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13
Analysis of the binary Euclidean algorithm
 Directions and Recent Results in Algorithms and Complexity
, 1976
"... The binary Euclidean algorithm is a variant of the classical Euclidean algorithm. It avoids multiplications and divisions, except by powers of two, so is potentially faster than the classical algorithm on a binary machine. We describe the binary algorithm and consider its average case behaviour. In ..."
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Cited by 28 (2 self)
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The binary Euclidean algorithm is a variant of the classical Euclidean algorithm. It avoids multiplications and divisions, except by powers of two, so is potentially faster than the classical algorithm on a binary machine. We describe the binary algorithm and consider its average case behaviour. In particular, we correct some errors in the literature, discuss some recent results of Vallée, and describe a numerical computation which supports a conjecture of Vallée. 1
On a parallel Lehmer–Euclid GCD algorithm
 in: Proceedings of the International Symposium on Symbolic and Algebraic Computation ISSAC’2001
"... A new version of Euclid’s GCD algorithm is proposed. It matches the best existing parallel integer GCD algorithms since it can be achieved in Oɛ(n / log n) time using at most n 1+ɛ processors on CRCW PRAM. 1. ..."
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Cited by 7 (1 self)
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A new version of Euclid’s GCD algorithm is proposed. It matches the best existing parallel integer GCD algorithms since it can be achieved in Oɛ(n / log n) time using at most n 1+ɛ processors on CRCW PRAM. 1.
A parallel extended GCD algorithm
, 2008
"... A new parallel extended GCD algorithm is proposed. It matches the best existing parallel integer GCD algorithms of Sorenson and Chor and Goldreich, since it can be achieved in Oɛ(n / log n) time using at most n 1+ɛ processors on CRCW PRAM. Sorenson and Chor and Goldreich both use a modular approach ..."
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Cited by 4 (2 self)
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A new parallel extended GCD algorithm is proposed. It matches the best existing parallel integer GCD algorithms of Sorenson and Chor and Goldreich, since it can be achieved in Oɛ(n / log n) time using at most n 1+ɛ processors on CRCW PRAM. Sorenson and Chor and Goldreich both use a modular approach which consider the least significant bits. By contrast, our algorithm only deals with the leading bits of the integers u and v, with u � v. This approach is more suitable for extended GCD algorithms since the coefficients of the extended version a and b, such that au + bv = gcd(u, v), are deeply linked with the order of magnitude of the rational v/u and its continuants. Consequently, the computation of such coefficients is much easier.
Comparing several GCD algorithms
 In 11th IEEE Symposium on Computer Arithmetic
, 1993
"... Abstract 0 binary, Ibinary: The binary GCD algorithm We compare the executron times of several algoixtliiiis for computing the G‘C‘U of arbitrary precasion iirlegers. These algorithms are the known ones (Euclidean, brnary, plusmrnus), and the improved variants of these for multidigit compzltatio ..."
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Abstract 0 binary, Ibinary: The binary GCD algorithm We compare the executron times of several algoixtliiiis for computing the G‘C‘U of arbitrary precasion iirlegers. These algorithms are the known ones (Euclidean, brnary, plusmrnus), and the improved variants of these for multidigit compzltation (Lehmer and similar), as well as new algorithms introduced by the author: an improved Lehmer algorithm using two digits in partial cosequence computation, and a generalization of the binary algorithm using a new concept of “m.0dalar conjugates”. The last two algorithms prove to be the fastest of all, giving a speed,up of 6 to 8 times over th.e classical Euclidean scheme, and 2 times over the best currently known algorathins. Also, the generalized binary algorithm is suitable for systolic parallelization, an “leastsignificant digits first ” pipelined manner. 1
On a Parallel Extended Euclidean Algorithm
"... A new parallelization of Euclid’s greatest common divisor algorithm is proposed. It matches the best existing integer GCD algorithms since it can be achieved in parallel Oε(n/log n) time using only n 1+ε processors on a Priority CRCW PRAM. ..."
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Cited by 2 (0 self)
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A new parallelization of Euclid’s greatest common divisor algorithm is proposed. It matches the best existing integer GCD algorithms since it can be achieved in parallel Oε(n/log n) time using only n 1+ε processors on a Priority CRCW PRAM.
A modular reduction for GCD computation
, 2002
"... Most of integer GCD algorithms use one or several basic transformations which reduce at each step the size of the inputs integers u and v.These transformations called reductions are studied in a general framework.Our investigations lead to many applications such as a new integer division and a new r ..."
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Most of integer GCD algorithms use one or several basic transformations which reduce at each step the size of the inputs integers u and v.These transformations called reductions are studied in a general framework.Our investigations lead to many applications such as a new integer division and a new reduction called Modular Reduction or MR for short.This reduction is, at least theoretically, optimal on some subset of reductions, if we consider the number of bits chopped by each reductions.Although its computation is rather di cult, we suggest, as a rst attempt, a weaker version which is more e cient in time.Sequential and parallel integer GCD algorithms are designed based on this new reduction and our experiments show that it performs as well as the Weber’s version of the Sorenson’s kary reduction. c ○ 2003 Elsevier B.V. All rights reserved. 1.
Outline
"... Abstract The binary Euclidean algorithm is a variant of the classical Euclidean algorithm. It avoids divisions and multiplications, except by powers of two, so is potentially faster than the classical algorithm on a binary machine. In this talk I will describe the classical and binary algorithms, an ..."
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Abstract The binary Euclidean algorithm is a variant of the classical Euclidean algorithm. It avoids divisions and multiplications, except by powers of two, so is potentially faster than the classical algorithm on a binary machine. In this talk I will describe the classical and binary algorithms, and compare their worst case and average case behaviour. In particular, I will correct some small but significant errors in the literature, discuss some recent results of Brigitte Vall'ee, and describe a numerical computation which verifies Vall'ee's conjecture to 44 decimal places.