We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds.For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich [25]. The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

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The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their n-bit input strings is large (i.e., at least n/2 + √ n) or small (i.e., at most n/2 − √ n); they do not care if it is neither large nor small. This ( √ n) gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm.Thus far, for randomized communication, an (n) lower bound on this problem was known only in the one-way setting. We prove an (n) lower bound for randomized protocols that use any constant number of rounds.As a consequence we conclude, for instance, that ε-approximately counting the number of distinct elements in a data stream requires (1/ε 2 ) space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream.In the process, we also obtain tight n − ( √ n log n) lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an (n) lower bound on the one-way randomized communication complexity.

We consider the following cryptographic secret leaking problem. A group of players communicate with the goal of learning (and perhaps revealing) a secret held initially by one of them. Their conversation is monitored by a computationally unlimited eavesdropper, who wants to learn the identity of the secret-holder. Despite the unavailability of key, some protection can be provided to the identity of the secretholder. We call the study of such communication problems, either from the group's or the eavesdropper's point of view, cryptogenography.We introduce a basic cryptogenography problem and show that two players can force the eavesdropper to missguess the origin of a secret bit with probability 1/3; we complement this with a hardness result showing that they cannot do better than than 3/8. We prove that larger numbers of players can do better than 0.5644, but no group of any size can achieve 0.75.