Optimizing and Parallelizing Brown’s Modular GCD Algorithm

Consider the multivariate polynomial problem over the integers; that is, Gcd(A,B) where A,B ∈ Z[x1, x2, . . . xn]. We can solve this problem by solving the related Gcd problem in Zp[x1, x2, . . . xn] for several primes p, and then reconstructing the solution in the integers using Chinese Remaindering. The question we address in this paper is how fast can we solve the problem Gcd(A,B) in Zp[x1, x2, . . . xn] using either 31 or 63 bit primes, and how well can we make use of parallel processing to do so? To this end, we implemented a modular algorithm using evaluations and interpolations, and parallelized it using the CILK framework. Several optimizations for the algorithm were found, and a few parallelization strategies were attempted. Our final implementation preformed significantly better than both maple and magma on most test cases without using multiple processors, and with parallelization we acheive a speedup of a factor of about 11 on 16 processors. The modular method of solving the GCD problem was first explained by Brown in 1971 [1]. Further, the methods used here are very similar to those used in a 2000 paper by Monagan and Witkopf [2]. A proof of Brown’s algorithm will be included in this paper to aid in the explenations of the optimizations developed. However, one may wish to reference either the 1971 or 2000 papers for an alternative and more detailed proof. The MGCD (modular GCD) algorithm is going to compute the GCD G and the corresponding co-factors Ā, B̄ of the inputs A,B ∈ Zp[x1, x2 . . . xn]. The algorithm is modular and recursive. We will use an evaluation homomorphism to remove the variable x1, solve the problem in Zp[x2 . . . xn] several times recursively, and use the results to reconstruct the solution to the original n-variable problem.