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Non-Local Box Complexity and Secure Function Evaluation

Marc Kaplan, Iordanis Kerenidis, Sophie Laplante, Jérémie Roland, Marc Herbstritt

2009
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Foundations of Software Technology and Theoretical Computer Science
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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 a ⊕ b = x ∧ y. 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
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... rms of the communication complexity of the function both in the deterministic and randomized case. We show that non-local box complexity has interesting applications to classical cryptography, in particular to secure function evaluation, and study the question posed by Beimel and Malkin [4] 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. Introduction Communication complexity. Communication complexity is a central model of computation, which was first defined by Yao in 1979 [35] and has since found numerous applications. In this model Alice and Bob receive inputs x and y respectively and are allowed to communicate in order to compute a function f (x, y). The goal is to find the minimum amount of communication needed for this task. In different variants of the model, we allow Alice and Bob to err with some probability, and to share common resources in an attempt to enable them to solve their task more efficiently. One such resource is shared randomness. When Alice and Bob are not allowed any errors, shared randomness does not reduce the communication complexity. On the other hand, when they are allowed to err, a common random string can reduce the amount of communication needed. However, Newman's result tells us that shared randomness can be replaced by private randomness at an additional cost logarithmic in the input size. Another very powerful shared resource is entanglement. Using teleportation, Alice and Bob can transmit quantum messages by using their entanglement and only classical communication. This model has been proven to be very powerful, in some cases exponentially more efficient than the classical one. Another way to understand the power of entanglement is by looking at the CHSH game [13] , where Alice and Bob receive uniformly random bits x and y respectively and their goal is to output bits a and b resp. such that a ⊕ b = x ∧ y without communicating. It is not hard to conclude that even if Alice and Bob share randomness, their optimal strategy will be successful with probability 0.75 over the inputs. However, if

doi:10.4230/lipics.fsttcs.2009.2322
dblp:conf/fsttcs/KaplanKLR09
fatcat:54js7gjnsbem3hmjzh6efvqcsq