A modular reduction for GCD computation

Abstract

Most of integer GCD algorithms use one or several basic transformations which reduce at each step the size of the inputs integers u and v. These transformations called reductions are studied in a general framework. Our investigations lead to many applications such as a new integer division and a new reduction called Modular Reduction or MR for short. This reduction is, at least theoretically, optimal on some subset of reductions, if we consider the number of bits chopped by each reductions. Although its computation is rather difficult, we suggest, as a first attempt, a weaker version which is more efficient in time. Sequential and parallel integer GCD algorithms are designed based on this new reduction and our experiments show that it performs as well as the Weber's version of the Sorenson's k-ary reduction.