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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.
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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Cited by 2 (0 self)
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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.
The Mixed Binary Euclid Algorithm
"... Abstract We present a new GCD algorithm for two integers that combines both the Euclidean and the binary gcd approaches. We give its worst case time analysis and prove that its bittime complexity is still O(n 2) for two nbit integers. However, our preliminar experiments show that it is very fast f ..."
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Abstract We present a new GCD algorithm for two integers that combines both the Euclidean and the binary gcd approaches. We give its worst case time analysis and prove that its bittime complexity is still O(n 2) for two nbit integers. However, our preliminar experiments show that it is very fast for small integers. A parallel version of this algorithm matches the best presently known time complexity, namely O ( n log n) time with n1+ɛ, for any constant ɛ> 0.
A straight line program . . . (Extended Abstract)
"... While NC algorithms have been discovered for the basic arithmetic operations, the parallel complexity of some fundamental problems as integer gcd is still open, since first being raised in a paper of Cook [2]. Many authors attempt to design fast parallel integer GCD algorithms. Chor and Goldreich [1 ..."
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While NC algorithms have been discovered for the basic arithmetic operations, the parallel complexity of some fundamental problems as integer gcd is still open, since first being raised in a paper of Cook [2]. Many authors attempt to design fast parallel integer GCD algorithms. Chor and Goldreich [1] proposed O(n / log n)ɛ parallel time with O(n 1+ɛ) number of processors, for any ɛ> 0. Sorenson [4] and the author [3] also suggest other parallel algorithms with the same parallel performance. Since then, no major improvements have been made. In this paper, we propose a straight line program computing the integer GCD. It has polynomial size, but the outputs are polynomials with exponential degree. This work is a first attempt to improve the parallel integer GCD, thanks to Valiant et al. [5] contraction method, and, as far as we know, it is the first straight line program for computing the integer GCD. Throuhough this paper, we represent the input integers as formal strings of bits.
A Sublinear Time Parallel GCD Algorithm for the EREW PRAM
, 2009
"... We present a parallel algorithm that computes the greatest common divisor of two integers of n bits in length that takes O(n log log n / logn) expected time using n 6+ǫ processors on the EREW PRAM parallel model of computation. We believe this to be the first sublinear time algorithm on the EREW PRA ..."
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We present a parallel algorithm that computes the greatest common divisor of two integers of n bits in length that takes O(n log log n / logn) expected time using n 6+ǫ processors on the EREW PRAM parallel model of computation. We believe this to be the first sublinear time algorithm on the EREW PRAM for this problem.