Even faster integer multiplication

We give a new algorithm for the multiplication of n-bit integers in the bit complexity model, which is asymptotically faster than all previously known algorithms. More precisely, we prove that two n-bit integers can be multiplied in time O(nlognKlog∗n), where K=8 and log∗n=min{k∈N:log…k×logn⩽1}. Assuming standard conjectures about the distribution of Mersenne primes, we give yet another algorithm that achieves K=4. The fastest previously known algorithm was due to Fürer, who proved the existence of a complexity bound of the above form for some finite K. We show that an optimised variant of Fürer’s algorithm achieves only K=16, suggesting that our new algorithm is faster than Fürer’s by a factor of 2log∗n.