Topics in computational algebraic number theory

Karim Belabas[1]

  • [1] Mathématiques–Bâtiment 425 Université Paris–Sud F–91405 Orsay Cedex

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 1, page 19-63
  • ISSN: 1246-7405

Abstract

top
We describe practical algorithms from computational algebraic number theory, with applications to class field theory. These include basic arithmetic, approximation and uniformizers, discrete logarithms and computation of class fields. All algorithms have been implemented in the Pari/Gp system.

How to cite

top

Belabas, Karim. "Topics in computational algebraic number theory." Journal de Théorie des Nombres de Bordeaux 16.1 (2004): 19-63. <http://eudml.org/doc/249264>.

@article{Belabas2004,
abstract = {We describe practical algorithms from computational algebraic number theory, with applications to class field theory. These include basic arithmetic, approximation and uniformizers, discrete logarithms and computation of class fields. All algorithms have been implemented in the Pari/Gp system.},
affiliation = {Mathématiques–Bâtiment 425 Université Paris–Sud F–91405 Orsay Cedex},
author = {Belabas, Karim},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {class field theory; algorithmic number theory; computational methods for number fields; survey; ray class group; discrete logarithms; class field; PARI/GP},
language = {eng},
number = {1},
pages = {19-63},
publisher = {Université Bordeaux 1},
title = {Topics in computational algebraic number theory},
url = {http://eudml.org/doc/249264},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Belabas, Karim
TI - Topics in computational algebraic number theory
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 1
SP - 19
EP - 63
AB - We describe practical algorithms from computational algebraic number theory, with applications to class field theory. These include basic arithmetic, approximation and uniformizers, discrete logarithms and computation of class fields. All algorithms have been implemented in the Pari/Gp system.
LA - eng
KW - class field theory; algorithmic number theory; computational methods for number fields; survey; ray class group; discrete logarithms; class field; PARI/GP
UR - http://eudml.org/doc/249264
ER -

