[tzs]

The author is not saying that they don't know how to prove rigorously that their algorithm is O(n log(n)). They have rigorously proven that.

What they are saying is that they don't know how to prove that you can't do better than O(n log(n)).

Schönhage and Strassen conjectured that O(n log(n)) was the lower limit for this problem, and it seems that most experts believe they were right abut that. Hence, the expectation that with the discovery of an actual O(n log(n)) algorithm we are at the end of the road for this problem.

[bakul]

Thanks for the correction. Should've read the paper when less tired!