GCD of Random Linear Forms

  • Joachim von zur Gathen
  • Igor E. Shparlinski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3341)


We show that for arbitrary positive integers a 1, ..., a m , with probability at least 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 at least 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). Naturally, this algorithm can be repeated to achieve any desired confidence level.


Prime Divisor Integer Vector Random Integer Probabilistic Algorithm Arbitrary Positive Integer 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Joachim von zur Gathen
    • 1
  • Igor E. Shparlinski
    • 2
  1. 1.Fakultät für Elektrotechnik, Informatik und MathematikUniversität PaderbornPaderbornGermany
  2. 2.Department of ComputingMacquarie UniversityAustralia

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