For anyone else just looking for an outline of the new algorithm: last two paragraphs of p4, first two paragraphs of p5.
Coming from a position of a few abstract algebra classes in college many years ago, all the words and notation are familiar but I am a long way from being able to follow it.
> Integer multiplication in time O(n log n)
https://hal.archives-ouvertes.fr/hal-02070778/document