Topics in computational algebraic number theory
- [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
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topBelabas, 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- B. Allombert, An efficient algorithm for the computation of Galois automorphisms. Math. Comp. To appear. Zbl1094.11045MR2034127
- E. Bach, J. Driscoll, & J. Shallit, Factor refinement. J. Algorithms 15 (1993), no. 2, 199–222. Zbl0784.11058MR1231441
- K. Belabas, A relative van Hoeij algorithm over number fields. J. Symbolic Comput. To appear. Zbl1137.11360MR2094619
- K. Belabas & H. Gangl, Generators and relations for . K-Theory. To appear. Zbl1090.11074
- D. J. Bernstein, Factoring into coprimes in essentially linear time. J. Algorithms. To appear. Zbl1134.11352MR2108417
- 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
- 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
- J. Buchmann & F. Eisenbrand, On factor refinement in number fields. Math. Comp. 68 (1999), no. 225, 345–350. Zbl0916.11067MR1613766
- H. Cohen, A course in computational algebraic number theory, third ed., Springer-Verlag, 1996. Zbl0786.11071MR1228206
- H. Cohen, Advanced topics in computational number theory, Springer-Verlag, 2000. Zbl0977.11056MR1728313
- H. Cohen & F. Diaz y Diaz, A polynomial reduction algorithm. Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 2, 351–360. Zbl0758.11053MR1149802
- 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
- H. Cohen & X.-F. Roblot, Computing the Hilbert class field of real quadratic fields Math. Comp. 69 (2000), no. 231, 1229–1244. Zbl1042.11075MR1651747
- 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
- L. Ducos, Optimizations of the subresultant algorithm. J. Pure Appl. Algebra 145 (2000), no. 2, 149–163. Zbl0941.65044MR1733249
- M. Elkenbracht-Huizing, An implementation of the number field sieve Experiment. Math. 5 (1996), no. 3, 231–253. Zbl0869.11101MR1426450
- C. Fieker, Computing class fields via the Artin map. Math. Comp. 70 (2001), no. 235, 1293–1303. Zbl0982.11074MR1826583
- C. Fieker & C. Friedrichs, On reconstruction of algebraic numbers. Algorithmic number theory (Leiden, 2000), LNCS, vol. 1838, Springer, 2000, 285–296. Zbl0990.11084MR1850612
- 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
- D. Ford, S. Pauli, & X.-F. Roblot, A fast algorithm for polynomial factorization over . J. Théor. Nombres Bordeaux 14 (2002), no. 1, 151–169. Zbl1032.11053MR1925995
- X. Gourdon, Algorithmique du théorème fondamental de l’algèbre. Rapport de recherche 1852, INRIA, 1993.
- 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
- J. Klüners, On computing subfields. A detailed description of the algorithm. J. Théor. Nombres Bordeaux 10 (1998), no. 2, 243–271. Zbl0935.11047MR1828244
- 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
- H. Koy & C.-P. Schnorr, Segment LLL-Reduction with Floating Point Orthogonalization. CaLC, LNCS, vol. 2146, Springer, 2001, 81–96. Zbl1053.94013MR1903889
- A. K. Lenstra, H. W. Lenstra, Jr., & L. Lovász, Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), no. 4, 515–534. Zbl0488.12001MR682664
- 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
- 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
- PARI/GP, version 2.1.5, Bordeaux, 2003, http://pari.math.u-bordeaux.fr/.
- F. Rouillier & P. Zimmermann, Efficient isolation of a polynomial real roots. Journal of Computational and Applied Mathematics. To appear. Zbl1040.65041
- 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
- J. von zur Gathen & J. Gerhard, Modern computer algebra, Cambridge University Press, New York, 1999. Zbl0936.11069MR1689167
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