Accusation probabilities in Tardos codes: beyond the Gaussian approximation

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 κ 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

False positive probabilities in q-ary Tardos codes: comparison of attacks

“…However, these arguments are not very precise. In [14] and [13] an in-depth analytical and numerical investigation of the Gaussian approximation was given in the RDM case. The approach is based on the convolution rule for characteristic functions, and a way to express the false accusation probability as a power series expansion in the small parameter 1/m.…”

“…For instance, if the false positive rate is around 10 −10 , then a number of simulations of order at least 10 10 is required to measure this rate; in contrast, the computational effort in the CSE method does not depend on the error rate. The work of [14,13] showed, for various parameter settings and attack strategies, how the false positive probability has a transition from Gaussian behaviour in the central region to worse-thanGaussian behaviour outside the center. However, a number of questions were left unanswered.…”

“…The pre-computation is a weighted sum over the attack parameters, and the result is referred to as 'K b ' in [14,13]. The K b was pre-computed for the following attacks: majority voting, minority voting, interleaving, and the attack that minimizes the coalition score.…”

“…Full equations for this pdf were already given in [14], but the graphs better illustrate the differences between the various strategies.…”

“…Two important marginals were given in [14]. First, the marginal probability P 1 (b) = Pr[σ α = b] for one arbitrary component α,…”