Designs, Codes and Cryptography

, Volume 63, Issue 3, pp 379–412

Accusation probabilities in Tardos codes: beyond the Gaussian approximation

Open Access
Article

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.

Keywords

Traitor tracing Tardos fingerprinting Collusion resistance 

List of symbols

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

The alphabet

q

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

n

Number of users

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

Set of colluding users

c

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

c0

Coalition size that the code can resist

m

Code length (number of q-ary symbols)

Xji

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 (pα)

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

yi

Symbol in segment i of attacked content

\({{\theta}_{y|{\sigma}}}\)

Prob. that attackers output symbol y, given σ

Sj

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

Rm

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

Tm

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 σy = b and the rest of σ is equal to x

Kb

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

Ω(x)

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

Mathematics Subject Classification (2000)

94B60 60G35 60G50 

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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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