Peyman Afshani, Casper Benjamin Freksen, Lior Kamma, Kasper Green Larsen
2019 · arXiv
Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, by Fürer, shows that two $n$-bit numbers can be multiplied via a boolean circuit of size $O(n \lg n \cdot 4^{\lg^*n})$, where $\lg^*n$ is the very slowly growing iterated logarithm. 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 $Ω(n \lg n)$, 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.
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According to the article, it rests on a plausible conjecture. I need to read further. [edit] According to the abstract of the paper referred to in the article (https://arxiv.org/abs/1902.10935) it rests on a "central conjecture in the area of network coding". Make of that what you will.
The article says that it may be the fastest: > At the end of February, a team of computer scientists at Aarhus University posted a paper [ https://arxiv.org/abs/1902.10935 ] arguing that if another unproven conjecture is also true, this is indeed the fastest way multiplication can be done.
This very recent paper shows a matching (conditional) lower bound of \Omega(n\log n): https://arxiv.org/abs/1902.10935 (the hardness is from a conjecture in network coding).
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