Abstract
We study the probability distribution of user accusations in the q-ary Tardos fingerprinting system under the Marking Assumption, in the restricted digit model. In particular, we look at the applicability of the so-called Gaussian approximation, which states that accusation probabilities tend to the normal distribution when the fingerprinting code is long. We introduce a novel parametrization of the attack strategy which enables a significant speedup of numerical evaluations. We set up a method, based on power series expansions, to systematically compute the probability of accusing innocent users. The ‘small parameter’ in the power series is 1/m, where m is the code length. We use our method to semi-analytically study the performance of the Tardos code against majority voting and interleaving attacks. The bias function ‘shape’ parameter \({{\kappa}}\) strongly influences the distance between the actual probabilities and the asymptotic Gaussian curve. The impact on the collusion-resilience of the code is shown. For some realistic parameter values, the false accusation probability is even lower than the Gaussian approximation predicts.
Article PDF
Similar content being viewed by others
Abbreviations
- \({{\mathcal {Q}}}\) :
-
The alphabet
- q :
-
Alphabet size \({|{\mathcal {Q}}|}\)
- n :
-
Number of users
- \({{\mathcal {C}}}\) :
-
Set of colluding users
- c :
-
Number of colluders \({|{\mathcal {C}}|}\)
- c 0 :
-
Coalition size that the code can resist
- m :
-
Code length (number of q-ary symbols)
- X ji :
-
Embedded symbol in segment i for user j
- p (i) :
-
Bias vector for column i
- F :
-
Distribution function of the bias vector, p (i)~ F
- f (p α):
-
Marginal distribution of F for one component
- \({{\kappa}}\) :
-
Shape parameter contained in F
- \({{\sigma}_{\alpha}^{(i)}}\) :
-
Number of occurrences of symbol α in attackers’ segment i
- \({\mathbb {P}}\) :
-
Probability distribution for σ
- \({\mathbb {P}_1}\) :
-
Marginal distribution for one component of σ
- \({\mathbb {P}_{q-1}}\) :
-
Marginal distribution for q − 1 components of σ
- y i :
-
Symbol in segment i of attacked content
- \({{\theta}_{y|{\sigma}}}\) :
-
Prob. that attackers output symbol y, given σ
- S j :
-
Accusation sum of user j
- S :
-
Coalition accusation sum, \({S=\sum_{j\in{\mathcal {C}}}S_j}\)
- Z :
-
Accusation threshold
- \({\tilde Z}\) :
-
\({Z/\sqrt m}\)
- \({{\mathcal {L}}}\) :
-
List of accused users
- \({{\varepsilon}_1}\) :
-
Max. tolerable prob. of fixed innocent user getting accused
- \({{\varepsilon}_2}\) :
-
Max. tolerable prob. of not catching any attacker
- FP :
-
False positive
- FN :
-
False negative
- \({\tilde\mu}\) :
-
\({\mathbb {E}[S]/m}\) ; does not depend on m
- \({{\rho}_m}\) :
-
Prob. distribution of \({S_j/\sqrt m}\) for innocent j
- R m :
-
Area function for the right-hand tail of \({{\rho}_m}\)
- \({{\tau}_m}\) :
-
Prob. distribution of \({S/(c\sqrt m)}\) , normalized to zero mean and variance 1
- T m :
-
Cumulative distribution function for \({{\tau}_m}\)
- \({{\varphi}}\) :
-
Prob. distribution of one-segment contribution to innocent’s accusation
- \({{\psi}_b({\bf x})}\) :
-
\({{\theta}_{y|{\sigma}}}\) when σ y = b and the rest of σ is equal to x
- K b :
-
Quantity derived from \({{\psi}_b({\bf x})}\)
- Ω(x):
-
Probability mass in the right tail of a Gaussian, beyond x
References
Alfaro P.C.: Side-informed data hiding: robustness and security analysis. PhD thesis, Universidade de Vigo (2006).
Amiri E., Tardos G.: High rate fingerprinting codes and the fingerprinting capacity. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 336–345 (2009).
