Designs, Codes and Cryptography

, Volume 63, Issue 3, pp 379–412 | Cite as

Accusation probabilities in Tardos codes: beyond the Gaussian approximation

  • Antonino Simone
  • Boris Škorić
Open Access


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.


Traitor tracing Tardos fingerprinting Collusion resistance 

List of symbols

\({{\mathcal {Q}}}\)

The alphabet


Alphabet size \({|{\mathcal {Q}}|}\)


Number of users

\({{\mathcal {C}}}\)

Set of colluding users


Number of colluders \({|{\mathcal {C}}|}\)


Coalition size that the code can resist


Code length (number of q-ary symbols)


Embedded symbol in segment i for user j


Bias vector for column i


Distribution function of the bias vector, p (i)~ F

f (pα)

Marginal distribution of F for one component


Shape parameter contained in F


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


Symbol in segment i of attacked content


Prob. that attackers output symbol y, given σ


Accusation sum of user j


Coalition accusation sum, \({S=\sum_{j\in{\mathcal {C}}}S_j}\)


Accusation threshold

\({\tilde Z}\)

\({Z/\sqrt m}\)

\({{\mathcal {L}}}\)

List of accused users


Max. tolerable prob. of fixed innocent user getting accused


Max. tolerable prob. of not catching any attacker


False positive


False negative


\({\mathbb {E}[S]/m}\) ; does not depend on m


Prob. distribution of \({S_j/\sqrt m}\) for innocent j


Area function for the right-hand tail of \({{\rho}_m}\)


Prob. distribution of \({S/(c\sqrt m)}\) , normalized to zero mean and variance 1


Cumulative distribution function for \({{\tau}_m}\)


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


Quantity derived from \({{\psi}_b({\bf x})}\)


Probability mass in the right tail of a Gaussian, beyond x

Mathematics Subject Classification (2000)

94B60 60G35 60G50 



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.


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Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands

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