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.
In 2003, Tardos @cite_6 showed that the optimal length of such codes (i.e., the number of segments needed) is of the order @math with @math , where @math is an upper bound on the probability of catching one or more innocent users. Note that @math , commonly used for an upper bound on the probability of not catching any pirates, does not appear in the leading term of the code length for most practical values of @math and @math . In the same paper, Tardos gave a construction of a scheme with @math , which is widely known as the Tardos scheme. This shows that @math is optimal, and that the Tardos scheme has the optimal order code length.
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We construct binary codes for fingerprinting digital documents. Our codes for n users that are e-secure against c pirates have length O(c2log(n e)). This improves the codes proposed by Boneh and Shaw l1998r whose length is approximately the square of this length. The improvement carries over to works using the Boneh--Shaw code as a primitive, for example, to the dynamic traitor tracing scheme of Tassa l2005r. By proving matching lower bounds we establish that the length of our codes is best within a constant factor for reasonable error probabilities. This lower bound generalizes the bound found independently by l2003r that applies to a limited class of codes. Our results also imply that randomized fingerprint codes over a binary alphabet are as powerful as over an arbitrary alphabet and the equal strength of two distinct models for fingerprinting.