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Real and imaginary quadratic representations of hyperelliptic function fields
Author(s):
Sachar
Paulus;
Hans-Georg
Rück.
Journal:
Math. Comp.
68
(1999),
1233-1241.
MSC (1991):
Primary 11R58, 14Q05;
Secondary 11R65, 14H05, 14H40
Posted:
February 15, 1999
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Abstract:
A hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field. If at least one of the rational prime divisors is rational over the field of constants, then it also can be represented as an imaginary quadratic extension of the rational function field. The arithmetic in the divisor class group can be realized in the second case by Cantor's algorithm. We show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved. Furthermore, we show how the two representations are connected and compare the computational complexity.
References:
- 1.
- W. W. Adams, M. J. Razar: Multiples of points on elliptic curves and continued fractions. Proc. London Math. Soc. 41 (1980). pp. 481 - 498. MR 82c:14031
- 2.
- E. Artin: Quadratische Körper im Gebiete der höheren Kongruenzen I. Mathematische Zeitschrift 19 (1924). pp. 153 - 206; reprinted in S. Lang, J. Tate (eds.): The collected papers of Emil Artin. Reading, Mass.: Addison Wesley 1965. MR 31:1159
- 3.
- I. Biehl, J. Buchmann, C. Thiel: Cryptographic protocols based on discrete logarithms in real quadratic orders. Proceedings of CRYPTO '94. New York: Springer 1995.
- 4.
- D. G. Cantor: Computing in the Jacobian of a hyperelliptic curve. Mathematics of Computation 48 (1987). pp. 95 - 101. MR 88f:11118
- 5.
- H. W. Lenstra, Jr.: On the calculation of regulators and class numbers of quadratic fields. Number Theory Days (Exeter, 1980; J. V. Armitage, ed.), London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, 1982, pp. 123 - 150. MR 86g:11080
- 6.
- D. Mumford: Tata Lectures on Theta I, II. Boston: Birkhäuser Verlag 1983/84. MR 85h:14026; MR 86b:14017
- 7.
- R. Scheidler, A. Stein, H. C. Williams: Key-exchange in real quadratic congruence function fields. Designs, Codes and Cryptography 7 (1996). pp. 153 - 174. MR 97d:94009
- 8.
- D. Shanks: The infrastructure of a real quadratic field and its applications. Proc. Number Theory Conf., Univ. of Colorado, Boulder, CO, 1972, pp. 217 - 224. MR 52:10672
- 9.
- A. Stein: Equivalences between elliptic curves and real quadratic congruence function fields. Journal de Théorie des Nombres de Bordeaux 9. 1997. pp. 79 - 95. MR 98d:11144
- 10.
- H. Stichtenoth: Algebraic Function Fields and Codes. Berlin; Heidelberg: Springer 1993. MR 94k:14016
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Additional Information:
Sachar
Paulus
Affiliation:
Institut für Theoretische Informatik, TU Darmstadt, Alexanderstraße 10, 64283 Darmstadt (Germany)
Email:
sachar@cdc.informatik.th-darmstadt.de
Hans-Georg
Rück
Affiliation:
Institut für Experimentelle Mathematik, Universität GH Essen, Ellernstr.29, 45326 Essen (Germany)
Email:
rueck@exp-math.uni-essen.de
DOI:
10.1090/S0025-5718-99-01066-2
PII:
S 0025-5718(99)01066-2
Keywords:
Hyperelliptic curves,
divisor class groups,
real quadratic model
Received by editor(s):
July 24, 1997
Received by editor(s) in revised form:
November 3, 1997 and January 20, 1998
Posted:
February 15, 1999
Copyright of article:
Copyright
1999,
American Mathematical Society
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