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Two efficient algorithms for the computation of ideal sums in quadratic orders

Author: André Weilert
Journal: Math. Comp. 75 (2006), 941-981
MSC (2000): Primary 54C40, 14E20; Secondary 46E25, 20C20
Published electronically: December 8, 2005
MathSciNet review: 2197002
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Abstract: This paper deals with two different asymptotically fast algorithms for the computation of ideal sums in quadratic orders. If the class number of the quadratic number field is equal to 1, these algorithms can be used to calculate the GCD in the quadratic order. We show that the calculation of an ideal sum in a fixed quadratic order can be done as fast as in $ \mathbf{Z}$ up to a constant factor, i.e., in $ O(\mu(n) \log n),$ where $ n$ bounds the size of the operands and $ \mu(n)$ denotes an upper bound for the multiplication time of $ n$-bit integers. Using Schönhage-Strassen's asymptotically fast multiplication for $ n$-bit integers, we achieve $ \mu(n)=O(n\log n\log\log n).$

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  • 1. Tom M. Apostol, Modular functions and Dirichlet series in number theory, 2nd ed., Graduate Texts in Mathematics, vol. 41, Springer-Verlag, New York, 1990. MR 1027834
  • 2. Richard Brauer, On the zeta-functions of algebraic number fields, Amer. J. Math. 69 (1947), 243–250. MR 0020597,
  • 3. Richard Brauer, On the zeta-functions of algebraic number fields. II, Amer. J. Math. 72 (1950), 739–746. MR 0039009,
  • 4. Johannes Buchmann, Christoph Thiel, and Hugh Williams, Short representation of quadratic integers, Computational algebra and number theory (Sydney, 1992) Math. Appl., vol. 325, Kluwer Acad. Publ., Dordrecht, 1995, pp. 159–185. MR 1344929
  • 5. Duncan A. Buell, Binary quadratic forms, Springer-Verlag, New York, 1989. Classical theory and modern computations. MR 1012948
  • 6. B. F. Caviness, A Lehmer-Type Greatest Common Divisor Algorithm for Gaussian Integers, SIAM Rev. 15 (1973), no. 2, 414.
  • 7. B. F. Caviness and G. E. Collins, Algorithms for Gaussian Integer Arithmetic, Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation SYMSAC'76 (Yorktown Heights) (R. D. Jenks, ed.), 1976, pp. 36-45.
  • 8. Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206
  • 9. Henri Cohen, Hermite and Smith normal form algorithms over Dedekind domains, Math. Comp. 65 (1996), no. 216, 1681–1699. MR 1361805,
  • 10. Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313
  • 11. George E. Collins, A fast Euclidean algorithm for Gaussian integers, J. Symbolic Comput. 33 (2002), no. 4, 385–392. MR 1890576,
  • 12. Carl Friedrich Gauss, Untersuchungen über höhere Arithmetik, Deutsch herausgegeben von H. Maser, Chelsea Publishing Co., New York, 1965 (German). MR 0188045
  • 13. James L. Hafner and Kevin S. McCurley, Asymptotically fast triangularization of matrices over rings, SIAM J. Comput. 20 (1991), no. 6, 1068–1083. MR 1135749,
  • 14. Euclid, The thirteen books of Euclid’s Elements translated from the text of Heiberg. Vol. I: Introduction and Books I, II. Vol. II: Books III–IX. Vol. III: Books X–XIII and Appendix, Dover Publications, Inc., New York, 1956. Translated with introduction and commentary by Thomas L. Heath; 2nd ed. MR 0075873
  • 15. Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716
  • 16. E. Kaltofen and H. Rolletschek, Arithmetic in Quadratic Fields with Unique Factorization, Proceedings of the EUROCAL'85 Conference on Computer Algebra (Linz, Austria, April 1-3, 1985) B. F. Caviness, ed., Lecture Notes in Comput. Sci., vol. 204, Springer-Verlag, Berlin, 1985, pp. 279-288. MR 0826569 (87c:11099)
  • 17. -, Computing Greatest Common Divisors and Factorizations in Quadratic Number Fields, Math. Comp. 53 (1989), no. 188, 697-720. MR 0982367 (90a:11154)
  • 18. Donald E. Knuth, The analysis of algorithms, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 269–274. MR 0423865
  • 19. -, Seminumerical Algorithms, third ed., The Art of Computer Programming, vol. 2, Addison-Wesley, Reading, MA, 1998.MR 0633878 (83i:68003)
