Testing fast convolution of tensor sketches

Today I have been working on the final unit test of my code and the changes necessary to run  it. I am testing my first implementations last and changing the code a lot in the progress. It's fun  to see how different my C++ looks now. The final beast to slay is my implementation of the fast convulution of tensor sketches and checking that it actually keeps the promise of finding the Count Sketch of an outer product fast. Some things are bugging me:

I convert a CS into a polynomial.
I still use a discrete fourier transform. Somehow my CS(vector) represent the polynomial well enough. The conversion seems to be a math showpiece. I  have been spending a lot of time understanding this today and it's still not really clear.