Combining an information-theoretic approach to fingerprinting with a more constructive, statistical approach, we derive new results on the fingerprinting capacities for various informed settings, as well as new log-likelihood decoders with provable code lengths that asymptotically match these capacities. The simple decoder built against the interleaving attack is further shown to achieve the simple capacity for unknown attacks, and is argued to be an improved version of the recently proposed decoder of With this new universal decoder, cut-offs on the bias distribution function can finally be dismissed. Besides the application of these results to fingerprinting, a direct consequence of our results to group testing is that (i) a simple decoder asymptotically requires a factor 1.44 more tests to find defectives than a joint decoder, and (ii) the simple decoder presented in this paper provably achieves this bound.
Later work on the constructive side of fingerprinting focused on improving upon Tardos' result by sharpening the bounds @cite_10 @cite_16 , optimizing the distribution functions @cite_12 , improving the score function @cite_27 , tightening the bounds again with this improved score function @cite_4 @cite_26 @cite_17 @cite_40 @cite_20 @cite_43 , optimizing the score function @cite_39 , and again tightening the bounds with this optimized score function @cite_23 @cite_22 to finally end up with a sufficient asymptotic code length of @math for large @math . This construction can be extended to larger alphabets, in which case the code length scales as @math . Other work on practical constructions focused on joint decoders, which are computationally more involved but may work with shorter codes @cite_29 @cite_6 @cite_42 , and side-informed fingerprinting games @cite_31 @cite_45 @cite_32 @cite_0 , where estimating the collusion channel @math was considered to get an improved performance.
@cite_22 @cite_29 @cite_42 @cite_43 @cite_10 @cite_20 @cite_4 @cite_39 @cite_23 @cite_17 @cite_26 @cite_32 @cite_6 @cite_27 @cite_40 @cite_16 @cite_12 @cite_0 @cite_45 @cite_31
2083594366 2087879325 1969848549 1977832049
We study a fingerprinting game in which the number of colluders and the collusion channel are unknown. The encoder embeds fingerprints into a host sequence and provides the decoder with the capability to trace back pirated copies to the colluders. Fingerprinting capacity has recently been derived as the limit value of a sequence of maximin games with mutual information as their payoff functions. However, these games generally do not admit saddle-point solutions and are very hard to solve numerically. Here under the so-called Boneh-Shaw marking assumption, we reformulate the capacity as the value of a single two-person zero-sum game, and show that it is achieved by a saddle-point solution. If the maximal coalition size is k and the fingerprinting alphabet is binary, we show that capacity decays quadratically with k. Furthermore, we prove rigorously that the asymptotic capacity is 1 (k221n2) and we confirm our earlier conjecture that Tardos' choice of the arcsine distribution asymptotically maximizes the mutual information payoff function while the interleaving attack minimizes it. Along with the asymptotics, numerical solutions to the game for small k are also presented. 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. We study the asymptotic-capacity-achieving score function that was recently proposed by for bias-based traitor tracing codes. For the bias function, we choose the Dirichlet distribution with a cutoff. Using Bernstein's inequality and Bennett's inequality, we upper bound the false-positive and false-negative error probabilities. From these bounds we derive sufficient conditions for the scheme parameters. We solve these conditions in the limit of large coalition size c0 and obtain asymptotic solutions for the cutoff, the sufficient code length, and the corresponding accusation threshold. We find that the code length converges to its asymptote approximately as c0?1 2 @math , which is faster than the c0?1 3 @math of Tardos' score function. MSC:94B60 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.