The Tardos scheme is a well-known traitor tracing scheme to protect copyrighted content against collusion attacks. The original scheme contained some suboptimal design choices, such as the score function and the distribution function used for generating the biases. previously showed that a symbol-symmetric score function leads to shorter codes, while obtained the optimal distribution functions for arbitrary coalition sizes. Later, showed that combining these results leads to even shorter codes when the coalition size is small. We extend their analysis to the case of large coalitions and prove that these optimal distributions converge to the arcsine distribution, thus showing that the arcsine distribution is asymptotically optimal in the symmetric Tardos scheme. We also present a new, practical alternative to the discrete distributions of and give a comparison of the estimated lengths of the fingerprinting codes for each of these distributions.
Over the last ten years, improvements to the Tardos scheme have lead to a significant decrease in the code length parameter @math . We previously showed @cite_7 that combining the symbol-symmetric score function of S kori ' c et al @cite_4 with the improved analysis of Blayer and Tassa @cite_11 leads to an asymptotic code length constant of @math for large @math . For small coalitions, @cite_3 showed that even smaller values @math can be obtained by combining the symmetric score function with the optimized, discrete distribution functions previously obtained by @cite_0 . For large @math , this lead to an asymptotic code length constant of about @math .
@cite_4 @cite_7 @cite_3 @cite_0 @cite_11
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Fingerprinting provides a means of tracing unauthorized redistribution of digital data by individually marking each authorized copy with a personalized serial number. In order to prevent a group of users from collectively escaping identification, collusion-secure fingerprinting codes have been proposed. In this paper, we introduce a new construction of a collusion-secure fingerprinting code which is similar to a recent construction by Tardos but achieves shorter code lengths and allows for codes over arbitrary alphabets. We present results for symmetric' coalition strategies. For binary alphabets and a false accusation probability @math , a code length of @math symbols is provably sufficient, for large c 0, to withstand collusion attacks of up to c 0 colluders. This improves Tardos' construction by a factor of 10. Furthermore, invoking the Central Limit Theorem in the case of sufficiently large c 0, we show that even a code length of @math is adequate. Assuming the restricted digit model, the code length can be further reduced by moving from a binary alphabet to a q-ary alphabet. Numerical results show that a reduction of 35 is achievable for q = 3 and 80 for q = 10. For the Tardos traitor tracing scheme, we show that by combining the symbol-symmetric accusation function of ? with the improved analysis of Blayer and Tassa we get further improvements. Our construction gives codes that are up to four times shorter than Blayer and Tassa's, and up to two times shorter than the codes from ? Asymptotically, we achieve the theoretical optimal codelength for Tardos' distribution function and the symmetric score function. For large coalitions, our codelengths are asymptotically about 4.93 of Tardos' original codelengths, which also improves upon results from It has been proven that the code lengths of Tardos's collusion-secure fingerprinting codes are of theoretically minimal order with respect to the number of adversarial users (pirates). However, the code lengths can be further reduced as some preceding studies have revealed. In this article we improve a recent discrete variant of Tardos's codes, and give a security proof of our codes under an assumption weaker than the original Marking Assumption. Our analysis shows that our codes have significantly shorter lengths than Tardos's codes. For example, when c = 8, our code length is about 4.94 of Tardos's code in a practical setting and about 4.62 in a certain limit case. Our code lengths for large c are asymptotically about 5.35 of Tardos's codes. It is known that Tardos's collusion-secure probabilistic fingerprinting code (Tardos code) has length of theoretically minimal order. However, Tardos code uses certain continuous probability distribution, which causes that huge amount of extra memory is required in a practical use. An essential solution is to replace the continuous distributions with finite discrete ones, preserving the security. In this paper, we determine the optimal finite distribution for the purpose of reducing memory amount; the required extra memory is reduced to less than 1 32 of the original in some practical setting. Moreover, the code length is also reduced (to, asymptotically, about 20.6 of Tardos code), and some further practical problems such as approximation errors are also considered. We study the Tardos' probabilistic fingerprinting scheme and show that its codeword length may be shortened by a factor of approximately 4. We achieve this by retracing Tardos' analysis of the scheme and extracting from it all constants that were arbitrarily selected. We replace those constants with parameters and derive a set of inequalities that those parameters must satisfy so that the desired security properties of the scheme still hold. Then we look for a solution of those inequalities in which the parameter that governs the codeword length is minimal. A further reduction in the codeword length is achieved by decoupling the error probability of falsely accusing innocent users from the error probability of missing all colluding pirates. Finally, we simulate the Tardos scheme and show that, in practice, one may use codewords that are shorter than those in the original Tardos scheme by a factor of at least 16.