## A modular reduction for GCD computation (2002)

Citations: | 3 - 0 self |

### BibTeX

@MISC{Sedjelmaci02amodular,

author = {Sidi Mohammed Sedjelmaci},

title = {A modular reduction for GCD computation},

year = {2002}

}

### OpenURL

### Abstract

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 k-ary reduction. c ○ 2003 Elsevier B.V. All rights reserved. 1.

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Citation Context ...l rights reserved. doi:10.1016/j.cam.2003.08.014s18 S.M. Sedjelmaci / Journalof Computationaland Applied Mathematics 162 (2004) 17–31 serial integer GCD algorithm O(n(log n) 2 log log n) of Schonhage =-=[1,10]-=-, their method still requires n iterations; the parallelism only reduces the bit operations per iteration. In 1987, Kannan, Miller and Rudolph (KMR) [6] gave the rst sublinear time parallel integer GC... |

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Citation Context ....Moreover, we suggest to use Priority CRCW PRAM model in order to solve the write concurrency for such index i: In this CRCW PRAM model, the processor with the smallest index is allowed to write (see =-=[7]-=-).Otherwise, if Test 3 is false the calculation is stopped.Moreover, only the i − 1+ rst leading bits of iu and the rst leading bits of v are needed for computing the value of q ′ i.Taking ¡m yields i... |

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Citation Context ...l rights reserved. doi:10.1016/j.cam.2003.08.014s18 S.M. Sedjelmaci / Journalof Computationaland Applied Mathematics 162 (2004) 17–31 serial integer GCD algorithm O(n(log n) 2 log log n) of Schonhage =-=[1,10]-=-, their method still requires n iterations; the parallelism only reduces the bit operations per iteration. In 1987, Kannan, Miller and Rudolph (KMR) [6] gave the rst sublinear time parallel integer GC... |

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Citation Context ...r transformation for integer GCD algorithms is the linear combination of u and v; i.e., R(u; v)=au + bv, where a and b are assumed to be rational numbers. Sorenson’s k-ary GCD algorithms and the like =-=[4,11,13,16]-=- are practical and e cient; they use the k-ary reduction technique.Given an integer parameter k¿0 and two integers u¿v¿0 relatively prime to k (i.e., (u; k) and (v; k) are coprime), pairs of integers ... |

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Citation Context ...ormations (u; v) ↦→ (v; u mod v). Although there have been results in the computation of the GCD of polynomials, the integer case still appeared to be inherently serial.Indeed, in 1983 Brent and Kung =-=[2]-=- achieved a running time of O(n) with n processors arranged in a systolic array, where n is the number of bits required to represent the larger of the two input numbers.Although it is an improvement o... |

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Citation Context ...time parallel integer GCD algorithm on a common CRCW PRAM model.Their time bound is O(n log log n=log n) assuming there are n 2 (log n) 2 processors working in parallel.Since 1990, Chor and Goldreich =-=[3]-=- currently have the fastest parallel GCD algorithm; it is based on the systolic array GCD algorithm of Brent and Kung.The time complexity of their algorithm achieves O (n=log n) using only n 1+ proces... |

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Citation Context ...gorithm O(n(log n) 2 log log n) of Schonhage [1,10], their method still requires n iterations; the parallelism only reduces the bit operations per iteration. In 1987, Kannan, Miller and Rudolph (KMR) =-=[6]-=- gave the rst sublinear time parallel integer GCD algorithm on a common CRCW PRAM model.Their time bound is O(n log log n=log n) assuming there are n 2 (log n) 2 processors working in parallel.Since 1... |

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Citation Context ...r transformation for integer GCD algorithms is the linear combination of u and v; i.e., R(u; v)=au + bv, where a and b are assumed to be rational numbers. Sorenson’s k-ary GCD algorithms and the like =-=[4,11,13,16]-=- are practical and e cient; they use the k-ary reduction technique.Given an integer parameter k¿0 and two integers u¿v¿0 relatively prime to k (i.e., (u; k) and (v; k) are coprime), pairs of integers ... |

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Citation Context ...r transformation for integer GCD algorithms is the linear combination of u and v; i.e., R(u; v)=au + bv, where a and b are assumed to be rational numbers. Sorenson’s k-ary GCD algorithms and the like =-=[4,11,13,16]-=- are practical and e cient; they use the k-ary reduction technique.Given an integer parameter k¿0 and two integers u¿v¿0 relatively prime to k (i.e., (u; k) and (v; k) are coprime), pairs of integers ... |

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Citation Context ...h positive (see the example in Section 3.5).Note that this case (ab ¿ 0) never occurs with MR. 5.2. Experiments The implementation is written in C with GNU C Compiler gcc, version 2.7 (Stallman, 1991 =-=[15]-=-) on a Pentium III 667 MHz PC running Unix System 5.We have compared sequential algorithms computing the Weber’s reduction WEB in version [16], MR2 version of MR, bmod and Rho.The variable = l(v) − l(... |

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Citation Context ...e precision approximations of the inputs.KMR proposed [6] to compute the integer linear reduction R(u; v)=au mod v, in parallel for a =1; 2;:::;n with n = ‘2(u).Some improvements of Lehmer’s approach =-=[14,12]-=- have been proposed later to compute more e cient integer reductions. 2.4. Rationalreductions It is easy to see that all the rational linear reductions are based on a modular relation au + bu = 0 (mod... |

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Citation Context ...e precision approximations of the inputs.KMR proposed [6] to compute the integer linear reduction R(u; v)=au mod v, in parallel for a =1; 2;:::;n with n = ‘2(u).Some improvements of Lehmer’s approach =-=[14,12]-=- have been proposed later to compute more e cient integer reductions. 2.4. Rationalreductions It is easy to see that all the rational linear reductions are based on a modular relation au + bu = 0 (mod... |

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Citation Context ...u + bv|=k au + bv ≡ 0 (mod k) However, for multiprecision inputs, EEA is not the best choice to compute the coe cients a and b in practice.In order to avoid expensive long divisions, Lehmer suggested =-=[9]-=- to extract the leading digits u1 and v1 of u and v, and run EEA on these single precision approximations of the inputs.KMR proposed [6] to compute the integer linear reduction R(u; v)=au mod v, in pa... |

1 |
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Citation Context ... CRCW PRAM.By varying the main parameter to the algorithm, they also obtain a polylog time, subexponential processor algorithm. More recently (1994), Sorenson’s right- and left-shift k-ary algorithms =-=[13]-=- take O (n=log n) time using at most n 1+ processors on a CRCW PRAM, matching Chor and Goldreich’s.Although the k-ary algorithms seem more involved than, say, the Euclidean and binary algorithms, a st... |