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

Abstract

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.

Phase Transitions in Group Testing

“…Early studies of this type were performed by Malyutov [9], and more recent studies include those of Atia and Saligrama [12], Aldridge et al [11,14], and Laarhoven [15]. In the case that the number of defective items k does not scale with p, the fundamental limits are well-understood for both the noiseless and noisy settings [9,12,15]; for example, in the noiseless case, the smallest possible number of measurements with vanishing error probability behaves as k log 2 p k (1 + o(1)), which is in fact the same threshold as that for optimal adaptive measurements [16] (i.e., designs for which each test may depend on previous outcomes). Surprisingly, there remain significant gaps in the best known upper and lower bounds on n when k scales with p, which is of considerable interest in applications where the number of defective items is "not too small".…”

“…For fixed s ∈ S and a corresponding pair (s dif , s eq ), we introduce the notation 15) where P Y |Xs is the marginal distribution of (3.12).…”

Tally-Based Simple Decoders for Traitor Tracing and Group Testing

“…2 we show an example of Receiver Operating Characteristic (ROC) curves 2 obtained from simulations. Even at n = 1000, a rather small number of users, we see that there is little performance difference between the score (20) proposed in this paper and the Laarhoven score. An exception is the case of the Minority Voting attack, which favours low p y values; at low p y the statistical fluctuations in t y are more pronounced than at large p y , as already mentioned in Section III.…”

“…(On the other hand, an accurate estimate of P FP , which may be as small as 10 −6 , takes millions of simulation runs of Experiments 0 and 1). The one-user false positive probability P FP1 as a function of the threshold Z , for Experiment 1 with parameters n = 100000, c = 12, = 14000 and use of the score function (20). In the simulation the P FP1 was estimated by doing a single run of Experiment 0 and Experiment 1 and then counting how many innocent users had a score exceeding Z .…”

How little does non-exact recovery help in group testing?

“…The information-theoretic limits of group testing have been studied for decades (e.g., see [8,9]), and have recently become increasingly well-understood [10][11][12][13][14][15]. In particular, when the number of defectives scales as k = O(p 1/3 ), it is known that the number of tests n * required to exactly identify the defective set satisfies [14] …”