非局部盒复杂性与安全函数计算
Non-Local Box Complexity and Secure Function Evaluation
摘要：
非本地盒是一种抽象设备，Alice和Bob分别向其中输入比特x和y，并分别接收输出a和b，其中a，b均匀分布，a+b的奇偶性等于xy。这种盒子一直是研究量子或广义非定域性以及模拟非信号分布的核心。在本文中，我们首先研究Alice和Bob需要多少个非局部盒子来计算布尔函数f。我们提供了严格的上限和下限方面的通信复杂性的函数在确定性和随机的情况下。然后，我们继续表明，非局部盒复杂性的研究有有趣的应用经典计算。特别是，我们看看安全的函数评估，并研究了Beimel和Malkin提出的问题，Alice和Bob需要多少不经意传输调用才能安全地计算函数f。我们证明了这个问题与函数的非局部盒复杂性有关，并通过大大提高它们的边界得出结论。最后，我们的结果的另一个结果是，最大纠缠态的无痕双结果测量可以用3个非局域盒来模拟，而以前没有有限的限制是已知的。
英文摘要：
A non-local box is an abstract device into which Alice and Bob input bits x and y respectively and receive outputs a and b respectively, where a, b are uniformly distributed and the parity of a+b equals xy. Such boxes have been central to the study of quantum or generalized non-locality as well as the simulation of non-signaling distributions. In this paper, we start by studying how many non-local boxes Alice and Bob need in order to compute a Boolean function f. We provide tight upper and lower bounds in terms of the communication complexity of the function both in the deterministic and randomized case. We then proceed to show that the study of non-local box complexity has interesting applications for classical computation as well. In particular, we look at secure function evaluation, and study the question posed by Beimel and Malkin of how many Oblivious Transfer calls Alice and Bob need in order to securely compute a function f. We show that this question is related to the non-local box complexity of the function and conclude by greatly improving their bounds. Finally, another consequence of our results is that traceless two-outcome measurements on maximally entangled states can be simulated with 3 non-local boxes, while no finite bound was previously known.
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