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An analysis of Lehmer's Euclidean GCD algorithm

1995
*
Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95
*

*Lehmer's*greatest common divisor

*algorithm*will compute

*gcd*(u, v) (with u > u) using at most o{(log u log v)/k + k log v + log u + ,%2 } bit operations and O(log u + k22k ) space, where k is the number ...

*of*bits in the multiprecision base

*of*the

*algorithm*. ... The

*Algorithm*In this section, we present our modified version

*of*

*Lehmer's*

*Euclidean*

*GCD*

*algorithm*. We begin by briefly describing the main loop. ...

##
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Analysis of fast versions of the Euclid Algorithm

2007
*
2007 Proceedings of the Fourth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)
*

This work has been already presented in the ANALCO'07 conference, and a larger abstract already appeared in the proceedings

doi:10.1137/1.9781611972979.14
dblp:conf/analco/CesarattoDLMV07
fatcat:gviclhfz7ze7vazxc3ms7vykt4
*of*this conference [5] . A long paper on this subject is in preparation. ... In [7] , the authors perform*an*average-case*analysis**of**Lehmer's**algorithm*, and exhibit the average speed-up obtained using these techniques. ... Since 90, the Caen Group [16, 18, 17] has performed*an*average-case*analysis**of*various parameters*of*a large class*of**Euclidean**algorithms*. ...##
###
Parallel implementation of Schönhage's integer GCD algorithm
[chapter]

1998
*
Lecture Notes in Computer Science
*

We present a parallel implementation

doi:10.1007/bfb0054852
fatcat:zxyar22r3fbk3m5hnqzv7nthk4
*of*Schönhage's integer*GCD**algorithm*on distributed memory architectures. Results are generalized for the extended*GCD**algorithm*. ... Schönhage's parallel*algorithm*is analyzed by using a message-passing model*of*computation. Experimental results on distributed memory architectures, such as the Intel Paragon, confirm the*analysis*. ... CLN implements*an*improved version*of**Lehmer's**GCD*and GMP the accelerated*GCD*. ...##
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(1 +i) - ary GCD Computation inZ[ i ]as an Analogue to the Binary GCD Algorithm

2000
*
Journal of symbolic computation
*

Thus one can calculate the generator

doi:10.1006/jsco.2000.0422
fatcat:3xtu4jbnmjgpfmee5r2qturs5i
*of*a finite generated ideal as the*GCD**of*the generators. the*Euclidean**algorithm*and Lehmer-type*GCD**algorithms*Since Euclid (cf. ... Using asymptotically fast*algorithms*for the calculation*of*the least remainders, this*GCD**algorithm*has a running time*of*O(n·µ(n)) bit operations, where µ(n) denotes*an*upper bound for the multiplication ... with three other*GCD**algorithms*in Z[i] (least remainder*Euclidean**algorithm*, Lehmer-type*algorithm*and*an*asymptotically fast*GCD**algorithm*). ...##
###
On Schönhage's algorithm and subquadratic integer gcd computation

2008
*
Mathematics of Computation
*

by Knuth and Schönhage, and to the binary recursive

doi:10.1090/s0025-5718-07-02017-0
fatcat:6k5nck57yff55eh5xow76s6brm
*gcd**algorithm**of*Stehlé and Zimmermann. ... We describe a new subquadratic left-to-right*gcd**algorithm*, inspired by Schönhage's*algorithm*for reduction*of*binary quadratic forms, and compare it to the first subquadratic*gcd**algorithm*discovered ... , for writing gmp and for his zealous testing, and Damien Stehlé, for helping me understand the binary recursive*algorithm*, and for many helpful comments during the work with this paper. ...##
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A Fast Euclidean Algorithm for Gaussian Integers

2002
*
Journal of symbolic computation
*

A new version

doi:10.1006/jsco.2001.0518
fatcat:bkwzsvctereglnc6bynskcamqm
*of*the*Euclidean**algorithm*is developed for computing the greatest common divisor*of*two Gaussian integers. ... The*algorithm*is compared with the new (1+i)ary*algorithm**of*Weilert and found to be somewhat faster if properly implemented. ... In that paper he also briefly described*an**algorithm*for Gaussian integers that he referred to as "Lehmer-type". This*algorithm*was also*an*approximative*Euclidean**algorithm*in the above sense. ...##
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The Mixed Binary Euclid Algorithm

2009
*
Electronic Notes in Discrete Mathematics
*

Usually, the

doi:10.1016/j.endm.2009.11.029
fatcat:io7eettyt5d37goe3si4xcko6m
*euclidean*and the binary*GCD**algorithms*work very well in practice for this range*of*integers. ... We present a new*GCD**algorithm*for two integers that combines both the*Euclidean*and the binary*gcd*approaches. ... Complexity*Analysis*The complexity*analysis**of*the parallel*GCD**algorithm*based on Par-MBE reduction is similar to that*of*Par-ILE in [15] . ...##
###
On a parallel Lehmer-Euclid GCD algorithm

2001
*
Proceedings of the 2001 international symposium on Symbolic and algebraic computation - ISSAC '01
*

