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 generaliz...

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Bibliographic Details
Published in:Journal of algorithms Vol. 54; no. 2; pp. 152 - 167
Main Authors: Weber, Kenneth, Trevisan, Vilmar, Martins, Luiz Felipe
Format: Journal Article
Language:English
Published: San Diego, CA Elsevier Inc 01.02.2005
Elsevier
Subjects:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Computer science; control theory; systems
Exact sciences and technology
Integer GCD
Modular representation
Parallel algorithm
Residue arithmetic
Theoretical computing
Integer GCD
Parallel algorithm
Residue arithmetic
Modular representation
Integer GCD algorithm
Modular programming
Symbolic computation
Algorithm analysis
ISSN:0196-6774, 1090-2678
Online Access:Get full text
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Summary: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 − b V ) / 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 ≡ U V −1 ( mod   p ) . When the algorithm is executed on a bit common CRCW PRAM with O ( n log n log log log n ) processors, it takes O ( n ) time in the worst case. A heuristic model of the average case yields O ( n / log n ) time on the same number of processors.
ISSN:0196-6774
1090-2678
DOI:10.1016/j.jalgor.2004.06.006