Faster integer multiplication using short lattice vectors

Abstract

We prove that $n$-bit integers may be multiplied in $O(n\log n{4}^{{\log }^{\ast }n})$ 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.

16 pages