposed to estimate the secret key in the known-message attack (KMA) scenario. However, the estimation accuracies of existing estimators are not satisfactory when the number of observations is not large enough. Currently, it is still a challenging and open problem to design more effective estimators. In this paper, we propose an equivalent keys (EK)-based estimator to estimate the secret key for both the traditional and more secure SS watermarking methods. Equivalent keys form an equivalent region, which is the intersection of a unit hypersphere and a hypercone. According to the Monte Carlo simulation, we find that the secret key can be estimated by adding up the equivalent keys uniformly sampled from the equivalent region. Thus, the proposed estimator selects equivalent keys from randomly-generated vectors by exploiting the pairs of watermarked signals and their embedded messages. A theoretical analysis is performed for the proposed estimator to evaluate the estimation accuracy. Experimental results verify the theoretical analysis and show the superiority of the proposed estimator over existing estimation methods. Furthermore, this paper also shows the insecurity of the more secure SS watermarking methods in the KMA scenario from a practical perspective for the first time.