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Practical Improvements to Class Group and Regulator Computation of Real Quadratic Fields

  • Jean-François Biasse
  • Michael J. JacobsonJr.
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)

Abstract

We present improvements to the index-calculus algorithm for the computation of the ideal class group and regulator of a real quadratic field. Our improvements consist of applying the double large prime strategy, an improved structured Gaussian elimination strategy, and the use of Bernstein’s batch smoothness algorithm. We achieve a significant speed-up and are able to compute the ideal class group structure and the regulator corresponding to a number field with a 110-decimal digit discriminant.

Keywords

Prime Ideal Class Group Gaussian Elimination Relation Matrix Ideal Class Group 
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.

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References

  1. 1.
    Bach, E.: Explicit bounds for primality testing and related problems. Math. Comp. 55(191), 355–380 (1990)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bach, E.: Improved approximations for Euler products. In: Number Theory: CMS Proc., vol. 15, pp. 13–28. Amer. Math. Soc., Providence (1995)Google Scholar
  3. 3.
    Bernstein, D.: How to find smooth parts of integers. Mathematics of Computation (submited)Google Scholar
  4. 4.
    Biasse, J.-F.: Improvements in the computation of ideal class groups of imaginary quadratic number fields. In: Advances in Mathematics of Communications (to appear 2010)Google Scholar
  5. 5.
    Biasse, J.-F., Jacobson Jr., M.J., Silvester, A.K.: Security estimates for quadratic field based cryptosystems. In: ACISP (to appear 2010)Google Scholar
  6. 6.
    Buchmann, J.: A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In: Séminaire de Théorie des Nombres (Paris), pp. 27–41 (1988-1989)Google Scholar
  7. 7.
    Buchmann, J., Düllmann, S.: Distributed class group computation. In: Festschrift aus Anlaß des sechzigsten Geburtstages von Herrn Prof. Dr. G. Hotz, pp. 69–79. Universität des Saarlandes (1991), Teubner, Stuttgart (1992)Google Scholar
  8. 8.
    Cavallar, S.: Strategies in filtering in the number field sieve. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 209–232. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Chen, Z., Storjohann, A., Fletcher, C.: IML: Integer Matrix Library. Software (2010), http://www.cs.uwaterloo.ca/~astorjoh/iml.html
  10. 10.
    Dodson, B., Leyland, P.C., Lenstra, A.K., Muffett, A., Wagstaff, S.: MPQS with three large primes. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 446–460. Springer, Heidelberg (2002)Google Scholar
  11. 11.
    GMP, The GNU multiple precision bignum library. Software (2010), http://gmp-lib.org/
  12. 12.
    Gower, J.E., Wagstaff, S.: Square form factorization. Mathematics of Computation 77, 551–588 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Havas, G., Majewski, B.S.: Integer matrix diagonalization. Journal of Symbolic Computing 24, 399–408 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jacobson Jr., M.J.: Subexponential class group computation in quadratic orders, Ph.D. thesis, Technische Universitt Darmstadt, Darmstadt, Germany (1999)Google Scholar
  15. 15.
    Jacobson Jr., M.J., Scheidler, R., Williams, H.C.: The efficiency and security of a real quadratic field based key exchange protocol. In: Public-Key Cryptography and Computational Number Theory, Warsaw, Poland, pp. 89–112. de Gruyter (2001)Google Scholar
  16. 16.
    Jacobson Jr., M.J., Williams, H.C.: Solving the Pell equation. CMS Books in Mathematics. Springer, Heidelberg (2009) ISBN 978-0-387-84922-5zbMATHGoogle Scholar
  17. 17.
    Lenstra, A.K., Manasse, M.S.: Factoring with two large primes (extended abstract). In: Damgård, I.B. (ed.) EUROCRYPT 1990. LNCS, vol. 473, pp. 72–82. Springer, Heidelberg (1991)Google Scholar
  18. 18.
    LiDIA Group, LiDIA: a c++ library for computational number theory. Software, Technische Universität Darmstadt, Germany (1997), http://www.informatik.tu-darmstadt.de/TI/LiDIA
  19. 19.
    LinBox, Project LinBox: Exact computational linear algebra. Software (2010), http://www.linalg.org/
  20. 20.
    Louboutin, S.: Computation of class numbers of quadratic number fields. Math. Comp. 71(240), 1735–1743 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Maurer, M.: Regulator approximation and fundamental unit computation for real quadratic orders, Ph.D. thesis, Technische Universitt Darmstadt, Darmstadt, Germany (1999)Google Scholar
  22. 22.
  23. 23.
    Shoup, V.: NTL: A Library for doing Number Theory. Software (2010), http://www-shoup.net/ntl
  24. 24.
    Vollmer, U.: An accelerated Buchmann algorithm for regulator computation in real quadratic fields. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 148–162. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jean-François Biasse
    • 1
  • Michael J. JacobsonJr.
    • 2
  1. 1.École PolytechniquePalaiseauFrance
  2. 2.Department of Computer ScienceUniversity of CalgaryCalgaryCanada

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