PERFORMANCE OF RANDOM FINGERPRINTING CODES UNDER ARBITRARY NONLINEAR ATTACKS Pierre Moulin and Negar Kiyavash Beckman Inst., Coord. Sci. Lab and ECE Department University of Illinois at Urbana-Champaign, USA ABSTRACT This paper analyzes the performance of arbitrary nonlinear collusion attacks on random fingerprinting codes. We derive the error exponent of the fingerprinting system, which determines the exponential decay of the error probability. A Gaussian ensemble and an expurgated Gaussian ensemble of codes are considered. The collusion attacks include order-statistics attacks as special cases. In our model, a correlation detector is used. The colluders create a noisefree forgery by applying an arbitrary nonlinear mapping to their individual copies, and next they add a Gaussian noise sequence to form the final forgery. The colluders are subject to a mean-squared distortion constraint between host and forgery. We prove that the uniform linear averaging attack outperforms all others.

Index Terms: Digital fingerprinting, coding, detection performance, nonlinear signal processing. 1. INTRODUCTION Digital fingerprinting systems can be used for traitor tracing or digital rights management applications. A length- realusers. valued signal is to be protected and distributed to Some of the users ( of them) may collude and process their copies to create a forgery that contains only weak traces of their fingerprints. This problem was first posed by Cox et al. [1] who proposed the use of Gaussian fingerprints for this purpose. Specifically, their fingerprints were drawn randomly from an i.i.d. Gaussian distribution; the fingerprint code is shared with the detector but not revealed to the users. A fundamental question is what are the optimal performance limits for detection of colluders. To make the problem nontrivial, one may assume embedding distortion constraints on the fingerprinter and the colluders. Example of this analysis include [2, 3] for the case of signals defined over finite alphabets, and [4, 5, 6] for the case of real-valued signals. In the latter case, an obvious (but not necessarily optimal) strategy for the colluders is to perform a uniform linear average of their copies and add i.i.d. Gaussian noise; this strategy was examined in the above papers. Possible improvements for the attackers consist of developing (nonlinear) order-statistics attacks, as proposed by Stone [7]. Computer simulation results 



This research was supported in part by NSF grant CCR 03-25924.

1­4244­0728­1/07/$20.00 ©2007 IEEE

for seven order-statistics collusion attacks have been reported in [7, 8, 9], sometimes with conflicting findings. Our study aims at developing a comprehensive detectiontheoretic analysis of collusion attacks and identifying an optimal strategy for the colluders. The analysis is rooted in largedeviations theory. Initial results were reported in [10] for the class of order-statistics attacks, assuming a correlation detector and constraining the mean-squared distance between the host and the forgery. Under those assumptions, we proved that the uniform linear averaging strategy is optimal for the colluders in the class of order-statistics attacks. The analysis is extended in this paper to a broader class of nonlinear attacks. In our problem setup, two random ensembles of fingerprinting codes are considered. The first one is the same as the one used by Cox [1] and other researchers and is shown to be less performant than the second one, which is an expurgated ensemble (bad codes are eliminated). The detector has access to a forgery as well as to the host signal (nonblind detection) and performs a binary hypothesis test on each user to determine whether that user was involved in the forgery. The cost functions in this problem are the detector’s type-I and type-II probabilities of error, which the colluders want to maximize. Throughout this paper, we use boldface uppercase letters to denote random vectors, uppercase letters for the components of the vectors, and calligraphic fonts for sets. We use the symbol to denote mathematical expectation. For any collection of samples , we denote by the restriction of this collection to its elements . and (asymptotic The symbols and equality) mean that , respectively. The symbol denotes asymp. totic equality on the exponential scale: Of course, one may have and simultaneously. The Gaussian distribution with mean zero is denoted by . and variance 

















































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Collusion Forensics of Multimedia Fingerprinting Using Orthogonal Modulation,” IEEE T-IP, Vol. 14, No. 6, pp. 804—821, June 2005.

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[7] H. S. Stone, “Analysis of Attacks on Image Watermarks With Randomized Coefficients,” NEC TR 96-045, Princeton, NJ, 1996.

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[4] J. Kilian, F. T. Leighton, L. R. Matheson, T. G. Shamoon, R. E. Tarjan, and F. Zane, “Resistance of digital watermarks to collusive attacks,” Proc. ISIT, p. 271, Cambridge, MA, 1998.

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[3] A. Somekh-Baruch and N. Merhav, “On the Capacity Game of Private Fingerprinting Systems Under Collusion Attacks,” Proc. IEEE Int. Symp. on Information Theory, Yokohama, Japan, p. 191, July 2003.

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[2] P. Moulin and J. A. O’Sullivan, “Information-Theoretic Analysis of Information Hiding,” IEEE T-IT, Vol. 49, No. 3, pp. 563—593, 2003.

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[1] I. J. Cox, J. Killian, F. T. Leighton and T. Shamoon, “Secure Spread Spectrum Watermarking for Multimedia,” IEEE T-IP, Vol. 6, pp. 1673—1687, Dec. 1997. (Also NEC Tech. Rep. 95-10, 1995).



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[11] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed., Springer-Verlag, New York, 1998. [12] R. G. Gallager, Information Theory and Reliable Communication, Wiley, NY, 1968.

II ­ 160

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University of Illinois at Urbana-Champaign, USA. ABSTRACT. This paper analyzes the performance of arbitrary nonlinear collu- sion attacks on random fingerprinting codes. We derive the error exponent of the fingerprinting system, which determines the expo- nential decay of the error probability. A Gaussian ensemble and ...

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