References

top
  1. B. Allombert, An efficient algorithm for the computation of Galois automorphisms. Math. Comp. To appear. Zbl1094.11045MR2034127
  2. E. Bach, J. Driscoll, & J. Shallit, Factor refinement. J. Algorithms 15 (1993), no. 2, 199–222. Zbl0784.11058MR1231441
  3. K. Belabas, A relative van Hoeij algorithm over number fields. J. Symbolic Comput. To appear. Zbl1137.11360MR2094619
  4. K. Belabas & H. Gangl, Generators and relations for K 2 𝒪 F . K-Theory. To appear. Zbl1090.11074
  5. D. J. Bernstein, Factoring into coprimes in essentially linear time. J. Algorithms. To appear. Zbl1134.11352MR2108417
  6. J. Buchmann & H. W. Lenstra, Jr., Approximating rings of integers in number fields. J. Théor. Nombres Bordeaux 6 (1994), no. 2,  221–260. Zbl0828.11075MR1360644
  7. J. 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, 1990, 27–41. Zbl0727.11059MR1104698
  8. J. Buchmann & F. Eisenbrand, On factor refinement in number fields. Math. Comp. 68 (1999), no. 225, 345–350. Zbl0916.11067MR1613766
  9. H. Cohen, A course in computational algebraic number theory, third ed., Springer-Verlag, 1996. Zbl0786.11071MR1228206
  10. H. Cohen, Advanced topics in computational number theory, Springer-Verlag, 2000. Zbl0977.11056MR1728313
  11. H. Cohen & F. Diaz y Diaz, A polynomial reduction algorithm. Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 2, 351–360. Zbl0758.11053MR1149802
  12. H. Cohen, F. Diaz y Diaz, & M. Olivier, Subexponential algorithms for class group and unit computations. J. Symbolic Comput. 24 (1997), no. 3-4, 433–441, Computational algebra and number theory (London, 1993). Zbl0880.68067MR1484490
  13. H. Cohen & X.-F. Roblot, Computing the Hilbert class field of real quadratic fields Math. Comp. 69 (2000), no. 231, 1229–1244. Zbl1042.11075MR1651747
  14. M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, & K. Wildanger, KANT V4, J. Symbolic Comput. 24 (1997), no. 3-4, 267–283, Computational algebra and number theory (London, 1993). Zbl0886.11070MR1484479
  15. L. Ducos, Optimizations of the subresultant algorithm. J. Pure Appl. Algebra 145 (2000), no. 2, 149–163. Zbl0941.65044MR1733249
  16. M. Elkenbracht-Huizing, An implementation of the number field sieve Experiment. Math. 5 (1996), no. 3, 231–253. Zbl0869.11101MR1426450
  17. C. Fieker, Computing class fields via the Artin map. Math. Comp. 70 (2001), no. 235, 1293–1303. Zbl0982.11074MR1826583
  18. C. Fieker & C. Friedrichs, On reconstruction of algebraic numbers. Algorithmic number theory (Leiden, 2000), LNCS, vol. 1838, Springer, 2000, 285–296. Zbl0990.11084MR1850612
  19. U. Fincke & M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Math. Comp. 44 (1985), no. 170,  463–471. Zbl0556.10022MR777278
  20. D. Ford, S. Pauli, & X.-F. Roblot, A fast algorithm for polynomial factorization over p . J. Théor. Nombres Bordeaux 14 (2002), no. 1, 151–169. Zbl1032.11053MR1925995
  21. X. Gourdon, Algorithmique du théorème fondamental de l’algèbre. Rapport de recherche 1852, INRIA, 1993. 
  22. J. L. Hafner & K. S. McCurley, A rigorous subexponential algorithm for computation of class groups. J. Amer. Math. Soc. 2 (1989), no. 4, 837–850. Zbl0702.11088MR1002631
  23. J. Klüners, On computing subfields. A detailed description of the algorithm. J. Théor. Nombres Bordeaux 10 (1998), no. 2, 243–271. Zbl0935.11047MR1828244
  24. D. E. Knuth, The art of computer programming. Vol. 2, second ed., Addison-Wesley Publishing Co., Reading, Mass., 1981, Seminumerical algorithms, Addison-Wesley Series in Computer Science and Information Processing. Zbl0477.65002MR633878
  25. H. Koy & C.-P. Schnorr, Segment LLL-Reduction with Floating Point Orthogonalization. CaLC, LNCS, vol. 2146, Springer, 2001, 81–96. Zbl1053.94013MR1903889
  26. A. K. Lenstra, H. W. Lenstra, Jr., & L. Lovász, Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), no. 4,  515–534. Zbl0488.12001MR682664
  27. P. L. Montgomery, Square roots of products of algebraic numbers. Mathematics of Computation 1943–1993: a half-century of computational mathematics (Vancouver, BC, 1993). Proc. Sympos. Appl. Math., vol. 48, Amer. Math. Soc., 1994,  567–571. Zbl0819.11069MR1314892
  28. P. Nguyen, A Montgomery-like square root for the number field sieve. Algorithmic number theory (Portland, OR, 1998), Lecture Notes in Comput. Sci., vol. 1423, Springer, 1998, 151–168. Zbl0983.11075MR1726068
  29. PARI/GP, version 2.1.5, Bordeaux, 2003, http://pari.math.u-bordeaux.fr/. 
  30. F. Rouillier & P. Zimmermann, Efficient isolation of a polynomial real roots. Journal of Computational and Applied Mathematics. To appear. Zbl1040.65041
  31. C.-P. Schnorr & M. Euchner, Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math. Programming 66 (1994), no. 2, Ser. A, 181–199. Zbl0829.90099MR1297061
  32. J. von zur Gathen & J. Gerhard, Modern computer algebra, Cambridge University Press, New York, 1999. Zbl0936.11069MR1689167

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.