Asymptotics of Fingerprinting and Group Testing: Tight Bounds From Channel Capacities

In this paper, we consider the large-coalition asymptotics of various fingerprinting and group testing games, and derive explicit expressions for the capacities for each of these models. We do this both for simple (fast but suboptimal) and arbitrary, joint decoders (slow but optimal). For fingerprinting, we show that if the pirate strategy is known, the capacity often decreases linearly with the number of colluders, instead of quadratically as in the uninformed fingerprinting game. For all considered attacks, the joint capacity is shown to be strictly higher than the simple capacity, including the interleaving attack. This last result contradicts the results of Huang and Moulin regarding the joint fingerprinting capacity, which implies that finding the fingerprinting capacity without restrictions on the tracer's capabilities remains an open problem. For group testing, we improve upon the previous work about the joint capacities, and derive new explicit asymptotics for the simple capacities of various models. These show that the existing simple group testing algorithms of Chan et al. are suboptimal, and that simple decoders cannot asymptotically be as efficient as joint decoders. For the traditional group testing model, we show that the gap between the simple and joint capacities is a factor log
₂
(e) ≈ 1.44 for large numbers of defectives.