Abstract: 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 Hoven shows that two $n$-bit numbers can be multiplied via a boolean circuit of size $O(n \lg n)$.
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 (conditioned on the conjecture) 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.
Joint work with Peyman Afshani, Casper Freksen and Kasper Green Larsen