Faster integer multiplication using short lattice vectors

TL;DR

This paper presents an unconditional method for multiplying n-bit integers efficiently, achieving a complexity bound similar to previous results that relied on unproven hypotheses, by leveraging Minkowski's theorem on lattice vectors.

Contribution

It provides an unconditional proof of a near-optimal integer multiplication algorithm using short lattice vectors, improving upon prior conditional results.

Findings

Achieves integer multiplication in O(n log n 4^{log^* n}) bit operations

Provides an unconditional proof relying on Minkowski's theorem

Bridges number theory and lattice geometry for algorithmic improvements

Abstract

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