A fast carry-free algorithm and hardware design for extended integer GCD computation

Abstract

I$ is well known that finding the greatest common divisor (GCD) of two integers is one of the fundamental computations in exact rational arithmetic, factorization and cryptography. Euclid3 algorithm and its variants are the widely used for GCD computations [Knu 811. However they are not suitable in the parallel computation. Since the whole-word comparisons is required. G.B. Purdy /Pur 831 proposed a different way to compute GCD which requires no comparison. The advantage of the Purdy’s algorithm is provided a possible way to speed up the period of each iteration time by using carry save technique. However, it requires 0( n2) iterations in its worst case where n denotes the number of bits of two inputs. In addition, it requires the additional hardware support to handle the overflow problem. R. P. Brent and H. T. Kung [B&K 85) have developed a plus-minus (PM) algorithm that test only the two least significant bits of two integers. The advantage of the PM algorithm is that the number of the iteration is at most 3.012*n units. In particular, this gives a linear time implementation on a systolic array [B&K 851. Although reaaonabl efficient in its use of silicon area, the delay between first input and first output of a computation for the serial-in-serial-out GCD is great than 3 n time units which may be undesirable long depending on the application. The basic idea in our algorithm is to combine two sequence operations of PM algorithm of BrenbKung into one basic operation, and also to avoid swap operations during the iterations to achieve higher parallelism. It has been proved that for any two n bit integers, the number of iterations of the new algorithm is less than 1.51*n+ 1 time units. A preliminary hardware design shows that the algorithm can be implemented in a simple way which consists of several conventional computer components such ss shift registers, borrow save adder, counter and a small PLA as controller. The algorithm can be extended to find not only the greatest common divisor of two numbers A and B, but also to find a pair of integers (2, y) such that AZ + By =GCD(A,B) with the same time complexity. A scheme to cascade a number of such GCD chips to compute very large GCD’s is also at hand, which alleviates a critical difficulty in such fields as cryptography.