Sublinear Parallel Algorithm for Computing the Greatest Common Divisor of Two Integers

The atdvent of practical parallel processors has caused a reexamination of many existing algorithms with'the hope of discovering a parallel implementation. One of the oldest and best know algorithms is Euclid's algorithm for computing the greatest common divisor (GCD). In this paper we present a parallel algorithm to compute the GCD of two integers. Although there have been results in the parallel computation of the GCD 9f polynomials (Borodin, von zur Gathen, and Hopcroft (1983)), the integer case still appeared to be inherently serial. To our knowledge, the best result to date for computing the GCD of two integers is by Brent and Kung (1983), who achieve a running time of O(n) with n processors arranged in a systolic array, where n is the number of bits required to represent the larger of the two inpnt numbers. Although their method is an improvement on the best know serial integer GCD algorithm O(n log2 n log log n) by Schonhage (1971), it still requires n iterations; the parallelism only reduces the bit operations per iteration. In this paper we present a subline? time parallel algorithm to compute the integer GCD of two numbers on a weak CRCW model of parallel computation allowb g concurrent reads but only concurrent writes of the same value. The time bound is O(n log log n/ log n) assume there are n210g2n processors working in parallel. This is computed assuming unit time for each elementary hit operation. There is a nice duality &eorem that we mention in passing. Note that the GCD(a,b) equals both (i) the mhximum d such that d divides a and b, and (ii) the minimum d > 0 expressible as pa + qb, where p and q are integers. Both these properties can be demonstrated in O(1ogn) parallel time. Thus, if the language L = ('(k,a, b) I the kth binary digit of GCD(a,b) equals 1 } were complete for polynomial time with respect to logn parallel time reductions, then all problems in P would have a logn parallel time duality theorem. Given this fact, we conjecture that the GCD problem is not complete for P. An Overview of the Algori thm