On acceleration of the k-ary GCD algorithm

Abstract

© 2019, Kazan Federal University. All rights reserved. In this paper, methods of acceleration of GCD algorithms for natural numbers based on the k-ary GCD algorithm have been studied. The k-ary algorithm was elaborated by J. Sorenson in 1990. Its main idea is to find for given numbers A, B and a parameter k, co-prime to both A and B, integers x and y satisfying the equation Ax + By ≡ 0 mod k. Then, integer C = (Ax+By)/k takes a value less than A. At the next iteration, a new pair (B, C) is formed. The k-ary GCD algorithm ensures a significant diminishing of the number of iterations against the classical Euclidian scheme, but the common productivity of the k-ary algorithm is less than the Euclidian method. We have suggested a method of acceleration for the k-ary algorithm based on application of preliminary calculated tables of parameters like as inverse by module k . We have shown that the k-ary GCD algorithm overcomes the classical Euclidian algorithm at a sufficiently large k when such tables are used.