  • 20. Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556
  • 21. Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723
  • 22. D. H. Lehmer, Euclid's Algorithm for Large Numbers, Amer. Math. Monthly 45 (1938), 227-233.
  • 23. Franz Lemmermeyer, The Euclidean algorithm in algebraic number fields, Exposition. Math. 13 (1995), no. 5, 385–416. MR 1362867
  • 24. H. W. Lenstra, Jr, On the Computation of Regulators and Class Numbers of Quadratic Fields, London Math. Soc. Lecture Note Ser. 56 (1982), 123-150. MR 0697260 (86g:11080)
  • 25. Udi Manber, Introduction to algorithms, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1989. A creative approach. MR 1091251
  • 26. Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859
  • 27. Donald J. Newman, Analytic number theory, Graduate Texts in Mathematics, vol. 177, Springer-Verlag, New York, 1998. MR 1488421
  • 28. A. Schönhage, Schnelle Berechnung von Kettenbruchentwicklungen, Acta Inform. 1 (1971), 139-144.
  • 29. -, IGCDOC, Computation of Integer GCD's, Unpublished Manuscript, 1987.
  • 30. -, Fast Reduction and Composition of Binary Quadratic Forms, Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation ISSAC'91 (Bonn, Germany, July 15-17, 1991) S. M. Watt, ed., ACM Press, New York, 1991, pp. 128-133.
  • 31. Arnold Schönhage, Andreas F. W. Grotefeld, and Ekkehart Vetter, Fast algorithms, Bibliographisches Institut, Mannheim, 1994. A multitape Turing machine implementation. MR 1290996
  • 32. A. Schönhage and V. Strassen, Schnelle Multiplikation grosser Zahlen, Computing (Arch. Elektron. Rechnen) 7 (1971), 281–292 (German, with English summary). MR 0292344
  • 33. M. A. Shokrollahi and V. Stemann, Approximation of Complex Numbers by Cyclotomic Integers, Technical Report TR-96-033, International Computer Science Institute, Berkeley, September 1996.
  • 34. C. L. Siegel, Über die Classenzahl quadratischer Zahlkörper, Acta Arith. 1 (1935), 83-86.
  • 35. D. Stehle and P. Zimmermann, A Binary Recursive GCD Algorithm, Proceedings of the Sixth International Algorithmic Number Theory Symposium ANTS VI (Burlington, VT, June 13-18, 2004) D. Buell, ed., Lecture Notes in Comput. Sci., vol. 3076, Springer-Verlag, Berlin, 2004, pp. 411-425.
  • 36. J. Stein, Computational Problems Associated with Racah Algebra, J. Comput. Phys. 1 (1967), 397-405.
  • 37. Joachim von zur Gathen and Jürgen Gerhard, Modern computer algebra, Cambridge University Press, New York, 1999. MR 1689167
  • 38. André Weilert, (1+𝑖)-ary GCD computation in 𝑍[𝑖] is an analogue to the binary GCD algorithm, J. Symbolic Comput. 30 (2000), no. 5, 605–617. MR 1797272,
  • 39. André Weilert, Asymptotically fast GCD computation in ℤ[𝕚], Algorithmic number theory (Leiden, 2000) Lecture Notes in Comput. Sci., vol. 1838, Springer, Berlin, 2000, pp. 595–613. MR 1850636,
  • 40. -, Effiziente Algorithmen zur Berechnung von Idealsummen in quadratischen Ordnungen, Dissertation, Mathematisch-Naturwissenschaftliche Fakultät der Rheinischen-Friedrich-Wilhelms Universität Bonn, Juli 2000.
  • 41. André Weilert, Fast computation of the biquadratic residue symbol, J. Number Theory 96 (2002), no. 1, 133–151. MR 1931197

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Additional Information

André Weilert
Affiliation: Department of Computer Science II, University of Bonn, Römerstraße 164, 53117 Bonn, Germany
Address at time of publication: Liliencronstr. 8, 12167 Berlin, Germany

Keywords: Computational number theory, quadratic number fields, GCD computation, Euclidean algorithm
Received by editor(s): July 20, 2003
Received by editor(s) in revised form: January 7, 2005
Published electronically: December 8, 2005
Additional Notes: This paper deals with the main results of my doctoral thesis [40]. I would like to thank my academic teachers Arnold Schönhage and Jens Franke (both at the University of Bonn, Germany).
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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