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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Vydáno v:Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki Ročník 161; číslo 1; s. 110 - 118
Hlavní autoři: Amer, I., Ishmukhametov, S.T.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Kazan Federal University 01.01.2019
Témata:
binary gcd algorithm
euclidian gcd algorithm
greatest common divisor for natural numbers
k-ary gcd algorithm
ISSN:2541-7746, 2500-2198
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Abstract 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.
AbstractList 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.
Author Amer, I.
Ishmukhametov, S.T.
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euclidian gcd algorithm
greatest common divisor for natural numbers
k-ary gcd algorithm
Title On Acceleration of the k-ary GCD Algorithm
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