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Algorithmica

, Volume 46, Issue 1, pp 137–148 | Cite as

GCD of Random Linear Combinations

  • Joachim von zur GathenEmail author
  • Igor E. ShparlinskiEmail author
Article

Abstract

We show that for arbitrary positive integers \(a_1, \ldots, a_m,\) with probability \(6/\pi^2 + o(1),\) the gcd of two linear combinations of these integers with rather small random integer coefficients coincides with \(\gcd(a_1, \ldots, a_m).\) This naturally leads to a probabilistic algorithm for computing the gcd of several integers, with probability \(6/\pi^2 + o(1),\) 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.

Keywords

Prime Divisor Probabilistic Algorithm Arbitrary Positive Integer Euler Function Desire Confidence Level 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.B-IT Computer Security, Universitat Bonn, 53113 BonnGermany
  2. 2.Department of Computing, Macquarie University, Sydney, NSW 2109Australia

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