Two Party Distribution Testing: Communication and Security

Authors Alexandr Andoni, Tal Malkin, Negev Shekel Nosatzki

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Alexandr Andoni
  • Columbia University, New York City, NY, USA
Tal Malkin
  • Columbia University, New York City, NY, USA
Negev Shekel Nosatzki
  • Columbia University, New York City, NY, USA


We thank Devanshi Nishit Vyas for her contribution to some of the initial work which led to this paper. We thank Clement Canonne for invaluable comments on an early draft of the manuscript. We thank Yuval Ishai for helpful discussions. Work supported in part by Simons Foundation (#491119), NSF grants CCF-1617955 and CCF-1740833.

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Alexandr Andoni, Tal Malkin, and Negev Shekel Nosatzki. Two Party Distribution Testing: Communication and Security. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We study the problem of discrete distribution testing in the two-party setting. For example, in the standard closeness testing problem, Alice and Bob each have t samples from, respectively, distributions a and b over [n], and they need to test whether a=b or a,b are epsilon-far (in the l_1 distance). This is in contrast to the well-studied one-party case, where the tester has unrestricted access to samples of both distributions. Despite being a natural constraint in applications, the two-party setting has previously evaded attention. We address two fundamental aspects of the two-party setting: 1) what is the communication complexity, and 2) can it be accomplished securely, without Alice and Bob learning extra information about each other’s input. Besides closeness testing, we also study the independence testing problem, where Alice and Bob have t samples from distributions a and b respectively, which may be correlated; the question is whether a,b are independent or epsilon-far from being independent. Our contribution is three-fold: 1) We show how to gain communication efficiency given more samples, beyond the information-theoretic bound on t. The gain is polynomially better than what one would obtain via adapting one-party algorithms. 2) We prove tightness of our trade-off for the closeness testing, as well as that the independence testing requires tight Omega(sqrt{m}) communication for unbounded number of samples. These lower bounds are of independent interest as, to the best of our knowledge, these are the first 2-party communication lower bounds for testing problems, where the inputs are a set of i.i.d. samples. 3) We define the concept of secure distribution testing, and provide secure versions of the above protocols with an overhead that is only polynomial in the security parameter.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Hypothesis testing and confidence interval computation
  • distribution testing
  • communication complexity
  • security


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