Parallel implementation of schönhage’s integer GCD algorithm

Abstract

We present a parallel implementation of Schönhage’s integer GCD algorithm on distributed memory architectures. Results are generalized for the extended GCD algorithm. Experiments on sequential architectures show that Schönhage’s algorithm overcomes other GCD algorithms implemented in two well known multiple-precision packages for input sizes larger than about 50000 bytes. In the extended case this threshold drops to 10000 bytes. In these input ranges a parallel implementation provides additional speed-up. Parallelization is achieved by distributing matrix operations and by using parallel implementations of the multiple-precision integer multiplication algorithms. We use parallel Karatsuba’s and parallel 3-primes FFT multiplication algorithms implemented in CALYPSO, a computer algebra library for parallel symbolic computation we have developed. SchSnhage’s parallel algorithm is analyzed by using a message-passing model of computation. Experimental results on distributed memory architectures, such as the Intel Paragon, confirm the analysis.