Risk-distortion analysis for video collusion attacks: a mouse-and-cat game

Copyright protection is a key issue for video sharing over public networks. To protect the video content from unauthorized redistribution, digital fingerprinting is commonly used. To develop an efficient collusion-resistant fingerprinting scheme, it is very important for the system designer to understand how the behavior dynamics of colluders affect the performance of collusion attack. In the literature, little effort has been made to explicitly study the relationship between risk, e.g., the probability of the colluders to be detected, and the distortion of the colluded signal. In this paper, we investigate the risk-distortion relationship for the linear video collusion attack with Gaussian fingerprint. We formulate the optimal linear collusion attack as an optimization problem of finding the optimal collusion parameters to minimize the distortion subject to a risk constraint. By varying the risk constraint and solving the corresponding optimization problem, we can derive the optimal risk-distortion curve. Moreover, based upon the observation that the detector/attacker can each improve the detection/attack performance with the knowledge of his/her opponent's strategy, we formulate the attack and detection problem as a dynamic mouse and cat game and study the optimal strategies for both the attacker and detector. We show that if the detector uses a fixed detection strategy, the attacker can estimate the detector's strategy and choose the corresponding optimal strategy to attack the fingerprinted video with a small distortion. However, if the detector is powerful, i.e., the detector can always estimate the attacker's strategy, the best strategy for the attacker is the min-max strategy. Finally, we conduct several experiments to verify the proposed risk-distortion model using real video data.