In this video(https://www.youtube.com/watch?v=KuXjwB4LzSA) it states you can use polynomial multiplication to find the convolution. The box method is metaphorical to convolution for multiplying the coefficients of the polynomial. So what can be done is you can take the two input lists and pretend they are the coefficients of polynomials. Then you can input arbitrary x-values into those polynomials and multiply the output to get a product. Then once you have enough points($N$ of them), you can solve for the coefficients. The big-O notation for this is $N^2$, because you have $N$ required outputs and $N$ terms in each product.
The video states that Fast Fourier Transform(FFT) can be used do this faster. You can take the x-values to be the $N$ $N$th roots of unity. This would allow "redundancy" to occur, which would make the computation faster. FFT would then be used over the set of products to determine the coefficients of the polynomial product and thus the convolution.
But I am confused by this. Sure, the roots of unity would allow for FFT. But what redundancy would the $N$th roots of unity give? The powers of $x$ in the polynomials would be at most $N$, so it wouldn't reset there.
Could someone please explain this to me? Thanks