Time complexity of Weil pairing
Miller's algorithm is a loop over the group size; let's say it has $n$ bits. (The group size is usually near the prime $p$ of the underlying field)

Each iteration of the loop computes a point doubling and maybe a point addition, and a bunch of field multiplications (or squarings). That sets a upper limit of a constant number of field multiplications in each iteration. The complexity of naive multiplication is $O(n^2)$; with Karatsuba it is $O(n^{\log_2{3}}) = O(n^{1.585})$. Therefore that's the complexity of each iteration.

Thus, since there are $n$ iterations, the complexity of Miller's algorithm is $O(n^3)$ with naive multiplication and $O(n^{2.585})$ with Karatsuba.