Fast arithmetic for faster integer multiplication

For almost 35 years, Schönhage-Strassen’s algorithm has been the fastest algorithm known for multiplying integers, with a time complexity O(n · log n · 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 · log n · Klog∗ ). Recent work showed that this complexity estimate can be made more precise with K = 8, and conjecturally K = 4. We obtain here the same result K = 4 using simple modular arithmetic as a building block, and a careful complexity analysis. We rely on a conjecture about the existence of sufficiently many primes of a certain form.