(with Joris van der Hoeven)
arXiv preprint (February 2018).
We prove that n-bit integers may be multiplied in O(n log n 4log* n) bit operations. This complexity bound had been achieved previously by several authors, assuming various unproved number-theoretic hypotheses. Our proof is unconditional, and depends in an essential way on Minkowski's theorem concerning lattice vectors in symmetric convex sets.