As expected, the "news article" is pretty much useless, comparing to the "school" long multiplication while it wasn't the efficient thing to compare to since 1962.
Fortunately, clicking through the "source" links does get to the actual paper
And the actual paper does go into modern FFT-based algorithms.
Unfortunately, a lot of it is way over my head, though I did skim through it. I'm not sure how implementable (or theoretical) the new algorithm is.