An Improved Protocol for the Exactly-N Problem

Authors Nati Linial, Adi Shraibman

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Author Details

Nati Linial
  • Hebrew University of Jerusalem, Israel
Adi Shraibman
  • The Academic College of Tel-Aviv-Yaffo, Israel


We thank Noga Alon, Aya Bernstine and Alex Samorodnitsky for insightful discussions.

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Nati Linial and Adi Shraibman. An Improved Protocol for the Exactly-N Problem. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 2:1-2:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In the 3-players exactly-N problem the players need to decide whether x+y+z = N for inputs x,y,z and fixed N. This is the first problem considered in the multiplayer Number On the Forehead (NOF) model. Even though this is such a basic problem, no progress has been made on it throughout the years. Only recently have explicit protocols been found for the first time, yet no improvement in complexity has been achieved to date. The present paper offers the first improved protocol for the exactly-N problem. This improved protocol has also interesting consequences in additive combinatorics. As we explain below, it yields a higher lower bound on the possible density of corner-free sets in [N]×[N].

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Communication complexity
  • Number-On-the-Forehead
  • Corner-free sets


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