Fast integer multiplication using generalized Fermat primes

TL;DR

This paper introduces a new integer multiplication algorithm based on generalized Fermat primes, potentially achieving the fastest known complexity of O(n log n) with a conjectural constant K=4, improving upon previous methods.

Contribution

It presents a novel approach using generalized Fermat primes that could theoretically match the fastest known multiplication complexity with a deterministic algorithm.

Findings

Complexity potentially reduced to O(n log n)

Achieves conjectural constant K=4 in complexity

Provides a deterministic algorithm based on number theory

Abstract

For almost 35 years, Sch{\"o}nhage-Strassen's algorithm has been the fastest algorithm known for multiplying integers, with a time complexity O(n $\times$ log n $\times$ log log n) for multiplying n-bit inputs. In 2007, F{\"u}rer proved that there exists K > 1 and an algorithm performing this operation in O(n $\times$ log n $\times$ K log n). Recent work by Harvey, van der Hoeven, and Lecerf showed that this complexity estimate can be improved in order to get K = 8, and conjecturally K = 4. Using an alternative algorithm, which relies on arithmetic modulo generalized Fermat primes, we obtain conjecturally the same result K = 4 via a careful complexity analysis in the deterministic multitape Turing model.