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Shared vs Private Randomness in Distributed Interactive Proofs

Authors Pedro Montealegre, Diego Ramírez-Romero, Ivan Rapaport

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Author Details

Pedro Montealegre
  • Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile
Diego Ramírez-Romero
  • Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile
Ivan Rapaport
  • DIM-CMM (UMI 2807 CNRS), Universidad de Chile, Santiago, Chile

Funding

Partially supported by CONICYT via PIA / Apoyo a Centros Científicos y Tecnológicos de Excelencia AFB 170001 (P.M. and I.R.), FONDECYT 1170021 (D.R. and I.R.), FONDECYT 11190482 (P.M.) and PAI + Convocatoria Nacional Subvención a la Incorporación en la Academia Año 2017 + PAI77170068 (P.M.).

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Pedro Montealegre, Diego Ramírez-Romero, and Ivan Rapaport. Shared vs Private Randomness in Distributed Interactive Proofs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 51:1-51:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ISAAC.2020.51

Abstract

In distributed interactive proofs, the nodes of a graph G interact with a powerful but untrustable prover who tries to convince them, in a small number of rounds and through short messages, that G satisfies some property. This series of interactions is followed by a phase of distributed verification, which may be either deterministic or randomized, where nodes exchange messages with their neighbors.
The nature of this last verification round defines the two types of interactive protocols. We say that the protocol is of Arthur-Merlin type if the verification round is deterministic. We say that the protocol is of Merlin-Arthur type if, in the verification round, the nodes are allowed to use a fresh set of random bits.
In the original model introduced by Kol, Oshman, and Saxena [PODC 2018], the randomness was private in the sense that each node had only access to an individual source of random coins. Crescenzi, Fraigniaud, and Paz [DISC 2019] initiated the study of the impact of shared randomness (the situation where the coin tosses are visible to all nodes) in the distributed interactive model.
In this work, we continue that research line by showing that the impact of the two forms of randomness is very different depending on whether we are considering Arthur-Merlin protocols or Merlin-Arthur protocols. While private randomness gives more power to the first type of protocols, shared randomness provides more power to the second. Our results also connect shared randomness in distributed interactive proofs with distributed verification, and new lower bounds are obtained.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed computing models
  • Theory of computation → Interactive proof systems
  • Theory of computation → Distributed algorithms
Keywords
  • Distributed interactive proofs
  • Distributed verification
  • Shared randomness
  • Private randomness

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