On Acceleration of the k-ary GCD Algorithm

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 sa...

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Bibliographic Details
Published in:	Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki Vol. 161; no. 1; pp. 110 - 118
Main Authors:	Amer, I., Ishmukhametov, S.T.
Format:	Journal Article
Language:	English
Published:	Kazan Federal University 01.01.2019
Subjects:	binary gcd algorithm euclidian gcd algorithm greatest common divisor for natural numbers k-ary gcd algorithm
ISSN:	2541-7746, 2500-2198
Online Access:	Get full text
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Summary:	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 ЃЯ 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.
ISSN:	2541-7746 2500-2198
DOI:	10.26907/2541-7746.2019.1.110-118