# Evidence integer multiplication is in linear time?

After millenia of quest we have identified two $$n$$ bit integers can be multiplied in $$O(n\log n)$$ time. Please refer details in https://www.quantamagazine.org/mathematicians-discover-the-perfect-way-to-multiply-20190411/. The problem of placing multiplication in $$O(n)$$ time is still unsolved.

1. What evidence do we have that this can/cannot be brought to $$O(n)$$ time on uniform Turning model?

2. What evidence do we have that this can/cannot be brought to $$O(n)$$ complexity in a reasonable non-uniform model?

In regards to (2), conditional super-linear lower bounds are known. A recent preprint by Afshani, Freksen, Kamma, and Larsen proves an $$\Omega(n \log n)$$ lower bound for the size of Boolean circuits computing integer multiplication, assuming a certain conjecture on network coding in undirected graphs. (See also this blog post and a follow-up post.)

From the linked post on major unsolved problems, a super-linear lower bound follows from the Heartmanis-Stearns conjecture.