(1 + i) - ary GCD Computation inZ[ i ]as an Analogue to the Binary GCD Algorithm

Journal of Symbolic Computation

Volumn 30, Issue 5, 2000, Pages 605-617

  Weilert, André ^a  

^a UNIVERSITY OF BONN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001026996     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1006/jsco.2000.0422     Document Type: Article

References (34)

1. Bach E., Shallit J. Algorithmic Number Theory, Volume I: Efficient Algorithms. 1996;MIT Press, Cambridge, MA, USA.
2. Brent R. P. Analysis of the binary Euclidean algorithm. Traub J. F. Algorithms and Complexity: New Directions and Recent Results. 1976;321-355 Academic Press, New York.
3. Buhler J. P., ed.,. Proceedings of the Third International Algorithmic Number Theory Symposium ANTS III (Reed College, Portland, Oregon, USA, June 21-25, 1998). 1998;Springer-Verlag, Berlin.
4. Caviness B. F. A Lehmer-type greatest common divisor algorithm for Gaussian integers. SIAM Rev. 15:1973;414.
5. Caviness B. F., Collins G. E. Algorithms for Gaussian integer arithmetic. Jenks R. D. Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation SYMSAC'76 (Yorktown Heights). 1976;36-45.
6. Cesari G. Parallel implementation of Schönhage's integer GCD algorithm. In Buhler (1998), ed. 1998;. p. 64-76.
7. Heath T. L. The Thirteen Books of Euclid's Elements. 1956;Cambridge University Press, New York.
8. Ireland K., Rosen M. A Classical Introduction to Modern Number Theory. 1990;Springer-Verlag, Berlin.
9. Jebelean T. A generalization of the binary GCD algorithm. Bronstein M. Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation ISSAC'93 (Kiev, Ukraine, July 6-8, 1993). 1993a;111-116 ACM Press, New York.
10. Jebelean T. Comparing several GCD algorithms. Swartzlander E. Jr, Irwin M. J., Jullien G. Proceedings of the 11th Symposium on Computer Arithmetic (Windsor, Ontario). 1993b;180-185 IEEE Press.
11. Kaltofen E., Rolletschek H. Arithmetic in quadratic fields with unique factorization. Caviness B. F. Proceedings of the EUROCAL'85 Conference on Computer Algebra (Linz, Austria, April 1-3, 1985). 1985;279-288 Springer-Verlag, Berlin.
12. Kaltofen E., Rolletschek H. Computing greatest common divisors and factorizations in quadratic number fields. Math. Comput. 53:1989;697-720.
13. Knopfmacher A., Knopfmacher J. The number of steps in the Euclidean algorithm over complex quadratic fields. BIT. 31:1991;286-292.
14. Knuth D. E. Seminumerical Algorithms. 1998;Addison-Wesley, Reading, MA, USA.
15. Lehmer D. H. Euclid's algorithm for large numbers. Amer. Math. Monthly. 45:1938;227-233.
16. Neukirch J. Algebraic Number Theory. 1999;Springer-Verlag, Berlin.
17. Norton G. H. Extending the binary GCD algorithm. Calmet J. Proceedings of the 3rd International Conference on Algebraic Algorithms and Error-Correcting Codes AAECC-3 (Grenoble, France, July 15-19, 1985). 1985;363-372 Springer-Verlag, Berlin.
18. Norton G. H. A shift-remainder GCD algorithm. Huguet L., Poli A. Proceedings of the 5th International Conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes AAECC-5 (Menorca, Spain, June 15-19, 1987). 1987;350-356 Springer-Verlag, Berlin.
19. Rolletschek H. The Euclidean algorithm for Gaussian integers. van Hulzen J. A. Proceedings of the EUROCAL'83 Conference on Computer Algebra. 1983;12-23 Springer-Verlag, Berlin.
20. Rolletschek H. On the number of divisions of the Euclidean algorithm applied to Gaussian integers. J. Symb. Comput. 2:1986;261-291.
21. Rolletschek H. Shortest division chains in imaginary quadratic number Fields. Gianni P. Proceedings of the 1988 International Symposium on Symbolic and Algebraic Computation ISSAC'88 (Rome, Italy, July 4-8, 1988, Conference held jointly with AAECC-6). 1989;231-243 Springer-Verlag, Berlin.
22. Rolletschek H. Shortest division chains in imaginary quadratic number fields. J. Symb. Comput. 9:1990;321-354.
23. Schönhage A. Schnelle Berechnung von Kettenbruchentwicklungen. Acta Inform. 1:1971;139-144.
24. A. Schönhage.
25. Schönhage A., Grotefeld A. F. W., Vetter E. Fast Algorithms - A Multitape Turing Machine Implementation. 1994;BI Wissenschaftsverlag, Mannheim, Germany.
26. Schönhage A., Strassen V. Schnelle Multiplikation großer Zahlen. Computing. 7:1971;281-292.
27. J. Sorenson, 1990, University of Wisconsin, Madison.
28. Sorenson J. Two fast GCD algorithms. J. Algorithms. 16:1994;110-144.
29. Stein J. Computational problems associated with Racah algebra. J. Comput. Phys. 1:1967;397-405.
30. Vallée B. The complete analysis of the binary euclidean algorithm. In Buhler (1998), ed. 1998;. p. 77-94.
31. Weber K. The accelerated integer GCD algorithm. ACM Trans. Math. Software. 21:1995;111-122.
32. Weber K. Parallel implementation of the accelerated integer GCD algorithm. J. Symb. Comput. 21:1996;457-466.
33. A. Weilert, Z, i.
34. Weilert A. Asymptotically fast GCD computation inZ[ i ]. Bosma W. Proceedings of the Fourth International Algorithmic Number Theory Symposium ANTS IV (Leiden, The Netherlands, July 2-7, 2000). 2000;595-613 Springer-Verlag, Berlin.