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

The last years have seen a growing interest in collaborative systems like electronic marketplaces and P2P file sharing systems where people are intended to interact with other people. Those systems, however, are subject to security and operational risks because of their open and distributed nature. Reputation systems provide a mechanism to reduce such risks by building trust relationships among entities and identifying malicious entities. A popular reputation model is the so called flow-based model. Most existing reputation systems based on such a model provide only a ranking, without absolute reputation values; this makes it difficult to determine whether entities are actually trustworthy or untrustworthy. In addition, those systems ignore a significant part of the available information; as a consequence, reputation values may not be accurate. In this paper, we present a flow-based reputation metric that gives absolute values instead of merely a ranking. Our metric makes use of all the available information. We study, both analytically and numerically, the properties of the proposed metric and the effect of attacks on reputation values.

We use a method recently introduced by Simone andŠkorić to study accusation probabilities for non-binary Tardos fingerprinting codes. We generalize the pre-computation steps in this approach to include a broad class of collusion attack strategies. We analytically derive properties of a special attack that asymptotically maximizes false accusation probabilities. We present numerical results on sufficient code lengths for this attack, and explain the abrupt transitions that occur in these results.

We investigate False Positive (FP) accusation probabilities for q-ary Tardos codes in the Restricted Digit Model. We employ a computation method recently introduced by us, to which we refer as Convolution and Series Expansion (CSE). We present a comparison of several collusion attacks on q-ary codes: majority voting, minority voting, Interleaving,μ-minimizing and Random Symbol (the q-ary equivalent of the Coin Flip strategy). The comparison is made by looking at the FP rate at approximately fixed False Negative rate. In nearly all cases we find that the strongest attack is either minority voting orμ-minimizing, depending on the exact setting of parameters such as alphabet size, code length, and coalition size.Furthermore, we present results on the convergence speed of the CSE method, and we show how FP rate computations for the Random Symbol strategy can be sped up by a precomputation step.

Forensic watermarking is the application of digital watermarks for the purpose of tracing unauthorized redistribution of content. The most powerful type of attack on watermarks is the collusion attack, in which multiple users compare their differently watermarked versions of the same content. Collusionresistant codes have been developed against these attacks. One of the most famous such codes is the Tardos code. It has the asymptotically optimal property that it can resist c attackers with a code of length proportional to c 2 .Determining error rates for the Tardos code and its various extensions and generalizations turns out to be a nontrivial problem. In recent work we developed an approach called the Convolution and Series Expansion (CSE) method to accurately compute false positive accusation probabilities. In this paper we extend the CSE method in order to make it possible to compute false negative accusation probabilities as well.