Even faster integer multiplication

We give a new proof of F"urer’s bound for the cost of multiplying n-bit integers in the bit complexity model. Unlike F"urer, our method does not require constructing special coefficient rings with “fast” roots of unity. Moreover, we prove the more explicit bound O(n log n K^(log^* n))$ with K = 8. We show that an optimised variant of F"urer’s algorithm achieves only K = 16, suggesting that the new algorithm is faster than F"urer’s by a factor of 2^(log^* n). Assuming standard conjectures about the distribution of Mersenne primes, we give yet another algorithm that achieves K = 4.