Karatsuba multiplication is an algorithm invented by Anatoly Karatsuba in 1960 to multiply n-digit numbers in less than O(n^2) time. The algorithm was invented for use on integers, but can be applied to polynomials represented as coefficient arrays.
Time complexity: O(n^log3) ~ O(n^1,585)
[1] https://cse.hkust.edu.hk/mjg_lib/Classes/COMP3711H_Fall14/lectures/DandC_Multiplication_Handout.pdf