GCD of Random Linear Forms

Gathen, Joachim von zur; Shparlinski, Igor E.

doi:10.1007/978-3-540-30551-4_41

Cited by 2 publications

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“…A drawback of that algorithm is that for its proof of correctness, the arguments given to the gcd computations are substantially larger than the original inputs. In [4] we give an asymptotic lower bound on ρ a (M) (of the expected order ζ(2) −1 ) which holds starting with very small values of M. Here we present a completely explicit and slightly stronger form of that result. More importantly, we obtain an asymptotic formula for ρ a (M) which holds in a wide range of parameters.…”

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confidence: 77%

GCD of Random Linear Combinations

Gathen

Shparlinski

2006

Algorithmica

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We show that for arbitrary positive integers a 1 , . . . , a m , with probability 6/π 2 + o(1), the gcd of two linear combinations of these integers with rather small random integer coefficients coincides with gcd(a 1 , . . . , a m ). This naturally leads to a probabilistic algorithm for computing the gcd of several integers, with probability 6/π 2 + o(1), via just one gcd of two numbers with about the same size as the initial data (namely the above linear combinations). This algorithm can be repeated to achieve any desired confidence level.

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confidence: 77%

GCD of Random Linear Combinations

Gathen

Shparlinski

2006

Algorithmica

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“…, a r ) = gcd(S, T ). (2) This was improved recently by von zur Gathen and Shparlinski [6] and they gave the following strong result: with probability at least 6/π 2 + o(1), gcd(a 1 , a 2 , . .…”

Section: Chinese Remainder Representationmentioning

confidence: 88%

Fast arithmetics using Chinese remaindering

Davida

Litow

2009

Information Processing Letters

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In this paper, some issues concerning the Chinese remaindering representation are discussed. A new converting method, which is an efficient probabilistic algorithm based on a recent result of von zur Gathen and Shparlinski [J. von zur Gathen, I. Shparlinski, GCD of random linear forms, Algorithmica 46 (2006) 137-148], is described. An efficient refinement of the NC 1 division algorithm of Chiu, Davida and Litow [A. Chiu, G. Davida, B. Litow, Division in logspace-uniform NC 1 , Theoret. Informatics Appl. 35 (2001) 259-275] is given, where the number of moduli is reduced by a factor of log n.

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GCD of Random Linear Forms

Cited by 2 publications

References 1 publication

GCD of Random Linear Combinations

GCD of Random Linear Combinations

Fast arithmetics using Chinese remaindering

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