Advertisement

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)

Abstract

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.

Keywords

Prime Divisor Integer Vector Random Integer Probabilistic Algorithm Arbitrary Positive Integer 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Apostol, T.M.: Introduction to analytic number theory. Springer, Heidelberg (1976)zbMATHGoogle Scholar
  2. 2.
    Cooperman, G., Feisel, S., von zur Gathen, J., Havas, G.: GCD of many integers. In: Asano, T., Imai, H., Lee, D.T., Nakano, S.-i., Tokuyama, T. (eds.) COCOON 1999. LNCS, vol. 1627, pp. 310–317. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. 3.
    von zur Gathen, J., Gerhard, J.: Modern computer algebra. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  4. 4.
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford Univ. Press, Oxford (1979)zbMATHGoogle Scholar

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

Personalised recommendations