Non-Local Box Complexity and Secure Function Evaluation
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 $a \oplus b = x \wedge y$. 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 terms 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 \cite{BM} 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 \nlbs, while
no finite bound was previously known.
Communication complexity
non-locality
non-local boxes
secure function evaluation
239-250
Regular Paper
Marc
Kaplan
Marc Kaplan
Iordanis
Kerenidis
Iordanis Kerenidis
Sophie
Laplante
Sophie Laplante
Jérémie
Roland
Jérémie Roland
10.4230/LIPIcs.FSTTCS.2009.2322
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license
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