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Méthodes Combinatoires et Algébriques en Complexité de la Communication

Abstract : Communication complexity was introduced in 1979 by Andrew Chi-Chi Yao. It has since become a central model of complexity. The model addresses problems whose inputs are shared among different players, who have to communicate in order to solve it. We first use Kolmogorov complexity, an algorithmic characterization of randomness, to prove lower bounds on communication complexity. Our method is a generalization of the well-known incompressibility method. One important aspect of our method is to emphasize the combinatorial structure of the proofs. We then focus on the simulation of non-signaling distributions using communication. This model encompasses traditionnal communication complexity and the simulation of distributions arising from bipartite measurement of quantum states. We show upper and lower bounds on this problem. In the case of boolean functions, the lower bound we prove is equivalent to the factorization norm, a powerful method introduced by Linial and Shraibman in 2006. Finally, we study non-local box complexity. This resource has been introduced by Popescu and Rohrlich to study non-locality. The problem is to quantify the number of boxes required to compute a boolean function or simulate a distribution. We prove upper and lower bounds for these problems, and show applications to secure function evaluation, an important cryptographic task.
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https://tel.archives-ouvertes.fr/tel-00439929
Contributor : Marc Kaplan <>
Submitted on : Wednesday, December 9, 2009 - 2:14:18 AM
Last modification on : Wednesday, October 14, 2020 - 3:59:18 AM
Long-term archiving on: : Thursday, June 17, 2010 - 9:23:15 PM

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Marc Kaplan. Méthodes Combinatoires et Algébriques en Complexité de la Communication. Informatique [cs]. Université Paris Sud - Paris XI, 2009. Français. ⟨tel-00439929⟩

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