A new version

doi:10.1145/384101.384142
dblp:conf/issac/Sedjelmaci01
fatcat:3j4hhb3o75htnguzxqors2pkqu
*of*Euclid's*GCD**algorithm*is proposed. ... It matches the best existing parallel integer*GCD**algorithms*since it can be achieved in O (n/ log n) time using at most n 1+ processors on CRCW PRAM. ... COMPLEXITY*ANALYSIS*We give below the complexity*analysis**of*the parallel ILE-*GCD**Algorithm*. ...##
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Computing greatest common divisors and factorizations in quadratic number fields

1989
*
Mathematics of Computation
*

We then provide two

doi:10.1090/s0025-5718-1989-0982367-2
fatcat:n4v7s6ics5bfvi6eaiv3nzo3cy
*algorithms*for computing the*GCD**of*algebraic integers in quadratic number fields Q(s/D). ... In a quadratic number field Q(V~D), D a squarefree integer, with class number 1, any algebraic integer can be decomposed uniquely into primes, but for only 21 domains*Euclidean**algorithms*are known. ... The*Euclidean**algorithm*(EA) consists*of*computing for any two elements po, p\ € R a sequence*of**Euclidean*divisions, Pi = Pi-2 -li-lPi-\, i > 2, such that N(pi) < N(pi-i) or p, = 0, in which case*GCD*...##
###
Page 8767 of Mathematical Reviews Vol. , Issue 99m
[page]

1999
*
Mathematical Reviews
*

In this paper, the authors study

*an*average-case*analysis**of*the FF*algorithm*in the discrete uniform model. A sharp upper bound is presented. Moreover,*an*open problem posed a decade ago is solved. ... (English summary) Average-case*analysis**of**algorithms*(Dagstuhl, 1995). Random Structures*Algorithms*10 (1997), no. 1-2, 69-101. ...##
###
A parallel extended GCD algorithm

2008
*
Journal of Discrete Algorithms
*

This approach is more suitable for extended

doi:10.1016/j.jda.2006.12.009
fatcat:nogvjkb6e5benm5ywlpph3h7pe
*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 ... A new parallel extended*GCD**algorithm*is proposed. ... Let k be*an*integer parameter s.t. k = 2 m with m 2 and m = O(log n). EEA denotes the Extended*Euclidean**Algorithm*. ...##
###
A modular reduction for GCD computation

2004
*
Journal of Computational and Applied Mathematics
*

Most

doi:10.1016/j.cam.2003.08.014
fatcat:xwy4grairjbfzg3gd5sxjf2ray
*of*integer*GCD**algorithms*use one or several basic transformations which reduce at each step the size*of*the inputs integers u and v. ... 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. ... Although the k-ary*algorithms*seem more involved than, say, the*Euclidean*and binary*algorithms*, a straightforward parallelization is su cient to rival the best previous parallel integer*GCD**algorithms*...##
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CopyCat: Controlled Instruction-Level Attacks on Enclaves
[article]

2020
*
arXiv
*
pre-print

We demonstrate the improved resolution and practicality

arXiv:2002.08437v3
fatcat:5epnn447mjfq3hk6mqk35hlewm
*of*CopyCat on Intel SGX in*an*extensive study*of*single-trace and deterministic attacks against cryptographic implementations, and give novel*algorithmic*... We propose*an*innovative controlled-channel attack, named CopyCat, that deterministically counts the number*of*instructions executed within a single enclave code page. ... Jo Van Bulck is supported by a grant*of*the Research Foundation -Flanders (FWO). ...##
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An LLL-reduction algorithm with quasi-linear time complexity

2011
*
Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11
*

We devise

doi:10.1145/1993636.1993691
dblp:conf/stoc/NovocinSV11
fatcat:qplf3df6nvgrverml4bzzkfi6e
*an**algorithm*, L 1 , with the following specifications: It takes as input*an*arbitrary basis B = (bi)i ∈ Z d×d*of*a*Euclidean*lattice L; It computes a basis*of*L which is reduced for a mild modification ... The backbone structure*of*L 1 is able to mimic the Knuth-Schönhage fast*gcd**algorithm*thanks to a combination*of*cutting-edge ingredients. ... Part*of*this work was done while Damien Stehlé was hosted by Macquarie University and the University*of*Sydney, whose hospitalities are gratefully acknowledged. ...##
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Primality Testing
[chapter]

2016
*
Computational Number Theory
*

Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity

doi:10.1201/b14045-8
fatcat:6sslltlq65e2bj2q6a7qcyyh3a
*of*all materials or the consequences*of*their use. ... Jonathan Sorenson, Two fast*GCD**algorithms*, Journal*of**Algorithms*, 16(1), 110-144, 1994.11 Jonathan Sorenson,*An**analysis**of**Lehmer's**Euclidean**gcd**algorithm*, ISSAC, 254-258, 1995. 12 The concept*of*congruences ...*GCD**Euclidean**GCD**Algorithm*At*an*early age, we have learned the repeated*Euclidean*division procedure for computing the*gcd**of*two integers. ...
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