Binary GCD Like Algorithms for Some Complex Quadratic Rings

  • Saurabh Agarwal
  • Gudmund Skovbjerg Frandsen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3076)


On the lines of the binary gcd algorithm for rational integers, algorithms for computing the gcd are presented for the ring of integers in \(\mathbb{Q}(\sqrt{d})\) where d ∈ { − 2, − 7, − 11, − 19}. Thus a binary gcd like algorithm is presented for a unique factorization domain which is not Euclidean (case d=-19). Together with the earlier known binary gcd like algorithms for the ring of integers in \(\mathbb{Q}(\sqrt{-1})\) and \(\mathbb{Q}(\sqrt{-3})\), one now has binary gcd like algorithms for all complex quadratic Euclidean domains. The running time of our algorithms is O(n 2) in each ring. While there exists an O(n 2) algorithm for computing the gcd in quadratic number rings by Erich Kaltofen and Heinrich Rolletschek, it has large constants hidden under the big-oh notation and it is not practical for medium sized inputs. On the other hand our algorithms are quite fast and very simple to implement.


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  1. 1.
    Agarwal, S.: Binary gcd like algorithms in some number rings. Department of Computer Science, University of Aarhus (2004)Google Scholar
  2. 2.
    Bach, E., Shallit, J.: Algorithmic number theory. Foundations of Computing Series, vol. 1. MIT Press, Cambridge (1996)MATHGoogle Scholar
  3. 3.
    Cohen, H.: Hermite and Smith normal form algorithms over Dedekind domains. Math. Comp. 65(216), 1681–1699 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Collins, G.E.: A fast Euclidean algorithm for Gaussian integers. J. Symbolic Comput. 33(4), 385–392 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Damgård, I.B., Frandsen, G.S.: Efficient algorithms for gcd and cubic residuosity in the ring of Eisenstein integers. In: Lingas, A., Nilsson, B.J. (eds.) FCT 2003. LNCS, vol. 2751, pp. 109–117. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Hungerford, T.W.: Algebra. Graduate Texts in Mathematics, vol. 73. Springer, New York (1980); Reprint of the 1974 originalMATHGoogle Scholar
  7. 7.
    Ireland, K., Rosen, M.: A classical introduction to modern number theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, New York (1990)MATHGoogle Scholar
  8. 8.
    Kaltofen, E., Rolletschek, H.: Computing greatest common divisors and factorizations in quadratic number fields. Math. Comp. 53(188), 697–720 (1989)MATHMathSciNetGoogle Scholar
  9. 9.
    Knuth, D.E.: The art of computer programming, 2nd edn., vol. 2. Addison-Wesley Publishing Co., Reading (1981)MATHGoogle Scholar
  10. 10.
    Lang, S.: Algebra, 3rd edn. Addison-Wesley Publishing Company, Reading (1993)MATHGoogle Scholar
  11. 11.
    Lehmer, D.H.: Euclid’s algorithm for large numbers.  45, 227–233 (1938)Google Scholar
  12. 12.
    Lemmermeyer, F.: The Euclidean algorithm in algebraic number fields. Exposition. Math. 13(5), 385–416 (1995); An updated version is available at the webpage, MATHMathSciNetGoogle Scholar
  13. 13.
    Lenstra Jr., H.W.: Euclid’s algorithm in cyclotomic fields. J. London Math. Soc (2) 10(4), 457–465 (1975)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Rolletschek, H.: On the number of divisions of the Euclidean algorithm applied to Gaussian integers. J. Symbolic Comput. 2(3), 261–291 (1986)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Scheidler, R., Williams, H.C.: A public-key cryptosystem utilizing cyclotomic fields. Des. Codes Cryptogr. 6(2), 117–131 (1995)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Schönhage, A.: Schnelle Berechnung von Kettenbruchentwilungen. Acta Informatica 1, 139–144 (1971)MATHCrossRefGoogle Scholar
  17. 17.
    Sorenson, J.: Two fast GCD algorithms. J. Algorithms 16(1), 110–144 (1994)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Stark, H.M.: A complete determination of the complex quadratic fields of classnumber one. Michigan Math. J. 14, 1–27 (1967)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Stein, J.: Computational problems associated with Racah algebra. J. Comput. Phys. (1), 397–405 (1967)Google Scholar
  20. 20.
    Weilert, A.: Effiziente Algorithmen zur Berechnung von Idealsummen in Quadratischen Ornungen. Dissertation, Universitaet Bonn (2000)Google Scholar
  21. 21.
    Weilert, A.: (1 + i)-ary GCD computation in Z[i] as an analogue to the binary GCD algorithm. J. Symbolic Comput. 30(5), 605–617 (2000)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Weilert, A.: Asymptotically fast GCD computation in Z[i]. In: Algorithmic number theory (Leiden, 2000). LNCS, vol. 1838, pp. 595–613. Springer, Berlin (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Saurabh Agarwal
    • 1
  • Gudmund Skovbjerg Frandsen
    • 1
  1. 1.BRICS, Department of Computer ScienceUniversity of AarhusAarhus NDenmark

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