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Lower Bounds for Multiplication via Network Coding

Authors Peyman Afshani, Casper Benjamin Freksen, Lior Kamma, Kasper Green Larsen

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Peyman Afshani
  • Computer Science Department, Aarhus University, Denmark
Casper Benjamin Freksen
  • Computer Science Department, Aarhus University, Denmark
Lior Kamma
  • Computer Science Department, Aarhus University, Denmark
Kasper Green Larsen
  • Computer Science Department, Aarhus University, Denmark

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Peyman Afshani, Casper Benjamin Freksen, Lior Kamma, and Kasper Green Larsen. Lower Bounds for Multiplication via Network Coding. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 10:1-10:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, very recently proved by Harvey and van der Hoeven (2019), shows that two n-bit numbers can be multiplied via a boolean circuit of size O(n lg n). In this work, we prove that if a central conjecture in the area of network coding is true, then any constant degree boolean circuit for multiplication must have size Omega(n lg n), thus almost completely settling the complexity of multiplication circuits. We additionally revisit classic conjectures in circuit complexity, due to Valiant, and show that the network coding conjecture also implies one of Valiant’s conjectures.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Circuit complexity
  • Circuit Complexity
  • Circuit Lower Bounds
  • Multiplication
  • Network Coding
  • Fine-Grained Complexity


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