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An analysis of Lehmer's Euclidean GCD algorithm

Jonathan Sorenson
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.  ... 
doi:10.1145/220346.220378 dblp:conf/issac/Sorenson95 fatcat:vqfm75bj25fvbklzph6mjnh2zi

Analysis of fast versions of the Euclid Algorithm

Eda Cesaratto, Julien Clement, Benoît Daireaux, Loïck Lhote, Véronique Maume-Deschamps, Brigitte Vallée
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 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.  ... 
doi:10.1137/1.9781611972979.14 dblp:conf/analco/CesarattoDLMV07 fatcat:gviclhfz7ze7vazxc3ms7vykt4

Parallel implementation of Schönhage's integer GCD algorithm [chapter]

Giovanni Cesari
1998 Lecture Notes in Computer Science  
We present a parallel implementation 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.  ... 
doi:10.1007/bfb0054852 fatcat:zxyar22r3fbk3m5hnqzv7nthk4

(1 +i) - ary GCD Computation inZ[ i ]as an Analogue to the Binary GCD Algorithm

André Weilert
2000 Journal of symbolic computation  
Thus one can calculate the generator 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).  ... 
doi:10.1006/jsco.2000.0422 fatcat:3xtu4jbnmjgpfmee5r2qturs5i

On Schönhage's algorithm and subquadratic integer gcd computation

Niels Möller
2008 Mathematics of Computation  
by Knuth and Schönhage, and to the binary recursive 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.  ... 
doi:10.1090/s0025-5718-07-02017-0 fatcat:6k5nck57yff55eh5xow76s6brm

A Fast Euclidean Algorithm for Gaussian Integers

George E. Collins
2002 Journal of symbolic computation  
A new version 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.  ... 
doi:10.1006/jsco.2001.0518 fatcat:bkwzsvctereglnc6bynskcamqm

The Mixed Binary Euclid Algorithm

Sidi Mohamed Sedjelmaci
2009 Electronic Notes in Discrete Mathematics  
Usually, the 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] .  ... 
doi:10.1016/j.endm.2009.11.029 fatcat:io7eettyt5d37goe3si4xcko6m

On a parallel Lehmer-Euclid GCD algorithm

Sidi Mohammed Sedjelmaci
2001 Proceedings of the 2001 international symposium on Symbolic and algebraic computation - ISSAC '01  
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 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.  ... 
doi:10.1145/384101.384142 dblp:conf/issac/Sedjelmaci01 fatcat:3j4hhb3o75htnguzxqors2pkqu

Computing greatest common divisors and factorizations in quadratic number fields

Erich Kaltofen, Heinrich Rolletschek
1989 Mathematics of Computation  
We then provide two 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  ... 
doi:10.1090/s0025-5718-1989-0982367-2 fatcat:n4v7s6ics5bfvi6eaiv3nzo3cy

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

Sidi Mohamed Sedjelmaci
2008 Journal of Discrete Algorithms  
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  ...  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.  ... 
doi:10.1016/j.jda.2006.12.009 fatcat:nogvjkb6e5benm5ywlpph3h7pe

A modular reduction for GCD computation

Sidi Mohammed Sedjelmaci
2004 Journal of Computational and Applied Mathematics  
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.  ...  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  ... 
doi:10.1016/ fatcat:xwy4grairjbfzg3gd5sxjf2ray

CopyCat: Controlled Instruction-Level Attacks on Enclaves [article]

Daniel Moghimi, Jo Van Bulck, Nadia Heninger, Frank Piessens, Berk Sunar
2020 arXiv   pre-print
We demonstrate the improved resolution and practicality 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).  ... 
arXiv:2002.08437v3 fatcat:5epnn447mjfq3hk6mqk35hlewm

An LLL-reduction algorithm with quasi-linear time complexity

Andrew Novocin, Damien Stehlé, Gilles Villard
2011 Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11  
We devise 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.  ... 
doi:10.1145/1993636.1993691 dblp:conf/stoc/NovocinSV11 fatcat:qplf3df6nvgrverml4bzzkfi6e

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 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.  ... 
doi:10.1201/b14045-8 fatcat:6sslltlq65e2bj2q6a7qcyyh3a
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