An efficient algorithm for doing this based on the convolution theorem uses a Fast Fourier Transform and an inverse Fast Fourier Transform before a combination step; that algorithm was slightly improved by Martin Furer. The "big-Oh of n to the log base 2 of 3" running time of one method for doing this task made it faster than the "big-Oh of n squared" naive approach since 
An efficient algorithm for doing this based on the convolution theorem uses a Fast Fourier Transform and an inverse Fast Fourier Transform before a combination step; that algorithm was slightly improved by Martin Furer. The "big-Oh of n to the log base 2 of 3" running time of one method for doing this task made it faster than the "big-Oh of n squared" naive approach since it reduced four operations to three. Algorithms that do this include the (*) Schonhage-Strassen algorithm and the Karatsuba algorithm, which is asymptotically slower than the more general Toom-Cook algorithm. In two's complement, this operation can be done by sign extending to 2n length to avoid overflow errors, and this operation with a 2 in binary is equivalent to shifting left once. For 10 points, name this operation whose identity is 1 and that is unfortunately explained as repeated addition.
An efficient algorithm for doing this based on the convolution theorem uses a Fast Fourier Transform and an inverse Fast Fourier Transform before a combination step; that algorithm was slightly improved by Martin Furer.
multiplication [accept word forms]