Applying sieving to the computation of quadratic class groups

Jacobson, Michael, Jr.

doi:10.1090/S0025-5718-99-01003-0

Applying sieving to the computation
of quadratic class groups

Author: Michael J. Jacobson Jr.
Journal: Math. Comp. 68 (1999), 859-867
MSC (1991): Primary 11Y40; Secondary 11Y16
DOI: https://doi.org/10.1090/S0025-5718-99-01003-0
MathSciNet review: 1604324
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Abstract: We present a new algorithm for computing the ideal class group of an imaginary quadratic order which is based on the multiple polynomial version of the quadratic sieve factoring algorithm. Although no formal analysis is given, we conjecture that our algorithm has sub-exponential complexity, and computational experience shows that it is significantly faster in practice than existing algorithms.

1. C.S. Abel, Ein Algorithmus zur Berechnung der Klassenzahl und des Regulators reellquadratischer Ordnungen, Ph.D. thesis, Universität des Saarlandes, Saarbrücken, Germany, 1994.
2. Eric Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), no. 191, 355–380. MR 1023756, https://doi.org/10.1090/S0025-5718-1990-1023756-8
3. Eric Bach, Improved approximations for Euler products, Number theory (Halifax, NS, 1994) CMS Conf. Proc., vol. 15, Amer. Math. Soc., Providence, RI, 1995, pp. 13–28. MR 1353917
4. I. Biehl, J. Buchmann, and T. Papanikolaou, LiDIA - a library for computational number theory, The LiDIA Group, Universität des Saarlandes, Saarbrücken, Germany, 1995.
5. Johannes Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, Séminaire de Théorie des Nombres, Paris 1988–1989, Progr. Math., vol. 91, Birkhäuser Boston, Boston, MA, 1990, pp. 27–41. MR 1104698
6. Johannes Buchmann and Stephan Düllmann, Distributed class group computation, Informatik, Teubner-Texte Inform., vol. 1, Teubner, Stuttgart, 1992, pp. 69–79. MR 1182565, https://doi.org/10.1007/978-3-322-95233-2_5
7. J. Buchmann and S. Düllmann, A probabilistic class group and regulator algorithm and its implementation, Computational number theory (Debrecen, 1989) de Gruyter, Berlin, 1991, pp. 53–72. MR 1151855
8. Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206
9. H. Cohen, F. Diaz y Diaz, and M. Olivier, Calculs de nombres de classes et de régulateurs de corps quadratiques en temps sous-exponentiel, Séminaire de Théorie des Nombres, Paris, 1990–91, Progr. Math., vol. 108, Birkhäuser Boston, Boston, MA, 1993, pp. 35–46 (French). MR 1263522
10. S. Düllmann, Ein Algorithmus zur Bestimmung der Klassengruppe positiv definiter binärer quadratischer Formen, Ph.D. thesis, Universität des Saarlandes, Saarbrücken, Germany, 1991.
11. James L. Hafner and Kevin S. McCurley, A rigorous subexponential algorithm for computation of class groups, J. Amer. Math. Soc. 2 (1989), no. 4, 837–850. MR 1002631, https://doi.org/10.1090/S0894-0347-1989-1002631-0
12. M.J. Jacobson, Jr., Some experimental results on ideal class groups of quadratic fields, Unpublished MS., 1997.
13. S. Paulus, An algorithm of subexponential type computing the class group of quadratic orders over principal ideal domains, Algorithmic Number Theory (Université Bordeaux I, Talence, France), Lecture Notes in Computer Sci., vol. 1122, Springer-Verlag, Berlin, 1996, pp. 243-257. MR 98e:11143
14. Martin Seysen, A probabilistic factorization algorithm with quadratic forms of negative discriminant, Math. Comp. 48 (1987), no. 178, 757–780. MR 878705, https://doi.org/10.1090/S0025-5718-1987-0878705-X
15. Robert D. Silverman, The multiple polynomial quadratic sieve, Math. Comp. 48 (1987), no. 177, 329–339. MR 866119, https://doi.org/10.1090/S0025-5718-1987-0866119-8