Using the Parallel Karatsuba Algorithm for Long Integer Multiplication and Division

We experiment with sequential and parallel versions of the Karatsuba multiplication algorithm implemented under the paclib computer algebra system on a Sequent Symmetry shared-memory architecture. In comparison with the classical multiplication algorithm, the sequential version gives a speed-up of 2 at 50 words, up to 5 at 500 words. On 9 processors, the parallel Karatsuba algorithm exhibits a combined speed-up of 10 (50 words) up to 40 (500 words). Moreover, we use the Karatsuba algorithm within long integer division with remainder, using a recent divide-and-conquer technique which delays part of the dividend updates until they can be performed by multiplication between large operands. The sequential algorithm is about two times slower than Karatsuba multiplication and shows a speed-up of 2 at 200 words and of 3 at 500 words, when compared to the classical division method. Using parallel multiplication on 9 processors leads to a combined speed-up of almost 3 at 100 words and more than 10 at 500 words.