Beaulieu N.C.: An infinite series for the computation of the complementary probability distribution function of a sum of independent random variables and its application to the sum of Rayleigh random variables. IEEE Trans. Commun. 38(9), 1463–1474 (1990)
Blayer O., Tassa T.: Improved versions of Tardos’ fingerprinting scheme. Des. Codes Cryptogr. 48(1), 79–103 (2008)
Boneh D., Shaw J.: Collusion-secure fingerprinting for digital data. IEEE Trans. Inf. Theory 44(5), 1897–1905 (1998)
Furon T., Guyader A., Cérou, F.: On the design and optimization of Tardos probabilistic fingerprinting codes. In: Information Hiding, volume 5284 of Lecture Notes in Computer Science, pp. 341–356. Springer (2008).
Furon T., Pérez-Freire L., Guyader A., Cérou, F.: Estimating the minimal length of Tardos code. In: Information Hiding, volume 5806 of Lecture Notes in Computer Science, pp. 176–190. Springer (2009).
Gradshteyn I.S., Ryzhik I.M.: Table of Integrals, Series, and Products, 5th edn. Academic Press, San Diego (1994)
He S., Wu M.: Joint coding and embedding techniques for multimedia fingerprinting. IEEE Trans. Inf. Forensics Secur. 1, 231–248 (2006)
Huang Y.-W., Moulin P.: Maximin optimality of the arcsine fingerprinting distribution and the interleaving attack for large coalitions. In: IEEE International Workshop on Information Forensics and Security, pp. 1–6 (2010).
Huang Y.W., Moulin P.: Saddle-point solution of the fingerprinting capacity game under the marking assumption. In: IEEE International Symposium on Information Theory (2009).
Huang Y.W., Moulin P.: Saddle-point solution of the fingerprinting capacity game under the marking assumption. http://arxiv.org/abs/0905.1375 (2009).
Kilian J., Leighton F.T., Matheson L.R., Shamoon T.G., Tarjan R.E., Zane F.: Resistance of digital watermarks to collusive attacks. In: IEEE International Symposium on Information Theory, p. 271 (1998).
Kuribayashi M., Akashi N., Morii M.: On the systematic generation of Tardos’s fingerprinting codes. In: International Workshop on Multimedia Signal Processing, pp. 748–753 (2008).
Lukacs E.: Characteristic Functions. Statistical monographs & courses. Griffin, London (1960)
Moulin P.: Universal fingerprinting: capacity and random-coding exponents. Preprint arXiv:0801.3837v2, available at http://arxiv.org/abs/0801.3837 (2008).
Nuida K., Hagiwara M., Watanabe H., Imai H.: Optimal probabilistic fingerprinting codes using optimal finite random variables related to numerical quadrature. CoRR, abs/cs/0610036, http://arxiv.org/abs/cs/0610036 (2006).
Prudnikov A.P., Brychkov Yu.A., Marichev O.I.: Integrals and Series, 4th printing, vol. 1. CRC Press, Boca Raton (1998)
Schaathun H.G.: On error-correcting fingerprinting codes for use with watermarking. Multimedia Syst. 13(5–6), 331–344 (2008)
Škorić B., Katzenbeisser S., Celik M.U.: Symmetric Tardos fingerprinting codes for arbitrary alphabet sizes. Des. Codes Cryptogr. 46(2), 137–166 (2008)
Škorić B., Vladimirova T.U., Celik M.U., Talstra J.C.: Tardos fingerprinting is better than we thought. IEEE Trans. Inf. Theory 54(8), 3663–3676 (2008).
Somekh-Baruch A., Merhav N.: On the capacity game of private fingerprinting systems under collusion attacks. IEEE Trans. Inf. Theory 51, 884–899 (2005)
Tardos G.: Optimal probabilistic fingerprint codes. In: ACM Symposium on Theory of Computing, pp. 116–125 (2003)
Acknowledgment
We kindly thank Benne de Weger, Dion Boesten, Jan-Jaap Oosterwijk and Guido Janssen for useful discussions.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. van Tilborg.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Simone, A., Škorić, B. Accusation probabilities in Tardos codes: beyond the Gaussian approximation. Des. Codes Cryptogr. 63, 379–412 (2012). https://doi.org/10.1007/s10623-011-9563-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-011-9563-4