两方分布测试：通信和安全
Two Party Distribution Testing: Communication and Security
摘要：
研究了两方情形下离散分布的检验问题。例如，在标准的接近度测试问题中，Alice和Bob各自具有分别来自分布$a$和分布$b$ over $[n]$的$t$样本，并且他们需要测试$a=b$或$a,b$是否是$ε$ -far（对于某个固定的$ε>0$）。这与研究充分的一方情况相反，在这种情况下，测试人员可以不受限制地访问两种分布的样本，对于这些样本，最佳边界对于许多变化是已知的。尽管这是应用中的一个自然限制，但两党设置迄今为止一直没有引起注意。我们解决两个基本方面：1）什么是通信复杂性，以及2）它可以安全地完成，没有爱丽丝和鲍勃学习额外的信息，对方的输入。除了接近性测试，我们还研究了独立性测试问题，其中Alice和Bob分别具有来自分布$a$和$b$的$t$样本，它们可能是相关的;问题是$a,b$是否独立于$ε$ -远非独立。我们的贡献是三方面的：1）通信：我们展示了如何获得更多的样本，超越$t$上的信息理论约束的通信效率。增益是多项式优于通过适应标准算法获得的。2)下限：我们证明了我们的协议的紧密性测试，并为独立性测试时，样本的数量是无限的。这些下限是独立的利益，因为这些是第一个2方通信的测试问题的下限。3)安全性：我们定义了安全分发测试，并认为它必须至少泄漏一些最小的信息。然后，我们提供安全版本的上述协议的开销，只有多项式的安全参数。
英文摘要：
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 $ε$-far for some fixed $ε>0$. This is in contrast to the well-studied one-party case, where the tester has unrestricted access to samples of both distributions, for which optimal bounds are known for a number of variations. Despite being a natural constraint in applications, the two-party setting has evaded attention so far. We address two fundamental aspects: 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 of $ε$-far from being independent. Our contribution is three-fold: 1) Communication: we show how to gain communication efficiency with more samples, beyond the information-theoretic bound on $t$. The gain is polynomially better than what one obtains by adapting standard algorithms. 2) Lower bounds: we prove tightness of our protocols for the closeness testing, and for the independence testing when the number of samples is unbounded. These lower bounds are of independent interest as these are the first 2-party communication lower bounds for testing problems. 3) Security: we define secure distribution testing and argue that it must leak at least some minimal information. We then provide secure versions of the above protocols with an overhead that is only polynomial in the security parameter.
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