We consider the problem of detecting a small subset of defective items from a large set via non-adaptive "random pooling" group tests. We consider both the case when the measurements are noiseless, and the case2 when the measurements are noisy (the outcome of each group test may be independently faulty with probability q). Order-optimal results for these scenarios are known in the literature. We give information-theoretic lower bounds on the query complexity of these problems, and provide corresponding computationally efficient algorithms that match the lower bounds up to a constant factor. To the best of our knowledge this work is the first to explicitly estimate such a constant that characterizes the gap between the upper and lower bounds for these problems.

Alice may wish to reliably send a message to Bob over a binary symmetric channel (BSC) while ensuring that her transmission is deniable from an eavesdropper Willie. That is, if Willie observes a "significantly noisier" transmission than Bob does, he should be unable to estimate even whether Alice is transmitting or not. Even when Alice's (potential) communication scheme is publicly known to Willie (with no common randomness between Alice and Bob), we prove that over n channel uses Alice can transmit a message of length O( √ n) bits to Bob, deniably from Willie. We also prove information-theoretically order-optimality of our results.

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Abstract-Alice wishes to potentially communicate covertly with Bob over a Binary Symmetric Channel while Willie the wiretapper listens in over a channel that is noisier than Bob's. We show that Alice can send her messages reliably to Bob while ensuring that even whether or not she is actively communicating is (a) deniable to Willie, and (b) optionally, her message is also hidable from Willie. We consider two different variants of the problem depending on the Alice's "default" behavior, i.e., her transmission statistics when she has no covert message to send: 1) When Alice has no covert message, she stays "silent", i.e., her transmission is 0; 2) When has no covert message, she transmits "innocently", i.e., her transmission is drawn uniformly from an innocent random codebook; We prove that the best rate at which Alice can communicate both deniably and hidably in model 1 is O(1/ √ n). On the other hand, in model 2, Alice can communicate at a constant rate.

We survey here recent work pertaining to "deniable" communicationi.e., talking without being detected. We first highlight connections to other related notions (anonymity and secrecy). We then contrast the notions of deniability and secrecy. We highlight similarities and distinctions of deniability with a variety of related notions (LPD communications, stealth, channel resolvability) extant in the literature.