Large integer multiplication on hypercubes

Abstract

Previous work has reported on the use of polynomial transforms to compute exact convolution and to perform multiplication of large integers on a massively parallel processor. We now present results of an improved technique, using the Fermat Number Transform. When the Fermat Number Transform was first proposed, word-length constraints limited its effectiveness. Despite the development of multidimensional techniques to extend the length of the FNT, the relatively small word length of existing machines made the transform of little more than academic interest. The emergence of new computer architectures, however, has made the Fermat Number Transform more attractive. On machines like the Connection Machine, for example, we may implement the Fermat Number Transform without regard to word-length considerations. We present a convolution algorithm on a massively parallel processor, based on the Fermat Number Transform. We present some examples of the trade-offs between modulus, interprocessor communication steps, and input size. We then discuss the application of this algorithm in the multiplication of large integers and report performance results on a Connection Machine. Our results show multiplication times ranging from about 50 ms for 2-kbit integers to 2 s for 8-Mbit integers, an improvement of about five times over previous work.