Fast integer multiplication using generalized Fermat primes

Abstract

For almost 35 years, Schönhage-Strassen’s algorithm has been the fastest algorithm known for multiplying integers, with a time complexity $O(n \cdot \log n \cdot \log \log n)$ for multiplying $n$-bit inputs. In 2007, Fürer proved that there exists $K>1$ and an algorithm performing this operation in $O(n \cdot \log n \cdot K^{\log ^* n})$. 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 (of the form $r^{2^{\lambda }}+1$), we obtain conjecturally the same result $K=4$ via a careful complexity analysis in the deterministic multitape Turing model.