A modular integer GCD algorithm

This paper describes the first algorithm to compute the greatest common divisor (GCD) of two n-bit integers using a modular representation for intermediate values U , V and also for the result. It is based on a reduction step, similar to one used in the accelerated algorithm [T. Jebelean, A generalization of the binary GCD algorithm, in: ISSAC ’93: International Symposium on Symbolic and Algebraic Computation, Kiev, Ukraine, 1993, pp. 111–116; K. Weber, The accelerated integer GCD algorithm, ACM Trans. Math. Softw. 21 (1995) 111–122] when U and V are close to the same size, that replaces U by (U −bV )/p, where p is one of the prime moduli and b is the unique integer in the interval (−p/2,p/2) such that b ≡ UV−1 (mod p). When the algorithm is executed on a bit common CRCW PRAM with O(n logn log log logn) processors, it takes O(n) time in the worst case. A heuristic model of the average case yields O(n/ logn) time on the same number of processors.  2004 Elsevier Inc. All rights reserved.