We revisit recent results from the area of collusion-resistant traitor tracing, and show how they can be combined and improved to obtain more efficient dynamic traitor tracing schemes. In particular, we show how the dynamic Tardos scheme of can be combined with the optimized score functions of to trace coalitions much faster. If the attack strategy is known, in many cases the order of the code length goes down from quadratic to linear in the number of colluders, while if the attack is not known, we show how the interleaving defense may be used to catch all colluders about twice as fast as in the dynamic Tardos scheme. Some of these results also apply to the static traitor tracing setting where the attack strategy is known in advance, and to group testing.
Fiat and Tassa @cite_5 showed that if the alphabet size @math satisfies @math , then with @math segments, one can find and disconnect all colluders deterministically, i.e., with probability of error @math . This scheme is very efficient, and the only drawback is that it requires a large alphabet size, which may not be possible in practice. Tassa @cite_11 later showed that combining the binary ( @math ) static scheme of Boneh and Shaw @cite_4 with the dynamic scheme of Fiat and Tassa @cite_5 leads to a binary dynamic traitor tracing scheme that uses @math content segments. Due to the large required length of the code, this scheme is not practical.
@cite_5 @cite_4 @cite_11
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We give a generic divide-and-conquer approach for constructing collusion-resistant probabilistic dynamic traitor tracing schemes with larger alphabets from schemes with smaller alphabets. This construction offers a linear tradeoff between the alphabet size and the codelength. In particular, we show that applying our results to the binary dynamic Tardos scheme of leads to schemes that are shorter by a factor equal to half the alphabet size. Asymptotically, these codelengths correspond, up to a constant factor, to the fingerprinting capacity for static probabilistic schemes. This gives a hierarchy of probabilistic dynamic traitor tracing schemes, and bridges the gap between the low bandwidth, high codelength scheme of and the high bandwidth, low codelength scheme of Fiat and Tassa. Dynamic traitor tracing schemes were introduced by Fiat and Tassa in order to combat piracy in active broadcast scenarios. In such settings the data provider supplies access control keys to its legal customers on a periodical basis. A number of users may collude in order to publish those keys via the Internet or any other network. Dynamic traitor tracing schemes rely on the feedback from the pirate network in order to modify their key allocation until they are able either to incriminate and disconnect all traitors or force them to stop their illegal activity. Those schemes are deterministic in the sense that incrimination is always certain. As such deterministic schemes must multiply the critical data by at least p + 1, where p is the number of traitors, they may impose a too large toll on bandwidth. We suggest here probabilistic schemes that enable one to trace all traitors with almost certainty, where the critical data is multiplied by two, regardless of the number of traitors. These techniques are obtained by combining dynamic traitor tracing schemes with binary fingerprinting techniques, such as those proposed by Boneh and Shaw. We construct binary dynamic traitor tracing schemes, where the number of watermark bits needed to trace and disconnect any coalition of pirates is quadratic in the number of pirates, and logarithmic in the total number of users and the error probability. Our results improve upon results of Tassa, and our schemes have several other advantages, such as being able to generate all codewords in advance, a simple accusation method, and flexibility when the feedback from the pirate network is delayed. 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.