Faster Integer Multiplication

For more than 35 years, the fastest known method for integer multiplication has been the Schonhage-Strassen algorithm running in time O(n log n log log n). Under certain restrictive conditions there is a corresponding Q(n log n) lower bound. The prevailing conjecture has always been that the complexity of an optimal algorithm is Theta(n log n). We present a major step towards closing the gap from above by presenting an algorithm running in time n log n 2(O(log* n)).The main result is for boolean circuits as well as for multitape Turing machines, but it has consequences to other models of computation as well.