Tree search for digital fingerprinting

In this work we consider the application of tree codes in designing digital fingerprinting schemes under collusion attacks. Our setting is the Boneh-Shaw´s marking assumption which is a common model for software fingerprinting. In a previous work we studied minimum distance decoding for the problem and proposed a low complexity scheme capable of catching one pirate for a coalition size equal to 2. The previous contribution was not extendable to the case of larger coalition sizes. In this paper we extend our work on designing low complexity fingerprinting schemes to the general case of arbitrary coalition sizes. Moreover, here we aim to identify the whole coalition instead of just one colluder. In order to do so we assume that the attack is fair the validity of which hinges on the fact that all pirates in the coalition will want to take the same chance of being caught as others. The decoding is done by searching for a coalition which satisfies the marking assumption sequentially on the Cartesian product of the tree code with its replicas. Using a random coding argument, the decoder´s cutoff rate is characterized for a general t. Average error probability for the random ensemble is also proved to be vanishing for rates less than the cutoff rate. Finally, using an expurgation argument we establish the existence of a tree code with vanishing error probability and finite complexity. Using explicit constructions we obtain vanishing probabilities of decoding failure with linear complexity at rates 1/8 and 1/40 when t = 2 and t = 3 respectively. Our results are the first of its kind in the design of low complexity fingerprinting schemes for a general t.