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The arithmetic of certain cubic function fields

Author: Mark L. Bauer
Journal: Math. Comp. 73 (2004), 387-413
MSC (2000): Primary 11R58, 94A60
Published electronically: June 17, 2003
MathSciNet review: 2034129
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Abstract: In this paper, we discuss the properties of curves of the form $y^3=f(x)$ over a given field K of characteristic different from 3. If $f(x)$ satisfies certain properties, then the Jacobian of such a curve is isomorphic to the ideal class group of the maximal order in the corresponding function field. We seek to make this connection concrete and then use it to develop an explicit arithmetic for the Jacobian of such curves. From a purely mathematical perspective, this provides explicit and efficient techniques for performing arithmetic in certain ideal class groups which are of fundamental interest in algebraic number theory. At the same time, it provides another source of groups which are suitable for Diffie-Hellman type protocols in cryptographic applications.

References [Enhancements On Off] (What's this?)

  • [C] Cantor, David G. Computing in the Jacobian of a hyperelliptic curve. Math. Comp. 48 (1987), no. 177, 95-101. MR 88f:11118
  • [GPS] Galbraith, Paulus, Smart. Arithmetic of Superelliptic Curves. Math. Comp. 71 (2002), 393-405. MR 2002h:14102
  • [H] Hartshorne, Robin. Algebraic Geometry. Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York - Heidelberg, 1997. MR 57:3116
  • [L] Lenstra, A. K. Factoring multivariate polynomials over finite fields. J. of Comput. System Sci. 30 (1985),no. 2, 235-248. MR 87a:11124
  • [Sch] Scheidler, R. Ideal arithmetic and infrastructure in purely cubic function fields. J. Théor. Nombres Bordeaux 13 (2002), 609-631. MR 2002k:11209
  • [SchSt] Scheidler, R., Stein, A. Unit computation in purely cubic function fields of unit rank 1. Algorithmic number theory (Portland, OR, 1998) 592-606, Lecture Notes in Comput. Sci., 1423, Springer-Verlag, Berlin, 1998. MR 2000k:11145
  • [St] Stichtenoth, Henning. Algebraic Function Fields and Codes. Universitext. Springer-Verlag, Berlin, 1993. MR 94k:14016

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Additional Information

Mark L. Bauer
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Address at time of publication: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2G 3L1 Canada

Received by editor(s): April 10, 2001
Received by editor(s) in revised form: July 18, 2002
Published electronically: June 17, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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