## Comparing several GCD algorithms (1993)

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Venue: | In 11th IEEE Symposium on Computer Arithmetic |

Citations: | 2 - 0 self |

### BibTeX

@INPROCEEDINGS{Jebelean93comparingseveral,

author = {T. Jebelean},

title = {Comparing several GCD algorithms},

booktitle = {In 11th IEEE Symposium on Computer Arithmetic},

year = {1993}

}

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### Abstract

Abstract 0 binary, I-binary: The binary GCD algorithm We compare the executron times of several algo-ixtliiiis for computing the G‘C‘U of arbitrary precasion iirlegers. These algorithms are the known ones (Eucli-dean, brnary, plus-mrnus), and the improved variants of these for multidigit compzltation (Lehmer and simi-lar), as well as new algorithms introduced by the aut-hor: an improved Lehmer algorithm using two digits in partial cosequence computation, and a generalization of the binary algorithm using a new concept of “m.0-dalar 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 “least-significant digits first ” pipelined manner. 1

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Citation Context ... and a further improvement, by the author using two digits in partial cosequence computation ([SI). 1063-6889/93 $03.00 0 1993 IEEE 180 ([lS]) and its improvement for multidigit integers (Gosper, see =-=[12]-=-). 0 PlusMinus, I-PlusMinus: The plus-minus scheme introduced in [2] and its improvement for multidigit computation. G-binary, I-G-binary: A new algorithm for multiprecision GCD, which generalizes the... |

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Citation Context ...algorithms. Accordiiig to our experiments [SI, in typical algebraic comput,ations more than half of the time is spent. for calculating GCD of long integers. For instance, in Grobner Bases computation =-=[4]-=-, calculating GCD takes 53% of the total time if the length of the input coefficients is 5 decimal digits and 70% if the length is 50. We report here on the computing time of multiprecis ion G C: D co... |

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Citation Context ...ther scheme - for instance, by division. However, division is not suitable for parallelization, and it is also a relatively slow operation. We applied instead the “exact division” scheme described in =-=[9]-=-, which works like this: Let be d = length(A) - length(B) and u,b the trailing d bits of A, B. Set c = (U * b-’) mod 2d. Then C = (A - c * B)/2d is (roughly) d bits shorter than A. Hence, the generali... |

30 | Analysis of the binary Euclidean algorithm, Algorithms and Complexity, New directions and recent
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Citation Context ...t most one word (32 bits) long. A more detailed description of the theoretical background and of the implementation can be found in [SI. 2.4 Binary and PlusMinus The binary GCD algorithm ([16], [12], =-=[3]-=-) is based on the relations: GCD(A, B) = GCD(A - B, B), If A odd, B even, then (11) GCD(A, B) = GCD(A, B/2). The algorithm begins by shifting Ao, A1 rightwise as many positions as there are zero trail... |

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Citation Context ..., Vk+l are at most one word (32 bits) long. A more detailed description of the theoretical background and of the implementation can be found in [SI. 2.4 Binary and PlusMinus The binary GCD algorithm (=-=[16]-=-, [12], [3]) is based on the relations: GCD(A, B) = GCD(A - B, B), If A odd, B even, then (11) GCD(A, B) = GCD(A, B/2). The algorithm begins by shifting Ao, A1 rightwise as many positions as there are... |

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Citation Context ... the decisions on the procedure are taken using only the lowest digits of the operands. 3 Experiment settings and results We implemented the algorithms using the GNU multiprecision arithmetic library =-=[7]-=-, under the GNU optimizing C compiler. The experiments were done using a the Digital DECstation 5000/200 (RISC architecture). For each length, each of the algorithms was applied to 1000 pairs of rando... |

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Citation Context ... cosequence computation ([SI). 1063-6889/93 $03.00 0 1993 IEEE 180 ([lS]) and its improvement for multidigit integers (Gosper, see [12]). 0 PlusMinus, I-PlusMinus: The plus-minus scheme introduced in =-=[2]-=- and its improvement for multidigit computation. G-binary, I-G-binary: A new algorithm for multiprecision GCD, which generalizes the binary and plus-minus GCD algorithms and its improvement by using t... |

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Citation Context ...re recovered using (2), and the cycle can start again. The algorithm needs a sufficient condition for qi+l = q k. We have used: ak+l 2 vk+l and (ak - ak+l) 2 ( vk -k vk+l), (4) which was developed in =-=[6]-=-. Recovering A I, Ak+1 involves 4 multiplications of a single digit number by a multidigit number, and this is the most time consuming part of the whole computation. Experimentally, one notices that f... |

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Citation Context ...nd of parallelization. Also, these three algorithms work “least-significant digits first,” (LSF), hence they are suitable for pipelined aggregation with other LSF systolic algorithms (multiplication: =-=[l]-=-, exact division: [ll]).s2 Description of algorithms We present here the outline of the GCD algorithms which were measured, and indicate the appropriate literature for the readers which are interested... |

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1 |
A generalization of the binary
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Citation Context ...n step, and the combined noise must be eliminated after finding G’ by : GCD(A, B) = GCD(G’, A, B) = GCD(GCD(G’, A mod GI), B mod G’)). (14) This “noise” is nevertheless small in the average case (see =-=[lo]-=-), and we experimentally noticed that the operations (14) take less than 5% of the total GCD computation time, in average. We also note that the operation 3:-’ mod22m, which is quite costly when perfo... |

1 | Euclid’s algorithmfor large numbers”, Am - Lehmer - 1938 |

1 |
Schnelle Berechung von Kettenbruchentwicklugen
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Citation Context ...neralizes the binary and plus-minus GCD algorithms and its improvement by using two digits in computation of cofactors [IO]. We did not consider the GCD algorithms based on FFT multiplication scheme (=-=[15]-=-, [14]), which are asymptotically faster, but are not expected to give a practical speed-up for the range of integers we are interested in (up to 100 words of 32 bits). The results of the experiment s... |