Non-Local Box Complexity and Secure Function Evaluation

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

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Marc Kaplan
Iordanis Kerenidis
Sophie Laplante
Jérémie Roland

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Marc Kaplan, Iordanis Kerenidis, Sophie Laplante, and Jérémie Roland. Non-Local Box Complexity and Secure Function Evaluation. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 239-250, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)


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


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